#+TITLE: Simscape Model - Micro Station :DRAWER: #+LANGUAGE: en #+EMAIL: dehaeze.thomas@gmail.com #+AUTHOR: Dehaeze Thomas #+HTML_LINK_HOME: ../index.html #+HTML_LINK_UP: ../index.html #+HTML_HEAD: #+HTML_HEAD: #+BIND: org-latex-image-default-option "scale=1" #+BIND: org-latex-image-default-width "" #+LaTeX_CLASS: scrreprt #+LaTeX_CLASS_OPTIONS: [a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc] #+LATEX_HEADER: \input{preamble.tex} #+LATEX_HEADER_EXTRA: \input{preamble_extra.tex} #+LATEX_HEADER_EXTRA: \bibliography{simscape-micro-station.bib} #+BIND: org-latex-bib-compiler "biber" #+PROPERTY: header-args:matlab :session *MATLAB* #+PROPERTY: header-args:matlab+ :comments org #+PROPERTY: header-args:matlab+ :exports none #+PROPERTY: header-args:matlab+ :results none #+PROPERTY: header-args:matlab+ :eval no-export #+PROPERTY: header-args:matlab+ :noweb yes #+PROPERTY: header-args:matlab+ :mkdirp yes #+PROPERTY: header-args:matlab+ :output-dir figs #+PROPERTY: header-args:matlab+ :tangle no #+PROPERTY: header-args:latex :headers '("\\usepackage{tikz}" "\\usepackage{import}" "\\import{$HOME/Cloud/tikz/org/}{config.tex}") #+PROPERTY: header-args:latex+ :imagemagick t :fit yes #+PROPERTY: header-args:latex+ :iminoptions -scale 100% -density 150 #+PROPERTY: header-args:latex+ :imoutoptions -quality 100 #+PROPERTY: header-args:latex+ :results file raw replace #+PROPERTY: header-args:latex+ :buffer no #+PROPERTY: header-args:latex+ :tangle no #+PROPERTY: header-args:latex+ :eval no-export #+PROPERTY: header-args:latex+ :exports results #+PROPERTY: header-args:latex+ :mkdirp yes #+PROPERTY: header-args:latex+ :output-dir figs #+PROPERTY: header-args:latex+ :post pdf2svg(file=*this*, ext="png") #+latex: \clearpage :END: * Build :noexport: #+NAME: startblock #+BEGIN_SRC emacs-lisp :results none :tangle no (add-to-list 'org-latex-classes '("scrreprt" "\\documentclass{scrreprt}" ("\\chapter{%s}" . "\\chapter*{%s}") ("\\section{%s}" . "\\section*{%s}") ("\\subsection{%s}" . "\\subsection*{%s}") ("\\paragraph{%s}" . "\\paragraph*{%s}") )) ;; Remove automatic org heading labels (defun my-latex-filter-removeOrgAutoLabels (text backend info) "Org-mode automatically generates labels for headings despite explicit use of `#+LABEL`. This filter forcibly removes all automatically generated org-labels in headings." (when (org-export-derived-backend-p backend 'latex) (replace-regexp-in-string "\\\\label{sec:org[a-f0-9]+}\n" "" text))) (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) ;; Do not include the subtitle inside the title (setq org-latex-subtitle-separate t) (setq org-latex-subtitle-format "\\subtitle{%s}") (setq org-export-before-parsing-hook '(org-ref-glossary-before-parsing org-ref-acronyms-before-parsing)) #+END_SRC * Notes :noexport: ** Notes Prefix is =ustation= From modal analysis: validation of the multi-body model. *Goals*: - *Modelling of the micro-station*: Kinematics + Dynamics + Disturbances - Kinematics of each stage - Modelling: solid bodies + joints. Show what is used for each stage - Correlation with the dynamical measurements - Inclusion of disturbances (correlation with measurements) *Notes*: - Do not talk about Nano-Hexapod yet - Do not talk about external metrology - Therefore, do not talk about computation of the sample position / error Based on: - [X] [[file:~/Cloud/work-projects/ID31-NASS/matlab/nass-simscape/org/kinematics.org][kinematics]]: compute sample's motion from each stage motion - [X] [[file:~/Cloud/work-projects/ID31-NASS/matlab/nass-simscape/org/simscape_subsystems.org][simscape_subsystems]]: general presentation of the micro-station. Used model: solid body + joints. Presentation of each stage. - [X] [[file:~/Cloud/work-projects/ID31-NASS/documents/work-package-1/work-package-1.org::*Specification of requirements][Specification of requirements]] - [X] [[file:~/Cloud/work-projects/ID31-NASS/matlab/nass-simscape/org/identification.org][identification]]: comparison of measurements and simscape model (not so good?) - [ ] file:/home/thomas/Cloud/meetings/esrf-meetings/2018-04-24-Simscape-Model/2018-04-24-Simscape-Model.pdf - [ ] [[file:~/Cloud/work-projects/ID31-NASS/matlab/nass-simscape/org/experiments.org][experiments]]: simulation of experiments with the Simscape model - [ ] Disturbances: Similar to what was done for the uniaxial model (the same?) - [[file:~/Cloud/work-projects/ID31-NASS/matlab/nass-simscape/org/disturbances.org::+TITLE: Identification of the disturbances]] - [ ] Measurement of disturbances / things that will have to be corrected using the NASS: - [ ] [[file:~/Cloud/work-projects/ID31-NASS/matlab/nass-measurements/static-to-dynamic/index.org]] - [ ] [[file:~/Cloud/work-projects/ID31-NASS/matlab/nass-measurements/disturbance-control-system/index.org]] - [ ] [[file:~/Cloud/work-projects/ID31-NASS/matlab/nass-measurements/disturbance-sr-rz/index.org]] - [ ] [[file:~/Cloud/work-projects/ID31-NASS/matlab/nass-measurements/ground-motion/index.org]] - [ ] [[file:~/Cloud/work-projects/ID31-NASS/matlab/nass-measurements/static-spindle/index.org]] - [ ] Check [[file:~/Cloud/work-projects/ID31-NASS/specifications/id-31-spindle-meas][this directory]] and [[file:~/Cloud/work-projects/ID31-NASS/specifications/stage-by-stage][this directory]] ** DONE [#B] Put colors for each different stage CLOSED: [2024-10-30 Wed 14:07] ** DONE [#B] Delete Gravity compensation Stage CLOSED: [2024-10-30 Wed 13:51] ** DONE [#B] Maybe make a simpler Simscape model for this report CLOSED: [2024-10-30 Wed 13:51] ** DONE [#A] Open the Simscape model and verify it all works CLOSED: [2024-10-30 Wed 13:33] SCHEDULED: <2024-10-30 Wed> #+begin_src matlab initializeSimscapeConfiguration('gravity', false); initializeLoggingConfiguration('log', 'none'); initializeGround( 'type', 'rigid'); initializeGranite( 'type', 'flexible'); initializeTy( 'type', 'flexible'); initializeRy( 'type', 'flexible'); initializeRz( 'type', 'flexible'); initializeMicroHexapod('type', 'flexible'); initializeReferences(); initializeDisturbances('enable', false); #+end_src #+begin_src matlab % initializeAxisc( 'type', 'flexible'); % initializeMirror( 'type', 'none'); % initializeNanoHexapod( 'type', 'none'); % initializeSample( 'type', 'none'); initializeController( 'type', 'open-loop'); #+end_src ** TODO [#B] Make good "init" for the Simscape model ** TODO [#B] Just keep smallest number of variants for each stage - [ ] none - [ ] rigid - [ ] flexible - [X] Init => Removed - [X] modal analysis => Removed ** TODO [#A] Verify that we get "correct" compliance SCHEDULED: <2024-10-30 Wed> - [ ] Find the compliance measurements - [ ] See if it matches somehow the current model - [ ] If not, see if model parameters can be tuned to have better match For instance from values here: file:/home/thomas/Cloud/meetings/esrf-meetings/2018-04-24-Simscape-Model/2018-04-24-Simscape-Model.pdf * Introduction :ignore: #+name: tab:ustation_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: | =ustation_1_.m= | * Micro-Station Kinematics :PROPERTIES: :HEADER-ARGS:matlab+: :tangle matlab/.m :END: <> ** Introduction :ignore: [[file:~/Cloud/work-projects/ID31-NASS/matlab/nass-simscape/org/kinematics.org]] # - Small overview of each stage and associated stiffnesses / inertia # - schematic that shows to considered DoF # - import from CAD ** Matlab Init :noexport:ignore: #+begin_src matlab %% .m #+end_src #+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) <> #+end_src #+begin_src matlab :exports none :results silent :noweb yes <> #+end_src #+begin_src matlab :tangle no :noweb yes <> #+end_src #+begin_src matlab :eval no :noweb yes <> #+end_src #+begin_src matlab :noweb yes <> #+end_src #+begin_src matlab :noweb yes <> #+end_src ** Motion Stages *** Translation Stage *** Tilt Stage *** Spindle *** Micro-Hexapod ** Specification for each stage <> For each stage, after a quick presentation, the specifications (stroke, precision, ...) of the stage, the metrology used with that stage and the parasitic motions associated to it are given. *** Translation along Y **** Presentation This is the first stage, it is fixed directly on the granite. In order to minimize the parasitic motions, two cylinders are used to guide the stage. The motor used to drive the stage is one linear motor. The 3D model of this translation stage is shown figure ref:fig:stage_translation. #+name: fig:stage_translation #+caption: Translation stage [[./figures/stages/stage_translation.pdf]] **** Specifications #+attr_latex: :align |c|c|c|c|c| :placement [!htpb] #+CAPTION: Specifications for the translation stage along Y #+NAME: fig:spec-t-y |--------+------------------------+--------------------------+--------------+--------------| | Motion | Mode | Stroke | Repetability | MIM[fn:3] | |--------+------------------------+--------------------------+--------------+--------------| | $T_y$ | P[fn:1]/S[fn:2] | $\pm5 mm$ | $0.04 \mu m$ | $0.02 \mu m$ | | | rotation stage along Y | ($\pm10 mm$ if possible) | | | |--------+------------------------+--------------------------+--------------+--------------| **** Metrology A linear digital encoder is used to measure the displacement of the translation stage along the y axis. **** Parasitic motions #+attr_latex: :align |c|c|c| :placement [!htpb] #+CAPTION: Parasitic motions for the translation stage along Y #+NAME: fig:parasitic-motions-t-y |--------------------+-----------------------+------------------------------| | Axial Motion ($y$) | Radial Motion ($y-z$) | Rotation motion ($\theta-z$) | |--------------------+-----------------------+------------------------------| | $10nm$ | $20nm$ | $< 1.7 \mu rad$ | |--------------------+-----------------------+------------------------------| *** Rotation around Y **** Presentation This stage is mainly use for the reflectivity experiment. This permit to make the sample rotates around the y axis. The rotation axis should be parallel to the y-axis and cross the X-ray. As it is located below the Spindle, it make the axis of rotation of the spindle to also tilt with respect to the X-ray. 4 joints are used in order to pre-stress the system. The model of the stage is shown figure ref:fig:stage_tilt. #+name: fig:stage_tilt #+caption: Tilt Stage [[./figures/stages/stage_tilt.pdf]] **** Specifications #+attr_latex: :align |c|c|c|c|c| :placement [!htpb] #+CAPTION: Specifications for the rotation stage along Y #+NAME: fig:spec-r-y |------------+------+----------------------+--------------+-------------| | Motion | Mode | Stroke | Repetability | MIM | |------------+------+----------------------+--------------+-------------| | $\theta_y$ | S | $\pm51 mrad$ | $5 \mu rad$ | $2 \mu rad$ | | | | Speed: $\pm 3^o/min$ | | | |------------+------+----------------------+--------------+-------------| **** Metrology Two linear digital encoders are used (one on each side). **** Parasitic motions #+attr_latex: :align |c|c|c| :placement [!htpb] #+CAPTION: Parasitic motions for the rotation stage along Y #+NAME: fig:parasitic-motions-r-y |-------------------+--------------------+---------------| | Axial Error ($y$) | Radial Error ($z$) | Tilt error () | |-------------------+--------------------+---------------| | $0.5\mu m$ | $10nm$ | $1.5 \mu rad$ | |-------------------+--------------------+---------------| *** Spindle **** Presentation The Spindle consist of one stator (attached to the tilt stage) and one rotor. The rotor is separated from the stator thanks to an air bearing. The spindle is represented figure ref:fig:stage_spindle. It has been developed by Lab-Leuven[fn:5]. #+name: fig:stage_spindle #+caption: Spindle [[./figures/stages/stage_spindle.pdf]] **** Specifications #+attr_latex: :align |c|c|c|c|c| :placement [!htpb] #+CAPTION: Specifications for the Spindle #+NAME: fig:spec-r-y |------------+------+----------------------+--------------+-------------| | Motion | Mode | Stroke | Repetability | MIM | |------------+------+----------------------+--------------+-------------| | $\theta_z$ | S | $\pm51 mrad$ | $5 \mu rad$ | $2 \mu rad$ | | | | Speed: $\pm 3^o/min$ | | | |------------+------+----------------------+--------------+-------------| **** Metrology There is an incremental rotation encoder. **** Parasitic motions #+attr_latex: :align |c|c| :placement [!htpb] #+CAPTION: Parasitic motions for the Spindle #+NAME: fig:parasitic-motions-r-z |------------------------+----------------------| | Radial Error ($x$-$y$) | Vertical Error ($z$) | |------------------------+----------------------| | $0.33\mu m$ | $0.07\mu m$ | |------------------------+----------------------| More complete measurements have been conducted at the PEL[fn:7]. *** Long Stroke Hexapod **** Presentation The long stroke hexapod consists of 6 legs, each composed of: - one linear actuators - one limit switch - one absolute linear encoder - one ball joint at each side, one is fixed on the fixed platform, the other on the mobile platform This long stroke hexapod has been developed by Symetrie[fn:4] and is shown figure ref:fig:stage_hexapod. #+name: fig:stage_hexapod #+caption: Hexapod Symetrie [[./figures/stages/stage_hexapod.pdf]] **** Specifications #+attr_latex: :align |c|c|c|c|c| :placement [!htpb] #+CAPTION: Specifications for the long stroke hexapod #+NAME: fig:spec-hexa |------------------+--------------+----------------+--------------+---------------| | Motion | Stroke | Repetability | MIM | Stiffness | |------------------+--------------+----------------+--------------+---------------| | $T_{\mu_x}$ | $\pm10mm$ | $\pm1\mu m$ | $0.5\mu m$ | $>12N/\mu m$ | | $T_{\mu_y}$ | $\pm10mm$ | $\pm1\mu m$ | $0.5\mu m$ | $>12N/\mu m$ | | $T_{\mu_z}$ | $\pm10mm$ | $\pm1\mu m$ | $0.5\mu m$ | $>135N/\mu m$ | | $\theta_{\mu_x}$ | $\pm3 deg$ | $\pm5 \mu rad$ | $2.5\mu rad$ | | | $\theta_{\mu_y}$ | $\pm3 deg$ | $\pm5 \mu rad$ | $2.5\mu rad$ | | | $\theta_{\mu_z}$ | $\pm0.5 deg$ | $\pm5 \mu rad$ | $2.5\mu rad$ | | |------------------+--------------+----------------+--------------+---------------| **** Metrology The metrology consist of one absolute linear encoder in each leg. **** Parasitic motions #+attr_latex: :align |c|c|c| :placement [!htpb] #+CAPTION: Parasitic motions for the Spindle #+NAME: fig:parasitic-motions-r-z |------------------+--------------+----------------| | Movement | Resolution | Repeatability | |------------------+--------------+----------------| | $T_{\mu_x}$ | $0.5\mu m$ | $\pm 1\mu m$ | | $T_{\mu_y}$ | $0.5\mu m$ | $\pm 1\mu m$ | | $T_{\mu_z}$ | $0.1\mu m$ | $\pm 1\mu m$ | | $\theta_{\mu_x}$ | $2.5\mu rad$ | $\pm 4\mu rad$ | | $\theta_{\mu_y}$ | $2.5\mu rad$ | $\pm 4\mu rad$ | |------------------+--------------+----------------| *** Short Stroke Hexapod **** Presentation This last stage is located just below the sample to study. This is the only stage that is not yet build nor designed. **** Specifications #+attr_latex: :align |c|c|c|c|c| :placement [!htpb] #+CAPTION: Specifications for the nano-stage #+NAME: fig:spec-nano |------------------+----------------+--------------+-------| | Motion | Stroke | Repetability | MIM | |------------------+----------------+--------------+-------| | $T_{\nu_x}$ | $\pm10\mu m$ | $10nm$ | $3nm$ | | $T_{\nu_y}$ | $\pm10\mu m$ | $10nm$ | $3nm$ | | $T_{\nu_z}$ | $\pm10\mu m$ | $10nm$ | $3nm$ | | $\theta_{\nu_x}$ | $\pm10\mu rad$ | $1.7\mu rad$ | | | $\theta_{\nu_y}$ | $\pm10\mu rad$ | $1.7\mu rad$ | | | $\theta_{\nu_z}$ | $\pm10\mu rad$ | | | |------------------+----------------+--------------+-------| **** Dimensions - Bottom plate : - Inner diameter: $>150mm$ - External diameter: $<305mm$ - Top plate - External diameter: $<300mm$ - Height: $90mm$ - Location of the rotation point: Centered, at $175mm$ above the top platform ** Compute sample's motion from stage motion *** Introduction :ignore: #+begin_quote Rx = [1 0 0; 0 cos(t) -sin(t); 0 sin(t) cos(t)]; Ry = [ cos(t) 0 sin(t); 0 1 0; -sin(t) 0 cos(t)]; Rz = [cos(t) -sin(t) 0; sin(t) cos(t) 0; 0 0 1]; #+end_quote Let's define the following frames: - $\{W\}$ the frame that is *fixed to the granite* and its origin at the theoretical meeting point between the X-ray and the spindle axis. - $\{S\}$ the frame *attached to the sample* (in reality attached to the top platform of the nano-hexapod) with its origin at 175mm above the top platform of the nano-hexapod. Its origin is $O_S$. - $\{T\}$ the theoretical wanted frame that correspond to the wanted pose of the frame $\{S\}$. $\{T\}$ is computed from the wanted position of each stage. It is thus theoretical and does not correspond to a real position. The origin of $T$ is $O_T$ and is the wanted position of the sample. Thus: - the *measurement* of the position of the sample corresponds to ${}^W O_S = \begin{bmatrix} {}^WP_{x,m} & {}^WP_{y,m} & {}^WP_{z,m} \end{bmatrix}^T$ in translation and to $\theta_m {}^W\bm{s}_m = \theta_m \cdot \begin{bmatrix} {}^Ws_{x,m} & {}^Ws_{y,m} & {}^Ws_{z,m} \end{bmatrix}^T$ in rotations - the *wanted position* of the sample expressed w.r.t. the granite is ${}^W O_T = \begin{bmatrix} {}^WP_{x,r} & {}^WP_{y,r} & {}^WP_{z,r} \end{bmatrix}^T$ in translation and to $\theta_r {}^W\bm{s}_r = \theta_r \cdot \begin{bmatrix} {}^Ws_{x,r} & {}^Ws_{y,r} & {}^Ws_{z,r} \end{bmatrix}^T$ in rotations *** Wanted Position of the Sample with respect to the Granite Let's define the wanted position of each stage. #+begin_src matlab Ty = 0; % [m] Ry = 3*pi/180; % [rad] Rz = 180*pi/180; % [rad] % Hexapod (first consider only translations) Thx = 0; % [m] Thy = 0; % [m] Thz = 0; % [m] #+end_src Now, we compute the corresponding wanted translation and rotation of the sample with respect to the granite frame $\{W\}$. This corresponds to ${}^WO_T$ and $\theta_m {}^Ws_m$. To do so, we have to define the homogeneous transformation for each stage. #+begin_src matlab % Translation Stage Rty = [1 0 0 0; 0 1 0 Ty; 0 0 1 0; 0 0 0 1]; % Tilt Stage - Pure rotating aligned with Ob Rry = [ cos(Ry) 0 sin(Ry) 0; 0 1 0 0; -sin(Ry) 0 cos(Ry) 0; 0 0 0 1]; % Spindle - Rotation along the Z axis Rrz = [cos(Rz) -sin(Rz) 0 0 ; sin(Rz) cos(Rz) 0 0 ; 0 0 1 0 ; 0 0 0 1 ]; % Micro-Hexapod (only rotations first) Rh = [1 0 0 Thx ; 0 1 0 Thy ; 0 0 1 Thz ; 0 0 0 1 ]; #+end_src We combine the individual homogeneous transformations into one homogeneous transformation for all the station. #+begin_src matlab Ttot = Rty*Rry*Rrz*Rh; #+end_src Using this homogeneous transformation, we can compute the wanted position and orientation of the sample with respect to the granite. Translation. #+begin_src matlab WOr = Ttot*[0;0;0;1]; WOr = WOr(1:3); #+end_src Rotation. #+begin_src matlab thetar = acos((trace(Ttot(1:3, 1:3))-1)/2) if thetar == 0 WSr = [0; 0; 0]; else [V, D] = eig(Ttot(1:3, 1:3)); WSr = thetar*V(:, abs(diag(D) - 1) < eps(1)); end #+end_src #+begin_src matlab WPr = [WOr ; WSr]; #+end_src *** Measured Position of the Sample with respect to the Granite The measurement of the position of the sample using the metrology system gives the position and orientation of the sample with respect to the granite. #+begin_src matlab % Measurements: Xm, Ym, Zm, Rx, Ry, Rz Dxm = 0; % [m] Dym = 0; % [m] Dzm = 0; % [m] Rxm = 0*pi/180; % [rad] Rym = 0*pi/180; % [rad] Rzm = 180*pi/180; % [rad] #+end_src Let's compute the corresponding orientation using screw axis. #+begin_src matlab Trxm = [1 0 0; 0 cos(Rxm) -sin(Rxm); 0 sin(Rxm) cos(Rxm)]; Trym = [ cos(Rym) 0 sin(Rym); 0 1 0; -sin(Rym) 0 cos(Rym)]; Trzm = [cos(Rzm) -sin(Rzm) 0; sin(Rzm) cos(Rzm) 0; 0 0 1]; STw = [[ Trym*Trxm*Trzm , [Dxm; Dym; Dzm]]; 0 0 0 1]; #+end_src We then obtain the orientation measurement in the form of screw coordinate $\theta_m ({}^Ws_{x,m},\ {}^Ws_{y,m},\ {}^Ws_{z,m})^T$ where: - $\theta_m = \cos^{-1} \frac{\text{Tr}(R) - 1}{2}$ - ${}^W\bm{s}_m$ is the eigen vector of the rotation matrix $R$ corresponding to the eigen value $\lambda = 1$ #+begin_src matlab thetam = acos((trace(STw(1:3, 1:3))-1)/2); % [rad] if thetam == 0 WSm = [0; 0; 0]; else [V, D] = eig(STw(1:3, 1:3)); WSm = thetam*V(:, abs(diag(D) - 1) < eps(1)); end #+end_src #+begin_src matlab WPm = [Dxm ; Dym ; Dzm ; WSm]; #+end_src *** Positioning Error with respect to the Granite The wanted position expressed with respect to the granite is ${}^WO_T$ and the measured position with respect to the granite is ${}^WO_S$, thus the *position error* expressed in $\{W\}$ is \[ {}^W E = {}^W O_T - {}^W O_S \] The same is true for rotations: \[ \theta_\epsilon {}^W\bm{s}_\epsilon = \theta_r {}^W\bm{s}_r - \theta_m {}^W\bm{s}_m \] #+begin_src matlab WPe = WPr - WPm; #+end_src #+begin_quote Now we want to express this error in a frame attached to the *base of the nano-hexapod* with its origin at the same point where the Jacobian of the nano-hexapod is computed (175mm above the top platform + 90mm of total height of the nano-hexapod). Or maybe should we want to express this error with respect to the *top platform of the nano-hexapod*? We are measuring the position of the top-platform, and we don't know exactly the position of the bottom platform. We could compute the position of the bottom platform in two ways: - from the encoders of each stage - from the measurement of the nano-hexapod top platform + the internal metrology in the nano-hexapod (capacitive sensors e.g) A third option is to say that the maximum stroke of the nano-hexapod is so small that the error should no change to much by the change of base. #+end_quote *** Position Error Expressed in the Nano-Hexapod Frame We now want the position error to be expressed in $\{S\}$ (the frame attach to the sample) for control: \[ {}^S E = {}^S T_W \cdot {}^W E \] Thus we need to compute the homogeneous transformation ${}^ST_W$. Fortunately, this homogeneous transformation can be computed from the measurement of the sample position and orientation with respect to the granite. #+begin_src matlab Trxm = [1 0 0; 0 cos(Rxm) -sin(Rxm); 0 sin(Rxm) cos(Rxm)]; Trym = [ cos(Rym) 0 sin(Rym); 0 1 0; -sin(Rym) 0 cos(Rym)]; Trzm = [cos(Rzm) -sin(Rzm) 0; sin(Rzm) cos(Rzm) 0; 0 0 1]; STw = [[ Trym*Trxm*Trzm , [Dxm; Dym; Dzm]]; 0 0 0 1]; #+end_src Translation Error. #+begin_src matlab SEm = STw * [WPe(1:3); 0]; SEm = SEm(1:3); #+end_src Rotation Error. #+begin_src matlab SEr = STw * [WPe(4:6); 0]; SEr = SEr(1:3); #+end_src #+begin_src matlab Etot = [SEm ; SEr] #+end_src *** Another try Let's denote: - $\{W\}$ the initial fixed frame - $\{R\}$ the reference frame corresponding to the wanted pose of the sample - $\{M\}$ the frame corresponding to the measured pose of the sample We have then computed: - ${}^WT_R$ - ${}^WT_M$ We have: \begin{align} {}^MT_R &= {}^MT_W {}^WT_R \\ &= {}^WT_M^t {}^WT_R \end{align} #+begin_src matlab MTr = STw'*Ttot; #+end_src Position error: #+begin_src matlab MTr(1:3, 1:4)*[0; 0; 0; 1] #+end_src Orientation error: #+begin_src matlab MTr(1:3, 1:3) #+end_src *** Verification How can we verify that the computation is correct? Options: - Test with simscape multi-body - Impose motion on each stage - Measure the position error w.r.t. the NASS - Compare with the computation * Stage Modeling :PROPERTIES: :HEADER-ARGS:matlab+: :tangle matlab/.m :END: <> ** Introduction :ignore: The goal here is to tune the Simscape model of the station in order to have a good dynamical representation of the real system. In order to do so, we reproduce the Modal Analysis done on the station using the Simscape model. We can then compare the measured Frequency Response Functions with the identified dynamics of the model. Finally, this should help to tune the parameters of the model such that the dynamics is closer to the measured FRF. # Validation of the Model # - Most important metric: support compliance # - Compare model and measurement ** Matlab Init :noexport:ignore: #+begin_src matlab %% .m #+end_src #+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) <> #+end_src #+begin_src matlab :exports none :results silent :noweb yes <> #+end_src #+begin_src matlab :tangle no :noweb yes <> #+end_src #+begin_src matlab :eval no :noweb yes <> #+end_src #+begin_src matlab :noweb yes <> #+end_src #+begin_src matlab :noweb yes <> #+end_src ** Some notes about the Simscape Model The Simscape Model of the micro-station consists of several solid bodies: - Bottom Granite - Top Granite - Translation Stage - Tilt Stage - Spindle - Hexapod Each solid body has some characteristics: Center of Mass, mass, moment of inertia, etc... These parameters are automatically computed from the geometry and from the density of the materials. Then, the solid bodies are connected with springs and dampers. Some of the springs and dampers values can be estimated from the joints/stages specifications, however, we here prefer to tune these values based on the measurements. ** Compare with measurements at the CoM of each element #+begin_src matlab %% We load the configuration. load('conf_simulink.mat'); % We set a small =StopTime=. set_param(conf_simulink, 'StopTime', '0.5'); %% We initialize all the stages. initializeGround( 'type', 'rigid'); initializeGranite( 'type', 'flexible'); initializeTy( 'type', 'flexible'); initializeRy( 'type', 'flexible'); initializeRz( 'type', 'flexible'); initializeMicroHexapod('type', 'flexible'); initializeLoggingConfiguration('log', 'none'); initializeReferences(); initializeDisturbances('enable', false); #+end_src We now use a new Simscape Model where 6DoF inertial sensors are located at the Center of Mass of each solid body. We use the =linearize= function in order to estimate the dynamics from forces applied on the Translation stage at the same position used for the real modal analysis to the inertial sensors. #+begin_src matlab %% Identification % Input/Output definition clear io; io_i = 1; io(io_i) = linio([mdl, '/Micro-Station/Translation Stage/Flexible/F_hammer'], 1, 'openinput'); io_i = io_i + 1; io(io_i) = linio([mdl, '/Micro-Station/Granite/Flexible/accelerometer'], 1, 'openoutput'); io_i = io_i + 1; io(io_i) = linio([mdl, '/Micro-Station/Translation Stage/Flexible/accelerometer'], 1, 'openoutput'); io_i = io_i + 1; io(io_i) = linio([mdl, '/Micro-Station/Tilt Stage/Flexible/accelerometer'], 1, 'openoutput'); io_i = io_i + 1; io(io_i) = linio([mdl, '/Micro-Station/Spindle/Flexible/accelerometer'], 1, 'openoutput'); io_i = io_i + 1; io(io_i) = linio([mdl, '/Micro-Station/Micro Hexapod/Flexible/accelerometer'], 1, 'openoutput'); io_i = io_i + 1; % Run the linearization G_ms = linearize(mdl, io, 0); % Input/Output definition G_ms.InputName = {'Fx', 'Fy', 'Fz'}; G_ms.OutputName = {'gtop_x', 'gtop_y', 'gtop_z', 'gtop_rx', 'gtop_ry', 'gtop_rz', ... 'ty_x', 'ty_y', 'ty_z', 'ty_rx', 'ty_ry', 'ty_rz', ... 'ry_x', 'ry_y', 'ry_z', 'ry_rx', 'ry_ry', 'ry_rz', ... 'rz_x', 'rz_y', 'rz_z', 'rz_rx', 'rz_ry', 'rz_rz', ... 