From b46fc734f9f9da98a76b26df79d7bc48ecb43b0e Mon Sep 17 00:00:00 2001 From: Thomas Dehaeze Date: Wed, 9 Jun 2021 11:46:28 +0200 Subject: [PATCH] Compare IFF/DVF plant with Simscape model --- .gitmodules | 3 + figs/enc_struts_dvf_comp_offdiag_simscape.pdf | Bin 0 -> 711313 bytes figs/enc_struts_dvf_comp_offdiag_simscape.png | Bin 0 -> 128281 bytes figs/enc_struts_dvf_comp_simscape.pdf | Bin 0 -> 484840 bytes figs/enc_struts_dvf_comp_simscape.png | Bin 0 -> 160295 bytes figs/enc_struts_iff_comp_offdiag_simscape.pdf | Bin 0 -> 710587 bytes figs/enc_struts_iff_comp_offdiag_simscape.png | Bin 0 -> 117694 bytes figs/enc_struts_iff_comp_simscape.pdf | Bin 0 -> 473898 bytes figs/enc_struts_iff_comp_simscape.png | Bin 0 -> 128983 bytes figs/nano_hexapod_signals.pdf | Bin 0 -> 70930 bytes figs/nano_hexapod_signals.png | Bin 0 -> 25495 bytes figs/nano_hexapod_signals.svg | Bin 0 -> 71359 bytes matlab/nano_hexapod_simscape.slx | Bin 0 -> 41769 bytes matlab/nass-simscape | 1 + test-bench-nano-hexapod.org | 418 +++++++++++++++++- test-bench-nano-hexapod.pdf | Bin 6951111 -> 6995985 bytes 16 files changed, 416 insertions(+), 6 deletions(-) create mode 100644 .gitmodules create mode 100644 figs/enc_struts_dvf_comp_offdiag_simscape.pdf create mode 100644 figs/enc_struts_dvf_comp_offdiag_simscape.png create mode 100644 figs/enc_struts_dvf_comp_simscape.pdf create mode 100644 figs/enc_struts_dvf_comp_simscape.png create mode 100644 figs/enc_struts_iff_comp_offdiag_simscape.pdf create mode 100644 figs/enc_struts_iff_comp_offdiag_simscape.png create mode 100644 figs/enc_struts_iff_comp_simscape.pdf create mode 100644 figs/enc_struts_iff_comp_simscape.png create mode 100644 figs/nano_hexapod_signals.pdf create mode 100644 figs/nano_hexapod_signals.png create mode 100644 figs/nano_hexapod_signals.svg create mode 100644 matlab/nano_hexapod_simscape.slx create mode 160000 matlab/nass-simscape diff --git a/.gitmodules b/.gitmodules new file mode 100644 index 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zP`(;)Jplr>AK3$HSbIlv<3D04!JzCde8pw}a36pHftG)?KPylJYWf92y8LgHpw>H= z8UTO?^k@AqlsX{$_(znqD{zt=0G00VP7vpuqIEUEx(zUY%CvvBKg-hqj4J+-scdXz zY~W!0*8(!|L646=s3D_-K+XdBoxscA+G9YD_KWIh=7cn~HZZrkZEI}`W`M)Ye?tDO zED4a({*onwqs_q>aIo|VMw96`n3I@kFbJI2djgUJme~EO=U>u$!9;Mr>j@DFTwW0I z|M{By_O=CQr=B2<*#3t6?<7@lUEpZ969^%o2>(?VIOq+G0cXXWU?2j2!~8n1Kayj> z5U|h72_#tkcaVR&Bf<77CmbQbrQp{w{oafPTol-M;{+B4=+S?