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rootLocusPolesSorted - + % + % Syntax: [poles] = rootLocusPolesSorted(G, K, gains, args) + % + % Inputs: + % - G, K, gains, args - + % + % Outputs: + % - poles - + + arguments + G + K + gains + args.minreal double {mustBeNumericOrLogical} = false + args.p_half double {mustBeNumericOrLogical} = false + args.d_max double {mustBeNumeric} = -1 + end + + if args.minreal + p1 = pole(minreal(feedback(G, gains(1)*K))); + [~, i_uniq] = uniquetol([real(p1), imag(p1)], 1e-10, 'ByRows', true); + p1 = p1(i_uniq); + + poles = zeros(length(p1), length(gains)); + poles(:, 1) = p1; + else + p1 = pole(feedback(G, gains(1)*K)); + [~, i_uniq] = uniquetol([real(p1), imag(p1)], 1e-10, 'ByRows', true); + p1 = p1(i_uniq); + + poles = zeros(length(p1), length(gains)); + poles(:, 1) = p1; + end + + if args.minreal + p2 = pole(minreal(feedback(G, gains(2)*K))); + [~, i_uniq] = uniquetol([real(p2), imag(p2)], 1e-10, 'ByRows', true); + p2 = p2(i_uniq); + poles(:, 2) = p2; + else + p2 = pole(feedback(G, gains(2)*K)); + [~, i_uniq] = uniquetol([real(p2), imag(p2)], 1e-10, 'ByRows', true); + p2 = p2(i_uniq); + poles(:, 2) = p2; + end + + for g_i = 3:length(gains) + % Estimated value of the poles + poles_est = poles(:, g_i-1) + (poles(:, g_i-1) - poles(:, g_i-2))*(gains(g_i) - gains(g_i-1))/(gains(g_i-1) - gains(g_i - 2)); + + % New values for the poles + poles_gi = pole(feedback(G, gains(g_i)*K)); + [~, i_uniq] = uniquetol([real(poles_gi), imag(poles_gi)], 1e-10, 'ByRows', true); + poles_gi = poles_gi(i_uniq); + + % Array of distances between all the poles + poles_dist = sqrt((poles_est-poles_gi.').*conj(poles_est-poles_gi.')); + + % Get indices corresponding to distances from lowest to highest + [~, c] = sort(min(poles_dist)); + + as = 1:length(poles_gi); + + % for each column of poles_dist corresponding to the i'th pole + % with closest previous poles + for p_i = c + % Get the indice a_i of the previous pole that is the closest + % to pole c(p_i) + [~, a_i] = min(poles_dist(:, p_i)); + + poles(as(a_i), g_i) = poles_gi(p_i); + + % Remove old poles that are already matched + % poles_gi(as(a_i), :) = []; + poles_dist(a_i, :) = []; + as(a_i) = []; + end + end + + + if args.d_max > 0 + poles = poles(max(abs(poles(:, 2:end) - poles(:, 1:end-1))') > args.d_max, :); + end + + if args.p_half + poles = poles(1:round(end/2), :); + end + + [~, s_p] = sort(imag(poles(:,1)), 'descend'); + poles = poles(s_p, :); + + poles = poles.'; diff --git a/nass-rotating-3dof-model.bib b/nass-rotating-3dof-model.bib index 8a901e6..5f18543 100644 --- a/nass-rotating-3dof-model.bib +++ b/nass-rotating-3dof-model.bib @@ -1,70 +1,3 @@ -@article{collette15_sensor_fusion_method_high_perfor, - author = {C. Collette and F. Matichard}, - title = {Sensor Fusion Methods for High Performance Active Vibration Isolation Systems}, - journal = {Journal of Sound and Vibration}, - volume = {342}, - pages = {1-21}, - year = {2015}, - doi = {10.1016/j.jsv.2015.01.006}, - url = {https://doi.org/10.1016/j.jsv.2015.01.006}, - keywords = {}, -} - -@inproceedings{collette14_vibrat, - author = {Collette, C. and Matichard, F}, - title = {Vibration control of flexible structures using fusion of inertial sensors and hyper-stable actuator-sensor pairs}, - booktitle = {International Conference on Noise and Vibration Engineering (ISMA2014)}, - year = {2014}, - keywords = {}, -} - -@article{oomen18_advan_motion_contr_precis_mechat, - author = {Tom Oomen}, - title = {Advanced Motion Control for Precision Mechatronics: Control, Identification, and Learning of Complex Systems}, - journal = {IEEJ Journal of Industry Applications}, - volume = {7}, - number = {2}, - pages = {127-140}, - year = {2018}, - doi = {10.1541/ieejjia.7.127}, - url = {https://doi.org/10.1541/ieejjia.7.127}, -} - -@book{skogestad07_multiv_feedb_contr, - author = {Skogestad, Sigurd and Postlethwaite, Ian}, - title = {Multivariable Feedback Control: Analysis and Design}, - year = {2007}, - publisher = {John Wiley}, - keywords = {favorite}, -} - -@phdthesis{hua05_low_ligo, - author = {Hua, Wensheng}, - school = {stanford university}, - title = {Low frequency vibration isolation and alignment system for - advanced LIGO}, - year = 2005, -} - -@book{lurie12_class, - author = {Lurie, B. J}, - title = {Classical feedback control : with MATLAB and Simulink}, - year = 2012, - publisher = {CRC Press}, - address = {Boca Raton, FL}, - isbn = 9781439897461, - keywords = {favorite}, -} - -@techreport{bibel92_guidel_h, - author = {Bibel, John E and Malyevac, D Stephen}, - institution = {NAVAL SURFACE WARFARE CENTER DAHLGREN DIV VA}, - title = {Guidelines for the selection of weighting functions for - H-infinity control}, - year = 1992, - keywords = {}, -} - @inproceedings{dehaeze20_activ_dampin_rotat_platf_integ_force_feedb, author = {Dehaeze, T. and Collette, C.}, title = {Active Damping of Rotating Platforms using Integral Force @@ -74,72 +7,66 @@ year = 2020, } -@misc{dehaeze20_activ_dampin_rotat_posit_platf, - author = {Thomas Dehaeze}, - howpublished = {Source Code on Zonodo}, - month = 07, - title = {Active Damping of Rotating Positioning Platforms}, - url = {https://doi.org/10.5281/zenodo.3894342}, - doi = {10.5281/zenodo.3894342}, - year = 2020, -} + @article{dehaeze21_activ_dampin_rotat_platf_using, - author = {Thomas Dehaeze and Christophe Collette}, - title = {Active Damping of Rotating Platforms Using Integral Force Feedback}, - journal = {Engineering Research Express}, - year = {2021}, - url = {http://iopscience.iop.org/article/10.1088/2631-8695/abe803}, + author = {Thomas Dehaeze and Christophe Collette}, + title = {Active Damping of Rotating Platforms Using Integral Force + Feedback}, + journal = {Engineering Research Express}, + year = 2021, + doi = {10.1088/2631-8695/abe803}, + url = {https://doi.org/10.1088/2631-8695/abe803}, + month = {Feb}, + keywords = {nass, esrf}, } -@misc{dehaeze20_activ_dampin_rotat_posit_platf, - author = {Thomas Dehaeze}, - howpublished = {Source Code on Zonodo}, - month = 07, - title = {Active Damping of Rotating Positioning Platforms}, - url = {https://doi.org/10.5281/zenodo.3894342}, - doi = {10.5281/zenodo.3894342}, - year = 2020, + + +@article{collette11_review_activ_vibrat_isolat_strat, + author = {Christophe Collette and Stef Janssens and Kurt Artoos}, + title = {Review of Active Vibration Isolation Strategies}, + journal = {Recent Patents on Mechanical Engineeringe}, + volume = 4, + number = 3, + pages = {212-219}, + year = 2011, + doi = {10.2174/2212797611104030212}, + url = {https://doi.org/10.2174/2212797611104030212}, + keywords = {favorite}, } -@inproceedings{dehaeze18_sampl_stabil_for_tomog_exper, - author = {Thomas Dehaeze and M. Magnin Mattenet and Christophe - Collette}, - title = {Sample Stabilization For Tomography Experiments In Presence - Of Large Plant Uncertainty}, - booktitle = {MEDSI'18}, - year = 2018, - number = 10, - pages = {153--157}, - doi = {10.18429/JACoW-MEDSI2018-WEOAMA02}, - url = {https://doi.org/10.18429/JACoW-MEDSI2018-WEOAMA02}, - address = {Geneva, Switzerland}, - isbn = {978-3-95450-207-3}, - language = {english}, - month = {Dec}, - publisher = {JACoW Publishing}, - series = {Mechanical Engineering Design of Synchrotron Radiation - Equipment and Instrumentation}, - venue = {Paris, France}, + + +@article{lin06_distur_atten_precis_hexap_point, + author = {Haomin Lin and John E. McInroy}, + title = {Disturbance Attenuation in Precise Hexapod Pointing Using + Positive Force Feedback}, + journal = {Control Engineering Practice}, + volume = 14, + number = 11, + pages = {1377-1386}, + year = 2006, + doi = {10.1016/j.conengprac.2005.10.002}, + url = {https://doi.org/10.1016/j.conengprac.2005.10.002}, + keywords = {parallel robot}, } -@book{skogestad07_multiv_feedb_contr, - author = {Skogestad, Sigurd and Postlethwaite, Ian}, - title = {Multivariable Feedback Control: Analysis and Design}, - year = 2007, - publisher = {John Wiley}, - isbn = 9780470011683, + + +@article{fanson90_posit_posit_feedb_contr_large_space_struc, + author = {Fanson, JL and Caughey, T Kv}, + title = {Positive Position Feedback Control for Large Space + Structures}, + journal = {AIAA journal}, + volume = 28, + number = 4, + pages = {717--724}, + year = 1990, + keywords = {active damping}, } -@book{preumont18_vibrat_contr_activ_struc_fourt_edition, - author = {Andre Preumont}, - title = {Vibration Control of Active Structures - Fourth Edition}, - year = 2018, - publisher = {Springer International Publishing}, - url = {https://doi.org/10.1007/978-3-319-72296-2}, - doi = {10.1007/978-3-319-72296-2}, - series = {Solid Mechanics and Its Applications}, -} + @inproceedings{preumont91_activ, author = {Andre Preumont and Jean-Paul Dufour and Christian Malekian}, @@ -152,25 +79,41 @@ url = {https://doi.org/10.2514/6.1991-989}, month = {apr}, publisher = {American Institute of Aeronautics and Astronautics}, + keywords = {active damping}, } -@article{preumont08_trans_zeros_struc_contr_with, - author = {Preumont, Andr{\'e} and De Marneffe, Bruno and Krenk, - Steen}, - title = {Transmission Zeros in Structural Control With Collocated - Multi-Input/multi-Output Pairs}, - journal = {Journal of guidance, control, and dynamics}, - volume = 31, - number = 2, - pages = {428--432}, - year = 2008, - doi = {10.2514/1.31529}, - url = {https://doi.org/10.2514/1.31529}, + + +@article{karnopp74_vibrat_contr_using_semi_activ_force_gener, + author = {Karnopp, Dean and Crosby, Michael J and Harwood, RA}, + title = {Vibration Control Using Semi-Active Force Generators}, + journal = {Journal of Engineering for Industry}, + volume = 96, + pages = {619-626}, + year = 1974, + doi = {10.1115/1.3438373}, + url = {https://doi.org/10.1115/1.3438373}, } + + +@article{serrand00_multic_feedb_contr_isolat_base_excit_vibrat, + author = {Serrand, M and Elliott, SJ}, + title = {Multichannel Feedback Control for the Isolation of + Base-Excited Vibration}, + journal = {Journal of Sound and Vibration}, + volume = 234, + number = 4, + pages = {681--704}, + year = 2000, + publisher = {Elsevier}, +} + + + @article{preumont02_force_feedb_versus_accel_feedb, - author = {Preumont, Andr{\'e} and A. Fran{\c{c}}ois and F. Bossens - and A. Abu-Hanieh}, + author = {A. Preumont and A. Fran{\c{c}}ois and F. Bossens and A. + Abu-Hanieh}, title = {Force Feedback Versus Acceleration Feedback in Active Vibration Isolation}, journal = {Journal of Sound and Vibration}, @@ -182,20 +125,36 @@ url = {https://doi.org/10.1006/jsvi.2002.5047}, } -@article{preumont92_activ_dampin_by_local_force, - author = {Preumont, Andre and Dufour, Jean-Paul and Malekian, - Christian}, - title = {Active Damping By a Local Force Feedback With Piezoelectric - Actuators}, + + +@article{preumont08_trans_zeros_struc_contr_with, + author = {Preumont, Andr{\'e} and De Marneffe, Bruno and Krenk, + Steen}, + title = {Transmission Zeros in Structural Control With Collocated + Multi-Input/multi-Output Pairs}, journal = {Journal of guidance, control, and dynamics}, - volume = 15, + volume = 31, number = 2, - pages = {390--395}, - year = 1992, - doi = 10.2514/3.20848, - url = {https://doi.org/10.2514/3.20848}, + pages = {428--432}, + year = 2008, } + + +@article{preumont08_trans_zeros_struc_contr_with, + author = {Preumont, Andr{\'e} and De Marneffe, Bruno and Krenk, + Steen}, + title = {Transmission Zeros in Structural Control With Collocated + Multi-Input/multi-Output Pairs}, + journal = {Journal of guidance, control, and dynamics}, + volume = 31, + number = 2, + pages = {428--432}, + year = 2008, +} + + + @article{teo15_optim_integ_force_feedb_activ_vibrat_contr, author = {Yik R. Teo and Andrew J. Fleming}, title = {Optimal Integral Force Feedback for Active Vibration @@ -208,195 +167,10 @@ url = {https://doi.org/10.1016/j.jsv.2015.06.046}, month = {nov}, publisher = {Elsevier {BV}}, + keywords = {iff}, } -@phdthesis{hanieh03_activ_stewar, - author = {Hanieh, Ahmed Abu}, - school = {Universit{\'e} Libre de Bruxelles, Brussels, Belgium}, - title = {Active isolation and damping of vibrations via Stewart - platform}, - year = 2003, -} -@article{hauge04_sensor_contr_space_based_six, - author = {G.S. Hauge and M.E. Campbell}, - title = {Sensors and Control of a Space-Based Six-Axis Vibration - Isolation System}, - journal = {Journal of Sound and Vibration}, - volume = 269, - number = {3-5}, - pages = {913-931}, - year = 2004, - doi = {10.1016/s0022-460x(03)00206-2}, - url = {https://doi.org/10.1016/s0022-460x(03)00206-2}, -} - -@article{souleille18_concep_activ_mount_space_applic, - author = {Souleille, Adrien and Lampert, Thibault and Lafarga, V and - Hellegouarch, Sylvain and Rondineau, Alan and Rodrigues, - Gon{\c{c}}alo and Collette, Christophe}, - title = {A Concept of Active Mount for Space Applications}, - journal = {CEAS Space Journal}, - volume = 10, - number = 2, - pages = {157--165}, - year = 2018, - doi = {10.1007/s12567-017-0180-6}, - url = {https://doi.org/10.1007/s12567-017-0180-6}, - publisher = {Springer}, -} - -@article{yoshioka01_activ_microv_isolat_system_hi, - author = {H. Yoshioka and Y. Takahashi and K. Katayama and T. Imazawa - and N. Murai}, - title = {An Active Microvibration Isolation System for Hi-Tech - Manufacturing Facilities}, - journal = {Journal of Vibration and Acoustics}, - volume = 123, - number = 2, - pages = 269, - year = 2001, - doi = {10.1115/1.1350566}, - url = {https://doi.org/10.1115/1.1350566}, -} - -@article{lan08_activ_vibrat_isolat_long_range, - author = {Kuo-Jung Lan and Jia-Yush Yen and John A. Kramar}, - title = {Active Vibration Isolation for a Long Range Scanning - Tunneling Microscope}, - journal = {Asian Journal of Control}, - volume = 6, - number = 2, - pages = {179-186}, - year = 2008, - doi = {10.1111/j.1934-6093.2004.tb00196.x}, - url = {https://doi.org/10.1111/j.1934-6093.2004.tb00196.x}, -} - -@article{fleming15_low_order_dampin_track_contr, - author = {Andrew J. Fleming and Yik Ren Teo and Kam K. Leang}, - title = {Low-Order Damping and Tracking Control for Scanning Probe - Systems}, - journal = {Frontiers in Mechanical Engineering}, - volume = 1, - year = 2015, - doi = {10.3389/fmech.2015.00014}, - url = {https://doi.org/10.3389/fmech.2015.00014}, -} - -@article{matichard15_seism_isolat_advan_ligo, - author = {Matichard, F and Lantz, B and Mittleman, R and Mason, K and - Kissel, J and Abbott, B and Biscans, S and McIver, J and - Abbott, R and Abbott, S and others}, - title = {Seismic Isolation of Advanced Ligo: Review of Strategy, - Instrumentation and Performance}, - journal = {Classical and Quantum Gravity}, - volume = 32, - number = 18, - pages = 185003, - year = 2015, - doi = 10.1088/0264-9381/32/18/185003, - url = {https://doi.org/10.1088/0264-9381/32/18/185003}, - publisher = {IOP Publishing}, -} - -@article{collette10_activ_quadr_stabil_futur_linear_partic_collid, - author = {Collette, Christophe and Artoos, Kurt and Kuzmin, A and - Janssens, S and Sylte, Magnus and Guinchard, Michael and - Hauviller, Claude}, - title = {Active Quadrupole Stabilization for Future Linear Particle - Colliders}, - journal = {Nuclear Instruments and Methods in Physics Research Section - A: Accelerators, Spectrometers, Detectors and Associated - Equipment}, - volume = 621, - number = {1-3}, - pages = {71--78}, - year = 2010, - doi = 10.1016/j.nima.2010.05.020, - url = {https://doi.org/10.1016/j.nima.2010.05.020}, - publisher = {Elsevier}, -} - -@misc{reilly06_critic, - author = {Reilly, S P and Leach, R K}, - note = {NPL Report}, - title = {Critical review of seismic vibration isolation techniques}, - year = 2006, -} - -@phdthesis{poel10_explor_activ_hard_mount_vibrat, - author = {van der Poel, Gerrit Wijnand}, - doi = {10.3990/1.9789036530163}, - isbn = {978-90-365-3016-3}, - keywords = {parallel robot}, - school = {University of Twente}, - title = {An Exploration of Active Hard Mount Vibration Isolation for - Precision Equipment}, - url = {https://doi.org/10.3990/1.9789036530163}, - year = 2010, -} - -@article{collette11_review_activ_vibrat_isolat_strat, - author = {Christophe Collette and Stef Janssens and Kurt Artoos}, - title = {Review of Active Vibration Isolation Strategies}, - journal = {Recent Patents on Mechanical Engineeringe}, - volume = 4, - number = 3, - pages = {212-219}, - year = 2011, - doi = {10.2174/2212797611104030212}, - url = {https://doi.org/10.2174/2212797611104030212}, -} - -@article{lin06_distur_atten_precis_hexap_point, - author = {Haomin Lin and John E. McInroy}, - title = {Disturbance Attenuation in Precise Hexapod Pointing Using - Positive Force Feedback}, - journal = {Control Engineering Practice}, - volume = 14, - number = 11, - pages = {1377-1386}, - year = 2006, - doi = {10.1016/j.conengprac.2005.10.002}, - url = {https://doi.org/10.1016/j.conengprac.2005.10.002}, -} - -@article{fanson90_posit_posit_feedb_contr_large_space_struc, - author = {Fanson, JL and Caughey, T Kv}, - title = {Positive Position Feedback Control for Large Space - Structures}, - journal = {AIAA journal}, - volume = 28, - number = 4, - pages = {717--724}, - year = 1990, - doi = 10.2514/3.10451, - url = {https://doi.org/10.2514/3.10451}, -} - -@article{karnopp74_vibrat_contr_using_semi_activ_force_gener, - author = {Karnopp, Dean and Crosby, Michael J and Harwood, RA}, - title = {Vibration Control Using Semi-Active Force Generators}, - journal = {Journal of Engineering for Industry}, - year = 1974, - doi = {10.1115/1.3438373}, - url = {https://doi.org/10.1115/1.3438373}, -} - -@article{serrand00_multic_feedb_contr_isolat_base_excit_vibrat, - author = {Serrand, M and Elliott, SJ}, - title = {Multichannel Feedback Control for the Isolation of - Base-Excited Vibration}, - journal = {Journal of Sound and Vibration}, - volume = 234, - number = 4, - pages = {681--704}, - year = 2000, - doi = 10.1006/jsvi.2000.2891, - url = {https://doi.org/10.1006/jsvi.2000.2891}, - publisher = {Elsevier}, -} @article{chesne16_enhan_dampin_flexib_struc_using_force_feedb, author = {Simon Chesn{\'e} and Ariston Milhomem and Christophe @@ -410,8 +184,11 @@ year = 2016, doi = {10.2514/1.g001620}, url = {https://doi.org/10.2514/1.g001620}, + keywords = {active damping, integral force feedback}, } + + @article{zhao19_optim_integ_force_feedb_contr, author = {Zhao, Guoying and Paknejad, A and Deraemaeker, Arnaud and Collette, Christophe}, @@ -422,32 +199,43 @@ number = 17, pages = {2330--2339}, year = 2019, - doi = 10.1177/1077546319853165, - url = {https://doi.org/10.1177/1077546319853165}, publisher = {SAGE Publications Sage UK: London, England}, + keywords = {iff}, } + + +@book{skogestad07_multiv_feedb_contr, + author = {Skogestad, Sigurd and Postlethwaite, Ian}, + title = {Multivariable Feedback Control: Analysis and Design - + Second Edition}, + year = 2007, + publisher = {John Wiley}, + isbn = 978-0470011683, + keywords = {favorite}, +} + + + @phdthesis{marneffe07_activ_passiv_vibrat_isolat_dampin_shunt_trans, author = {de Marneffe, Bruno}, school = {Universit{\'e} Libre de Bruxelles, Brussels, Belgium}, title = {Active and Passive Vibration Isolation and Damping via Shunted Transducers}, year = 2007, + keywords = {parallel robot}, } -@book{matlab20, - author = {MATLAB}, - title = {version 9.9.0 (R2020b)}, - year = 2020, - publisher = {The MathWorks Inc.}, - address = {Natick, Massachusetts}, + + +@book{preumont18_vibrat_contr_activ_struc_fourt_edition, + author = {Andre Preumont}, + title = {Vibration Control of Active Structures - Fourth Edition}, + year = 2018, + publisher = {Springer International Publishing}, + url = {https://doi.org/10.1007/978-3-319-72296-2}, + doi = {10.1007/978-3-319-72296-2}, + keywords = {favorite, parallel robot}, + series = {Solid Mechanics and Its Applications}, } -@inproceedings{dehaeze20_activ_dampin_rotat_platf_integ_force_feedb, - author = {Dehaeze, T. and Collette, C.}, - title = {Active Damping of Rotating Platforms using Integral Force - Feedback}, - booktitle = {Proceedings of the International Conference on Modal - Analysis Noise and Vibration Engineering (ISMA)}, - year = 2020, -} diff --git a/nass-rotating-3dof-model.org b/nass-rotating-3dof-model.org index dc3dfa8..09b4d47 100644 --- a/nass-rotating-3dof-model.org +++ b/nass-rotating-3dof-model.org @@ -1,4 +1,4 @@ -#+TITLE: NASS - Effect of rotation +#+TITLE: Nano Active Stabilization System - Effect of rotation :DRAWER: #+LANGUAGE: en #+EMAIL: dehaeze.thomas@gmail.com @@ -15,13 +15,8 @@ #+LATEX_CLASS: scrreprt #+LATEX_CLASS_OPTIONS: [a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc] -#+LATEX_HEADER: \usepackage{siunitx} -#+LATEX_HEADER: \usepackage{tikz} -#+LATEX_HEADER: \usetikzlibrary{shapes.misc,arrows,arrows.meta} -#+LATEX_HEADER: \usepackage{bm} -#+LATEX_HEADER: \usepackage{amsmath} -#+LATEX_HEADER_EXTRA: \usepackage{amssymb} -#+LATEX_HEADER_EXTRA: \input{preamble.tex} +#+LATEX_HEADER: \input{preamble.tex} +#+LATEX_HEADER_EXTRA: \input{preamble_extra.tex} #+LATEX_HEADER_EXTRA: \bibliography{nass-rotating-3dof-model.bib} #+BIND: org-latex-bib-compiler "biber" @@ -79,16 +74,6 @@ (add-to-list 'org-export-filter-headline-functions 'my-latex-filter-removeOrgAutoLabels) -;; Remove all org comments in the output LaTeX file -(defun delete-org-comments (backend) - (loop for comment in (reverse (org-element-map (org-element-parse-buffer) - 'comment 'identity)) - do - (setf (buffer-substring (org-element-property :begin comment) - (org-element-property :end comment)) - ""))) -(add-hook 'org-export-before-processing-hook 'delete-org-comments) - ;; Use no package by default (setq org-latex-packages-alist nil) (setq org-latex-default-packages-alist nil) @@ -104,38 +89,61 @@ * Notes :noexport: Prefix is =rotating= +* Glossary and Acronyms - Tables :ignore: + +#+name: glossary +| label | name | description | +|-------+-------------------------+---------------------------------------------| +| psdx | \ensuremath{\Phi_{x}} | Power spectral density of signal $x$ | +| asdx | \ensuremath{\Gamma_{x}} | Amplitude spectral density of signal $x$ | +| cpsx | \ensuremath{\Phi_{x}} | Cumulative Power Spectrum of signal $x$ | +| casx | \ensuremath{\Gamma_{x}} | Cumulative Amplitude Spectrum of signal $x$ | + +#+name: acronyms +| key | abbreviation | full form | +|--------+--------------+------------------------------------------------| +| haclac | HAC-LAC | High Authority Control - Low Authority Control | +| hac | HAC | High Authority Control | +| lac | LAC | Low Authority Control | +| nass | NASS | Nano Active Stabilization System | +| asd | ASD | Amplitude Spectral Density | +| psd | PSD | Power Spectral Density | +| cps | CPS | Cumulative Power Spectrum | +| cas | CAS | Cumulative Amplitude Spectrum | +| frf | FRF | Frequency Response Function | +| iff | IFF | Integral Force Feedback | +| rdc | RDC | Relative Damping Control | +| drga | DRGA | Dynamical Relative Gain Array | + * Introduction :ignore: -An important aspect of the Nano Active Stabilization System (NASS) is that the nano-hexapod is continuously rotating around a vertical axis while the external metrology is not. -Such rotation induces gyroscopic effects that may impact the system dynamics and obtained performances. +An important aspect of the acrfull:nass is that the nano-hexapod is continuously rotating around a vertical axis while the external metrology is not. +Such rotation induces gyroscopic effects that may impact the system dynamics and obtained performance. +To study these effects, a model of a rotating suspended platform is first presented (Section ref:sec:rotating_system_description) +This model is simple enough to be able to derive its dynamics analytically and to well understand its behavior, while still allowing to capture the important physical effects in play. -In this report, this rotating aspect of the NASS is studied. -It is structured in several sections: -- Section ref:sec:rotating_system_description: a simple model of a rotating suspended platform that will be used throughout this study is presented. The effect of the rotation velocity on the system dynamics is shown. -- Section ref:sec:rotating_iff_pure_int: Integral Force Feedback (IFF) is applied to the rotating platform, and it is shown that the unconditional stability of IFF is lost due to Gyroscopic effects induced by the rotation. -- Section ref:sec:rotating_iff_pseudo_int: A first modification of the IFF control law is proposed such that damping can be added to the suspension modes in a robust way. This modification consists of adding an high pass filter to the IFF controller. Optimal high pass filter cut-off frequency is computed. -- Section ref:sec:rotating_iff_parallel_stiffness: A second modification is proposed to regain the unconditional stability of IFF. This modification consists of adding stiffness in parallel to the force sensors. Optimal parallel stiffness is computed. -- Section ref:sec:rotating_relative_damp_control: Relative damping control is applied to the rotating system. -- Section ref:sec:rotating_comp_act_damp: Once the optimal control parameters for the three tested active damping techniques are obtained, they are compared in terms of achievable damping, obtained damped plant and closed-loop compliance and transmissibility. -- Section ref:sec:rotating_nano_hexapod: the previous analysis is applied on three nano-hexapod stiffnesses and optimal active damping controller are obtained. -- Section ref:sec:rotating_nass: up until this section, the study was performed on a very simplistic model that just captures the rotation aspect and the model parameters were not tuned to corresponds to the NASS. In this last section, a model of the micro-station is added below the suspended platform (i.e. the nano-hexapod) with a rotating spindle and parameters tuned to match the NASS dynamics. The goal is to determine if the rotation imposes performance limitation for the NASS. +acrfull:iff is then applied to the rotating platform, and it is shown that the unconditional stability of acrshort:iff is lost due to gyroscopic effects induced by the rotation (Section ref:sec:rotating_iff_pure_int). +Two modifications of the Integral Force Feedback are then proposed. +The first one consists of adding an high pass filter to the acrshort:iff controller (Section ref:sec:rotating_iff_pseudo_int). +It is shown that the acrshort:iff controller is stable for some values of the gain, and that damping can be added to the suspension modes. +Optimal high pass filter cut-off frequency is computed. +The second modification consists of adding a stiffness in parallel to the force sensors (Section ref:sec:rotating_iff_parallel_stiffness). +Under a certain condition, the unconditional stability of the the IFF controller is regained. +Optimal parallel stiffness is then computed. +This study of adapting acrshort:iff for the damping of rotating platforms was the subject of two published papers [[cite:&dehaeze20_activ_dampin_rotat_platf_integ_force_feedb;&dehaeze21_activ_dampin_rotat_platf_using]]. -To run the Matlab code, go in the =matlab= directory and run the Matlab files corresponding to each section (see Table ref:tab:section_matlab_code). +It is then shown that acrfull:rdc is less affected by gyroscopic effects (Section ref:sec:rotating_relative_damp_control). +Once the optimal control parameters for the three tested active damping techniques are obtained, they are compared in terms of achievable damping, obtained damped plant and closed-loop compliance and transmissibility (Section ref:sec:rotating_comp_act_damp). -#+name: tab:section_matlab_code -#+caption: Report sections and corresponding Matlab files -#+attr_latex: :environment tabularx :width 0.6\linewidth :align lX -#+attr_latex: :center t :booktabs t -| *Sections* | *Matlab File* | -|-------------------------------------------------+------------------------------------| -| Section ref:sec:rotating_system_description | =rotating_1_system_description.m= | -| Section ref:sec:rotating_iff_pure_int | =rotating_2_iff_pure_int.m= | -| Section ref:sec:rotating_iff_pseudo_int | =rotating_3_iff_hpf.m= | -| Section ref:sec:rotating_iff_parallel_stiffness | =rotating_4_iff_kp.m= | -| Section ref:sec:rotating_relative_damp_control | =rotating_5_rdc.m= | -| Section ref:sec:rotating_comp_act_damp | =rotating_5_act_damp_comparison.m= | -| Section ref:sec:rotating_nano_hexapod | =rotating_6_nano_hexapod.m= | -| Section ref:sec:rotating_nass | =rotating_6_nass.m= | +The previous analysis is applied on three considered nano-hexapod stiffnesses ($k_n = 0.01\,N/\mu m$, $k_n = 1\,N/\mu m$ and $k_n = 100\,N/\mu m$) and optimal active damping controller are obtained in each case (Section ref:sec:rotating_nano_hexapod). +Up until this section, the study was performed on a very simplistic model that just captures the rotation aspect and the model parameters were not tuned to corresponds to the NASS. +In the last section (Section ref:sec:rotating_nass), a model of the micro-station is added below the suspended platform (i.e. the nano-hexapod) with a rotating spindle and parameters tuned to match the NASS dynamics. +The goal is to determine if the rotation imposes performance limitation for the NASS. + +#+name: fig:rotating_overview +#+caption: Overview of this report on rotating effects +#+attr_latex: :width \linewidth +[[file:figs/rotating_overview.png]] * System Description and Analysis :PROPERTIES: @@ -145,14 +153,12 @@ To run the Matlab code, go in the =matlab= directory and run the Matlab files co ** Introduction :ignore: The studied system consists of a 2 degree of freedom translation stage on top of a rotating stage (Figure ref:fig:rotating_3dof_model_schematic). - The rotating stage is supposed to be ideal, meaning it induces a perfect rotation $\theta(t) = \Omega t$ where $\Omega$ is the rotational speed in $\si{\radian\per\s}$. - The suspended platform consists of two orthogonal actuators each represented by three elements in parallel: a spring with a stiffness $k$ in $\si{\newton\per\meter}$, a dashpot with a damping coefficient $c$ in $\si{\newton\per(\meter\per\second)}$ and an ideal force source $F_u, F_v$. A payload with a mass $m$ in $\si{\kilo\gram}$, is mounted on the (rotating) suspended platform. - -Two reference frames are used: an inertial frame $(\vec{i}_x, \vec{i}_y, \vec{i}_z)$ and a uniform rotating frame $(\vec{i}_u, \vec{i}_v, \vec{i}_w)$ rigidly fixed on top of the rotating stage with $\vec{i}_w$ aligned with the rotation axis. +Two reference frames are used: an /inertial/ frame $(\vec{i}_x, \vec{i}_y, \vec{i}_z)$ and a /uniform rotating/ frame $(\vec{i}_u, \vec{i}_v, \vec{i}_w)$ rigidly fixed on top of the rotating stage with $\vec{i}_w$ aligned with the rotation axis. The position of the payload is represented by $(d_u, d_v, 0)$ expressed in the rotating frame. +After the dynamics of this system is studied, the objective will be to damp the two suspension modes of the payload while the rotating stage performs a constant rotation. #+begin_src latex :file rotating_3dof_model_schematic.pdf \begin{tikzpicture} @@ -215,6 +221,7 @@ The position of the payload is represented by $(d_u, d_v, 0)$ expressed in the r #+name: fig:rotating_3dof_model_schematic #+caption: Schematic of the studied system +#+attr_latex: :scale 0.8 #+RESULTS: [[file:figs/rotating_3dof_model_schematic.png]] @@ -245,63 +252,58 @@ mdl = 'rotating_model'; #+end_src ** Equations of motion -To obtain the equations of motion for the system represented in Figure ref:fig:rotating_3dof_model_schematic, the Lagrangian equations are used: -#+name: eq:lagrangian_equations -\begin{equation} +To obtain the equations of motion for the system represented in Figure ref:fig:rotating_3dof_model_schematic, the Lagrangian equation eqref:eq:rotating_lagrangian_equations is used. +$L = T - V$ is the Lagrangian, $T$ the kinetic coenergy, $V$ the potential energy, $D$ the dissipation function, and $Q_i$ the generalized force associated with the generalized variable $\begin{bmatrix}q_1 & q_2\end{bmatrix} = \begin{bmatrix}d_u & d_v\end{bmatrix}$. +These terms are derived in eqref:eq:rotating_energy_functions_lagrange. +Note that the equation of motion corresponding to the constant rotation along $\vec{i}_w$ is disregarded as this motion is considered to be imposed by the rotation stage. + +\begin{equation}\label{eq:rotating_lagrangian_equations} \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}_i} \right) + \frac{\partial D}{\partial \dot{q}_i} - \frac{\partial L}{\partial q_i} = Q_i \end{equation} -with $L = T - V$ the Lagrangian, $T$ the kinetic coenergy, $V$ the potential energy, $D$ the dissipation function, and $Q_i$ the generalized force associated with the generalized variable $\begin{bmatrix}q_1 & q_2\end{bmatrix} = \begin{bmatrix}d_u & d_v\end{bmatrix}$. -The equation of motion corresponding to the constant rotation along $\vec{i}_w$ is disregarded as this motion is considered to be imposed by the rotation stage. -#+name: eq:energy_functions_lagrange -\begin{equation} + +\begin{equation} \label{eq:rotating_energy_functions_lagrange} \begin{aligned} - T &= \frac{1}{2} m \left( ( \dot{d}_u - \Omega d_v )^2 + ( \dot{d}_v + \Omega d_u )^2 \right), \\ - V &= \frac{1}{2} k \big( {d_u}^2 + {d_v}^2 \big), \ Q_1 = F_u, \\ - D &= \frac{1}{2} c \big( \dot{d}_u{}^2 + \dot{d}_v{}^2 \big), \ Q_2 = F_v + T &= \frac{1}{2} m \left( ( \dot{d}_u - \Omega d_v )^2 + ( \dot{d}_v + \Omega d_u )^2 \right), \quad Q_1 = F_u, \quad Q_2 = F_v, \\ + V &= \frac{1}{2} k \big( {d_u}^2 + {d_v}^2 \big), \quad D = \frac{1}{2} c \big( \dot{d}_u{}^2 + \dot{d}_v{}^2 \big) \end{aligned} \end{equation} -Substituting equations eqref:eq:energy_functions_lagrange into equation eqref:eq:lagrangian_equations for both generalized coordinates gives two coupled differential equations eqref:eq:eom_coupled_1 and eqref:eq:eom_coupled_2. -#+name: eq:eom_coupled -\begin{subequations} +Substituting equations eqref:eq:rotating_energy_functions_lagrange into equation eqref:eq:rotating_lagrangian_equations for both generalized coordinates gives two coupled differential equations eqref:eq:rotating_eom_coupled_1 and eqref:eq:rotating_eom_coupled_2. + +\begin{subequations} \label{eq:rotating_eom_coupled} \begin{align} - m \ddot{d}_u + c \dot{d}_u + ( k - m \Omega^2 ) d_u &= F_u + 2 m \Omega \dot{d}_v \label{eq:eom_coupled_1} \\ - m \ddot{d}_v + c \dot{d}_v + ( k \underbrace{-\,m \Omega^2}_{\text{Centrif.}} ) d_v &= F_v \underbrace{-\,2 m \Omega \dot{d}_u}_{\text{Coriolis}} \label{eq:eom_coupled_2} + m \ddot{d}_u + c \dot{d}_u + ( k - m \Omega^2 ) d_u &= F_u + 2 m \Omega \dot{d}_v \label{eq:rotating_eom_coupled_1} \\ + m \ddot{d}_v + c \dot{d}_v + ( k \underbrace{-\,m \Omega^2}_{\text{Centrif.}} ) d_v &= F_v \underbrace{-\,2 m \Omega \dot{d}_u}_{\text{Coriolis}} \label{eq:rotating_eom_coupled_2} \end{align} \end{subequations} -The uniform rotation of the system induces *two gyroscopic effects* as shown in equation eqref:eq:eom_coupled: -- *Centrifugal forces*: that can been seen as an added *negative stiffness* $- m \Omega^2$ along $\vec{i}_u$ and $\vec{i}_v$ -- *Coriolis Forces*: that adds *coupling* between the two orthogonal directions. - -One can verify that without rotation ($\Omega = 0$) the system becomes equivalent to two uncoupled one degree of freedom mass-spring-damper systems. +The uniform rotation of the system induces two /gyroscopic effects/ as shown in equation eqref:eq:rotating_eom_coupled: +- /Centrifugal forces/: that can been seen as an added /negative stiffness/ $- m \Omega^2$ along $\vec{i}_u$ and $\vec{i}_v$ +- /Coriolis forces/: that adds /coupling/ between the two orthogonal directions. +One can verify that without rotation ($\Omega = 0$) the system becomes equivalent to two /uncoupled/ one degree of freedom mass-spring-damper systems. ** Transfer Functions in the Laplace domain -To study the dynamics of the system, the differential equations of motions eqref:eq:eom_coupled are converted into the Laplace domain and the $2 \times 2$ transfer function matrix $\mathbf{G}_d$ from $\begin{bmatrix}F_u & F_v\end{bmatrix}$ to $\begin{bmatrix}d_u & d_v\end{bmatrix}$ in equation eqref:eq:Gd_mimo_tf is obtained. -Its elements are shown in equation eqref:eq:Gd_indiv_el. +To study the dynamics of the system, the two differential equations of motions eqref:eq:rotating_eom_coupled are converted into the Laplace domain and the $2 \times 2$ transfer function matrix $\mathbf{G}_d$ from $\begin{bmatrix}F_u & F_v\end{bmatrix}$ to $\begin{bmatrix}d_u & d_v\end{bmatrix}$ in equation eqref:eq:rotating_Gd_mimo_tf is obtained. +The four transfer functions in $\mathbf{G}_d$ are shown in equation eqref:eq:rotating_Gd_indiv_el. -#+name: eq:Gd_mimo_tf -\begin{equation} +\begin{equation}\label{eq:rotating_Gd_mimo_tf} \begin{bmatrix} d_u \\ d_v \end{bmatrix} = \mathbf{G}_d \begin{bmatrix} F_u \\ F_v \end{bmatrix} \end{equation} -#+name: eq:Gd_indiv_el -\begin{subequations} +\begin{subequations}\label{eq:rotating_Gd_indiv_el} \begin{align} \mathbf{G}_{d}(1,1) &= \mathbf{G}_{d}(2,2) = \frac{ms^2 + cs + k - m \Omega^2}{\left( m s^2 + cs + k - m \Omega^2 \right)^2 + \left( 2 m \Omega s \right)^2} \\ \mathbf{G}_{d}(1,2) &= -\mathbf{G}_{d}(2,1) = \frac{2 m \Omega s}{\left( m s^2 + cs + k - m \Omega^2 \right)^2 + \left( 2 m \Omega s \right)^2} \end{align} \end{subequations} -To simplify the analysis, the undamped natural frequency $\omega_0$ and the damping ratio $\xi$ are used as in equation eqref:eq:xi_and_omega. -#+name: eq:xi_and_omega -\begin{equation} +To simplify the analysis, the undamped natural frequency $\omega_0$ and the damping ratio $\xi$ defined in eqref:eq:rotating_xi_and_omega are used instead. +The elements of transfer function matrix $\mathbf{G}_d$ are now described by equation eqref:eq:rotating_Gd_w0_xi_k. +\begin{equation} \label{eq:rotating_xi_and_omega} \omega_0 = \sqrt{\frac{k}{m}} \text{ in } \si{\radian\per\second}, \quad \xi = \frac{c}{2 \sqrt{k m}} \end{equation} -The elements of transfer function matrix $\mathbf{G}_d$ are now described by equation eqref:eq:Gd_w0_xi_k. -#+name: eq:Gd_w0_xi_k -\begin{subequations} +\begin{subequations} \label{eq:rotating_Gd_w0_xi_k} \begin{align} \mathbf{G}_{d}(1,1) &= \frac{\frac{1}{k} \left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)}{\left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2} \\ \mathbf{G}_{d}(1,2) &= \frac{\frac{1}{k} \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)}{\left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2} @@ -309,17 +311,15 @@ The elements of transfer function matrix $\mathbf{G}_d$ are now described by equ \end{subequations} ** System Poles: Campbell Diagram -The poles of $\mathbf{G}_d$ are the complex solutions $p$ of equation eqref:eq:poles. +The poles of $\mathbf{G}_d$ are the complex solutions $p$ of equation eqref:eq:rotating_poles (i.e. the roots of its denominator). -#+name: eq:poles -\begin{equation} +\begin{equation}\label{eq:rotating_poles} \left( \frac{p^2}{{\omega_0}^2} + 2 \xi \frac{p}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{p}{\omega_0} \right)^2 = 0 \end{equation} -Supposing small damping ($\xi \ll 1$), two pairs of complex conjugate poles are obtained as shown in equation eqref:eq:pole_values. +Supposing small damping ($\xi \ll 1$), two pairs of complex conjugate poles $[p_{+}, p_{-}]$ are obtained as shown in equation eqref:eq:rotating_pole_values. -#+name: eq:pole_values -\begin{subequations} +\begin{subequations} \label{eq:rotating_pole_values} \begin{align} p_{+} &= - \xi \omega_0 \left( 1 + \frac{\Omega}{\omega_0} \right) \pm j \omega_0 \left( 1 + \frac{\Omega}{\omega_0} \right) \\ p_{-} &= - \xi \omega_0 \left( 1 - \frac{\Omega}{\omega_0} \right) \pm j \omega_0 \left( 1 - \frac{\Omega}{\omega_0} \right) @@ -334,9 +334,7 @@ cn = 0.05; % Actuator Damping [N/(m/s)] xin = cn/(2*sqrt(kn*mn)); % Modal Damping [-] w0n = sqrt(kn/mn); % Natural Frequency [rad/s] -#+end_src -#+begin_src matlab %% Computation of the poles as a function of the rotating velocity Wzs = linspace(0, 2, 51); % Vector of rotation speeds [rad/s] @@ -354,16 +352,13 @@ clear pole_G; #+end_src The real and complex parts of these two pairs of complex conjugate poles are represented in Figure ref:fig:rotating_campbell_diagram as a function of the rotational speed $\Omega$. -As the rotational speed increases, $p_{+}$ goes to higher frequencies and $p_{-}$ goes to lower frequencies. -The system becomes unstable for $\Omega > \omega_0$ as the real part of $p_{-}$ is positive. +As the rotational speed increases, $p_{+}$ goes to higher frequencies and $p_{-}$ goes to lower frequencies (Figure ref:fig:rotating_campbell_diagram_imag). +The system becomes unstable for $\Omega > \omega_0$ as the real part of $p_{-}$ is positive (Figure ref:fig:rotating_campbell_diagram_real). Physically, the negative stiffness term $-m\Omega^2$ induced by centrifugal forces exceeds the spring stiffness $k$. #+begin_src matlab :results none %% Campbell diagram - Real and Imaginary parts of the poles as a function of the rotating velocity figure; -tiledlayout(1, 2, 'TileSpacing', 'Compact', 'Padding', 'None'); - -ax1 = nexttile(); hold on; plot(Wzs, real(p_ws(1, :)), '-', 'color', colors(1,:), 'DisplayName', '$p_{+}$') plot(Wzs, real(p_ws(4, :)), '-', 'color', colors(1,:), 'HandleVisibility', 'off') @@ -380,8 +375,14 @@ xticklabels({'$0$', '', '$\omega_0$', '', '$2 \omega_0$'}) ylim([-3*xin, 3*xin]); yticks([-3*xin, -2*xin, -xin, 0, xin, 2*xin, 3*xin]) yticklabels({'', '', '$-\xi\omega_0$', '$0$', ''}) +#+end_src -ax2 = nexttile(); +#+begin_src matlab :tangle no :exports results :results file none +exportFig('figs/rotating_campbell_diagram_real.pdf', 'width', 450, 'height', 400); +#+end_src + +#+begin_src matlab :results none +figure hold on; plot(Wzs, imag(p_ws(1, :)), '-', 'color', colors(1,:)) plot(Wzs, imag(p_ws(4, :)), '-', 'color', colors(1,:)) @@ -398,33 +399,45 @@ yticks([-3*w0n, -2*w0n, -w0n, 0, w0n, 2*w0n, 3*w0n]) yticklabels({'', '', '$-\omega_0$', '$0$', '$\omega_0$', '', ''}) #+end_src -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/rotating_campbell_diagram.pdf', 'width', 'full', 'height', 'normal'); +#+begin_src matlab :tangle no :exports results :results file none +exportFig('figs/rotating_campbell_diagram_imag.pdf', 'width', 450, 'height', 400); #+end_src #+name: fig:rotating_campbell_diagram -#+caption: Campbell diagram - Real and Imaginary parts of the poles as a function of the rotating velocity -#+RESULTS: -[[file:figs/rotating_campbell_diagram.png]] +#+caption: Campbell diagram - Real (\subref{fig:rotating_campbell_diagram_real}) and Imaginary (\subref{fig:rotating_campbell_diagram_imag}) parts of the poles as a function of the rotating velocity $\Omega$. +#+attr_latex: :options [htbp] +#+begin_figure +#+attr_latex: :caption \subcaption{\label{fig:rotating_campbell_diagram_real}Real part} +#+attr_latex: :options {0.49\textwidth} +#+begin_subfigure +#+attr_latex: :scale 1 +[[file:figs/rotating_campbell_diagram_real.png]] +#+end_subfigure +#+attr_latex: :caption \subcaption{\label{fig:rotating_campbell_diagram_imag}Imaginary part} +#+attr_latex: :options {0.49\textwidth} +#+begin_subfigure +#+attr_latex: :scale 1 +[[file:figs/rotating_campbell_diagram_imag.png]] +#+end_subfigure +#+end_figure ** Identify Generic Dynamics :noexport: #+begin_src matlab -%% Sample +%% Identify the dynamics for several rotating velocities +% Sample ms = 0.5; % Sample mass [kg] -%% Tuv Stage +% Tuv Stage kn = 1; % Stiffness [N/m] mn = 0.5; % Tuv mass [kg] cn = 0.01*2*sqrt(kn*(mn+ms)); % Damping [N/(m/s)] -%% General Configuration +% General Configuration model_config = struct(); model_config.controller = "open_loop"; % Default: Open-Loop model_config.Tuv_type = "normal"; % Default: 2DoF stage -#+end_src -#+begin_src matlab -%% Input/Output definition +% Input/Output definition clear io; io_i = 1; io(io_i) = linio([mdl, '/controller'], 1, 'openinput'); io_i = io_i + 1; % [Fu, Fv] io(io_i) = linio([mdl, '/fd'], 1, 'openinput'); io_i = io_i + 1; % [Fdu, Fdv] @@ -432,10 +445,8 @@ io(io_i) = linio([mdl, '/xf'], 1, 'openinput'); io_i = io_i + 1; io(io_i) = linio([mdl, '/translation_stage'], 1, 'openoutput'); io_i = io_i + 1; % [Fmu, Fmv] io(io_i) = linio([mdl, '/translation_stage'], 2, 'openoutput'); io_i = io_i + 1; % [Du, Dv] io(io_i) = linio([mdl, '/ext_metrology'], 1, 'openoutput'); io_i = io_i + 1; % [Dx, Dy] -#+end_src -#+begin_src matlab -%% Tested rotating velocities [rad/s] +% Tested rotating velocities [rad/s] Wzs = [0, 0.1, 0.2, 0.7, 1.2]; % Vector of rotation speeds [rad/s] Gs = {zeros(2, 2, length(Wzs))}; % Direct terms @@ -455,29 +466,28 @@ end #+end_src #+begin_src matlab :tangle no -%% Save All Identified Plants +% Save All Identified Plants save('./matlab/mat/rotating_generic_plants.mat', 'Gs', 'Wzs'); #+end_src #+begin_src matlab :eval no -%% Save All Identified Plants +% Save All Identified Plants save('./mat/rotating_generic_plants.mat', 'Gs', 'Wzs'); #+end_src ** System Dynamics: Effect of rotation The system dynamics from actuator forces $[F_u, F_v]$ to the relative motion $[d_u, d_v]$ is identified for several rotating velocities. - -Looking at the transfer function matrix $\mathbf{G}_d$ in equation eqref:eq:Gd_w0_xi_k, one can see that the two diagonal (direct) terms are equal and that the two off-diagonal (coupling) terms are opposite. -The bode plot of these two terms are shown in Figure ref:fig:rotating_direct_coupling_bode_plot for several rotational speeds $\Omega$. +Looking at the transfer function matrix $\mathbf{G}_d$ in equation eqref:eq:rotating_Gd_w0_xi_k, one can see that the two diagonal (direct) terms are equal and that the two off-diagonal (coupling) terms are opposite. +The bode plot of these two terms are shown in Figure ref:fig:rotating_bode_plot for several rotational speeds $\Omega$. These plots confirm the expected behavior: the frequency of the two pairs of complex conjugate poles are further separated as $\Omega$ increases. -For $\Omega > \omega_0$, the low frequency pair of complex conjugate poles $p_{-}$ becomes unstable. +For $\Omega > \omega_0$, the low frequency pair of complex conjugate poles $p_{-}$ becomes unstable (shown be the 180 degrees phase lead instead of phase lag). #+begin_src matlab :results none %% Bode plot of the direct and coupling terms for several rotating velocities freqs = logspace(-1, 1, 1000); figure; -tiledlayout(3, 2, 'TileSpacing', 'Compact', 'Padding', 'None'); +tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); % Magnitude ax1 = nexttile([2, 1]); @@ -488,26 +498,11 @@ end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); set(gca, 'XTickLabel',[]); ylabel('Magnitude [m/N]'); -title('Direct terms: $d_u/F_u$, $d_v/F_v$'); ylim([1e-2, 1e2]); yticks([1e-2,1e-1,1,1e1,1e2]) yticklabels({'$0.01/k$', '$0.1/k$', '$1/k$', '$10/k$', '$100/k$'}) -ax2 = nexttile([2, 1]); -hold on; -for i = 1:length(Wzs) - plot(freqs, abs(squeeze(freqresp(Gs{i}('dv', 'Fu'), freqs, 'rad/s'))), '-', 'color', colors(i,:), ... - 'DisplayName', sprintf('$\\Omega = %.1f \\omega_0$', Wzs(i))) -end -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]); -title('Coupling Terms: $d_u/F_v$, $d_v/F_u$'); -ldg = legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1); -ldg.ItemTokenSize = [10, 1]; -ylim([1e-2, 1e2]); - -ax3 = nexttile; +ax2 = nexttile; hold on; for i = 1:length(Wzs) plot(freqs, 180/pi*angle(squeeze(freqresp(Gs{i}('du', 'Fu'), freqs, 'rad/s'))), '-', 'color', colors(i,:)) @@ -520,31 +515,72 @@ ylim([-180 180]); xticks([1e-1,1,1e1]) xticklabels({'$0.1 \omega_0$', '$\omega_0$', '$10 \omega_0$'}) -ax4 = nexttile; +linkaxes([ax1,ax2],'x'); +xlim([freqs(1), freqs(end)]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/rotating_bode_plot_direct.pdf', 'width', 'half', 'height', 600); +#+end_src + +#+begin_src matlab :results none +figure; +tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); + +ax1 = nexttile([2, 1]); +hold on; +for i = 1:length(Wzs) + plot(freqs, abs(squeeze(freqresp(Gs{i}('dv', 'Fu'), freqs, 'rad/s'))), '-', 