'hexa_x', 'hexa_y', 'hexa_z', 'hexa_rx', 'hexa_ry', 'hexa_rz'}; % Put the output of G in displacements instead of acceleration G_ms = G_ms/s^2; #+end_src #+begin_src matlab %% Load estimated FRF at CoM load('/home/thomas/Cloud/work-projects/ID31-NASS/phd-thesis-chapters/A3-micro-station-modal-analysis/matlab/mat/frf_matrix.mat', 'freqs'); load('/home/thomas/Cloud/work-projects/ID31-NASS/phd-thesis-chapters/A3-micro-station-modal-analysis/matlab/mat/frf_com.mat', 'frfs_CoM'); #+end_src #+begin_src matlab :exports none %% Compare the two dirs = {'x', 'y', 'z', 'rx', 'ry', 'rz'}; stages = {'gbot', 'gtop', 'ty', 'ry', 'rz', 'hexa'} n_stg = 3; n_dir = 6; % x, y, z, Rx, Ry, Rz n_exc = 2; % x, y, z f = logspace(0, 3, 1000); figure; hold on; plot(freqs, abs(squeeze(frfs_CoM(6*(n_stg-1) + n_dir, n_exc, :)))./((2*pi*freqs).^2)'); plot(f, abs(squeeze(freqresp(G_ms([stages{n_stg}, '_', dirs{n_dir}], ['F', dirs{n_exc}]), f, 'Hz')))); set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); ylabel('Amplitude [m/N]'); hold off; xlim([1, 200]); #+end_src #+begin_src matlab :exports none dirs = {'x', 'y', 'z', 'rx', 'ry', 'rz'}; stages = {'gtop', 'ty', 'ry', 'rz', 'hexa'} f = logspace(1, 3, 1000); figure; for n_stg = 1:2 for n_dir = 1:3 subplot(3, 2, (n_dir-1)*2 + n_stg); title(['F ', dirs{n_dir}, ' to ', stages{n_stg}, ' ', dirs{n_dir}]); hold on; plot(freqs, abs(squeeze(frfs_CoM(6*(n_stg) + n_dir, n_dir, :)))./((2*pi*freqs).^2)'); plot(f, abs(squeeze(freqresp(G_ms([stages{n_stg}, '_', dirs{n_dir}], ['F', dirs{n_dir}]), f, 'Hz')))); set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); ylabel('Amplitude [m/N]'); if n_dir == 3 xlabel('Frequency [Hz]'); end hold off; xlim([10, 1000]); ylim([1e-12, 1e-6]); end end #+end_src #+begin_src matlab :exports none dirs = {'x', 'y', 'z', 'rx', 'ry', 'rz'}; stages = {'ry', 'rz', 'hexa'} f = logspace(1, 3, 1000); figure; for n_stg = 1:2 for n_dir = 1:3 subplot(3, 2, (n_dir-1)*2 + n_stg); title(['F ', dirs{n_dir}, ' to ', stages{n_stg}, ' ', dirs{n_dir}]); hold on; plot(freqs, abs(squeeze(frfs_CoM(6*(n_stg+2) + n_dir, n_dir, :)))./((2*pi*freqs).^2)'); plot(f, abs(squeeze(freqresp(G_ms([stages{n_stg}, '_', dirs{n_dir}], ['F', dirs{n_dir}]), f, 'Hz')))); set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); ylabel('Amplitude [m/N]'); if n_dir == 3 xlabel('Frequency [Hz]'); end hold off; xlim([10, 1000]); ylim([1e-12, 1e-6]); end end #+end_src #+begin_src matlab :exports none dirs = {'x', 'y', 'z', 'rx', 'ry', 'rz'}; stages = {'hexa'} f = logspace(1, 3, 1000); figure; for n_stg = 1 for n_dir = 1:3 subplot(3, 1, (n_dir-1) + n_stg); title(['F ', dirs{n_dir}, ' to ', stages{n_stg}, ' ', dirs{n_dir}]); hold on; plot(freqs, abs(squeeze(frfs_CoM(6*(n_stg+4) + n_dir, n_dir, :)))./((2*pi*freqs).^2)'); plot(f, abs(squeeze(freqresp(G_ms([stages{n_stg}, '_', dirs{n_dir}], ['F', dirs{n_dir}]), f, 'Hz')))); set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); ylabel('Amplitude [m/N]'); if n_dir == 3 xlabel('Frequency [Hz]'); end hold off; xlim([10, 1000]); ylim([1e-12, 1e-6]); end end #+end_src ** Obtained Compliance of the Micro-Station #+begin_src matlab %% Initialize simulation with default parameters (flexible elements) initializeGround(); initializeGranite(); initializeTy(); initializeRy(); initializeRz(); initializeMicroHexapod(); initializeReferences(); initializeDisturbances(); initializeSimscapeConfiguration(); initializeLoggingConfiguration(); #+end_src And we identify the dynamics from forces/torques applied on the micro-hexapod top platform to the motion of the micro-hexapod top platform at the same point. #+begin_src matlab %% Identification of the compliance % Input/Output definition clear io; io_i = 1; io(io_i) = linio([mdl, '/Micro-Station/Micro Hexapod/Flexible/Fm'], 1, 'openinput'); io_i = io_i + 1; % Direct Forces/Torques applied on the micro-hexapod top platform io(io_i) = linio([mdl, '/Micro-Station/Micro Hexapod/Flexible/Dm'], 1, 'output'); io_i = io_i + 1; % Absolute displacement of the top platform % Run the linearization Gm = linearize(mdl, io, 0); Gm.InputName = {'Fmx', 'Fmy', 'Fmz', 'Mmx', 'Mmy', 'Mmz'}; Gm.OutputName = {'Dx', 'Dy', 'Dz', 'Drx', 'Dry', 'Drz'}; #+end_src #+begin_src matlab :exports none %% Plot of the compliance curves labels = {'$D_x/F_{x}$', '$D_y/F_{y}$', '$D_z/F_{z}$', '$R_{x}/M_{x}$', '$R_{y}/M_{y}$', '$R_{R}/M_{z}$'}; freqs = logspace(1, 3, 1000); figure; hold on; for i = 1:6 plot(freqs, abs(squeeze(freqresp(Gm(i, i), freqs, 'Hz'))), 'DisplayName', labels{i}); end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); xlabel('Frequency [Hz]'); ylabel('Compliance'); legend('location', 'northwest'); #+end_src - [ ] Comparison with the estimated (or measured) compliance ** Conclusion For such a complex system, we believe that the Simscape Model represents the dynamics of the system with enough fidelity. * Measurement of Positioning Errors :PROPERTIES: :HEADER-ARGS:matlab+: :tangle matlab/ustation_2_kinematics.m :END: <> ** Introduction :ignore: [[file:~/Cloud/work-projects/ID31-NASS/matlab/nass-simscape/org/kinematics.org]] ** Matlab Init :noexport:ignore: #+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) <> #+end_src #+begin_src matlab :exports none :results silent :noweb yes <> #+end_src #+begin_src matlab :tangle no :noweb yes <> #+end_src #+begin_src matlab :eval no :noweb yes <> #+end_src #+begin_src matlab :noweb yes <> #+end_src * Simulation of Scientific Experiments :PROPERTIES: :HEADER-ARGS:matlab+: :tangle matlab/.m :END: <> ** Introduction :ignore: ** Matlab Init :noexport:ignore: #+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) <> #+end_src #+begin_src matlab :exports none :results silent :noweb yes <> #+end_src #+begin_src matlab :tangle no :noweb yes <> #+end_src #+begin_src matlab :eval no :noweb yes <> #+end_src #+begin_src matlab :noweb yes <> #+end_src * Estimation of disturbances * Conclusion <> * Bibliography :ignore: #+latex: \printbibliography[heading=bibintoc,title={Bibliography}] * Matlab Functions :noexport: ** Simscape Configuration :PROPERTIES: :header-args:matlab+: :tangle matlab/src/initializeSimscapeConfiguration.m :header-args:matlab+: :comments none :mkdirp yes :eval no :END: *** Function description :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab function [] = initializeSimscapeConfiguration(args) #+end_src *** Optional Parameters :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab arguments args.gravity logical {mustBeNumericOrLogical} = true end #+end_src *** Structure initialization :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab conf_simscape = struct(); #+end_src *** Add Type :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab if args.gravity conf_simscape.type = 1; else conf_simscape.type = 2; end #+end_src *** Save the Structure #+begin_src matlab if exist('./mat', 'dir') if exist('./mat/conf_simscape.mat', 'file') save('mat/conf_simscape.mat', 'conf_simscape', '-append'); else save('mat/conf_simscape.mat', 'conf_simscape'); end elseif exist('./matlab', 'dir') if exist('./matlab/mat/conf_simscape.mat', 'file') save('matlab/mat/conf_simscape.mat', 'conf_simscape', '-append'); else save('matlab/mat/conf_simscape.mat', 'conf_simscape'); end end #+end_src ** Logging Configuration :PROPERTIES: :header-args:matlab+: :tangle matlab/src/initializeLoggingConfiguration.m :header-args:matlab+: :comments none :mkdirp yes :eval no :END: *** Function description :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab function [] = initializeLoggingConfiguration(args) #+end_src *** Optional Parameters :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab arguments args.log char {mustBeMember(args.log,{'none', 'all', 'forces'})} = 'none' args.Ts (1,1) double {mustBeNumeric, mustBePositive} = 1e-3 end #+end_src *** Structure initialization :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab conf_log = struct(); #+end_src *** Add Type :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab switch args.log case 'none' conf_log.type = 0; case 'all' conf_log.type = 1; case 'forces' conf_log.type = 2; end #+end_src *** Sampling Time :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab conf_log.Ts = args.Ts; #+end_src *** Save the Structure #+begin_src matlab if exist('./mat', 'dir') if exist('./mat/conf_log.mat', 'file') save('mat/conf_log.mat', 'conf_log', '-append'); else save('mat/conf_log.mat', 'conf_log'); end elseif exist('./matlab', 'dir') if exist('./matlab/mat/conf_log.mat', 'file') save('matlab/mat/conf_log.mat', 'conf_log', '-append'); else save('matlab/mat/conf_log.mat', 'conf_log'); end end #+end_src ** Ground :PROPERTIES: :header-args:matlab+: :tangle matlab/src/initializeGround.m :header-args:matlab+: :comments none :mkdirp yes :eval no :END: *** Function description :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab function [ground] = initializeGround(args) #+end_src *** Optional Parameters :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab arguments args.type char {mustBeMember(args.type,{'none', 'rigid'})} = 'rigid' args.rot_point (3,1) double {mustBeNumeric} = zeros(3,1) % Rotation point for the ground motion [m] end #+end_src *** Structure initialization :PROPERTIES: :UNNUMBERED: t :END: First, we initialize the =granite= structure. #+begin_src matlab ground = struct(); #+end_src *** Add Type :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab switch args.type case 'none' ground.type = 0; case 'rigid' ground.type = 1; end #+end_src *** Ground Solid properties :PROPERTIES: :UNNUMBERED: t :END: We set the shape and density of the ground solid element. #+begin_src matlab ground.shape = [2, 2, 0.5]; % [m] ground.density = 2800; % [kg/m3] #+end_src *** Rotation Point :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab ground.rot_point = args.rot_point; #+end_src *** Save the Structure #+begin_src matlab if exist('./mat', 'dir') if exist('./mat/nass_stages.mat', 'file') save('mat/nass_stages.mat', 'ground', '-append'); else save('mat/nass_stages.mat', 'ground'); end elseif exist('./matlab', 'dir') if exist('./matlab/mat/nass_stages.mat', 'file') save('matlab/mat/nass_stages.mat', 'ground', '-append'); else save('matlab/mat/nass_stages.mat', 'ground'); end end #+end_src ** Granite :PROPERTIES: :header-args:matlab+: :tangle matlab/src/initializeGranite.m :header-args:matlab+: :comments none :mkdirp yes :eval no :END: *** Function description :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab function [granite] = initializeGranite(args) #+end_src *** Optional Parameters :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab arguments args.type char {mustBeMember(args.type,{'rigid', 'flexible', 'none'})} = 'flexible' args.density (1,1) double {mustBeNumeric, mustBeNonnegative} = 2800 % Density [kg/m3] args.K (3,1) double {mustBeNumeric, mustBeNonnegative} = [4e9; 3e8; 8e8] % [N/m] args.C (3,1) double {mustBeNumeric, mustBeNonnegative} = [4.0e5; 1.1e5; 9.0e5] % [N/(m/s)] args.x0 (1,1) double {mustBeNumeric} = 0 % Rest position of the Joint in the X direction [m] args.y0 (1,1) double {mustBeNumeric} = 0 % Rest position of the Joint in the Y direction [m] args.z0 (1,1) double {mustBeNumeric} = 0 % Rest position of the Joint in the Z direction [m] args.sample_pos (1,1) double {mustBeNumeric} = 0.8 % Height of the measurment point [m] end #+end_src *** Structure initialization :PROPERTIES: :UNNUMBERED: t :END: First, we initialize the =granite= structure. #+begin_src matlab granite = struct(); #+end_src *** Add Granite Type :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab switch args.type case 'none' granite.type = 0; case 'rigid' granite.type = 1; case 'flexible' granite.type = 2; end #+end_src *** Material and Geometry :PROPERTIES: :UNNUMBERED: t :END: Properties of the Material and link to the geometry of the granite. #+begin_src matlab granite.density = args.density; % [kg/m3] granite.STEP = 'granite.STEP'; #+end_src Z-offset for the initial position of the sample with respect to the granite top surface. #+begin_src matlab granite.sample_pos = args.sample_pos; % [m] #+end_src *** Stiffness and Damping properties :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab granite.K = args.K; % [N/m] granite.C = args.C; % [N/(m/s)] #+end_src *** Save the Structure #+begin_src matlab if exist('./mat', 'dir') if exist('./mat/nass_stages.mat', 'file') save('mat/nass_stages.mat', 'granite', '-append'); else save('mat/nass_stages.mat', 'granite'); end elseif exist('./matlab', 'dir') if exist('./matlab/mat/nass_stages.mat', 'file') save('matlab/mat/nass_stages.mat', 'granite', '-append'); else save('matlab/mat/nass_stages.mat', 'granite'); end end #+end_src ** Translation Stage :PROPERTIES: :header-args:matlab+: :tangle matlab/src/initializeTy.m :header-args:matlab+: :comments none :mkdirp yes :eval no :END: *** Function description :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab function [ty] = initializeTy(args) #+end_src *** Optional Parameters :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab arguments args.type char {mustBeMember(args.type,{'none', 'rigid', 'flexible'})} = 'flexible' end #+end_src *** Structure initialization :PROPERTIES: :UNNUMBERED: t :END: First, we initialize the =ty= structure. #+begin_src matlab ty = struct(); #+end_src *** Add Translation Stage Type :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab switch args.type case 'none' ty.type = 0; case 'rigid' ty.type = 1; case 'flexible' ty.type = 2; end #+end_src *** Material and Geometry :PROPERTIES: :UNNUMBERED: t :END: Define the density of the materials as well as the geometry (STEP files). #+begin_src matlab % Ty Granite frame ty.granite_frame.density = 7800; % [kg/m3] => 43kg ty.granite_frame.STEP = 'Ty_Granite_Frame.STEP'; % Guide Translation Ty ty.guide.density = 7800; % [kg/m3] => 76kg ty.guide.STEP = 'Ty_Guide.STEP'; % Ty - Guide_Translation12 