'color', colors(i,:), ... + 'DisplayName', sprintf('$\\Omega = %.1f \\omega_0$', Wzs(i))) +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +set(gca, 'XTickLabel',[]); ylabel('Magnitude [m/N]'); +ldg = legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1); +ldg.ItemTokenSize = [10, 1]; +ylim([1e-2, 1e2]); +yticks([1e-2,1e-1,1,1e1,1e2]) +yticklabels({'$0.01/k$', '$0.1/k$', '$1/k$', '$10/k$', '$100/k$'}) + + +ax2 = nexttile; hold on; for i = 1:length(Wzs) plot(freqs, 180/pi*angle(squeeze(freqresp(Gs{i}('dv', 'Fu'), freqs, 'rad/s'))), '-', 'color', colors(i,:)); end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); -xlabel('Frequency [rad/s]'); set(gca, 'YTickLabel',[]); +xlabel('Frequency [rad/s]'); ylabel('Phase [deg]'); +yticks(-180:90:180); +ylim([-180 180]); xticks([1e-1,1,1e1]) xticklabels({'$0.1 \omega_0$', '$\omega_0$', '$10 \omega_0$'}) -linkaxes([ax1,ax2,ax3,ax4],'x'); +linkaxes([ax1,ax2],'x'); xlim([freqs(1), freqs(end)]); - -linkaxes([ax1,ax2],'y'); #+end_src #+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/rotating_direct_coupling_bode_plot.pdf', 'width', 'full', 'height', 'tall'); +exportFig('figs/rotating_bode_plot_coupling.pdf', 'width', 'half', 'height', 600); #+end_src -#+name: fig:rotating_direct_coupling_bode_plot -#+caption: Bode plot of the direct and coupling terms for several rotating velocities -#+RESULTS: -[[file:figs/rotating_direct_coupling_bode_plot.png]] +#+name: fig:rotating_bode_plot +#+caption: Bode plot of the direct (\subref{fig:rotating_bode_plot_direct}) and coupling (\subref{fig:rotating_bode_plot_direct}) terms for several rotating velocities +#+attr_latex: :options [htbp] +#+begin_figure +#+attr_latex: :caption \subcaption{\label{fig:rotating_bode_plot_direct}Direct terms: $d_u/F_u$, $d_v/F_v$} +#+attr_latex: :options {0.49\textwidth} +#+begin_subfigure +#+attr_latex: :width 0.95\linewidth +[[file:figs/rotating_bode_plot_direct.png]] +#+end_subfigure +#+attr_latex: :caption \subcaption{\label{fig:rotating_bode_plot_coupling}Coupling terms: $d_u/F_v$, $d_v/F_u$} +#+attr_latex: :options {0.49\textwidth} +#+begin_subfigure +#+attr_latex: :width 0.95\linewidth +[[file:figs/rotating_bode_plot_coupling.png]] +#+end_subfigure +#+end_figure * Integral Force Feedback :PROPERTIES: @@ -553,15 +589,15 @@ exportFig('figs/rotating_direct_coupling_bode_plot.pdf', 'width', 'full', 'heigh <> ** Introduction :ignore: -In order to further decrease the residual vibrations, active damping can be used for reducing the magnification of the response in the vicinity of the resonances cite:collette11_review_activ_vibrat_isolat_strat. +The goal is now to damp the two suspension modes of the payload using an active damping strategy while the rotating stage performs a constant rotation. +As was explained with the uniaxial model, such active damping strategy is key to both reducing the magnification of the response in the vicinity of the resonances cite:collette11_review_activ_vibrat_isolat_strat and to make the plant easier to control for the high authority controller. -Many active damping techniques have been developed over the years such as Positive Position Feedback (PPF) cite:lin06_distur_atten_precis_hexap_point,fanson90_posit_posit_feedb_contr_large_space_struc, Integral Force Feedback (IFF) cite:preumont91_activ and Direct Velocity Feedback (DVF) cite:karnopp74_vibrat_contr_using_semi_activ_force_gener,serrand00_multic_feedb_contr_isolat_base_excit_vibrat,preumont02_force_feedb_versus_accel_feedb. \par +Many active damping techniques have been developed over the years such as Positive Position Feedback (PPF) cite:lin06_distur_atten_precis_hexap_point,fanson90_posit_posit_feedb_contr_large_space_struc, Integral Force Feedback (IFF) cite:preumont91_activ and Direct Velocity Feedback (DVF) cite:karnopp74_vibrat_contr_using_semi_activ_force_gener,serrand00_multic_feedb_contr_isolat_base_excit_vibrat,preumont02_force_feedb_versus_accel_feedb. +In cite:preumont92_activ_dampin_by_local_force, the IFF control scheme has been proposed, where a force sensor, a force actuator and an integral controller are used to increase the damping of a mechanical system. +When the force sensor is collocated with the actuator, the open-loop transfer function has alternating poles and zeros which facilitates to guarantee the stability of the closed loop system cite:preumont02_force_feedb_versus_accel_feedb. +It was latter shown that this property holds for multiple collated actuator/sensor pairs cite:preumont08_trans_zeros_struc_contr_with. -In cite:preumont92_activ_dampin_by_local_force, the IFF control scheme has been proposed, where a force sensor, a force actuator and an integral controller are used to directly augment the damping of a mechanical system. -When the force sensor is collocated with the actuator, the open-loop transfer function has alternating poles and zeros which facilitate to guarantee the stability of the closed loop system cite:preumont02_force_feedb_versus_accel_feedb. -It was latter shown that this property holds for multiple collated actuator/sensor pairs cite:preumont08_trans_zeros_struc_contr_with. \par - -The main advantages of IFF over other active damping techniques are the guaranteed stability even in presence of flexible dynamics, good performances and robustness properties cite:preumont02_force_feedb_versus_accel_feedb. \par +The main advantages of IFF over other active damping techniques are the guaranteed stability even in presence of flexible dynamics, good performance and robustness properties cite:preumont02_force_feedb_versus_accel_feedb. Several improvements of the classical IFF have been proposed, such as adding a feed-through term to increase the achievable damping cite:teo15_optim_integ_force_feedb_activ_vibrat_contr or adding an high pass filter to recover the loss of compliance at low frequency cite:chesne16_enhan_dampin_flexib_struc_using_force_feedb. Recently, an $\mathcal{H}_\infty$ optimization criterion has been used to derive optimal gains for the IFF controller cite:zhao19_optim_integ_force_feedb_contr. \par @@ -602,227 +638,85 @@ load('rotating_generic_plants.mat', 'Gs', 'Wzs'); ** System and Equations of motion In order to apply Integral Force Feedback, two force sensors are added in series with the actuators (Figure ref:fig:rotating_3dof_model_schematic_iff). -Two identical controllers $K_F$ are then used to feedback each of the sensed force to its associated actuator: -\begin{equation} +Two identical controllers $K_F$ described by eqref:eq:rotating_iff_controller are then used to feedback each of the sensed force to its associated actuator. + +\begin{equation}\label{eq:rotating_iff_controller} K_{F}(s) = g \cdot \frac{1}{s} \end{equation} #+begin_src latex :file rotating_3dof_model_schematic_iff.pdf - \begin{tikzpicture} - % Angle - \def\thetau{25} +\begin{tikzpicture} + % Angle + \def\thetau{25} - % Rotational Stage - \draw[fill=black!60!white] (0, 0) circle (4.3); - \draw[fill=black!40!white] (0, 0) circle (3.8); + % Rotational Stage + \draw[fill=black!60!white] (0, 0) circle (4.3); + \draw[fill=black!40!white] (0, 0) circle (3.8); + % Label + \node[anchor=north west, rotate=\thetau] at (-2.5, 2.5) {\small Rotating Stage}; + + % Rotating Scope + \begin{scope}[rotate=\thetau] + % Rotating Frame + \draw[fill=black!20!white] (-2.6, -2.6) rectangle (2.6, 2.6); % Label - \node[anchor=north west, rotate=\thetau] at (-2.5, 2.5) {\small Rotating Stage}; + \node[anchor=north west, rotate=\thetau] at (-2.6, 2.6) {\small Suspended Platform}; - % Rotating Scope - \begin{scope}[rotate=\thetau] - % Rotating Frame - \draw[fill=black!20!white] (-2.6, -2.6) rectangle (2.6, 2.6); - % Label - \node[anchor=north west, rotate=\thetau] at (-2.6, 2.6) {\small Suspended Platform}; + % Mass + \draw[fill=white] (-1, -1) rectangle (1, 1); + % Label + \node[anchor=south west, rotate=\thetau] at (-1, -1) {\small Payload}; - % Mass - \draw[fill=white] (-1, -1) rectangle (1, 1); - % Label - \node[anchor=south west, rotate=\thetau] at (-1, -1) {\small Payload}; + % Attached Points + \node[] at (-1, 0){$\bullet$}; + \draw[] (-1, 0) -- ++(-0.2, 0) coordinate(au); + \node[] at (0, -1){$\bullet$}; + \draw[] (0, -1) -- ++(0, -0.2) coordinate(av); - % Attached Points - \node[] at (-1, 0){$\bullet$}; - \draw[] (-1, 0) -- ++(-0.2, 0) coordinate(au); - \node[] at (0, -1){$\bullet$}; - \draw[] (0, -1) -- ++(0, -0.2) coordinate(av); + % Force Sensors + \draw[draw=colorblue, fill=colorblue!10!white] ($(au) + (-0.2, -0.5)$) rectangle ($(au) + (0, 0.5)$); + \draw[draw=colorblue] ($(au) + (-0.2, -0.5)$)coordinate(actu) -- ($(au) + (0, 0.5)$); + \draw[draw=colorblue] ($(au) + (-0.2, 0.5)$)coordinate(ku) -- ($(au) + (0, -0.5)$); - % Force Sensors - \draw[draw=colorblue, fill=colorblue!10!white] ($(au) + (-0.2, -0.5)$) rectangle ($(au) + (0, 0.5)$); - \draw[draw=colorblue] ($(au) + (-0.2, -0.5)$)coordinate(actu) -- ($(au) + (0, 0.5)$); - \draw[draw=colorblue] ($(au) + (-0.2, 0.5)$)coordinate(ku) -- ($(au) + (0, -0.5)$); + \draw[draw=colorblue, fill=colorblue!10!white] ($(av) + (-0.5, -0.2)$) rectangle ($(av) + (0.5, 0)$); + \draw[draw=colorblue] ($(av) + ( 0.5, -0.2)$)coordinate(actv) -- ($(av) + (-0.5, 0)$); + \draw[draw=colorblue] ($(av) + (-0.5, -0.2)$)coordinate(kv) -- ($(av) + ( 0.5, 0)$); - \draw[draw=colorblue, fill=colorblue!10!white] ($(av) + (-0.5, -0.2)$) rectangle ($(av) + (0.5, 0)$); - \draw[draw=colorblue] ($(av) + ( 0.5, -0.2)$)coordinate(actv) -- ($(av) + (-0.5, 0)$); - \draw[draw=colorblue] ($(av) + (-0.5, -0.2)$)coordinate(kv) -- ($(av) + ( 0.5, 0)$); + % Spring and Actuator for U + \draw[actuator={0.6}{0.2}{black}] (actu) -- coordinate[midway](actumid) (actu-|-2.6,0); + \draw[spring=0.2] (ku) -- node[above=0.1, rotate=\thetau]{$k$} (ku-|-2.6,0); - % Spring and Actuator for U - \draw[actuator={0.6}{0.2}{black}] (actu) -- coordinate[midway](actumid) (actu-|-2.6,0); - \draw[spring=0.2] (ku) -- node[above=0.1, rotate=\thetau]{$k$} (ku-|-2.6,0); + % \draw[actuator={0.6}{0.2}] (actv) -- node[right, rotate=\thetau]{$F_v$} (actv|-0,-2.6); + \draw[actuator={0.6}{0.2}{black}] (actv) -- coordinate[midway](actvmid) (actv|-0,-2.6); + \draw[spring=0.2] (kv) -- node[left, rotate=\thetau]{$k$} (kv|-0,-2.6); - % \draw[actuator={0.6}{0.2}] (actv) -- node[right, rotate=\thetau]{$F_v$} (actv|-0,-2.6); - \draw[actuator={0.6}{0.2}{black}] (actv) -- coordinate[midway](actvmid) (actv|-0,-2.6); - \draw[spring=0.2] (kv) -- node[left, rotate=\thetau]{$k$} (kv|-0,-2.6); + \node[color=colorblue, block={0.8cm}{0.6cm}, fill=colorblue!10!white, rotate=\thetau] (Ku) at ($(actumid) + (0, -1.2)$) {$K_{F}$}; + \draw[->, draw=colorblue] ($(au) + (-0.1, -0.5)$) |- (Ku.east) node[below right, rotate=\thetau]{$f_{u}$}; + \draw[->, draw=colorblue] (Ku.north) -- ($(actumid) + (0, -0.1)$) node[below left, rotate=\thetau]{$F_u$}; - \node[color=colorblue, block={0.8cm}{0.6cm}, fill=colorblue!10!white, rotate=\thetau] (Ku) at ($(actumid) + (0, -1.2)$) {$K_{F}$}; - \draw[->, draw=colorblue] ($(au) + (-0.1, -0.5)$) |- (Ku.east) node[below right, rotate=\thetau]{$f_{u}$}; - \draw[->, draw=colorblue] (Ku.north) -- ($(actumid) + (0, -0.1)$) node[below left, rotate=\thetau]{$F_u$}; + \node[color=colorblue, block={0.8cm}{0.6cm}, fill=colorblue!10!white, rotate=\thetau] (Kv) at ($(actvmid) + (1.2, 0)$) {$K_{F}$}; + \draw[->, draw=colorblue] ($(av) + (0.5, -0.1)$) -| (Kv.north) node[above right, rotate=\thetau]{$f_{v}$}; + \draw[->, draw=colorblue] (Kv.west) -- ($(actvmid) + (0.1, 0)$) node[below right, rotate=\thetau]{$F_v$}; + \end{scope} - \node[color=colorblue, block={0.8cm}{0.6cm}, fill=colorblue!10!white, rotate=\thetau] (Kv) at ($(actvmid) + (1.2, 0)$) {$K_{F}$}; - \draw[->, draw=colorblue] ($(av) + (0.5, -0.1)$) -| (Kv.north) node[above right, rotate=\thetau]{$f_{v}$}; - \draw[->, draw=colorblue] (Kv.west) -- ($(actvmid) + (0.1, 0)$) node[below right, rotate=\thetau]{$F_v$}; - \end{scope} + % Inertial Frame + \draw[->] (-4, -4) -- ++(2, 0) node[below]{$\vec{i}_x$}; + \draw[->] (-4, -4) -- ++(0, 2) node[left]{$\vec{i}_y$}; + \draw[fill, color=black] (-4, -4) circle (0.06); + \node[draw, circle, inner sep=0pt, minimum size=0.3cm, label=left:$\vec{i}_z$] at (-4, -4){}; - % Inertial Frame - \draw[->] (-4, -4) -- ++(2, 0) node[below]{$\vec{i}_x$}; - \draw[->] (-4, -4) -- ++(0, 2) node[left]{$\vec{i}_y$}; - \draw[fill, color=black] (-4, -4) circle (0.06); - \node[draw, circle, inner sep=0pt, minimum size=0.3cm, label=left:$\vec{i}_z$] at (-4, -4){}; + \node[draw, circle, inner sep=0pt, minimum size=0.3cm] at (0, 0){}; + \draw[->] (0, 0) node[above left, rotate=\thetau]{$\vec{i}_w$} -- ++(\thetau:2) node[above, rotate=\thetau]{$\vec{i}_u$}; + \draw[->] (0, 0) -- ++(\thetau+90:2) node[left, rotate=\thetau]{$\vec{i}_v$}; + \draw[dashed] (0, 0) -- ++(2, 0); + \draw[] (1.5, 0) arc (0:\thetau:1.5) node[midway, right]{$\theta$}; + \node[] at (0,0) {$\bullet$}; - \node[draw, circle, inner sep=0pt, minimum size=0.3cm] at (0, 0){}; - \draw[->] (0, 0) node[above left, rotate=\thetau]{$\vec{i}_w$} -- ++(\thetau:2) node[above, rotate=\thetau]{$\vec{i}_u$}; - \draw[->] (0, 0) -- ++(\thetau+90:2) node[left, rotate=\thetau]{$\vec{i}_v$}; - \draw[dashed] (0, 0) -- ++(2, 0); - \draw[] (1.5, 0) arc (0:\thetau:1.5) node[midway, right]{$\theta$}; - \node[] at (0,0) {$\bullet$}; - - \draw[->] (3.5, 0) arc (0:40:3.5) node[midway, left]{$\Omega$}; - \end{tikzpicture} + \draw[->] (3.5, 0) arc (0:40:3.5) node[midway, left]{$\Omega$}; +\end{tikzpicture} #+end_src -#+name: fig:rotating_3dof_model_schematic_iff -#+caption: System with added Force Sensor in series with the actuators (shown in blue with the associated controllers) -#+RESULTS: -[[file:figs/rotating_3dof_model_schematic_iff.png]] - - -The forces $\begin{bmatrix}f_u & f_v\end{bmatrix}$ measured by the two force sensors represented in Figure ref:fig:rotating_3dof_model_schematic_iff are described by equation eqref:eq:measured_force. -#+name: eq:measured_force -\begin{equation} - \begin{bmatrix} f_{u} \\ f_{v} \end{bmatrix} = - \begin{bmatrix} F_u \\ F_v \end{bmatrix} - (c s + k) - \begin{bmatrix} d_u \\ d_v \end{bmatrix} -\end{equation} - -The transfer function matrix $\mathbf{G}_{f}$ from actuator forces to measured forces in equation eqref:eq:Gf_mimo_tf can be obtained by inserting equation eqref:eq:Gd_w0_xi_k into equation eqref:eq:measured_force. -Its elements are shown in equation eqref:eq:Gf_indiv_el. - -#+name: eq:Gf_mimo_tf -\begin{equation} - \begin{bmatrix} f_{u} \\ f_{v} \end{bmatrix} = \mathbf{G}_{f} \begin{bmatrix} F_u \\ F_v \end{bmatrix} -\end{equation} - -#+name: eq:Gf_indiv_el -\begin{subequations} -\label{eq:Gf} - \begin{align} - \mathbf{G}_{f}(1,1) &= \mathbf{G}_{f}(2,2) = \frac{\left( \frac{s^2}{{\omega_0}^2} - \frac{\Omega^2}{{\omega_0}^2} \right) \left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right) + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2}{\left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2} \label{eq:Gf_diag_tf} \\ - \mathbf{G}_{f}(1,2) &= -\mathbf{G}_{f}(2,1) = \frac{- \left( 2 \xi \frac{s}{\omega_0} + 1 \right) \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)}{\left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2} \label{eq:Gf_off_diag_tf} - \end{align} -\end{subequations} - -The zeros of the diagonal terms of $\mathbf{G}_f$ in equation eqref:eq:Gf_diag_tf are computed, and neglecting the damping for simplicity, *two complex conjugated zeros* $z_{c}$ are obtained in equation eqref:eq:iff_zero_cc, and *two real zeros* $z_{r}$ in equation eqref:eq:iff_zero_real. -\begin{subequations} - \begin{align} - z_c &= \pm j \omega_0 \sqrt{\frac{1}{2} \sqrt{8 \frac{\Omega^2}{{\omega_0}^2} + 1} + \frac{\Omega^2}{{\omega_0}^2} + \frac{1}{2} } \label{eq:iff_zero_cc} \\ - z_r &= \pm \omega_0 \sqrt{\frac{1}{2} \sqrt{8 \frac{\Omega^2}{{\omega_0}^2} + 1} - \frac{\Omega^2}{{\omega_0}^2} - \frac{1}{2} } \label{eq:iff_zero_real} - \end{align} -\end{subequations} - -It is interesting to see that the frequency of the pair of complex conjugate zeros $z_c$ in equation eqref:eq:iff_zero_cc always lies between the frequency of the two pairs of complex conjugate poles $p_{-}$ and $p_{+}$ in equation eqref:eq:pole_values. -This is what usually gives the unconditional stability of IFF when collocated force sensors are used. - -However, for non-null rotational speeds, the two real zeros $z_r$ in equation eqref:eq:iff_zero_real are inducing a *non-minimum phase behavior*. -This can be seen in the Bode plot of the diagonal terms (Figure ref:fig:rotating_iff_bode_plot_effect_rot) where the low frequency gain is no longer zero while the phase stays at $\SI{180}{\degree}$. - -The low frequency gain of $\mathbf{G}_f$ increases with the rotational speed $\Omega$ as shown in equation eqref:eq:low_freq_gain_iff_plan. -#+name: eq:low_freq_gain_iff_plan -\begin{equation} - \lim_{\omega \to 0} \left| \mathbf{G}_f (j\omega) \right| = \begin{bmatrix} - \frac{\Omega^2}{{\omega_0}^2 - \Omega^2} & 0 \\ - 0 & \frac{\Omega^2}{{\omega_0}^2 - \Omega^2} -\end{bmatrix} -\end{equation} - -This can be explained as follows: a constant actuator force $F_u$ induces a small displacement of the mass $d_u = \frac{F_u}{k - m\Omega^2}$ (Hooke's law taking into account the negative stiffness induced by the rotation). -This small displacement then increases the centrifugal force $m\Omega^2d_u = \frac{\Omega^2}{{\omega_0}^2 - \Omega^2} F_u$ which is then measured by the force sensors. - -** Effect of the rotation speed on the IFF plant dynamics -The transfer functions from actuator forces $[F_u,\ F_v]$ to the measured force sensors $[f_u,\ f_v]$ are identified for several rotating velocities and shown in Figure ref:fig:rotating_iff_bode_plot_effect_rot. - -As was expected from the derived equations of motion: -- when $0 < \Omega < \omega_0$: the low frequency gain is no longer zero and two (non-minimum phase) real zero appears at low frequency. - The low frequency gain increases with $\Omega$. - A pair of (minimum phase) complex conjugate zeros appears between the two complex conjugate poles that are split further apart as $\Omega$ increases. -- when $\omega_0 < \Omega$: the low frequency pole becomes unstable. - -#+begin_src matlab :results none -%% Bode plot of the direct and coupling term for Integral Force Feedback - Effect of rotation -freqs = logspace(-2, 1, 1000); - -figure; -tiledlayout(3, 2, 'TileSpacing', 'Compact', 'Padding', 'None'); - -% Magnitude -ax1 = nexttile([2, 1]); -hold on; -for i = 1:length(Wzs) - plot(freqs, abs(squeeze(freqresp(Gs{i}('fu', 'Fu'), freqs, 'rad/s'))), '-', 'color', colors(i,:)) -end -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -set(gca, 'XTickLabel',[]); ylabel('Magnitude [N/N]'); -title('Direct terms: $f_u/F_u$, $f_v/F_v$'); -ylim([1e-3, 1e2]); - -ax2 = nexttile([2, 1]); -hold on; -for i = 1:length(Wzs) - plot(freqs, abs(squeeze(freqresp(Gs{i}('fv', 'Fu'), freqs, 'rad/s'))), '-', 'color', colors(i,:), ... - 'DisplayName', sprintf('$\\Omega = %.1f \\omega_0$', Wzs(i))) -end -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]); -title('Coupling Terms: $f_u/F_v$, $f_v/F_u$'); -ldg = legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 1); -ldg.ItemTokenSize = [10, 1]; -ylim([1e-3, 1e2]); - -ax3 = nexttile; -hold on; -for i = 1:length(Wzs) - plot(freqs, 180/pi*angle(squeeze(freqresp(Gs{i}('fu', 'Fu'), freqs, 'rad/s'))), '-', 'color', colors(i,:)) -end -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); -xlabel('Frequency [rad/s]'); ylabel('Phase [deg]'); -yticks(-180:90:180); -ylim([-180 180]); -xticks([1e-2,1e-1,1,1e1]) -xticklabels({'$0.01 \omega_0$', '$0.1 \omega_0$', '$\omega_0$', '$10 \omega_0$'}) - -ax4 = nexttile; -hold on; -for i = 1:length(Wzs) - plot(freqs, 180/pi*angle(squeeze(freqresp(Gs{i}('fv', 'Fu'), freqs, 'rad/s'))), '-', 'color', colors(i,:)); -end -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); -xlabel('Frequency [rad/s]'); set(gca, 'YTickLabel',[]); -yticks(-180:90:180); -ylim([-180 180]); -xticks([1e-2,1e-1,1,1e1]) -xticklabels({'$0.01 \omega_0$', '$0.1 \omega_0$', '$\omega_0$', '$10 \omega_0$'}) - -linkaxes([ax1,ax2,ax3,ax4],'x'); -xlim([freqs(1), freqs(end)]); - -linkaxes([ax1,ax2],'y'); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/rotating_iff_bode_plot_effect_rot.pdf', 'width', 'full', 'height', 'tall'); -#+end_src - -#+name: fig:rotating_iff_bode_plot_effect_rot -#+caption: Bode plot of the direct and coupling term for Integral Force Feedback - Effect of rotation -#+RESULTS: -[[file:figs/rotating_iff_bode_plot_effect_rot.png]] - -** Decentralized Integral Force Feedback -The control diagram for decentralized Integral Force Feedback is shown in Figure ref:fig:rotating_iff_diagram. - #+begin_src latex :file rotating_iff_diagram.pdf \tikzset{block/.default={0.8cm}{0.8cm}} \tikzset{addb/.append style={scale=0.7}} @@ -857,14 +751,146 @@ The control diagram for decentralized Integral Force Feedback is shown in Figure \end{tikzpicture} #+end_src -#+name: fig:rotating_iff_diagram -#+caption: Control diagram for decentralized Integral Force Feedback -#+RESULTS: +#+name: fig:rotating_iff_pure_int +#+caption: Integral Force Feedback applied to the suspended rotating platform. The damper $c$ in (\subref{fig:rotating_3dof_model_schematic_iff}) is omitted for readability. +#+attr_latex: :options [htbp] +#+begin_figure +#+attr_latex: :caption \subcaption{\label{fig:rotating_3dof_model_schematic_iff}System with added Force Sensor in series with the actuators} +#+attr_latex: :options {0.49\textwidth} +#+begin_subfigure +#+attr_latex: :width 0.95\linewidth +[[file:figs/rotating_3dof_model_schematic_iff.png]] +#+end_subfigure +#+attr_latex: :caption \subcaption{\label{fig:rotating_iff_diagram}Control diagram} +#+attr_latex: :options {0.49\textwidth} +#+begin_subfigure +#+attr_latex: :scale 1 [[file:figs/rotating_iff_diagram.png]] +#+end_subfigure +#+end_figure + +The forces $\begin{bmatrix}f_u & f_v\end{bmatrix}$ measured by the two force sensors represented in Figure ref:fig:rotating_3dof_model_schematic_iff are described by equation eqref:eq:rotating_measured_force. + +\begin{equation}\label{eq:rotating_measured_force} + \begin{bmatrix} f_{u} \\ f_{v} \end{bmatrix} = + \begin{bmatrix} F_u \\ F_v \end{bmatrix} - (c s + k) + \begin{bmatrix} d_u \\ d_v \end{bmatrix} +\end{equation} + +The transfer function matrix $\mathbf{G}_{f}$ from actuator forces to measured forces in equation eqref:eq:rotating_Gf_mimo_tf can be obtained by inserting equation eqref:eq:rotating_Gd_w0_xi_k into equation eqref:eq:rotating_measured_force. +Its elements are shown in equation eqref:eq:rotating_Gf. + +\begin{equation}\label{eq:rotating_Gf_mimo_tf} + \begin{bmatrix} f_{u} \\ f_{v} \end{bmatrix} = \mathbf{G}_{f} \begin{bmatrix} F_u \\ F_v \end{bmatrix} +\end{equation} + +\begin{subequations}\label{eq:rotating_Gf} + \begin{align} + \mathbf{G}_{f}(1,1) &= \mathbf{G}_{f}(2,2) = \frac{\left( \frac{s^2}{{\omega_0}^2} - \frac{\Omega^2}{{\omega_0}^2} \right) \left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right) + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2}{\left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2} \label{eq:rotating_Gf_diag_tf} \\ + \mathbf{G}_{f}(1,2) &= -\mathbf{G}_{f}(2,1) = \frac{- \left( 2 \xi \frac{s}{\omega_0} + 1 \right) \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)}{\left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2} \label{eq:rotating_Gf_off_diag_tf} + \end{align} +\end{subequations} + +The zeros of the diagonal terms of $\mathbf{G}_f$ in equation eqref:eq:rotating_Gf_diag_tf are computed, and neglecting the damping for simplicity, two complex conjugated zeros $z_{c}$ eqref:eq:rotating_iff_zero_cc, and two real zeros $z_{r}$ eqref:eq:rotating_iff_zero_real are obtained. + +\begin{subequations} + \begin{align} + z_c &= \pm j \omega_0 \sqrt{\frac{1}{2} \sqrt{8 \frac{\Omega^2}{{\omega_0}^2} + 1} + \frac{\Omega^2}{{\omega_0}^2} + \frac{1}{2} } \label{eq:rotating_iff_zero_cc} \\ + z_r &= \pm \omega_0 \sqrt{\frac{1}{2} \sqrt{8 \frac{\Omega^2}{{\omega_0}^2} + 1} - \frac{\Omega^2}{{\omega_0}^2} - \frac{1}{2} } \label{eq:rotating_iff_zero_real} + \end{align} +\end{subequations} + +It is interesting to see that the frequency of the pair of complex conjugate zeros $z_c$ in equation eqref:eq:rotating_iff_zero_cc always lies between the frequency of the two pairs of complex conjugate poles $p_{-}$ and $p_{+}$ in equation eqref:eq:rotating_pole_values. +This is what usually gives the unconditional stability of IFF when collocated force sensors are used. + +However, for non-null rotational speeds, the two real zeros $z_r$ in equation eqref:eq:rotating_iff_zero_real are inducing a /non-minimum phase behavior/. +This can be seen in the Bode plot of the diagonal terms (Figure ref:fig:rotating_iff_bode_plot_effect_rot) where the low frequency gain is no longer zero while the phase stays at $\SI{180}{\degree}$. + +The low frequency gain of $\mathbf{G}_f$ increases with the rotational speed $\Omega$ as shown in equation eqref:eq:rotating_low_freq_gain_iff_plan. +This can be explained as follows: a constant actuator force $F_u$ induces a small displacement of the mass $d_u = \frac{F_u}{k - m\Omega^2}$ (Hooke's law taking into account the negative stiffness induced by the rotation). +This small displacement then increases the centrifugal force $m\Omega^2d_u = \frac{\Omega^2}{{\omega_0}^2 - \Omega^2} F_u$ which is then measured by the force sensors. + +\begin{equation}\label{eq:rotating_low_freq_gain_iff_plan} + \lim_{\omega \to 0} \left| \mathbf{G}_f (j\omega) \right| = \begin{bmatrix} + \frac{\Omega^2}{{\omega_0}^2 - \Omega^2} & 0 \\ + 0 & \frac{\Omega^2}{{\omega_0}^2 - \Omega^2} +\end{bmatrix} +\end{equation} + +** Effect of the rotation speed on the IFF plant dynamics +The transfer functions from actuator forces $[F_u,\ F_v]$ to the measured force sensors $[f_u,\ f_v]$ are identified for several rotating velocities and are shown in Figure ref:fig:rotating_iff_bode_plot_effect_rot. +As was expected from the derived equations of motion: +- when $\Omega < \omega_0$: the low frequency gain is no longer zero and two (non-minimum phase) real zero appears at low frequency. + The low frequency gain increases with $\Omega$. + A pair of (minimum phase) complex conjugate zeros appears between the two complex conjugate poles that are split further apart as $\Omega$ increases. +- when $\omega_0 < \Omega$: the low frequency pole becomes unstable. + +#+begin_src matlab :results none +%% Bode plot of the direct and coupling term for Integral Force Feedback - Effect of rotation +freqs = logspace(-2, 1, 1000); + +Wz_i = [1,3,4]; + +figure; +tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); + +% Magnitude +ax1 = nexttile([2, 1]); +hold on; +for i = 1:length(Wz_i) + plot(freqs, abs(squeeze(freqresp(Gs{Wz_i(i)}('fu', 'Fu'), freqs, 'rad/s'))), '-', 'color', colors(i,:), ... + 'DisplayName', sprintf('$\\Omega = %.1f \\omega_0 $', Wzs(Wz_i(i))),'MarkerSize',8); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +set(gca, 'XTickLabel',[]); ylabel('Magnitude [N/N]'); +ylim([1e-3, 1e2]); +leg = legend('location', 'northwest', 'FontSize', 8); + +ax2 = nexttile; +hold on; +for i = 1:length(Wz_i) + plot(freqs, 180/pi*angle(squeeze(freqresp(Gs{Wz_i(i)}('fu', 'Fu'), freqs, 'rad/s'))), '-', 'color', colors(i,:)) +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +xlabel('Frequency [rad/s]'); ylabel('Phase [deg]'); +yticks(-180:90:180); +ylim([0 180]); +xticks([1e-2,1e-1,1,1e1]) +xticklabels({'$0.01 \omega_0$', '$0.1 \omega_0$', '$\omega_0$', '$10 \omega_0$'}) + +linkaxes([ax1,ax2],'x'); +xlim([freqs(1), freqs(end)]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file none +exportFig('figs/rotating_iff_bode_plot_effect_rot_direct.pdf', 'width', 'half', 'height', 600); +#+end_src + +#+name: fig:rotating_iff_bode_plot_effect_rot +#+caption: Effect of the rotation velocity on the bode plot of the direct terms (\subref{fig:rotating_iff_bode_plot_effect_rot_direct}) and on the IFF root locus (\subref{fig:rotating_root_locus_iff_pure_int}) +#+attr_latex: :options [htbp] +#+begin_figure +#+attr_latex: :caption \subcaption{\label{fig:rotating_iff_bode_plot_effect_rot_direct}Direct terms: $d_u/F_u$, $d_v/F_v$} +#+attr_latex: :options {0.49\textwidth} +#+begin_subfigure +#+attr_latex: :width 0.95\linewidth +[[file:figs/rotating_iff_bode_plot_effect_rot_direct.png]] +#+end_subfigure +#+attr_latex: :caption \subcaption{\label{fig:rotating_root_locus_iff_pure_int}Root Locus} +#+attr_latex: :options {0.49\textwidth} +#+begin_subfigure +#+attr_latex: :scale 1 +[[file:figs/rotating_root_locus_iff_pure_int.png]] +#+end_subfigure +#+end_figure + +** Decentralized Integral Force Feedback +The control diagram for decentralized Integral Force Feedback is shown in Figure ref:fig:rotating_iff_diagram. The decentralized IFF controller $\bm{K}_F$ corresponds to a diagonal controller with integrators: -#+name: eq:Kf_pure_int -\begin{equation} +\begin{equation} \label{eq:rotating_Kf_pure_int} \begin{aligned} \mathbf{K}_{F}(s) &= \begin{bmatrix} K_{F}(s) & 0 \\ 0 & K_{F}(s) \end{bmatrix} \\ K_{F}(s) &= g \cdot \frac{1}{s} @@ -874,13 +900,10 @@ The decentralized IFF controller $\bm{K}_F$ corresponds to a diagonal controller In order to see how the IFF controller affects the poles of the closed loop system, a Root Locus plot (Figure ref:fig:rotating_root_locus_iff_pure_int) is constructed as follows: the poles of the closed-loop system are drawn in the complex plane as the controller gain $g$ varies from $0$ to $\infty$ for the two controllers $K_{F}$ simultaneously. As explained in cite:preumont08_trans_zeros_struc_contr_with,skogestad07_multiv_feedb_contr, the closed-loop poles start at the open-loop poles (shown by $\tikz[baseline=-0.6ex] \node[cross out, draw=black, minimum size=1ex, line width=2pt, inner sep=0pt, outer sep=0pt] at (0, 0){};$) for $g = 0$ and coincide with the transmission zeros (shown by $\tikz[baseline=-0.6ex] \draw[line width=2pt, inner sep=0pt, outer sep=0pt] (0,0) circle[radius=3pt];$) as $g \to \infty$. -#+begin_important Whereas collocated IFF is usually associated with unconditional stability cite:preumont91_activ, this property is lost due to gyroscopic effects as soon as the rotation velocity in non-null. This can be seen in the Root Locus plot (Figure ref:fig:rotating_root_locus_iff_pure_int) where poles corresponding to the controller are bound to the right half plane implying closed-loop system instability. -#+end_important - Physically, this can be explained like so: at low frequency, the loop gain is very large due to the pure integrator in $K_{F}$ and the finite gain of the plant (Figure ref:fig:rotating_iff_bode_plot_effect_rot). -The control system is thus canceling the spring forces which makes the suspended platform no able to hold the payload against centrifugal forces, hence the instability. +The control system is thus canceling the spring forces which makes the suspended platform not capable to hold the payload against centrifugal forces, hence the instability. #+begin_src matlab %% Root Locus for the Decentralized Integral Force Feedback controller @@ -915,15 +938,10 @@ leg = legend('location', 'northwest', 'FontSize', 8); leg.ItemTokenSize(1) = 8; #+end_src -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/rotating_root_locus_iff_pure_int.pdf', 'width', 'wide', 'height', 'tall'); +#+begin_src matlab :tangle no :exports results :results file none +exportFig('figs/rotating_root_locus_iff_pure_int.pdf', 'width', 600, 'height', 600); #+end_src -#+name: fig:rotating_root_locus_iff_pure_int -#+caption: Root Locus for the Decentralized Integral Force Feedback controller. Several rotating speed are shown. -#+RESULTS: -[[file:figs/rotating_root_locus_iff_pure_int.png]] - * Integral Force Feedback with an High Pass Filter :PROPERTIES: :header-args:matlab+: :tangle matlab/rotating_3_iff_hpf.m @@ -931,20 +949,16 @@ exportFig('figs/rotating_root_locus_iff_pure_int.pdf', 'width', 'wide', 'height' <> ** Introduction :ignore: -As was explained in the previous section, the instability of the IFF controller applied on the rotating system comes in part from the high gain at low frequency caused by the pure integrators. +As was explained in the previous section, the instability of the IFF controller applied on the rotating system is due to the high gain of the integrator at low frequency. +In order to limit the low frequency controller gain, an High Pass Filter (HPF) can be added to the controller as shown in equation eqref:eq:rotating_iff_lhf. +This is equivalent to slightly shifting the controller pole to the left along the real axis. +This modification of the IFF controller is typically done to avoid saturation associated with the pure integrator cite:preumont91_activ,marneffe07_activ_passiv_vibrat_isolat_dampin_shunt_trans. +This is however not the reason why this high pass filter is added here. -In order to limit the low frequency controller gain, an High Pass Filter (HPF) can be added to the controller as shown in equation eqref:eq:iff_lhf. - -#+name: eq:iff_lhf -\begin{equation} +\begin{equation}\label{eq:rotating_iff_lhf} \boxed{K_{F}(s) = g \cdot \frac{1}{s} \cdot \underbrace{\frac{s/\omega_i}{1 + s/\omega_i}}_{\text{HPF}} = g \cdot \frac{1}{s + \omega_i}} \end{equation} -This is equivalent as to slightly *shifting the controller pole to the left along the real axis*. - -This modification of the IFF controller is typically done to avoid saturation associated with the pure integrator cite:preumont91_activ,marneffe07_activ_passiv_vibrat_isolat_dampin_shunt_trans. -This is however not why this high pass filter is added here. - ** Matlab Init :noexport:ignore: #+begin_src matlab :tangle no :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) <> @@ -977,17 +991,19 @@ load('rotating_generic_plants.mat', 'Gs', 'Wzs'); #+end_src ** Modified Integral Force Feedback Controller -The Integral Force Feedback Controller is modified such that instead of using pure integrators, pseudo integrators (i.e. low pass filters) are used: -\begin{equation} - K_{\text{IFF}}(s) = g\frac{1}{\omega_i + s} \begin{bmatrix} - 1 & 0 \\ - 0 & 1 -\end{bmatrix} -\end{equation} -where $\omega_i$ characterize down to which frequency the signal is integrated. +The Integral Force Feedback Controller is modified such that instead of using pure integrators, pseudo integrators (i.e. low pass filters) are used eqref:eq:rotating_iff_lhf where $\omega_i$ characterize the frequency down to which the signal is integrated. +The loop gains ($K_F(s)$ times the direct dynamics $f_u/F_u$) with and without the added HPF are shown in Figure ref:fig:rotating_iff_modified_loop_gain. +The effect of the added HPF limits the low frequency gain to finite values as expected. -Let's arbitrary choose the following control parameters: -#+begin_src matlab :exports code +The Root Locus plots for the decentralized IFF with and without the HPF are displayed in Figure ref:fig:rotating_iff_root_locus_hpf_large. +With the added HPF, the poles of the closed loop system are shown to be stable up to some value of the gain $g_\text{max}$ given by equation eqref:eq:rotating_gmax_iff_hpf. +It is interesting to note that $g_{\text{max}}$ also corresponds to the controller gain at which the low frequency loop gain reaches one (for instance the gain $g$ can be increased by a factor $5$ in Figure ref:fig:rotating_iff_modified_loop_gain before the system becomes unstable). + +\begin{equation}\label{eq:rotating_gmax_iff_hpf} + \boxed{g_{\text{max}} = \omega_i \left( \frac{{\omega_0}^2}{\Omega^2} - 1 \right)} +\end{equation} + +#+begin_src matlab %% Modified IFF - parameters g = 2; % Controller gain wi = 0.1; % HPF Cut-Off frequency [rad/s] @@ -996,9 +1012,6 @@ Kiff = (g/s)*eye(2); % Pure IFF Kiff_hpf = (g/(wi+s))*eye(2); % IFF with added HPF #+end_src -The loop gains ($K_F(s)$ times the direct dynamics $f_u/F_u$) with and without the added HPF are shown in Figure ref:fig:rotating_iff_modified_loop_gain. -The effect of the added HPF limits the low frequency gain to finite values as expected. - #+begin_src matlab :results none %% Loop gain for the IFF with pure integrator and modified IFF with added high pass filter freqs = logspace(-2, 1, 1000); @@ -1020,228 +1033,98 @@ set(gca, 'XTickLabel',[]); ylabel('Loop Gain'); ax2 = nexttile; hold on; plot(freqs, 180/pi*angle(squeeze(freqresp(Gs{Wz_i}('fu', 'Fu')*Kiff(1,1), freqs, 'rad/s'))), 'DisplayName', 'IFF') -plot(freqs, 180/pi*angle(squeeze(freqresp(Gs{Wz_i}('fu', 'Fu')*Kiff_hpf(1,1), freqs, 'rad/s'))), 'DisplayName', 'IFF + HPF') +plot(freqs, 180/pi*angle(squeeze(freqresp(Gs{Wz_i}('fu', 'Fu')*Kiff_hpf(1,1), freqs, 'rad/s'))), 'DisplayName', 'IFF,HPF') hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); xlabel('Frequency [rad/s]'); ylabel('Phase [deg]'); yticks(-180:90:180); -ylim([-180 180]); +ylim([-90 180]); xticks([1e-2,1e-1,1,1e1]) -xticklabels({'$0.01 \omega_0$', '$0.1 \omega_0$', '$\omega_0$', '$10 \omega_0$'}) +xticklabels({'', '$0.1 \omega_0$', '$\omega_0$', '$10 \omega_0$'}) leg = legend('location', 'southwest', 'FontSize', 8); -leg.ItemTokenSize(1) = 20; +leg.ItemTokenSize(1) = 15; linkaxes([ax1,ax2],'x'); xlim([freqs(1), freqs(end)]); #+end_src -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/rotating_iff_modified_loop_gain.pdf', 'width', 'wide', 'height', 'tall'); -#+end_src - -#+name: fig:rotating_iff_modified_loop_gain -#+caption: Loop gain for the IFF with pure integrator and modified IFF with added high pass filter ($\Omega = 0.1\omega_0$) -#+RESULTS: +#+name: fig:rotating_iff_modified_loop_gain_root_locus +#+caption: Comparison of the IFF with pure integrator and modified IFF with added high pass filter ($\Omega = 0.1\omega_0$). Loop gain is shown in (\subref{fig:rotating_iff_modified_loop_gain}) with $\omega_i = 0.1 \omega_0$ and $g = 2$. Root Locus is shown in (\subref{fig:rotating_iff_root_locus_hpf_large}) +#+attr_latex: :options [htbp] +#+begin_figure +#+attr_latex: :caption \subcaption{\label{fig:rotating_iff_modified_loop_gain}Loop gain} +#+attr_latex: :options {0.49\textwidth} +#+begin_subfigure +#+attr_latex: :scale 0.8 [[file:figs/rotating_iff_modified_loop_gain.png]] - -The Root Locus plots for the decentralized IFF with and without the HPF are displayed in Figure ref:fig:rotating_iff_root_locus_hpf. -With the added HPF, the poles of the closed loop system are shown to be *stable up to some value of the gain* $g_\text{max}$ given by equation eqref:eq:gmax_iff_hpf. - -#+name: eq:gmax_iff_hpf -\begin{equation} - \boxed{g_{\text{max}} = \omega_i \left( \frac{{\omega_0}^2}{\Omega^2} - 1 \right)} -\end{equation} - -It is interesting to note that $g_{\text{max}}$ also corresponds to the controller gain at which the low frequency loop gain (Figure ref:fig:rotating_iff_modified_loop_gain) reaches one. - -#+begin_src matlab :results none -%% Root Locus for the initial IFF and the modified IFF -gains = logspace(-2, 4, 200); - -figure; -tiledlayout(1, 3, 'TileSpacing', 'Compact', 'Padding', 'None'); - -ax1 = nexttile([1,2]); -hold on; -for g = gains - clpoles = pole(feedback(Gs{Wz_i}({'fu', 'fv'}, {'Fu', 'Fv'}), g*Kiff)); - plot(real(clpoles), imag(clpoles), '.', 'color', colors(1,:), 'HandleVisibility', 'off','MarkerSize',4); -end -for g = gains - clpoles = pole(feedback(Gs{Wz_i}({'fu', 'fv'}, {'Fu', 'Fv'}), g*Kiff_hpf)); - plot(real(clpoles), imag(clpoles), '.', 'color', colors(2,:), 'HandleVisibility', 'off','MarkerSize',4); -end -% Pure Integrator -plot(real(pole(Gs{Wz_i}({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff)), ... - imag(pole(Gs{Wz_i}({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff)), ... - 'x', 'color', colors(1,:), 'DisplayName', 'IFF','MarkerSize',8); -plot(real(tzero(Gs{Wz_i}({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff)), ... - imag(tzero(Gs{Wz_i}({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff)), ... - 'o', 'color', colors(1,:), 'HandleVisibility', 'off','MarkerSize',8); -% Modified IFF -plot(real(pole(Gs{Wz_i}({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_hpf)), ... - imag(pole(Gs{Wz_i}({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_hpf)), ... - 'x', 'color', colors(2,:), 'DisplayName', 'IFF + HPF','MarkerSize',8); -plot(real(tzero(Gs{Wz_i}({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_hpf)), ... - imag(tzero(Gs{Wz_i}({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_hpf)), ... - 'o', 'color', colors(2,:), 'HandleVisibility', 'off','MarkerSize',8); -hold off; -axis square; -leg = legend('location', 'northwest', 'FontSize', 8); -leg.ItemTokenSize(1) = 8; -xlabel('Real Part'); ylabel('Imaginary Part'); - -xlim([-2.25, 0.25]); ylim([-1.25, 1.25]); -xticks([-2, -1, 0]) -xticklabels({'$-2\omega_0$', '$-\omega_0$', '$0$'}) -yticks([-1, 0, 1]) -yticklabels({'$-\omega_0$', '$0$', '$\omega_0$'}) - -ax2 = nexttile(); -hold on; -for g = gains - clpoles = pole(feedback(Gs{Wz_i}({'fu', 'fv'}, {'Fu', 'Fv'}), g*Kiff)); - plot(real(clpoles), imag(clpoles), '.', 'color', colors(1,:),'MarkerSize',4); -end -for g = gains - clpoles = pole(feedback(Gs{Wz_i}({'fu', 'fv'}, {'Fu', 'Fv'}), g*Kiff_hpf)); - plot(real(clpoles), imag(clpoles), '.', 'color', colors(2,:),'MarkerSize',4); -end -% Pure Integrator -plot(real(pole(Gs{Wz_i}({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff)), ... - imag(pole(Gs{Wz_i}({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff)), ... - 'x', 'color', colors(1,:),'MarkerSize',8); -plot(real(tzero(Gs{Wz_i}({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff)), ... - imag(tzero(Gs{Wz_i}({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff)), ... - 'o', 'color', colors(1,:),'MarkerSize',8); -% Modified IFF -plot(real(pole(Gs{Wz_i}({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_hpf)), ... - imag(pole(Gs{Wz_i}({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_hpf)), ... - 'x', 'color', colors(2,:),'MarkerSize',8); -plot(real(tzero(Gs{Wz_i}({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_hpf)), ... - imag(tzero(Gs{Wz_i}({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_hpf)), ... - 'o', 'color', colors(2,:),'MarkerSize',8); -hold off; -axis square; -xlabel('Real Part'); ylabel('Imaginary Part'); -title('Zoom near the origin'); - -xlim([-0.15, 0.1]); ylim([-0.15, 0.15]); -xticks([-0.1, 0, 0.1]) -xticklabels({'$-0.1\omega_0$', '$0$', '$0.1\omega_0$'}) -yticks([-0.1, 0, 0.1]) -yticklabels({'$-0.1\omega_0$', '$0$', '$0.1\omega_0$'}) -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/rotating_iff_root_locus_hpf.pdf', 'width', 'full', 'height', 700); -#+end_src - -#+name: fig:rotating_iff_root_locus_hpf -#+caption: Root Locus for the initial IFF and the modified IFF -#+RESULTS: -[[file:figs/rotating_iff_root_locus_hpf.png]] +#+end_subfigure +#+attr_latex: :caption \subcaption{\label{fig:rotating_iff_root_locus_hpf_large}Root Locus} +#+attr_latex: :options {0.49\textwidth} +#+begin_subfigure +#+attr_latex: :scale 0.8 +[[file:figs/rotating_iff_root_locus_hpf_large.png]] +#+end_subfigure +#+end_figure ** Optimal IFF with HPF parameters $\omega_i$ and $g$ -Two parameters can be tuned for the modified controller in equation eqref:eq:iff_lhf: the gain $g$ and the pole's location $\omega_i$. +Two parameters can be tuned for the modified controller in equation eqref:eq:rotating_iff_lhf: the gain $g$ and the pole's location $\omega_i$. The optimal values of $\omega_i$ and $g$ are here considered as the values for which the damping of all the closed-loop poles are simultaneously maximized. In order to visualize how $\omega_i$ does affect the attainable damping, the Root Locus plots for several $\omega_i$ are displayed in Figure ref:fig:rotating_root_locus_iff_modified_effect_wi. -It is shown that even though small $\omega_i$ seem to allow more damping to be added to the suspension modes, the control gain $g$ may be limited to small values due to equation eqref:eq:gmax_iff_hpf. - -#+begin_src matlab -%% High Pass Filter Cut-Off Frequency -wis = [0.01, 0.1, 0.5, 1]*Wzs(2); % [rad/s] -#+end_src - -#+begin_src matlab :results none -%% Root Locus for the initial IFF and the modified IFF -gains = logspace(-2, 4, 200); - -figure; -tiledlayout(1, 2, 'TileSpacing', 'Compact', 'Padding', 'None'); - -ax1 = nexttile(); -hold on; -for wi_i = 1:length(wis) - wi = wis(wi_i); - Kiff = (1/(wi+s))*eye(2); - L(wi_i) = plot(nan, nan, '.', 'color', colors(wi_i,:), 'DisplayName', sprintf('$\\omega_i = %.2f \\omega_0$', wi./Wzs(2))); - for g = gains - clpoles = pole(feedback(Gs{2}({'fu', 'fv'}, {'Fu', 'Fv'}), g*Kiff)); - plot(real(clpoles), imag(clpoles), '.', 'color', colors(wi_i,:),'MarkerSize',4); - end - plot(real(pole(Gs{2}({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff)), ... - imag(pole(Gs{2}({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff)), ... - 'x', 'color', colors(wi_i,:),'MarkerSize',8); - plot(real(tzero(Gs{2}({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff)), ... - imag(tzero(Gs{2}({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff)), ... - 'o', 'color', colors(wi_i,:),'MarkerSize',8); -end -hold off; -axis square; -xlim([-2.3, 0.1]); ylim([-1.2, 1.2]); -xticks([-2:1:2]); yticks([-2:1:2]); -leg = legend(L, 'location', 'northwest', 'FontSize', 8); -leg.ItemTokenSize(1) = 8; -xlabel('Real Part'); ylabel('Imaginary Part'); -clear L - -xlim([-2.25, 0.25]); ylim([-1.25, 1.25]); -xticks([-2, -1, 0]) -xticklabels({'$-2\omega_0$', '$-\omega_0$', '$0$'}) -yticks([-1, 0, 1]) -yticklabels({'$-\omega_0$', '$0$', '$\omega_0$'}) - -ax2 = nexttile(); -hold on; -for wi_i = 1:length(wis) - wi = wis(wi_i); - Kiff = (1/(wi+s))*eye(2); - L(wi_i) = plot(nan, nan, '.', 'color', colors(wi_i,:)); - for g = gains - clpoles = pole(feedback(Gs{2}({'fu', 'fv'}, {'Fu', 'Fv'}), g*Kiff)); - plot(real(clpoles), imag(clpoles), '.', 'color', colors(wi_i,:),'MarkerSize',4); - end - plot(real(pole(Gs{2}({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff)), ... - imag(pole(Gs{2}({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff)), ... - 'x', 'color', colors(wi_i,:),'MarkerSize',8); - plot(real(tzero(Gs{2}({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff)), ... - imag(tzero(Gs{2}({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff)), ... - 'o', 'color', colors(wi_i,:),'MarkerSize',8); -end -hold off; -axis square; -xlim([-2.3, 0.1]); ylim([-1.2, 1.2]); -xticks([-2:1:2]); yticks([-2:1:2]); -xlabel('Real Part'); ylabel('Imaginary Part'); -title('Zoom near the origin'); -clear L - -xlim([-0.15, 0.1]); ylim([-0.15, 0.15]); -xticks([-0.1, 0, 0.1]) -xticklabels({'$-0.1\omega_0$', '$0$', '$0.1\omega_0$'}) -yticks([-0.1, 0, 0.1]) -yticklabels({'$-0.1\omega_0$', '$0$', '$0.1\omega_0$'}) -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/rotating_root_locus_iff_modified_effect_wi.pdf', 'width', 'full', 'height', 'tall'); -#+end_src - -#+name: fig:rotating_root_locus_iff_modified_effect_wi -#+caption: Root Locus for several high pass filter cut-off frequency -#+RESULTS: -[[file:figs/rotating_root_locus_iff_modified_effect_wi.png]] - +It is shown that even though small $\omega_i$ seem to allow more damping to be added to the suspension modes (see Root locus in Figure ref:fig:rotating_root_locus_iff_modified_effect_wi_large), the control gain $g$ may be limited to small values due to equation eqref:eq:rotating_gmax_iff_hpf. In order to study this trade off, the attainable closed-loop damping ratio $\xi_{\text{cl}}$ is computed as a function of $\omega_i/\omega_0$. The gain $g_{\text{opt}}$ at which this maximum damping is obtained is also displayed and compared with the gain $g_{\text{max}}$ at which the system becomes unstable (Figure ref:fig:rotating_iff_hpf_optimal_gain). +# TODO - Maybe comment on these "regions" + Three regions can be observed: - $\omega_i/\omega_0 < 0.02$: the added damping is limited by the maximum allowed control gain $g_{\text{max}}$ - $0.02 < \omega_i/\omega_0 < 0.2$: the attainable damping ratio is maximized and is reached for $g \approx 2$ - $0.2 < \omega_i/\omega_0$: the added damping decreases as $\omega_i/\omega_0$ increases. +#+begin_src matlab +%% High Pass Filter Cut-Off Frequency +wis = [0.01, 0.05, 0.1]; % [rad/s] +#+end_src + +#+begin_src matlab :results none +%% Root Locus for the initial IFF and the modified IFF +gains = logspace(-2, 4, 200); + +figure; +hold on; +for wi_i = 1:length(wis) + wi = wis(wi_i); + Kiff = (1/(wi+s))*eye(2); + L(wi_i) = plot(nan, nan, '.', 'color', colors(wi_i,:), 'DisplayName', sprintf('$\\omega_i = %.2f \\omega_0$', wi)); + for g = gains + clpoles = pole(feedback(Gs{2}({'fu', 'fv'}, {'Fu', 'Fv'}), g*Kiff)); + plot(real(clpoles), imag(clpoles), '.', 'color', colors(wi_i,:),'MarkerSize',4); + end + plot(real(pole(Gs{2}({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff)), ... + imag(pole(Gs{2}({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff)), ... + 'x', 'color', colors(wi_i,:),'MarkerSize',8); + plot(real(tzero(Gs{2}({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff)), ... + imag(tzero(Gs{2}({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff)), ... + 'o', 'color', colors(wi_i,:),'MarkerSize',8); +end +hold off; +axis equal; +xlim([-2.3, 0.1]); ylim([-1.2, 1.2]); +xticks([-2:1:2]); yticks([-2:1:2]); +leg = legend(L, 'location', 'southwest', 'FontSize', 8); +leg.ItemTokenSize(1) = 8; +xlabel('Real Part'); ylabel('Imaginary Part'); +clear L + +xlim([-1.25, 0.25]); ylim([-1.25, 1.25]); +xticks([-1, 0]) +xticklabels({'$-\omega_0$', '$0$'}) +yticks([-1, 0, 1]) +yticklabels({'$-\omega_0$', '$0$', '$\omega_0$'}) +ytickangle(90) +#+end_src + #+begin_src matlab %% Compute the optimal control gain wis = logspace(-2, 1, 100); % [rad/s] @@ -1274,6 +1157,7 @@ yyaxis right hold on; plot(wis, opt_gain, '-', 'DisplayName', '$g_{opt}$'); plot(wis, wis*((1/Wzs(2))^2 - 1), '--', 'DisplayName', '$g_{max}$'); +hold off; set(gca, 'YScale', 'lin'); ylim([0,10]); ylabel('Controller gain $g$'); @@ -1283,90 +1167,32 @@ set(gca, 'XScale', 'log'); legend('location', 'northeast', 'FontSize', 8); #+end_src -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/rotating_iff_hpf_optimal_gain.pdf', 'width', 'wide', 'height', 'normal'); +#+begin_src matlab :tangle no :exports results :results file none +exportFig('figs/rotating_iff_hpf_optimal_gain.pdf', 'width', 'half', 'height', 450); #+end_src -#+name: fig:rotating_iff_hpf_optimal_gain -#+caption: Attainable damping ratio $\xi_\text{cl}$ as a function of $\omega_i/\omega_0$. Corresponding control gain $g_\text{opt}$ and $g_\text{max}$ are also shown -#+RESULTS: +#+name: fig:rotating_iff_modified_effect_wi +#+caption: Root Locus for several high pass filter cut-off frequency (\subref{fig:rotating_root_locus_iff_modified_effect_wi_large}). +#+attr_latex: :options [htbp] +#+begin_figure +#+attr_latex: :caption \subcaption{\label{fig:rotating_root_locus_iff_modified_effect_wi}Root Locus} +#+attr_latex: :options {0.49\textwidth} +#+begin_subfigure +#+attr_latex: :width 0.95\linewidth +[[file:figs/rotating_root_locus_iff_modified_effect_wi.png]] +#+end_subfigure +#+attr_latex: :caption \subcaption{\label{fig:rotating_iff_hpf_optimal_gain}Attainable damping ratio $\xi_\text{cl}$ as a function of $\omega_i/\omega_0$. Corresponding control gain $g_\text{opt}$ and $g_\text{max}$ are also shown} +#+attr_latex: :options {0.49\textwidth} +#+begin_subfigure +#+attr_latex: :width 0.95\linewidth [[file:figs/rotating_iff_hpf_optimal_gain.png]] +#+end_subfigure +#+end_figure ** Obtained Damped Plant - -Let's choose $\omega_i = 0.1 \cdot \omega_0$ and compute the damped plant. -The undamped and damped plants are compared in Figure ref:fig:rotating_iff_hpf_damped_plant in blue and red respectively. -A well damped plant is indeed obtained. - -However, the magnitude of the coupling term ($d_v/F_u$) is larger then IFF is applied. - -#+begin_src matlab -%% Compute damped plant -wi = 0.1; % [rad/s] -g = 2; % Gain -Kiff_hpf = (g/(wi+s))*eye(2); % IFF with added HPF -Kiff_hpf.InputName = {'fu', 'fv'}; -Kiff_hpf.OutputName = {'Fu', 'Fv'}; - -G_iff_hpf = feedback(Gs{2}, Kiff_hpf, 'name'); - -isstable(G_iff_hpf) % Verify stability -#+end_src - -#+begin_src matlab :results none -%% Damped plant with IFF and added HPF - Transfer function from $F_u$ to $d_u$ -freqs = logspace(-2, 1, 1000); - -figure; -tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); - -% Magnitude -ax1 = nexttile([2, 1]); -hold on; -plot(freqs, abs(squeeze(freqresp(Gs{2}('Du', 'Fu'), freqs, 'rad/s'))), '-', 'color', [zeros(1,3)], ... - 'DisplayName', '$d_u/F_u$, OL') -plot(freqs, abs(squeeze(freqresp(G_iff_hpf('Du', 'Fu'), freqs, 'rad/s'))), '-', 'color', [colors(1,:)], ... - 'DisplayName', '$d_u/F_u$, IFF') -plot(freqs, abs(squeeze(freqresp(Gs{2}('Dv', 'Fu'), freqs, 'rad/s'))), '-', 'color', [zeros(1,3), 0.5], ... - 'DisplayName', '$d_v/F_u$, OL') -plot(freqs, abs(squeeze(freqresp(G_iff_hpf('Dv', 'Fu'), freqs, 'rad/s'))), '-', 'color', [colors(1,:), 0.5], ... - 'DisplayName', '$d_v/F_u$, IFF') -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -set(gca, 'XTickLabel',[]); ylabel('Magnitude [m/N]'); -leg = legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 2); -leg.ItemTokenSize(1) = 20; - -ax2 = nexttile; -hold on; -plot(freqs, 180/pi*angle(squeeze(freqresp(Gs{2}('Du', 'Fu'), freqs, 'rad/s'))), '-', 'color', zeros(1,3)) -plot(freqs, 180/pi*angle(squeeze(freqresp(G_iff_hpf('Du', 'Fu'), freqs, 'rad/s'))), '-', 'color', colors(1,:)) -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); -xlabel('Frequency [rad/s]'); ylabel('Phase [deg]'); -yticks(-180:90:180); -ylim([-180 180]); -xticks([1e-2,1e-1,1,1e1]) -xticklabels({'$0.01 \omega_0$', '$0.1 \omega_0$', '$\omega_0$', '$10 \omega_0$'}) - -linkaxes([ax1,ax2],'x'); -xlim([freqs(1), freqs(end)]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/rotating_iff_hpf_damped_plant.pdf', 'width', 'wide', 'height', 'tall'); -#+end_src - -#+name: fig:rotating_iff_hpf_damped_plant -#+caption: Damped plant with IFF and added HPF - Transfer function from $F_u$ to $d_u$, $\omega_i = 0.1 \cdot \omega_0$, $\Omega = 0.1 \cdot \omega_0$ -#+RESULTS: -[[file:figs/rotating_iff_hpf_damped_plant.png]] - -In order to study how $\omega_i$ affects the coupling of the damped plant, the closed-loop plant is identified for several $\omega_i$. -The direct and coupling terms of the plants are shown in Figure ref:fig:rotating_iff_hpf_damped_plant_effect_wi_coupling (left) and the ratio between the two (i.e. the coupling ratio) is shown in Figure ref:fig:rotating_iff_hpf_damped_plant_effect_wi_coupling (right). - -The coupling ratio is decreasing as $\omega_i$ increases. -There is therefore a *trade-off between achievable damping and coupling ratio* for the choice of $\omega_i$. +In order to study how the parameter $\omega_i$ affects the damped plant, the obtained damped plants for several $\omega_i$ are compared in Figure ref:fig:rotating_iff_hpf_damped_plant_effect_wi_plant. +It can be seen that the low frequency coupling increases as $\omega_i$ increases. +There is therefore a trade-off between achievable damping and added coupling when tuning $\omega_i$. The same trade-off can be seen between achievable damping and loss of compliance at low frequency (see Figure ref:fig:rotating_iff_hpf_effect_wi_compliance). #+begin_src matlab @@ -1390,7 +1216,7 @@ end freqs = logspace(-2, 1, 1000); figure; -tiledlayout(3, 2, 'TileSpacing', 'Compact', 'Padding', 'None'); +tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); % Magnitude ax1 = nexttile([2, 1]); @@ -1407,20 +1233,6 @@ set(gca, 'XTickLabel',[]); ylabel('Magnitude [m/N]'); leg = legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 1); leg.ItemTokenSize(1) = 20; -ax3 = nexttile([3,1]); -hold on; -for i = 1:length(wis) - plot(freqs, abs(squeeze(freqresp(Gs_iff_hpf{i}('Dv', 'Fu')/Gs_iff_hpf{i}('Du', 'Fu'), freqs, 'rad/s'))), '-', 'color', [colors(i,:)], ... - 'DisplayName', sprintf('$\\omega_i = %.2f \\omega_0$', wis(i))) -end -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -xlabel('Frequency [rad/s]'); ylabel('Coupling Ratio'); -leg = legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1); -leg.ItemTokenSize(1) = 20; -xticks([1e-2,1e-1,1,1e1]) -xticklabels({'$0.01 \omega_0$', '$0.1 \omega_0$', '$\omega_0$', '$10 \omega_0$'}) - ax2 = nexttile; hold on; for i = 1:length(wis) @@ -1430,7 +1242,7 @@ hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); xlabel('Frequency [rad/s]'); ylabel('Phase [deg]'); yticks(-180:90:180); -ylim([-180 180]); +ylim([-180 0]); xticks([1e-2,1e-1,1,1e1]) xticklabels({'$0.01 \omega_0$', '$0.1 \omega_0$', '$\omega_0$', '$10 \omega_0$'}) @@ -1438,15 +1250,10 @@ linkaxes([ax1,ax2],'x'); xlim([freqs(1), freqs(end)]); #+end_src -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/rotating_iff_hpf_damped_plant_effect_wi_coupling.pdf', 'width', 'full', 'height', 'tall'); +#+begin_src matlab :tangle no :exports results :results file none +exportFig('figs/rotating_iff_hpf_damped_plant_effect_wi_plant.pdf', 'width', 'half', 'height', 600); #+end_src -#+name: fig:rotating_iff_hpf_damped_plant_effect_wi_coupling -#+caption: Effect of $\omega_i$ on the damped plant coupling -#+RESULTS: -[[file:figs/rotating_iff_hpf_damped_plant_effect_wi_coupling.png]] - #+begin_src matlab :exports none :results none %% Effect of $\omega_i$ on the obtained compliance freqs = logspace(-2, 1, 1000); @@ -1473,13 +1280,26 @@ xticklabels({'$0.01 \omega_0$', '$0.1 \omega_0$', '$\omega_0$', '$10 \omega_0$'} #+end_src #+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/rotating_iff_hpf_effect_wi_compliance.pdf', 'width', 'wide', 'height', 'normal'); +exportFig('figs/rotating_iff_hpf_effect_wi_compliance.pdf', 'width', 'half', 'height', 600); #+end_src -#+name: fig:rotating_iff_hpf_effect_wi_compliance -#+caption: Effect of $\omega_i$ on the obtained compliance -#+RESULTS: +#+name: fig:rotating_iff_hpf_damped_plant_effect_wi +#+caption: Effect of $\omega_i$ on the damped plant coupling +#+attr_latex: :options [htbp] +#+begin_figure +#+attr_latex: :caption \subcaption{\label{fig:rotating_iff_hpf_damped_plant_effect_wi_plant}Obtained plants} +#+attr_latex: :options {0.49\textwidth} +#+begin_subfigure +#+attr_latex: :width 0.95\linewidth +[[file:figs/rotating_iff_hpf_damped_plant_effect_wi_plant.png]] +#+end_subfigure +#+attr_latex: :caption \subcaption{\label{fig:rotating_iff_hpf_effect_wi_compliance}Effect of $\omega_i$ on the compliance} +#+attr_latex: :options {0.49\textwidth} +#+begin_subfigure +#+attr_latex: :width 0.95\linewidth [[file:figs/rotating_iff_hpf_effect_wi_compliance.png]] +#+end_subfigure +#+end_figure * IFF with a stiffness in parallel with the force sensor :PROPERTIES: @@ -1489,7 +1309,6 @@ exportFig('figs/rotating_iff_hpf_effect_wi_compliance.pdf', 'width', 'wide', 'he ** Introduction :ignore: In this section it is proposed to add springs in parallel with the force sensors to counteract the negative stiffness induced by the gyroscopic effects. - Such springs are schematically shown in Figure ref:fig:rotating_3dof_model_schematic_iff_parallel_springs where $k_a$ is the stiffness of the actuator and $k_p$ the added stiffness in parallel with the actuator and force sensor. #+begin_src latex :file rotating_3dof_model_schematic_iff_parallel_springs.pdf @@ -1560,6 +1379,7 @@ Such springs are schematically shown in Figure ref:fig:rotating_3dof_model_schem #+name: fig:rotating_3dof_model_schematic_iff_parallel_springs #+caption: Studied system with additional springs in parallel with the actuators and force sensors (shown in red) +#+attr_latex: :scale 0.8 #+RESULTS: [[file:figs/rotating_3dof_model_schematic_iff_parallel_springs.png]] @@ -1595,52 +1415,44 @@ load('rotating_generic_plants.mat', 'Gs', 'Wzs'); #+end_src ** Equations -The forces measured by the two force sensors represented in Figure ref:fig:rotating_3dof_model_schematic_iff_parallel_springs are described by Eq. eqref:eq:measured_force_kp. +The forces measured by the two force sensors represented in Figure ref:fig:rotating_3dof_model_schematic_iff_parallel_springs are described by eqref:eq:rotating_measured_force_kp. -#+name: eq:measured_force_kp -\begin{equation} +\begin{equation}\label{eq:rotating_measured_force_kp} \begin{bmatrix} f_{u} \\ f_{v} \end{bmatrix} = \begin{bmatrix} F_u \\ F_v \end{bmatrix} - (c s + k_a) \begin{bmatrix} d_u \\ d_v \end{bmatrix} \end{equation} -In order to keep the overall stiffness $k = k_a + k_p$ constant, thus not modifying the open-loop poles as $k_p$ is changed, a scalar parameter $\alpha$ ($0 \le \alpha < 1$) is defined to describe the fraction of the total stiffness in parallel with the actuator and force sensor as in Eq. eqref:eq:kp_alpha. +In order to keep the overall stiffness $k = k_a + k_p$ constant, thus not modifying the open-loop poles as $k_p$ is changed, a scalar parameter $\alpha$ ($0 \le \alpha < 1$) is defined to describe the fraction of the total stiffness in parallel with the actuator and force sensor as in eqref:eq:rotating_kp_alpha. -#+name: eq:kp_alpha -\begin{equation} +\begin{equation}\label{eq:rotating_kp_alpha} k_p = \alpha k, \quad k_a = (1 - \alpha) k \end{equation} -After the equations of motion derived and transformed in the Laplace domain, the transfer function matrix $\mathbf{G}_k$ in Eq. eqref:eq:Gk_mimo_tf is computed. -Its elements are shown in Eq. eqref:eq:Gk_diag and eqref:eq:Gk_off_diag. +After the equations of motion derived and transformed in the Laplace domain, the transfer function matrix $\mathbf{G}_k$ in Eq. eqref:eq:rotating_Gk_mimo_tf is computed. +Its elements are shown in Eq. eqref:eq:rotating_Gk_diag and eqref:eq:rotating_Gk_off_diag. -#+name: eq:Gk_mimo_tf -\begin{equation} +\begin{equation}\label{eq:rotating_Gk_mimo_tf} \begin{bmatrix} f_u \\ f_v \end{bmatrix} = \mathbf{G}_k \begin{bmatrix} F_u \\ F_v \end{bmatrix} \end{equation} -#+name: eq:Gk -\begin{subequations} +\begin{subequations}\label{eq:rotating_Gk} \begin{align} -\mathbf{G}_{k}(1,1) &= \mathbf{G}_{k}(2,2) = \frac{\big( \frac{s^2}{{\omega_0}^2} - \frac{\Omega^2}{{\omega_0}^2} + \alpha \big) \big( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \big) + \big( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \big)^2}{\big( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \big)^2 + \big( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \big)^2} \label{eq:Gk_diag} \\ -\mathbf{G}_{k}(1,2) &= -\mathbf{G}_{k}(2,1) = \frac{- \left( 2 \xi \frac{s}{\omega_0} + 1 - \alpha \right) \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)}{\left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2} \label{eq:Gk_off_diag} +\mathbf{G}_{k}(1,1) &= \mathbf{G}_{k}(2,2) = \frac{\big( \frac{s^2}{{\omega_0}^2} - \frac{\Omega^2}{{\omega_0}^2} + \alpha \big) \big( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \big) + \big( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \big)^2}{\big( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \big)^2 + \big( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \big)^2} \label{eq:rotating_Gk_diag} \\ +\mathbf{G}_{k}(1,2) &= -\mathbf{G}_{k}(2,1) = \frac{- \left( 2 \xi \frac{s}{\omega_0} + 1 - \alpha \right) \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)}{\left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2} \label{eq:rotating_Gk_off_diag} \end{align} \end{subequations} -Comparing $\mathbf{G}_k$ in Eq. eqref:eq:Gk with $\mathbf{G}_f$ in Eq. eqref:eq:Gf shows that while the poles of the system are kept the same, the zeros of the diagonal terms have changed. -The two real zeros $z_r$ in Eq. eqref:eq:iff_zero_real that were inducing a non-minimum phase behavior are transformed into two complex conjugate zeros if the condition in Eq. eqref:eq:kp_cond_cc_zeros holds. +Comparing $\mathbf{G}_k$ in eqref:eq:rotating_Gk with $\mathbf{G}_f$ in eqref:eq:rotating_Gf shows that while the poles of the system are kept the same, the zeros of the diagonal terms have changed. +The two real zeros $z_r$ in eqref:eq:rotating_iff_zero_real that were inducing a non-minimum phase behavior are transformed into two complex conjugate zeros if the condition in eqref:eq:rotating_kp_cond_cc_zeros holds. +Thus, if the added /parallel stiffness/ $k_p$ is higher than the /negative stiffness/ induced by centrifugal forces $m \Omega^2$, the dynamics from actuator to its collocated force sensor will show /minimum phase behavior/. -#+name: eq:kp_cond_cc_zeros -\begin{equation} +\begin{equation}\label{eq:rotating_kp_cond_cc_zeros} \boxed{\alpha > \frac{\Omega^2}{{\omega_0}^2} \quad \Leftrightarrow \quad k_p > m \Omega^2} \end{equation} -#+begin_important -Thus, if the added *parallel stiffness* $k_p$ is *higher than the negative stiffness induced by centrifugal forces* $m \Omega^2$, the dynamics from actuator to its collocated force sensor will show minimum phase behavior. -#+end_important - ** Identify plant with parallel stiffnesses :noexport: #+begin_src matlab %% Tuv Stage @@ -1666,10 +1478,13 @@ io(io_i) = linio([mdl, '/ext_metrology'], 1, 'openoutput'); io_i = io_i + 1; #+end_src ** Effect of the parallel stiffness on the IFF plant -The IFF plant (transfer function from $[F_u, F_v]$ to $[f_u, f_v]$) is identified in three different cases: -- without parallel stiffness $k_p = 0$ -- with a small parallel stiffness $k_p < m \Omega^2$ -- with a large parallel stiffness $k_p > m \Omega^2$ +The IFF plant (transfer function from $[F_u, F_v]$ to $[f_u, f_v]$) is identified without parallel stiffness $k_p = 0$, with a small parallel stiffness $k_p < m \Omega^2$ and with a large parallel stiffness $k_p > m \Omega^2$. +The Bode plots of the obtained dynamics are shown in Figure ref:fig:rotating_iff_effect_kp. +One can see that the the two real zeros for $k_p < m \Omega^2$ are transformed into two complex conjugate zeros for $k_p > m \Omega^2$. +In that case, the systems shows alternating complex conjugate poles and zeros as what is the case in the non-rotating case. + +Figure ref:fig:rotating_iff_kp_root_locus shows the Root Locus plots for $k_p = 0$, $k_p < m \Omega^2$ and $k_p > m \Omega^2$ when $K_F$ is a pure integrator as in Eq. eqref:eq:rotating_Kf_pure_int. +It is shown that if the added stiffness is higher than the maximum negative stiffness, the poles of the closed-loop system are bounded on the (stable) left half-plane, and hence the unconditional stability of IFF is recovered. #+begin_src matlab Wz = 0.1; % The rotation speed [rad/s] @@ -1705,9 +1520,6 @@ G_high_kp.InputName = {'Fu', 'Fv', 'Fdx', 'Fdy'}; G_high_kp.OutputName = {'fu', 'fv', 'Du', 'Dv', 'Dx', 'Dy'}; #+end_src -The Bode plots of the obtained dynamics are shown in Figure ref:fig:rotating_iff_effect_kp. -One can see that for $k_p > m \Omega^2$, the two real zeros with $k_p < m \Omega^2$ are transformed into two complex conjugate zeros and the systems shows alternating complex conjugate poles and zeros. - #+begin_src matlab :results none %% Effect of the parallel stiffness on the IFF plant freqs = logspace(-2, 1, 1000); @@ -1727,7 +1539,7 @@ plot(freqs, abs(squeeze(freqresp(G_high_kp('fu', 'Fu'), freqs, 'rad/s'))), '-', hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); set(gca, 'XTickLabel',[]); ylabel('Magnitude [N/N]'); -ylim([1e-5, 5e1]); +ylim([1e-4, 5e1]); legend('location', 'southeast', 'FontSize', 8); % Phase @@ -1739,7 +1551,7 @@ plot(freqs, 180/pi*angle(squeeze(freqresp(G_high_kp('fu', 'Fu'), freqs, 'rad/s') set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); xlabel('Frequency [rad/s]'); ylabel('Phase [deg]'); yticks(-180:90:180); -ylim([-180 180]); +ylim([0 180]); hold off; xticks([1e-2,1e-1,1,1e1]) xticklabels({'$0.01 \omega_0$', '$0.1 \omega_0$', '$\omega_0$', '$10 \omega_0$'}) @@ -1749,25 +1561,14 @@ xlim([freqs(1), freqs(end)]); #+end_src #+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/rotating_iff_effect_kp.pdf', 'width', 'wide', 'height', 'tall'); +exportFig('figs/rotating_iff_effect_kp.pdf', 'width', 'half', 'height', 500); #+end_src -#+name: fig:rotating_iff_effect_kp -#+caption: Effect of the parallel stiffness on the IFF plant: Bode plot of $G_{k}(1,1) = f_u/F_u$ without parallel spring, with parallel spring stiffness $k_p < m \Omega^2$ and $k_p > m \Omega^2$, $\Omega = 0.1 \omega_0$ -#+RESULTS: -[[file:figs/rotating_iff_effect_kp.png]] - -Figure ref:fig:rotating_iff_kp_root_locus shows the Root Locus plots for $k_p = 0$, $k_p < m \Omega^2$ and $k_p > m \Omega^2$ when $K_F$ is a pure integrator as in Eq. eqref:eq:Kf_pure_int. -It is shown that if the added stiffness is higher than the maximum negative stiffness, the poles of the closed-loop system are bounded on the (stable) left half-plane, and hence the *unconditional stability of IFF is recovered*. - #+begin_src matlab :results none %% Root Locus for IFF without parallel spring, with small parallel spring and with large parallel spring gains = logspace(-2, 2, 200); figure; -tiledlayout(1, 3, 'TileSpacing', 'Compact', 'Padding', 'None'); - -ax1 = nexttile([1,2]); hold on; plot(real(pole(G_no_kp({'fu','fv'},{'Fu','Fv'})*(1/s))), imag(pole(G_no_kp({'fu','fv'},{'Fu','Fv'})*(1/s))), 'x', 'color', colors(1,:), ... 'DisplayName', '$k_p = 0$','MarkerSize',8); @@ -1809,57 +1610,33 @@ yticklabels({'$-\omega_0$', '$0$', '$\omega_0$'}) xlabel('Real Part'); ylabel('Imaginary Part'); leg = legend('location', 'northwest', 'FontSize', 8); leg.ItemTokenSize(1) = 8; - -ax2 = nexttile(); -hold on; -plot(real(pole(G_no_kp( {'fu','fv'},{'Fu','Fv'})*(1/s))), imag(pole(G_no_kp( {'fu','fv'},{'Fu','Fv'})*(1/s))), 'x', 'color', colors(1,:), 'MarkerSize',8); -plot(real(tzero(G_no_kp({'fu','fv'},{'Fu','Fv'})*(1/s))), imag(tzero(G_no_kp({'fu','fv'},{'Fu','Fv'})*(1/s))), 'o', 'color', colors(1,:), 'MarkerSize',8); -for g = gains - cl_poles = pole(feedback(G_no_kp({'fu','fv'},{'Fu','Fv'}), (g/s)*eye(2))); - plot(real(cl_poles), imag(cl_poles), '.', 'color', colors(1,:), 'MarkerSize', 4); -end - -plot(real(pole(G_low_kp( {'fu','fv'},{'Fu','Fv'})*(1/s))), imag(pole(G_low_kp( {'fu','fv'},{'Fu','Fv'})*(1/s))), 'x', 'color', colors(2,:), 'MarkerSize',8); -plot(real(tzero(G_low_kp({'fu','fv'},{'Fu','Fv'})*(1/s))), imag(tzero(G_low_kp({'fu','fv'},{'Fu','Fv'})*(1/s))), 'o', 'color', colors(2,:), 'MarkerSize',8); -for g = gains - cl_poles = pole(feedback(G_low_kp({'fu','fv'},{'Fu','Fv'}), (g/s)*eye(2))); - plot(real(cl_poles), imag(cl_poles), '.', 'color', colors(2,:), 'MarkerSize', 4); -end - -plot(real(pole(G_high_kp( {'fu','fv'},{'Fu','Fv'})*(1/s))), imag(pole(G_high_kp( {'fu','fv'},{'Fu','Fv'})*(1/s))), 'x', 'color', colors(3,:), 'MarkerSize',8); -plot(real(tzero(G_high_kp({'fu','fv'},{'Fu','Fv'})*(1/s))), imag(tzero(G_high_kp({'fu','fv'},{'Fu','Fv'})*(1/s))), 'o', 'color', colors(3,:), 'MarkerSize',8); -for g = gains - cl_poles = pole(feedback(G_high_kp({'fu','fv'},{'Fu','Fv'}), (g/s)*eye(2))); - plot(real(cl_poles), imag(cl_poles), '.', 'color', colors(3,:), 'MarkerSize',4); -end -hold off; -axis equal; -xlim([-0.04, 0.04]); ylim([-0.08, 0.08]); -set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]); -xlabel('Real Part'); ylabel('Imaginary Part'); -title('Zoom on controller pole') #+end_src -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/rotating_iff_kp_root_locus.pdf', 'width', 'full', 'height', 'tall'); -#+end_src - -#+name: fig:rotating_iff_kp_root_locus -#+caption: Root Locus for IFF without parallel spring, with small parallel spring and with large parallel spring -#+RESULTS: +#+name: fig:rotating_iff_plant_effect_kp +#+caption: Effect of the parallel stiffness on the IFF plant +#+attr_latex: :options [htbp] +#+begin_figure +#+attr_latex: :caption \subcaption{\label{fig:rotating_iff_effect_kp}Bode plot of $G_{k}(1,1) = f_u/F_u$ without parallel spring, with parallel spring stiffness $k_p < m \Omega^2$ and $k_p > m \Omega^2$, $\Omega = 0.1 \omega_0$} +#+attr_latex: :options {0.55\linewidth} +#+begin_subfigure +#+attr_latex: :width 0.95\linewidth +[[file:figs/rotating_iff_effect_kp.png]] +#+end_subfigure +#+attr_latex: :caption \subcaption{\label{fig:rotating_iff_kp_root_locus}Root Locus for IFF without parallel spring, with small parallel spring and with large parallel spring} +#+attr_latex: :options {0.44\linewidth} +#+begin_subfigure +#+attr_latex: :width 0.95\linewidth [[file:figs/rotating_iff_kp_root_locus.png]] +#+end_subfigure +#+end_figure ** Effect of $k_p$ on the attainable damping - Even though the parallel stiffness $k_p$ has no impact on the open-loop poles (as the overall stiffness $k$ is kept constant), it has a large impact on the transmission zeros. Moreover, as the attainable damping is generally proportional to the distance between poles and zeros cite:preumont18_vibrat_contr_activ_struc_fourt_edition, the parallel stiffness $k_p$ is foreseen to have some impact on the attainable damping. - To study this effect, Root Locus plots for several parallel stiffnesses $k_p > m \Omega^2$ are shown in Figure ref:fig:rotating_iff_kp_root_locus_effect_kp. The frequencies of the transmission zeros of the system are increasing with an increase of the parallel stiffness $k_p$ (thus getting closer to the poles) and the associated attainable damping is reduced. - -#+begin_important Therefore, even though the parallel stiffness $k_p$ should be larger than $m \Omega^2$ for stability reasons, it should not be taken too large as this would limit the attainable damping. -#+end_important +This is confirmed by the Figure ref:fig:rotating_iff_kp_optimal_gain where the attainable closed-loop damping ratio $\xi_{\text{cl}}$ and the associated optimal control gain $g_\text{opt}$ are computed as a function of the parallel stiffness. #+begin_src matlab %% Tested parallel stiffnesses @@ -1907,17 +1684,6 @@ leg = legend('location', 'northwest', 'FontSize', 8); leg.ItemTokenSize(1) = 12; #+end_src -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/rotating_iff_kp_root_locus_effect_kp.pdf', 'width', 'wide', 'height', 'tall'); -#+end_src - -#+name: fig:rotating_iff_kp_root_locus_effect_kp -#+caption: Root Locus: Effect of the parallel stiffness on the attainable damping, $\Omega = 0.1 \omega_0$ -#+RESULTS: -[[file:figs/rotating_iff_kp_root_locus_effect_kp.png]] - -This is confirmed by the Figure ref:fig:rotating_iff_kp_optimal_gain where the attainable closed-loop damping ratio $\xi_{\text{cl}}$ and the associated optimal control gain $g_\text{opt}$ are computed as a function of the parallel stiffness. - #+begin_src matlab %% Computes the optimal parameters and attainable simultaneous damping alphas = logspace(-2, 0, 100); @@ -1972,22 +1738,48 @@ xticks([0.01, 0.1, 1]) xticklabels({'$m\Omega^2$', '$10m\Omega^2$', '$100m\Omega^2$'}) #+end_src -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/rotating_iff_kp_optimal_gain.pdf', 'width', 'wide', 'height', 'normal'); +#+begin_src matlab :tangle no :exports results :results file none +exportFig('figs/rotating_iff_kp_optimal_gain.pdf', 'width', 'half', 'height', 450); #+end_src -#+name: fig:rotating_iff_kp_optimal_gain -#+caption: Attainable damping ratio $\xi_\text{cl}$ as a function of the parallel stiffness $k_p$. Corresponding control gain $g_\text{opt}$ is also shown. Values for $k_p < m\Omega^2$ are not shown as the system is unstable. -#+RESULTS: +#+name: fig:rotating_iff_optimal_kp +#+caption: Effect of the parallel stiffness on the IFF plant +#+attr_latex: :options [htbp] +#+begin_figure +#+attr_latex: :caption \subcaption{\label{fig:rotating_iff_kp_root_locus_effect_kp}Root Locus: Effect of the parallel stiffness on the attainable damping, $\Omega = 0.1 \omega_0$} +#+attr_latex: :options {0.49\linewidth} +#+begin_subfigure +#+attr_latex: :scale 1 +[[file:figs/rotating_iff_kp_root_locus_effect_kp.png]] +#+end_subfigure +#+attr_latex: :caption \subcaption{\label{fig:rotating_iff_kp_optimal_gain}Attainable damping ratio $\xi_\text{cl}$ as a function of the parallel stiffness $k_p$. Corresponding control gain $g_\text{opt}$ is also shown. Values for $k_p < m\Omega^2$ are not shown as the system is unstable.} +#+attr_latex: :options {0.49\linewidth} +#+begin_subfigure +#+attr_latex: :scale 0.9 [[file:figs/rotating_iff_kp_optimal_gain.png]] +#+end_subfigure +#+end_figure ** Damped plant Let's choose a parallel stiffness equal to $k_p = 2 m \Omega^2$ and compute the damped plant. The damped and undamped transfer functions from $F_u$ to $d_u$ are compared in Figure ref:fig:rotating_iff_kp_damped_plant. - Even though the two resonances are well damped, the IFF changes the low frequency behavior of the plant which is usually not wanted. This is due to the fact that "pure" integrators are used, and that the low frequency loop gains becomes large below some frequency. +In order to lower the low frequency gain, a high pass filter is added to the IFF controller (which is equivalent as shifting the controller pole to the left in the complex plane): +\begin{equation} + K_{\text{IFF}}(s) = g\frac{1}{\omega_i + s} \begin{bmatrix} + 1 & 0 \\ + 0 & 1 +\end{bmatrix} +\end{equation} + +In order to see how the high pass filter impacts the attainable damping, the controller gain $g$ is kept constant while $\omega_i$ is changed, and the minimum damping ratio of the damped plant is computed. +The obtained damping ratio as a function of $\omega_i/\omega_0$ (where $\omega_0$ is the resonance of the system without rotation) is shown in Figure ref:fig:rotating_iff_kp_added_hpf_effect_damping. +It is shown that the attainable damping ratio reduces as $\omega_i$ is increased (same conclusion than in Section ref:sec:rotating_iff_pseudo_int). +Let's choose $\omega_i = 0.1 \cdot \omega_0$ and compare the obtained damped plant again with the undamped and with the "pure" IFF in Figure ref:fig:rotating_iff_kp_added_hpf_damped_plant. +The added high pass filter gives almost the same damping properties to the suspension while giving good low frequency behavior. + #+begin_src matlab %% Identify dynamics with parallel stiffness = 2mW^2 Wz = 0.1; % [rad/s] @@ -2010,68 +1802,6 @@ Kiff_kp.OutputName = {'Fu', 'Fv'}; G_cl_iff_kp = feedback(G, Kiff_kp, 'name'); #+end_src -#+begin_src matlab :results none -%% Damped plant with IFF - Transfer function from $F_u$ to $d_u$ -freqs = logspace(-3, 1, 1000); - -figure; -tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); - -% Magnitude -ax1 = nexttile([2, 1]); -hold on; -plot(freqs, abs(squeeze(freqresp(G('Du', 'Fu'), freqs, 'rad/s'))), '-', 'color', zeros(1,3), ... - 'DisplayName', '$d_u/F_u$ - OL') -plot(freqs, abs(squeeze(freqresp(G_cl_iff_kp('Du', 'Fu'), freqs, 'rad/s'))), '-', 'color', colors(1,:), ... - 'DisplayName', '$d_u/F_u$ - IFF + $k_p$') -plot(freqs, abs(squeeze(freqresp(G('Dv', 'Fu'), freqs, 'rad/s'))), '-', 'color', [zeros(1,3), 0.5], ... - 'DisplayName', '$d_v/F_u$ - OL') -plot(freqs, abs(squeeze(freqresp(G_cl_iff_kp('Dv', 'Fu'), freqs, 'rad/s'))), '-', 'color', [colors(1,:), 0.5], ... - 'DisplayName', '$d_v/F_u$ - IFF + $k_p$') -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -set(gca, 'XTickLabel',[]); ylabel('Magnitude [m/N]'); -ldg = legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 2); -ldg.ItemTokenSize(1) = 20; -ylim([1e-6, 1e2]); - -ax2 = nexttile; -hold on; -plot(freqs, 180/pi*angle(squeeze(freqresp(G('Du', 'Fu'), freqs, 'rad/s'))), '-', 'color', zeros(1,3)) -plot(freqs, 180/pi*angle(squeeze(freqresp(G_cl_iff_kp('Du', 'Fu'), freqs, 'rad/s'))), '-', 'color', colors(1,:)) -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); -xlabel('Frequency [rad/s]'); ylabel('Phase [deg]'); -yticks(-180:90:180); -ylim([-180 180]); -xticks([1e-3,1e-2,1e-1,1,1e1]) -xticklabels({'$0.001 \omega_0$', '$0.01 \omega_0$', '$0.1 \omega_0$', '$\omega_0$', '$10 \omega_0$'}) - -linkaxes([ax1,ax2],'x'); -xlim([freqs(1), freqs(end)]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/rotating_iff_kp_damped_plant.pdf', 'width', 'wide', 'height', 'tall'); -#+end_src - -#+name: fig:rotating_iff_kp_damped_plant -#+caption: Damped plant with IFF - Transfer function from $F_u$ to $d_u$ -#+RESULTS: -[[file:figs/rotating_iff_kp_damped_plant.png]] - -In order to lower the low frequency gain, an high pass filter is added to the IFF controller (which is equivalent as shifting the controller pole to the left in the complex plane): -\begin{equation} - K_{\text{IFF}}(s) = g\frac{1}{\omega_i + s} \begin{bmatrix} - 1 & 0 \\ - 0 & 1 -\end{bmatrix} -\end{equation} - -Let's see how the high pass filter impacts the attainable damping. -The controller gain $g$ is kept constant while $\omega_i$ is changed, and the minimum damping ratio of the damped plant is computed. -The obtained damping ratio as a function of $\omega_i/\omega_0$ (where $\omega_0$ is the resonance of the system without rotation) is shown in Figure ref:fig:rotating_iff_kp_added_hpf_effect_damping. - #+begin_src matlab w0 = sqrt((kn+kp)/(mn+ms)); % Resonance frequency [rad/s] wis = w0*logspace(-2, 0, 100); % LPF cut-off [rad/s] @@ -2102,19 +1832,10 @@ ylabel('Damping Ratio $\xi$'); xlabel('$\omega_i/\omega_0$'); #+end_src -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/rotating_iff_kp_added_hpf_effect_damping.pdf', 'width', 'wide', 'height', 'normal'); +#+begin_src matlab :tangle no :exports results :results file none +exportFig('figs/rotating_iff_kp_added_hpf_effect_damping.pdf', 'width', 'third', 'height', 600); #+end_src -#+name: fig:rotating_iff_kp_added_hpf_effect_damping -#+caption: Effect of the high pass filter cut-off frequency on the obtained damping -#+RESULTS: -[[file:figs/rotating_iff_kp_added_hpf_effect_damping.png]] - -Let's choose $\omega_i = 0.1 \cdot \omega_0$ and compute the damped plant again. -The Bode plots of the undamped, damped with "pure" IFF, and with added high pass filters are shown in Figure ref:fig:rotating_iff_kp_added_hpf_damped_plant. -The added high pass filter gives almost the same damping properties while giving acceptable low frequency behavior. - #+begin_src matlab %% Compute the damped plant with added High Pass Filter Kiff_kp_hpf = (2.2/(s + 0.1*w0))*eye(2); @@ -2141,16 +1862,16 @@ plot(freqs, abs(squeeze(freqresp(G_cl_iff_kp( 'Du', 'Fu'), freqs, 'rad/s'))), plot(freqs, abs(squeeze(freqresp(G_cl_iff_hpf_kp('Du', 'Fu'), freqs, 'rad/s'))), '-', 'color', colors(2,:), ... 'DisplayName', '$d_u/F_u$ - IFF + $k_p$ + HPF') plot(freqs, abs(squeeze(freqresp(G( 'Dv', 'Fu'), freqs, 'rad/s'))), '-', 'color', [zeros( 1,3), 0.5], ... - 'DisplayName', '$d_v/F_u$ - OL') + 'HandleVisibility', 'off') plot(freqs, abs(squeeze(freqresp(G_cl_iff_kp( 'Dv', 'Fu'), freqs, 'rad/s'))), '-', 'color', [colors(1,:), 0.5], ... - 'DisplayName', '$d_v/F_u$ - IFF + $k_p$') + 'HandleVisibility', 'off') plot(freqs, abs(squeeze(freqresp(G_cl_iff_hpf_kp('Dv', 'Fu'), freqs, 'rad/s'))), '-', 'color', [colors(2,:), 0.5], ... - 'DisplayName', '$d_v/F_u$ - IFF + $k_p$ + HPF') + 'HandleVisibility', 'off') hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); set(gca, 'XTickLabel',[]); ylabel('Magnitude [m/N]'); -ldg = legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 2); -ldg.ItemTokenSize(1) = 20; +ldg = legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 1); +ldg.ItemTokenSize(1) = 10; ax2 = nexttile; hold on; @@ -2161,7 +1882,7 @@ hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); xlabel('Frequency [rad/s]'); ylabel('Phase [deg]'); yticks(-180:90:180); -ylim([-180 180]); +ylim([-180 90]); xticks([1e-3,1e-2,1e-1,1,1e1]) xticklabels({'$0.001 \omega_0$', '$0.01 \omega_0$', '$0.1 \omega_0$', '$\omega_0$', '$10 \omega_0$'}) @@ -2169,14 +1890,27 @@ linkaxes([ax1,ax2],'x'); xlim([freqs(1), freqs(end)]); #+end_src -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/rotating_iff_kp_added_hpf_damped_plant.pdf', 'width', 'wide', 'height', 'tall'); +#+begin_src matlab :tangle no :exports results :results file none +exportFig('figs/rotating_iff_kp_added_hpf_damped_plant.pdf', 'width', 700, 'height', 600); #+end_src -#+name: fig:rotating_iff_kp_added_hpf_damped_plant -#+caption: Damped plant with IFF - Transfer function from $F_u$ to $d_u$ -#+RESULTS: +#+name: fig:rotating_iff_optimal_kp +#+caption:Effect of the high pass filter cut-off frequency on the obtained damping +#+attr_latex: :options [htbp] +#+begin_figure +#+attr_latex: :caption \subcaption{\label{fig:rotating_iff_kp_added_hpf_effect_damping}Reduced damping ratio with increased cut-off frequency $\omega_i$} +#+attr_latex: :options {0.34\linewidth} +#+begin_subfigure +#+attr_latex: :scale 0.95 +[[file:figs/rotating_iff_kp_added_hpf_effect_damping.png]] +#+end_subfigure +#+attr_latex: :caption \subcaption{\label{fig:rotating_iff_kp_added_hpf_damped_plant}Damped plant with the parallel stiffness, effect of the added HPF} +#+attr_latex: :options {0.65\linewidth} +#+begin_subfigure +#+attr_latex: :scale 0.95 [[file:figs/rotating_iff_kp_added_hpf_damped_plant.png]] +#+end_subfigure +#+end_figure * Relative Damping Control :PROPERTIES: @@ -2185,20 +1919,14 @@ exportFig('figs/rotating_iff_kp_added_hpf_damped_plant.pdf', 'width', 'wide', 'h <> ** Introduction :ignore: -In order to apply a "relative damping control strategy", relative motion sensors are added in parallel with the actuators as shown in Figure ref:fig:rotating_3dof_model_schematic_rdc. - +In order to apply a "Relative Damping Control" strategy, relative motion sensors are added in parallel with the actuators as shown in Figure ref:fig:rotating_3dof_model_schematic_rdc. Two controllers $K_d$ are used to fed back the relative motion to the actuator. -$K_d$ is a derivator: -\begin{equation} -K_d(s) = s -\end{equation} +These controllers are in principle pure derivators ($K_d = s$), but to be implemented in practice they are usually replaced by a high pass filter eqref:eq:rotating_rdc_controller. -To be implemented in practice, it is usually replaced by a an high pass filter: -\begin{equation} -K_d(s) = \frac{s}{s + \omega_d} +\begin{equation}\label{eq:rotating_rdc_controller} +K_d(s) = g \cdot \frac{s}{s + \omega_d} \end{equation} - #+begin_src latex :file rotating_3dof_model_schematic_rdc.pdf \begin{tikzpicture} % Angle @@ -2234,19 +1962,19 @@ K_d(s) = \frac{s}{s + \omega_d} % Spring and Actuator for U \draw[actuator={0.6}{0.2}{black}] (cu) -- coordinate[midway, below=0.1](actumid) node[above=0.1, rotate=\thetau]{$F_u$} (cu-|-2.6,0); \draw[spring=0.2] (ku) -- node[above=0.1, rotate=\thetau]{$k$} (ku-|-2.6,0); - \draw[<->, dashed, draw=coloryellow] (actu) node[below=0.1, rotate=\thetau, color=coloryellow](du){$d_u$} -- (actu-|-2.6,0); + \draw[<->, dashed, draw=colorgreen] (actu) node[below=0.1, rotate=\thetau, color=colorgreen](du){$d_u$} -- (actu-|-2.6,0); - \node[color=coloryellow, block={0.6cm}{0.6cm}, fill=coloryellow!10!white, rotate=\thetau] (Ku) at ($(actumid) + (0, -1.4)$) {$K_{d}$}; - \draw[->, draw=coloryellow] (du.south) -- ++(0, -0.8) -| (Ku.south); - \draw[->, draw=coloryellow] (Ku.north) -- (actumid); + \node[color=colorgreen, block={0.6cm}{0.6cm}, fill=colorgreen!10!white, rotate=\thetau] (Ku) at ($(actumid) + (0, -1.4)$) {$K_{d}$}; + \draw[->, draw=colorgreen] (du.south) -- ++(0, -0.8) -| (Ku.south); + \draw[->, draw=colorgreen] (Ku.north) -- (actumid); \draw[actuator={0.6}{0.2}{black}] (cv) -- coordinate[midway, right=0.1](actvmid) node[left, rotate=\thetau]{$F_v$} (cv|-0,-2.6); \draw[spring=0.2] (kv) -- node[left, rotate=\thetau]{$k$} (kv|-0,-2.6); - \draw[<->, dashed, draw=coloryellow] (actv)node[right=0.1, rotate=\thetau, color=coloryellow](dv){$d_v$} -- (actv|-0,-2.6); + \draw[<->, dashed, draw=colorgreen] (actv)node[right=0.1, rotate=\thetau, color=colorgreen](dv){$d_v$} -- (actv|-0,-2.6); - \node[color=coloryellow, block={0.6cm}{0.6cm}, fill=coloryellow!10!white, rotate=\thetau] (Kv) at ($(actvmid) + (1.4, 0)$) {$K_{d}$}; - \draw[->, draw=coloryellow] (dv.east) -- ++(0.8, 0) |- (Kv.east); - \draw[->, draw=coloryellow] (Kv.west) -- (actvmid); + \node[color=colorgreen, block={0.6cm}{0.6cm}, fill=colorgreen!10!white, rotate=\thetau] (Kv) at ($(actvmid) + (1.4, 0)$) {$K_{d}$}; + \draw[->, draw=colorgreen] (dv.east) -- ++(0.8, 0) |- (Kv.east); + \draw[->, draw=colorgreen] (Kv.west) -- (actvmid); \end{scope} % Inertial Frame @@ -2268,6 +1996,7 @@ K_d(s) = \frac{s}{s + \omega_d} #+name: fig:rotating_3dof_model_schematic_rdc #+caption: System with relative motion sensor and decentralized "relative damping control" applied. +#+attr_latex: :scale 0.8 #+RESULTS: [[file:figs/rotating_3dof_model_schematic_rdc.png]] @@ -2303,111 +2032,37 @@ load('rotating_generic_plants.mat', 'Gs', 'Wzs'); #+end_src ** Equations of motion -Let's note $\bm{G}_d$ the transfer function between actuator forces and measured relative motion in parallel with the actuators: -\begin{equation} +Let's note $\bm{G}_d$ the transfer function between actuator forces and measured relative motion in parallel with the actuators eqref:eq:rotating_rdc_plant_matrix. +The elements of $\bm{G}_d$ were derived in Section ref:sec:rotating_system_description are shown in eqref:eq:rotating_rdc_plant_elements. + +\begin{equation}\label{eq:rotating_rdc_plant_matrix} \begin{bmatrix} d_u \\ d_v \end{bmatrix} = \mathbf{G}_d \begin{bmatrix} F_u \\ F_v \end{bmatrix} \end{equation} -With: -\begin{subequations} +\begin{subequations}\label{eq:rotating_rdc_plant_elements} \begin{align} \mathbf{G}_{d}(1,1) &= \mathbf{G}_{d}(2,2) = \frac{\frac{1}{k} \left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)}{\left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2} \\ \mathbf{G}_{d}(1,2) &= -\mathbf{G}_{d}(2,1) = \frac{\frac{1}{k} \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)}{\left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2} \end{align} \end{subequations} -Neglecting the damping for simplicity ($\xi \ll 1$), the direct terms have two complex conjugate zeros: -\begin{equation} - z = \pm j \sqrt{\omega_0^2 - \omega^2} +Neglecting the damping for simplicity ($\xi \ll 1$), the direct terms have two complex conjugate zeros which are between the two pairs of complex conjugate poles eqref:eq:rotating_rdc_zeros_poles. +Therefore, for $\Omega < \sqrt{k/m}$ (i.e. stable system), the transfer functions for Relative Damping Control have alternating complex conjugate poles and zeros. + +\begin{equation}\label{eq:rotating_rdc_zeros_poles} + z = \pm j \sqrt{\omega_0^2 - \omega^2}, \quad p_1 = \pm j (\omega_0 - \omega), \quad p_2 = \pm j (\omega_0 + \omega) \end{equation} -Which are between the two pairs of complex conjugate poles at: -\begin{align} - p_1 &= \pm j (\omega_0 - \omega) \\ - p_2 &= \pm j (\omega_0 + \omega) -\end{align} - -Therefore, for $\Omega < \sqrt{k/m}$ (i.e. stable system), the transfer functions for Relative Damping Control have *alternating complex conjugate poles and zeros*. - ** Decentralized Relative Damping Control -The transfer functions from $[F_u,\ F_v]$ to $[d_u,\ d_v]$ is identified and shown in Figure ref:fig:rotating_rdc_plant_effect_rot for several rotating velocities. - -#+begin_src matlab :results none -%% Bode plot of the direct and coupling term for the "relative damping control" plant - Effect of rotation -freqs = logspace(-2, 1, 1000); - -figure; -tiledlayout(3, 2, 'TileSpacing', 'Compact', 'Padding', 'None'); - -% Magnitude -ax1 = nexttile([2, 1]); -hold on; -for i = 1:length(Wzs) - plot(freqs, abs(squeeze(freqresp(Gs{i}('du', 'Fu'), freqs, 'rad/s'))), '-', 'color', colors(i,:)) -end -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -set(gca, 'XTickLabel',[]); ylabel('Magnitude [N/N]'); -title('Direct terms: $d_u/F_u$, $d_v/F_v$'); -ylim([1e-3, 1e2]); - -ax2 = nexttile([2, 1]); -hold on; -for i = 1:length(Wzs) - plot(freqs, abs(squeeze(freqresp(Gs{i}('dv', 'Fu'), freqs, 'rad/s'))), '-', 'color', colors(i,:), ... - 'DisplayName', sprintf('$\\Omega = %.1f \\omega_0$', Wzs(i))) -end -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]); -title('Coupling Terms: $d_u/F_v$, $d_v/F_u$'); -ldg = legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 1); -ldg.ItemTokenSize = [10, 1]; -ylim([1e-3, 1e2]); - -ax3 = nexttile; -hold on; -for i = 1:length(Wzs) - plot(freqs, 180/pi*angle(squeeze(freqresp(Gs{i}('du', 'Fu'), freqs, 'rad/s'))), '-', 'color', colors(i,:)) -end -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); -xlabel('Frequency [rad/s]'); ylabel('Phase [deg]'); -yticks(-180:90:180); -ylim([-180 180]); -xticks([1e-2,1e-1,1,1e1]) -xticklabels({'$0.01 \omega_0$', '$0.1 \omega_0$', '$\omega_0$', '$10 \omega_0$'}) - -ax4 = nexttile; -hold on; -for i = 1:length(Wzs) - plot(freqs, 180/pi*angle(squeeze(freqresp(Gs{i}('dv', 'Fu'), freqs, 'rad/s'))), '-', 'color', colors(i,:)); -end -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); -xlabel('Frequency [rad/s]'); set(gca, 'YTickLabel',[]); -yticks(-180:90:180); -ylim([-180 180]); -xticks([1e-2,1e-1,1,1e1]) -xticklabels({'$0.01 \omega_0$', '$0.1 \omega_0$', '$\omega_0$', '$10 \omega_0$'}) - -linkaxes([ax1,ax2,ax3,ax4],'x'); -xlim([freqs(1), freqs(end)]); - -linkaxes([ax1,ax2],'y'); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/rotating_rdc_plant_effect_rot.pdf', 'width', 'full', 'height', 'tall'); -#+end_src - -#+name: fig:rotating_rdc_plant_effect_rot -#+caption: Bode plot of the direct and coupling term for the "relative damping control" plant - Effect of rotation -#+RESULTS: -[[file:figs/rotating_rdc_plant_effect_rot.png]] +The transfer functions from $[F_u,\ F_v]$ to $[d_u,\ d_v]$ were identified for several rotating velocities in Section ref:sec:rotating_system_description and are shown in Figure ref:fig:rotating_bode_plot (page pageref:fig:rotating_bode_plot). In order to see if large damping can be added with Relative Damping Control, the root locus is computed (Figure ref:fig:rotating_rdc_root_locus). -The closed-loop system is unconditionally stable and the poles can be damped as much as wanted. +The closed-loop system is unconditionally stable as expected and the poles can be damped as much as wanted. + +Let's select a reasonable "Relative Damping Control" gain, and compute the closed-loop damped system. +The open-loop and damped plants are compared in Figure ref:fig:rotating_rdc_damped_plant. +The rotating aspect does not add any complexity for the use of Relative Damping Control. +It does not increase the low frequency coupling as compared to Integral Force Feedback. #+begin_src matlab :results none %% Root Locus for Relative Damping Control @@ -2442,21 +2097,43 @@ leg = legend('location', 'northwest', 'FontSize', 8); leg.ItemTokenSize(1) = 8; #+end_src -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/rotating_rdc_root_locus.pdf', 'width', 'wide', 'height', 'tall'); +#+begin_src matlab :tangle no :results none +%% Root Locus for Relative Damping Control +Krdc = s*eye(2); % Relative damping controller + +gains = logspace(-2, 2, 300); % Tested gains +Wz_i = [1,3,4]; + +figure; +hold on; +for i = 1:length(Wz_i) + plot(real(pole(Gs{Wz_i(i)}({'du', 'dv'}, {'Fu', 'Fv'})*Krdc)), imag(pole(Gs{Wz_i(i)}({'du', 'dv'}, {'Fu', 'Fv'})*Krdc)), 'x', 'color', colors(i,:), ... + 'DisplayName', sprintf('$\\Omega = %.1f \\omega_0 $', Wzs(Wz_i(i))),'MarkerSize',8); + plot(real(tzero(Gs{Wz_i(i)}({'du', 'dv'}, {'Fu', 'Fv'})*Krdc)), imag(tzero(Gs{Wz_i(i)}({'du', 'dv'}, {'Fu', 'Fv'})*Krdc)), 'o', 'color', colors(i,:), ... + 'HandleVisibility', 'off','MarkerSize',8); + + poles_rdc = rootLocusPolesSorted(Gs{Wz_i(i)}({'du', 'dv'}, {'Fu', 'Fv'}), Krdc, gains, 'd_max', 1e-4); + for p_i = 1:size(poles_rdc, 2) + plot(real(poles_rdc(:, p_i)), imag(poles_rdc(:, p_i)), '-', 'color', colors(i,:), ... + 'HandleVisibility', 'off'); + end +end +hold off; +axis equal; +xlim([-1.8, 0.2]); ylim([0, 2]); +xticks([-1, 0]) +xticklabels({'-$\omega_0$', '$0$'}) +yticks([0, 1, 2]) +yticklabels({'$0$', '$\omega_0$', '$2 \omega_0$'}) + +xlabel('Real Part'); ylabel('Imaginary Part'); +leg = legend('location', 'northwest', 'FontSize', 8); +leg.ItemTokenSize(1) = 8; #+end_src -#+name: fig:rotating_rdc_root_locus -#+caption: Root Locus for Relative Damping Control -#+RESULTS: -[[file:figs/rotating_rdc_root_locus.png]] - -** Damped Plant -Let's select a reasonable "Relative Damping Control" gain, and compute the closed-loop damped system. -The open-loop and damped plants are compared in