ty.guide12.density = 7800; % [kg/m3] ty.guide12.STEP = 'Ty_Guide_12.STEP'; % Ty - Guide_Translation11 ty.guide11.density = 7800; % [kg/m3] ty.guide11.STEP = 'Ty_Guide_11.STEP'; % Ty - Guide_Translation22 ty.guide22.density = 7800; % [kg/m3] ty.guide22.STEP = 'Ty_Guide_22.STEP'; % Ty - Guide_Translation21 ty.guide21.density = 7800; % [kg/m3] ty.guide21.STEP = 'Ty_Guide_21.STEP'; % Ty - Plateau translation ty.frame.density = 7800; % [kg/m3] ty.frame.STEP = 'Ty_Stage.STEP'; % Ty Stator Part ty.stator.density = 5400; % [kg/m3] ty.stator.STEP = 'Ty_Motor_Stator.STEP'; % Ty Rotor Part ty.rotor.density = 5400; % [kg/m3] ty.rotor.STEP = 'Ty_Motor_Rotor.STEP'; #+end_src *** Stiffness and Damping properties :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab ty.K = [2e8; 1e8; 2e8; 6e7; 9e7; 6e7]; % [N/m, N*m/rad] ty.C = [8e4; 5e4; 8e4; 2e4; 3e4; 2e4]; % [N/(m/s), N*m/(rad/s)] #+end_src *** Save the Structure #+begin_src matlab if exist('./mat', 'dir') if exist('./mat/nass_stages.mat', 'file') save('mat/nass_stages.mat', 'ty', '-append'); else save('mat/nass_stages.mat', 'ty'); end elseif exist('./matlab', 'dir') if exist('./matlab/mat/nass_stages.mat', 'file') save('matlab/mat/nass_stages.mat', 'ty', '-append'); else save('matlab/mat/nass_stages.mat', 'ty'); end end #+end_src ** Tilt Stage :PROPERTIES: :header-args:matlab+: :tangle matlab/src/initializeRy.m :header-args:matlab+: :comments none :mkdirp yes :eval no :END: *** Function description :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab function [ry] = initializeRy(args) #+end_src *** Optional Parameters :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab arguments args.type char {mustBeMember(args.type,{'none', 'rigid', 'flexible'})} = 'flexible' args.Ry_init (1,1) double {mustBeNumeric} = 0 end #+end_src *** Structure initialization :PROPERTIES: :UNNUMBERED: t :END: First, we initialize the =ry= structure. #+begin_src matlab ry = struct(); #+end_src *** Add Tilt Type :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab switch args.type case 'none' ry.type = 0; case 'rigid' ry.type = 1; case 'flexible' ry.type = 2; end #+end_src *** Material and Geometry :PROPERTIES: :UNNUMBERED: t :END: Properties of the Material and link to the geometry of the Tilt stage. #+begin_src matlab % Ry - Guide for the tilt stage ry.guide.density = 7800; % [kg/m3] ry.guide.STEP = 'Tilt_Guide.STEP'; % Ry - Rotor of the motor ry.rotor.density = 2400; % [kg/m3] ry.rotor.STEP = 'Tilt_Motor_Axis.STEP'; % Ry - Motor ry.motor.density = 3200; % [kg/m3] ry.motor.STEP = 'Tilt_Motor.STEP'; % Ry - Plateau Tilt ry.stage.density = 7800; % [kg/m3] ry.stage.STEP = 'Tilt_Stage.STEP'; #+end_src Z-Offset so that the center of rotation matches the sample center; #+begin_src matlab ry.z_offset = 0.58178; % [m] #+end_src #+begin_src matlab ry.Ry_init = args.Ry_init; % [rad] #+end_src *** Stiffness and Damping properties :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab ry.K = [3.8e8; 4e8; 3.8e8; 1.2e8; 6e4; 1.2e8]; ry.C = [1e5; 1e5; 1e5; 3e4; 1e3; 3e4]; #+end_src *** Save the Structure #+begin_src matlab if exist('./mat', 'dir') if exist('./mat/nass_stages.mat', 'file') save('mat/nass_stages.mat', 'ry', '-append'); else save('mat/nass_stages.mat', 'ry'); end elseif exist('./matlab', 'dir') if exist('./matlab/mat/nass_stages.mat', 'file') save('matlab/mat/nass_stages.mat', 'ry', '-append'); else save('matlab/mat/nass_stages.mat', 'ry'); end end #+end_src ** Spindle :PROPERTIES: :header-args:matlab+: :tangle matlab/src/initializeRz.m :header-args:matlab+: :comments none :mkdirp yes :eval no :END: *** Function description :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab function [rz] = initializeRz(args) #+end_src *** Optional Parameters :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab arguments args.type char {mustBeMember(args.type,{'none', 'rigid', 'flexible'})} = 'flexible' end #+end_src *** Structure initialization :PROPERTIES: :UNNUMBERED: t :END: First, we initialize the =rz= structure. #+begin_src matlab rz = struct(); #+end_src *** Add Spindle Type :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab switch args.type case 'none' rz.type = 0; case 'rigid' rz.type = 1; case 'flexible' rz.type = 2; end #+end_src *** Material and Geometry :PROPERTIES: :UNNUMBERED: t :END: Properties of the Material and link to the geometry of the spindle. #+begin_src matlab % Spindle - Slip Ring rz.slipring.density = 7800; % [kg/m3] rz.slipring.STEP = 'Spindle_Slip_Ring.STEP'; % Spindle - Rotor rz.rotor.density = 7800; % [kg/m3] rz.rotor.STEP = 'Spindle_Rotor.STEP'; % Spindle - Stator rz.stator.density = 7800; % [kg/m3] rz.stator.STEP = 'Spindle_Stator.STEP'; #+end_src *** Stiffness and Damping properties :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab rz.K = [7e8; 7e8; 2e9; 1e7; 1e7; 1e7]; rz.C = [4e4; 4e4; 7e4; 1e4; 1e4; 1e4]; #+end_src *** Save the Structure #+begin_src matlab if exist('./mat', 'dir') if exist('./mat/nass_stages.mat', 'file') save('mat/nass_stages.mat', 'rz', '-append'); else save('mat/nass_stages.mat', 'rz'); end elseif exist('./matlab', 'dir') if exist('./matlab/mat/nass_stages.mat', 'file') save('matlab/mat/nass_stages.mat', 'rz', '-append'); else save('matlab/mat/nass_stages.mat', 'rz'); end end #+end_src ** Micro Hexapod :PROPERTIES: :header-args:matlab+: :tangle matlab/src/initializeMicroHexapod.m :header-args:matlab+: :comments none :mkdirp yes :eval no :END: *** Function description :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab function [micro_hexapod] = initializeMicroHexapod(args) #+end_src *** Optional Parameters :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab arguments args.type char {mustBeMember(args.type,{'none', 'rigid', 'flexible'})} = 'flexible' % initializeFramesPositions args.H (1,1) double {mustBeNumeric, mustBePositive} = 350e-3 args.MO_B (1,1) double {mustBeNumeric} = 270e-3 % generateGeneralConfiguration args.FH (1,1) double {mustBeNumeric, mustBePositive} = 50e-3 args.FR (1,1) double {mustBeNumeric, mustBePositive} = 175.5e-3 args.FTh (6,1) double {mustBeNumeric} = [-10, 10, 120-10, 120+10, 240-10, 240+10]*(pi/180) args.MH (1,1) double {mustBeNumeric, mustBePositive} = 45e-3 args.MR (1,1) double {mustBeNumeric, mustBePositive} = 118e-3 args.MTh (6,1) double {mustBeNumeric} = [-60+10, 60-10, 60+10, 180-10, 180+10, -60-10]*(pi/180) % initializeStrutDynamics args.Ki (6,1) double {mustBeNumeric, mustBeNonnegative} = 2e7*ones(6,1) args.Ci (6,1) double {mustBeNumeric, mustBeNonnegative} = 1.4e3*ones(6,1) % initializeCylindricalPlatforms args.Fpm (1,1) double {mustBeNumeric, mustBePositive} = 10 args.Fph (1,1) double {mustBeNumeric, mustBePositive} = 26e-3 args.Fpr (1,1) double {mustBeNumeric, mustBePositive} = 207.5e-3 args.Mpm (1,1) double {mustBeNumeric, mustBePositive} = 10 args.Mph (1,1) double {mustBeNumeric, mustBePositive} = 26e-3 args.Mpr (1,1) double {mustBeNumeric, mustBePositive} = 150e-3 % initializeCylindricalStruts args.Fsm (1,1) double {mustBeNumeric, mustBePositive} = 1 args.Fsh (1,1) double {mustBeNumeric, mustBePositive} = 100e-3 args.Fsr (1,1) double {mustBeNumeric, mustBePositive} = 25e-3 args.Msm (1,1) double {mustBeNumeric, mustBePositive} = 1 args.Msh (1,1) double {mustBeNumeric, mustBePositive} = 100e-3 args.Msr (1,1) double {mustBeNumeric, mustBePositive} = 25e-3 % inverseKinematics args.AP (3,1) double {mustBeNumeric} = zeros(3,1) args.ARB (3,3) double {mustBeNumeric} = eye(3) end #+end_src *** Function content :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab stewart = initializeStewartPlatform(); stewart = initializeFramesPositions(stewart, ... 'H', args.H, ... 'MO_B', args.MO_B); stewart = generateGeneralConfiguration(stewart, ... 'FH', args.FH, ... 'FR', args.FR, ... 'FTh', args.FTh, ... 'MH', args.MH, ... 'MR', args.MR, ... 'MTh', args.MTh); stewart = computeJointsPose(stewart); #+end_src #+begin_src matlab stewart = initializeStrutDynamics(stewart, ... 'K', args.Ki, ... 'C', args.Ci); stewart = initializeJointDynamics(stewart, ... 'type_F', 'universal_p', ... 'type_M', 'spherical_p'); #+end_src #+begin_src matlab stewart = initializeCylindricalPlatforms(stewart, ... 'Fpm', args.Fpm, ... 'Fph', args.Fph, ... 'Fpr', args.Fpr, ... 'Mpm', args.Mpm, ... 'Mph', args.Mph, ... 'Mpr', args.Mpr); stewart = initializeCylindricalStruts(stewart, ... 'Fsm', args.Fsm, ... 'Fsh', args.Fsh, ... 'Fsr', args.Fsr, ... 'Msm', args.Msm, ... 'Msh', args.Msh, ... 'Msr', args.Msr); stewart = computeJacobian(stewart); stewart = initializeStewartPose(stewart, ... 'AP', args.AP, ... 'ARB', args.ARB); #+end_src #+begin_src matlab stewart = initializeInertialSensor(stewart, 'type', 'none'); #+end_src *** Add Type :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab switch args.type case 'none' stewart.type = 0; case 'rigid' stewart.type = 1; case 'flexible' stewart.type = 2; end #+end_src *** Save the Structure #+begin_src matlab micro_hexapod = stewart; if exist('./mat', 'dir') if exist('./mat/nass_stages.mat', 'file') save('mat/nass_stages.mat', 'micro_hexapod', '-append'); else save('mat/nass_stages.mat', 'micro_hexapod'); end elseif exist('./matlab', 'dir') if exist('./matlab/mat/nass_stages.mat', 'file') save('matlab/mat/nass_stages.mat', 'micro_hexapod', '-append'); else save('matlab/mat/nass_stages.mat', 'micro_hexapod'); end end #+end_src ** Generate Reference Signals :PROPERTIES: :header-args:matlab+: :tangle matlab/src/initializeReferences.m :header-args:matlab+: :comments none :mkdirp yes :eval no :END: *** Function Declaration and Documentation :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab function [ref] = initializeReferences(args) #+end_src *** Optional Parameters :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab arguments % Sampling Frequency [s] args.Ts (1,1) double {mustBeNumeric, mustBePositive} = 1e-3 % Maximum simulation time [s] args.Tmax (1,1) double {mustBeNumeric, mustBePositive} = 100 % Either "constant" / "triangular" / "sinusoidal" args.Dy_type char {mustBeMember(args.Dy_type,{'constant', 'triangular', 'sinusoidal'})} = 'constant' % Amplitude of the displacement [m] args.Dy_amplitude (1,1) double {mustBeNumeric} = 0 % Period of the displacement [s] args.Dy_period (1,1) double {mustBeNumeric, mustBePositive} = 1 % Either "constant" / "triangular" / "sinusoidal" args.Ry_type char {mustBeMember(args.Ry_type,{'constant', 'triangular', 'sinusoidal'})} = 'constant' % Amplitude [rad] args.Ry_amplitude (1,1) double {mustBeNumeric} = 0 % Period of the displacement [s] args.Ry_period (1,1) double {mustBeNumeric, mustBePositive} = 1 % Either "constant" / "rotating" args.Rz_type char {mustBeMember(args.Rz_type,{'constant', 'rotating', 'rotating-not-filtered'})} = 'constant' % Initial angle [rad] args.Rz_amplitude (1,1) double {mustBeNumeric} = 0 % Period of the rotating [s] args.Rz_period (1,1) double {mustBeNumeric, mustBePositive} = 1 % For now, only constant is implemented args.Dh_type char {mustBeMember(args.Dh_type,{'constant'})} = 'constant' % Initial position [m,m,m,rad,rad,rad] of the top platform (Pitch-Roll-Yaw Euler angles) args.Dh_pos (6,1) double {mustBeNumeric} = zeros(6, 1), ... % For now, only constant is implemented args.Rm_type char {mustBeMember(args.Rm_type,{'constant'})} = 'constant' % Initial position of the two masses args.Rm_pos (2,1) double {mustBeNumeric} = [0; pi] % For now, only constant is implemented args.Dn_type char {mustBeMember(args.Dn_type,{'constant'})} = 'constant' % Initial position [m,m,m,rad,rad,rad] of the top platform args.Dn_pos (6,1) double {mustBeNumeric} = zeros(6,1) end #+end_src *** Initialize Parameters :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab %% Set Sampling Time Ts = args.Ts; Tmax = args.Tmax; %% Low Pass Filter to filter out the references s = zpk('s'); w0 = 2*pi*10; xi = 1; H_lpf = 1/(1 + 2*xi/w0*s + s^2/w0^2); #+end_src *** Translation Stage :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab %% Translation stage - Dy t = 0:Ts:Tmax; % Time Vector [s] Dy = zeros(length(t), 1); Dyd = zeros(length(t), 1); Dydd = zeros(length(t), 1); switch args.Dy_type case 'constant' Dy(:) = args.Dy_amplitude; Dyd(:) = 0; Dydd(:) = 0; case 'triangular' % This is done to unsure that we start with no displacement Dy_raw = args.Dy_amplitude*sawtooth(2*pi*t/args.Dy_period,1/2); i0 = find(t>=args.Dy_period/4,1); Dy(1:end-i0+1) = Dy_raw(i0:end); Dy(end-i0+2:end) = Dy_raw(end); % we fix the last value % The signal is filtered out Dy = lsim(H_lpf, Dy, t); Dyd = lsim(H_lpf*s, Dy, t); Dydd = lsim(H_lpf*s^2, Dy, t); case 'sinusoidal' Dy(:) = args.Dy_amplitude*sin(2*pi/args.Dy_period*t); Dyd = args.Dy_amplitude*2*pi/args.Dy_period*cos(2*pi/args.Dy_period*t); Dydd = -args.Dy_amplitude*(2*pi/args.Dy_period)^2*sin(2*pi/args.Dy_period*t); otherwise warning('Dy_type is not set correctly'); end Dy = struct('time', t, 'signals', struct('values', Dy), 'deriv', Dyd, 'dderiv', Dydd); #+end_src *** Tilt Stage :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab %% Tilt Stage - Ry t = 0:Ts:Tmax; % Time Vector [s] Ry = zeros(length(t), 1); Ryd = zeros(length(t), 1); Rydd = zeros(length(t), 1); switch args.Ry_type case 'constant' Ry(:) = args.Ry_amplitude; Ryd(:) = 0; Rydd(:) = 0; case 'triangular' Ry_raw = args.Ry_amplitude*sawtooth(2*pi*t/args.Ry_period,1/2); i0 = find(t>=args.Ry_period/4,1); Ry(1:end-i0+1) = Ry_raw(i0:end); Ry(end-i0+2:end) = Ry_raw(end); % we fix the last value % The signal is filtered out Ry = lsim(H_lpf, Ry, t); Ryd = lsim(H_lpf*s, Ry, t); Rydd = lsim(H_lpf*s^2, Ry, t); case 