Figure ref:fig:rotating_rdc_damped_plant. - -The rotating aspect does not add any complexity for the use of Relative Damping Control. -It does not increase the low frequency coupling as compared to Integral Force Feedback. +#+begin_src matlab :tangle no :exports results :results file none +exportFig('figs/rotating_rdc_root_locus_inkscape.pdf', 'width', 'half', 'height', 500, 'png', false, 'pdf', false, 'svg', true); +#+end_src #+begin_src matlab i = 2; @@ -2493,7 +2170,7 @@ set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); set(gca, 'XTickLabel',[]); ylabel('Magnitude [m/N]'); ldg = legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 2); ldg.ItemTokenSize(1) = 20; -ylim([1e-6, 1e2]); +ylim([1e-4, 1e2]); ax2 = nexttile; hold on; @@ -2503,20 +2180,33 @@ hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); xlabel('Frequency [rad/s]'); ylabel('Phase [deg]'); yticks(-180:90:180); -ylim([-180 180]); +ylim([-180 0]); linkaxes([ax1,ax2],'x'); xlim([freqs(1), freqs(end)]); #+end_src -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/rotating_rdc_damped_plant.pdf', 'width', 'wide', 'height', 'tall'); +#+begin_src matlab :tangle no :exports results :results file none +exportFig('figs/rotating_rdc_damped_plant.pdf', 'width', 'half', 'height', 500); #+end_src -#+name: fig:rotating_rdc_damped_plant -#+caption: Damped plant using Relative Damping Control -#+RESULTS: +#+name: fig:rotating_rdc_result +#+caption: Relative Damping Control. Root Locus (\subref{fig:rotating_rdc_root_locus}) and obtained damped plant (\subref{rotating_rdc_damped_plant}) +#+attr_latex: :options [htbp] +#+begin_figure +#+attr_latex: :caption \subcaption{\label{fig:rotating_rdc_root_locus}Root Locus for Relative Damping Control} +#+attr_latex: :options {0.49\linewidth} +#+begin_subfigure +#+attr_latex: :scale 1 +[[file:figs/rotating_rdc_root_locus.png]] +#+end_subfigure +#+attr_latex: :caption \subcaption{\label{fig:rotating_rdc_damped_plant}Damped plant using Relative Damping Control} +#+attr_latex: :options {0.49\linewidth} +#+begin_subfigure +#+attr_latex: :scale 0.8 [[file:figs/rotating_rdc_damped_plant.png]] +#+end_subfigure +#+end_figure * Comparison of Active Damping Techniques :PROPERTIES: @@ -2526,7 +2216,6 @@ exportFig('figs/rotating_rdc_damped_plant.pdf', 'width', 'wide', 'height', 'tall ** Introduction :ignore: These two proposed IFF modifications as well as relative damping control are now compared in terms of added damping and closed-loop behavior. - For the following comparisons, the cut-off frequency for the added HPF is set to $\omega_i = 0.1 \omega_0$ and the stiffness of the parallel springs is set to $k_p = 5 m \Omega^2$ (corresponding to $\alpha = 0.05$). These values are chosen based on previous discussion about optimal parameters. @@ -2556,11 +2245,6 @@ These values are chosen based on previous discussion about optimal parameters. mdl = 'rotating_model'; #+end_src -#+begin_src matlab -%% Load "Generic" system dynamics -load('rotating_generic_plants.mat', 'Gs', 'Wzs'); -#+end_src - ** Identify plants :noexport: #+begin_src matlab %% The rotating speed is set to $\Omega = 0.1 \omega_0$. @@ -2588,7 +2272,6 @@ io(io_i) = linio([mdl, '/ext_metrology'], 1, 'openoutput'); io_i = io_i + 1; #+begin_src matlab %% Identifying plant with parallel stiffness - model_config.Tuv_type = "parallel_k"; % Parallel stiffness @@ -2607,7 +2290,6 @@ G_kp.OutputName = {'fu', 'fv', 'Du', 'Dv', 'Dx', 'Dy'}; #+begin_src matlab %% Identifying plant with no parallel stiffness - model_config.Tuv_type = "normal"; % Tuv Stage @@ -2642,15 +2324,13 @@ Krdc.OutputName = {'Fu', 'Fv'}; #+end_src ** Root Locus - Figure ref:fig:rotating_comp_techniques_root_locus shows the Root Locus plots for the two proposed IFF modifications as well as for relative damping control. While the two pairs of complex conjugate open-loop poles are identical for both IFF modifications, the transmission zeros are not. This means that the closed-loop behavior of both systems will differ when large control gains are used. One can observe that the closed loop poles corresponding to the system with added springs (in red) are bounded to the left half plane implying unconditional stability. This is not the case for the system where the controller is augmented with an HPF (in blue). - -It is interesting to note that the maximum added damping is very similar for both techniques. +It is interesting to note that the maximum added damping is very similar for both modified IFF techniques. #+begin_src matlab :exports none :results none %% Comparison of active damping techniques for rotating platform - Root Locus @@ -2697,22 +2377,84 @@ leg = legend('location', 'northwest', 'FontSize', 8); leg.ItemTokenSize(1) = 12; #+end_src -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/rotating_comp_techniques_root_locus.pdf', 'width', 'wide', 'height', 'tall'); +#+begin_src matlab :tangle no +gains = logspace(-2, 2, 500); + +poles_iff_hpf = rootLocusPolesSorted(G({'fu', 'fv'}, {'Fu', 'Fv'}), Kiff, gains, 'd_max', 1e-4); +poles_iff_kp = rootLocusPolesSorted(G_kp({'fu', 'fv'}, {'Fu', 'Fv'}), Kiff_kp, gains, 'd_max', 1e-4); +poles_rdc = rootLocusPolesSorted(G({'Du', 'Dv'}, {'Fu', 'Fv'}), Krdc, gains, 'd_max', 1e-4); + +figure; +hold on; +% IFF +plot(real(pole(G({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff)), imag(pole(G({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff)), 'x', 'color', colors(1,:), ... + 'DisplayName', 'IFF with HPF', 'MarkerSize', 8); +plot(real(tzero(G({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff)), imag(tzero(G({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff)), 'o', 'color', colors(1,:), ... + 'HandleVisibility', 'off', 'MarkerSize', 8); +for p_i = 1:size(poles_iff_hpf, 2) + plot(real(poles_iff_hpf(:, p_i)), imag(poles_iff_hpf(:, p_i)), '-', 'color', colors(1,:), ... + 'HandleVisibility', 'off'); +end + +% IFF with parallel stiffness +plot(real(pole(G_kp({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_kp)), imag(pole(G_kp({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_kp)), 'x', 'color', colors(2,:), ... + 'DisplayName', 'IFF with $k_p$', 'MarkerSize', 8); +plot(real(tzero(G_kp({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_kp)), imag(tzero(G_kp({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_kp)), 'o', 'color', colors(2,:), ... + 'HandleVisibility', 'off', 'MarkerSize', 8); +for p_i = 1:size(poles_iff_kp, 2) + plot(real(poles_iff_kp(:, p_i)), imag(poles_iff_kp(:, p_i)), '-', 'color', colors(2,:), ... + 'HandleVisibility', 'off'); +end +% RDC +plot(real(pole(G({'Du', 'Dv'}, {'Fu', 'Fv'})*Krdc)), imag(pole(G({'Du', 'Dv'}, {'Fu', 'Fv'})*Krdc)), 'x', 'color', colors(3,:), ... + 'DisplayName', 'RDC', 'MarkerSize', 8); +plot(real(tzero(G({'Du', 'Dv'}, {'Fu', 'Fv'})*Krdc)), imag(tzero(G({'Du', 'Dv'}, {'Fu', 'Fv'})*Krdc)), 'o', 'color', colors(3,:), ... + 'HandleVisibility', 'off', 'MarkerSize', 8); +for p_i = 1:size(poles_rdc, 2) + plot(real(poles_rdc(:, p_i)), imag(poles_rdc(:, p_i)), '-', 'color', colors(3,:), ... + 'HandleVisibility', 'off'); +end +hold off; +axis equal; +xlim([-1.15, 0.05]); ylim([0, 1.2]); +xticks([-1, -0.8, -0.6, -0.4, -0.2 , 0]); +xticklabels({'-$\omega_0$', '', '', '', '', '$0$'}); +yticks([0, 0.2, 0.4, 0.6, 0.8, 1]); +yticklabels({'$0$', '', '', '', '', '$\omega_0$'}); + +xlabel('Real Part'); ylabel('Imaginary Part'); +leg = legend('location', 'northwest', 'FontSize', 8); +leg.ItemTokenSize(1) = 12; + +exportFig('figs/rotating_comp_techniques_root_locus_inkscape.pdf', 'width', 600, 'height', 600, 'png', false, 'pdf', false, 'svg', true); + +xlim([-0.14, 0.02]); ylim([0, 0.2]); + +exportFig('figs/rotating_comp_techniques_root_locus_zoom.pdf', 'width', 600, 'height', 600, 'png', false, 'pdf', false, 'svg', true); #+end_src -#+name: fig:rotating_comp_techniques_root_locus -#+caption: Comparison of active damping techniques for rotating platform - Root Locus -#+RESULTS: +#+name: fig:rotating_comp_techniques +#+caption: Comparison of active damping techniques for rotating platform +#+attr_latex: :options [htbp] +#+begin_figure +#+attr_latex: :caption \subcaption{\label{fig:rotating_comp_techniques_root_locus}Root Locus} +#+attr_latex: :options {0.49\textwidth} +#+begin_subfigure +#+attr_latex: :width 0.95\linewidth [[file:figs/rotating_comp_techniques_root_locus.png]] +#+end_subfigure +#+attr_latex: :caption \subcaption{\label{fig:rotating_comp_techniques_dampled_plants}Damped plants} +#+attr_latex: :options {0.49\textwidth} +#+begin_subfigure +#+attr_latex: :width 0.95\linewidth +[[file:figs/rotating_comp_techniques_dampled_plants.png]] +#+end_subfigure +#+end_figure ** Obtained Damped Plant The actively damped plants are computed for the three techniques and compared in Figure ref:fig:rotating_comp_techniques_dampled_plants. - -#+begin_important -It is shown that while the diagonal (direct) terms of the damped plants are similar for the three active damping techniques, of off-diagonal (coupling) terms are not. -Integral Force Feedback strategy is adding some coupling at low frequency which may negatively impact the positioning performances. -#+end_important +It is shown that while the diagonal (direct) terms of the damped plants are similar for the three active damping techniques, the off-diagonal (coupling) terms are not. +Integral Force Feedback strategy is adding some coupling at low frequency which may negatively impact the positioning performance. #+begin_src matlab %% Compute Damped plants @@ -2723,7 +2465,7 @@ G_cl_rdc = feedback(G, Krdc, 'name'); #+begin_src matlab :exports none :results none %% Comparison of the damped plants obtained with the three active damping techniques -freqs = logspace(-3, 2, 1000); +freqs = logspace(-2, 2, 1000); figure; tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); @@ -2764,48 +2506,36 @@ hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); xlabel('Frequency [rad/s]'); ylabel('Phase [deg]'); yticks(-180:90:180); -ylim([-180 180]); +ylim([-180 15]); linkaxes([ax1,ax2],'x'); xlim([freqs(1), freqs(end)]); +xticks([1e-2,1e-1,1,1e1,1e2]) +xticklabels({'$0.01 \omega_0$', '$0.1 \omega_0$', '$\omega_0$', '$10 \omega_0$', '$100 \omega_0$'}) #+end_src -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/rotating_comp_techniques_dampled_plants.pdf', 'width', 'wide', 'height', 'tall'); +#+begin_src matlab :tangle no :exports results :results file none +exportFig('figs/rotating_comp_techniques_dampled_plants.pdf', 'width', 'half', 'height', 600); #+end_src -#+name: fig:rotating_comp_techniques_dampled_plants -#+caption: Comparison of the damped plants obtained with the three active damping techniques -#+RESULTS: -[[file:figs/rotating_comp_techniques_dampled_plants.png]] - - ** Transmissibility And Compliance The proposed active damping techniques are now compared in terms of closed-loop transmissibility and compliance. - The transmissibility is here defined as the transfer function from a displacement of the rotating stage along $\vec{i}_x$ to the displacement of the payload along the same direction. It is used to characterize how much vibration is transmitted through the suspended platform to the payload. - The compliance describes the displacement response of the payload to external forces applied to it. This is a useful metric when disturbances are directly applied to the payload. It is here defined as the transfer function from external forces applied on the payload along $\vec{i}_x$ to the displacement of the payload along the same direction. -Very similar results are obtained for the two proposed IFF modifications in terms of transmissibility and compliance (Figure ref:fig:rotating_comp_techniques_transmissibility_compliance). - -#+begin_important +Very similar results are obtained for the two proposed IFF modifications in terms of transmissibility and compliance (Figure ref:fig:rotating_comp_techniques_trans_compliance). Using IFF degrades the compliance at low frequency while using relative damping control degrades the transmissibility at high frequency. This is very well known characteristics of these common active damping techniques that holds when applied to rotating platforms. -#+end_important #+begin_src matlab :exports none :results none %% Comparison of the obtained transmissibilty and compliance for the three tested active damping techniques freqs = logspace(-2, 2, 1000); +% transmissibility figure; -tiledlayout(1, 2, 'TileSpacing', 'Compact', 'Padding', 'None'); - -% Transmissibility -ax1 = nexttile(); hold on; plot(freqs, abs(squeeze(freqresp(G( 'Dx', 'Dfx'), freqs, 'rad/s'))), '-', 'color', zeros(1,3), ... 'DisplayName', 'OL') @@ -2819,8 +2549,15 @@ hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); xlabel('Frequency [rad/s]'); ylabel('Transmissibility [m/m]'); xlim([freqs(1), freqs(end)]); +#+end_src +#+begin_src matlab :tangle no :exports results :results file none +exportFig('figs/rotating_comp_techniques_transmissibility.pdf', 'width', 'half', 'height', 450); +#+end_src + +#+begin_src matlab :exports none :results none % Compliance +figure; ax1 = nexttile(); hold on; plot(freqs, abs(squeeze(freqresp(G( 'Dx', 'Fdx'), freqs, 'rad/s'))), '-', 'color', zeros(1,3), ... @@ -2839,14 +2576,27 @@ ldg.ItemTokenSize = [10, 1]; xlim([freqs(1), freqs(end)]); #+end_src -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/rotating_comp_techniques_transmissibility_compliance.pdf', 'width', 'full', 'height', 'normal'); +#+begin_src matlab :tangle no :exports results :results file none +exportFig('figs/rotating_comp_techniques_compliance.pdf', 'width', 'half', 'height', 450); #+end_src -#+name: fig:rotating_comp_techniques_transmissibility_compliance -#+caption: Comparison of the obtained transmissibilty and compliance for the three tested active damping techniques -#+RESULTS: -[[file:figs/rotating_comp_techniques_transmissibility_compliance.png]] +#+name: fig:rotating_comp_techniques_trans_compliance +#+caption: Comparison of the obtained transmissibilty (\subref{fig:rotating_comp_techniques_transmissibility}) and compliance (\subref{fig:rotating_comp_techniques_compliance}) for the three tested active damping techniques +#+attr_latex: :options [htbp] +#+begin_figure +#+attr_latex: :caption \subcaption{\label{fig:rotating_comp_techniques_transmissibility}Transmissibility} +#+attr_latex: :options {0.49\textwidth} +#+begin_subfigure +#+attr_latex: :width 0.95\linewidth +[[file:figs/rotating_comp_techniques_transmissibility.png]] +#+end_subfigure +#+attr_latex: :caption \subcaption{\label{fig:rotating_comp_techniques_compliance}Compliance} +#+attr_latex: :options {0.49\textwidth} +#+begin_subfigure +#+attr_latex: :width 0.95\linewidth +[[file:figs/rotating_comp_techniques_compliance.png]] +#+end_subfigure +#+end_figure * Rotating Nano-Hexapod :PROPERTIES: @@ -2854,11 +2604,9 @@ exportFig('figs/rotating_comp_techniques_transmissibility_compliance.pdf', 'widt :END: <> ** Introduction :ignore: -The current analysis is now applied on a model representing the rotating nano-hexapod. - -Three nano-hexapod stiffnesses are tested: $k_n = \SI{0.01}{\N\per\mu\m}$, $k_n = \SI{1}{\N\per\mu\m}$ and $k_n = \SI{100}{\N\per\mu\m}$. - -Only the maximum rotating velocity is considered ($\Omega = \SI{60}{rpm}$) with the light sample ($m_s = \SI{1}{kg}$) as this is the worst identified case scenario. +The previous analysis is now applied on a model representing the rotating nano-hexapod. +Three nano-hexapod stiffnesses are tested as for the uniaxial model: $k_n = \SI{0.01}{\N\per\mu\m}$, $k_n = \SI{1}{\N\per\mu\m}$ and $k_n = \SI{100}{\N\per\mu\m}$. +Only the maximum rotating velocity is here considered ($\Omega = \SI{60}{rpm}$) with the light sample ($m_s = \SI{1}{kg}$) as this is the worst identified case scenario in terms of gyroscopic effects. ** Matlab Init :noexport:ignore: #+begin_src matlab :tangle no :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) @@ -2887,8 +2635,6 @@ mdl = 'rotating_model'; #+end_src ** Identify NASS dynamics :noexport: -First, the dynamics is identified for all the considered cases. - #+begin_src matlab %% Nano-Hexapod mn = 15; % Nano-Hexapod mass [kg] @@ -2967,16 +2713,13 @@ Kneg_light = (15+1)*(2*pi)^2; #+end_src The transfer functions from nano-hexapod actuator force $F_u$ to the displacement of the nano-hexapod in the same direction $d_u$ as well as in the orthogonal direction $d_v$ (coupling) are shown in Figure ref:fig:rotating_nano_hexapod_dynamics for all three considered nano-hexapod stiffnesses. - -#+begin_important -It is shown that the rotation has the largest effect on the soft nano-hexapod: -- larger coupling (the ratio of the coupling term to the direct term is larger for the sort nano-hexapod) -- larger shift of poles as a function of the rotating velocity -#+end_important +The soft nano-hexapod is the most affected by the rotation. +This can be seen by the large shift of the resonance frequencies, and by the induced coupling which is larger than for the stiffer nano-hexapods. +The coupling (or interaction) in a MIMO $2 \times 2$ system can be visually estimated as the ratio between the diagonal term and the off-diagonal terms (see corresponding Appendix). #+begin_src matlab :results none %% Effect of rotation on the nano-hexapod dynamics -freqs = logspace(0, 3, 1000); +freqs = logspace(0, 1, 1000); figure; tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); @@ -2984,21 +2727,84 @@ tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); ax1 = nexttile([2,1]); hold on; plot(freqs, abs(squeeze(freqresp(G_vc_norot('Du', 'Fu'), freqs, 'Hz'))), '--', 'color', colors(1,:), ... - 'DisplayName', '$k_n = 0.01\,N/\mu m$'); + 'DisplayName', '$\Omega = 0\,$rpm, $D_u/F_u$'); plot(freqs, abs(squeeze(freqresp(G_vc_fast( 'Du', 'Fu'), freqs, 'Hz'))), '-' , 'color', colors(1,:), ... 'DisplayName', '$\Omega = 60\,$rpm, $D_u/F_u$'); plot(freqs, abs(squeeze(freqresp(G_vc_fast( 'Dv', 'Fu'), freqs, 'Hz'))), '-' , 'color', [colors(1,:), 0.5], ... 'DisplayName', '$\Omega = 60\,$rpm, $D_v/F_u$'); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Magnitude [m/N]'); set(gca, 'XTickLabel',[]); +ylim([1e-6, 1e-2]) +ax2 = nexttile; +hold on; +plot(freqs, 180/pi*angle(squeeze(freqresp(G_vc_norot('Du', 'Fu'), freqs, 'Hz'))), '--', 'color', colors(1,:)); +plot(freqs, 180/pi*angle(squeeze(freqresp(G_vc_fast( 'Du', 'Fu'), freqs, 'Hz'))), '-' , 'color', colors(1,:)); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); +hold off; +yticks(-360:90:360); +ylim([-180, 0]); + +linkaxes([ax1,ax2],'x'); +% xlim([1, 1e3]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file none +exportFig('figs/rotating_nano_hexapod_dynamics_vc.pdf', 'width', 'third', 'height', 600); +#+end_src + +#+begin_src matlab :results none +%% Effect of rotation on the nano-hexapod dynamics +freqs = logspace(1, 2, 1000); + +figure; +tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; plot(freqs, abs(squeeze(freqresp(G_md_norot('Du', 'Fu'), freqs, 'Hz'))), '--', 'color', colors(2,:), ... - 'DisplayName', '$k_n = 1\,N/\mu m$'); + 'DisplayName', '$\Omega = 0\,$rpm, $D_u/F_u$'); plot(freqs, abs(squeeze(freqresp(G_md_fast( 'Du', 'Fu'), freqs, 'Hz'))), '-' , 'color', colors(2,:), ... 'DisplayName', '$\Omega = 60\,$rpm, $D_u/F_u$'); plot(freqs, abs(squeeze(freqresp(G_md_fast( 'Dv', 'Fu'), freqs, 'Hz'))), '-' , 'color', [colors(2,:), 0.5], ... 'DisplayName', '$\Omega = 60\,$rpm, $D_v/F_u$'); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Magnitude [m/N]'); set(gca, 'XTickLabel',[]); +ylim([1e-8, 1e-4]) +ax2 = nexttile; +hold on; +plot(freqs, 180/pi*angle(squeeze(freqresp(G_md_norot('Du', 'Fu'), freqs, 'Hz'))), '--', 'color', colors(2,:)); +plot(freqs, 180/pi*angle(squeeze(freqresp(G_md_fast( 'Du', 'Fu'), freqs, 'Hz'))), '-' , 'color', colors(2,:)); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); +hold off; +yticks(-360:90:360); +ylim([-180, 0]); + +linkaxes([ax1,ax2],'x'); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file none +exportFig('figs/rotating_nano_hexapod_dynamics_md.pdf', 'width', 'third', 'height', 600); +#+end_src + +#+begin_src matlab :results none +%% Effect of rotation on the nano-hexapod dynamics +freqs = logspace(2, 3, 1000); + +figure; +tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; plot(freqs, abs(squeeze(freqresp(G_pz_norot('Du', 'Fu'), freqs, 'Hz'))), '--', 'color', colors(3,:), ... - 'DisplayName', '$k_n = 100\,N/\mu m$'); + 'DisplayName', '$\Omega = 0\,$rpm, $D_u/F_u$'); plot(freqs, abs(squeeze(freqresp(G_pz_fast( 'Du', 'Fu'), freqs, 'Hz'))), '-' , 'color', colors(3,:), ... 'DisplayName', '$\Omega = 60\,$rpm, $D_u/F_u$'); plot(freqs, abs(squeeze(freqresp(G_pz_fast( 'Dv', 'Fu'), freqs, 'Hz'))), '-' , 'color', [colors(3,:), 0.5], ... @@ -3006,18 +2812,10 @@ plot(freqs, abs(squeeze(freqresp(G_pz_fast( 'Dv', 'Fu'), freqs, 'Hz'))), '-' , ' hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); ylabel('Magnitude [m/N]'); set(gca, 'XTickLabel',[]); -ylim([1e-12, 1e-2]) -ldg = legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 3); -ldg.ItemTokenSize = [20, 1]; +ylim([1e-10, 1e-6]) ax2 = nexttile; hold on; -plot(freqs, 180/pi*angle(squeeze(freqresp(G_vc_norot('Du', 'Fu'), freqs, 'Hz'))), '--', 'color', colors(1,:)); -plot(freqs, 180/pi*angle(squeeze(freqresp(G_vc_fast( 'Du', 'Fu'), freqs, 'Hz'))), '-' , 'color', colors(1,:)); - -plot(freqs, 180/pi*angle(squeeze(freqresp(G_md_norot('Du', 'Fu'), freqs, 'Hz'))), '--', 'color', colors(2,:)); -plot(freqs, 180/pi*angle(squeeze(freqresp(G_md_fast( 'Du', 'Fu'), freqs, 'Hz'))), '-' , 'color', colors(2,:)); - plot(freqs, 180/pi*angle(squeeze(freqresp(G_pz_norot('Du', 'Fu'), freqs, 'Hz'))), '--', 'color', colors(3,:)); plot(freqs, 180/pi*angle(squeeze(freqresp(G_pz_fast( 'Du', 'Fu'), freqs, 'Hz'))), '-' , 'color', colors(3,:)); hold off; @@ -3028,67 +2826,45 @@ yticks(-360:90:360); ylim([-180, 0]); linkaxes([ax1,ax2],'x'); -xlim([1, 1e3]); #+end_src -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/rotating_nano_hexapod_dynamics.pdf', 'width', 'full', 'height', 'tall'); +#+begin_src matlab :tangle no :exports results :results file none +exportFig('figs/rotating_nano_hexapod_dynamics_pz.pdf', 'width', 'third', 'height', 600); #+end_src #+name: fig:rotating_nano_hexapod_dynamics -#+caption: Effect of rotation on the nano-hexapod dynamics - Dashed lines are the plants without rotation, solid lines are plants at maximum rotating velocity, and shaded lines are coupling terms at maximum rotating velocity -#+RESULTS: -[[file:figs/rotating_nano_hexapod_dynamics.png]] - -** Coupling :noexport: -Let's define the /coupling ratio/ $\mathcal{C}$ in the system as the ratio of the magnitude of the coupling term $G_c$ to the magnitude of the direct term $G_d$: -\begin{equation} -\mathcal{C}(\omega) = \frac{|G_c(j\omega)|}{|G_d(j\omega)|} -\end{equation} - -This gives some information in the coupling in the system. -This is quite important as high coupling can affect the closed-loop stability. - -The coupling ratio for the three nano-hexapod stiffnesses are shown in Figure ref:fig:rotating_coupling_ratio_nano_hexapod for the maximum rotating velocity $\Omega = 60\,\text{rpm}$. - -#+begin_important -It is shown that the low frequency coupling is inversely proportional to the nano-hexapod stiffness (Figure ref:fig:rotating_coupling_ratio_nano_hexapod). -High nano-hexapod stiffness makes the system better decoupled and therefore easier to control. -#+end_important - -#+begin_src matlab :results none -%% Coupling ratio of the nano-hexapod at maximum velocity -figure; -tiledlayout(1, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); -hold on; -plot(freqs, abs(squeeze(freqresp(G_vc_fast('Dv', 'Fu')/G_vc_fast('Du', 'Fu'), freqs, 'Hz'))), ... - 'DisplayName', '$k_n = 0.01\,N/\mu m$'); -plot(freqs, abs(squeeze(freqresp(G_md_fast('Dv', 'Fu')/G_md_fast('Du', 'Fu'), freqs, 'Hz'))), ... - 'DisplayName', '$k_n = 1\,N/\mu m$'); -plot(freqs, abs(squeeze(freqresp(G_pz_fast('Dv', 'Fu')/G_pz_fast('Du', 'Fu'), freqs, 'Hz'))), ... - 'DisplayName', '$k_n = 100\,N/\mu m$'); -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -xlabel('Frequency [Hz]'); ylabel('Coupling ratio'); -xlim([1, 500]); -legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 1); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/rotating_coupling_ratio_nano_hexapod.pdf', 'width', 'wide', 'height', 'normal'); -#+end_src - -#+name: fig:rotating_coupling_ratio_nano_hexapod -#+caption: Coupling ratio of the nano-hexapod at maximum velocity ($\Omega = 60\,\text{rpm}$) -#+RESULTS: -[[file:figs/rotating_coupling_ratio_nano_hexapod.png]] +#+caption: Effect of rotation on the nano-hexapod dynamics. Dashed lines are the plants without rotation, solid lines are plants at maximum rotating velocity ($\Omega = 60\,\text{rpm}$), and shaded lines are coupling terms at maximum rotating velocity +#+attr_latex: :options [htbp] +#+begin_figure +#+attr_latex: :caption \subcaption{\label{fig:uniaxial_damped_plant_three_active_damping_techniques_vc}$k_n = 0.01\,N/\mu m$} +#+attr_latex: :options {0.33\textwidth} +#+begin_subfigure +#+attr_latex: :width 0.95\linewidth +[[file:figs/rotating_nano_hexapod_dynamics_vc.png]] +#+end_subfigure +#+attr_latex: :caption \subcaption{\label{fig:rotating_nano_hexapod_dynamics_md}$k_n = 1\,N/\mu m$} +#+attr_latex: :options {0.33\textwidth} +#+begin_subfigure +#+attr_latex: :width 0.95\linewidth +[[file:figs/rotating_nano_hexapod_dynamics_md.png]] +#+end_subfigure +#+attr_latex: :caption \subcaption{\label{fig:rotating_nano_hexapod_dynamics_pz}$k_n = 100\,N/\mu m$} +#+attr_latex: :options {0.33\textwidth} +#+begin_subfigure +#+attr_latex: :width 0.95\linewidth +[[file:figs/rotating_nano_hexapod_dynamics_pz.png]] +#+end_subfigure +#+end_figure ** Optimal IFF with High Pass Filter -Let's apply Integral Force Feedback with an added High Pass Filter to the three nano-hexapods. - -First, let's find the parameters of the IFF controller that yield best simultaneous damping. -The results are shown in Figure ref:fig:rotating_iff_hpf_nass_optimal_gain. -The added damping for the soft nano-hexapod is quite low and is limited by the maximum usable gain. +Integral Force Feedback with an added High Pass Filter is applied to the three nano-hexapods. +First, the parameters ($\omega_i$ and $g$) of the IFF controller that yield best simultaneous damping are determined from Figure ref:fig:rotating_iff_hpf_nass_optimal_gain. +The IFF parameters are chosen as follow: +- for $k_n = \SI{0.01}{\N\per\mu\m}$ (Figure ref:fig:rotating_iff_hpf_nass_optimal_gain): $\omega_i$ is chosen such that the maximum damping is achieved while the gain is less than half of the maximum gain at which the system is unstable. + This is done to have some control robustness. +- for $k_n = \SI{1}{\N\per\mu\m}$ and $k_n = \SI{100}{\N\per\mu\m}$ (Figure ref:fig:rotating_iff_hpf_nass_optimal_gain_md and ref:fig:rotating_iff_hpf_nass_optimal_gain_pz): the largest $\omega_i$ is chosen such that obtained damping is $\SI{95}{\percent}$ of the maximum achievable damping. + Large $\omega_i$ is chosen here to limit the loss of compliance and the increase of coupling at low frequency as was shown in Section ref:sec:rotating_iff_pseudo_int. +The obtained IFF parameters and the achievable damping are visually shown by large dots in Figure ref:fig:rotating_iff_hpf_nass_optimal_gain and are summarized in Table ref:tab:rotating_iff_hpf_opt_iff_hpf_params_nass. #+begin_src matlab %% Compute the optimal control gain @@ -3127,92 +2903,6 @@ for wi_i = 1:length(wis) end #+end_src -#+begin_src matlab :results none -%% Optimal modified IFF parameters that yields maximum simultaneous damping -figure; -tiledlayout(1, 3, 'TileSpacing', 'Compact', 'Padding', 'None'); - -ax1 = nexttile(); -yyaxis left -plot(wis, opt_iff_hpf_xi_vc, '-', 'DisplayName', '$\xi_{cl}$'); -set(gca, 'YScale', 'lin'); -ylim([0,1]); -ylabel('Damping Ratio $\xi$'); - -yyaxis right -hold on; -plot(wis, opt_iff_hpf_gain_vc, '-', 'DisplayName', '$g_{opt}$'); -plot(wis, wis*((sqrt(1e4/16)/(2*pi))^2 - 1), '--', 'DisplayName', '$g_{max}$'); -set(gca, 'YScale', 'lin'); -ylim([0,200]); -xlabel('$\omega_i$ [rad/s]'); -set(gca, 'YTickLabel',[]); -% ylabel('Controller gain $g$'); -set(gca, 'XScale', 'log'); -xticks([1e-2,1,1e2]) -legend('location', 'northwest', 'FontSize', 8); -title('$k_n = 0.01\,N/\mu m$'); - -ax2 = nexttile(); -yyaxis left -plot(wis, opt_iff_hpf_xi_md, '-'); -set(gca, 'YScale', 'lin'); -ylim([0,1]); -% ylabel('Damping Ratio $\xi$'); -set(gca, 'YTickLabel',[]); - -yyaxis right -hold on; -plot(wis, opt_iff_hpf_gain_md, '-'); -plot(wis, wis*((sqrt(1e6/16)/(2*pi))^2 - 1), '--'); -set(gca, 'YScale', 'lin'); -ylim([0,1000]); -xlabel('$\omega_i$ [rad/s]'); -% ylabel('Controller gain $g$'); -set(gca, 'YTickLabel',[]); -set(gca, 'XScale', 'log'); -xticks([1e-2,1,1e2]) -title('$k_n = 1\,N/\mu m$'); - -ax3 = nexttile(); -yyaxis left -plot(wis, opt_iff_hpf_xi_pz, '-'); -set(gca, 'YScale', 'lin'); -ylim([0,1]); -set(gca, 'YTickLabel',[]); -% ylabel('Damping Ratio $\xi$'); - -yyaxis right -hold on; -plot(wis, opt_iff_hpf_gain_pz, '-'); -plot(wis, wis*((sqrt(1e8/16)/(2*pi))^2 - 1), '--'); -set(gca, 'YScale', 'lin'); -ylim([0,10000]); -xlabel('$\omega_i$ [rad/s]'); -set(gca, 'YTickLabel',[]); -ylabel('Controller gain $g$'); -set(gca, 'XScale', 'log'); -xticks([1e-2,1,1e2]) -title('$k_n = 100\,N/\mu m$'); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/rotating_iff_hpf_nass_optimal_gain.pdf', 'width', 'full', 'height', 'normal'); -#+end_src - -#+name: fig:rotating_iff_hpf_nass_optimal_gain -#+caption: Optimal high pass filter cut-off frequency $\omega_i$ that yields maximum simultaneous damping -#+RESULTS: -[[file:figs/rotating_iff_hpf_nass_optimal_gain.png]] - -The IFF parameters are chosen as follow: -- for $k_n = \SI{0.01}{\N\per\mu\m}$: $\omega_i$ is chosen such that the maximum damping is achieved while the gain is less than half of the maximum gain at which the system is unstable. - This is done to have some control robustness. -- for $k_n = \SI{1}{\N\per\mu\m}$ and $k_n = \SI{100}{\N\per\mu\m}$: the largest $\omega_i$ is chosen such that obtained damping is $\SI{95}{\percent}$ of the maximum achievable damping. - Large $\omega_i$ is chosen here to limit the loss of compliance and the increase of coupling at low frequency as was shown in Section ref:sec:rotating_iff_pseudo_int. - -The obtained IFF parameters and the achievable damping are summarized in Table ref:tab:iff_hpf_opt_iff_hpf_params_nass. - #+begin_src matlab %% Find optimal parameters with at least a gain margin of 2 i_iff_hpf_vc = find(opt_iff_hpf_gain_vc < 0.5*(wis*((sqrt(1e4/16)/(2*pi))^2 - 1))); @@ -3225,196 +2915,147 @@ i_iff_hpf_pz = find(opt_iff_hpf_xi_pz > 0.95*max(opt_iff_hpf_xi_pz)); i_iff_hpf_pz = i_iff_hpf_pz(end)+1; #+end_src +#+begin_src matlab :results none +%% Optimal modified IFF parameters that yields maximum simultaneous damping +figure; +yyaxis left +hold on; +plot(wis, opt_iff_hpf_xi_vc, '-', 'DisplayName', '$\xi_{cl}$'); +plot(wis(i_iff_hpf_vc), opt_iff_hpf_xi_vc(i_iff_hpf_vc), '.', 'MarkerSize', 15, 'HandleVisibility', 'off'); +hold off; +set(gca, 'YScale', 'lin'); +ylim([0,1]); +ylabel('Damping Ratio $\xi$'); + +yyaxis right +hold on; +plot(wis, opt_iff_hpf_gain_vc, '-', 'DisplayName', '$g_{opt}$'); +plot(wis, wis*((sqrt(1e4/16)/(2*pi))^2 - 1), '--', 'DisplayName', '$g_{max}$'); +plot(wis(i_iff_hpf_vc), opt_iff_hpf_gain_vc(i_iff_hpf_vc), '.', 'MarkerSize', 15, 'HandleVisibility', 'off'); +set(gca, 'YScale', 'lin'); +ylim([0,200]); +xlabel('$\omega_i$ [rad/s]'); +set(gca, 'YTickLabel',[]); +ylabel('Controller gain $g$'); +set(gca, 'XScale', 'log'); +xticks([1e-2,1,1e2]) +legend('location', 'northwest', 'FontSize', 8); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/rotating_iff_hpf_nass_optimal_gain_vc.pdf', 'width', 'third', 'height', 450); +#+end_src + +#+begin_src matlab :results none +figure; +yyaxis left +hold on; +plot(wis, opt_iff_hpf_xi_md, '-'); +plot(wis(i_iff_hpf_md), opt_iff_hpf_xi_md(i_iff_hpf_md), '.', 'MarkerSize', 15, 'HandleVisibility', 'off'); +hold off; +set(gca, 'YScale', 'lin'); +ylim([0,1]); +ylabel('Damping Ratio $\xi$'); + +yyaxis right +hold on; +plot(wis, opt_iff_hpf_gain_md, '-'); +plot(wis(i_iff_hpf_md), opt_iff_hpf_gain_md(i_iff_hpf_md), '.', 'MarkerSize', 15, 'HandleVisibility', 'off'); +plot(wis, wis*((sqrt(1e6/16)/(2*pi))^2 - 1), '--'); +set(gca, 'YScale', 'lin'); +ylim([0,1000]); +xlabel('$\omega_i$ [rad/s]'); +ylabel('Controller gain $g$'); +set(gca, 'YTickLabel',[]); +set(gca, 'XScale', 'log'); +xticks([1e-2,1,1e2]) +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/rotating_iff_hpf_nass_optimal_gain_md.pdf', 'width', 'third', 'height', 450); +#+end_src + +#+begin_src matlab :results none +figure; +yyaxis left +hold on; +plot(wis, opt_iff_hpf_xi_pz, '-'); +plot(wis(i_iff_hpf_pz), opt_iff_hpf_xi_pz(i_iff_hpf_pz), '.', 'MarkerSize', 15, 'HandleVisibility', 'off'); +hold off; +set(gca, 'YScale', 'lin'); +ylim([0,1]); +ylabel('Damping Ratio $\xi$'); + +yyaxis right +hold on; +plot(wis, opt_iff_hpf_gain_pz, '-'); +plot(wis(i_iff_hpf_pz), opt_iff_hpf_gain_pz(i_iff_hpf_pz), '.', 'MarkerSize', 15, 'HandleVisibility', 'off'); +plot(wis, wis*((sqrt(1e8/16)/(2*pi))^2 - 1), '--'); +set(gca, 'YScale', 'lin'); +ylim([0,10000]); +xlabel('$\omega_i$ [rad/s]'); +set(gca, 'YTickLabel',[]); +ylabel('Controller gain $g$'); +set(gca, 'XScale', 'log'); +xticks([1e-2,1,1e2]) +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/rotating_iff_hpf_nass_optimal_gain_pz.pdf', 'width', 'third', 'height', 450); +#+end_src + +#+name: fig:rotating_iff_hpf_nass_optimal_gain +#+caption: For each value of $\omega_i$, the maximum damping ratio $\xi$ is computed (blue) and the corresponding controller gain is shown (in red). The choosen controller parameters used for further analysis are shown by the large dots. +#+attr_latex: :options [htbp] +#+begin_figure +#+attr_latex: :caption \subcaption{\label{fig:rotating_iff_hpf_nass_optimal_gain_vc}$k_n = 0.01\,N/\mu m$} +#+attr_latex: :options {0.33\textwidth} +#+begin_subfigure +#+attr_latex: :width 0.95\linewidth +[[file:figs/rotating_iff_hpf_nass_optimal_gain_vc.png]] +#+end_subfigure +#+attr_latex: :caption \subcaption{\label{fig:rotating_iff_hpf_nass_optimal_gain_md}$k_n = 1\,N/\mu m$} +#+attr_latex: :options {0.33\textwidth} +#+begin_subfigure +#+attr_latex: :width 0.95\linewidth +[[file:figs/rotating_iff_hpf_nass_optimal_gain_md.png]] +#+end_subfigure +#+attr_latex: :caption \subcaption{\label{fig:rotating_iff_hpf_nass_optimal_gain_pz}$k_n = 100\,N/\mu m$} +#+attr_latex: :options {0.33\textwidth} +#+begin_subfigure +#+attr_latex: :width 0.95\linewidth +[[file:figs/rotating_iff_hpf_nass_optimal_gain_pz.png]] +#+end_subfigure +#+end_figure + #+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*) data2orgtable([wis(i_iff_hpf_vc), opt_iff_hpf_gain_vc(i_iff_hpf_vc), opt_iff_hpf_xi_vc(i_iff_hpf_vc); wis(i_iff_hpf_md), opt_iff_hpf_gain_md(i_iff_hpf_md), opt_iff_hpf_xi_md(i_iff_hpf_md); wis(i_iff_hpf_pz), opt_iff_hpf_gain_pz(i_iff_hpf_pz), opt_iff_hpf_xi_pz(i_iff_hpf_pz)], {'$k_n = 0.01\,N/\mu m$', '$k_n = 1\,N/\mu m$', '$k_n = 100\,N/\mu m$'}, {'$\omega_i$', '$g$', '$\xi$'}, ' %.2f '); #+end_src -#+name: tab:iff_hpf_opt_iff_hpf_params_nass -#+caption: Obtained optimal parameters for the modified IFF controller -#+attr_latex: :environment tabularx :width 0.5\linewidth :align lXXX +#+name: tab:rotating_iff_hpf_opt_iff_hpf_params_nass +#+caption: Obtained optimal parameters ($\omega_i$ and $g$) for the modified IFF controller including a high pass filter. The corresponding achievable simultaneous damping of the two modes $\xi$ is also shown. +#+attr_latex: :environment tabularx :width 0.4\linewidth :align Xccc #+attr_latex: :center t :booktabs t #+RESULTS: -| | $\omega_i$ | $g$ | $\xi$ | -|-----------------------+------------+---------+-------| -| $k_n = 0.01\,N/\mu m$ | 7.32 | 51.13 | 0.45 | -| $k_n = 1\,N/\mu m$ | 39.17 | 426.95 | 0.93 | -| $k_n = 100\,N/\mu m$ | 499.45 | 3774.63 | 0.94 | - -The Root Locus for all three nano-hexapods are shown in Figure ref:fig:rotating_root_locus_iff_hpf_nass with included optimal chosen gains. - -#+begin_src matlab -%% Optimal IFF with added HPF -Kiff_hpf_vc = opt_iff_hpf_gain_vc(i_iff_hpf_vc)/(s + wis(i_iff_hpf_vc))*eye(2); -Kiff_hpf_vc.InputName = {'fu', 'fv'}; -Kiff_hpf_vc.OutputName = {'Fu', 'Fv'}; - -Kiff_hpf_md = opt_iff_hpf_gain_md(i_iff_hpf_md)/(s + wis(i_iff_hpf_md))*eye(2); -Kiff_hpf_md.InputName = {'fu', 'fv'}; -Kiff_hpf_md.OutputName = {'Fu', 'Fv'}; - -Kiff_hpf_pz = opt_iff_hpf_gain_pz(i_iff_hpf_pz)/(s + wis(i_iff_hpf_pz))*eye(2); -Kiff_hpf_pz.InputName = {'fu', 'fv'}; -Kiff_hpf_pz.OutputName = {'Fu', 'Fv'}; -#+end_src - -#+begin_src matlab :results none -%% Root Locus for optimal parameters - Comparison of attainable damping with the soft and moderately stiff nano-hexapods -gains = logspace(-2, 3, 200); - -figure; -tiledlayout(1, 3, 'TileSpacing', 'Compact', 'Padding', 'None'); - -% Voice coil Nano-Hexapod -ax1 = nexttile(); -hold on; -for g = gains - clpoles = pole(feedback(G_vc_norot({'fu', 'fv'}, {'Fu', 'Fv'}), g*Kiff_hpf_vc)); - plot(real(clpoles), imag(clpoles), '.', 'color', colors(1,:),'MarkerSize',4, ... - 'HandleVisibility', 'off'); -end -clpoles = pole(feedback(G_vc_norot({'fu', 'fv'}, {'Fu', 'Fv'}), Kiff_hpf_vc)); -plot(real(clpoles), imag(clpoles), '.', 'color', colors(1,:),'MarkerSize', 15, ... - 'DisplayName', '$\Omega = 0$'); -plot(real(pole(G_vc_norot({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_hpf_vc)), ... - imag(pole(G_vc_norot({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_hpf_vc)), ... - 'x', 'color', colors(1,:),'MarkerSize',8, ... - 'HandleVisibility', 'off'); -plot(real(tzero(G_vc_norot({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_hpf_vc)), ... - imag(tzero(G_vc_norot({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_hpf_vc)), ... - 'o', 'color', colors(1,:),'MarkerSize',8, ... - 'HandleVisibility', 'off'); - -for g = gains - clpoles = pole(feedback(G_vc_fast({'fu', 'fv'}, {'Fu', 'Fv'}), g*Kiff_hpf_vc)); - plot(real(clpoles), imag(clpoles), '.', 'color', colors(2,:),'MarkerSize',4, ... - 'HandleVisibility', 'off'); -end -clpoles = pole(feedback(G_vc_fast({'fu', 'fv'}, {'Fu', 'Fv'}), Kiff_hpf_vc)); -plot(real(clpoles), imag(clpoles), '.', 'color', colors(2,:),'MarkerSize', 15, ... - 'DisplayName', '$\Omega = 60$ rpm'); -plot(real(pole(G_vc_fast({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_hpf_vc)), ... - imag(pole(G_vc_fast({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_hpf_vc)), ... - 'x', 'color', colors(2,:),'MarkerSize',8, ... - 'HandleVisibility', 'off'); -plot(real(tzero(G_vc_fast({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_hpf_vc)), ... - imag(tzero(G_vc_fast({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_hpf_vc)), ... - 'o', 'color', colors(2,:),'MarkerSize',8, ... - 'HandleVisibility', 'off'); - -plot([0, -1e2*opt_iff_hpf_xi_vc(i_iff_hpf_vc)], [0, 1e2*cos(asin(opt_iff_hpf_xi_vc(i_iff_hpf_vc)))], '-', ... - 'DisplayName', sprintf('$\\xi = %.2f$', opt_iff_hpf_xi_vc(i_iff_hpf_vc)), 'color', [zeros(1,3), 0.5]) -hold off; -axis square; -xlabel('Real Part'); ylabel('Imaginary Part'); -xlim([-65, 5]); ylim([-35, 35]); -set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]); -title('$k_n = 0.01\,N/\mu m$'); -ldg = legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 1); -ldg.ItemTokenSize = [10, 1]; - -% APA Nano-Hexapod -ax2 = nexttile(); -hold on; -for g = gains - clpoles = pole(feedback(G_md_norot({'fu', 'fv'}, {'Fu', 'Fv'}), g*Kiff_hpf_md)); - plot(real(clpoles), imag(clpoles), '.', 'color', colors(1,:),'MarkerSize',4); -end -clpoles = pole(feedback(G_md_norot({'fu', 'fv'}, {'Fu', 'Fv'}), Kiff_hpf_md)); -plot(real(clpoles), imag(clpoles), '.', 'color', colors(1,:),'MarkerSize', 15); -plot(real(pole(G_md_norot({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_hpf_md)), ... - imag(pole(G_md_norot({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_hpf_md)), ... - 'x', 'color', colors(1,:),'MarkerSize',8); -plot(real(tzero(G_md_norot({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_hpf_md)), ... - imag(tzero(G_md_norot({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_hpf_md)), ... - 'o', 'color', colors(1,:),'MarkerSize',8); - -for g = gains - clpoles = pole(feedback(G_md_fast({'fu', 'fv'}, {'Fu', 'Fv'}), g*Kiff_hpf_md)); - plot(real(clpoles), imag(clpoles), '.', 'color', colors(2,:),'MarkerSize',4); -end -clpoles = pole(feedback(G_md_fast({'fu', 'fv'}, {'Fu', 'Fv'}), Kiff_hpf_md)); -plot(real(clpoles), imag(clpoles), '.', 'color', colors(2,:),'MarkerSize', 15); -plot(real(pole(G_md_fast({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_hpf_md)), ... - imag(pole(G_md_fast({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_hpf_md)), ... - 'x', 'color', colors(2,:),'MarkerSize',8); -plot(real(tzero(G_md_fast({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_hpf_md)), ... - imag(tzero(G_md_fast({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_hpf_md)), ... - 'o', 'color', colors(2,:),'MarkerSize',8); - -L = plot([0, -1e3*opt_iff_hpf_xi_md(i_iff_hpf_md)], [0, 1e3*cos(asin(opt_iff_hpf_xi_md(i_iff_hpf_md)))], '-', ... - 'DisplayName', sprintf('$\\xi = %.2f$', opt_iff_hpf_xi_md(i_iff_hpf_md)), 'color', [zeros(1,3), 0.5]); -leg = legend(L, 'location', 'northwest', 'FontSize', 8); -leg.ItemTokenSize(1) = 10; -hold off; -axis square; -xlabel('Real Part'); ylabel('Imaginary Part'); -xlim([-520, 20]); ylim([-270, 270]); -set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]); -title('$k_n = 1\,N/\mu m$'); - -% Piezo Nano-Hexapod -ax3 = nexttile(); -hold on; -for g = gains - clpoles = pole(feedback(G_pz_norot({'fu', 'fv'}, {'Fu', 'Fv'}), g*Kiff_hpf_pz)); - plot(real(clpoles), imag(clpoles), '.', 'color', colors(1,:),'MarkerSize',4); -end -clpoles = pole(feedback(G_pz_norot({'fu', 'fv'}, {'Fu', 'Fv'}), Kiff_hpf_pz)); -plot(real(clpoles), imag(clpoles), '.', 'color', colors(1,:),'MarkerSize', 15); -plot(real(pole(G_pz_norot({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_hpf_pz)), ... - imag(pole(G_pz_norot({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_hpf_pz)), ... - 'x', 'color', colors(1,:),'MarkerSize',8); -plot(real(tzero(G_pz_norot({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_hpf_pz)), ... - imag(tzero(G_pz_norot({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_hpf_pz)), ... - 'o', 'color', colors(1,:),'MarkerSize',8); - -for g = gains - clpoles = pole(feedback(G_pz_fast({'fu', 'fv'}, {'Fu', 'Fv'}), g*Kiff_hpf_pz)); - plot(real(clpoles), imag(clpoles), '.', 'color', colors(2,:),'MarkerSize',4); -end -clpoles = pole(feedback(G_pz_fast({'fu', 'fv'}, {'Fu', 'Fv'}), Kiff_hpf_pz)); -plot(real(clpoles), imag(clpoles), '.', 'color', colors(2,:),'MarkerSize', 15); -plot(real(pole(G_pz_fast({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_hpf_pz)), ... - imag(pole(G_pz_fast({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_hpf_pz)), ... - 'x', 'color', colors(2,:),'MarkerSize',8); -plot(real(tzero(G_pz_fast({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_hpf_pz)), ... - imag(tzero(G_pz_fast({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_hpf_pz)), ... - 'o', 'color', colors(2,:),'MarkerSize',8); - -L = plot([0, -1e4*opt_iff_hpf_xi_pz(i_iff_hpf_pz)], [0, 1e4*cos(asin(opt_iff_hpf_xi_pz(i_iff_hpf_pz)))], '-', ... - 'DisplayName', sprintf('$\\xi = %.2f$', opt_iff_hpf_xi_pz(i_iff_hpf_pz)), 'color', [zeros(1,3), 0.5]); -leg = legend(L, 'location', 'northwest', 'FontSize', 8); -leg.ItemTokenSize(1) = 10; -hold off; -axis square; -xlabel('Real Part'); ylabel('Imaginary Part'); -xlim([-5200, 20]); ylim([-2700, 2700]); -set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]); -title('$k_n = 100\,N/\mu m$'); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/rotating_root_locus_iff_hpf_nass.pdf', 'width', 'full', 'height', 'tall'); -#+end_src - -#+name: fig:rotating_root_locus_iff_hpf_nass -#+caption: Root Locus for modified IFF with high pass filter. Optimal $\omega_i$ is used. The three nano-hexapod stiffnesses are compared. The grey line indicates the minimum damping obtained with the optimal chosen control parameters. -#+RESULTS: -[[file:figs/rotating_root_locus_iff_hpf_nass.png]] +| $k_n$ | $\omega_i$ | $g$ | $\xi_\text{opt}$ | +|-----------------+------------+------+------------------| +| $0.01\,N/\mu m$ | 7.3 | 51 | 0.45 | +| $1\,N/\mu m$ | 39 | 427 | 0.93 | +| $100\,N/\mu m$ | 500 | 3775 | 0.94 | ** Optimal IFF with Parallel Stiffness -For each considered nano-hexapod stiffness, the parallel stiffness $k_p$ is varied from $k_{p,\text{min}} = m\Omega^2$ (the minimum stiffness to have unconditional stability) to $k_{p,\text{max}} = k_n$ (the total nano-hexapod stiffness). -In order to keep the overall stiffness constant, the actuator stiffness $k_a$ is decreased when $k_p$ is increased: -\begin{equation} -k_a = k_n - k_p -\end{equation} -With $k_n$ the total nano-hexapod stiffness. +For each considered nano-hexapod stiffness, the parallel stiffness $k_p$ is varied from $k_{p,\text{min}} = m\Omega^2$ (the minimum stiffness that yields unconditional stability) to $k_{p,\text{max}} = k_n$ (the total nano-hexapod stiffness). +In order to keep the overall stiffness constant, the actuator stiffness $k_a$ is decreased when $k_p$ is increased ($k_a = k_n - k_p$, with $k_n$ the total nano-hexapod stiffness). +A high pass filter is also added to limit the low frequency gain with a cut-off frequency $\omega_i$ equal to one tenth of the system resonance ($\omega_i = \omega_0/10$). -An high pass filter is also added to limit the low frequency gain. -The cut-off frequency $\omega_i$ is chosen to be one tenth of the system resonance: -\begin{equation} -\omega_i = \omega_0/10 -\end{equation} +The achievable maximum simultaneous damping of all the modes is computed as a function of the parallel stiffnesses (Figure ref:fig:rotating_iff_kp_nass_optimal_gain). +It is shown that the soft nano-hexapod cannot yield good damping as the parallel stiffness cannot be made large enough compared to the negative stiffness induced by the rotation. +For the two stiff options, the achievable damping decreases when the parallel stiffness is chosen too high as explained in Section ref:sec:rotating_iff_parallel_stiffness. +Such behavior can be explain by the fact that the achievable damping can be approximated by the distance between the open-loop pole and the open-loop zero [[cite:&preumont18_vibrat_contr_activ_struc_fourt_edition chapt 7.2]]. +This distance is larger for stiff nano-hexapod as the open-loop pole will be at higher frequencies while the open-loop zero, which depends on the value of the parallel stiffness, can only be made large for stiff nano-hexapods. + +Let's choose $k_p = 1\,N/mm$, $k_p = 0.01\,N/\mu m$ and $k_p = 1\,N/\mu m$ for the three considered nano-hexapods. +The corresponding optimal controller gains and achievable damping are summarized in Table ref:tab:rotating_iff_kp_opt_iff_kp_params_nass. #+begin_src matlab %% Maximum rotating velocity @@ -3422,9 +3063,7 @@ Wz = 2*pi; % [rad/s] %% Minimum parallel stiffness kp_min = (mn + ms) * Wz^2; % [N/m] -#+end_src -#+begin_src matlab %% Parameters for simulation mn = 15; % Nano-Hexapod mass [kg] ms = 1; % Sample Mass [kg] @@ -3438,9 +3077,7 @@ Kiff_pz = 1/(s + 0.1*sqrt(1e8/(mn+ms)))*eye(2); % IFF model_config = struct(); model_config.controller = "open_loop"; % Default: Open-Loop model_config.Tuv_type = "parallel_k"; % Default: 2DoF stage -#+end_src -#+begin_src matlab %% Computes the optimal parameters and attainable simultaneous damping - Voice Coil nano-hexapod kps_vc = logspace(log10(kp_min), log10(1e4), 100); % Tested parallel stiffnesses [N/m] kps_vc(end) = []; @@ -3466,9 +3103,7 @@ for kp_i = 1:length(kps_vc) opt_iff_kp_xi_vc(kp_i) = 1/xi_opt; opt_iff_kp_gain_vc(kp_i) = g_opt; end -#+end_src -#+begin_src matlab %% Computes the optimal parameters and attainable simultaneous damping - APA nano-hexapod kps_md = logspace(log10(kp_min), log10(1e6), 100); % Tested parallel stiffnesses [N/m] kps_md(end) = []; @@ -3495,9 +3130,7 @@ for kp_i = 1:length(kps_md) opt_iff_kp_xi_md(kp_i) = 1/xi_opt; opt_iff_kp_gain_md(kp_i) = g_opt; end -#+end_src -#+begin_src matlab %% Computes the optimal parameters and attainable simultaneous damping - Piezo nano-hexapod kps_pz = logspace(log10(kp_min), log10(1e8), 100); % Tested parallel stiffnesses [N/m] kps_pz(end) = []; @@ -3526,89 +3159,17 @@ for kp_i = 1:length(kps_pz) end #+end_src -The achievable maximum simultaneous damping of all the modes is computed as a function of the parallel stiffnesses. -The comparison for the nano-hexapod stiffnesses is done in Figure ref:fig:rotating_iff_kp_nass_optimal_gain. -It is shown that *the soft nano-hexapod cannot yield good damping*. -For the two stiff options, the achievable damping starts to significantly decrease when the parallel stiffness is one tenth of the total stiffness $k_p = k_n/10$. - -#+begin_src matlab :results none -%% Optimal IFF gain and associated simultaneous damping as a function of the parallel stiffness -figure; -hold on; -plot(kps_vc, opt_iff_kp_xi_vc, '-', 'DisplayName', '$k_n = 0.01\,N/\mu m$'); -plot(kps_md, opt_iff_kp_xi_md, '-', 'DisplayName', '$k_n = 1\,N/\mu m$'); -plot(kps_pz, opt_iff_kp_xi_pz, '-', 'DisplayName', '$k_n = 100\,N/\mu m$'); -hold off; -xlabel('$k_p [N/m]$'); -ylabel('Damping Ratio $\xi$'); -set(gca, 'XScale', 'log'); -set(gca, 'YScale', 'lin'); -ylim([0,1]); -legend('location', 'southeast', 'FontSize', 8); -xlim([kps_pz(1), kps_pz(end)]) -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/rotating_iff_kp_nass_optimal_gain.pdf', 'width', 'wide', 'height', 'normal'); -#+end_src - -#+name: fig:rotating_iff_kp_nass_optimal_gain -#+caption: Maximum achievable simultaneous damping with IFF as a function of the parallel stiffness for all three nano-hexapod stiffnesses -#+RESULTS: -[[file:figs/rotating_iff_kp_nass_optimal_gain.png]] - -Let's choose $k_p = \SI{e3}{\newton\per\m}$, $k_p = \SI{e4}{\newton\per\m}$ and $k_p = \SI{e6}{\newton\per\m}$ for the three considered nano-hexapods respectively based on Figure ref:fig:rotating_iff_kp_nass_optimal_gain. - -The corresponding optimal controller gains are shown in Table ref:tab:iff_kp_opt_iff_kp_params_nass. - #+begin_src matlab %% Find result with wanted parallel stiffness [~, i_kp_vc] = min(abs(kps_vc - 1e3)); [~, i_kp_md] = min(abs(kps_md - 1e4)); [~, i_kp_pz] = min(abs(kps_pz - 1e6)); -#+end_src -#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*) -data2orgtable([opt_iff_kp_gain_vc(i_kp_vc), opt_iff_kp_xi_vc(i_kp_vc); opt_iff_kp_gain_md(i_kp_md), opt_iff_kp_xi_md(i_kp_md); opt_iff_kp_gain_pz(i_kp_pz), opt_iff_kp_xi_pz(i_kp_pz)], {'$k_n = 0.01\,N/\mu m$', '$k_n = 1\,N/\mu m$', '$k_n = 100\,N/\mu m$'}, {'$g$', '$\xi_{\text{opt}}$'}, ' %.2f '); -#+end_src +%% Identify plants with choosen Parallel stiffnesses +model_config.Tuv_type = "parallel_k"; % Default: 2DoF stage -#+name: tab:iff_kp_opt_iff_kp_params_nass -#+caption: Obtained optimal parameters for the modified IFF controller -#+attr_latex: :environment tabularx :width 0.4\linewidth :align lXX -#+attr_latex: :center t :booktabs t -#+RESULTS: -| | $g$ | $\xi_{\text{opt}}$ | -|-----------------------+---------+--------------------| -| $k_n = 0.01\,N/\mu m$ | 47.9 | 0.44 | -| $k_n = 1\,N/\mu m$ | 465.57 | 0.97 | -| $k_n = 100\,N/\mu m$ | 4624.25 | 1.0 | - -The root locus for the three nano-hexapod with parallel stiffnesses are shown in Figure ref:fig:rotating_root_locus_iff_kp_nass. - -#+begin_important -Similarly to what was found with the IFF and added High Pass Filter: -- the stiff nano-hexapod is less affected by the rotation than the soft one -- the achievable damping is much larger with the stiff nano-hexapods -#+end_important - -#+begin_src matlab -%% Optimal IFF with added parallel stiffness -Kiff_kp_vc = opt_iff_kp_gain_vc(i_kp_vc)/(s + 0.1*sqrt(1e4/(ms+mn)))*eye(2); -Kiff_kp_vc.InputName = {'fu', 'fv'}; -Kiff_kp_vc.OutputName = {'Fu', 'Fv'}; - -Kiff_kp_md = opt_iff_kp_gain_md(i_kp_md)/(s + 0.1*sqrt(1e6/(ms+mn)))*eye(2); -Kiff_kp_md.InputName = {'fu', 'fv'}; -Kiff_kp_md.OutputName = {'Fu', 'Fv'}; - -Kiff_kp_pz = opt_iff_kp_gain_pz(i_kp_pz)/(s + 0.1*sqrt(1e8/(ms+mn)))*eye(2); -Kiff_kp_pz.InputName = {'fu', 'fv'}; -Kiff_kp_pz.OutputName = {'Fu', 'Fv'}; -#+end_src - -#+begin_src matlab -%% Identify plant with optimal parallel stiffness - Soft nano-hexapod -kp = kps_vc(i_kp_vc); +% Voice Coil +kp = 1e3; cp = 2*0.001*sqrt((ms + mn)*kp); kn = 1e4-kp; % Nano-Hexapod Stiffness [N/m] cn = 2*0.01*sqrt((ms + mn)*kn); % Nano-Hexapod Damping [N/(m/s)] @@ -3624,8 +3185,8 @@ G_vc_kp_norot = linearize(mdl, io, 0); G_vc_kp_norot.InputName = {'Fu', 'Fv', 'Fdx', 'Fdy', 'Dfx', 'Dfy'}; G_vc_kp_norot.OutputName = {'fu', 'fv', 'Du', 'Dv', 'Dx', 'Dy'}; -%% Identify plant with optimal parallel stiffness - Stiff nano-hexapod -kp = kps_md(i_kp_md); +% APA +kp = 1e4; cp = 2*0.001*sqrt((ms + mn)*kp); kn = 1e6 - kp; % Nano-Hexapod Stiffness [N/m] cn = 2*0.01*sqrt((ms + mn)*kn); % Nano-Hexapod Damping [N/(m/s)] @@ -3641,8 +3202,8 @@ G_md_kp_norot = linearize(mdl, io, 0); G_md_kp_norot.InputName = {'Fu', 'Fv', 'Fdx', 'Fdy', 'Dfx', 'Dfy'}; G_md_kp_norot.OutputName = {'fu', 'fv', 'Du', 'Dv', 'Dx', 'Dy'}; -%% Identify plant with optimal parallel stiffness - Stiff nano-hexapod -kp = kps_pz(i_kp_pz); +% Piezo +kp = 1e6; cp = 2*0.001*sqrt((ms + mn)*kp); kn = 1e8 - kp; % Nano-Hexapod Stiffness [N/m] cn = 2*0.01*sqrt((ms + mn)*kn); % Nano-Hexapod Damping [N/(m/s)] @@ -3660,153 +3221,65 @@ G_pz_kp_norot.OutputName = {'fu', 'fv', 'Du', 'Dv', 'Dx', 'Dy'}; #+end_src #+begin_src matlab :results none -%% Root Locus for optimal parameters - Comparison of attainable damping with the soft and moderately stiff nano-hexapods -gains = logspace(-2, 2, 400); - +%% Optimal IFF gain and associated simultaneous damping as a function of the parallel stiffness figure; -tiledlayout(1, 3, 'TileSpacing', 'Compact', 'Padding', 'None'); - -ax1 = nexttile(); hold on; -% Soft Nano-Hexapod - No Rotation -for g = gains - clpoles = pole(feedback(G_vc_kp_norot({'fu', 'fv'}, {'Fu', 'Fv'}), g*Kiff_kp_vc)); - plot(real(clpoles), imag(clpoles), '.', 'color', colors(1,:),'MarkerSize',4, ... - 'HandleVisibility', 'off'); -end -clpoles = pole(feedback(G_vc_kp_norot({'fu', 'fv'}, {'Fu', 'Fv'}), Kiff_kp_vc)); -plot(real(clpoles), imag(clpoles), '.', 'color', colors(1,:),'MarkerSize', 15, ... - 'DisplayName', '$\Omega = 0$'); -plot(real(pole(G_vc_kp_norot({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_kp_vc)), ... - imag(pole(G_vc_kp_norot({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_kp_vc)), ... - 'x', 'color', colors(1,:),'MarkerSize',8, ... - 'HandleVisibility', 'off'); -plot(real(tzero(G_vc_kp_norot({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_kp_vc)), ... - imag(tzero(G_vc_kp_norot({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_kp_vc)), ... - 'o', 'color', colors(1,:),'MarkerSize',8, ... - 'HandleVisibility', 'off'); - -% Soft Nano-Hexapod - High Speed Rotation -for g = gains - clpoles = pole(feedback(G_vc_kp_fast({'fu', 'fv'}, {'Fu', 'Fv'}), g*Kiff_kp_vc)); - plot(real(clpoles), imag(clpoles), '.', 'color', colors(2,:),'MarkerSize',4, ... - 'HandleVisibility', 'off'); -end -clpoles = pole(feedback(G_vc_kp_fast({'fu', 'fv'}, {'Fu', 'Fv'}), Kiff_kp_vc)); -plot(real(clpoles), imag(clpoles), '.', 'color', colors(2,:),'MarkerSize', 15, ... - 'DisplayName', '$\Omega = 60$ rpm'); -plot(real(pole(G_vc_kp_fast({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_kp_vc)), ... - imag(pole(G_vc_kp_fast({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_kp_vc)), ... - 'x', 'color', colors(2,:),'MarkerSize',8, ... - 'HandleVisibility', 'off'); -plot(real(tzero(G_vc_kp_fast({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_kp_vc)), ... - imag(tzero(G_vc_kp_fast({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_kp_vc)), ... - 'o', 'color', colors(2,:),'MarkerSize',8, ... - 'HandleVisibility', 'off'); -plot([0, -1e2*opt_iff_kp_xi_vc(i_kp_vc)], [0, 1e2*cos(asin(opt_iff_kp_xi_vc(i_kp_vc)))], '-', ... - 'DisplayName', sprintf('$\\xi = %.2f$', opt_iff_kp_xi_vc(i_kp_vc)), 'color', [zeros(1,3), 0.5]); +plot(kps_vc, opt_iff_kp_xi_vc, '-', ... + 'color', colors(1,:), 'DisplayName', '$k_n = 0.01\,N/\mu m$'); +plot(kps_vc(i_kp_vc), opt_iff_kp_xi_vc(i_kp_vc), '.', ... + 'color', colors(1,:), 'MarkerSize', 15, 'HandleVisibility', 'off'); +plot(kps_md, opt_iff_kp_xi_md, '-', ... + 'color', colors(2,:), 'DisplayName', '$k_n = 1\,N/\mu m$'); +plot(kps_md(i_kp_md), opt_iff_kp_xi_md(i_kp_md), '.', ... + 'color', colors(2,:), 'MarkerSize', 15, 'HandleVisibility', 'off'); +plot(kps_pz, opt_iff_kp_xi_pz, '-', ... + 'color', colors(3,:), 'DisplayName', '$k_n = 100\,N/\mu m$'); +plot(kps_pz(i_kp_pz), opt_iff_kp_xi_pz(i_kp_pz), '.', ... + 'color', colors(3,:), 'MarkerSize', 15, 'HandleVisibility', 'off'); hold off; -axis square; -xlabel('Real Part'); ylabel('Imaginary Part'); -xlim([-65, 5]); ylim([-35, 35]); -set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]); -title('$k_n = 0.01\,N/\mu m$'); -ldg = legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 1); -ldg.ItemTokenSize = [10, 1]; - -ax2 = nexttile(); -hold on; -% Stiff Nano-Hexapod - No Rotation -for g = gains - clpoles = pole(feedback(G_md_kp_norot({'fu', 'fv'}, {'Fu', 'Fv'}), g*Kiff_kp_md)); - plot(real(clpoles), imag(clpoles), '.', 'color', colors(1,:),'MarkerSize',4); -end -clpoles = pole(feedback(G_md_kp_norot({'fu', 'fv'}, {'Fu', 'Fv'}), Kiff_kp_md)); -plot(real(clpoles), imag(clpoles), '.', 'color', colors(1,:),'MarkerSize', 15); -plot(real(pole(G_md_kp_norot({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_kp_md)), ... - imag(pole(G_md_kp_norot({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_kp_md)), ... - 'x', 'color', colors(1,:),'MarkerSize',8); -plot(real(tzero(G_md_kp_norot({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_kp_md)), ... - imag(tzero(G_md_kp_norot({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_kp_md)), ... - 'o', 'color', colors(1,:),'MarkerSize',8); - -% Stiff Nano-Hexapod - High Speed Rotation -for g = gains - clpoles = pole(feedback(G_md_kp_fast({'fu', 'fv'}, {'Fu', 'Fv'}), g*Kiff_kp_md)); - plot(real(clpoles), imag(clpoles), '.', 'color', colors(2,:),'MarkerSize',4); -end -clpoles = pole(feedback(G_md_kp_fast({'fu', 'fv'}, {'Fu', 'Fv'}), Kiff_kp_md)); -plot(real(clpoles), imag(clpoles), '.', 'color', colors(2,:),'MarkerSize', 15); -plot(real(pole(G_md_kp_fast({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_kp_md)), ... - imag(pole(G_md_kp_fast({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_kp_md)), ... - 'x', 'color', colors(2,:),'MarkerSize',8); -plot(real(tzero(G_md_kp_fast({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_kp_md)), ... - imag(tzero(G_md_kp_fast({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_kp_md)), ... - 'o', 'color', colors(2,:),'MarkerSize',8); -L = plot([0, -1e3*opt_iff_kp_xi_md(i_kp_md)], [0, 1e3*cos(asin(opt_iff_kp_xi_md(i_kp_md)))], '-', ... - 'DisplayName', sprintf('$\\xi = %.2f$', opt_iff_kp_xi_md(i_kp_md)), 'color', [zeros(1,3), 0.5]); -leg = legend(L, 'location', 'northwest', 'FontSize', 8); -leg.ItemTokenSize(1) = 10; -hold off; -axis square; -xlabel('Real Part'); ylabel('Imaginary Part'); -set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]); -xlim([-520, 20]); ylim([-270, 270]); -title('$k_n = 1\,N/\mu m$'); - -ax3 = nexttile(); -hold on; -% Stiff Nano-Hexapod - No Rotation -for g = gains - clpoles = pole(feedback(G_pz_kp_norot({'fu', 'fv'}, {'Fu', 'Fv'}), g*Kiff_kp_pz)); - plot(real(clpoles), imag(clpoles), '.', 'color', colors(1,:),'MarkerSize',4); -end -clpoles = pole(feedback(G_pz_kp_norot({'fu', 'fv'}, {'Fu', 'Fv'}), Kiff_kp_pz)); -plot(real(clpoles), imag(clpoles), '.', 'color', colors(1,:),'MarkerSize', 15); -plot(real(pole(G_pz_kp_norot({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_kp_pz)), ... - imag(pole(G_pz_kp_norot({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_kp_pz)), ... - 'x', 'color', colors(1,:),'MarkerSize',8); -plot(real(tzero(G_pz_kp_norot({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_kp_pz)), ... - imag(tzero(G_pz_kp_norot({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_kp_pz)), ... - 'o', 'color', colors(1,:),'MarkerSize',8); - -% Stiff Nano-Hexapod - High Speed Rotation -for g = gains - clpoles = pole(feedback(G_pz_kp_fast({'fu', 'fv'}, {'Fu', 'Fv'}), g*Kiff_kp_pz)); - plot(real(clpoles), imag(clpoles), '.', 'color', colors(2,:),'MarkerSize',4); -end -clpoles = pole(feedback(G_pz_kp_fast({'fu', 'fv'}, {'Fu', 'Fv'}), Kiff_kp_pz)); -plot(real(clpoles), imag(clpoles), '.', 'color', colors(2,:),'MarkerSize', 15); -plot(real(pole(G_pz_kp_fast({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_kp_pz)), ... - imag(pole(G_pz_kp_fast({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_kp_pz)), ... - 'x', 'color', colors(2,:),'MarkerSize',8); -plot(real(tzero(G_pz_kp_fast({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_kp_pz)), ... - imag(tzero(G_pz_kp_fast({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_kp_pz)), ... - 'o', 'color', colors(2,:),'MarkerSize',8); -L = plot([0, -1e4*opt_iff_kp_xi_pz(i_kp_pz)], [0, 1e4*cos(asin(opt_iff_kp_xi_pz(i_kp_pz)))], '-', ... - 'DisplayName', sprintf('$\\xi = %.2f$', opt_iff_kp_xi_pz(i_kp_pz)), 'color', [zeros(1,3), 0.5]); -leg = legend(L, 'location', 'northwest', 'FontSize', 8); -leg.ItemTokenSize(1) = 10; -hold off; -axis square; -xlabel('Real Part'); ylabel('Imaginary Part'); -set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]); -xlim([-5200, 200]); ylim([-2700, 2700]); -title('$k_n = 100\,N/\mu m$'); +xlabel('$k_p [N/m]$'); +ylabel('Damping Ratio $\xi$'); +set(gca, 'XScale', 'log'); +set(gca, 'YScale', 'lin'); +ylim([0,1]); +yticks([0:0.2:1]) +legend('location', 'southeast', 'FontSize', 8); +xlim([kps_pz(1), kps_pz(end)]) #+end_src -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/rotating_root_locus_iff_kp_nass.pdf', 'width', 'full', 'height', 'tall'); +#+begin_src matlab :tangle no :exports results :results file none +exportFig('figs/rotating_iff_kp_nass_optimal_gain.pdf', 'width', 'half', 'height', 350); #+end_src -#+name: fig:rotating_root_locus_iff_kp_nass -#+caption: Root Locus for optimal parameters (IFF + $k_p$ strategy) - Comparison of attainable damping with the three nano-hexapod stiffnesses +#+attr_latex: :options [t]{0.49\linewidth} +#+begin_minipage +#+name: fig:rotating_iff_kp_nass_optimal_gain +#+attr_latex: :width \linewidth :float nil +#+caption: Maximum damping $\xi$ as a function of the parallel stiffness $k_p$ +[[file:figs/rotating_iff_kp_nass_optimal_gain.png]] +#+end_minipage +\hfill +#+attr_latex: :options [b]{0.45\linewidth} +#+begin_minipage +#+caption: Obtained optimal parameters for the IFF controller when using parallel stiffnesses +#+name: tab:rotating_iff_kp_opt_iff_kp_params_nass +#+attr_latex: :environment tabularx :width \linewidth :placement [b] :align Xccc +#+attr_latex: :booktabs t :float nil #+RESULTS: -[[file:figs/rotating_root_locus_iff_kp_nass.png]] +| $k_n$ | $k_p$ | $g$ | $\xi_{\text{opt}}$ | +|-----------------+-----------------+---------+--------------------| +| $0.01\,N/\mu m$ | $1\,N/mm$ | 47.9 | 0.44 | +| $1\,N/\mu m$ | $0.01\,N/\mu m$ | 465.57 | 0.97 | +| $100\,N/\mu m$ | $1\,N/\mu m$ | 4624.25 | 0.99 | +#+end_minipage + +#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*) +data2orgtable([opt_iff_kp_gain_vc(i_kp_vc), opt_iff_kp_xi_vc(i_kp_vc); opt_iff_kp_gain_md(i_kp_md), opt_iff_kp_xi_md(i_kp_md); opt_iff_kp_gain_pz(i_kp_pz), opt_iff_kp_xi_pz(i_kp_pz)], {'$k_n = 0.01\,N/\mu m$', '$k_n = 1\,N/\mu m$', '$k_n = 100\,N/\mu m$'}, {'$g$', '$\xi_{\text{opt}}$'}, ' %.2f '); +#+end_src ** Optimal Relative Motion Control - -For each considered nano-hexapod stiffness, relative damping control is applied and the achievable damping ratio as a function of the controller gain is shown in Figure ref:fig:rotating_rdc_optimal_gain. +For each considered nano-hexapod stiffness, relative damping control is applied and the achievable damping ratio as a function of the controller gain is computed (Figure ref:fig:rotating_rdc_optimal_gain). +The gain is chosen is chosen such that 99% of modal damping is obtained (obtained gains are summarized in Table ref:tab:rotating_rdc_opt_params_nass). #+begin_src matlab %% Computes the optimal parameters and attainable simultaneous damping - Piezo nano-hexapod @@ -3832,40 +3305,6 @@ for g_i = 1:length(rdc_gains) end #+end_src -#+begin_src matlab :results none -%% Optimal IFF gain and associated simultaneous damping as a function of the parallel stiffness -figure; -hold on; -plot(rdc_gains, rdc_xi_vc, '-', 'DisplayName', '$k_n = 0.01\,N/\mu m$'); -plot(rdc_gains, rdc_xi_md, '-', 'DisplayName', '$k_n = 1\,N/\mu m$'); -plot(rdc_gains, rdc_xi_pz, '-', 'DisplayName', '$k_n = 100\,N/\mu m$'); -hold off; -xlabel('Relative Damping Controller gain $g$'); -ylabel('Damping Ratio $\xi$'); -set(gca, 'XScale', 'log'); -set(gca, 'YScale', 'lin'); -ylim([0,1]); -xlim([rdc_gains(1), rdc_gains(end)]) -legend('location', 'southeast', 'FontSize', 8); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/rotating_rdc_optimal_gain.pdf', 'width', 'wide', 'height', 'normal'); -#+end_src - -#+name: fig:rotating_rdc_optimal_gain -#+caption: Achievable simultaneous damping with "Relative Damping Control" as a function of the controller gain for all three nano-hexapod stiffnesses -#+RESULTS: -[[file:figs/rotating_rdc_optimal_gain.png]] - -The gain is chosen is chosen such that 99% of modal damping is obtained. -The root locus for all three nano-hexapod stiffnesses are shown in Figure ref:fig:rotating_root_locus_rdc_nass. - -#+begin_important -Relative damping control is much less impacted by gyroscopic effects. -It can be easily applied on the nano-hexapod with and without rotation without much differences. -#+end_important - #+begin_src matlab %% Optimal RDC [~, i_rdc_vc] = min(abs(rdc_xi_vc - 0.99)); @@ -3878,159 +3317,65 @@ Krdc_pz = rdc_gains(i_rdc_pz)*Krdc; #+end_src #+begin_src matlab :results none -%% Root Locus for optimal parameters - Comparison of attainable damping with the soft and moderately strdc nano-hexapods -gains = logspace(-2, 3, 200); - +%% Optimal IFF gain and associated simultaneous damping as a function of the parallel stiffness figure; -tiledlayout(1, 3, 'TileSpacing', 'Compact', 'Padding', 'None'); - -% Voice coil Nano-Hexapod -ax1 = nexttile(); hold on; -for g = gains - clpoles = pole(feedback(G_vc_norot({'Du', 'Dv'}, {'Fu', 'Fv'}), g*Krdc_vc)); - plot(real(clpoles), imag(clpoles), '.', 'color', colors(1,:),'MarkerSize',4, ... - 'HandleVisibility', 'off'); -end -clpoles = pole(feedback(G_vc_norot({'Du', 'Dv'}, {'Fu', 'Fv'}), Krdc_vc)); -plot(real(clpoles), imag(clpoles), '.', 'color', colors(1,:),'MarkerSize', 15, ... - 'DisplayName', '$\Omega = 0$'); -plot(real(pole(G_vc_norot({'Du', 'Dv'}, {'Fu', 'Fv'})*Krdc_vc)), ... - imag(pole(G_vc_norot({'Du', 'Dv'}, {'Fu', 'Fv'})*Krdc_vc)), ... - 'x', 'color', colors(1,:),'MarkerSize',8, ... - 'HandleVisibility', 'off'); -plot(real(tzero(G_vc_norot({'Du', 'Dv'}, {'Fu', 'Fv'})*Krdc_vc)), ... - imag(tzero(G_vc_norot({'Du', 'Dv'}, {'Fu', 'Fv'})*Krdc_vc)), ... - 'o', 'color', colors(1,:),'MarkerSize',8, ... - 'HandleVisibility', 'off'); - -for g = gains - clpoles = pole(feedback(G_vc_fast({'Du', 'Dv'}, {'Fu', 'Fv'}), g*Krdc_vc)); - plot(real(clpoles), imag(clpoles), '.', 'color', colors(2,:),'MarkerSize',4, ... - 'HandleVisibility', 'off'); -end -clpoles = pole(feedback(G_vc_fast({'Du', 'Dv'}, {'Fu', 'Fv'}), Krdc_vc)); -plot(real(clpoles), imag(clpoles), '.', 'color', colors(2,:),'MarkerSize', 15, ... - 'DisplayName', '$\Omega = 60$ rpm'); -plot(real(pole(G_vc_fast({'Du', 'Dv'}, {'Fu', 'Fv'})*Krdc_vc)), ... - imag(pole(G_vc_fast({'Du', 'Dv'}, {'Fu', 'Fv'})*Krdc_vc)), ... - 'x', 'color', colors(2,:),'MarkerSize',8, ... - 'HandleVisibility', 'off'); -plot(real(tzero(G_vc_fast({'Du', 'Dv'}, {'Fu', 'Fv'})*Krdc_vc)), ... - imag(tzero(G_vc_fast({'Du', 'Dv'}, {'Fu', 'Fv'})*Krdc_vc)), ... - 'o', 'color', colors(2,:),'MarkerSize',8, ... - 'HandleVisibility', 'off'); - -plot([0, -1e2*rdc_xi_vc(i_rdc_vc)], [0, 1e2*cos(asin(rdc_xi_vc(i_rdc_vc)))], '-', ... - 'DisplayName', sprintf('$\\xi = %.2f$', rdc_xi_vc(i_rdc_vc)), 'color', [zeros(1,3), 0.5]) +plot(rdc_gains, rdc_xi_vc, '-', ... + 'color', colors(1,:), 'DisplayName', '$k_n = 0.01\,N/\mu m$'); +plot(rdc_gains(i_rdc_vc), rdc_xi_vc(i_rdc_vc), '.', ... + 'color', colors(1,:), 'MarkerSize', 15, 'HandleVisibility', 'off'); +plot(rdc_gains, rdc_xi_md, '-', ... + 'color', colors(2,:), 'DisplayName', '$k_n = 1\,N/\mu m$'); +plot(rdc_gains(i_rdc_md), rdc_xi_md(i_rdc_md), '.', ... + 'color', colors(2,:), 'MarkerSize', 15, 'HandleVisibility', 'off'); +plot(rdc_gains, rdc_xi_pz, '-', ... + 'color', colors(3,:), 'DisplayName', '$k_n = 100\,N/\mu m$'); +plot(rdc_gains(i_rdc_pz), rdc_xi_pz(i_rdc_pz), '.', ... + 'color', colors(3,:), 'MarkerSize', 15, 'HandleVisibility', 'off'); hold off; -axis square; -xlabel('Real Part'); ylabel('Imaginary Part'); -xlim([-65, 5]); ylim([-35, 35]); -set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]); -title('$k_n = 0.01\,N/\mu m$'); -ldg = legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 1); -ldg.ItemTokenSize = [10, 1]; - -% APA Nano-Hexapod -ax2 = nexttile(); -hold on; -for g = gains - clpoles = pole(feedback(G_md_norot({'Du', 'Dv'}, {'Fu', 'Fv'}), g*Krdc_md)); - plot(real(clpoles), imag(clpoles), '.', 'color', colors(1,:),'MarkerSize',4); -end -clpoles = pole(feedback(G_md_norot({'Du', 'Dv'}, {'Fu', 'Fv'}), Krdc_md)); -plot(real(clpoles), imag(clpoles), '.', 'color', colors(1,:),'MarkerSize', 15); -plot(real(pole(G_md_norot({'Du', 'Dv'}, {'Fu', 'Fv'})*Krdc_md)), ... - imag(pole(G_md_norot({'Du', 'Dv'}, {'Fu', 'Fv'})*Krdc_md)), ... - 'x', 'color', colors(1,:),'MarkerSize',8); -plot(real(tzero(G_md_norot({'Du', 'Dv'}, {'Fu', 'Fv'})*Krdc_md)), ... - imag(tzero(G_md_norot({'Du', 'Dv'}, {'Fu', 'Fv'})*Krdc_md)), ... - 'o', 'color', colors(1,:),'MarkerSize',8); - -for g = gains - clpoles = pole(feedback(G_md_fast({'Du', 'Dv'}, {'Fu', 'Fv'}), g*Krdc_md)); - plot(real(clpoles), imag(clpoles), '.', 'color', colors(2,:),'MarkerSize',4); -end -clpoles = pole(feedback(G_md_fast({'Du', 'Dv'}, {'Fu', 'Fv'}), Krdc_md)); -plot(real(clpoles), imag(clpoles), '.', 'color', colors(2,:),'MarkerSize', 15); -plot(real(pole(G_md_fast({'Du', 'Dv'}, {'Fu', 'Fv'})*Krdc_md)), ... - imag(pole(G_md_fast({'Du', 'Dv'}, {'Fu', 'Fv'})*Krdc_md)), ... - 'x', 'color', colors(2,:),'MarkerSize',8); -plot(real(tzero(G_md_fast({'Du', 'Dv'}, {'Fu', 'Fv'})*Krdc_md)), ... - imag(tzero(G_md_fast({'Du', 'Dv'}, {'Fu', 'Fv'})*Krdc_md)), ... - 'o', 'color', colors(2,:),'MarkerSize',8); - -L = plot([0, -1e3*rdc_xi_md(i_rdc_md)], [0, 1e3*cos(asin(rdc_xi_md(i_rdc_md)))], '-', ... - 'DisplayName', sprintf('$\\xi = %.2f$', rdc_xi_md(i_rdc_md)), 'color', [zeros(1,3), 0.5]); -leg = legend(L, 'location', 'northwest', 'FontSize', 8); -leg.ItemTokenSize(1) = 10; -hold off; -axis square; -xlabel('Real Part'); ylabel('Imaginary Part'); -xlim([-520, 20]); ylim([-270, 270]); -set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]); -title('$k_n = 1\,N/\mu m$'); - -% Piezo Nano-Hexapod -ax3 = nexttile(); -hold on; -for g = gains - clpoles = pole(feedback(G_pz_norot({'Du', 'Dv'}, {'Fu', 'Fv'}), g*Krdc_pz)); - plot(real(clpoles), imag(clpoles), '.', 'color', colors(1,:),'MarkerSize',4); -end -clpoles = pole(feedback(G_pz_norot({'Du', 'Dv'}, {'Fu', 'Fv'}), Krdc_pz)); -plot(real(clpoles), imag(clpoles), '.', 'color', colors(1,:),'MarkerSize', 15); -plot(real(pole(G_pz_norot({'Du', 'Dv'}, {'Fu', 'Fv'})*Krdc_pz)), ... - imag(pole(G_pz_norot({'Du', 'Dv'}, {'Fu', 'Fv'})*Krdc_pz)), ... - 'x', 'color', colors(1,:),'MarkerSize',8); -plot(real(tzero(G_pz_norot({'Du', 'Dv'}, {'Fu', 'Fv'})*Krdc_pz)), ... - imag(tzero(G_pz_norot({'Du', 'Dv'}, {'Fu', 'Fv'})*Krdc_pz)), ... - 'o', 'color', colors(1,:),'MarkerSize',8); - -for g = gains - clpoles = pole(feedback(G_pz_fast({'Du', 'Dv'}, {'Fu', 'Fv'}), g*Krdc_pz)); - plot(real(clpoles), imag(clpoles), '.', 'color', colors(2,:),'MarkerSize',4); -end -clpoles = pole(feedback(G_pz_fast({'Du', 'Dv'}, {'Fu', 'Fv'}), Krdc_pz)); -plot(real(clpoles), imag(clpoles), '.', 'color', colors(2,:),'MarkerSize', 15); -plot(real(pole(G_pz_fast({'Du', 'Dv'}, {'Fu', 'Fv'})*Krdc_pz)), ... - imag(pole(G_pz_fast({'Du', 'Dv'}, {'Fu', 'Fv'})*Krdc_pz)), ... - 'x', 'color', colors(2,:),'MarkerSize',8); -plot(real(tzero(G_pz_fast({'Du', 'Dv'}, {'Fu', 'Fv'})*Krdc_pz)), ... - imag(tzero(G_pz_fast({'Du', 'Dv'}, {'Fu', 'Fv'})*Krdc_pz)), ... - 'o', 'color', colors(2,:),'MarkerSize',8); - -L = plot([0, -1e4*rdc_xi_pz(i_rdc_pz)], [0, 1e4*cos(asin(rdc_xi_pz(i_rdc_pz)))], '-', ... - 'DisplayName', sprintf('$\\xi = %.2f$', rdc_xi_pz(i_rdc_pz)), 'color', [zeros(1,3), 0.5]); -leg = legend(L, 'location', 'northwest', 'FontSize', 8); -leg.ItemTokenSize(1) = 10; -hold off; -axis square; -xlabel('Real Part'); ylabel('Imaginary Part'); -xlim([-5200, 20]); ylim([-2700, 2700]); -set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]); -title('$k_n = 100\,N/\mu m$'); +xlabel('Relative Damping Controller gain $g$'); +ylabel('Damping Ratio $\xi$'); +set(gca, 'XScale', 'log'); +set(gca, 'YScale', 'lin'); +ylim([0,1]); +yticks([0:0.2:1]) +xlim([rdc_gains(1), rdc_gains(end)]) +legend('location', 'southeast', 'FontSize', 8); #+end_src #+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/rotating_root_locus_rdc_nass.pdf', 'width', 'full', 'height', 'tall'); +exportFig('figs/rotating_rdc_optimal_gain.pdf', 'width', 'half', 'height', 350); #+end_src -#+name: fig:rotating_root_locus_rdc_nass -#+caption: Root Locus for optimal parameters - Comparison of attainable damping with the soft and moderately stiff nano-hexapods +#+attr_latex: :options [t]{0.49\linewidth} +#+begin_minipage +#+name: fig:rotating_rdc_optimal_gain +#+attr_latex: :width \linewidth :float nil +#+caption: Maximum damping $\xi$ as a function of the RDC gain $g$ +[[file:figs/rotating_rdc_optimal_gain.png]] +#+end_minipage +\hfill +#+attr_latex: :options [b]{0.45\linewidth} +#+begin_minipage +#+caption: Obtained optimal parameters for the RDC +#+name: tab:rotating_rdc_opt_params_nass +#+attr_latex: :environment tabularx :width 0.8\linewidth :placement [b] :align Xcc +#+attr_latex: :booktabs t :float nil #+RESULTS: -[[file:figs/rotating_root_locus_rdc_nass.png]] +| $k_n$ | $g$ | $\xi_{\text{opt}}$ | +|-----------------+-------+--------------------| +| $0.01\,N/\mu m$ | 1600 | 0.99 | +| $1\,N/\mu m$ | 8200 | 0.99 | +| $100\,N/\mu m$ | 80000 | 0.99 | +#+end_minipage ** Comparison of the obtained damped plants -Let's now compare the obtained damped plants for the three active damping techniques applied on the three nano-hexapod stiffnesses (Figure ref:fig:rotating_nass_damped_plant_comp). +Now that optimal parameters for the three considered active damping techniques have been determined, the obtained damped plants are computed and compared in Figure ref:fig:rotating_nass_damped_plant_comp. -#+begin_important Similarly to what was concluded in previous analysis: -- IFF adds coupling below the resonance frequency as compared to the open-loop and RDC cases -- Add three methods are yielding good damping, except for IFF applied on the soft nano-hexapod where things are more complicated +- acrshort:iff adds coupling below the resonance frequency as compared to the open-loop and acrshort:rdc cases +- All three methods are yielding good damping, except for acrshort:iff applied on the soft nano-hexapod - Coupling is smaller for stiff nano-hexapods -#+end_important #+begin_src matlab :tangle no %% Saved controllers @@ -4080,11 +3425,9 @@ G_pz_fast_rdc = feedback(G_pz_fast, Krdc_pz, 'name'); #+begin_src matlab :exports none :results none %% Comparison of the damped plants (direct and coupling terms) for the three proposed active damping techniques (IFF with HPF, IFF with $k_p$ and RDC) applied on the three nano-hexapod stiffnesses freqs_vc = logspace(-1, 2, 1000); -freqs_md = logspace(0, 3, 1000); -freqs_pz = logspace(0, 3, 1000); figure; -tiledlayout(3, 3, 'TileSpacing', 'Compact', 'Padding', 'None'); +tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); ax1 = nexttile([2,1]); hold on; @@ -4099,9 +3442,36 @@ plot(freqs_vc, abs(squeeze(freqresp(G_vc_fast_rdc( 'Dv', 'Fu'), freqs_vc, 'Hz')) hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); ylabel('Magnitude [m/N]'); set(gca, 'XTickLabel',[]); -title('$k_n = 0.01\,N/\mu m$'); +ylim([1e-8, 1e-2]) -ax2 = nexttile([2,1]); +ax2 = nexttile; +hold on; +plot(freqs_vc, 180/pi*angle(squeeze(freqresp(G_vc_fast( 'Du', 'Fu'), freqs_vc, 'Hz'))), '-' , 'color', [zeros(1,3)]); +plot(freqs_vc, 180/pi*angle(squeeze(freqresp(G_vc_fast_iff_hpf( 'Du', 'Fu'), freqs_vc, 'Hz'))), '-' , 'color', [colors(1,:)]); +plot(freqs_vc, 180/pi*angle(squeeze(freqresp(G_vc_fast_iff_kp( 'Du', 'Fu'), freqs_vc, 'Hz'))), '-' , 'color', [colors(2,:)]); +plot(freqs_vc, 180/pi*angle(squeeze(freqresp(G_vc_fast_rdc( 'Du', 'Fu'), freqs_vc, 'Hz'))), '-' , 'color', [colors(3,:)]); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); +hold off; +yticks(-360:90:360); +ylim([-180, 0]); + +linkaxes([ax1,ax2],'x'); +xlim([freqs_vc(1), freqs_vc(end)]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/rotating_nass_damped_plant_comp_vc.pdf', 'width', 'third', 'height', 600); +#+end_src + +#+begin_src matlab :exports none :results none +freqs_md = logspace(0, 3, 1000); + +figure; +tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); + +ax1 = nexttile([2,1]); hold on; plot(freqs_md, abs(squeeze(freqresp(G_md_fast( 'Du', 'Fu'), freqs_md, 'Hz'))), '-' , 'color', [zeros(1,3)]); plot(freqs_md, abs(squeeze(freqresp(G_md_fast( 'Dv', 'Fu'), freqs_md, 'Hz'))), '-' , 'color', [zeros(1,3), 0.5]); @@ -4113,10 +3483,37 @@ plot(freqs_md, abs(squeeze(freqresp(G_md_fast_rdc( 'Du', 'Fu'), freqs_md, 'Hz')) plot(freqs_md, abs(squeeze(freqresp(G_md_fast_rdc( 'Dv', 'Fu'), freqs_md, 'Hz'))), '-' , 'color', [colors(3,:), 0.5]); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]); -title('$k_n = 1\,N/\mu m$'); +ylabel('Magnitude [m/N]'); set(gca, 'XTickLabel',[]); +ylim([1e-10, 1e-4]) -ax3 = nexttile([2,1]); +ax2 = nexttile; +hold on; +plot(freqs_md, 180/pi*angle(squeeze(freqresp(G_md_fast( 'Du', 'Fu'), freqs_md, 'Hz'))), '-' , 'color', [zeros(1,3)]); +plot(freqs_md, 180/pi*angle(squeeze(freqresp(G_md_fast_iff_hpf( 'Du', 'Fu'), freqs_md, 'Hz'))), '-' , 'color', [colors(1,:)]); +plot(freqs_md, 180/pi*angle(squeeze(freqresp(G_md_fast_iff_kp( 'Du', 'Fu'), freqs_md, 'Hz'))), '-' , 'color', [colors(2,:)]); +plot(freqs_md, 180/pi*angle(squeeze(freqresp(G_md_fast_rdc( 'Du', 'Fu'), freqs_md, 'Hz'))), '-' , 'color', [colors(3,:)]); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); +hold off; +yticks(-360:90:360); +ylim([-180, 0]); + +linkaxes([ax1,ax2],'x'); +xlim([freqs_md(1), freqs_md(end)]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/rotating_nass_damped_plant_comp_md.pdf', 'width', 'third', 'height', 600); +#+end_src + +#+begin_src matlab :exports none :results none +freqs_pz = logspace(0, 3, 1000); + +figure; +tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); + +ax1 = nexttile([2,1]); hold on; plot(freqs_pz, abs(squeeze(freqresp(G_pz_fast( 'Du', 'Fu'), freqs_pz, 'Hz'))), '-' , 'color', [zeros(1,3)], ... 'DisplayName', 'OL'); @@ -4136,41 +3533,12 @@ plot(freqs_pz, abs(squeeze(freqresp(G_pz_fast_rdc( 'Dv', 'Fu'), freqs_pz, 'Hz')) 'HandleVisibility', 'off'); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]); -ylim([1e-12, 1e-2]) +ylabel('Magnitude [m/N]'); set(gca, 'XTickLabel',[]); +ylim([1e-12, 1e-6]) ldg = legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 1); ldg.ItemTokenSize = [20, 1]; -title('$k_n = 100\,N/\mu m$'); -ax1b = nexttile; -hold on; -plot(freqs_vc, 180/pi*angle(squeeze(freqresp(G_vc_fast( 'Du', 'Fu'), freqs_vc, 'Hz'))), '-' , 'color', [zeros(1,3)]); -plot(freqs_vc, 180/pi*angle(squeeze(freqresp(G_vc_fast_iff_hpf( 'Du', 'Fu'), freqs_vc, 'Hz'))), '-' , 'color', [colors(1,:)]); -plot(freqs_vc, 180/pi*angle(squeeze(freqresp(G_vc_fast_iff_kp( 'Du', 'Fu'), freqs_vc, 'Hz'))), '-' , 'color', [colors(2,:)]); -plot(freqs_vc, 180/pi*angle(squeeze(freqresp(G_vc_fast_rdc( 'Du', 'Fu'), freqs_vc, 'Hz'))), '-' , 'color', [colors(3,:)]); -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); -xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); -hold off; -yticks(-360:90:360); -ylim([-180, 180]); -xlim([freqs_vc(1), freqs_vc(end)]); - -ax2b = nexttile; -hold on; -plot(freqs_md, 180/pi*angle(squeeze(freqresp(G_md_fast( 'Du', 'Fu'), freqs_md, 'Hz'))), '-' , 'color', [zeros(1,3)]); -plot(freqs_md, 180/pi*angle(squeeze(freqresp(G_md_fast_iff_hpf( 'Du', 'Fu'), freqs_md, 'Hz'))), '-' , 'color', [colors(1,:)]); -plot(freqs_md, 180/pi*angle(squeeze(freqresp(G_md_fast_iff_kp( 'Du', 'Fu'), freqs_md, 'Hz'))), '-' , 'color', [colors(2,:)]); -plot(freqs_md, 180/pi*angle(squeeze(freqresp(G_md_fast_rdc( 'Du', 'Fu'), freqs_md, 'Hz'))), '-' , 'color', [colors(3,:)]); -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); -xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]); -hold off; -yticks(-360:90:360); -ylim([-180, 180]); -xlim([freqs_md(1), freqs_md(end)]); - -ax3b = nexttile; +ax2 = nexttile; hold on; plot(freqs_pz, 180/pi*angle(squeeze(freqresp(G_pz_fast( 'Du', 'Fu'), freqs_pz, 'Hz'))), '-' , 'color', [zeros(1,3)]); plot(freqs_pz, 180/pi*angle(squeeze(freqresp(G_pz_fast_iff_hpf( 'Du', 'Fu'), freqs_pz, 'Hz'))), '-' , 'color', [colors(1,:)]); @@ -4178,27 +3546,42 @@ plot(freqs_pz, 180/pi*angle(squeeze(freqresp(G_pz_fast_iff_kp( 'Du', 'Fu'), freq plot(freqs_pz, 180/pi*angle(squeeze(freqresp(G_pz_fast_rdc( 'Du', 'Fu'), freqs_pz, 'Hz'))), '-' , 'color', [colors(3,:)]); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); -xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]); +xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); hold off; yticks(-360:90:360); -ylim([-180, 180]); -xlim([freqs_pz(1), freqs_pz(end)]); +ylim([-180, 0]); -linkaxes([ax1,ax2,ax3],'y'); -linkaxes([ax1b,ax2b,ax3b],'y'); -linkaxes([ax1,ax1b],'x'); -linkaxes([ax2,ax2b],'x'); -linkaxes([ax3,ax3b],'x'); +linkaxes([ax1,ax2],'x'); +xlim([freqs_pz(1), freqs_pz(end)]); #+end_src #+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/rotating_nass_damped_plant_comp.pdf', 'width', 'full', 'height', 'tall'); +exportFig('figs/rotating_nass_damped_plant_comp_pz.pdf', 'width', 'third', 'height', 600); #+end_src #+name: fig:rotating_nass_damped_plant_comp -#+caption: Comparison of the damped plants (direct and coupling terms) for the three proposed active damping techniques (IFF with HPF, IFF with $k_p$ and RDC) applied on the three nano-hexapod stiffnesses. $\Omega = 60\,\text{rmp}$ and $m_n + m_s = \SI{16}{\kg}$. -#+RESULTS: -[[file:figs/rotating_nass_damped_plant_comp.png]] +#+caption: Comparison of the damped plants for the three proposed active damping techniques (IFF with HPF in blue, IFF with $k_p$ in red and RDC in yellow). The direct terms are shown by the solid lines and coupling terms are shown by the shaded lines. Three nano-hexapod stiffnesses are considered. For this analysis the rotating velocity is $\Omega = 60\,\text{rpm}$ and the suspended mass is $m_n + m_s = \SI{16}{\kg}$. +#+attr_latex: :options [htbp] +#+begin_figure +#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_damped_plant_comp_vc}$k_n = 0.01\,N/\mu m$} +#+attr_latex: :options {0.33\textwidth} +#+begin_subfigure +#+attr_latex: :width 0.95\linewidth +[[file:figs/rotating_nass_damped_plant_comp_vc.png]] +#+end_subfigure +#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_damped_plant_comp_md}$k_n = 1\,N/\mu m$} +#+attr_latex: :options {0.33\textwidth} +#+begin_subfigure +#+attr_latex: :width 0.95\linewidth +[[file:figs/rotating_nass_damped_plant_comp_md.png]] +#+end_subfigure +#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_damped_plant_comp_pz}$k_n = 100\,N/\mu m$} +#+attr_latex: :options {0.33\textwidth} +#+begin_subfigure +#+attr_latex: :width 0.95\linewidth +[[file:figs/rotating_nass_damped_plant_comp_pz.png]] +#+end_subfigure +#+end_figure * Nano-Active-Stabilization-System with rotation :PROPERTIES: @@ -4206,10 +3589,8 @@ exportFig('figs/rotating_nass_damped_plant_comp.pdf', 'width', 'full', 'height', :END: <> ** Introduction :ignore: -Up until now, the model used consisted of an infinitely stiff vertical rotating stage with a X-Y suspended stage. - +Up until now, the model used to study gyroscopic effects consisted of an infinitely stiff rotating stage with a X-Y suspended stage on top. While quite simplistic, this allowed to study the effects of rotation and the associated limitations when active damping is to be applied. - In this section, the limited compliance of the micro-station is taken into account as well as the rotation of the spindle. ** Matlab Init :noexport:ignore: @@ -4244,38 +3625,42 @@ load('uniaxial_micro_station_parameters.mat') load('nass_controllers.mat'); #+end_src -** NASS model - -In order to be a bit closer to the NASS application, the 2DoF nano-hexapod (modelled as shown in Figure ref:fig:rotating_3dof_model_schematic) is now located on top of a model of the micro-station including (see Figure ref:fig:rotating_nass_model for a 3D view): +** Nano Active Stabilization System model +In order to have a more realistic dynamics model of the NASS, the 2-DoF nano-hexapod (modelled as shown in Figure ref:fig:rotating_3dof_model_schematic) is now located on top of a model of the micro-station including (see Figure ref:fig:rotating_nass_model for a 3D view): - the floor whose motion is imposed -- a 2DoF granite ($k_{g,x} = k_{g,y} = \SI{950}{\N\per\mu\m}$, $m_g = \SI{2500}{\kg}$) -- a 2DoF $T_y$ stage ($k_{t,x} = k_{t,y} = \SI{520}{\N\per\mu\m}$, $m_g = \SI{600}{\kg}$) +- a 2-DoF granite ($k_{g,x} = k_{g,y} = \SI{950}{\N\per\mu\m}$, $m_g = \SI{2500}{\kg}$) +- a 2-DoF $T_y$ stage ($k_{t,x} = k_{t,y} = \SI{520}{\N\per\mu\m}$, $m_g = \SI{600}{\kg}$) - a spindle (vertical rotation) stage whose rotation is imposed ($m_s = \SI{600}{\kg}$) -- a 2DoF micro-hexapod ($k_{h,x} = k_{h,y} = \SI{61}{\N\per\mu\m}$, $m_g = \SI{15}{\kg}$) +- a 2-DoF micro-hexapod ($k_{h,x} = k_{h,y} = \SI{61}{\N\per\mu\m}$, $m_g = \SI{15}{\kg}$) A payload is rigidly fixed to the nano-hexapod and the $x,y$ motion of the payload is measured with respect to the granite. #+name: fig:rotating_nass_model #+caption: 3D view of the Nano-Active-Stabilization-System model. +#+attr_latex: :scale 0.7 [[file:figs/rotating_nass_model.png]] ** System dynamics -#+begin_src matlab -%% Nano-Hexapod -mn = 15; % Nano-Hexapod mass [kg] +The dynamics of the un-damped and damped plants are identified using the optimal parameters found in Section ref:sec:rotating_nano_hexapod. +The obtained dynamics are compared in Figure ref:fig:rotating_nass_plant_comp_stiffness where the direct terms are shown by the solid curves while the coupling terms are shown by the shaded ones. +It can be observed that: +- Coupling (quantified by the ratio between the off-diagonal and direct terms) is higher for the soft nano-hexapod +- Damping added by the three proposed techniques is quite high and the obtained plant is rather easy to control +- There is some coupling between nano-hexapod and micro-station dynamics for the stiff nano-hexapod (mode at 200Hz) +- The two proposed IFF modification yields similar results -%% Light Sample +#+begin_src matlab +%% System parameters +mn = 15; % Nano-Hexapod mass [kg] ms = 1; % Sample Mass [kg] -%% General Configuration +% General Configuration model_config = struct(); model_config.controller = "open_loop"; % Default: Open-Loop model_config.Tuv_type = "normal"; % Default: 2DoF stage -#+end_src -#+begin_src matlab -%% Input/Output definition +% Input/Output definition clear io; io_i = 1; io(io_i) = linio([mdl, '/controller'], 1, 'openinput'); io_i = io_i + 1; % Actuator Forces [Fu, Fv] io(io_i) = linio([mdl, '/fd'], 1, 'openinput'); io_i = io_i + 1; % Direct Forces on Sample [Fdx, Fdy] @@ -4284,13 +3669,9 @@ io(io_i) = linio([mdl, '/ft'], 1, 'openinput'); io_i = io_i + 1; % Mi io(io_i) = linio([mdl, '/nano_hexapod'], 1, 'openoutput'); io_i = io_i + 1; % [Fmu, Fmv] io(io_i) = linio([mdl, '/nano_hexapod'], 2, 'openoutput'); io_i = io_i + 1; % [Du, Dv] io(io_i) = linio([mdl, '/ext_metrology'],1, 'openoutput'); io_i = io_i + 1; % [Dx, Dy] -#+end_src -The dynamics of the undamped and damped plants are identified. -The active damping parameters used are the optimal ones previously identified (i.e. for the rotating nano-hexapod fixed on a rigid platform). - -#+begin_src matlab -%% Voice Coil (i.e. soft) Nano-Hexapod +%% Identify plant without parallel stiffness +% Voice Coil (i.e. soft) Nano-Hexapod kn = 1e4; % Nano-Hexapod Stiffness [N/m] cn = 2*0.005*sqrt((ms + mn)*kn); % Nano-Hexapod Damping [N/(m/s)] @@ -4304,7 +3685,7 @@ G_vc_fast = linearize(mdl, io, 0.0); G_vc_fast.InputName = {'Fu', 'Fv', 'Fdx', 'Fdy', 'Dfx', 'Dfy', 'Ftx', 'Fty'}; G_vc_fast.OutputName = {'fu', 'fv', 'Du', 'Dv', 'Dx', 'Dy'}; -%% APA (i.e. relatively stiff) Nano-Hexapod +% APA (i.e. relatively stiff) Nano-Hexapod kn = 1e6; % Nano-Hexapod Stiffness [N/m] cn = 2*0.005*sqrt((ms + mn)*kn); % Nano-Hexapod Damping [N/(m/s)] @@ -4318,7 +3699,7 @@ G_md_fast = linearize(mdl, io, 0.0); G_md_fast.InputName = {'Fu', 'Fv', 'Fdx', 'Fdy', 'Dfx', 'Dfy', 'Ftx', 'Fty'}; G_md_fast.OutputName = {'fu', 'fv', 'Du', 'Dv', 'Dx', 'Dy'}; -%% Piezoelectric (i.e. stiff) Nano-Hexapod +% Piezoelectric (i.e. stiff) Nano-Hexapod kn = 1e8; % Nano-Hexapod Stiffness [N/m] cn = 2*0.005*sqrt((ms + mn)*kn); % Nano-Hexapod Damping [N/(m/s)] @@ -4331,9 +3712,7 @@ Wz = 2*pi; % Rotating Velocity [rad/s] G_pz_fast = linearize(mdl, io, 0.0); G_pz_fast.InputName = {'Fu', 'Fv', 'Fdx', 'Fdy', 'Dfx', 'Dfy', 'Ftx', 'Fty'}; G_pz_fast.OutputName = {'fu', 'fv', 'Du', 'Dv', 'Dx', 'Dy'}; -#+end_src -#+begin_src matlab %% Identify plants with Parallel stiffnesses model_config.Tuv_type = "parallel_k"; % Default: 2DoF stage @@ -4387,10 +3766,9 @@ Wz = 0; % [rad/s] G_pz_kp_norot = linearize(mdl, io, 0); G_pz_kp_norot.InputName = {'Fu', 'Fv', 'Fdx', 'Fdy', 'Dfx', 'Dfy', 'Ftx', 'Fty'}; G_pz_kp_norot.OutputName = {'fu', 'fv', 'Du', 'Dv', 'Dx', 'Dy'}; -#+end_src -#+begin_src matlab -%% Closed Loop Plants - IFF with HPF +%% Compute dampepd plants +% Closed Loop Plants - IFF with HPF G_vc_norot_iff_hpf = feedback(G_vc_norot, Kiff_hpf_vc, 'name'); G_vc_fast_iff_hpf = feedback(G_vc_fast, Kiff_hpf_vc, 'name'); @@ -4400,7 +3778,7 @@ G_md_fast_iff_hpf = feedback(G_md_fast, Kiff_hpf_md, 'name'); G_pz_norot_iff_hpf = feedback(G_pz_norot, Kiff_hpf_pz, 'name'); G_pz_fast_iff_hpf = feedback(G_pz_fast, Kiff_hpf_pz, 'name'); -%% Closed Loop Plants - IFF with Parallel Stiffness +% Closed Loop Plants - IFF with Parallel Stiffness G_vc_norot_iff_kp = feedback(G_vc_kp_norot, Kiff_kp_vc, 'name'); G_vc_fast_iff_kp = feedback(G_vc_kp_fast, Kiff_kp_vc, 'name'); @@ -4410,7 +3788,7 @@ G_md_fast_iff_kp = feedback(G_md_kp_fast, Kiff_kp_md, 'name'); G_pz_norot_iff_kp = feedback(G_pz_kp_norot, Kiff_kp_pz, 'name'); G_pz_fast_iff_kp = feedback(G_pz_kp_fast, Kiff_kp_pz, 'name'); -%% Closed Loop Plants - RDC +% Closed Loop Plants - RDC G_vc_norot_rdc = feedback(G_vc_norot, Krdc_vc, 'name'); G_vc_fast_rdc = feedback(G_vc_fast, Krdc_vc, 'name'); @@ -4421,26 +3799,12 @@ G_pz_norot_rdc = feedback(G_pz_norot, Krdc_pz, 'name'); G_pz_fast_rdc = feedback(G_pz_fast, Krdc_pz, 'name'); #+end_src -The undamped and damped plants are shown in Figure ref:fig:rotating_nass_plant_comp_stiffness. -Three nano-hexapod velocities are shown (from left to right): $k_n = \SI{0.01}{\N\per\mu\m}$, $k_n = \SI{1}{\N\per\mu\m}$ and $k_n = \SI{100}{\N\per\mu\m}$. -The direct terms are shown by the solid curves while the coupling terms are shown by the shaded ones. - -#+begin_important -It can be observed on Figure ref:fig:rotating_nass_plant_comp_stiffness that: -- Coupling (ratio between the off-diagonal and direct terms) is larger for the soft nano-hexapod -- Damping added by the three proposed techniques is quite high and the obtained plant is rather easy to control -- There is some coupling between nano-hexapod and micro-station dynamics for the stiff nano-hexapod (mode at 200Hz) -- The two proposed IFF modification yields similar results -#+end_important - #+begin_src matlab :exports none :results none %% Bode plot of the transfer function from nano-hexapod actuator to measured motion by the external metrology freqs_vc = logspace(-1, 2, 1000); -freqs_md = logspace(0, 3, 1000); -freqs_pz = logspace(0, 3, 1000); figure; -tiledlayout(3, 3, 'TileSpacing', 'Compact', 'Padding', 'None'); +tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); ax1 = nexttile([2,1]); hold on; @@ -4464,9 +3828,35 @@ hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); ylabel('Magnitude [m/N]'); set(gca, 'XTickLabel',[]); ylim([1e-12, 1e-2]) -title('$k_n = 0.01\,N/\mu m$'); -ax2 = nexttile([2,1]); +ax2 = nexttile; +hold on; +plot(freqs_vc, 180/pi*unwrap(angle(squeeze(freqresp(G_vc_fast('Dx', 'Fu'), freqs_vc, 'Hz')))), 'color', zeros(1,3)); +plot(freqs_vc, 180/pi*unwrap(angle(squeeze(freqresp(G_vc_fast_iff_hpf('Dx', 'Fu'), freqs_vc, 'Hz')))), 'color', colors(1,:)); +plot(freqs_vc, 180/pi*unwrap(angle(squeeze(freqresp(G_vc_fast_iff_kp('Dx', 'Fu'), freqs_vc, 'Hz')))), 'color', colors(2,:)); +plot(freqs_vc, 180/pi*unwrap(angle(squeeze(freqresp(G_vc_fast_rdc('Dx', 'Fu'), freqs_vc, 'Hz')))), 'color', colors(3,:)); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); +hold off; +yticks(-360:90:360); +ylim([-270, 90]); + +linkaxes([ax,ax2],'x'); +xlim([freqs_vc(1), freqs_vc(end)]); +xticks([1e-1, 1e0, 1e1]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file none +exportFig('figs/rotating_nass_plant_comp_stiffness_vc.pdf', 'width', 'third', 'height', 'tall'); +#+end_src + +#+begin_src matlab :exports none :results none +freqs_md = logspace(0, 3, 1000); +figure; +tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); + +ax1 = nexttile([2,1]); hold on; plot(freqs_md, abs(squeeze(freqresp(G_md_fast('Dx', 'Fu'), freqs_md, 'Hz'))), 'color', zeros(1,3), ... 'DisplayName', 'OL'); @@ -4486,11 +3876,37 @@ plot(freqs_md, abs(squeeze(freqresp(G_md_fast_rdc('Dy', 'Fu'), freqs_md, 'Hz'))) 'HandleVisibility', 'off'); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]); +ylabel('Magnitude [m/N]'); set(gca, 'XTickLabel',[]); ylim([1e-12, 1e-2]) -title('$k_n = 1\,N/\mu m$'); -ax3 = nexttile([2,1]); +ax2 = nexttile; +hold on; +plot(freqs_md, 180/pi*unwrap(angle(squeeze(freqresp(G_md_fast('Dx', 'Fu'), freqs_md, 'Hz')))), 'color', zeros(1,3)); +plot(freqs_md, 180/pi*unwrap(angle(squeeze(freqresp(G_md_fast_iff_hpf('Dx', 'Fu'), freqs_md, 'Hz')))), 'color', colors(1,:)); +plot(freqs_md, 180/pi*unwrap(angle(squeeze(freqresp(G_md_fast_iff_kp('Dx', 'Fu'), freqs_md, 'Hz')))), 'color', colors(2,:)); +plot(freqs_md, 180/pi*unwrap(angle(squeeze(freqresp(G_md_fast_rdc('Dx', 'Fu'), freqs_md, 'Hz')))), 'color', colors(3,:)); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); +hold off; +yticks(-360:90:360); +ylim([-270, 90]); + +linkaxes([ax1,ax2],'x'); +xlim([freqs_md(1), freqs_md(end)]); +xticks([1e0, 1e1, 1e2]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file none +exportFig('figs/rotating_nass_plant_comp_stiffness_md.pdf', 'width', 'third', 'height', 'tall'); +#+end_src + +#+begin_src matlab :exports none :results none +freqs_pz = logspace(0, 3, 1000); +figure; +tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); + +ax1 = nexttile([2,1]); hold on; plot(freqs_pz, abs(squeeze(freqresp(G_pz_fast('Dx', 'Fu'), freqs_pz, 'Hz'))), 'color', zeros(1,3), ... 