'sinusoidal' Ry(:) = args.Ry_amplitude*sin(2*pi/args.Ry_period*t); Ryd = args.Ry_amplitude*2*pi/args.Ry_period*cos(2*pi/args.Ry_period*t); Rydd = -args.Ry_amplitude*(2*pi/args.Ry_period)^2*sin(2*pi/args.Ry_period*t); otherwise warning('Ry_type is not set correctly'); end Ry = struct('time', t, 'signals', struct('values', Ry), 'deriv', Ryd, 'dderiv', Rydd); #+end_src *** Spindle :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab %% Spindle - Rz t = 0:Ts:Tmax; % Time Vector [s] Rz = zeros(length(t), 1); Rzd = zeros(length(t), 1); Rzdd = zeros(length(t), 1); switch args.Rz_type case 'constant' Rz(:) = args.Rz_amplitude; Rzd(:) = 0; Rzdd(:) = 0; case 'rotating-not-filtered' Rz(:) = 2*pi/args.Rz_period*t; % The signal is filtered out Rz(:) = 2*pi/args.Rz_period*t; Rzd(:) = 2*pi/args.Rz_period; Rzdd(:) = 0; % We add the angle offset Rz = Rz + args.Rz_amplitude; case 'rotating' Rz(:) = 2*pi/args.Rz_period*t; % The signal is filtered out Rz = lsim(H_lpf, Rz, t); Rzd = lsim(H_lpf*s, Rz, t); Rzdd = lsim(H_lpf*s^2, Rz, t); % We add the angle offset Rz = Rz + args.Rz_amplitude; otherwise warning('Rz_type is not set correctly'); end Rz = struct('time', t, 'signals', struct('values', Rz), 'deriv', Rzd, 'dderiv', Rzdd); #+end_src *** Micro Hexapod :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab %% Micro-Hexapod t = [0, Ts]; Dh = zeros(length(t), 6); Dhl = zeros(length(t), 6); switch args.Dh_type case 'constant' Dh = [args.Dh_pos, args.Dh_pos]; load('nass_stages.mat', 'micro_hexapod'); AP = [args.Dh_pos(1) ; args.Dh_pos(2) ; args.Dh_pos(3)]; tx = args.Dh_pos(4); ty = args.Dh_pos(5); tz = args.Dh_pos(6); ARB = [cos(tz) -sin(tz) 0; sin(tz) cos(tz) 0; 0 0 1]*... [ cos(ty) 0 sin(ty); 0 1 0; -sin(ty) 0 cos(ty)]*... [1 0 0; 0 cos(tx) -sin(tx); 0 sin(tx) cos(tx)]; [~, Dhl] = inverseKinematics(micro_hexapod, 'AP', AP, 'ARB', ARB); Dhl = [Dhl, Dhl]; otherwise warning('Dh_type is not set correctly'); end Dh = struct('time', t, 'signals', struct('values', Dh)); Dhl = struct('time', t, 'signals', struct('values', Dhl)); #+end_src *** Save the Structure #+begin_src matlab if exist('./mat', 'dir') if exist('./mat/nass_references.mat', 'file') save('mat/nass_references.mat', 'Dy', 'Ry', 'Rz', 'Dh', 'Dhl', 'args', 'Ts', '-append'); else save('mat/nass_references.mat', 'Dy', 'Ry', 'Rz', 'Dh', 'Dhl', 'args', 'Ts'); end elseif exist('./matlab', 'dir') if exist('./matlab/mat/nass_references.mat', 'file') save('matlab/mat/nass_references.mat', 'Dy', 'Ry', 'Rz', 'Dh', 'Dhl', 'args', 'Ts', '-append'); else save('matlab/mat/nass_references.mat', 'Dy', 'Ry', 'Rz', 'Dh', 'Dhl', 'args', 'Ts'); end end #+end_src ** Initialize Disturbances :PROPERTIES: :header-args:matlab+: :tangle matlab/src/initializeDisturbances.m :header-args:matlab+: :comments none :mkdirp yes :header-args:matlab+: :eval no :results none :END: *** Function Declaration and Documentation :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab function [] = initializeDisturbances(args) % initializeDisturbances - Initialize the disturbances % % Syntax: [] = initializeDisturbances(args) % % Inputs: % - args - #+end_src *** Optional Parameters :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab arguments % Global parameter to enable or disable the disturbances args.enable logical {mustBeNumericOrLogical} = true % Ground Motion - X direction args.Dwx logical {mustBeNumericOrLogical} = true % Ground Motion - Y direction args.Dwy logical {mustBeNumericOrLogical} = true % Ground Motion - Z direction args.Dwz logical {mustBeNumericOrLogical} = true % Translation Stage - X direction args.Fty_x logical {mustBeNumericOrLogical} = true % Translation Stage - Z direction args.Fty_z logical {mustBeNumericOrLogical} = true % Spindle - Z direction args.Frz_z logical {mustBeNumericOrLogical} = true end #+end_src *** Load Data :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab load('dist_psd.mat', 'dist_f'); #+end_src We remove the first frequency point that usually is very large. #+begin_src matlab :exports none dist_f.f = dist_f.f(2:end); dist_f.psd_gm = dist_f.psd_gm(2:end); dist_f.psd_ty = dist_f.psd_ty(2:end); dist_f.psd_rz = dist_f.psd_rz(2:end); #+end_src *** Parameters :PROPERTIES: :UNNUMBERED: t :END: We define some parameters that will be used in the algorithm. #+begin_src matlab Fs = 2*dist_f.f(end); % Sampling Frequency of data is twice the maximum frequency of the PSD vector [Hz] N = 2*length(dist_f.f); % Number of Samples match the one of the wanted PSD T0 = N/Fs; % Signal Duration [s] df = 1/T0; % Frequency resolution of the DFT [Hz] % Also equal to (dist_f.f(2)-dist_f.f(1)) t = linspace(0, T0, N+1)'; % Time Vector [s] Ts = 1/Fs; % Sampling Time [s] #+end_src *** Ground Motion :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab phi = dist_f.psd_gm; C = zeros(N/2,1); for i = 1:N/2 C(i) = sqrt(phi(i)*df); end #+end_src #+begin_src matlab if args.Dwx && args.enable rng(111); theta = 2*pi*rand(N/2,1); % Generate random phase [rad] Cx = [0 ; C.*complex(cos(theta),sin(theta))]; Cx = [Cx; flipud(conj(Cx(2:end)))];; Dwx = N/sqrt(2)*ifft(Cx); % Ground Motion - x direction [m] else Dwx = zeros(length(t), 1); end #+end_src #+begin_src matlab if args.Dwy && args.enable rng(112); theta = 2*pi*rand(N/2,1); % Generate random phase [rad] Cx = [0 ; C.*complex(cos(theta),sin(theta))]; Cx = [Cx; flipud(conj(Cx(2:end)))];; Dwy = N/sqrt(2)*ifft(Cx); % Ground Motion - y direction [m] else Dwy = zeros(length(t), 1); end #+end_src #+begin_src matlab if args.Dwy && args.enable rng(113); theta = 2*pi*rand(N/2,1); % Generate random phase [rad] Cx = [0 ; C.*complex(cos(theta),sin(theta))]; Cx = [Cx; flipud(conj(Cx(2:end)))];; Dwz = N/sqrt(2)*ifft(Cx); % Ground Motion - z direction [m] else Dwz = zeros(length(t), 1); end #+end_src *** Translation Stage - X direction :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab if args.Fty_x && args.enable phi = dist_f.psd_ty; % TODO - we take here the vertical direction which is wrong but approximate C = zeros(N/2,1); for i = 1:N/2 C(i) = sqrt(phi(i)*df); end rng(121); theta = 2*pi*rand(N/2,1); % Generate random phase [rad] Cx = [0 ; C.*complex(cos(theta),sin(theta))]; Cx = [Cx; flipud(conj(Cx(2:end)))];; u = N/sqrt(2)*ifft(Cx); % Disturbance Force Ty x [N] Fty_x = u; else Fty_x = zeros(length(t), 1); end #+end_src *** Translation Stage - Z direction :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab if args.Fty_z && args.enable phi = dist_f.psd_ty; C = zeros(N/2,1); for i = 1:N/2 C(i) = sqrt(phi(i)*df); end rng(122); theta = 2*pi*rand(N/2,1); % Generate random phase [rad] Cx = [0 ; C.*complex(cos(theta),sin(theta))]; Cx = [Cx; flipud(conj(Cx(2:end)))];; u = N/sqrt(2)*ifft(Cx); % Disturbance Force Ty z [N] Fty_z = u; else Fty_z = zeros(length(t), 1); end #+end_src *** Spindle - Z direction :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab if args.Frz_z && args.enable phi = dist_f.psd_rz; C = zeros(N/2,1); for i = 1:N/2 C(i) = sqrt(phi(i)*df); end rng(131); theta = 2*pi*rand(N/2,1); % Generate random phase [rad] Cx = [0 ; C.*complex(cos(theta),sin(theta))]; Cx = [Cx; flipud(conj(Cx(2:end)))];; u = N/sqrt(2)*ifft(Cx); % Disturbance Force Rz z [N] Frz_z = u; else Frz_z = zeros(length(t), 1); end #+end_src *** Direct Forces :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab u = zeros(length(t), 6); Fd = u; #+end_src *** Set initial value to zero :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab Dwx = Dwx - Dwx(1); Dwy = Dwy - Dwy(1); Dwz = Dwz - Dwz(1); Fty_x = Fty_x - Fty_x(1); Fty_z = Fty_z - Fty_z(1); Frz_z = Frz_z - Frz_z(1); #+end_src *** Save the Structure #+begin_src matlab if exist('./mat', 'dir') if exist('./mat/nass_disturbances.mat', 'file') save('mat/nass_disturbances.mat', 'Dwx', 'Dwy', 'Dwz', 'Fty_x', 'Fty_z', 'Frz_z', 'Fd', 'Ts', 't', 'args', '-append'); else save('mat/nass_disturbances.mat', 'Dwx', 'Dwy', 'Dwz', 'Fty_x', 'Fty_z', 'Frz_z', 'Fd', 'Ts', 't', 'args'); end elseif exist('./matlab', 'dir') if exist('./matlab/mat/nass_disturbances.mat', 'file') save('matlab/mat/nass_disturbances.mat', 'Dwx', 'Dwy', 'Dwz', 'Fty_x', 'Fty_z', 'Frz_z', 'Fd', 'Ts', 't', 'args', '-append'); else save('matlab/mat/nass_disturbances.mat', 'Dwx', 'Dwy', 'Dwz', 'Fty_x', 'Fty_z', 'Frz_z', 'Fd', 'Ts', 't', 'args'); end end #+end_src ** =initializeStewartPlatform=: Initialize the Stewart Platform structure :PROPERTIES: :header-args:matlab+: :tangle matlab/src/initializeStewartPlatform.m :header-args:matlab+: :comments none :mkdirp yes :eval no :END: *** Documentation :PROPERTIES: :UNNUMBERED: t :END: #+name: fig:stewart-frames-position #+caption: Definition of the position of the frames [[file:figs/stewart-frames-position.png]] *** Function description :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab function [stewart] = initializeStewartPlatform() % initializeStewartPlatform - Initialize the stewart structure % % Syntax: [stewart] = initializeStewartPlatform(args) % % Outputs: % - stewart - A structure with the following sub-structures: % - platform_F - % - platform_M - % - joints_F - % - joints_M - % - struts_F - % - struts_M - % - actuators - % - geometry - % - properties - #+end_src *** Initialize the Stewart structure :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab stewart = struct(); stewart.platform_F = struct(); stewart.platform_M = struct(); stewart.joints_F = struct(); stewart.joints_M = struct(); stewart.struts_F = struct(); stewart.struts_M = struct(); stewart.actuators = struct(); stewart.sensors = struct(); stewart.sensors.inertial = struct(); stewart.sensors.force = struct(); stewart.sensors.relative = struct(); stewart.geometry = struct(); stewart.kinematics = struct(); #+end_src ** =initializeFramesPositions=: Initialize the positions of frames {A}, {B}, {F} and {M} :PROPERTIES: :header-args:matlab+: :tangle matlab/src/initializeFramesPositions.m :header-args:matlab+: :comments none :mkdirp yes :eval no :END: *** Documentation :PROPERTIES: :UNNUMBERED: t :END: #+name: fig:stewart-frames-position #+caption: Definition of the position of the frames [[file:figs/stewart-frames-position.png]] *** Function description :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab function [stewart] = initializeFramesPositions(stewart, args) % initializeFramesPositions - Initialize the positions of frames {A}, {B}, {F} and {M} % % Syntax: [stewart] = initializeFramesPositions(stewart, args) % % Inputs: % - args - Can have the following fields: % - H [1x1] - Total Height of the Stewart Platform (height from {F} to {M}) [m] % - MO_B [1x1] - Height of the frame {B} with respect to {M} [m] % % Outputs: % - stewart - A structure with the following fields: % - geometry.H [1x1] - Total Height of the Stewart Platform [m] % - geometry.FO_M [3x1] - Position of {M} with respect to {F} [m] % - platform_M.MO_B [3x1] - Position of {B} with respect to {M} [m] % - platform_F.FO_A [3x1] - Position of {A} with respect to {F} [m] #+end_src *** Optional Parameters :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab arguments stewart args.H (1,1) double {mustBeNumeric, mustBePositive} = 90e-3 args.MO_B (1,1) double {mustBeNumeric} = 50e-3 end #+end_src *** Compute the position of each frame :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab H = args.H; % Total Height of the Stewart Platform [m] FO_M = [0; 0; H]; % Position of {M} with respect to {F} [m] MO_B = [0; 0; args.MO_B]; % Position of {B} with respect to {M} [m] FO_A = MO_B + FO_M; % Position of {A} with respect to {F} [m] #+end_src *** Populate the =stewart= structure :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab stewart.geometry.H = H; stewart.geometry.FO_M = FO_M; stewart.platform_M.MO_B = MO_B; stewart.platform_F.FO_A = FO_A; #+end_src ** =generateGeneralConfiguration=: Generate a Very General Configuration :PROPERTIES: :header-args:matlab+: :tangle matlab/src/generateGeneralConfiguration.m :header-args:matlab+: :comments none :mkdirp yes :eval no :END: *** Documentation :PROPERTIES: :UNNUMBERED: t :END: Joints are positions on a circle centered with the Z axis of {F} and {M} and at a chosen distance from {F} and {M}. The radius of the circles can be chosen as well as the angles where the joints are located (see Figure [[fig:joint_position_general]]). #+begin_src latex :file stewart_bottom_plate.pdf :tangle no \begin{tikzpicture} % Internal and external limit \draw[fill=white!80!black] (0, 0) circle [radius=3]; % Circle where the joints are located \draw[dashed] (0, 0) circle [radius=2.5]; % Bullets for the positions of the joints \node[] (J1) at ( 80:2.5){$\bullet$}; \node[] (J2) at (100:2.5){$\bullet$}; \node[] (J3) at (200:2.5){$\bullet$}; \node[] (J4) at (220:2.5){$\bullet$}; \node[] (J5) at (320:2.5){$\bullet$}; \node[] (J6) at (340:2.5){$\bullet$}; % Name of the points \node[above right] at (J1) {$a_{1}$}; \node[above left] at (J2) {$a_{2}$}; \node[above left] at (J3) {$a_{3}$}; \node[right ] at (J4) {$a_{4}$}; \node[left ] at (J5) {$a_{5}$}; \node[above right] at (J6) {$a_{6}$}; % First 2 angles \draw[dashed, ->] (0:1) arc [start angle=0, end angle=80, radius=1] node[below right]{$\theta_{1}$}; \draw[dashed, ->] (0:1.5) arc [start angle=0, end angle=100, radius=1.5] node[left ]{$\theta_{2}$}; % Division of 360 degrees by 3 \draw[dashed] (0, 0) -- ( 80:3.2); \draw[dashed] (0, 0) -- (100:3.2); \draw[dashed] (0, 0) -- (200:3.2); \draw[dashed] (0, 0) -- (220:3.2); \draw[dashed] (0, 0) -- (320:3.2); \draw[dashed] (0, 0) -- (340:3.2); % Radius for the position of the joints \draw[<->] (0, 0) --node[near end, above]{$R$} (180:2.5); \draw[->] (0, 0) -- ++(3.4, 0) node[above]{$x$}; \draw[->] (0, 0) -- ++(0, 3.4) node[left]{$y$}; \end{tikzpicture} #+end_src #+name: fig:joint_position_general #+caption: Position of the joints #+RESULTS: [[file:figs/stewart_bottom_plate.png]] *** Function description :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab function [stewart] = generateGeneralConfiguration(stewart, args) % generateGeneralConfiguration - Generate a Very General Configuration % % Syntax: [stewart] = generateGeneralConfiguration(stewart, args) % % Inputs: % - args - Can have the following fields: % - FH [1x1] - Height of the position of the fixed joints with respect to the frame {F} [m] % - FR [1x1] - Radius of the position of the fixed joints in the X-Y [m] % - FTh [6x1] - Angles of the fixed joints in the X-Y plane with respect to the X axis [rad] % - MH [1x1] - Height of the position of the mobile joints with respect to the frame {M} [m] % - FR [1x1] - Radius of the position of the mobile joints in the X-Y [m] % - MTh [6x1] - Angles of the mobile joints in the X-Y plane with respect to the X axis [rad] % % Outputs: % - stewart - updated Stewart structure with the added fields: % - platform_F.Fa [3x6] - Its i'th column is the position vector of joint ai with respect to {F} % - platform_M.Mb [3x6] - Its i'th column is the position vector of joint bi with respect to {M} #+end_src *** Optional Parameters :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab arguments stewart args.FH (1,1) double {mustBeNumeric, mustBePositive} = 15e-3 args.FR (1,1) double {mustBeNumeric, mustBePositive} = 115e-3; args.FTh (6,1) double {mustBeNumeric} = [-10, 10, 120-10, 120+10, 240-10, 240+10]*(pi/180); args.MH (1,1) double {mustBeNumeric, mustBePositive} = 15e-3 args.MR (1,1) double {mustBeNumeric, mustBePositive} = 90e-3; args.MTh (6,1) double {mustBeNumeric} = [-60+10, 60-10, 60+10, 180-10, 180+10, -60-10]*(pi/180); end #+end_src *** Compute the pose :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab Fa = zeros(3,6); Mb = zeros(3,6); #+end_src #+begin_src matlab for i = 1:6 Fa(:,i) = [args.FR*cos(args.FTh(i)); args.FR*sin(args.FTh(i)); args.FH]; Mb(:,i) = [args.MR*cos(args.MTh(i)); args.MR*sin(args.MTh(i)); -args.MH]; end #+end_src *** Populate the =stewart= structure :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab stewart.platform_F.Fa = Fa; stewart.platform_M.Mb = Mb; #+end_src ** =computeJointsPose=: Compute the Pose of the Joints :PROPERTIES: :header-args:matlab+: :tangle matlab/src/computeJointsPose.m :header-args:matlab+: :comments none :mkdirp yes :eval no :END: *** Documentation :PROPERTIES: :UNNUMBERED: t :END: #+name: fig:stewart-struts #+caption: Position and orientation of the struts [[file:figs/stewart-struts.png]] *** Function description :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab function [stewart] = computeJointsPose(stewart) % computeJointsPose - % % Syntax: [stewart] = computeJointsPose(stewart) % % Inputs: % - stewart - A structure with the following fields % - platform_F.Fa [3x6] - Its i'th column is the position vector of joint ai with respect to {F} % - platform_M.Mb [3x6] - Its i'th column is the position vector of joint bi with respect to {M} % - platform_F.FO_A [3x1] - Position of {A} with respect to {F} % - platform_M.MO_B [3x1] - Position of {B} with respect to {M} % - geometry.FO_M [3x1] - Position of {M} with respect to {F} % % Outputs: % - stewart - A structure with the following added fields % - geometry.Aa [3x6] - The i'th column is the position of ai with respect to {A} % - geometry.Ab [3x6] - The i'th column is the position of bi with respect to {A} % - geometry.Ba [3x6] - The i'th column is the position of ai with respect to {B} % - geometry.Bb [3x6] - The i'th column is the position of bi with respect to {B} % - geometry.l [6x1] - The i'th element is the initial length of strut i % - geometry.As [3x6] - The i'th column is the unit vector of strut i expressed in {A} % - geometry.Bs [3x6] - The i'th column is the unit vector of strut i expressed in {B} % - struts_F.l [6x1] - Length of the Fixed part of the i'th strut % - struts_M.l [6x1] - Length of the Mobile part of the i'th strut % - platform_F.FRa [3x3x6] - The i'th 3x3 array is the rotation matrix to orientate the bottom of the i'th strut from {F} % - platform_M.MRb [3x3x6] - The i'th 3x3 array is the rotation matrix to orientate the top of the i'th strut from {M} #+end_src *** Check the =stewart= structure elements :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab assert(isfield(stewart.platform_F, 'Fa'), 'stewart.platform_F should have attribute Fa') Fa = stewart.platform_F.Fa; assert(isfield(stewart.platform_M, 'Mb'), 'stewart.platform_M should have attribute Mb') Mb = stewart.platform_M.Mb; assert(isfield(stewart.platform_F, 'FO_A'), 'stewart.platform_F should have attribute FO_A') FO_A = stewart.platform_F.FO_A; assert(isfield(stewart.platform_M, 'MO_B'), 'stewart.platform_M should have attribute MO_B') MO_B = stewart.platform_M.MO_B; assert(isfield(stewart.geometry, 'FO_M'), 'stewart.geometry should have attribute FO_M') FO_M = stewart.geometry.FO_M; #+end_src *** Compute the position of the Joints :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab Aa = Fa - repmat(FO_A, [1, 6]); Bb = Mb - repmat(MO_B, [1, 6]); Ab = Bb - repmat(-MO_B-FO_M+FO_A, [1, 6]); Ba = Aa - repmat( MO_B+FO_M-FO_A, [1, 6]); #+end_src *** Compute the strut length and orientation :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab As = (Ab - Aa)./vecnorm(Ab - Aa); % As_i is the i'th vector of As l = vecnorm(Ab - Aa)'; #+end_src #+begin_src matlab Bs = (Bb - Ba)./vecnorm(Bb - Ba); #+end_src *** Compute the orientation of the Joints :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab FRa = zeros(3,3,6); MRb = zeros(3,3,6); for i = 1:6 FRa(:,:,i) = [cross([0;1;0], As(:,i)) , cross(As(:,i), cross([0;1;0], As(:,i))) , As(:,i)]; FRa(:,:,i) = FRa(:,:,i)./vecnorm(FRa(:,:,i)); MRb(:,:,i) = [cross([0;1;0], Bs(:,i)) , cross(Bs(:,i), cross([0;1;0], Bs(:,i))) , Bs(:,i)]; MRb(:,:,i) = MRb(:,:,i)./vecnorm(MRb(:,:,i)); end #+end_src *** Populate the =stewart= structure :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab stewart.geometry.Aa = Aa; stewart.geometry.Ab = Ab; stewart.geometry.Ba = Ba; stewart.geometry.Bb = Bb; stewart.geometry.As = As; stewart.geometry.Bs = Bs; stewart.geometry.l = l; stewart.struts_F.l = l/2; stewart.struts_M.l = l/2; stewart.platform_F.FRa = FRa; stewart.platform_M.MRb = MRb; #+end_src ** =initializeCylindricalPlatforms=: Initialize the geometry of the Fixed and Mobile Platforms :PROPERTIES: :header-args:matlab+: :tangle matlab/src/initializeCylindricalPlatforms.m :header-args:matlab+: :comments none :mkdirp yes :eval no :END: *** Function description :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab function [stewart] = initializeCylindricalPlatforms(stewart, args) % initializeCylindricalPlatforms - Initialize the geometry of the Fixed and Mobile Platforms % % Syntax: [stewart] = initializeCylindricalPlatforms(args) % % Inputs: % - args - Structure with the following fields: % - Fpm [1x1] - Fixed Platform Mass [kg] % - Fph [1x1] - Fixed Platform Height [m] % - Fpr [1x1] - Fixed Platform Radius [m] % - Mpm [1x1] - Mobile Platform Mass [kg] % - Mph [1x1] - Mobile Platform Height [m] % - Mpr [1x1] - Mobile Platform Radius [m] % % Outputs: % - stewart - updated Stewart structure with the added fields: % - platform_F [struct] - structure with the following fields: % - type = 1 % - M [1x1] - Fixed Platform Mass [kg] % - I [3x3] - Fixed Platform Inertia matrix [kg*m^2] % - H [1x1] - Fixed Platform Height [m] % - R [1x1] - Fixed Platform Radius [m] % - platform_M [struct] - structure with the following fields: % - M [1x1] - Mobile Platform Mass [kg] % - I [3x3] - Mobile Platform Inertia matrix [kg*m^2] % - H [1x1] - Mobile Platform Height [m] % - R [1x1] - Mobile Platform Radius [m] #+end_src *** Optional Parameters :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab arguments stewart args.Fpm (1,1) double {mustBeNumeric, mustBePositive} = 1 args.Fph (1,1) double {mustBeNumeric, mustBePositive} = 10e-3 args.Fpr (1,1) double {mustBeNumeric, mustBePositive} = 125e-3 args.Mpm (1,1) double {mustBeNumeric, mustBePositive} = 1 args.Mph (1,1) double {mustBeNumeric, mustBePositive} = 10e-3 args.Mpr (1,1) double {mustBeNumeric, mustBePositive} = 100e-3 end #+end_src *** Compute the Inertia matrices of platforms :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab I_F = diag([1/12*args.Fpm * (3*args.Fpr^2 + args.Fph^2), ... 1/12*args.Fpm * (3*args.Fpr^2 + args.Fph^2), ... 1/2 *args.Fpm * args.Fpr^2]); #+end_src #+begin_src matlab I_M = diag([1/12*args.Mpm * (3*args.Mpr^2 + args.Mph^2), ... 1/12*args.Mpm * (3*args.Mpr^2 + args.Mph^2), ... 1/2 *args.Mpm * args.Mpr^2]); #+end_src *** Populate the =stewart= structure :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab stewart.platform_F.type = 1; stewart.platform_F.I = I_F; stewart.platform_F.M = args.Fpm; stewart.platform_F.R = args.Fpr; stewart.platform_F.H = args.Fph; #+end_src #+begin_src matlab stewart.platform_M.type = 1; stewart.platform_M.I = I_M; stewart.platform_M.M = args.Mpm; stewart.platform_M.R = args.Mpr; stewart.platform_M.H = args.Mph; #+end_src ** =initializeCylindricalStruts=: Define the inertia of cylindrical struts :PROPERTIES: :header-args:matlab+: :tangle matlab/src/initializeCylindricalStruts.m :header-args:matlab+: :comments none :mkdirp yes :eval no :END: *** Function description :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab function [stewart] = initializeCylindricalStruts(stewart, args) % initializeCylindricalStruts - Define the mass and moment of inertia of cylindrical struts % % Syntax: [stewart] = initializeCylindricalStruts(args) % % Inputs: % - args - Structure with the following fields: % - Fsm [1x1] - Mass of the Fixed part of the struts [kg] % - Fsh [1x1] - Height of cylinder for the Fixed part of the struts [m] % - Fsr [1x1] - Radius of cylinder for the Fixed part of the struts [m] % - Msm [1x1] - Mass of the Mobile part of the struts [kg] % - Msh [1x1] - Height of cylinder for the Mobile part of the struts [m] % - Msr [1x1] - Radius of cylinder for the Mobile part of the struts [m] % % Outputs: % - stewart - updated Stewart structure with the added fields: % - struts_F [struct] - structure with the following fields: % - M [6x1] - Mass of the Fixed part of the struts [kg] % - I [3x3x6] - Moment of Inertia for the Fixed part of the struts [kg*m^2] % - H [6x1] - Height of cylinder for the Fixed part of the struts [m] % - R [6x1] - Radius of cylinder for the Fixed part of the struts [m] % - struts_M [struct] - structure with the following fields: % - M [6x1] - Mass of the Mobile part of the struts [kg] % - I [3x3x6] - Moment of Inertia for the Mobile part of the struts [kg*m^2] % - H [6x1] - Height of cylinder for the Mobile part of the struts [m] % - R [6x1] - Radius of cylinder for the Mobile part of the struts [m] #+end_src *** Optional Parameters :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab arguments stewart args.Fsm (1,1) double {mustBeNumeric, mustBePositive} = 0.1 args.Fsh (1,1) double {mustBeNumeric, mustBePositive} = 50e-3 args.Fsr (1,1) double {mustBeNumeric, mustBePositive} = 5e-3 args.Msm (1,1) double {mustBeNumeric, mustBePositive} = 0.1 args.Msh (1,1) double {mustBeNumeric, mustBePositive} = 50e-3 args.Msr (1,1) double {mustBeNumeric, mustBePositive} = 5e-3 end #+end_src *** Compute the properties of the cylindrical struts :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab Fsm = ones(6,1).*args.Fsm; Fsh = ones(6,1).*args.Fsh; Fsr = ones(6,1).*args.Fsr; Msm = ones(6,1).*args.Msm; Msh = ones(6,1).*args.Msh; Msr = ones(6,1).*args.Msr; #+end_src #+begin_src matlab I_F = zeros(3, 3, 6); % Inertia of the "fixed" part of the strut I_M = zeros(3, 3, 6); % Inertia of the "mobile" part of the strut for i = 1:6 I_F(:,:,i) = diag([1/12 * Fsm(i) * (3*Fsr(i)^2 + Fsh(i)^2), ... 1/12 * Fsm(i) * (3*Fsr(i)^2 + Fsh(i)^2), ... 1/2 * Fsm(i) * Fsr(i)^2]); I_M(:,:,i) = diag([1/12 * Msm(i) * (3*Msr(i)^2 + Msh(i)^2), ... 1/12 * Msm(i) * (3*Msr(i)^2 + Msh(i)^2), ... 