'DisplayName', 'OL'); @@ -4510,39 +3926,12 @@ plot(freqs_pz, abs(squeeze(freqresp(G_pz_fast_rdc('Dy', 'Fu'), freqs_pz, 'Hz'))) 'HandleVisibility', 'off'); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]); +ylabel('Magnitude [m/N]'); set(gca, 'XTickLabel',[]); ylim([1e-12, 1e-2]) ldg = legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1); ldg.ItemTokenSize = [20, 1]; -title('$k_n = 100\,N/\mu m$'); -ax1b = nexttile; -hold on; -plot(freqs_vc, 180/pi*unwrap(angle(squeeze(freqresp(G_vc_fast('Dx', 'Fu'), freqs_vc, 'Hz')))), 'color', zeros(1,3)); -plot(freqs_vc, 180/pi*unwrap(angle(squeeze(freqresp(G_vc_fast_iff_hpf('Dx', 'Fu'), freqs_vc, 'Hz')))), 'color', colors(1,:)); -plot(freqs_vc, 180/pi*unwrap(angle(squeeze(freqresp(G_vc_fast_iff_kp('Dx', 'Fu'), freqs_vc, 'Hz')))), 'color', colors(2,:)); -plot(freqs_vc, 180/pi*unwrap(angle(squeeze(freqresp(G_vc_fast_rdc('Dx', 'Fu'), freqs_vc, 'Hz')))), 'color', colors(3,:)); -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); -xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); -hold off; -yticks(-360:90:360); -ylim([-270, 90]); - -ax2b = nexttile; -hold on; -plot(freqs_md, 180/pi*unwrap(angle(squeeze(freqresp(G_md_fast('Dx', 'Fu'), freqs_md, 'Hz')))), 'color', zeros(1,3)); -plot(freqs_md, 180/pi*unwrap(angle(squeeze(freqresp(G_md_fast_iff_hpf('Dx', 'Fu'), freqs_md, 'Hz')))), 'color', colors(1,:)); -plot(freqs_md, 180/pi*unwrap(angle(squeeze(freqresp(G_md_fast_iff_kp('Dx', 'Fu'), freqs_md, 'Hz')))), 'color', colors(2,:)); -plot(freqs_md, 180/pi*unwrap(angle(squeeze(freqresp(G_md_fast_rdc('Dx', 'Fu'), freqs_md, 'Hz')))), 'color', colors(3,:)); -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); -xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]); -hold off; -yticks(-360:90:360); -ylim([-270, 90]); - -ax3b = nexttile; +ax2 = nexttile; hold on; plot(freqs_pz, 180/pi*unwrap(angle(squeeze(freqresp(G_pz_fast('Dx', 'Fu'), freqs_pz, 'Hz')))), 'color', zeros(1,3)); plot(freqs_pz, 180/pi*unwrap(angle(squeeze(freqresp(G_pz_fast_iff_hpf('Dx', 'Fu'), freqs_pz, 'Hz')))), 'color', colors(1,:)); @@ -4550,42 +3939,50 @@ plot(freqs_pz, 180/pi*unwrap(angle(squeeze(freqresp(G_pz_fast_iff_kp('Dx', 'Fu') plot(freqs_pz, 180/pi*unwrap(angle(squeeze(freqresp(G_pz_fast_rdc('Dx', 'Fu'), freqs_pz, 'Hz')))), 'color', colors(3,:)); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); -xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]); +xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); hold off; yticks(-360:90:360); ylim([-270, 90]); -linkaxes([ax1,ax1b],'x'); -xlim([freqs_vc(1), freqs_vc(end)]); - -linkaxes([ax2,ax2b],'x'); -xlim([freqs_md(1), freqs_md(end)]); - -linkaxes([ax3,ax3b],'x'); +linkaxes([ax1,ax2],'x'); xlim([freqs_pz(1), freqs_pz(end)]); +xticks([1e0, 1e1, 1e2]); #+end_src -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/rotating_nass_plant_comp_stiffness.pdf', 'width', 'full', 'height', 'tall'); +#+begin_src matlab :tangle no :exports results :results file none +exportFig('figs/rotating_nass_plant_comp_stiffness_pz.pdf', 'width', 'third', 'height', 'tall'); #+end_src #+name: fig:rotating_nass_plant_comp_stiffness #+caption: Bode plot of the transfer function from nano-hexapod actuator to measured motion by the external metrology -#+RESULTS: -[[file:figs/rotating_nass_plant_comp_stiffness.png]] +#+attr_latex: :options [htbp] +#+begin_figure +#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_plant_comp_stiffness_vc}$k_n = 0.01\,N/\mu m$} +#+attr_latex: :options {0.33\textwidth} +#+begin_subfigure +#+attr_latex: :width 0.95\linewidth +[[file:figs/rotating_nass_plant_comp_stiffness_vc.png]] +#+end_subfigure +#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_plant_comp_stiffness_md}$k_n = 1\,N/\mu m$} +#+attr_latex: :options {0.33\textwidth} +#+begin_subfigure +#+attr_latex: :width 0.95\linewidth +[[file:figs/rotating_nass_plant_comp_stiffness_md.png]] +#+end_subfigure +#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_plant_comp_stiffness_pz}$k_n = 100\,N/\mu m$} +#+attr_latex: :options {0.33\textwidth} +#+begin_subfigure +#+attr_latex: :width 0.95\linewidth +[[file:figs/rotating_nass_plant_comp_stiffness_pz.png]] +#+end_subfigure +#+end_figure To confirm that the coupling is smaller when the stiffness of the nano-hexapod is increase, the /coupling ratio/ for the three nano-hexapod stiffnesses are shown in Figure ref:fig:rotating_nass_plant_coupling_comp. #+begin_src matlab :exports none :results none %% Coupling ratio for the proposed active damping techniques evaluated for the three nano-hexapod stiffnesses freqs_vc = logspace(-1, 2, 1000); -freqs_md = logspace(0, 3, 1000); -freqs_pz = logspace(0, 3, 1000); - figure; -tiledlayout(1, 3, 'TileSpacing', 'Compact', 'Padding', 'None'); - -ax1 = nexttile(); hold on; plot(freqs_vc, abs(squeeze(freqresp(G_vc_fast('Dy', 'Fu')/G_vc_fast('Dx', 'Fu'), freqs_vc, 'Hz'))), 'color', zeros(1,3), ... 'DisplayName', 'OL'); @@ -4598,9 +3995,17 @@ plot(freqs_vc, abs(squeeze(freqresp(G_vc_fast_rdc('Dy', 'Fu')/G_vc_fast_rdc('Dx' hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); xlabel('Frequency [Hz]'); ylabel('Coupling Ratio'); -title('$k_n = 0.01\,N/\mu m$'); +ylim([1e-4, 1e2]); +xticks([1e-1, 1e0, 1e1]); +#+end_src -ax2 = nexttile(); +#+begin_src matlab :tangle no :exports results :results file none +exportFig('figs/rotating_nass_plant_coupling_comp_vc.pdf', 'width', 'third', 'height', 'normal'); +#+end_src + +#+begin_src matlab :exports none :results none +freqs_md = logspace(0, 3, 1000); +figure; hold on; plot(freqs_md, abs(squeeze(freqresp(G_md_fast('Dy', 'Fu')/G_md_fast('Dx', 'Fu'), freqs_md, 'Hz'))), 'color', zeros(1,3), ... 'DisplayName', 'OL'); @@ -4612,10 +4017,18 @@ plot(freqs_md, abs(squeeze(freqresp(G_md_fast_rdc('Dy', 'Fu')/G_md_fast_rdc('Dx' 'DisplayName', 'RDC'); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]); -title('$k_n = 1\,N/\mu m$'); +xlabel('Frequency [Hz]'); ylabel('Coupling Ratio'); +ylim([1e-4, 1e2]); +xticks([1e0, 1e1, 1e2]); +#+end_src -ax3 = nexttile(); +#+begin_src matlab :tangle no :exports results :results file none +exportFig('figs/rotating_nass_plant_coupling_comp_md.pdf', 'width', 'third', 'height', 'normal'); +#+end_src + +#+begin_src matlab :exports none :results none +freqs_pz = logspace(0, 3, 1000); +figure; hold on; plot(freqs_pz, abs(squeeze(freqresp(G_pz_fast('Dy', 'Fu')/G_pz_fast('Dx', 'Fu'), freqs_pz, 'Hz'))), 'color', zeros(1,3), ... 'DisplayName', 'OL'); @@ -4627,22 +4040,40 @@ plot(freqs_pz, abs(squeeze(freqresp(G_pz_fast_rdc('Dy', 'Fu')/G_pz_fast_rdc('Dx' 'DisplayName', 'RDC'); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]); -title('$k_n = 100\,N/\mu m$'); +xlabel('Frequency [Hz]'); ylabel('Coupling Ratio'); ldg = legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 1); ldg.ItemTokenSize = [20, 1]; - -linkaxes([ax1,ax2,ax3], 'y') +ylim([1e-4, 1e2]); +xticks([1e0, 1e1, 1e2]); #+end_src -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/rotating_nass_plant_coupling_comp.pdf', 'width', 'full', 'height', 'normal'); +#+begin_src matlab :tangle no :exports results :results file none +exportFig('figs/rotating_nass_plant_coupling_comp_pz.pdf', 'width', 'third', 'height', 'normal'); #+end_src #+name: fig:rotating_nass_plant_coupling_comp #+caption: Coupling ratio for the proposed active damping techniques evaluated for the three nano-hexapod stiffnesses -#+RESULTS: -[[file:figs/rotating_nass_plant_coupling_comp.png]] +#+attr_latex: :options [htbp] +#+begin_figure +#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_plant_coupling_comp_vc}$k_n = 0.01\,N/\mu m$} +#+attr_latex: :options {0.33\textwidth} +#+begin_subfigure +#+attr_latex: :width 0.95\linewidth +[[file:figs/rotating_nass_plant_coupling_comp_vc.png]] +#+end_subfigure +#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_plant_coupling_comp_md}$k_n = 1\,N/\mu m$} +#+attr_latex: :options {0.33\textwidth} +#+begin_subfigure +#+attr_latex: :width 0.95\linewidth +[[file:figs/rotating_nass_plant_coupling_comp_md.png]] +#+end_subfigure +#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_plant_coupling_comp_pz}$k_n = 100\,N/\mu m$} +#+attr_latex: :options {0.33\textwidth} +#+begin_subfigure +#+attr_latex: :width 0.95\linewidth +[[file:figs/rotating_nass_plant_coupling_comp_pz.png]] +#+end_subfigure +#+end_figure ** Effect of disturbances @@ -4655,7 +4086,7 @@ The effect of three disturbances are considered: Conclusions are similar than with the uniaxial (non-rotating) model: - Regarding the effect of floor motion and forces applied on the payload: - The stiffer, the better (magnitudes are lower for the right curves, Figures ref:fig:rotating_nass_effect_floor_motion and ref:fig:rotating_nass_effect_direct_forces) - - Integral Force Feedback degrades the performances at low frequency compared to relative damping control + - Integral Force Feedback degrades the performance at low frequency compared to relative damping control - Regarding the effect of micro-station vibrations: - Having a soft nano-hexapod allows to filter these vibrations between the suspensions modes of the nano-hexapod and some flexible modes of the micro-station. Using relative damping control reduce this filtering (Figure ref:fig:rotating_nass_effect_stage_vibration, left). #+end_important @@ -4665,9 +4096,6 @@ Conclusions are similar than with the uniaxial (non-rotating) model: freqs = logspace(-1, 3, 1000); figure; -tiledlayout(1, 3, 'TileSpacing', 'Compact', 'Padding', 'None'); - -ax1 = nexttile(); hold on; plot(freqs, abs(squeeze(freqresp(G_vc_fast('Dx', 'Dfx'), freqs, 'Hz'))), 'color', zeros(1,3), ... 'DisplayName', 'OL'); @@ -4682,9 +4110,15 @@ set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); xlabel('Frequency [Hz]'); ylabel('Magnitude $d_x/D_{f,x}$ [m/N]'); xticks([1e-1, 1e0, 1e1, 1e2, 1e3]); xtickangle(0) -title('$k_n = 0.01\,N/\mu m$'); +ylim([1e-4, 1e2]); +#+end_src -ax2 = nexttile(); +#+begin_src matlab :tangle no :exports results :results file none +exportFig('figs/rotating_nass_effect_floor_motion_vc.pdf', 'width', 'third', 'height', 'normal'); +#+end_src + +#+begin_src matlab :exports none :results none +figure; hold on; plot(freqs, abs(squeeze(freqresp(G_md_fast('Dx', 'Dfx'), freqs, 'Hz'))), 'color', zeros(1,3), ... 'DisplayName', 'OL'); @@ -4696,12 +4130,18 @@ plot(freqs, abs(squeeze(freqresp(G_md_fast_rdc('Dx', 'Dfx'), freqs, 'Hz'))), 'co 'DisplayName', 'RDC'); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]); +xlabel('Frequency [Hz]'); ylabel('Magnitude $d_x/D_{f,x}$ [m/N]'); xticks([1e-1, 1e0, 1e1, 1e2, 1e3]); xtickangle(0) -title('$k_n = 1\,N/\mu m$'); +ylim([1e-4, 1e2]); +#+end_src -ax3 = nexttile(); +#+begin_src matlab :tangle no :exports results :results file none +exportFig('figs/rotating_nass_effect_floor_motion_md.pdf', 'width', 'third', 'height', 'normal'); +#+end_src + +#+begin_src matlab :exports none :results none +figure; hold on; plot(freqs, abs(squeeze(freqresp(G_pz_fast('Dx', 'Dfx'), freqs, 'Hz'))), 'color', zeros(1,3), ... 'DisplayName', 'OL'); @@ -4713,31 +4153,45 @@ plot(freqs, abs(squeeze(freqresp(G_pz_fast_rdc('Dx', 'Dfx'), freqs, 'Hz'))), 'co 'DisplayName', 'RDC'); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]); +xlabel('Frequency [Hz]'); ylabel('Magnitude $d_x/D_{f,x}$ [m/N]'); xticks([1e-1, 1e0, 1e1, 1e2, 1e3]); xtickangle(0) -title('$k_n = 100\,N/\mu m$'); -ldg = legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 1); -ldg.ItemTokenSize = [20, 1]; - -linkaxes([ax1,ax2,ax3], 'y') +ldg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 1); +ldg.ItemTokenSize = [15, 1]; +ylim([1e-4, 1e2]); #+end_src -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/rotating_nass_effect_floor_motion.pdf', 'width', 'full', 'height', 'normal'); +#+begin_src matlab :tangle no :exports results :results file none +exportFig('figs/rotating_nass_effect_floor_motion_pz.pdf', 'width', 'third', 'height', 'normal'); #+end_src #+name: fig:rotating_nass_effect_floor_motion #+caption: Effect of Floor motion on the position error - Comparison of active damping techniques for the three nano-hexapod stiffnesses -#+RESULTS: -[[file:figs/rotating_nass_effect_floor_motion.png]] +#+attr_latex: :options [htbp] +#+begin_figure +#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_effect_floor_motion_vc}$k_n = 0.01\,N/\mu m$} +#+attr_latex: :options {0.33\textwidth} +#+begin_subfigure +#+attr_latex: :width 0.95\linewidth +[[file:figs/rotating_nass_effect_floor_motion_vc.png]] +#+end_subfigure +#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_effect_floor_motion_md}$k_n = 1\,N/\mu m$} +#+attr_latex: :options {0.33\textwidth} +#+begin_subfigure +#+attr_latex: :width 0.95\linewidth +[[file:figs/rotating_nass_effect_floor_motion_md.png]] +#+end_subfigure +#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_effect_floor_motion_pz}$k_n = 100\,N/\mu m$} +#+attr_latex: :options {0.33\textwidth} +#+begin_subfigure +#+attr_latex: :width 0.95\linewidth +[[file:figs/rotating_nass_effect_floor_motion_pz.png]] +#+end_subfigure +#+end_figure #+begin_src matlab :exports none :results none %% Effect of micro-station vibrations on the position error - Comparison of active damping techniques for the three nano-hexapod stiffnesses figure; -tiledlayout(1, 3, 'TileSpacing', 'Compact', 'Padding', 'None'); - -ax1 = nexttile(); hold on; plot(freqs, abs(squeeze(freqresp(G_vc_fast('Dx', 'Ftx'), freqs, 'Hz'))), 'color', zeros(1,3), ... 'DisplayName', 'OL'); @@ -4752,9 +4206,15 @@ set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); xlabel('Frequency [Hz]'); ylabel('Magnitude $d_x/f_{t,x}$ [m/N]'); xticks([1e-1, 1e0, 1e1, 1e2, 1e3]); xtickangle(0) -title('$k_n = 0.01\,N/\mu m$'); +ylim([1e-12, 2e-7]); +#+end_src -ax2 = nexttile(); +#+begin_src matlab :tangle no :exports results :results file none +exportFig('figs/rotating_nass_effect_stage_vibration_vc.pdf', 'width', 'third', 'height', 'normal'); +#+end_src + +#+begin_src matlab :exports none :results none +figure; hold on; plot(freqs, abs(squeeze(freqresp(G_md_fast('Dx', 'Ftx'), freqs, 'Hz'))), 'color', zeros(1,3), ... 'DisplayName', 'OL'); @@ -4766,12 +4226,18 @@ plot(freqs, abs(squeeze(freqresp(G_md_fast_rdc('Dx', 'Ftx'), freqs, 'Hz'))), 'co 'DisplayName', 'RDC'); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]); +xlabel('Frequency [Hz]'); ylabel('Magnitude $d_x/f_{t,x}$ [m/N]'); xticks([1e-1, 1e0, 1e1, 1e2, 1e3]); xtickangle(0) -title('$k_n = 1\,N/\mu m$'); +ylim([1e-12, 2e-7]); +#+end_src -ax3 = nexttile(); +#+begin_src matlab :tangle no :exports results :results file none +exportFig('figs/rotating_nass_effect_stage_vibration_md.pdf', 'width', 'third', 'height', 'normal'); +#+end_src + +#+begin_src matlab :exports none :results none +figure; hold on; plot(freqs, abs(squeeze(freqresp(G_pz_fast('Dx', 'Ftx'), freqs, 'Hz'))), 'color', zeros(1,3), ... 'DisplayName', 'OL'); @@ -4783,31 +4249,46 @@ plot(freqs, abs(squeeze(freqresp(G_pz_fast_rdc('Dx', 'Ftx'), freqs, 'Hz'))), 'co 'DisplayName', 'RDC'); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]); +xlabel('Frequency [Hz]'); ylabel('Magnitude $d_x/f_{t,x}$ [m/N]'); xticks([1e-1, 1e0, 1e1, 1e2, 1e3]); xtickangle(0) -title('$k_n = 100\,N/\mu m$'); -ldg = legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 1); +ldg = legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 1); ldg.ItemTokenSize = [20, 1]; - -linkaxes([ax1,ax2,ax3], 'y') +ylim([1e-12, 2e-7]); #+end_src -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/rotating_nass_effect_stage_vibration.pdf', 'width', 'full', 'height', 'normal'); +#+begin_src matlab :tangle no :exports results :results file none +exportFig('figs/rotating_nass_effect_stage_vibration_pz.pdf', 'width', 'third', 'height', 'normal'); #+end_src #+name: fig:rotating_nass_effect_stage_vibration #+caption: Effect of micro-station vibrations on the position error - Comparison of active damping techniques for the three nano-hexapod stiffnesses -#+RESULTS: -[[file:figs/rotating_nass_effect_stage_vibration.png]] +#+attr_latex: :options [htbp] +#+begin_figure +#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_effect_stage_vibration_vc}$k_n = 0.01\,N/\mu m$} +#+attr_latex: :options {0.33\textwidth} +#+begin_subfigure +#+attr_latex: :width 0.95\linewidth +[[file:figs/rotating_nass_effect_stage_vibration_vc.png]] +#+end_subfigure +#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_effect_stage_vibration_md}$k_n = 1\,N/\mu m$} +#+attr_latex: :options {0.33\textwidth} +#+begin_subfigure +#+attr_latex: :width 0.95\linewidth +[[file:figs/rotating_nass_effect_stage_vibration_md.png]] +#+end_subfigure +#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_effect_stage_vibration_pz}$k_n = 100\,N/\mu m$} +#+attr_latex: :options {0.33\textwidth} +#+begin_subfigure +#+attr_latex: :width 0.95\linewidth +[[file:figs/rotating_nass_effect_stage_vibration_pz.png]] +#+end_subfigure +#+end_figure + #+begin_src matlab :exports none :results none %% Effect of sample forces on the position error - Comparison of active damping techniques for the three nano-hexapod stiffnesses figure; -tiledlayout(1, 3, 'TileSpacing', 'Compact', 'Padding', 'None'); - -ax1 = nexttile(); hold on; plot(freqs, abs(squeeze(freqresp(G_vc_fast('Dx', 'Fdx'), freqs, 'Hz'))), 'color', zeros(1,3), ... 'DisplayName', 'OL'); @@ -4822,9 +4303,15 @@ set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); xlabel('Frequency [Hz]'); ylabel('Magnitude $d_x/f_{s,x}$ [m/N]'); xticks([1e-1, 1e0, 1e1, 1e2, 1e3]); xtickangle(0) -title('$k_n = 0.01\,N/\mu m$'); +ylim([1e-9, 1e-2]) +#+end_src -ax2 = nexttile(); +#+begin_src matlab :tangle no :exports results :results file none +exportFig('figs/rotating_nass_effect_direct_forces_vc.pdf', 'width', 'third', 'height', 'normal'); +#+end_src + +#+begin_src matlab :exports none :results none +figure; hold on; plot(freqs, abs(squeeze(freqresp(G_md_fast('Dx', 'Fdx'), freqs, 'Hz'))), 'color', zeros(1,3), ... 'DisplayName', 'OL'); @@ -4836,12 +4323,18 @@ plot(freqs, abs(squeeze(freqresp(G_md_fast_rdc('Dx', 'Fdx'), freqs, 'Hz'))), 'co 'DisplayName', 'RDC'); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]); +xlabel('Frequency [Hz]'); ylabel('Magnitude $d_x/f_{s,x}$ [m/N]'); xticks([1e-1, 1e0, 1e1, 1e2, 1e3]); xtickangle(0) -title('$k_n = 1\,N/\mu m$'); +ylim([1e-9, 1e-2]) +#+end_src -ax3 = nexttile(); +#+begin_src matlab :tangle no :exports results :results file none +exportFig('figs/rotating_nass_effect_direct_forces_md.pdf', 'width', 'third', 'height', 'normal'); +#+end_src + +#+begin_src matlab :exports none :results none +figure; hold on; plot(freqs, abs(squeeze(freqresp(G_pz_fast('Dx', 'Fdx'), freqs, 'Hz'))), 'color', zeros(1,3), ... 'DisplayName', 'OL'); @@ -4853,33 +4346,60 @@ plot(freqs, abs(squeeze(freqresp(G_pz_fast_rdc('Dx', 'Fdx'), freqs, 'Hz'))), 'co 'DisplayName', 'RDC'); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]); +xlabel('Frequency [Hz]'); ylabel('Magnitude $d_x/f_{s,x}$ [m/N]'); xticks([1e-1, 1e0, 1e1, 1e2, 1e3]); xtickangle(0) -title('$k_n = 100\,N/\mu m$'); ldg = legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 1); ldg.ItemTokenSize = [20, 1]; linkaxes([ax1,ax2,ax3], 'y') +ylim([1e-9, 1e-2]) #+end_src -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/rotating_nass_effect_direct_forces.pdf', 'width', 'full', 'height', 'normal'); +#+begin_src matlab :tangle no :exports results :results file none +exportFig('figs/rotating_nass_effect_direct_forces_pz.pdf', 'width', 'third', 'height', 'normal'); #+end_src #+name: fig:rotating_nass_effect_direct_forces #+caption: Effect of sample forces on the position error - Comparison of active damping techniques for the three nano-hexapod stiffnesses -#+RESULTS: -[[file:figs/rotating_nass_effect_direct_forces.png]] +#+attr_latex: :options [htbp] +#+begin_figure +#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_effect_direct_forces_vc}$k_n = 0.01\,N/\mu m$} +#+attr_latex: :options {0.33\textwidth} +#+begin_subfigure +#+attr_latex: :width 0.95\linewidth +[[file:figs/rotating_nass_effect_direct_forces_vc.png]] +#+end_subfigure +#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_effect_direct_forces_md}$k_n = 1\,N/\mu m$} +#+attr_latex: :options {0.33\textwidth} +#+begin_subfigure +#+attr_latex: :width 0.95\linewidth +[[file:figs/rotating_nass_effect_direct_forces_md.png]] +#+end_subfigure +#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_effect_direct_forces_pz}$k_n = 100\,N/\mu m$} +#+attr_latex: :options {0.33\textwidth} +#+begin_subfigure +#+attr_latex: :width 0.95\linewidth +[[file:figs/rotating_nass_effect_direct_forces_pz.png]] +#+end_subfigure +#+end_figure * Conclusion +:PROPERTIES: +:UNNUMBERED: t +:END: + +# - problem with voice coil actuator +# - Two solutions: add parallel stiffness, or change controller +# - Conclusion: minimum stiffness is required +# - APA is a nice architecture for parallel stiffness + integrated force sensor (have to speak about IFF before that) In this study, the gyroscopic effects induced by the spindle's rotation have been studied using a spindle model (Section ref:sec:rotating_system_description). Decentralized IFF with pure integrators was shown to be unstable when applied to rotating platforms (Section ref:sec:rotating_iff_pure_int). Two modifications of the classical IFF control have been proposed to overcome this issue. -The first modification concerns the controller and consists of adding an high pass filter to the pure integrators. -This is equivalent as to moving the controller pole to the left along the real axis. +The first modification concerns the controller and consists of adding a high pass filter to the pure integrators. +This is equivalent to moving the controller pole to the left along the real axis. This allows the closed loop system to be stable up to some value of the controller gain (Section ref:sec:rotating_iff_pseudo_int). The second proposed modification concerns the mechanical system. @@ -4893,31 +4413,12 @@ Then, this study has been applied to a rotating system that corresponds to the n To be closer to the real system dynamics, the limited compliance of the micro-station has been taken into account. Results show that the two proposed IFF modifications can be applied for the NASS even in the presence of spindle rotation. -* Notations :noexport: -<> - -| | Mathematical Notation | Matlab | Unit | -|-----------------------------------+------------------------------+---------------+---------| -| Actuator Stiffness | $k$ | =k= | N/m | -| Actuator Damping | $c$ | =c= | N/(m/s) | -| Payload Mass | $m$ | =m= | kg | -| Damping Ratio | $\xi = \frac{c}{2\sqrt{km}}$ | =xi= | | -| Actuator Force | $\bm{F}, F_u, F_v$ | =F= =Fu= =Fv= | N | -| Force Sensor signal | $\bm{f}, f_u, f_v$ | =f= =fu= =fv= | N | -| Relative Displacement | $\bm{d}, d_u, d_v$ | =d= =du= =dv= | m | -| Resonance freq. when $\Omega = 0$ | $\omega_0$ | =w0= | rad/s | -| Rotation Speed | $\Omega = \dot{\theta}$ | =W= | rad/s | -| Low Pass Filter corner frequency | $\omega_i$ | =wi= | rad/s | - -| | Mathematical Notation | Matlab | Unit | -|------------------+-----------------------+--------+---------| -| Laplace variable | $s$ | =s= | | -| Complex number | $j$ | =j= | | -| Frequency | $\omega$ | =w= | [rad/s] | - * Bibliography :ignore: #+latex: \printbibliography[heading=bibintoc,title={Bibliography}] +* Glossary :ignore: +[[printglossaries:]] + * Helping Functions :noexport: ** Initialize Path #+NAME: m-init-path @@ -4942,6 +4443,104 @@ addpath('./src/'); % Path for Functions colors = colororder; #+END_SRC +** =rootLocusPolesSorted= +#+begin_src matlab :tangle matlab/src/rootLocusPolesSorted.m + function [poles] = rootLocusPolesSorted(G, K, gains, args) + % rootLocusPolesSorted - + % + % Syntax: [poles] = rootLocusPolesSorted(G, K, gains, args) + % + % Inputs: + % - G, K, gains, args - + % + % Outputs: + % - poles - + + arguments + G + K + gains + args.minreal double {mustBeNumericOrLogical} = false + args.p_half double {mustBeNumericOrLogical} = false + args.d_max double {mustBeNumeric} = -1 + end + + if args.minreal + p1 = pole(minreal(feedback(G, gains(1)*K))); + [~, i_uniq] = uniquetol([real(p1), imag(p1)], 1e-10, 'ByRows', true); + p1 = p1(i_uniq); + + poles = zeros(length(p1), length(gains)); + poles(:, 1) = p1; + else + p1 = pole(feedback(G, gains(1)*K)); + [~, i_uniq] = uniquetol([real(p1), imag(p1)], 1e-10, 'ByRows', true); + p1 = p1(i_uniq); + + poles = zeros(length(p1), length(gains)); + poles(:, 1) = p1; + end + + if args.minreal + p2 = pole(minreal(feedback(G, gains(2)*K))); + [~, i_uniq] = uniquetol([real(p2), imag(p2)], 1e-10, 'ByRows', true); + p2 = p2(i_uniq); + poles(:, 2) = p2; + else + p2 = pole(feedback(G, gains(2)*K)); + [~, i_uniq] = uniquetol([real(p2), imag(p2)], 1e-10, 'ByRows', true); + p2 = p2(i_uniq); + poles(:, 2) = p2; + end + + for g_i = 3:length(gains) + % Estimated value of the poles + poles_est = poles(:, g_i-1) + (poles(:, g_i-1) - poles(:, g_i-2))*(gains(g_i) - gains(g_i-1))/(gains(g_i-1) - gains(g_i - 2)); + + % New values for the poles + poles_gi = pole(feedback(G, gains(g_i)*K)); + [~, i_uniq] = uniquetol([real(poles_gi), imag(poles_gi)], 1e-10, 'ByRows', true); + poles_gi = poles_gi(i_uniq); + + % Array of distances between all the poles + poles_dist = sqrt((poles_est-poles_gi.').*conj(poles_est-poles_gi.')); + + % Get indices corresponding to distances from lowest to highest + [~, c] = sort(min(poles_dist)); + + as = 1:length(poles_gi); + + % for each column of poles_dist corresponding to the i'th pole + % with closest previous poles + for p_i = c + % Get the indice a_i of the previous pole that is the closest + % to pole c(p_i) + [~, a_i] = min(poles_dist(:, p_i)); + + poles(as(a_i), g_i) = poles_gi(p_i); + + % Remove old poles that are already matched + % poles_gi(as(a_i), :) = []; + poles_dist(a_i, :) = []; + as(a_i) = []; + end + end + + + if args.d_max > 0 + poles = poles(max(abs(poles(:, 2:end) - poles(:, 1:end-1))') > args.d_max, :); + end + + if args.p_half + poles = poles(1:round(end/2), :); + end + + [~, s_p] = sort(imag(poles(:,1)), 'descend'); + poles = poles(s_p, :); + + poles = poles.'; +#+end_src + ** =computeSimultaneousDamping= #+begin_src matlab :tangle matlab/src/computeSimultaneousDamping.m function [xi_min] = computeSimultaneousDamping(g, G, K) diff --git a/nass-rotating-3dof-model.pdf b/nass-rotating-3dof-model.pdf index 9c21504..39c8aa9 100644 Binary files a/nass-rotating-3dof-model.pdf and b/nass-rotating-3dof-model.pdf differ diff --git a/nass-rotating-3dof-model.tex b/nass-rotating-3dof-model.tex index 7bdbda3..a32d22a 100644 --- a/nass-rotating-3dof-model.tex +++ b/nass-rotating-3dof-model.tex @@ -1,24 +1,34 @@ -% Created 2024-03-21 Thu 18:28 +% Created 2024-04-16 Tue 22:57 % Intended LaTeX compiler: pdflatex \documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt} -\usepackage{siunitx} -\usepackage{tikz} -\usetikzlibrary{shapes.misc,arrows,arrows.meta} -\usepackage{bm} -\usepackage{amsmath} -\usepackage{amssymb} \input{preamble.tex} +\newacronym{haclac}{HAC-LAC}{High Authority Control - Low Authority Control} +\newacronym{hac}{HAC}{High Authority Control} +\newacronym{lac}{LAC}{Low Authority Control} +\newacronym{nass}{NASS}{Nano Active Stabilization System} +\newacronym{asd}{ASD}{Amplitude Spectral Density} +\newacronym{psd}{PSD}{Power Spectral Density} +\newacronym{cps}{CPS}{Cumulative Power Spectrum} +\newacronym{cas}{CAS}{Cumulative Amplitude Spectrum} +\newacronym{frf}{FRF}{Frequency Response Function} +\newacronym{iff}{IFF}{Integral Force Feedback} +\newacronym{rdc}{RDC}{Relative Damping Control} +\newglossaryentry{psdx}{name=\ensuremath{\Phi_{x}},description={{Power spectral density of signal $x$}}} +\newglossaryentry{asdx}{name=\ensuremath{\Gamma_{x}},description={{Amplitude spectral density of signal $x$}}} +\newglossaryentry{cpsx}{name=\ensuremath{\Phi_{x}},description={{Cumulative Power Spectrum of signal $x$}}} +\newglossaryentry{casx}{name=\ensuremath{\Gamma_{x}},description={{Cumulative Amplitude Spectrum of signal $x$}}} +\input{preamble_extra.tex} \bibliography{nass-rotating-3dof-model.bib} \author{Dehaeze Thomas} \date{\today} -\title{NASS - Effect of rotation} +\title{Nano Active Stabilization System - Effect of rotation} \hypersetup{ pdfauthor={Dehaeze Thomas}, - pdftitle={NASS - Effect of rotation}, + pdftitle={Nano Active Stabilization System - Effect of rotation}, pdfkeywords={}, pdfsubject={}, - pdfcreator={Emacs 29.2 (Org mode 9.7)}, + pdfcreator={Emacs 29.3 (Org mode 9.6)}, pdflang={English}} \usepackage{biblatex} @@ -28,136 +38,123 @@ \tableofcontents \clearpage -An important aspect of the Nano Active Stabilization System (NASS) is that the nano-hexapod is continuously rotating around a vertical axis while the external metrology is not. -Such rotation induces gyroscopic effects that may impact the system dynamics and obtained performances. -In this report, this rotating aspect of the NASS is studied. -It is structured in several sections: -\begin{itemize} -\item Section \ref{sec:rotating_system_description}: a simple model of a rotating suspended platform that will be used throughout this study is presented. The effect of the rotation velocity on the system dynamics is shown. -\item Section \ref{sec:rotating_iff_pure_int}: Integral Force Feedback (IFF) is applied to the rotating platform, and it is shown that the unconditional stability of IFF is lost due to Gyroscopic effects induced by the rotation. -\item Section \ref{sec:rotating_iff_pseudo_int}: A first modification of the IFF control law is proposed such that damping can be added to the suspension modes in a robust way. This modification consists of adding an high pass filter to the IFF controller. Optimal high pass filter cut-off frequency is computed. -\item Section \ref{sec:rotating_iff_parallel_stiffness}: A second modification is proposed to regain the unconditional stability of IFF. This modification consists of adding stiffness in parallel to the force sensors. Optimal parallel stiffness is computed. -\item Section \ref{sec:rotating_relative_damp_control}: Relative damping control is applied to the rotating system. -\item Section \ref{sec:rotating_comp_act_damp}: Once the optimal control parameters for the three tested active damping techniques are obtained, they are compared in terms of achievable damping, obtained damped plant and closed-loop compliance and transmissibility. -\item Section \ref{sec:rotating_nano_hexapod}: the previous analysis is applied on three nano-hexapod stiffnesses and optimal active damping controller are obtained. -\item Section \ref{sec:rotating_nass}: up until this section, the study was performed on a very simplistic model that just captures the rotation aspect and the model parameters were not tuned to corresponds to the NASS. In this last section, a model of the micro-station is added below the suspended platform (i.e. the nano-hexapod) with a rotating spindle and parameters tuned to match the NASS dynamics. The goal is to determine if the rotation imposes performance limitation for the NASS. -\end{itemize} +An important aspect of the \acrfull{nass} is that the nano-hexapod is continuously rotating around a vertical axis while the external metrology is not. +Such rotation induces gyroscopic effects that may impact the system dynamics and obtained performance. +To study these effects, a model of a rotating suspended platform is first presented (Section \ref{sec:rotating_system_description}) +This model is simple enough to be able to derive its dynamics analytically and to well understand its behavior, while still allowing to capture the important physical effects in play. -To run the Matlab code, go in the \texttt{matlab} directory and run the Matlab files corresponding to each section (see Table \ref{tab:section_matlab_code}). +\acrfull{iff} is then applied to the rotating platform, and it is shown that the unconditional stability of \acrshort{iff} is lost due to gyroscopic effects induced by the rotation (Section \ref{sec:rotating_iff_pure_int}). +Two modifications of the Integral Force Feedback are then proposed. +The first one consists of adding an high pass filter to the \acrshort{iff} controller (Section \ref{sec:rotating_iff_pseudo_int}). +It is shown that the \acrshort{iff} controller is stable for some values of the gain, and that damping can be added to the suspension modes. +Optimal high pass filter cut-off frequency is computed. +The second modification consists of adding a stiffness in parallel to the force sensors (Section \ref{sec:rotating_iff_parallel_stiffness}). +Under a certain condition, the unconditional stability of the the IFF controller is regained. +Optimal parallel stiffness is then computed. +This study of adapting \acrshort{iff} for the damping of rotating platforms was the subject of two published papers \cite{dehaeze20_activ_dampin_rotat_platf_integ_force_feedb,dehaeze21_activ_dampin_rotat_platf_using}. -\begin{table}[htbp] -\caption{\label{tab:section_matlab_code}Report sections and corresponding Matlab files} -\centering -\begin{tabularx}{0.6\linewidth}{lX} -\toprule -\textbf{Sections} & \textbf{Matlab File}\\ -\midrule -Section \ref{sec:rotating_system_description} & \texttt{rotating\_1\_system\_description.m}\\ -Section \ref{sec:rotating_iff_pure_int} & \texttt{rotating\_2\_iff\_pure\_int.m}\\ -Section \ref{sec:rotating_iff_pseudo_int} & \texttt{rotating\_3\_iff\_hpf.m}\\ -Section \ref{sec:rotating_iff_parallel_stiffness} & \texttt{rotating\_4\_iff\_kp.m}\\ -Section \ref{sec:rotating_relative_damp_control} & \texttt{rotating\_5\_rdc.m}\\ -Section \ref{sec:rotating_comp_act_damp} & \texttt{rotating\_5\_act\_damp\_comparison.m}\\ -Section \ref{sec:rotating_nano_hexapod} & \texttt{rotating\_6\_nano\_hexapod.m}\\ -Section \ref{sec:rotating_nass} & \texttt{rotating\_6\_nass.m}\\ -\bottomrule -\end{tabularx} -\end{table} -\chapter{System Description and Analysis} -\label{sec:rotating_system_description} +It is then shown that \acrfull{rdc} is less affected by gyroscopic effects (Section \ref{sec:rotating_relative_damp_control}). +Once the optimal control parameters for the three tested active damping techniques are obtained, they are compared in terms of achievable damping, obtained damped plant and closed-loop compliance and transmissibility (Section \ref{sec:rotating_comp_act_damp}). -The studied system consists of a 2 degree of freedom translation stage on top of a rotating stage (Figure \ref{fig:rotating_3dof_model_schematic}). - -The rotating stage is supposed to be ideal, meaning it induces a perfect rotation \(\theta(t) = \Omega t\) where \(\Omega\) is the rotational speed in \(\si{\radian\per\s}\). - -The suspended platform consists of two orthogonal actuators each represented by three elements in parallel: a spring with a stiffness \(k\) in \(\si{\newton\per\meter}\), a dashpot with a damping coefficient \(c\) in \(\si{\newton\per(\meter\per\second)}\) and an ideal force source \(F_u, F_v\). -A payload with a mass \(m\) in \(\si{\kilo\gram}\), is mounted on the (rotating) suspended platform. - -Two reference frames are used: an inertial frame \((\vec{i}_x, \vec{i}_y, \vec{i}_z)\) and a uniform rotating frame \((\vec{i}_u, \vec{i}_v, \vec{i}_w)\) rigidly fixed on top of the rotating stage with \(\vec{i}_w\) aligned with the rotation axis. -The position of the payload is represented by \((d_u, d_v, 0)\) expressed in the rotating frame. +The previous analysis is applied on three considered nano-hexapod stiffnesses (\(k_n = 0.01\,N/\mu m\), \(k_n = 1\,N/\mu m\) and \(k_n = 100\,N/\mu m\)) and optimal active damping controller are obtained in each case (Section \ref{sec:rotating_nano_hexapod}). +Up until this section, the study was performed on a very simplistic model that just captures the rotation aspect and the model parameters were not tuned to corresponds to the NASS. +In the last section (Section \ref{sec:rotating_nass}), a model of the micro-station is added below the suspended platform (i.e. the nano-hexapod) with a rotating spindle and parameters tuned to match the NASS dynamics. +The goal is to determine if the rotation imposes performance limitation for the NASS. \begin{figure}[htbp] \centering -\includegraphics[scale=1]{figs/rotating_3dof_model_schematic.png} +\includegraphics[scale=1,width=\linewidth]{figs/rotating_overview.png} +\caption{\label{fig:rotating_overview}Overview of this report on rotating effects} +\end{figure} + +\chapter{System Description and Analysis} +\label{sec:rotating_system_description} +The studied system consists of a 2 degree of freedom translation stage on top of a rotating stage (Figure \ref{fig:rotating_3dof_model_schematic}). +The rotating stage is supposed to be ideal, meaning it induces a perfect rotation \(\theta(t) = \Omega t\) where \(\Omega\) is the rotational speed in \(\si{\radian\per\s}\). +The suspended platform consists of two orthogonal actuators each represented by three elements in parallel: a spring with a stiffness \(k\) in \(\si{\newton\per\meter}\), a dashpot with a damping coefficient \(c\) in \(\si{\newton\per(\meter\per\second)}\) and an ideal force source \(F_u, F_v\). +A payload with a mass \(m\) in \(\si{\kilo\gram}\), is mounted on the (rotating) suspended platform. +Two reference frames are used: an \emph{inertial} frame \((\vec{i}_x, \vec{i}_y, \vec{i}_z)\) and a \emph{uniform rotating} frame \((\vec{i}_u, \vec{i}_v, \vec{i}_w)\) rigidly fixed on top of the rotating stage with \(\vec{i}_w\) aligned with the rotation axis. +The position of the payload is represented by \((d_u, d_v, 0)\) expressed in the rotating frame. +After the dynamics of this system is studied, the objective will be to damp the two suspension modes of the payload while the rotating stage performs a constant rotation. + +\begin{figure}[htbp] +\centering +\includegraphics[scale=1,scale=0.8]{figs/rotating_3dof_model_schematic.png} \caption{\label{fig:rotating_3dof_model_schematic}Schematic of the studied system} \end{figure} + \section{Equations of motion} -To obtain the equations of motion for the system represented in Figure \ref{fig:rotating_3dof_model_schematic}, the Lagrangian equations are used: -\begin{equation} -\label{eq:lagrangian_equations} +To obtain the equations of motion for the system represented in Figure \ref{fig:rotating_3dof_model_schematic}, the Lagrangian equation \eqref{eq:rotating_lagrangian_equations} is used. +\(L = T - V\) is the Lagrangian, \(T\) the kinetic coenergy, \(V\) the potential energy, \(D\) the dissipation function, and \(Q_i\) the generalized force associated with the generalized variable \(\begin{bmatrix}q_1 & q_2\end{bmatrix} = \begin{bmatrix}d_u & d_v\end{bmatrix}\). +These terms are derived in \eqref{eq:rotating_energy_functions_lagrange}. +Note that the equation of motion corresponding to the constant rotation along \(\vec{i}_w\) is disregarded as this motion is considered to be imposed by the rotation stage. + +\begin{equation}\label{eq:rotating_lagrangian_equations} \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}_i} \right) + \frac{\partial D}{\partial \dot{q}_i} - \frac{\partial L}{\partial q_i} = Q_i \end{equation} -with \(L = T - V\) the Lagrangian, \(T\) the kinetic coenergy, \(V\) the potential energy, \(D\) the dissipation function, and \(Q_i\) the generalized force associated with the generalized variable \(\begin{bmatrix}q_1 & q_2\end{bmatrix} = \begin{bmatrix}d_u & d_v\end{bmatrix}\). -The equation of motion corresponding to the constant rotation along \(\vec{i}_w\) is disregarded as this motion is considered to be imposed by the rotation stage. -\begin{equation} -\label{eq:energy_functions_lagrange} + +\begin{equation} \label{eq:rotating_energy_functions_lagrange} \begin{aligned} - T &= \frac{1}{2} m \left( ( \dot{d}_u - \Omega d_v )^2 + ( \dot{d}_v + \Omega d_u )^2 \right), \\ - V &= \frac{1}{2} k \big( {d_u}^2 + {d_v}^2 \big), \ Q_1 = F_u, \\ - D &= \frac{1}{2} c \big( \dot{d}_u{}^2 + \dot{d}_v{}^2 \big), \ Q_2 = F_v + T &= \frac{1}{2} m \left( ( \dot{d}_u - \Omega d_v )^2 + ( \dot{d}_v + \Omega d_u )^2 \right), \quad Q_1 = F_u, \quad Q_2 = F_v, \\ + V &= \frac{1}{2} k \big( {d_u}^2 + {d_v}^2 \big), \quad D = \frac{1}{2} c \big( \dot{d}_u{}^2 + \dot{d}_v{}^2 \big) \end{aligned} \end{equation} -Substituting equations \eqref{eq:energy_functions_lagrange} into equation \eqref{eq:lagrangian_equations} for both generalized coordinates gives two coupled differential equations \eqref{eq:eom_coupled_1} and \eqref{eq:eom_coupled_2}. -\begin{subequations} -\label{eq:eom_coupled} +Substituting equations \eqref{eq:rotating_energy_functions_lagrange} into equation \eqref{eq:rotating_lagrangian_equations} for both generalized coordinates gives two coupled differential equations \eqref{eq:rotating_eom_coupled_1} and \eqref{eq:rotating_eom_coupled_2}. + +\begin{subequations} \label{eq:rotating_eom_coupled} \begin{align} - m \ddot{d}_u + c \dot{d}_u + ( k - m \Omega^2 ) d_u &= F_u + 2 m \Omega \dot{d}_v \label{eq:eom_coupled_1} \\ - m \ddot{d}_v + c \dot{d}_v + ( k \underbrace{-\,m \Omega^2}_{\text{Centrif.}} ) d_v &= F_v \underbrace{-\,2 m \Omega \dot{d}_u}_{\text{Coriolis}} \label{eq:eom_coupled_2} + m \ddot{d}_u + c \dot{d}_u + ( k - m \Omega^2 ) d_u &= F_u + 2 m \Omega \dot{d}_v \label{eq:rotating_eom_coupled_1} \\ + m \ddot{d}_v + c \dot{d}_v + ( k \underbrace{-\,m \Omega^2}_{\text{Centrif.}} ) d_v &= F_v \underbrace{-\,2 m \Omega \dot{d}_u}_{\text{Coriolis}} \label{eq:rotating_eom_coupled_2} \end{align} \end{subequations} -The uniform rotation of the system induces \textbf{two gyroscopic effects} as shown in equation \eqref{eq:eom_coupled}: +The uniform rotation of the system induces two \emph{gyroscopic effects} as shown in equation \eqref{eq:rotating_eom_coupled}: \begin{itemize} -\item \textbf{Centrifugal forces}: that can been seen as an added \textbf{negative stiffness} \(- m \Omega^2\) along \(\vec{i}_u\) and \(\vec{i}_v\) -\item \textbf{Coriolis Forces}: that adds \textbf{coupling} between the two orthogonal directions. +\item \emph{Centrifugal forces}: that can been seen as an added \emph{negative stiffness} \(- m \Omega^2\) along \(\vec{i}_u\) and \(\vec{i}_v\) +\item \emph{Coriolis forces}: that adds \emph{coupling} between the two orthogonal directions. \end{itemize} +One can verify that without rotation (\(\Omega = 0\)) the system becomes equivalent to two \emph{uncoupled} one degree of freedom mass-spring-damper systems. -One can verify that without rotation (\(\Omega = 0\)) the system becomes equivalent to two uncoupled one degree of freedom mass-spring-damper systems. \section{Transfer Functions in the Laplace domain} -To study the dynamics of the system, the differential equations of motions \eqref{eq:eom_coupled} are converted into the Laplace domain and the \(2 \times 2\) transfer function matrix \(\mathbf{G}_d\) from \(\begin{bmatrix}F_u & F_v\end{bmatrix}\) to \(\begin{bmatrix}d_u & d_v\end{bmatrix}\) in equation \eqref{eq:Gd_mimo_tf} is obtained. -Its elements are shown in equation \eqref{eq:Gd_indiv_el}. +To study the dynamics of the system, the two differential equations of motions \eqref{eq:rotating_eom_coupled} are converted into the Laplace domain and the \(2 \times 2\) transfer function matrix \(\mathbf{G}_d\) from \(\begin{bmatrix}F_u & F_v\end{bmatrix}\) to \(\begin{bmatrix}d_u & d_v\end{bmatrix}\) in equation \eqref{eq:rotating_Gd_mimo_tf} is obtained. +The four transfer functions in \(\mathbf{G}_d\) are shown in equation \eqref{eq:rotating_Gd_indiv_el}. -\begin{equation} -\label{eq:Gd_mimo_tf} +\begin{equation}\label{eq:rotating_Gd_mimo_tf} \begin{bmatrix} d_u \\ d_v \end{bmatrix} = \mathbf{G}_d \begin{bmatrix} F_u \\ F_v \end{bmatrix} \end{equation} -\begin{subequations} -\label{eq:Gd_indiv_el} +\begin{subequations}\label{eq:rotating_Gd_indiv_el} \begin{align} \mathbf{G}_{d}(1,1) &= \mathbf{G}_{d}(2,2) = \frac{ms^2 + cs + k - m \Omega^2}{\left( m s^2 + cs + k - m \Omega^2 \right)^2 + \left( 2 m \Omega s \right)^2} \\ \mathbf{G}_{d}(1,2) &= -\mathbf{G}_{d}(2,1) = \frac{2 m \Omega s}{\left( m s^2 + cs + k - m \Omega^2 \right)^2 + \left( 2 m \Omega s \right)^2} \end{align} \end{subequations} -To simplify the analysis, the undamped natural frequency \(\omega_0\) and the damping ratio \(\xi\) are used as in equation \eqref{eq:xi_and_omega}. -\begin{equation} -\label{eq:xi_and_omega} +To simplify the analysis, the undamped natural frequency \(\omega_0\) and the damping ratio \(\xi\) defined in \eqref{eq:rotating_xi_and_omega} are used instead. +The elements of transfer function matrix \(\mathbf{G}_d\) are now described by equation \eqref{eq:rotating_Gd_w0_xi_k}. +\begin{equation} \label{eq:rotating_xi_and_omega} \omega_0 = \sqrt{\frac{k}{m}} \text{ in } \si{\radian\per\second}, \quad \xi = \frac{c}{2 \sqrt{k m}} \end{equation} -The elements of transfer function matrix \(\mathbf{G}_d\) are now described by equation \eqref{eq:Gd_w0_xi_k}. -\begin{subequations} -\label{eq:Gd_w0_xi_k} +\begin{subequations} \label{eq:rotating_Gd_w0_xi_k} \begin{align} \mathbf{G}_{d}(1,1) &= \frac{\frac{1}{k} \left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)}{\left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2} \\ \mathbf{G}_{d}(1,2) &= \frac{\frac{1}{k} \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)}{\left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2} \end{align} \end{subequations} -\section{System Poles: Campbell Diagram} -The poles of \(\mathbf{G}_d\) are the complex solutions \(p\) of equation \eqref{eq:poles}. -\begin{equation} -\label{eq:poles} +\section{System Poles: Campbell Diagram} +The poles of \(\mathbf{G}_d\) are the complex solutions \(p\) of equation \eqref{eq:rotating_poles} (i.e. the roots of its denominator). + +\begin{equation}\label{eq:rotating_poles} \left( \frac{p^2}{{\omega_0}^2} + 2 \xi \frac{p}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{p}{\omega_0} \right)^2 = 0 \end{equation} -Supposing small damping (\(\xi \ll 1\)), two pairs of complex conjugate poles are obtained as shown in equation \eqref{eq:pole_values}. +Supposing small damping (\(\xi \ll 1\)), two pairs of complex conjugate poles \([p_{+}, p_{-}]\) are obtained as shown in equation \eqref{eq:rotating_pole_values}. -\begin{subequations} -\label{eq:pole_values} +\begin{subequations} \label{eq:rotating_pole_values} \begin{align} p_{+} &= - \xi \omega_0 \left( 1 + \frac{\Omega}{\omega_0} \right) \pm j \omega_0 \left( 1 + \frac{\Omega}{\omega_0} \right) \\ p_{-} &= - \xi \omega_0 \left( 1 - \frac{\Omega}{\omega_0} \right) \pm j \omega_0 \left( 1 - \frac{\Omega}{\omega_0} \right) @@ -165,138 +162,170 @@ Supposing small damping (\(\xi \ll 1\)), two pairs of complex conjugate poles ar \end{subequations} The real and complex parts of these two pairs of complex conjugate poles are represented in Figure \ref{fig:rotating_campbell_diagram} as a function of the rotational speed \(\Omega\). -As the rotational speed increases, \(p_{+}\) goes to higher frequencies and \(p_{-}\) goes to lower frequencies. -The system becomes unstable for \(\Omega > \omega_0\) as the real part of \(p_{-}\) is positive. +As the rotational speed increases, \(p_{+}\) goes to higher frequencies and \(p_{-}\) goes to lower frequencies (Figure \ref{fig:rotating_campbell_diagram_imag}). +The system becomes unstable for \(\Omega > \omega_0\) as the real part of \(p_{-}\) is positive (Figure \ref{fig:rotating_campbell_diagram_real}). Physically, the negative stiffness term \(-m\Omega^2\) induced by centrifugal forces exceeds the spring stiffness \(k\). \begin{figure}[htbp] -\centering -\includegraphics[scale=1]{figs/rotating_campbell_diagram.png} -\caption{\label{fig:rotating_campbell_diagram}Campbell diagram - Real and Imaginary parts of the poles as a function of the rotating velocity} +\begin{subfigure}{0.49\textwidth} +\begin{center} +\includegraphics[scale=1,scale=1]{figs/rotating_campbell_diagram_real.png} +\end{center} +\subcaption{\label{fig:rotating_campbell_diagram_real}Real part} +\end{subfigure} +\begin{subfigure}{0.49\textwidth} +\begin{center} +\includegraphics[scale=1,scale=1]{figs/rotating_campbell_diagram_imag.png} +\end{center} +\subcaption{\label{fig:rotating_campbell_diagram_imag}Imaginary part} +\end{subfigure} +\caption{\label{fig:rotating_campbell_diagram}Campbell diagram - Real (\subref{fig:rotating_campbell_diagram_real}) and Imaginary (\subref{fig:rotating_campbell_diagram_imag}) parts of the poles as a function of the rotating velocity \(\Omega\).