1/2 * Msm(i) * Msr(i)^2]); end #+end_src *** Populate the =stewart= structure :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab stewart.struts_M.type = 1; stewart.struts_M.I = I_M; stewart.struts_M.M = Msm; stewart.struts_M.R = Msr; stewart.struts_M.H = Msh; #+end_src #+begin_src matlab stewart.struts_F.type = 1; stewart.struts_F.I = I_F; stewart.struts_F.M = Fsm; stewart.struts_F.R = Fsr; stewart.struts_F.H = Fsh; #+end_src ** =initializeStrutDynamics=: Add Stiffness and Damping properties of each strut :PROPERTIES: :header-args:matlab+: :tangle matlab/src/initializeStrutDynamics.m :header-args:matlab+: :comments none :mkdirp yes :eval no :END: *** Documentation :PROPERTIES: :UNNUMBERED: t :END: #+name: fig:piezoelectric_stack #+attr_html: :width 500px #+caption: Example of a piezoelectric stach actuator (PI) [[file:figs/piezoelectric_stack.jpg]] A simplistic model of such amplified actuator is shown in Figure [[fig:actuator_model_simple]] where: - $K$ represent the vertical stiffness of the actuator - $C$ represent the vertical damping of the actuator - $F$ represents the force applied by the actuator - $F_{m}$ represents the total measured force - $v_{m}$ represents the absolute velocity of the top part of the actuator - $d_{m}$ represents the total relative displacement of the actuator #+begin_src latex :file actuator_model_simple.pdf :tangle no \begin{tikzpicture} \draw (-1, 0) -- (1, 0); % Spring, Damper, and Actuator \draw[spring] (-1, 0) -- (-1, 1.5) node[midway, left=0.1]{$K$}; \draw[damper] ( 0, 0) -- ( 0, 1.5) node[midway, left=0.2]{$C$}; \draw[actuator] ( 1, 0) -- ( 1, 1.5) node[midway, left=0.1](F){$F$}; \node[forcesensor={2}{0.2}] (fsens) at (0, 1.5){}; \node[left] at (fsens.west) {$F_{m}$}; \draw[dashed] (1, 0) -- ++(0.4, 0); \draw[dashed] (1, 1.7) -- ++(0.4, 0); \draw[->] (0, 1.7)node[]{$\bullet$} -- ++(0, 0.5) node[right]{$v_{m}$}; \draw[<->] (1.4, 0) -- ++(0, 1.7) node[midway, right]{$d_{m}$}; \end{tikzpicture} #+end_src #+name: fig:actuator_model_simple #+caption: Simple model of an Actuator #+RESULTS: [[file:figs/actuator_model_simple.png]] *** Function description :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab function [stewart] = initializeStrutDynamics(stewart, args) % initializeStrutDynamics - Add Stiffness and Damping properties of each strut % % Syntax: [stewart] = initializeStrutDynamics(args) % % Inputs: % - args - Structure with the following fields: % - K [6x1] - Stiffness of each strut [N/m] % - C [6x1] - Damping of each strut [N/(m/s)] % % Outputs: % - stewart - updated Stewart structure with the added fields: % - actuators.type = 1 % - actuators.K [6x1] - Stiffness of each strut [N/m] % - actuators.C [6x1] - Damping of each strut [N/(m/s)] #+end_src *** Optional Parameters :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab arguments stewart args.type char {mustBeMember(args.type,{'classical', 'amplified'})} = 'classical' args.K (6,1) double {mustBeNumeric, mustBeNonnegative} = 20e6*ones(6,1) args.C (6,1) double {mustBeNumeric, mustBeNonnegative} = 2e1*ones(6,1) args.k1 (6,1) double {mustBeNumeric} = 1e6*ones(6,1) args.ke (6,1) double {mustBeNumeric} = 5e6*ones(6,1) args.ka (6,1) double {mustBeNumeric} = 60e6*ones(6,1) args.c1 (6,1) double {mustBeNumeric} = 10*ones(6,1) args.F_gain (6,1) double {mustBeNumeric} = 1*ones(6,1) args.me (6,1) double {mustBeNumeric} = 0.01*ones(6,1) args.ma (6,1) double {mustBeNumeric} = 0.01*ones(6,1) end #+end_src *** Add Stiffness and Damping properties of each strut :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab if strcmp(args.type, 'classical') stewart.actuators.type = 1; elseif strcmp(args.type, 'amplified') stewart.actuators.type = 2; end stewart.actuators.K = args.K; stewart.actuators.C = args.C; stewart.actuators.k1 = args.k1; stewart.actuators.c1 = args.c1; stewart.actuators.ka = args.ka; stewart.actuators.ke = args.ke; stewart.actuators.F_gain = args.F_gain; stewart.actuators.ma = args.ma; stewart.actuators.me = args.me; #+end_src ** =initializeJointDynamics=: Add Stiffness and Damping properties for spherical joints :PROPERTIES: :header-args:matlab+: :tangle matlab/src/initializeJointDynamics.m :header-args:matlab+: :comments none :mkdirp yes :eval no :END: *** Function description :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab function [stewart] = initializeJointDynamics(stewart, args) % initializeJointDynamics - Add Stiffness and Damping properties for the spherical joints % % Syntax: [stewart] = initializeJointDynamics(args) % % Inputs: % - args - Structure with the following fields: % - type_F - 'universal', 'spherical', 'universal_p', 'spherical_p' % - type_M - 'universal', 'spherical', 'universal_p', 'spherical_p' % - Kf_M [6x1] - Bending (Rx, Ry) Stiffness for each top joints [(N.m)/rad] % - Kt_M [6x1] - Torsion (Rz) Stiffness for each top joints [(N.m)/rad] % - Cf_M [6x1] - Bending (Rx, Ry) Damping of each top joint [(N.m)/(rad/s)] % - Ct_M [6x1] - Torsion (Rz) Damping of each top joint [(N.m)/(rad/s)] % - Kf_F [6x1] - Bending (Rx, Ry) Stiffness for each bottom joints [(N.m)/rad] % - Kt_F [6x1] - Torsion (Rz) Stiffness for each bottom joints [(N.m)/rad] % - Cf_F [6x1] - Bending (Rx, Ry) Damping of each bottom joint [(N.m)/(rad/s)] % - Cf_F [6x1] - Torsion (Rz) Damping of each bottom joint [(N.m)/(rad/s)] % % Outputs: % - stewart - updated Stewart structure with the added fields: % - stewart.joints_F and stewart.joints_M: % - type - 1 (universal), 2 (spherical), 3 (universal perfect), 4 (spherical perfect) % - Kx, Ky, Kz [6x1] - Translation (Tx, Ty, Tz) Stiffness [N/m] % - Kf [6x1] - Flexion (Rx, Ry) Stiffness [(N.m)/rad] % - Kt [6x1] - Torsion (Rz) Stiffness [(N.m)/rad] % - Cx, Cy, Cz [6x1] - Translation (Rx, Ry) Damping [N/(m/s)] % - Cf [6x1] - Flexion (Rx, Ry) Damping [(N.m)/(rad/s)] % - Cb [6x1] - Torsion (Rz) Damping [(N.m)/(rad/s)] #+end_src *** Optional Parameters :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab arguments stewart args.type_F char {mustBeMember(args.type_F,{'universal', 'spherical', 'universal_p', 'spherical_p', 'universal_3dof', 'spherical_3dof', 'flexible'})} = 'universal' args.type_M char {mustBeMember(args.type_M,{'universal', 'spherical', 'universal_p', 'spherical_p', 'universal_3dof', 'spherical_3dof', 'flexible'})} = 'spherical' args.Kf_M (6,1) double {mustBeNumeric, mustBeNonnegative} = 33*ones(6,1) args.Cf_M (6,1) double {mustBeNumeric, mustBeNonnegative} = 1e-4*ones(6,1) args.Kt_M (6,1) double {mustBeNumeric, mustBeNonnegative} = 236*ones(6,1) args.Ct_M (6,1) double {mustBeNumeric, mustBeNonnegative} = 1e-3*ones(6,1) args.Kf_F (6,1) double {mustBeNumeric, mustBeNonnegative} = 33*ones(6,1) args.Cf_F (6,1) double {mustBeNumeric, mustBeNonnegative} = 1e-4*ones(6,1) args.Kt_F (6,1) double {mustBeNumeric, mustBeNonnegative} = 236*ones(6,1) args.Ct_F (6,1) double {mustBeNumeric, mustBeNonnegative} = 1e-3*ones(6,1) args.Ka_F (6,1) double {mustBeNumeric, mustBeNonnegative} = 1.2e8*ones(6,1) args.Ca_F (6,1) double {mustBeNumeric, mustBeNonnegative} = 1e1*ones(6,1) args.Kr_F (6,1) double {mustBeNumeric, mustBeNonnegative} = 1.1e7*ones(6,1) args.Cr_F (6,1) double {mustBeNumeric, mustBeNonnegative} = 1e1*ones(6,1) args.Ka_M (6,1) double {mustBeNumeric, mustBeNonnegative} = 1.2e8*ones(6,1) args.Ca_M (6,1) double {mustBeNumeric, mustBeNonnegative} = 1e1*ones(6,1) args.Kr_M (6,1) double {mustBeNumeric, mustBeNonnegative} = 1.1e7*ones(6,1) args.Cr_M (6,1) double {mustBeNumeric, mustBeNonnegative} = 1e1*ones(6,1) args.K_M double {mustBeNumeric} = zeros(6,6) args.M_M double {mustBeNumeric} = zeros(6,6) args.n_xyz_M double {mustBeNumeric} = zeros(2,3) args.xi_M double {mustBeNumeric} = 0.1 args.step_file_M char {} = '' args.K_F double {mustBeNumeric} = zeros(6,6) args.M_F double {mustBeNumeric} = zeros(6,6) args.n_xyz_F double {mustBeNumeric} = zeros(2,3) args.xi_F double {mustBeNumeric} = 0.1 args.step_file_F char {} = '' end #+end_src *** Add Actuator Type :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab switch args.type_F case 'universal' stewart.joints_F.type = 1; case 'spherical' stewart.joints_F.type = 2; case 'universal_p' stewart.joints_F.type = 3; case 'spherical_p' stewart.joints_F.type = 4; case 'flexible' stewart.joints_F.type = 5; case 'universal_3dof' stewart.joints_F.type = 6; case 'spherical_3dof' stewart.joints_F.type = 7; end switch args.type_M case 'universal' stewart.joints_M.type = 1; case 'spherical' stewart.joints_M.type = 2; case 'universal_p' stewart.joints_M.type = 3; case 'spherical_p' stewart.joints_M.type = 4; case 'flexible' stewart.joints_M.type = 5; case 'universal_3dof' stewart.joints_M.type = 6; case 'spherical_3dof' stewart.joints_M.type = 7; end #+end_src *** Add Stiffness and Damping in Translation of each strut :PROPERTIES: :UNNUMBERED: t :END: Axial and Radial (shear) Stiffness #+begin_src matlab stewart.joints_M.Ka = args.Ka_M; stewart.joints_M.Kr = args.Kr_M; stewart.joints_F.Ka = args.Ka_F; stewart.joints_F.Kr = args.Kr_F; #+end_src Translation Damping #+begin_src matlab stewart.joints_M.Ca = args.Ca_M; stewart.joints_M.Cr = args.Cr_M; stewart.joints_F.Ca = args.Ca_F; stewart.joints_F.Cr = args.Cr_F; #+end_src *** Add Stiffness and Damping in Rotation of each strut :PROPERTIES: :UNNUMBERED: t :END: Rotational Stiffness #+begin_src matlab stewart.joints_M.Kf = args.Kf_M; stewart.joints_M.Kt = args.Kt_M; stewart.joints_F.Kf = args.Kf_F; stewart.joints_F.Kt = args.Kt_F; #+end_src Rotational Damping #+begin_src matlab stewart.joints_M.Cf = args.Cf_M; stewart.joints_M.Ct = args.Ct_M; stewart.joints_F.Cf = args.Cf_F; stewart.joints_F.Ct = args.Ct_F; #+end_src *** Stiffness and Mass matrices for flexible joint :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab stewart.joints_F.M = args.M_F; stewart.joints_F.K = args.K_F; stewart.joints_F.n_xyz = args.n_xyz_F; stewart.joints_F.xi = args.xi_F; stewart.joints_F.xi = args.xi_F; stewart.joints_F.step_file = args.step_file_F; stewart.joints_M.M = args.M_M; stewart.joints_M.K = args.K_M; stewart.joints_M.n_xyz = args.n_xyz_M; stewart.joints_M.xi = args.xi_M; stewart.joints_M.step_file = args.step_file_M; #+end_src ** =initializeStewartPose=: Determine the initial stroke in each leg to have the wanted pose :PROPERTIES: :header-args:matlab+: :tangle matlab/src/initializeStewartPose.m :header-args:matlab+: :comments none :mkdirp yes :eval no :END: *** Function description :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab function [stewart] = initializeStewartPose(stewart, args) % initializeStewartPose - Determine the initial stroke in each leg to have the wanted pose % It uses the inverse kinematic % % Syntax: [stewart] = initializeStewartPose(stewart, args) % % Inputs: % - stewart - A structure with the following fields % - Aa [3x6] - The positions ai expressed in {A} % - Bb [3x6] - The positions bi expressed in {B} % - args - Can have the following fields: % - AP [3x1] - The wanted position of {B} with respect to {A} % - ARB [3x3] - The rotation matrix that gives the wanted orientation of {B} with respect to {A} % % Outputs: % - stewart - updated Stewart structure with the added fields: % - actuators.Leq [6x1] - The 6 needed displacement of the struts from the initial position in [m] to have the wanted pose of {B} w.r.t. {A} #+end_src *** Optional Parameters :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab arguments stewart args.AP (3,1) double {mustBeNumeric} = zeros(3,1) args.ARB (3,3) double {mustBeNumeric} = eye(3) end #+end_src *** Use the Inverse Kinematic function :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab [Li, dLi] = inverseKinematics(stewart, 'AP', args.AP, 'ARB', args.ARB); #+end_src *** Populate the =stewart= structure :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab stewart.actuators.Leq = dLi; #+end_src ** =computeJacobian=: Compute the Jacobian Matrix :PROPERTIES: :header-args:matlab+: :tangle matlab/src/computeJacobian.m :header-args:matlab+: :comments none :mkdirp yes :eval no :END: *** Function description :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab function [stewart] = computeJacobian(stewart) % computeJacobian - % % Syntax: [stewart] = computeJacobian(stewart) % % Inputs: % - stewart - With at least the following fields: % - geometry.As [3x6] - The 6 unit vectors for each strut expressed in {A} % - geometry.Ab [3x6] - The 6 position of the joints bi expressed in {A} % - actuators.K [6x1] - Total stiffness of the actuators % % Outputs: % - stewart - With the 3 added field: % - kinematics.J [6x6] - The Jacobian Matrix % - kinematics.K [6x6] - The Stiffness Matrix % - kinematics.C [6x6] - The Compliance Matrix #+end_src *** Check the =stewart= structure elements :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab assert(isfield(stewart.geometry, 'As'), 'stewart.geometry should have attribute As') As = stewart.geometry.As; assert(isfield(stewart.geometry, 'Ab'), 'stewart.geometry should have attribute Ab') Ab = stewart.geometry.Ab; assert(isfield(stewart.actuators, 'K'), 'stewart.actuators should have attribute K') Ki = stewart.actuators.K; #+end_src *** Compute Jacobian Matrix :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab J = [As' , cross(Ab, As)']; #+end_src *** Compute Stiffness Matrix :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab K = J'*diag(Ki)*J; #+end_src *** Compute Compliance Matrix :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab C = inv(K); #+end_src *** Populate the =stewart= structure :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab stewart.kinematics.J = J; stewart.kinematics.K = K; stewart.kinematics.C = C; #+end_src ** =initializeInertialSensor=: Initialize the inertial sensor in each strut :PROPERTIES: :header-args:matlab+: :tangle matlab/src/initializeInertialSensor.m :header-args:matlab+: :comments none :mkdirp yes :eval no :END: *** Geophone - Working Principle :PROPERTIES: :UNNUMBERED: t :END: From the schematic of the Z-axis geophone shown in Figure [[fig:z_axis_geophone]], we can write the transfer function from the support velocity $\dot{w}$ to the relative velocity of the inertial mass $\dot{d}$: \[ \frac{\dot{d}}{\dot{w}} = \frac{-\frac{s^2}{{\omega_0}^2}}{\frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1} \] with: - $\omega_0 = \sqrt{\frac{k}{m}}$ - $\xi = \frac{1}{2} \sqrt{\frac{m}{k}}$ #+name: fig:z_axis_geophone #+caption: Schematic of a Z-Axis geophone [[file:figs/inertial_sensor.png]] We see that at frequencies above $\omega_0$: \[ \frac{\dot{d}}{\dot{w}} \approx -1 \] And thus, the measurement of the relative velocity of the mass with respect to its support gives the absolute velocity of the support. We generally want to have the smallest resonant frequency $\omega_0$ to measure low frequency absolute velocity, however there is a trade-off between $\omega_0$ and the mass