} \end{figure} + \section{System Dynamics: Effect of rotation} The system dynamics from actuator forces \([F_u, F_v]\) to the relative motion \([d_u, d_v]\) is identified for several rotating velocities. - -Looking at the transfer function matrix \(\mathbf{G}_d\) in equation \eqref{eq:Gd_w0_xi_k}, one can see that the two diagonal (direct) terms are equal and that the two off-diagonal (coupling) terms are opposite. -The bode plot of these two terms are shown in Figure \ref{fig:rotating_direct_coupling_bode_plot} for several rotational speeds \(\Omega\). +Looking at the transfer function matrix \(\mathbf{G}_d\) in equation \eqref{eq:rotating_Gd_w0_xi_k}, one can see that the two diagonal (direct) terms are equal and that the two off-diagonal (coupling) terms are opposite. +The bode plot of these two terms are shown in Figure \ref{fig:rotating_bode_plot} for several rotational speeds \(\Omega\). These plots confirm the expected behavior: the frequency of the two pairs of complex conjugate poles are further separated as \(\Omega\) increases. -For \(\Omega > \omega_0\), the low frequency pair of complex conjugate poles \(p_{-}\) becomes unstable. +For \(\Omega > \omega_0\), the low frequency pair of complex conjugate poles \(p_{-}\) becomes unstable (shown be the 180 degrees phase lead instead of phase lag). \begin{figure}[htbp] -\centering -\includegraphics[scale=1]{figs/rotating_direct_coupling_bode_plot.png} -\caption{\label{fig:rotating_direct_coupling_bode_plot}Bode plot of the direct and coupling terms for several rotating velocities} +\begin{subfigure}{0.49\textwidth} +\begin{center} +\includegraphics[scale=1,width=0.95\linewidth]{figs/rotating_bode_plot_direct.png} +\end{center} +\subcaption{\label{fig:rotating_bode_plot_direct}Direct terms: $d_u/F_u$, $d_v/F_v$} +\end{subfigure} +\begin{subfigure}{0.49\textwidth} +\begin{center} +\includegraphics[scale=1,width=0.95\linewidth]{figs/rotating_bode_plot_coupling.png} +\end{center} +\subcaption{\label{fig:rotating_bode_plot_coupling}Coupling terms: $d_u/F_v$, $d_v/F_u$} +\end{subfigure} +\caption{\label{fig:rotating_bode_plot}Bode plot of the direct (\subref{fig:rotating_bode_plot_direct}) and coupling (\subref{fig:rotating_bode_plot_direct}) terms for several rotating velocities} \end{figure} + \chapter{Integral Force Feedback} \label{sec:rotating_iff_pure_int} +The goal is now to damp the two suspension modes of the payload using an active damping strategy while the rotating stage performs a constant rotation. +As was explained with the uniaxial model, such active damping strategy is key to both reducing the magnification of the response in the vicinity of the resonances \cite{collette11_review_activ_vibrat_isolat_strat} and to make the plant easier to control for the high authority controller. -In order to further decrease the residual vibrations, active damping can be used for reducing the magnification of the response in the vicinity of the resonances \cite{collette11_review_activ_vibrat_isolat_strat}. +Many active damping techniques have been developed over the years such as Positive Position Feedback (PPF) \cite{lin06_distur_atten_precis_hexap_point,fanson90_posit_posit_feedb_contr_large_space_struc}, Integral Force Feedback (IFF) \cite{preumont91_activ} and Direct Velocity Feedback (DVF) \cite{karnopp74_vibrat_contr_using_semi_activ_force_gener,serrand00_multic_feedb_contr_isolat_base_excit_vibrat,preumont02_force_feedb_versus_accel_feedb}. +In \cite{preumont92_activ_dampin_by_local_force}, the IFF control scheme has been proposed, where a force sensor, a force actuator and an integral controller are used to increase the damping of a mechanical system. +When the force sensor is collocated with the actuator, the open-loop transfer function has alternating poles and zeros which facilitates to guarantee the stability of the closed loop system \cite{preumont02_force_feedb_versus_accel_feedb}. +It was latter shown that this property holds for multiple collated actuator/sensor pairs \cite{preumont08_trans_zeros_struc_contr_with}. -Many active damping techniques have been developed over the years such as Positive Position Feedback (PPF) \cite{lin06_distur_atten_precis_hexap_point,fanson90_posit_posit_feedb_contr_large_space_struc}, Integral Force Feedback (IFF) \cite{preumont91_activ} and Direct Velocity Feedback (DVF) \cite{karnopp74_vibrat_contr_using_semi_activ_force_gener,serrand00_multic_feedb_contr_isolat_base_excit_vibrat,preumont02_force_feedb_versus_accel_feedb}. \par - -In \cite{preumont92_activ_dampin_by_local_force}, the IFF control scheme has been proposed, where a force sensor, a force actuator and an integral controller are used to directly augment the damping of a mechanical system. -When the force sensor is collocated with the actuator, the open-loop transfer function has alternating poles and zeros which facilitate to guarantee the stability of the closed loop system \cite{preumont02_force_feedb_versus_accel_feedb}. -It was latter shown that this property holds for multiple collated actuator/sensor pairs \cite{preumont08_trans_zeros_struc_contr_with}. \par - -The main advantages of IFF over other active damping techniques are the guaranteed stability even in presence of flexible dynamics, good performances and robustness properties \cite{preumont02_force_feedb_versus_accel_feedb}. \par +The main advantages of IFF over other active damping techniques are the guaranteed stability even in presence of flexible dynamics, good performance and robustness properties \cite{preumont02_force_feedb_versus_accel_feedb}. Several improvements of the classical IFF have been proposed, such as adding a feed-through term to increase the achievable damping \cite{teo15_optim_integ_force_feedb_activ_vibrat_contr} or adding an high pass filter to recover the loss of compliance at low frequency \cite{chesne16_enhan_dampin_flexib_struc_using_force_feedb}. Recently, an \(\mathcal{H}_\infty\) optimization criterion has been used to derive optimal gains for the IFF controller \cite{zhao19_optim_integ_force_feedb_contr}. \par However, none of these study have been applied to a rotating system. In this section, Integral Force Feedback strategy is applied on the rotating suspended platform, and it is shown that gyroscopic effects alters the system dynamics and that IFF cannot be applied as is. + \section{System and Equations of motion} In order to apply Integral Force Feedback, two force sensors are added in series with the actuators (Figure \ref{fig:rotating_3dof_model_schematic_iff}). -Two identical controllers \(K_F\) are then used to feedback each of the sensed force to its associated actuator: -\begin{equation} +Two identical controllers \(K_F\) described by \eqref{eq:rotating_iff_controller} are then used to feedback each of the sensed force to its associated actuator. + +\begin{equation}\label{eq:rotating_iff_controller} K_{F}(s) = g \cdot \frac{1}{s} \end{equation} \begin{figure}[htbp] -\centering -\includegraphics[scale=1]{figs/rotating_3dof_model_schematic_iff.png} -\caption{\label{fig:rotating_3dof_model_schematic_iff}System with added Force Sensor in series with the actuators (shown in blue with the associated controllers)} +\begin{subfigure}{0.49\textwidth} +\begin{center} +\includegraphics[scale=1,width=0.95\linewidth]{figs/rotating_3dof_model_schematic_iff.png} +\end{center} +\subcaption{\label{fig:rotating_3dof_model_schematic_iff}System with added Force Sensor in series with the actuators} +\end{subfigure} +\begin{subfigure}{0.49\textwidth} +\begin{center} +\includegraphics[scale=1,scale=1]{figs/rotating_iff_diagram.png} +\end{center} +\subcaption{\label{fig:rotating_iff_diagram}Control diagram} +\end{subfigure} +\caption{\label{fig:rotating_iff_pure_int}Integral Force Feedback applied to the suspended rotating platform. The damper \(c\) in (\subref{fig:rotating_3dof_model_schematic_iff}) is omitted for readability.} \end{figure} +The forces \(\begin{bmatrix}f_u & f_v\end{bmatrix}\) measured by the two force sensors represented in Figure \ref{fig:rotating_3dof_model_schematic_iff} are described by equation \eqref{eq:rotating_measured_force}. -The forces \(\begin{bmatrix}f_u & f_v\end{bmatrix}\) measured by the two force sensors represented in Figure \ref{fig:rotating_3dof_model_schematic_iff} are described by equation \eqref{eq:measured_force}. -\begin{equation} -\label{eq:measured_force} +\begin{equation}\label{eq:rotating_measured_force} \begin{bmatrix} f_{u} \\ f_{v} \end{bmatrix} = \begin{bmatrix} F_u \\ F_v \end{bmatrix} - (c s + k) \begin{bmatrix} d_u \\ d_v \end{bmatrix} \end{equation} -The transfer function matrix \(\mathbf{G}_{f}\) from actuator forces to measured forces in equation \eqref{eq:Gf_mimo_tf} can be obtained by inserting equation \eqref{eq:Gd_w0_xi_k} into equation \eqref{eq:measured_force}. -Its elements are shown in equation \eqref{eq:Gf_indiv_el}. +The transfer function matrix \(\mathbf{G}_{f}\) from actuator forces to measured forces in equation \eqref{eq:rotating_Gf_mimo_tf} can be obtained by inserting equation \eqref{eq:rotating_Gd_w0_xi_k} into equation \eqref{eq:rotating_measured_force}. +Its elements are shown in equation \eqref{eq:rotating_Gf}. -\begin{equation} -\label{eq:Gf_mimo_tf} +\begin{equation}\label{eq:rotating_Gf_mimo_tf} \begin{bmatrix} f_{u} \\ f_{v} \end{bmatrix} = \mathbf{G}_{f} \begin{bmatrix} F_u \\ F_v \end{bmatrix} \end{equation} -\begin{subequations} -\label{eq:Gf_indiv_el} -\label{eq:Gf} +\begin{subequations}\label{eq:rotating_Gf} \begin{align} - \mathbf{G}_{f}(1,1) &= \mathbf{G}_{f}(2,2) = \frac{\left( \frac{s^2}{{\omega_0}^2} - \frac{\Omega^2}{{\omega_0}^2} \right) \left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right) + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2}{\left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2} \label{eq:Gf_diag_tf} \\ - \mathbf{G}_{f}(1,2) &= -\mathbf{G}_{f}(2,1) = \frac{- \left( 2 \xi \frac{s}{\omega_0} + 1 \right) \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)}{\left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2} \label{eq:Gf_off_diag_tf} + \mathbf{G}_{f}(1,1) &= \mathbf{G}_{f}(2,2) = \frac{\left( \frac{s^2}{{\omega_0}^2} - \frac{\Omega^2}{{\omega_0}^2} \right) \left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right) + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2}{\left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2} \label{eq:rotating_Gf_diag_tf} \\ + \mathbf{G}_{f}(1,2) &= -\mathbf{G}_{f}(2,1) = \frac{- \left( 2 \xi \frac{s}{\omega_0} + 1 \right) \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)}{\left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2} \label{eq:rotating_Gf_off_diag_tf} \end{align} \end{subequations} -The zeros of the diagonal terms of \(\mathbf{G}_f\) in equation \eqref{eq:Gf_diag_tf} are computed, and neglecting the damping for simplicity, \textbf{two complex conjugated zeros} \(z_{c}\) are obtained in equation \eqref{eq:iff_zero_cc}, and \textbf{two real zeros} \(z_{r}\) in equation \eqref{eq:iff_zero_real}. +The zeros of the diagonal terms of \(\mathbf{G}_f\) in equation \eqref{eq:rotating_Gf_diag_tf} are computed, and neglecting the damping for simplicity, two complex conjugated zeros \(z_{c}\) \eqref{eq:rotating_iff_zero_cc}, and two real zeros \(z_{r}\) \eqref{eq:rotating_iff_zero_real} are obtained. + \begin{subequations} \begin{align} - z_c &= \pm j \omega_0 \sqrt{\frac{1}{2} \sqrt{8 \frac{\Omega^2}{{\omega_0}^2} + 1} + \frac{\Omega^2}{{\omega_0}^2} + \frac{1}{2} } \label{eq:iff_zero_cc} \\ - z_r &= \pm \omega_0 \sqrt{\frac{1}{2} \sqrt{8 \frac{\Omega^2}{{\omega_0}^2} + 1} - \frac{\Omega^2}{{\omega_0}^2} - \frac{1}{2} } \label{eq:iff_zero_real} + z_c &= \pm j \omega_0 \sqrt{\frac{1}{2} \sqrt{8 \frac{\Omega^2}{{\omega_0}^2} + 1} + \frac{\Omega^2}{{\omega_0}^2} + \frac{1}{2} } \label{eq:rotating_iff_zero_cc} \\ + z_r &= \pm \omega_0 \sqrt{\frac{1}{2} \sqrt{8 \frac{\Omega^2}{{\omega_0}^2} + 1} - \frac{\Omega^2}{{\omega_0}^2} - \frac{1}{2} } \label{eq:rotating_iff_zero_real} \end{align} \end{subequations} -It is interesting to see that the frequency of the pair of complex conjugate zeros \(z_c\) in equation \eqref{eq:iff_zero_cc} always lies between the frequency of the two pairs of complex conjugate poles \(p_{-}\) and \(p_{+}\) in equation \eqref{eq:pole_values}. +It is interesting to see that the frequency of the pair of complex conjugate zeros \(z_c\) in equation \eqref{eq:rotating_iff_zero_cc} always lies between the frequency of the two pairs of complex conjugate poles \(p_{-}\) and \(p_{+}\) in equation \eqref{eq:rotating_pole_values}. This is what usually gives the unconditional stability of IFF when collocated force sensors are used. -However, for non-null rotational speeds, the two real zeros \(z_r\) in equation \eqref{eq:iff_zero_real} are inducing a \textbf{non-minimum phase behavior}. +However, for non-null rotational speeds, the two real zeros \(z_r\) in equation \eqref{eq:rotating_iff_zero_real} are inducing a \emph{non-minimum phase behavior}. This can be seen in the Bode plot of the diagonal terms (Figure \ref{fig:rotating_iff_bode_plot_effect_rot}) where the low frequency gain is no longer zero while the phase stays at \(\SI{180}{\degree}\). -The low frequency gain of \(\mathbf{G}_f\) increases with the rotational speed \(\Omega\) as shown in equation \eqref{eq:low_freq_gain_iff_plan}. -\begin{equation} -\label{eq:low_freq_gain_iff_plan} +The low frequency gain of \(\mathbf{G}_f\) increases with the rotational speed \(\Omega\) as shown in equation \eqref{eq:rotating_low_freq_gain_iff_plan}. +This can be explained as follows: a constant actuator force \(F_u\) induces a small displacement of the mass \(d_u = \frac{F_u}{k - m\Omega^2}\) (Hooke's law taking into account the negative stiffness induced by the rotation). +This small displacement then increases the centrifugal force \(m\Omega^2d_u = \frac{\Omega^2}{{\omega_0}^2 - \Omega^2} F_u\) which is then measured by the force sensors. + +\begin{equation}\label{eq:rotating_low_freq_gain_iff_plan} \lim_{\omega \to 0} \left| \mathbf{G}_f (j\omega) \right| = \begin{bmatrix} \frac{\Omega^2}{{\omega_0}^2 - \Omega^2} & 0 \\ 0 & \frac{\Omega^2}{{\omega_0}^2 - \Omega^2} \end{bmatrix} \end{equation} -This can be explained as follows: a constant actuator force \(F_u\) induces a small displacement of the mass \(d_u = \frac{F_u}{k - m\Omega^2}\) (Hooke's law taking into account the negative stiffness induced by the rotation). -This small displacement then increases the centrifugal force \(m\Omega^2d_u = \frac{\Omega^2}{{\omega_0}^2 - \Omega^2} F_u\) which is then measured by the force sensors. \section{Effect of the rotation speed on the IFF plant dynamics} -The transfer functions from actuator forces \([F_u,\ F_v]\) to the measured force sensors \([f_u,\ f_v]\) are identified for several rotating velocities and shown in Figure \ref{fig:rotating_iff_bode_plot_effect_rot}. - +The transfer functions from actuator forces \([F_u,\ F_v]\) to the measured force sensors \([f_u,\ f_v]\) are identified for several rotating velocities and are shown in Figure \ref{fig:rotating_iff_bode_plot_effect_rot}. As was expected from the derived equations of motion: \begin{itemize} -\item when \(0 < \Omega < \omega_0\): the low frequency gain is no longer zero and two (non-minimum phase) real zero appears at low frequency. +\item when \(\Omega < \omega_0\): the low frequency gain is no longer zero and two (non-minimum phase) real zero appears at low frequency. The low frequency gain increases with \(\Omega\). A pair of (minimum phase) complex conjugate zeros appears between the two complex conjugate poles that are split further apart as \(\Omega\) increases. \item when \(\omega_0 < \Omega\): the low frequency pole becomes unstable. \end{itemize} \begin{figure}[htbp] -\centering -\includegraphics[scale=1]{figs/rotating_iff_bode_plot_effect_rot.png} -\caption{\label{fig:rotating_iff_bode_plot_effect_rot}Bode plot of the direct and coupling term for Integral Force Feedback - Effect of rotation} +\begin{subfigure}{0.49\textwidth} +\begin{center} +\includegraphics[scale=1,width=0.95\linewidth]{figs/rotating_iff_bode_plot_effect_rot_direct.png} +\end{center} +\subcaption{\label{fig:rotating_iff_bode_plot_effect_rot_direct}Direct terms: $d_u/F_u$, $d_v/F_v$} +\end{subfigure} +\begin{subfigure}{0.49\textwidth} +\begin{center} +\includegraphics[scale=1,scale=1]{figs/rotating_root_locus_iff_pure_int.png} +\end{center} +\subcaption{\label{fig:rotating_root_locus_iff_pure_int}Root Locus} +\end{subfigure} +\caption{\label{fig:rotating_iff_bode_plot_effect_rot}Effect of the rotation velocity on the bode plot of the direct terms (\subref{fig:rotating_iff_bode_plot_effect_rot_direct}) and on the IFF root locus (\subref{fig:rotating_root_locus_iff_pure_int})} \end{figure} + \section{Decentralized Integral Force Feedback} The control diagram for decentralized Integral Force Feedback is shown in Figure \ref{fig:rotating_iff_diagram}. -\begin{figure}[htbp] -\centering -\includegraphics[scale=1]{figs/rotating_iff_diagram.png} -\caption{\label{fig:rotating_iff_diagram}Control diagram for decentralized Integral Force Feedback} -\end{figure} - The decentralized IFF controller \(\bm{K}_F\) corresponds to a diagonal controller with integrators: -\begin{equation} -\label{eq:Kf_pure_int} +\begin{equation} \label{eq:rotating_Kf_pure_int} \begin{aligned} \mathbf{K}_{F}(s) &= \begin{bmatrix} K_{F}(s) & 0 \\ 0 & K_{F}(s) \end{bmatrix} \\ K_{F}(s) &= g \cdot \frac{1}{s} @@ -306,92 +335,58 @@ The decentralized IFF controller \(\bm{K}_F\) corresponds to a diagonal controll In order to see how the IFF controller affects the poles of the closed loop system, a Root Locus plot (Figure \ref{fig:rotating_root_locus_iff_pure_int}) is constructed as follows: the poles of the closed-loop system are drawn in the complex plane as the controller gain \(g\) varies from \(0\) to \(\infty\) for the two controllers \(K_{F}\) simultaneously. As explained in \cite{preumont08_trans_zeros_struc_contr_with,skogestad07_multiv_feedb_contr}, the closed-loop poles start at the open-loop poles (shown by \(\tikz[baseline=-0.6ex] \node[cross out, draw=black, minimum size=1ex, line width=2pt, inner sep=0pt, outer sep=0pt] at (0, 0){};\)) for \(g = 0\) and coincide with the transmission zeros (shown by \(\tikz[baseline=-0.6ex] \draw[line width=2pt, inner sep=0pt, outer sep=0pt] (0,0) circle[radius=3pt];\)) as \(g \to \infty\). -\begin{important} Whereas collocated IFF is usually associated with unconditional stability \cite{preumont91_activ}, this property is lost due to gyroscopic effects as soon as the rotation velocity in non-null. This can be seen in the Root Locus plot (Figure \ref{fig:rotating_root_locus_iff_pure_int}) where poles corresponding to the controller are bound to the right half plane implying closed-loop system instability. -\end{important} - Physically, this can be explained like so: at low frequency, the loop gain is very large due to the pure integrator in \(K_{F}\) and the finite gain of the plant (Figure \ref{fig:rotating_iff_bode_plot_effect_rot}). -The control system is thus canceling the spring forces which makes the suspended platform no able to hold the payload against centrifugal forces, hence the instability. +The control system is thus canceling the spring forces which makes the suspended platform not capable to hold the payload against centrifugal forces, hence the instability. -\begin{figure}[htbp] -\centering -\includegraphics[scale=1]{figs/rotating_root_locus_iff_pure_int.png} -\caption{\label{fig:rotating_root_locus_iff_pure_int}Root Locus for the Decentralized Integral Force Feedback controller. Several rotating speed are shown.} -\end{figure} \chapter{Integral Force Feedback with an High Pass Filter} \label{sec:rotating_iff_pseudo_int} +As was explained in the previous section, the instability of the IFF controller applied on the rotating system is due to the high gain of the integrator at low frequency. +In order to limit the low frequency controller gain, an High Pass Filter (HPF) can be added to the controller as shown in equation \eqref{eq:rotating_iff_lhf}. +This is equivalent to slightly shifting the controller pole to the left along the real axis. +This modification of the IFF controller is typically done to avoid saturation associated with the pure integrator \cite{preumont91_activ,marneffe07_activ_passiv_vibrat_isolat_dampin_shunt_trans}. +This is however not the reason why this high pass filter is added here. -As was explained in the previous section, the instability of the IFF controller applied on the rotating system comes in part from the high gain at low frequency caused by the pure integrators. - -In order to limit the low frequency controller gain, an High Pass Filter (HPF) can be added to the controller as shown in equation \eqref{eq:iff_lhf}. - -\begin{equation} -\label{eq:iff_lhf} +\begin{equation}\label{eq:rotating_iff_lhf} \boxed{K_{F}(s) = g \cdot \frac{1}{s} \cdot \underbrace{\frac{s/\omega_i}{1 + s/\omega_i}}_{\text{HPF}} = g \cdot \frac{1}{s + \omega_i}} \end{equation} -This is equivalent as to slightly \textbf{shifting the controller pole to the left along the real axis}. - -This modification of the IFF controller is typically done to avoid saturation associated with the pure integrator \cite{preumont91_activ,marneffe07_activ_passiv_vibrat_isolat_dampin_shunt_trans}. -This is however not why this high pass filter is added here. \section{Modified Integral Force Feedback Controller} -The Integral Force Feedback Controller is modified such that instead of using pure integrators, pseudo integrators (i.e. low pass filters) are used: -\begin{equation} - K_{\text{IFF}}(s) = g\frac{1}{\omega_i + s} \begin{bmatrix} - 1 & 0 \\ - 0 & 1 -\end{bmatrix} -\end{equation} -where \(\omega_i\) characterize down to which frequency the signal is integrated. - -Let's arbitrary choose the following control parameters: -\begin{minted}[]{matlab} -%% Modified IFF - parameters -g = 2; % Controller gain -wi = 0.1; % HPF Cut-Off frequency [rad/s] - -Kiff = (g/s)*eye(2); % Pure IFF -Kiff_hpf = (g/(wi+s))*eye(2); % IFF with added HPF -\end{minted} - +The Integral Force Feedback Controller is modified such that instead of using pure integrators, pseudo integrators (i.e. low pass filters) are used \eqref{eq:rotating_iff_lhf} where \(\omega_i\) characterize the frequency down to which the signal is integrated. The loop gains (\(K_F(s)\) times the direct dynamics \(f_u/F_u\)) with and without the added HPF are shown in Figure \ref{fig:rotating_iff_modified_loop_gain}. The effect of the added HPF limits the low frequency gain to finite values as expected. -\begin{figure}[htbp] -\centering -\includegraphics[scale=1]{figs/rotating_iff_modified_loop_gain.png} -\caption{\label{fig:rotating_iff_modified_loop_gain}Loop gain for the IFF with pure integrator and modified IFF with added high pass filter (\(\Omega = 0.1\omega_0\))} -\end{figure} +The Root Locus plots for the decentralized IFF with and without the HPF are displayed in Figure \ref{fig:rotating_iff_root_locus_hpf_large}. +With the added HPF, the poles of the closed loop system are shown to be stable up to some value of the gain \(g_\text{max}\) given by equation \eqref{eq:rotating_gmax_iff_hpf}. +It is interesting to note that \(g_{\text{max}}\) also corresponds to the controller gain at which the low frequency loop gain reaches one (for instance the gain \(g\) can be increased by a factor \(5\) in Figure \ref{fig:rotating_iff_modified_loop_gain} before the system becomes unstable). -The Root Locus plots for the decentralized IFF with and without the HPF are displayed in Figure \ref{fig:rotating_iff_root_locus_hpf}. -With the added HPF, the poles of the closed loop system are shown to be \textbf{stable up to some value of the gain} \(g_\text{max}\) given by equation \eqref{eq:gmax_iff_hpf}. - -\begin{equation} -\label{eq:gmax_iff_hpf} +\begin{equation}\label{eq:rotating_gmax_iff_hpf} \boxed{g_{\text{max}} = \omega_i \left( \frac{{\omega_0}^2}{\Omega^2} - 1 \right)} \end{equation} -It is interesting to note that \(g_{\text{max}}\) also corresponds to the controller gain at which the low frequency loop gain (Figure \ref{fig:rotating_iff_modified_loop_gain}) reaches one. - \begin{figure}[htbp] -\centering -\includegraphics[scale=1]{figs/rotating_iff_root_locus_hpf.png} -\caption{\label{fig:rotating_iff_root_locus_hpf}Root Locus for the initial IFF and the modified IFF} +\begin{subfigure}{0.49\textwidth} +\begin{center} +\includegraphics[scale=1,scale=0.8]{figs/rotating_iff_modified_loop_gain.png} +\end{center} +\subcaption{\label{fig:rotating_iff_modified_loop_gain}Loop gain} +\end{subfigure} +\begin{subfigure}{0.49\textwidth} +\begin{center} +\includegraphics[scale=1,scale=0.8]{figs/rotating_iff_root_locus_hpf_large.png} +\end{center} +\subcaption{\label{fig:rotating_iff_root_locus_hpf_large}Root Locus} +\end{subfigure} +\caption{\label{fig:rotating_iff_modified_loop_gain_root_locus}Comparison of the IFF with pure integrator and modified IFF with added high pass filter (\(\Omega = 0.1\omega_0\)). Loop gain is shown in (\subref{fig:rotating_iff_modified_loop_gain}) with \(\omega_i = 0.1 \omega_0\) and \(g = 2\). Root Locus is shown in (\subref{fig:rotating_iff_root_locus_hpf_large})} \end{figure} + \section{Optimal IFF with HPF parameters \(\omega_i\) and \(g\)} -Two parameters can be tuned for the modified controller in equation \eqref{eq:iff_lhf}: the gain \(g\) and the pole's location \(\omega_i\). +Two parameters can be tuned for the modified controller in equation \eqref{eq:rotating_iff_lhf}: the gain \(g\) and the pole's location \(\omega_i\). The optimal values of \(\omega_i\) and \(g\) are here considered as the values for which the damping of all the closed-loop poles are simultaneously maximized. In order to visualize how \(\omega_i\) does affect the attainable damping, the Root Locus plots for several \(\omega_i\) are displayed in Figure \ref{fig:rotating_root_locus_iff_modified_effect_wi}. -It is shown that even though small \(\omega_i\) seem to allow more damping to be added to the suspension modes, the control gain \(g\) may be limited to small values due to equation \eqref{eq:gmax_iff_hpf}. - -\begin{figure}[htbp] -\centering -\includegraphics[scale=1]{figs/rotating_root_locus_iff_modified_effect_wi.png} -\caption{\label{fig:rotating_root_locus_iff_modified_effect_wi}Root Locus for several high pass filter cut-off frequency} -\end{figure} - +It is shown that even though small \(\omega_i\) seem to allow more damping to be added to the suspension modes (see Root locus in Figure \ref{fig:rotating_root_locus_iff_modified_effect_wi_large}), the control gain \(g\) may be limited to small values due to equation \eqref{eq:rotating_gmax_iff_hpf}. In order to study this trade off, the attainable closed-loop damping ratio \(\xi_{\text{cl}}\) is computed as a function of \(\omega_i/\omega_0\). The gain \(g_{\text{opt}}\) at which this maximum damping is obtained is also displayed and compared with the gain \(g_{\text{max}}\) at which the system becomes unstable (Figure \ref{fig:rotating_iff_hpf_optimal_gain}). @@ -403,164 +398,149 @@ Three regions can be observed: \end{itemize} \begin{figure}[htbp] -\centering -\includegraphics[scale=1]{figs/rotating_iff_hpf_optimal_gain.png} -\caption{\label{fig:rotating_iff_hpf_optimal_gain}Attainable damping ratio \(\xi_\text{cl}\) as a function of \(\omega_i/\omega_0\). Corresponding control gain \(g_\text{opt}\) and \(g_\text{max}\) are also shown} +\begin{subfigure}{0.49\textwidth} +\begin{center} +\includegraphics[scale=1,width=0.95\linewidth]{figs/rotating_root_locus_iff_modified_effect_wi.png} +\end{center} +\subcaption{\label{fig:rotating_root_locus_iff_modified_effect_wi}Root Locus} +\end{subfigure} +\begin{subfigure}{0.49\textwidth} +\begin{center} +\includegraphics[scale=1,width=0.95\linewidth]{figs/rotating_iff_hpf_optimal_gain.png} +\end{center} +\subcaption{\label{fig:rotating_iff_hpf_optimal_gain}Attainable damping ratio $\xi_\text{cl}$ as a function of $\omega_i/\omega_0$. Corresponding control gain $g_\text{opt}$ and $g_\text{max}$ are also shown} +\end{subfigure} +\caption{\label{fig:rotating_iff_modified_effect_wi}Root Locus for several high pass filter cut-off frequency (\subref{fig:rotating_root_locus_iff_modified_effect_wi_large}).} \end{figure} + \section{Obtained Damped Plant} - -Let's choose \(\omega_i = 0.1 \cdot \omega_0\) and compute the damped plant. -The undamped and damped plants are compared in Figure \ref{fig:rotating_iff_hpf_damped_plant} in blue and red respectively. -A well damped plant is indeed obtained. - -However, the magnitude of the coupling term (\(d_v/F_u\)) is larger then IFF is applied. - -\begin{figure}[htbp] -\centering -\includegraphics[scale=1]{figs/rotating_iff_hpf_damped_plant.png} -\caption{\label{fig:rotating_iff_hpf_damped_plant}Damped plant with IFF and added HPF - Transfer function from \(F_u\) to \(d_u\), \(\omega_i = 0.1 \cdot \omega_0\), \(\Omega = 0.1 \cdot \omega_0\)} -\end{figure} - -In order to study how \(\omega_i\) affects the coupling of the damped plant, the closed-loop plant is identified for several \(\omega_i\). -The direct and coupling terms of the plants are shown in Figure \ref{fig:rotating_iff_hpf_damped_plant_effect_wi_coupling} (left) and the ratio between the two (i.e. the coupling ratio) is shown in Figure \ref{fig:rotating_iff_hpf_damped_plant_effect_wi_coupling} (right). - -The coupling ratio is decreasing as \(\omega_i\) increases. -There is therefore a \textbf{trade-off between achievable damping and coupling ratio} for the choice of \(\omega_i\). +In order to study how the parameter \(\omega_i\) affects the damped plant, the obtained damped plants for several \(\omega_i\) are compared in Figure \ref{fig:rotating_iff_hpf_damped_plant_effect_wi_plant}. +It can be seen that the low frequency coupling increases as \(\omega_i\) increases. +There is therefore a trade-off between achievable damping and added coupling when tuning \(\omega_i\). The same trade-off can be seen between achievable damping and loss of compliance at low frequency (see Figure \ref{fig:rotating_iff_hpf_effect_wi_compliance}). \begin{figure}[htbp] -\centering -\includegraphics[scale=1]{figs/rotating_iff_hpf_damped_plant_effect_wi_coupling.png} -\caption{\label{fig:rotating_iff_hpf_damped_plant_effect_wi_coupling}Effect of \(\omega_i\) on the damped plant coupling} +\begin{subfigure}{0.49\textwidth} +\begin{center} +\includegraphics[scale=1,width=0.95\linewidth]{figs/rotating_iff_hpf_damped_plant_effect_wi_plant.png} +\end{center} +\subcaption{\label{fig:rotating_iff_hpf_damped_plant_effect_wi_plant}Obtained plants} +\end{subfigure} +\begin{subfigure}{0.49\textwidth} +\begin{center} +\includegraphics[scale=1,width=0.95\linewidth]{figs/rotating_iff_hpf_effect_wi_compliance.png} +\end{center} +\subcaption{\label{fig:rotating_iff_hpf_effect_wi_compliance}Effect of $\omega_i$ on the compliance} +\end{subfigure} +\caption{\label{fig:rotating_iff_hpf_damped_plant_effect_wi}Effect of \(\omega_i\) on the damped plant coupling} \end{figure} -\begin{figure}[htbp] -\centering -\includegraphics[scale=1]{figs/rotating_iff_hpf_effect_wi_compliance.png} -\caption{\label{fig:rotating_iff_hpf_effect_wi_compliance}Effect of \(\omega_i\) on the obtained compliance} -\end{figure} \chapter{IFF with a stiffness in parallel with the force sensor} \label{sec:rotating_iff_parallel_stiffness} - In this section it is proposed to add springs in parallel with the force sensors to counteract the negative stiffness induced by the gyroscopic effects. - Such springs are schematically shown in Figure \ref{fig:rotating_3dof_model_schematic_iff_parallel_springs} where \(k_a\) is the stiffness of the actuator and \(k_p\) the added stiffness in parallel with the actuator and force sensor. \begin{figure}[htbp] \centering -\includegraphics[scale=1]{figs/rotating_3dof_model_schematic_iff_parallel_springs.png} +\includegraphics[scale=1,scale=0.8]{figs/rotating_3dof_model_schematic_iff_parallel_springs.png} \caption{\label{fig:rotating_3dof_model_schematic_iff_parallel_springs}Studied system with additional springs in parallel with the actuators and force sensors (shown in red)} \end{figure} -\section{Equations} -The forces measured by the two force sensors represented in Figure \ref{fig:rotating_3dof_model_schematic_iff_parallel_springs} are described by Eq. \eqref{eq:measured_force_kp}. -\begin{equation} -\label{eq:measured_force_kp} +\section{Equations} +The forces measured by the two force sensors represented in Figure \ref{fig:rotating_3dof_model_schematic_iff_parallel_springs} are described by \eqref{eq:rotating_measured_force_kp}. + +\begin{equation}\label{eq:rotating_measured_force_kp} \begin{bmatrix} f_{u} \\ f_{v} \end{bmatrix} = \begin{bmatrix} F_u \\ F_v \end{bmatrix} - (c s + k_a) \begin{bmatrix} d_u \\ d_v \end{bmatrix} \end{equation} -In order to keep the overall stiffness \(k = k_a + k_p\) constant, thus not modifying the open-loop poles as \(k_p\) is changed, a scalar parameter \(\alpha\) (\(0 \le \alpha < 1\)) is defined to describe the fraction of the total stiffness in parallel with the actuator and force sensor as in Eq. \eqref{eq:kp_alpha}. +In order to keep the overall stiffness \(k = k_a + k_p\) constant, thus not modifying the open-loop poles as \(k_p\) is changed, a scalar parameter \(\alpha\) (\(0 \le \alpha < 1\)) is defined to describe the fraction of the total stiffness in parallel with the actuator and force sensor as in \eqref{eq:rotating_kp_alpha}. -\begin{equation} -\label{eq:kp_alpha} +\begin{equation}\label{eq:rotating_kp_alpha} k_p = \alpha k, \quad k_a = (1 - \alpha) k \end{equation} -After the equations of motion derived and transformed in the Laplace domain, the transfer function matrix \(\mathbf{G}_k\) in Eq. \eqref{eq:Gk_mimo_tf} is computed. -Its elements are shown in Eq. \eqref{eq:Gk_diag} and \eqref{eq:Gk_off_diag}. +After the equations of motion derived and transformed in the Laplace domain, the transfer function matrix \(\mathbf{G}_k\) in Eq. \eqref{eq:rotating_Gk_mimo_tf} is computed. +Its elements are shown in Eq. \eqref{eq:rotating_Gk_diag} and \eqref{eq:rotating_Gk_off_diag}. -\begin{equation} -\label{eq:Gk_mimo_tf} +\begin{equation}\label{eq:rotating_Gk_mimo_tf} \begin{bmatrix} f_u \\ f_v \end{bmatrix} = \mathbf{G}_k \begin{bmatrix} F_u \\ F_v \end{bmatrix} \end{equation} -\begin{subequations} -\label{eq:Gk} +\begin{subequations}\label{eq:rotating_Gk} \begin{align} -\mathbf{G}_{k}(1,1) &= \mathbf{G}_{k}(2,2) = \frac{\big( \frac{s^2}{{\omega_0}^2} - \frac{\Omega^2}{{\omega_0}^2} + \alpha \big) \big( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \big) + \big( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \big)^2}{\big( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \big)^2 + \big( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \big)^2} \label{eq:Gk_diag} \\ -\mathbf{G}_{k}(1,2) &= -\mathbf{G}_{k}(2,1) = \frac{- \left( 2 \xi \frac{s}{\omega_0} + 1 - \alpha \right) \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)}{\left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2} \label{eq:Gk_off_diag} +\mathbf{G}_{k}(1,1) &= \mathbf{G}_{k}(2,2) = \frac{\big( \frac{s^2}{{\omega_0}^2} - \frac{\Omega^2}{{\omega_0}^2} + \alpha \big) \big( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \big) + \big( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \big)^2}{\big( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \big)^2 + \big( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \big)^2} \label{eq:rotating_Gk_diag} \\ +\mathbf{G}_{k}(1,2) &= -\mathbf{G}_{k}(2,1) = \frac{- \left( 2 \xi \frac{s}{\omega_0} + 1 - \alpha \right) \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)}{\left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2} \label{eq:rotating_Gk_off_diag} \end{align} \end{subequations} -Comparing \(\mathbf{G}_k\) in Eq. \eqref{eq:Gk} with \(\mathbf{G}_f\) in Eq. \eqref{eq:Gf} shows that while the poles of the system are kept the same, the zeros of the diagonal terms have changed. -The two real zeros \(z_r\) in Eq. \eqref{eq:iff_zero_real} that were inducing a non-minimum phase behavior are transformed into two complex conjugate zeros if the condition in Eq. \eqref{eq:kp_cond_cc_zeros} holds. +Comparing \(\mathbf{G}_k\) in \eqref{eq:rotating_Gk} with \(\mathbf{G}_f\) in \eqref{eq:rotating_Gf} shows that while the poles of the system are kept the same, the zeros of the diagonal terms have changed. +The two real zeros \(z_r\) in \eqref{eq:rotating_iff_zero_real} that were inducing a non-minimum phase behavior are transformed into two complex conjugate zeros if the condition in \eqref{eq:rotating_kp_cond_cc_zeros} holds. +Thus, if the added \emph{parallel stiffness} \(k_p\) is higher than the \emph{negative stiffness} induced by centrifugal forces \(m \Omega^2\), the dynamics from actuator to its collocated force sensor will show \emph{minimum phase behavior}. -\begin{equation} -\label{eq:kp_cond_cc_zeros} +\begin{equation}\label{eq:rotating_kp_cond_cc_zeros} \boxed{\alpha > \frac{\Omega^2}{{\omega_0}^2} \quad \Leftrightarrow \quad k_p > m \Omega^2} \end{equation} -\begin{important} -Thus, if the added \textbf{parallel stiffness} \(k_p\) is \textbf{higher than the negative stiffness induced by centrifugal forces} \(m \Omega^2\), the dynamics from actuator to its collocated force sensor will show minimum phase behavior. -\end{important} \section{Effect of the parallel stiffness on the IFF plant} -The IFF plant (transfer function from \([F_u, F_v]\) to \([f_u, f_v]\)) is identified in three different cases: -\begin{itemize} -\item without parallel stiffness \(k_p = 0\) -\item with a small parallel stiffness \(k_p < m \Omega^2\) -\item with a large parallel stiffness \(k_p > m \Omega^2\) -\end{itemize} - +The IFF plant (transfer function from \([F_u, F_v]\) to \([f_u, f_v]\)) is identified without parallel stiffness \(k_p = 0\), with a small parallel stiffness \(k_p < m \Omega^2\) and with a large parallel stiffness \(k_p > m \Omega^2\). The Bode plots of the obtained dynamics are shown in Figure \ref{fig:rotating_iff_effect_kp}. -One can see that for \(k_p > m \Omega^2\), the two real zeros with \(k_p < m \Omega^2\) are transformed into two complex conjugate zeros and the systems shows alternating complex conjugate poles and zeros. +One can see that the the two real zeros for \(k_p < m \Omega^2\) are transformed into two complex conjugate zeros for \(k_p > m \Omega^2\). +In that case, the systems shows alternating complex conjugate poles and zeros as what is the case in the non-rotating case. + +Figure \ref{fig:rotating_iff_kp_root_locus} shows the Root Locus plots for \(k_p = 0\), \(k_p < m \Omega^2\) and \(k_p > m \Omega^2\) when \(K_F\) is a pure integrator as in Eq. \eqref{eq:rotating_Kf_pure_int}. +It is shown that if the added stiffness is higher than the maximum negative stiffness, the poles of the closed-loop system are bounded on the (stable) left half-plane, and hence the unconditional stability of IFF is recovered. \begin{figure}[htbp] -\centering -\includegraphics[scale=1]{figs/rotating_iff_effect_kp.png} -\caption{\label{fig:rotating_iff_effect_kp}Effect of the parallel stiffness on the IFF plant: Bode plot of \(G_{k}(1,1) = f_u/F_u\) without parallel spring, with parallel spring stiffness \(k_p < m \Omega^2\) and \(k_p > m \Omega^2\), \(\Omega = 0.1 \omega_0\)} +\begin{subfigure}{0.55\linewidth} +\begin{center} +\includegraphics[scale=1,width=0.95\linewidth]{figs/rotating_iff_effect_kp.png} +\end{center} +\subcaption{\label{fig:rotating_iff_effect_kp}Bode plot of $G_{k}(1,1) = f_u/F_u$ without parallel spring, with parallel spring stiffness $k_p < m \Omega^2$ and $k_p > m \Omega^2$, $\Omega = 0.1 \omega_0$} +\end{subfigure} +\begin{subfigure}{0.44\linewidth} +\begin{center} +\includegraphics[scale=1,scale=0.9]{figs/rotating_iff_kp_root_locus.png} +\end{center} +\subcaption{\label{fig:rotating_iff_kp_root_locus}Root Locus for IFF without parallel spring, with small parallel spring and with large parallel spring} +\end{subfigure} +\caption{\label{fig:rotating_iff_plant_effect_kp}Effect of the parallel stiffness on the IFF plant} \end{figure} -Figure \ref{fig:rotating_iff_kp_root_locus} shows the Root Locus plots for \(k_p = 0\), \(k_p < m \Omega^2\) and \(k_p > m \Omega^2\) when \(K_F\) is a pure integrator as in Eq. \eqref{eq:Kf_pure_int}. -It is shown that if the added stiffness is higher than the maximum negative stiffness, the poles of the closed-loop system are bounded on the (stable) left half-plane, and hence the \textbf{unconditional stability of IFF is recovered}. - -\begin{figure}[htbp] -\centering -\includegraphics[scale=1]{figs/rotating_iff_kp_root_locus.png} -\caption{\label{fig:rotating_iff_kp_root_locus}Root Locus for IFF without parallel spring, with small parallel spring and with large parallel spring} -\end{figure} \section{Effect of \(k_p\) on the attainable damping} - Even though the parallel stiffness \(k_p\) has no impact on the open-loop poles (as the overall stiffness \(k\) is kept constant), it has a large impact on the transmission zeros. Moreover, as the attainable damping is generally proportional to the distance between poles and zeros \cite{preumont18_vibrat_contr_activ_struc_fourt_edition}, the parallel stiffness \(k_p\) is foreseen to have some impact on the attainable damping. - To study this effect, Root Locus plots for several parallel stiffnesses \(k_p > m \Omega^2\) are shown in Figure \ref{fig:rotating_iff_kp_root_locus_effect_kp}. The frequencies of the transmission zeros of the system are increasing with an increase of the parallel stiffness \(k_p\) (thus getting closer to the poles) and the associated attainable damping is reduced. - -\begin{important} Therefore, even though the parallel stiffness \(k_p\) should be larger than \(m \Omega^2\) for stability reasons, it should not be taken too large as this would limit the attainable damping. -\end{important} - -\begin{figure}[htbp] -\centering -\includegraphics[scale=1]{figs/rotating_iff_kp_root_locus_effect_kp.png} -\caption{\label{fig:rotating_iff_kp_root_locus_effect_kp}Root Locus: Effect of the parallel stiffness on the attainable damping, \(\Omega = 0.1 \omega_0\)} -\end{figure} - This is confirmed by the Figure \ref{fig:rotating_iff_kp_optimal_gain} where the attainable closed-loop damping ratio \(\xi_{\text{cl}}\) and the associated optimal control gain \(g_\text{opt}\) are computed as a function of the parallel stiffness. \begin{figure}[htbp] -\centering -\includegraphics[scale=1]{figs/rotating_iff_kp_optimal_gain.png} -\caption{\label{fig:rotating_iff_kp_optimal_gain}Attainable damping ratio \(\xi_\text{cl}\) as a function of the parallel stiffness \(k_p\). Corresponding control gain \(g_\text{opt}\) is also shown. Values for \(k_p < m\Omega^2\) are not shown as the system is unstable.} +\begin{subfigure}{0.49\linewidth} +\begin{center} +\includegraphics[scale=1,scale=1]{figs/rotating_iff_kp_root_locus_effect_kp.png} +\end{center} +\subcaption{\label{fig:rotating_iff_kp_root_locus_effect_kp}Root Locus: Effect of the parallel stiffness on the attainable damping, $\Omega = 0.1 \omega_0$} +\end{subfigure} +\begin{subfigure}{0.49\linewidth} +\begin{center} +\includegraphics[scale=1,scale=0.9]{figs/rotating_iff_kp_optimal_gain.png} +\end{center} +\subcaption{\label{fig:rotating_iff_kp_optimal_gain}Attainable damping ratio $\xi_\text{cl}$ as a function of the parallel stiffness $k_p$. Corresponding control gain $g_\text{opt}$ is also shown. Values for $k_p < m\Omega^2$ are not shown as the system is unstable.} +\end{subfigure} +\caption{\label{fig:rotating_iff_optimal_kp}Effect of the parallel stiffness on the IFF plant} \end{figure} + \section{Damped plant} Let's choose a parallel stiffness equal to \(k_p = 2 m \Omega^2\) and compute the damped plant. The damped and undamped transfer functions from \(F_u\) to \(d_u\) are compared in Figure \ref{fig:rotating_iff_kp_damped_plant}. - Even though the two resonances are well damped, the IFF changes the low frequency behavior of the plant which is usually not wanted. This is due to the fact that ``pure'' integrators are used, and that the low frequency loop gains becomes large below some frequency. -\begin{figure}[htbp] -\centering -\includegraphics[scale=1]{figs/rotating_iff_kp_damped_plant.png} -\caption{\label{fig:rotating_iff_kp_damped_plant}Damped plant with IFF - Transfer function from \(F_u\) to \(d_u\)} -\end{figure} - -In order to lower the low frequency gain, an high pass filter is added to the IFF controller (which is equivalent as shifting the controller pole to the left in the complex plane): +In order to lower the low frequency gain, a high pass filter is added to the IFF controller (which is equivalent as shifting the controller pole to the left in the complex plane): \begin{equation} K_{\text{IFF}}(s) = g\frac{1}{\omega_i + s} \begin{bmatrix} 1 & 0 \\ @@ -568,311 +548,299 @@ In order to lower the low frequency gain, an high pass filter is added to the IF \end{bmatrix} \end{equation} -Let's see how the high pass filter impacts the attainable damping. -The controller gain \(g\) is kept constant while \(\omega_i\) is changed, and the minimum damping ratio of the damped plant is computed. +In order to see how the high pass filter impacts the attainable damping, the controller gain \(g\) is kept constant while \(\omega_i\) is changed, and the minimum damping ratio of the damped plant is computed. The obtained damping ratio as a function of \(\omega_i/\omega_0\) (where \(\omega_0\) is the resonance of the system without rotation) is shown in Figure \ref{fig:rotating_iff_kp_added_hpf_effect_damping}. +It is shown that the attainable damping ratio reduces as \(\omega_i\) is increased (same conclusion than in Section \ref{sec:rotating_iff_pseudo_int}). +Let's choose \(\omega_i = 0.1 \cdot \omega_0\) and compare the obtained damped plant again with the undamped and with the ``pure'' IFF in Figure \ref{fig:rotating_iff_kp_added_hpf_damped_plant}. +The added high pass filter gives almost the same damping properties to the suspension while giving good low frequency behavior. \begin{figure}[htbp] -\centering -\includegraphics[scale=1]{figs/rotating_iff_kp_added_hpf_effect_damping.png} -\caption{\label{fig:rotating_iff_kp_added_hpf_effect_damping}Effect of the high pass filter cut-off frequency on the obtained damping} +\begin{subfigure}{0.34\linewidth} +\begin{center} +\includegraphics[scale=1,scale=0.95]{figs/rotating_iff_kp_added_hpf_effect_damping.png} +\end{center} +\subcaption{\label{fig:rotating_iff_kp_added_hpf_effect_damping}Reduced damping ratio with increased cut-off frequency $\omega_i$} +\end{subfigure} +\begin{subfigure}{0.65\linewidth} +\begin{center} +\includegraphics[scale=1,scale=0.95]{figs/rotating_iff_kp_added_hpf_damped_plant.png} +\end{center} +\subcaption{\label{fig:rotating_iff_kp_added_hpf_damped_plant}Damped plant with the parallel stiffness, effect of the added HPF} +\end{subfigure} +\caption{\label{fig:rotating_iff_optimal_kp}Effect of the high pass filter cut-off frequency on the obtained damping} \end{figure} -Let's choose \(\omega_i = 0.1 \cdot \omega_0\) and compute the damped plant again. -The Bode plots of the undamped, damped with ``pure'' IFF, and with added high pass filters are shown in Figure \ref{fig:rotating_iff_kp_added_hpf_damped_plant}. -The added high pass filter gives almost the same damping properties while giving acceptable low frequency behavior. - -\begin{figure}[htbp] -\centering -\includegraphics[scale=1]{figs/rotating_iff_kp_added_hpf_damped_plant.png} -\caption{\label{fig:rotating_iff_kp_added_hpf_damped_plant}Damped plant with IFF - Transfer function from \(F_u\) to \(d_u\)} -\end{figure} \chapter{Relative Damping Control} \label{sec:rotating_relative_damp_control} - -In order to apply a ``relative damping control strategy'', relative motion sensors are added in parallel with the actuators as shown in Figure \ref{fig:rotating_3dof_model_schematic_rdc}. - +In order to apply a ``Relative Damping Control'' strategy, relative motion sensors are added in parallel with the actuators as shown in Figure \ref{fig:rotating_3dof_model_schematic_rdc}. Two controllers \(K_d\) are used to fed back the relative motion to the actuator. -\(K_d\) is a derivator: -\begin{equation} -K_d(s) = s -\end{equation} +These controllers are in principle pure derivators (\(K_d = s\)), but to be implemented in practice they are usually replaced by a high pass filter \eqref{eq:rotating_rdc_controller}. -To be implemented in practice, it is usually replaced by a an high pass filter: -\begin{equation} -K_d(s) = \frac{s}{s + \omega_d} +\begin{equation}\label{eq:rotating_rdc_controller} +K_d(s) = g \cdot \frac{s}{s + \omega_d} \end{equation} - \begin{figure}[htbp] \centering -\includegraphics[scale=1]{figs/rotating_3dof_model_schematic_rdc.png} +\includegraphics[scale=1,scale=0.8]{figs/rotating_3dof_model_schematic_rdc.png} \caption{\label{fig:rotating_3dof_model_schematic_rdc}System with relative motion sensor and decentralized ``relative damping control'' applied.