of the inertial mass. *** Accelerometer - Working Principle :PROPERTIES: :UNNUMBERED: t :END: From the schematic of the Z-axis accelerometer shown in Figure [[fig:z_axis_accelerometer]], we can write the transfer function from the support acceleration $\ddot{w}$ to the relative position of the inertial mass $d$: \[ \frac{d}{\ddot{w}} = \frac{-\frac{1}{{\omega_0}^2}}{\frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1} \] with: - $\omega_0 = \sqrt{\frac{k}{m}}$ - $\xi = \frac{1}{2} \sqrt{\frac{m}{k}}$ #+name: fig:z_axis_accelerometer #+caption: Schematic of a Z-Axis geophone [[file:figs/inertial_sensor.png]] We see that at frequencies below $\omega_0$: \[ \frac{d}{\ddot{w}} \approx -\frac{1}{{\omega_0}^2} \] And thus, the measurement of the relative displacement of the mass with respect to its support gives the absolute acceleration of the support. Note that there is trade-off between: - the highest measurable acceleration $\omega_0$ - the sensitivity of the accelerometer which is equal to $-\frac{1}{{\omega_0}^2}$ *** Function description :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab function [stewart] = initializeInertialSensor(stewart, args) % initializeInertialSensor - Initialize the inertial sensor in each strut % % Syntax: [stewart] = initializeInertialSensor(args) % % Inputs: % - args - Structure with the following fields: % - type - 'geophone', 'accelerometer', 'none' % - mass [1x1] - Weight of the inertial mass [kg] % - freq [1x1] - Cutoff frequency [Hz] % % Outputs: % - stewart - updated Stewart structure with the added fields: % - stewart.sensors.inertial % - type - 1 (geophone), 2 (accelerometer), 3 (none) % - K [1x1] - Stiffness [N/m] % - C [1x1] - Damping [N/(m/s)] % - M [1x1] - Inertial Mass [kg] % - G [1x1] - Gain #+end_src *** Optional Parameters :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab arguments stewart args.type char {mustBeMember(args.type,{'geophone', 'accelerometer', 'none'})} = 'none' args.mass (1,1) double {mustBeNumeric, mustBeNonnegative} = 1e-2 args.freq (1,1) double {mustBeNumeric, mustBeNonnegative} = 1e3 end #+end_src *** Compute the properties of the sensor :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab sensor = struct(); switch args.type case 'geophone' sensor.type = 1; sensor.M = args.mass; sensor.K = sensor.M * (2*pi*args.freq)^2; sensor.C = 2*sqrt(sensor.M * sensor.K); case 'accelerometer' sensor.type = 2; sensor.M = args.mass; sensor.K = sensor.M * (2*pi*args.freq)^2; sensor.C = 2*sqrt(sensor.M * sensor.K); sensor.G = -sensor.K/sensor.M; case 'none' sensor.type = 3; end #+end_src *** Populate the =stewart= structure :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab stewart.sensors.inertial = sensor; #+end_src ** =inverseKinematics=: Compute Inverse Kinematics :PROPERTIES: :header-args:matlab+: :tangle matlab/src/inverseKinematics.m :header-args:matlab+: :comments none :mkdirp yes :eval no :END: *** Theory :PROPERTIES: :UNNUMBERED: t :END: For inverse kinematic analysis, it is assumed that the position ${}^A\bm{P}$ and orientation of the moving platform ${}^A\bm{R}_B$ are given and the problem is to obtain the joint variables, namely, $\bm{L} = [l_1, l_2, \dots, l_6]^T$. From the geometry of the manipulator, the loop closure for each limb, $i = 1, 2, \dots, 6$ can be written as \begin{align*} l_i {}^A\hat{\bm{s}}_i &= {}^A\bm{A} + {}^A\bm{b}_i - {}^A\bm{a}_i \\ &= {}^A\bm{A} + {}^A\bm{R}_b {}^B\bm{b}_i - {}^A\bm{a}_i \end{align*} To obtain the length of each actuator and eliminate $\hat{\bm{s}}_i$, it is sufficient to dot multiply each side by itself: \begin{equation} l_i^2 \left[ {}^A\hat{\bm{s}}_i^T {}^A\hat{\bm{s}}_i \right] = \left[ {}^A\bm{P} + {}^A\bm{R}_B {}^B\bm{b}_i - {}^A\bm{a}_i \right]^T \left[ {}^A\bm{P} + {}^A\bm{R}_B {}^B\bm{b}_i - {}^A\bm{a}_i \right] \end{equation} Hence, for $i = 1, 2, \dots, 6$, each limb length can be uniquely determined by: \begin{equation} l_i = \sqrt{{}^A\bm{P}^T {}^A\bm{P} + {}^B\bm{b}_i^T {}^B\bm{b}_i + {}^A\bm{a}_i^T {}^A\bm{a}_i - 2 {}^A\bm{P}^T {}^A\bm{a}_i + 2 {}^A\bm{P}^T \left[{}^A\bm{R}_B {}^B\bm{b}_i\right] - 2 \left[{}^A\bm{R}_B {}^B\bm{b}_i\right]^T {}^A\bm{a}_i} \end{equation} If the position and orientation of the moving platform lie in the feasible workspace of the manipulator, one unique solution to the limb length is determined by the above equation. Otherwise, when the limbs' lengths derived yield complex numbers, then the position or orientation of the moving platform is not reachable. *** Function description :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab function [Li, dLi] = inverseKinematics(stewart, args) % inverseKinematics - Compute the needed length of each strut to have the wanted position and orientation of {B} with respect to {A} % % Syntax: [stewart] = inverseKinematics(stewart) % % Inputs: % - stewart - A structure with the following fields % - geometry.Aa [3x6] - The positions ai expressed in {A} % - geometry.Bb [3x6] - The positions bi expressed in {B} % - geometry.l [6x1] - Length of each strut % - args - Can have the following fields: % - AP [3x1] - The wanted position of {B} with respect to {A} % - ARB [3x3] - The rotation matrix that gives the wanted orientation of {B} with respect to {A} % % Outputs: % - Li [6x1] - The 6 needed length of the struts in [m] to have the wanted pose of {B} w.r.t. {A} % - dLi [6x1] - The 6 needed displacement of the struts from the initial position in [m] to have the wanted pose of {B} w.r.t. {A} #+end_src *** Optional Parameters :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab arguments stewart args.AP (3,1) double {mustBeNumeric} = zeros(3,1) args.ARB (3,3) double {mustBeNumeric} = eye(3) end #+end_src *** Check the =stewart= structure elements :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab assert(isfield(stewart.geometry, 'Aa'), 'stewart.geometry should have attribute Aa') Aa = stewart.geometry.Aa; assert(isfield(stewart.geometry, 'Bb'), 'stewart.geometry should have attribute Bb') Bb = stewart.geometry.Bb; assert(isfield(stewart.geometry, 'l'), 'stewart.geometry should have attribute l') l = stewart.geometry.l; #+end_src *** Compute :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab Li = sqrt(args.AP'*args.AP + diag(Bb'*Bb) + diag(Aa'*Aa) - (2*args.AP'*Aa)' + (2*args.AP'*(args.ARB*Bb))' - diag(2*(args.ARB*Bb)'*Aa)); #+end_src #+begin_src matlab dLi = Li-l; #+end_src ** =describeMicroStationSetup= :PROPERTIES: :header-args:matlab+: :tangle matlab/src/describeMicroStationSetup.m :header-args:matlab+: :comments none :mkdirp yes :eval no :END: *** Function description :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab function [] = describeMicroStationSetup() % describeMicroStationSetup - % % Syntax: [] = describeMicroStationSetup() % % Inputs: % - - % % Outputs: % - - #+end_src *** Simscape Configuration :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab load('./mat/conf_simscape.mat', 'conf_simscape'); #+end_src #+begin_src matlab fprintf('Simscape Configuration:\n'); if conf_simscape.type == 1 fprintf('- Gravity is included\n'); else fprintf('- Gravity is not included\n'); end fprintf('\n'); #+end_src *** Disturbances :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab load('./mat/nass_disturbances.mat', 'args'); #+end_src #+begin_src matlab fprintf('Disturbances:\n'); if ~args.enable fprintf('- No disturbance is included\n'); else if args.Dwx && args.Dwy && args.Dwz fprintf('- Ground motion\n'); end if args.Fty_x && args.Fty_z fprintf('- Vibrations of the Translation Stage\n'); end if args.Frz_z fprintf('- Vibrations of the Spindle\n'); end end fprintf('\n'); #+end_src *** References :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab load('./mat/nass_references.mat', 'args'); #+end_src #+begin_src matlab fprintf('Reference Tracking:\n'); fprintf('- Translation Stage:\n'); switch args.Dy_type case 'constant' fprintf(' - Constant Position\n'); fprintf(' - Dy = %.0f [mm]\n', args.Dy_amplitude*1e3); case 'triangular' fprintf(' - Triangular Path\n'); fprintf(' - Amplitude = %.0f [mm]\n', args.Dy_amplitude*1e3); fprintf(' - Period = %.0f [s]\n', args.Dy_period); case 'sinusoidal' fprintf(' - Sinusoidal Path\n'); fprintf(' - Amplitude = %.0f [mm]\n', args.Dy_amplitude*1e3); fprintf(' - Period = %.0f [s]\n', args.Dy_period); end fprintf('- Tilt Stage:\n'); switch args.Ry_type case 'constant' fprintf(' - Constant Position\n'); fprintf(' - Ry = %.0f [mm]\n', args.Ry_amplitude*1e3); case 'triangular' fprintf(' - Triangular Path\n'); fprintf(' - Amplitude = %.0f [mm]\n', args.Ry_amplitude*1e3); fprintf(' - Period = %.0f [s]\n', args.Ry_period); case 'sinusoidal' fprintf(' - Sinusoidal Path\n'); fprintf(' - Amplitude = %.0f [mm]\n', args.Ry_amplitude*1e3); fprintf(' - Period = %.0f [s]\n', args.Ry_period); end fprintf('- Spindle:\n'); switch args.Rz_type case 'constant' fprintf(' - Constant Position\n'); fprintf(' - Rz = %.0f [deg]\n', 180/pi*args.Rz_amplitude); case { 'rotating', 'rotating-not-filtered' } fprintf(' - Rotating\n'); fprintf(' - Speed = %.0f [rpm]\n', 60/args.Rz_period); end fprintf('- Micro Hexapod:\n'); switch args.Dh_type case 'constant' fprintf(' - Constant Position\n'); fprintf(' - Dh = %.0f, %.0f, %.0f [mm]\n', args.Dh_pos(1), args.Dh_pos(2), args.Dh_pos(3)); fprintf(' - Rh = %.0f, %.0f, %.0f [deg]\n', args.Dh_pos(4), args.Dh_pos(5), args.Dh_pos(6)); end fprintf('\n'); #+end_src *** Controller :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab load('./mat/controller.mat', 'controller'); #+end_src #+begin_src matlab fprintf('Controller:\n'); fprintf('- %s\n', controller.name); fprintf('\n'); #+end_src *** Micro-Station :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab load('./mat/stages.mat', 'ground', 'granite', 'ty', 'ry', 'rz', 'micro_hexapod', 'axisc'); #+end_src #+begin_src matlab fprintf('Micro Station:\n'); if granite.type == 1 && ... ty.type == 1 && ... ry.type == 1 && ... rz.type == 1 && ... micro_hexapod.type == 1; fprintf('- All stages are rigid\n'); elseif granite.type == 2 && ... ty.type == 2 && ... ry.type == 2 && ... rz.type == 2 && ... micro_hexapod.type == 2; fprintf('- All stages are flexible\n'); else if granite.type == 1 || granite.type == 4 fprintf('- Granite is rigid\n'); else fprintf('- Granite is flexible\n'); end if ty.type == 1 || ty.type == 4 fprintf('- Translation Stage is rigid\n'); else fprintf('- Translation Stage is flexible\n'); end if ry.type == 1 || ry.type == 4 fprintf('- Tilt Stage is rigid\n'); else fprintf('- Tilt Stage is flexible\n'); end if rz.type == 1 || rz.type == 4 fprintf('- Spindle is rigid\n'); else fprintf('- Spindle is flexible\n'); end if micro_hexapod.type == 1 || micro_hexapod.type == 4 fprintf('- Micro Hexapod is rigid\n'); else fprintf('- Micro Hexapod is flexible\n'); end end fprintf('\n'); #+end_src ** =computeReferencePose= :PROPERTIES: :header-args:matlab+: :tangle matlab/src/computeReferencePose.m :header-args:matlab+: :comments none :mkdirp yes :eval no :END: #+begin_src matlab function [WTr] = computeReferencePose(Dy, Ry, Rz, Dh, Dn) % computeReferencePose - Compute the homogeneous transformation matrix corresponding to the wanted pose of the sample % % Syntax: [WTr] = computeReferencePose(Dy, Ry, Rz, Dh, Dn) % % Inputs: % - Dy - Reference of the Translation Stage [m] % - Ry - Reference of the Tilt Stage [rad] % - Rz - Reference of the Spindle [rad] % - Dh - Reference of the Micro Hexapod (Pitch, Roll, Yaw angles) [m, m, m, rad, rad, rad] % - Dn - Reference of the Nano Hexapod [m, m, m, rad, rad, rad] % % Outputs: % - WTr - %% Translation Stage Rty = [1 0 0 0; 0 1 0 Dy; 0 0 1 0; 0 0 0 1]; %% Tilt Stage - Pure rotating aligned with Ob Rry = [ cos(Ry) 0 sin(Ry) 0; 0 1 0 0; -sin(Ry) 0 cos(Ry) 0; 0 0 0 1]; %% Spindle - Rotation along the Z axis Rrz = [cos(Rz) -sin(Rz) 0 0 ; sin(Rz) cos(Rz) 0 0 ; 0 0 1 0 ; 0 0 0 1 ]; %% Micro-Hexapod Rhx = [1 0 0; 0 cos(Dh(4)) -sin(Dh(4)); 0 sin(Dh(4)) cos(Dh(4))]; Rhy = [ cos(Dh(5)) 0 sin(Dh(5)); 0 1 0; -sin(Dh(5)) 0 cos(Dh(5))]; Rhz = [cos(Dh(6)) -sin(Dh(6)) 0; sin(Dh(6)) cos(Dh(6)) 0; 0 0 1]; Rh = [1 0 0 Dh(1) ; 0 1 0 Dh(2) ; 0 0 1 Dh(3) ; 0 0 0 1 ]; Rh(1:3, 1:3) = Rhz*Rhy*Rhx; %% Nano-Hexapod Rnx = [1 0 0; 0 cos(Dn(4)) -sin(Dn(4)); 0 sin(Dn(4)) cos(Dn(4))]; Rny = [ cos(Dn(5)) 0 sin(Dn(5)); 0 1 0; -sin(Dn(5)) 0 cos(Dn(5))]; Rnz = [cos(Dn(6)) -sin(Dn(6)) 0; sin(Dn(6)) cos(Dn(6)) 0; 0 0 1]; Rn = [1 0 0 Dn(1) ; 0 1 0 Dn(2) ; 0 0 1 Dn(3) ; 0 0 0 1 ]; Rn(1:3, 1:3) = Rnz*Rny*Rnx; %% Total Homogeneous transformation WTr = Rty*Rry*Rrz*Rh*Rn; end #+end_src * Helping Functions :noexport: ** Initialize Path #+NAME: m-init-path #+BEGIN_SRC matlab addpath('./matlab/'); % Path for scripts %% Path for functions, data and scripts addpath('./matlab/mat/'); % Path for Computed FRF addpath('./matlab/src/'); % Path for functions addpath('./matlab/STEPS/'); % Path for STEPS addpath('./matlab/subsystems/'); % Path for Subsystems Simulink files #+END_SRC #+NAME: m-init-path-tangle #+BEGIN_SRC matlab %% Path for functions, data and scripts addpath('./mat/'); % Path for Data addpath('./src/'); % Path for functions addpath('./STEPS/'); % Path for STEPS addpath('./subsystems/'); % Path for Subsystems Simulink files #+END_SRC ** Initialize Simscape Model #+NAME: m-init-simscape #+begin_src matlab % Simulink Model name mdl = 'ustation_simscape'; #+end_src ** Initialize other elements #+NAME: m-init-other #+BEGIN_SRC matlab %% Colors for the figures colors = colororder; %% Frequency Vector freqs = logspace(log10(10), log10(2e3), 1000); #+END_SRC