} \end{figure} + \section{Equations of motion} -Let's note \(\bm{G}_d\) the transfer function between actuator forces and measured relative motion in parallel with the actuators: -\begin{equation} +Let's note \(\bm{G}_d\) the transfer function between actuator forces and measured relative motion in parallel with the actuators \eqref{eq:rotating_rdc_plant_matrix}. +The elements of \(\bm{G}_d\) were derived in Section \ref{sec:rotating_system_description} are shown in \eqref{eq:rotating_rdc_plant_elements}. + +\begin{equation}\label{eq:rotating_rdc_plant_matrix} \begin{bmatrix} d_u \\ d_v \end{bmatrix} = \mathbf{G}_d \begin{bmatrix} F_u \\ F_v \end{bmatrix} \end{equation} -With: -\begin{subequations} +\begin{subequations}\label{eq:rotating_rdc_plant_elements} \begin{align} \mathbf{G}_{d}(1,1) &= \mathbf{G}_{d}(2,2) = \frac{\frac{1}{k} \left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)}{\left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2} \\ \mathbf{G}_{d}(1,2) &= -\mathbf{G}_{d}(2,1) = \frac{\frac{1}{k} \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)}{\left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2} \end{align} \end{subequations} -Neglecting the damping for simplicity (\(\xi \ll 1\)), the direct terms have two complex conjugate zeros: -\begin{equation} - z = \pm j \sqrt{\omega_0^2 - \omega^2} +Neglecting the damping for simplicity (\(\xi \ll 1\)), the direct terms have two complex conjugate zeros which are between the two pairs of complex conjugate poles \eqref{eq:rotating_rdc_zeros_poles}. +Therefore, for \(\Omega < \sqrt{k/m}\) (i.e. stable system), the transfer functions for Relative Damping Control have alternating complex conjugate poles and zeros. + +\begin{equation}\label{eq:rotating_rdc_zeros_poles} + z = \pm j \sqrt{\omega_0^2 - \omega^2}, \quad p_1 = \pm j (\omega_0 - \omega), \quad p_2 = \pm j (\omega_0 + \omega) \end{equation} -Which are between the two pairs of complex conjugate poles at: -\begin{align} - p_1 &= \pm j (\omega_0 - \omega) \\ - p_2 &= \pm j (\omega_0 + \omega) -\end{align} - -Therefore, for \(\Omega < \sqrt{k/m}\) (i.e. stable system), the transfer functions for Relative Damping Control have \textbf{alternating complex conjugate poles and zeros}. \section{Decentralized Relative Damping Control} -The transfer functions from \([F_u,\ F_v]\) to \([d_u,\ d_v]\) is identified and shown in Figure \ref{fig:rotating_rdc_plant_effect_rot} for several rotating velocities. - -\begin{figure}[htbp] -\centering -\includegraphics[scale=1]{figs/rotating_rdc_plant_effect_rot.png} -\caption{\label{fig:rotating_rdc_plant_effect_rot}Bode plot of the direct and coupling term for the ``relative damping control'' plant - Effect of rotation} -\end{figure} +The transfer functions from \([F_u,\ F_v]\) to \([d_u,\ d_v]\) were identified for several rotating velocities in Section \ref{sec:rotating_system_description} and are shown in Figure \ref{fig:rotating_bode_plot} (page \pageref{fig:rotating_bode_plot}). In order to see if large damping can be added with Relative Damping Control, the root locus is computed (Figure \ref{fig:rotating_rdc_root_locus}). -The closed-loop system is unconditionally stable and the poles can be damped as much as wanted. +The closed-loop system is unconditionally stable as expected and the poles can be damped as much as wanted. -\begin{figure}[htbp] -\centering -\includegraphics[scale=1]{figs/rotating_rdc_root_locus.png} -\caption{\label{fig:rotating_rdc_root_locus}Root Locus for Relative Damping Control} -\end{figure} -\section{Damped Plant} Let's select a reasonable ``Relative Damping Control'' gain, and compute the closed-loop damped system. The open-loop and damped plants are compared in Figure \ref{fig:rotating_rdc_damped_plant}. - The rotating aspect does not add any complexity for the use of Relative Damping Control. It does not increase the low frequency coupling as compared to Integral Force Feedback. \begin{figure}[htbp] -\centering -\includegraphics[scale=1]{figs/rotating_rdc_damped_plant.png} -\caption{\label{fig:rotating_rdc_damped_plant}Damped plant using Relative Damping Control} +\begin{subfigure}{0.49\linewidth} +\begin{center} +\includegraphics[scale=1,scale=1]{figs/rotating_rdc_root_locus.png} +\end{center} +\subcaption{\label{fig:rotating_rdc_root_locus}Root Locus for Relative Damping Control} +\end{subfigure} +\begin{subfigure}{0.49\linewidth} +\begin{center} +\includegraphics[scale=1,scale=0.8]{figs/rotating_rdc_damped_plant.png} +\end{center} +\subcaption{\label{fig:rotating_rdc_damped_plant}Damped plant using Relative Damping Control} +\end{subfigure} +\caption{\label{fig:rotating_rdc_result}Relative Damping Control. Root Locus (\subref{fig:rotating_rdc_root_locus}) and obtained damped plant (\subref{rotating_rdc_damped_plant})} \end{figure} + \chapter{Comparison of Active Damping Techniques} \label{sec:rotating_comp_act_damp} - These two proposed IFF modifications as well as relative damping control are now compared in terms of added damping and closed-loop behavior. - For the following comparisons, the cut-off frequency for the added HPF is set to \(\omega_i = 0.1 \omega_0\) and the stiffness of the parallel springs is set to \(k_p = 5 m \Omega^2\) (corresponding to \(\alpha = 0.05\)). These values are chosen based on previous discussion about optimal parameters. -\section{Root Locus} +\section{Root Locus} Figure \ref{fig:rotating_comp_techniques_root_locus} shows the Root Locus plots for the two proposed IFF modifications as well as for relative damping control. While the two pairs of complex conjugate open-loop poles are identical for both IFF modifications, the transmission zeros are not. This means that the closed-loop behavior of both systems will differ when large control gains are used. One can observe that the closed loop poles corresponding to the system with added springs (in red) are bounded to the left half plane implying unconditional stability. This is not the case for the system where the controller is augmented with an HPF (in blue). - -It is interesting to note that the maximum added damping is very similar for both techniques. +It is interesting to note that the maximum added damping is very similar for both modified IFF techniques. \begin{figure}[htbp] -\centering -\includegraphics[scale=1]{figs/rotating_comp_techniques_root_locus.png} -\caption{\label{fig:rotating_comp_techniques_root_locus}Comparison of active damping techniques for rotating platform - Root Locus} +\begin{subfigure}{0.49\textwidth} +\begin{center} +\includegraphics[scale=1,width=0.95\linewidth]{figs/rotating_comp_techniques_root_locus.png} +\end{center} +\subcaption{\label{fig:rotating_comp_techniques_root_locus}Root Locus} +\end{subfigure} +\begin{subfigure}{0.49\textwidth} +\begin{center} +\includegraphics[scale=1,width=0.95\linewidth]{figs/rotating_comp_techniques_dampled_plants.png} +\end{center} +\subcaption{\label{fig:rotating_comp_techniques_dampled_plants}Damped plants} +\end{subfigure} +\caption{\label{fig:rotating_comp_techniques}Comparison of active damping techniques for rotating platform} \end{figure} + \section{Obtained Damped Plant} The actively damped plants are computed for the three techniques and compared in Figure \ref{fig:rotating_comp_techniques_dampled_plants}. +It is shown that while the diagonal (direct) terms of the damped plants are similar for the three active damping techniques, the off-diagonal (coupling) terms are not. +Integral Force Feedback strategy is adding some coupling at low frequency which may negatively impact the positioning performance. -\begin{important} -It is shown that while the diagonal (direct) terms of the damped plants are similar for the three active damping techniques, of off-diagonal (coupling) terms are not. -Integral Force Feedback strategy is adding some coupling at low frequency which may negatively impact the positioning performances. -\end{important} - -\begin{figure}[htbp] -\centering -\includegraphics[scale=1]{figs/rotating_comp_techniques_dampled_plants.png} -\caption{\label{fig:rotating_comp_techniques_dampled_plants}Comparison of the damped plants obtained with the three active damping techniques} -\end{figure} \section{Transmissibility And Compliance} The proposed active damping techniques are now compared in terms of closed-loop transmissibility and compliance. - The transmissibility is here defined as the transfer function from a displacement of the rotating stage along \(\vec{i}_x\) to the displacement of the payload along the same direction. It is used to characterize how much vibration is transmitted through the suspended platform to the payload. - The compliance describes the displacement response of the payload to external forces applied to it. This is a useful metric when disturbances are directly applied to the payload. It is here defined as the transfer function from external forces applied on the payload along \(\vec{i}_x\) to the displacement of the payload along the same direction. -Very similar results are obtained for the two proposed IFF modifications in terms of transmissibility and compliance (Figure \ref{fig:rotating_comp_techniques_transmissibility_compliance}). - -\begin{important} +Very similar results are obtained for the two proposed IFF modifications in terms of transmissibility and compliance (Figure \ref{fig:rotating_comp_techniques_trans_compliance}). Using IFF degrades the compliance at low frequency while using relative damping control degrades the transmissibility at high frequency. This is very well known characteristics of these common active damping techniques that holds when applied to rotating platforms. -\end{important} \begin{figure}[htbp] -\centering -\includegraphics[scale=1]{figs/rotating_comp_techniques_transmissibility_compliance.png} -\caption{\label{fig:rotating_comp_techniques_transmissibility_compliance}Comparison of the obtained transmissibilty and compliance for the three tested active damping techniques} +\begin{subfigure}{0.49\textwidth} +\begin{center} +\includegraphics[scale=1,width=0.95\linewidth]{figs/rotating_comp_techniques_transmissibility.png} +\end{center} +\subcaption{\label{fig:rotating_comp_techniques_transmissibility}Transmissibility} +\end{subfigure} +\begin{subfigure}{0.49\textwidth} +\begin{center} +\includegraphics[scale=1,width=0.95\linewidth]{figs/rotating_comp_techniques_compliance.png} +\end{center} +\subcaption{\label{fig:rotating_comp_techniques_compliance}Compliance} +\end{subfigure} +\caption{\label{fig:rotating_comp_techniques_trans_compliance}Comparison of the obtained transmissibilty (\subref{fig:rotating_comp_techniques_transmissibility}) and compliance (\subref{fig:rotating_comp_techniques_compliance}) for the three tested active damping techniques} \end{figure} + \chapter{Rotating Nano-Hexapod} \label{sec:rotating_nano_hexapod} -The current analysis is now applied on a model representing the rotating nano-hexapod. - -Three nano-hexapod stiffnesses are tested: \(k_n = \SI{0.01}{\N\per\mu\m}\), \(k_n = \SI{1}{\N\per\mu\m}\) and \(k_n = \SI{100}{\N\per\mu\m}\). - -Only the maximum rotating velocity is considered (\(\Omega = \SI{60}{rpm}\)) with the light sample (\(m_s = \SI{1}{kg}\)) as this is the worst identified case scenario. +The previous analysis is now applied on a model representing the rotating nano-hexapod. +Three nano-hexapod stiffnesses are tested as for the uniaxial model: \(k_n = \SI{0.01}{\N\per\mu\m}\), \(k_n = \SI{1}{\N\per\mu\m}\) and \(k_n = \SI{100}{\N\per\mu\m}\). +Only the maximum rotating velocity is here considered (\(\Omega = \SI{60}{rpm}\)) with the light sample (\(m_s = \SI{1}{kg}\)) as this is the worst identified case scenario in terms of gyroscopic effects. \section{Nano-Active-Stabilization-System - Plant Dynamics} For the NASS, the maximum rotating velocity is \(\Omega = \SI[parse-numbers=false]{2\pi}{\radian\per\s}\) for a suspended mass on top of the nano-hexapod's actuators equal to \(m_n + m_s = \SI{16}{\kilo\gram}\). The parallel stiffness corresponding to the centrifugal forces is \(m \Omega^2 \approx \SI{0.6}{\newton\per\mm}\). The transfer functions from nano-hexapod actuator force \(F_u\) to the displacement of the nano-hexapod in the same direction \(d_u\) as well as in the orthogonal direction \(d_v\) (coupling) are shown in Figure \ref{fig:rotating_nano_hexapod_dynamics} for all three considered nano-hexapod stiffnesses. - -\begin{important} -It is shown that the rotation has the largest effect on the soft nano-hexapod: -\begin{itemize} -\item larger coupling (the ratio of the coupling term to the direct term is larger for the sort nano-hexapod) -\item larger shift of poles as a function of the rotating velocity -\end{itemize} -\end{important} +The soft nano-hexapod is the most affected by the rotation. +This can be seen by the large shift of the resonance frequencies, and by the induced coupling (the ratio between the direct term and the coupling term) which is larger than for the stiffer nano-hexapods. \begin{figure}[htbp] -\centering -\includegraphics[scale=1]{figs/rotating_nano_hexapod_dynamics.png} -\caption{\label{fig:rotating_nano_hexapod_dynamics}Effect of rotation on the nano-hexapod dynamics - Dashed lines are the plants without rotation, solid lines are plants at maximum rotating velocity, and shaded lines are coupling terms at maximum rotating velocity} +\begin{subfigure}{0.33\textwidth} +\begin{center} +\includegraphics[scale=1,width=0.95\linewidth]{figs/rotating_nano_hexapod_dynamics_vc.png} +\end{center} +\subcaption{\label{fig:uniaxial_damped_plant_three_active_damping_techniques_vc}$k_n = 0.01\,N/\mu m$} +\end{subfigure} +\begin{subfigure}{0.33\textwidth} +\begin{center} +\includegraphics[scale=1,width=0.95\linewidth]{figs/rotating_nano_hexapod_dynamics_md.png} +\end{center} +\subcaption{\label{fig:rotating_nano_hexapod_dynamics_md}$k_n = 1\,N/\mu m$} +\end{subfigure} +\begin{subfigure}{0.33\textwidth} +\begin{center} +\includegraphics[scale=1,width=0.95\linewidth]{figs/rotating_nano_hexapod_dynamics_pz.png} +\end{center} +\subcaption{\label{fig:rotating_nano_hexapod_dynamics_pz}$k_n = 100\,N/\mu m$} +\end{subfigure} +\caption{\label{fig:rotating_nano_hexapod_dynamics}Effect of rotation on the nano-hexapod dynamics. Dashed lines are the plants without rotation, solid lines are plants at maximum rotating velocity (\(\Omega = 60\,\text{rpm}\)), and shaded lines are coupling terms at maximum rotating velocity} \end{figure} + \section{Optimal IFF with High Pass Filter} -Let's apply Integral Force Feedback with an added High Pass Filter to the three nano-hexapods. - -First, let's find the parameters of the IFF controller that yield best simultaneous damping. -The results are shown in Figure \ref{fig:rotating_iff_hpf_nass_optimal_gain}. -The added damping for the soft nano-hexapod is quite low and is limited by the maximum usable gain. - -\begin{figure}[htbp] -\centering -\includegraphics[scale=1]{figs/rotating_iff_hpf_nass_optimal_gain.png} -\caption{\label{fig:rotating_iff_hpf_nass_optimal_gain}Optimal high pass filter cut-off frequency \(\omega_i\) that yields maximum simultaneous damping} -\end{figure} - +In this section, Integral Force Feedback with an added High Pass Filter is applied to the three nano-hexapods. +First, the parameters (\(\omega_i\) and \(g\)) of the IFF controller that yield best simultaneous damping are determined from Figure \ref{fig:rotating_iff_hpf_nass_optimal_gain}. The IFF parameters are chosen as follow: \begin{itemize} -\item for \(k_n = \SI{0.01}{\N\per\mu\m}\): \(\omega_i\) is chosen such that the maximum damping is achieved while the gain is less than half of the maximum gain at which the system is unstable. +\item for \(k_n = \SI{0.01}{\N\per\mu\m}\) (Figure \ref{fig:rotating_iff_hpf_nass_optimal_gain}): \(\omega_i\) is chosen such that the maximum damping is achieved while the gain is less than half of the maximum gain at which the system is unstable. This is done to have some control robustness. -\item for \(k_n = \SI{1}{\N\per\mu\m}\) and \(k_n = \SI{100}{\N\per\mu\m}\): the largest \(\omega_i\) is chosen such that obtained damping is \(\SI{95}{\percent}\) of the maximum achievable damping. +\item for \(k_n = \SI{1}{\N\per\mu\m}\) and \(k_n = \SI{100}{\N\per\mu\m}\) (Figure \ref{fig:rotating_iff_hpf_nass_optimal_gain_md} and \ref{fig:rotating_iff_hpf_nass_optimal_gain_pz}): the largest \(\omega_i\) is chosen such that obtained damping is \(\SI{95}{\percent}\) of the maximum achievable damping. Large \(\omega_i\) is chosen here to limit the loss of compliance and the increase of coupling at low frequency as was shown in Section \ref{sec:rotating_iff_pseudo_int}. \end{itemize} +The obtained IFF parameters and the achievable damping are visually shown by large dots in Figure \ref{fig:rotating_iff_hpf_nass_optimal_gain} and are summarized in Table \ref{tab:rotating_iff_hpf_opt_iff_hpf_params_nass}. -The obtained IFF parameters and the achievable damping are summarized in Table \ref{tab:iff_hpf_opt_iff_hpf_params_nass}. +\begin{figure}[htbp] +\begin{subfigure}{0.33\textwidth} +\begin{center} +\includegraphics[scale=1,width=0.95\linewidth]{figs/rotating_iff_hpf_nass_optimal_gain_vc.png} +\end{center} +\subcaption{\label{fig:rotating_iff_hpf_nass_optimal_gain_vc}$k_n = 0.01\,N/\mu m$} +\end{subfigure} +\begin{subfigure}{0.33\textwidth} +\begin{center} +\includegraphics[scale=1,width=0.95\linewidth]{figs/rotating_iff_hpf_nass_optimal_gain_md.png} +\end{center} +\subcaption{\label{fig:rotating_iff_hpf_nass_optimal_gain_md}$k_n = 1\,N/\mu m$} +\end{subfigure} +\begin{subfigure}{0.33\textwidth} +\begin{center} +\includegraphics[scale=1,width=0.95\linewidth]{figs/rotating_iff_hpf_nass_optimal_gain_pz.png} +\end{center} +\subcaption{\label{fig:rotating_iff_hpf_nass_optimal_gain_pz}$k_n = 100\,N/\mu m$} +\end{subfigure} +\caption{\label{fig:rotating_iff_hpf_nass_optimal_gain}For each value of \(\omega_i\), the maximum damping ratio \(\xi\) is computed (blue) and the corresponding controller gain is shown (in red). The choosen controller parameters used for further analysis are shown by the large dots.} +\end{figure} \begin{table}[htbp] -\caption{\label{tab:iff_hpf_opt_iff_hpf_params_nass}Obtained optimal parameters for the modified IFF controller} +\caption{\label{tab:rotating_iff_hpf_opt_iff_hpf_params_nass}Obtained optimal parameters (\(\omega_i\) and \(g\)) for the modified IFF controller including a high pass filter. The corresponding achievable simultaneous damping of the two modes \(\xi\) is also shown.} \centering -\begin{tabularx}{0.5\linewidth}{lXXX} +\begin{tabularx}{0.4\linewidth}{Xccc} \toprule - & \(\omega_i\) & \(g\) & \(\xi\)\\ +\(k_n\) & \(\omega_i\) & \(g\) & \(\xi_\text{opt}\)\\ \midrule -\(k_n = 0.01\,N/\mu m\) & 7.32 & 51.13 & 0.45\\ -\(k_n = 1\,N/\mu m\) & 39.17 & 426.95 & 0.93\\ -\(k_n = 100\,N/\mu m\) & 499.45 & 3774.63 & 0.94\\ +\(0.01\,N/\mu m\) & 7.3 & 51 & 0.45\\ +\(1\,N/\mu m\) & 39 & 427 & 0.93\\ +\(100\,N/\mu m\) & 500 & 3775 & 0.94\\ \bottomrule \end{tabularx} \end{table} -The Root Locus for all three nano-hexapods are shown in Figure \ref{fig:rotating_root_locus_iff_hpf_nass} with included optimal chosen gains. - -\begin{figure}[htbp] -\centering -\includegraphics[scale=1]{figs/rotating_root_locus_iff_hpf_nass.png} -\caption{\label{fig:rotating_root_locus_iff_hpf_nass}Root Locus for modified IFF with high pass filter. Optimal \(\omega_i\) is used. The three nano-hexapod stiffnesses are compared. The grey line indicates the minimum damping obtained with the optimal chosen control parameters.} -\end{figure} \section{Optimal IFF with Parallel Stiffness} -For each considered nano-hexapod stiffness, the parallel stiffness \(k_p\) is varied from \(k_{p,\text{min}} = m\Omega^2\) (the minimum stiffness to have unconditional stability) to \(k_{p,\text{max}} = k_n\) (the total nano-hexapod stiffness). -In order to keep the overall stiffness constant, the actuator stiffness \(k_a\) is decreased when \(k_p\) is increased: -\begin{equation} -k_a = k_n - k_p -\end{equation} -With \(k_n\) the total nano-hexapod stiffness. +For each considered nano-hexapod stiffness, the parallel stiffness \(k_p\) is varied from \(k_{p,\text{min}} = m\Omega^2\) (the minimum stiffness that yields unconditional stability) to \(k_{p,\text{max}} = k_n\) (the total nano-hexapod stiffness). +In order to keep the overall stiffness constant, the actuator stiffness \(k_a\) is decreased when \(k_p\) is increased (\(k_a = k_n - k_p\), with \(k_n\) the total nano-hexapod stiffness). +A high pass filter is also added to limit the low frequency gain with a cut-off frequency \(\omega_i\) equal to one tenth of the system resonance (\(\omega_i = \omega_0/10\)). -An high pass filter is also added to limit the low frequency gain. -The cut-off frequency \(\omega_i\) is chosen to be one tenth of the system resonance: -\begin{equation} -\omega_i = \omega_0/10 -\end{equation} - -The achievable maximum simultaneous damping of all the modes is computed as a function of the parallel stiffnesses. -The comparison for the nano-hexapod stiffnesses is done in Figure \ref{fig:rotating_iff_kp_nass_optimal_gain}. -It is shown that \textbf{the soft nano-hexapod cannot yield good damping}. +The achievable maximum simultaneous damping of all the modes is computed as a function of the parallel stiffnesses (Figure \ref{fig:rotating_iff_kp_nass_optimal_gain}). +It is shown that the soft nano-hexapod cannot yield good damping. For the two stiff options, the achievable damping starts to significantly decrease when the parallel stiffness is one tenth of the total stiffness \(k_p = k_n/10\). +Let's choose \(k_p = 1\,N/mm\), \(k_p = 0.01\,N/\mu m\) and \(k_p = 1\,N/\mu m\) for the three considered nano-hexapods. +The corresponding optimal controller gains and achievable damping are summarized in Table \ref{tab:rotating_iff_kp_opt_iff_kp_params_nass}. -\begin{figure}[htbp] -\centering -\includegraphics[scale=1]{figs/rotating_iff_kp_nass_optimal_gain.png} -\caption{\label{fig:rotating_iff_kp_nass_optimal_gain}Maximum achievable simultaneous damping with IFF as a function of the parallel stiffness for all three nano-hexapod stiffnesses} -\end{figure} - -Let's choose \(k_p = \SI{e3}{\newton\per\m}\), \(k_p = \SI{e4}{\newton\per\m}\) and \(k_p = \SI{e6}{\newton\per\m}\) for the three considered nano-hexapods respectively based on Figure \ref{fig:rotating_iff_kp_nass_optimal_gain}. - -The corresponding optimal controller gains are shown in Table \ref{tab:iff_kp_opt_iff_kp_params_nass}. - -\begin{table}[htbp] -\caption{\label{tab:iff_kp_opt_iff_kp_params_nass}Obtained optimal parameters for the modified IFF controller} -\centering -\begin{tabularx}{0.4\linewidth}{lXX} +\begin{minipage}[t]{0.49\linewidth} +\begin{center} +\includegraphics[scale=1,width=\linewidth]{figs/rotating_iff_kp_nass_optimal_gain.png} +\captionof{figure}{\label{fig:rotating_iff_kp_nass_optimal_gain}Maximum damping \(\xi\) as a function of the parallel stiffness \(k_p\)} +\end{center} +\end{minipage} +\hfill +\begin{minipage}[b]{0.45\linewidth} +\begin{center} +\captionof{table}{\label{tab:rotating_iff_kp_opt_iff_kp_params_nass}Obtained optimal parameters for the IFF controller when using parallel stiffnesses} +\begin{tabularx}{\linewidth}{Xccc} \toprule - & \(g\) & \(\xi_{\text{opt}}\)\\ +\(k_n\) & \(k_p\) & \(g\) & \(\xi_{\text{opt}}\)\\ \midrule -\(k_n = 0.01\,N/\mu m\) & 47.9 & 0.44\\ -\(k_n = 1\,N/\mu m\) & 465.57 & 0.97\\ -\(k_n = 100\,N/\mu m\) & 4624.25 & 1.0\\ +\(0.01\,N/\mu m\) & \(1\,N/mm\) & 47.9 & 0.44\\ +\(1\,N/\mu m\) & \(0.01\,N/\mu m\) & 465.57 & 0.97\\ +\(100\,N/\mu m\) & \(1\,N/\mu m\) & 4624.25 & 0.99\\ \bottomrule \end{tabularx} -\end{table} +\end{center} +\end{minipage} -The root locus for the three nano-hexapod with parallel stiffnesses are shown in Figure \ref{fig:rotating_root_locus_iff_kp_nass}. - -\begin{important} -Similarly to what was found with the IFF and added High Pass Filter: -\begin{itemize} -\item the stiff nano-hexapod is less affected by the rotation than the soft one -\item the achievable damping is much larger with the stiff nano-hexapods -\end{itemize} -\end{important} - -\begin{figure}[htbp] -\centering -\includegraphics[scale=1]{figs/rotating_root_locus_iff_kp_nass.png} -\caption{\label{fig:rotating_root_locus_iff_kp_nass}Root Locus for optimal parameters (IFF + \(k_p\) strategy) - Comparison of attainable damping with the three nano-hexapod stiffnesses} -\end{figure} \section{Optimal Relative Motion Control} +For each considered nano-hexapod stiffness, relative damping control is applied and the achievable damping ratio as a function of the controller gain is computed (Figure \ref{fig:rotating_rdc_optimal_gain}). +The gain is chosen is chosen such that 99\% of modal damping is obtained (obtained gains are summarized in Table \ref{tab:rotating_rdc_opt_params_nass}). -For each considered nano-hexapod stiffness, relative damping control is applied and the achievable damping ratio as a function of the controller gain is shown in Figure \ref{fig:rotating_rdc_optimal_gain}. +\begin{minipage}[t]{0.49\linewidth} +\begin{center} +\includegraphics[scale=1,width=\linewidth]{figs/rotating_rdc_optimal_gain.png} +\captionof{figure}{\label{fig:rotating_rdc_optimal_gain}Maximum damping \(\xi\) as a function of the RDC gain \(g\)} +\end{center} +\end{minipage} +\hfill +\begin{minipage}[b]{0.45\linewidth} +\begin{center} +\captionof{table}{\label{tab:rotating_rdc_opt_params_nass}Obtained optimal parameters for the RDC} +\begin{tabularx}{0.8\linewidth}{Xcc} +\toprule +\(k_n\) & \(g\) & \(\xi_{\text{opt}}\)\\ +\midrule +\(0.01\,N/\mu m\) & 1600 & 0.99\\ +\(1\,N/\mu m\) & 8200 & 0.99\\ +\(100\,N/\mu m\) & 80000 & 0.99\\ +\bottomrule +\end{tabularx} +\end{center} +\end{minipage} -\begin{figure}[htbp] -\centering -\includegraphics[scale=1]{figs/rotating_rdc_optimal_gain.png} -\caption{\label{fig:rotating_rdc_optimal_gain}Achievable simultaneous damping with ``Relative Damping Control'' as a function of the controller gain for all three nano-hexapod stiffnesses} -\end{figure} - -The gain is chosen is chosen such that 99\% of modal damping is obtained. -The root locus for all three nano-hexapod stiffnesses are shown in Figure \ref{fig:rotating_root_locus_rdc_nass}. - -\begin{important} -Relative damping control is much less impacted by gyroscopic effects. -It can be easily applied on the nano-hexapod with and without rotation without much differences. -\end{important} - -\begin{figure}[htbp] -\centering -\includegraphics[scale=1]{figs/rotating_root_locus_rdc_nass.png} -\caption{\label{fig:rotating_root_locus_rdc_nass}Root Locus for optimal parameters - Comparison of attainable damping with the soft and moderately stiff nano-hexapods} -\end{figure} \section{Comparison of the obtained damped plants} Let's now compare the obtained damped plants for the three active damping techniques applied on the three nano-hexapod stiffnesses (Figure \ref{fig:rotating_nass_damped_plant_comp}). @@ -880,16 +848,33 @@ Let's now compare the obtained damped plants for the three active damping techni Similarly to what was concluded in previous analysis: \begin{itemize} \item IFF adds coupling below the resonance frequency as compared to the open-loop and RDC cases -\item Add three methods are yielding good damping, except for IFF applied on the soft nano-hexapod where things are more complicated +\item All three methods are yielding good damping, except for IFF applied on the soft nano-hexapod where things are more complicated \item Coupling is smaller for stiff nano-hexapods \end{itemize} \end{important} \begin{figure}[htbp] -\centering -\includegraphics[scale=1]{figs/rotating_nass_damped_plant_comp.png} +\begin{subfigure}{0.33\textwidth} +\begin{center} +\includegraphics[scale=1,width=0.95\linewidth]{figs/rotating_nass_damped_plant_comp_vc.png} +\end{center} +\subcaption{\label{fig:rotating_nass_damped_plant_comp_vc}$k_n = 0.01\,N/\mu m$} +\end{subfigure} +\begin{subfigure}{0.33\textwidth} +\begin{center} +\includegraphics[scale=1,width=0.95\linewidth]{figs/rotating_nass_damped_plant_comp_md.png} +\end{center} +\subcaption{\label{fig:rotating_nass_damped_plant_comp_md}$k_n = 1\,N/\mu m$} +\end{subfigure} +\begin{subfigure}{0.33\textwidth} +\begin{center} +\includegraphics[scale=1,width=0.95\linewidth]{figs/rotating_nass_damped_plant_comp_pz.png} +\end{center} +\subcaption{\label{fig:rotating_nass_damped_plant_comp_pz}$k_n = 100\,N/\mu m$} +\end{subfigure} \caption{\label{fig:rotating_nass_damped_plant_comp}Comparison of the damped plants (direct and coupling terms) for the three proposed active damping techniques (IFF with HPF, IFF with \(k_p\) and RDC) applied on the three nano-hexapod stiffnesses. \(\Omega = 60\,\text{rmp}\) and \(m_n + m_s = \SI{16}{\kg}\).} \end{figure} + \chapter{Nano-Active-Stabilization-System with rotation} \label{sec:rotating_nass} Up until now, the model used consisted of an infinitely stiff vertical rotating stage with a X-Y suspended stage. @@ -915,6 +900,7 @@ A payload is rigidly fixed to the nano-hexapod and the \(x,y\) motion of the pay \includegraphics[scale=1]{figs/rotating_nass_model.png} \caption{\label{fig:rotating_nass_model}3D view of the Nano-Active-Stabilization-System model.} \end{figure} + \section{System dynamics} The dynamics of the undamped and damped plants are identified. @@ -935,18 +921,51 @@ It can be observed on Figure \ref{fig:rotating_nass_plant_comp_stiffness} that: \end{important} \begin{figure}[htbp] -\centering -\includegraphics[scale=1]{figs/rotating_nass_plant_comp_stiffness.png} +\begin{subfigure}{0.33\textwidth} +\begin{center} +\includegraphics[scale=1,width=0.95\linewidth]{figs/rotating_nass_plant_comp_stiffness_vc.png} +\end{center} +\subcaption{\label{fig:rotating_nass_plant_comp_stiffness_vc}$k_n = 0.01\,N/\mu m$} +\end{subfigure} +\begin{subfigure}{0.33\textwidth} +\begin{center} +\includegraphics[scale=1,width=0.95\linewidth]{figs/rotating_nass_plant_comp_stiffness_md.png} +\end{center} +\subcaption{\label{fig:rotating_nass_plant_comp_stiffness_md}$k_n = 1\,N/\mu m$} +\end{subfigure} +\begin{subfigure}{0.33\textwidth} +\begin{center} +\includegraphics[scale=1,width=0.95\linewidth]{figs/rotating_nass_plant_comp_stiffness_pz.png} +\end{center} +\subcaption{\label{fig:rotating_nass_plant_comp_stiffness_pz}$k_n = 100\,N/\mu m$} +\end{subfigure} \caption{\label{fig:rotating_nass_plant_comp_stiffness}Bode plot of the transfer function from nano-hexapod actuator to measured motion by the external metrology} \end{figure} To confirm that the coupling is smaller when the stiffness of the nano-hexapod is increase, the \emph{coupling ratio} for the three nano-hexapod stiffnesses are shown in Figure \ref{fig:rotating_nass_plant_coupling_comp}. \begin{figure}[htbp] -\centering -\includegraphics[scale=1]{figs/rotating_nass_plant_coupling_comp.png} +\begin{subfigure}{0.33\textwidth} +\begin{center} +\includegraphics[scale=1,width=0.95\linewidth]{figs/rotating_nass_plant_coupling_comp_vc.png} +\end{center} +\subcaption{\label{fig:rotating_nass_plant_coupling_comp_vc}$k_n = 0.01\,N/\mu m$} +\end{subfigure} +\begin{subfigure}{0.33\textwidth} +\begin{center} +\includegraphics[scale=1,width=0.95\linewidth]{figs/rotating_nass_plant_coupling_comp_md.png} +\end{center} +\subcaption{\label{fig:rotating_nass_plant_coupling_comp_md}$k_n = 1\,N/\mu m$} +\end{subfigure} +\begin{subfigure}{0.33\textwidth} +\begin{center} +\includegraphics[scale=1,width=0.95\linewidth]{figs/rotating_nass_plant_coupling_comp_pz.png} +\end{center} +\subcaption{\label{fig:rotating_nass_plant_coupling_comp_pz}$k_n = 100\,N/\mu m$} +\end{subfigure} \caption{\label{fig:rotating_nass_plant_coupling_comp}Coupling ratio for the proposed active damping techniques evaluated for the three nano-hexapod stiffnesses} \end{figure} + \section{Effect of disturbances} The effect of three disturbances are considered: @@ -962,7 +981,7 @@ Conclusions are similar than with the uniaxial (non-rotating) model: \item Regarding the effect of floor motion and forces applied on the payload: \begin{itemize} \item The stiffer, the better (magnitudes are lower for the right curves, Figures \ref{fig:rotating_nass_effect_floor_motion} and \ref{fig:rotating_nass_effect_direct_forces}) -\item Integral Force Feedback degrades the performances at low frequency compared to relative damping control +\item Integral Force Feedback degrades the performance at low frequency compared to relative damping control \end{itemize} \item Regarding the effect of micro-station vibrations: \begin{itemize} @@ -972,30 +991,79 @@ Conclusions are similar than with the uniaxial (non-rotating) model: \end{important} \begin{figure}[htbp] -\centering -\includegraphics[scale=1]{figs/rotating_nass_effect_floor_motion.png} +\begin{subfigure}{0.33\textwidth} +\begin{center} +\includegraphics[scale=1,width=0.95\linewidth]{figs/rotating_nass_effect_floor_motion_vc.png} +\end{center} +\subcaption{\label{fig:rotating_nass_effect_floor_motion_vc}$k_n = 0.01\,N/\mu m$} +\end{subfigure} +\begin{subfigure}{0.33\textwidth} +\begin{center} +\includegraphics[scale=1,width=0.95\linewidth]{figs/rotating_nass_effect_floor_motion_md.png} +\end{center} +\subcaption{\label{fig:rotating_nass_effect_floor_motion_md}$k_n = 1\,N/\mu m$} +\end{subfigure} +\begin{subfigure}{0.33\textwidth} +\begin{center} +\includegraphics[scale=1,width=0.95\linewidth]{figs/rotating_nass_effect_floor_motion_pz.png} +\end{center} +\subcaption{\label{fig:rotating_nass_effect_floor_motion_pz}$k_n = 100\,N/\mu m$} +\end{subfigure} \caption{\label{fig:rotating_nass_effect_floor_motion}Effect of Floor motion on the position error - Comparison of active damping techniques for the three nano-hexapod stiffnesses} \end{figure} \begin{figure}[htbp] -\centering -\includegraphics[scale=1]{figs/rotating_nass_effect_stage_vibration.png} +\begin{subfigure}{0.33\textwidth} +\begin{center} +\includegraphics[scale=1,width=0.95\linewidth]{figs/rotating_nass_effect_stage_vibration_vc.png} +\end{center} +\subcaption{\label{fig:rotating_nass_effect_stage_vibration_vc}$k_n = 0.01\,N/\mu m$} +\end{subfigure} +\begin{subfigure}{0.33\textwidth} +\begin{center} +\includegraphics[scale=1,width=0.95\linewidth]{figs/rotating_nass_effect_stage_vibration_md.png} +\end{center} +\subcaption{\label{fig:rotating_nass_effect_stage_vibration_md}$k_n = 1\,N/\mu m$} +\end{subfigure} +\begin{subfigure}{0.33\textwidth} +\begin{center} +\includegraphics[scale=1,width=0.95\linewidth]{figs/rotating_nass_effect_stage_vibration_pz.png} +\end{center} +\subcaption{\label{fig:rotating_nass_effect_stage_vibration_pz}$k_n = 100\,N/\mu m$} +\end{subfigure} \caption{\label{fig:rotating_nass_effect_stage_vibration}Effect of micro-station vibrations on the position error - Comparison of active damping techniques for the three nano-hexapod stiffnesses} \end{figure} + \begin{figure}[htbp] -\centering -\includegraphics[scale=1]{figs/rotating_nass_effect_direct_forces.png} +\begin{subfigure}{0.33\textwidth} +\begin{center} +\includegraphics[scale=1,width=0.95\linewidth]{figs/rotating_nass_effect_direct_forces_vc.png} +\end{center} +\subcaption{\label{fig:rotating_nass_effect_direct_forces_vc}$k_n = 0.01\,N/\mu m$} +\end{subfigure} +\begin{subfigure}{0.33\textwidth} +\begin{center} +\includegraphics[scale=1,width=0.95\linewidth]{figs/rotating_nass_effect_direct_forces_md.png} +\end{center} +\subcaption{\label{fig:rotating_nass_effect_direct_forces_md}$k_n = 1\,N/\mu m$} +\end{subfigure} +\begin{subfigure}{0.33\textwidth} +\begin{center} +\includegraphics[scale=1,width=0.95\linewidth]{figs/rotating_nass_effect_direct_forces_pz.png} +\end{center} +\subcaption{\label{fig:rotating_nass_effect_direct_forces_pz}$k_n = 100\,N/\mu m$} +\end{subfigure} \caption{\label{fig:rotating_nass_effect_direct_forces}Effect of sample forces on the position error - Comparison of active damping techniques for the three nano-hexapod stiffnesses} \end{figure} -\chapter{Conclusion} +\chapter*{Conclusion} In this study, the gyroscopic effects induced by the spindle's rotation have been studied using a spindle model (Section \ref{sec:rotating_system_description}). Decentralized IFF with pure integrators was shown to be unstable when applied to rotating platforms (Section \ref{sec:rotating_iff_pure_int}). Two modifications of the classical IFF control have been proposed to overcome this issue. -The first modification concerns the controller and consists of adding an high pass filter to the pure integrators. -This is equivalent as to moving the controller pole to the left along the real axis. +The first modification concerns the controller and consists of adding a high pass filter to the pure integrators. +This is equivalent to moving the controller pole to the left along the real axis. This allows the closed loop system to be stable up to some value of the controller gain (Section \ref{sec:rotating_iff_pseudo_int}). The second proposed modification concerns the mechanical system. @@ -1008,5 +1076,8 @@ While having very different implementations, both proposed modifications were fo Then, this study has been applied to a rotating system that corresponds to the nano-hexapod parameters (Section \ref{sec:rotating_nano_hexapod}). To be closer to the real system dynamics, the limited compliance of the micro-station has been taken into account. Results show that the two proposed IFF modifications can be applied for the NASS even in the presence of spindle rotation. + \printbibliography[heading=bibintoc,title={Bibliography}] + +\printglossaries \end{document} diff --git a/preamble.tex b/preamble.tex index cf43ea7..00c5fc8 100644 --- a/preamble.tex +++ b/preamble.tex @@ -1,137 +1,23 @@ -\usepackage{float} +\usepackage[ % + acronym, % Separate acronyms and glossary + toc, % appear in ToC + automake, % auto-use the makeglossaries command (requires shell-escape) + nonumberlist, % don't back reference pages + nogroupskip, % don't group by letter + nopostdot % don't add a dot at the end of each element +]{glossaries} -\usepackage{caption,tabularx,booktabs} -\usepackage{bm} +\usepackage[stylemods=longextra]{glossaries-extra} -\usepackage{xpatch} % Recommanded for biblatex -\usepackage[ % use biblatex for bibliography - backend=biber, % use biber backend (bibtex replacement) or bibtex - style=ieee, % bib style - citestyle=numeric-comp, % - 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-\usepackage{inconsolata} - -\usepackage[newfloat=true, chapter]{minted} -\usemintedstyle{autumn} - -\setminted{frame=lines,breaklines=true,tabsize=4,fontsize=\scriptsize,autogobble=true,labelposition=topline,bgcolor=my-pale-grey} -\setminted[matlab]{label=Matlab} -\setminted[latex]{label=LaTeX} -\setminted[bash]{label=Bash} -\setminted[python]{label=Python} -\setminted[text]{label=Results} -\setminted[md]{label=Org Mode} - -\setmintedinline{fontsize=\normalsize,bgcolor=my-pale-grey} - -\usepackage[most]{tcolorbox} - -\tcbuselibrary{minted} - -\newtcolorbox{seealso}{ enhanced,breakable,colback=my-pale-grey,colframe=my-grey,fonttitle=\bfseries,title=See Also} -\newtcolorbox{hint}{ enhanced,breakable,colback=my-pale-grey,colframe=my-grey,fonttitle=\bfseries,title=Hint} -\newtcolorbox{definition}{enhanced,breakable,colback=my-pale-red, colframe=my-red, fonttitle=\bfseries,title=Definition} -\newtcolorbox{important}{ enhanced,breakable,colback=my-pale-red, colframe=my-red, fonttitle=\bfseries,title=Important} -\newtcolorbox{exampl}[1][]{ enhanced,breakable,colback=my-pale-green,colframe=my-green,fonttitle=\bfseries,title=Example,#1} -\newtcolorbox{exercice}{ enhanced,breakable,colback=my-pale-yellow,colframe=my-yellow,fonttitle=\bfseries,title=Exercice} -\newtcolorbox{question}{ enhanced,breakable,colback=my-pale-yellow,colframe=my-yellow,fonttitle=\bfseries,title=Question} -\newtcolorbox{answer}{ enhanced,breakable,colback=my-pale-turq,colframe=my-turq,fonttitle=\bfseries,title=Answer} -\newtcolorbox{summary}{ enhanced,breakable,colback=my-pale-blue,colframe=my-blue,fonttitle=\bfseries,title=Summary} -\newtcolorbox{note}{ enhanced,breakable,colback=my-pale-blue,colframe=my-blue,fonttitle=\bfseries,title=Note} -\newtcolorbox{caution}{ enhanced,breakable,colback=my-pale-orange,colframe=my-orange,fonttitle=\bfseries,title=Caution} -\newtcolorbox{warning}{ enhanced,breakable,colback=my-pale-orange,colframe=my-orange,fonttitle=\bfseries,title=Warning} - -\newtcbox{\mywboxtext}{on line,colback=my-pale-blue,colframe=my-blue,size=fbox} -\renewcommand{\boxed}[1]{\mywboxtext{$#1$}} - -\newtcolorbox{my-quote}[1]{% - colback=my-pale-grey, - grow to right by=-10mm, - grow to left by=-10mm, - boxrule=0pt, - boxsep=0pt, - breakable, - enhanced jigsaw, - borderline west={4pt}{0pt}{my-grey}} - -\renewenvironment{quote}{\begin{my-quote}}{\end{my-quote}} - -\newtcolorbox{my-verse}[1]{% - colback=my-pale-grey, - grow to right by=-10mm, - grow to left by=-10mm, - boxrule=0pt, - boxsep=0pt, - breakable, - enhanced jigsaw, - borderline west={4pt}{0pt}{my-grey}} - -\renewenvironment{verse}{\begin{my-verse}}{\end{my-verse}} - -\usepackage{environ}% http://ctan.org/pkg/environ -\NewEnviron{aside}{% - \marginpar{\BODY} -} - -\renewenvironment{verbatim}{\VerbatimEnvironment\begin{minted}[]{text}}{\end{minted}} - -\usepackage{soul} -\sethlcolor{my-pale-grey} - -\let\OldTexttt\texttt -\renewcommand{\texttt}[1]{{\ttfamily\hl{\mbox{\,#1\,}}}} - -\makeatletter -\preto\Gin@extensions{png,} -\DeclareGraphicsRule{.png}{pdf}{.pdf}{\noexpand\Gin@base.pdf} -\preto\Gin@extensions{gif,} -\DeclareGraphicsRule{.gif}{png}{.png}{\noexpand\Gin@base.png} -\makeatother - -\usepackage{hyperref} -\hypersetup{ - colorlinks = true, - allcolors = my-blue -} - -\usepackage{hypcap} +\usepackage{siunitx} +\usepackage{mathtools} +\usepackage{amssymb} diff --git a/preamble_extra.tex b/preamble_extra.tex new file mode 100644 index 0000000..98cfc04 --- /dev/null +++ b/preamble_extra.tex @@ -0,0 +1,134 @@ +\usepackage{float} +\usepackage{enumitem} + +\usepackage{caption,tabularx,booktabs} +\usepackage{bm} + +\usepackage{xpatch} % Recommanded for biblatex +\usepackage[ % use biblatex for bibliography + backend=biber, % use biber backend (bibtex replacement) or bibtex + style=ieee, % bib style + hyperref=true, % activate hyperref support + backref=true, % activate backrefs + isbn=false, % don't show isbn tags + url=false, % don't show url tags + doi=false, % don't show doi tags + urldate=long, % display type for dates + maxnames=3, % + minnames=1, % + maxbibnames=5, % + minbibnames=3, % + maxcitenames=2, % + mincitenames=1 % + ]{biblatex} + +\setlength\bibitemsep{1.1\itemsep} + +\usepackage{caption} +\usepackage{subcaption} + +\captionsetup[figure]{labelfont=bf} +\captionsetup[subfigure]{labelfont=bf} +\captionsetup[listing]{labelfont=bf} +\captionsetup[table]{labelfont=bf} + +\usepackage{xcolor} + +\definecolor{my-blue}{HTML}{6b7adb} +\definecolor{my-pale-blue}{HTML}{e6e9f9} +\definecolor{my-red}{HTML}{db6b6b} +\definecolor{my-pale-red}{HTML}{f9e6e6} +\definecolor{my-green}{HTML}{6bdbb6} +\definecolor{my-pale-green}{HTML}{e6f9f3} +\definecolor{my-yellow}{HTML}{dbd26b} +\definecolor{my-pale-yellow}{HTML}{f9f7e6} +\definecolor{my-orange}{HTML}{dba76b} +\definecolor{my-pale-orange}{HTML}{f9f0e6} +\definecolor{my-grey}{HTML}{a3a3a3} +\definecolor{my-pale-grey}{HTML}{f0f0f0} +\definecolor{my-turq}{HTML}{6bc7db} +\definecolor{my-pale-turq}{HTML}{e6f6f9} + +\usepackage{inconsolata} + +\usepackage[newfloat=true, chapter]{minted} +\usemintedstyle{autumn} + +\setminted{frame=lines,breaklines=true,tabsize=4,fontsize=\scriptsize,autogobble=true,labelposition=topline,bgcolor=my-pale-grey} +\setminted[matlab]{label=Matlab} +\setminted[latex]{label=LaTeX} +\setminted[bash]{label=Bash} +\setminted[python]{label=Python} +\setminted[text]{label=Results} +\setminted[md]{label=Org Mode} + +\setmintedinline{fontsize=\normalsize,bgcolor=my-pale-grey} + +\usepackage[most]{tcolorbox} + +\tcbuselibrary{minted} + +\newtcolorbox{seealso}{ enhanced,breakable,colback=my-pale-grey,colframe=my-grey,fonttitle=\bfseries,title=See Also} +\newtcolorbox{hint}{ enhanced,breakable,colback=my-pale-grey,colframe=my-grey,fonttitle=\bfseries,title=Hint} +\newtcolorbox{definition}{enhanced,breakable,colback=my-pale-red, colframe=my-red, fonttitle=\bfseries,title=Definition} +\newtcolorbox{important}{ enhanced,breakable,colback=my-pale-red, colframe=my-red, fonttitle=\bfseries,title=Important} +\newtcolorbox{exampl}[1][]{ enhanced,breakable,colback=my-pale-green,colframe=my-green,fonttitle=\bfseries,title=Example,#1} +\newtcolorbox{exercice}{ enhanced,breakable,colback=my-pale-yellow,colframe=my-yellow,fonttitle=\bfseries,title=Exercice} +\newtcolorbox{question}{ enhanced,breakable,colback=my-pale-yellow,colframe=my-yellow,fonttitle=\bfseries,title=Question} +\newtcolorbox{answer}{ enhanced,breakable,colback=my-pale-turq,colframe=my-turq,fonttitle=\bfseries,title=Answer} +\newtcolorbox{summary}{ enhanced,breakable,colback=my-pale-blue,colframe=my-blue,fonttitle=\bfseries,title=Summary} +\newtcolorbox{note}{ enhanced,breakable,colback=my-pale-blue,colframe=my-blue,fonttitle=\bfseries,title=Note} +\newtcolorbox{caution}{ enhanced,breakable,colback=my-pale-orange,colframe=my-orange,fonttitle=\bfseries,title=Caution} +\newtcolorbox{warning}{ enhanced,breakable,colback=my-pale-orange,colframe=my-orange,fonttitle=\bfseries,title=Warning} + +\newtcolorbox{my-quote}[1]{% + colback=my-pale-grey, + grow to right by=-10mm, + grow to left by=-10mm, + boxrule=0pt, + boxsep=0pt, + breakable, + enhanced jigsaw, + borderline west={4pt}{0pt}{my-grey}} + +\renewenvironment{quote}{\begin{my-quote}}{\end{my-quote}} + +\newtcolorbox{my-verse}[1]{% + colback=my-pale-grey, + grow to right by=-10mm, + grow to left by=-10mm, + boxrule=0pt, + boxsep=0pt, + breakable, + enhanced jigsaw, + borderline west={4pt}{0pt}{my-grey}} + +\renewenvironment{verse}{\begin{my-verse}}{\end{my-verse}} + +\usepackage{environ}% http://ctan.org/pkg/environ +\NewEnviron{aside}{% + \marginpar{\BODY} +} + +\renewenvironment{verbatim}{\VerbatimEnvironment\begin{minted}[]{text}}{\end{minted}} + +\usepackage{soul} +\sethlcolor{my-pale-grey} + +\let\OldTexttt\texttt +\renewcommand{\texttt}[1]{{\ttfamily\hl{\mbox{\,#1\,}}}} + +\makeatletter +\preto\Gin@extensions{png,} +\DeclareGraphicsRule{.png}{pdf}{.pdf}{\noexpand\Gin@base.pdf} +\preto\Gin@extensions{gif,} +\DeclareGraphicsRule{.gif}{png}{.png}{\noexpand\Gin@base.png} +\makeatother + +\usepackage{hyperref} +\hypersetup{ + colorlinks = true, + allcolors = my-blue +} + +\usepackage{hypcap}