{e&fOcCV{ObPDtBY zek)Kni}9}@awYtbu$0kQ~?YF`yiaarhW(ecUJ^39PC_h0!QinrqX}(F#vPHegh|5 zF8#l8f9pN~W`VW*CoBeozq9_gvL8$aE8 #+end_export +\clearpage + * Introduction :ignore: +In this document, the dynamics of the nano-hexapod shown in Figure [[fig:picture_bench_granite_nano_hexapod]] is identified. + #+begin_note Here are the documentation of the equipment used for this test bench: - Voltage Amplifier: PiezoDrive [[file:doc/PD200-V7-R1.pdf][PD200]] @@ -66,6 +70,73 @@ Here are the documentation of the equipment used for this test bench: #+attr_latex: :width \linewidth [[file:figs/IMG_20210608_154722.jpg]] +#+begin_src latex :file nano_hexapod_signals.pdf +\definecolor{instrumentation}{rgb}{0, 0.447, 0.741} +\definecolor{mechanics}{rgb}{0.8500, 0.325, 0.098} + +\begin{tikzpicture} + % Blocs + \node[block={4.0cm}{3.0cm}, fill=mechanics!20!white] (nano_hexapod) {Mechanics}; + \coordinate[] (inputF) at (nano_hexapod.west); + \coordinate[] (outputL) at ($(nano_hexapod.south east)!0.8!(nano_hexapod.north east)$); + \coordinate[] (outputF) at ($(nano_hexapod.south east)!0.2!(nano_hexapod.north east)$); + + \node[block, left= 0.8 of inputF, fill=instrumentation!20!white, align=center] (F_stack) {\tiny Actuator \\ \tiny stacks}; + \node[block, left= 0.8 of F_stack, fill=instrumentation!20!white] (PD200) {PD200}; + \node[DAC, left= 0.8 of PD200, fill=instrumentation!20!white] (F_DAC) {DAC}; + \node[block, right=0.8 of outputF, fill=instrumentation!20!white, align=center] (Fm_stack){\tiny Sensor \\ \tiny stack}; + \node[ADC, right=0.8 of Fm_stack,fill=instrumentation!20!white] (Fm_ADC) {ADC}; + \node[block, right=0.8 of outputL, fill=instrumentation!20!white] (encoder) {\tiny Encoder}; + + % Connections and labels + \draw[->] ($(F_DAC.west)+(-0.8,0)$) node[above right]{$\bm{u}$} node[below right]{$[V]$} -- node[sloped]{$/$} (F_DAC.west); + \draw[->] (F_DAC.east) -- node[midway, above]{$\tilde{\bm{u}}$}node[midway, below]{$[V]$} (PD200.west); + \draw[->] (PD200.east) -- node[midway, above]{$\bm{u}_a$}node[midway, below]{$[V]$} (F_stack.west); + \draw[->] (F_stack.east) -- (inputF) node[above left]{$\bm{\tau}$}node[below left]{$[N]$}; + + \draw[->] (outputF) -- (Fm_stack.west) node[above left]{$\bm{\epsilon}$} node[below left]{$[m]$}; + \draw[->] (Fm_stack.east) -- node[midway, above]{$\tilde{\bm{\tau}}_m$}node[midway, below]{$[V]$} (Fm_ADC.west); + \draw[->] (Fm_ADC.east) -- node[sloped]{$/$} ++(0.8, 0)coordinate(end) node[above left]{$\bm{\tau}_m$}node[below left]{$[V]$}; + + \draw[->] (outputL) -- (encoder.west) node[above left]{$d\bm{\mathcal{L}}$} node[below left]{$[m]$}; + \draw[->] (encoder.east) -- node[sloped]{$/$} (encoder-|end) node[above left]{$d\bm{\mathcal{L}}_m$}node[below left]{$[m]$}; + + % Nano-Hexapod + \begin{scope}[on background layer] + \node[fit={(F_stack.west|-nano_hexapod.south) (Fm_stack.east|-nano_hexapod.north)}, fill=black!20!white, draw, inner sep=2pt] (system) {}; + \node[above] at (system.north) {Nano-Hexapod}; + \end{scope} +\end{tikzpicture} +#+end_src + +#+name: fig:nano_hexapod_signals +#+caption: Block diagram of the system with named signals +#+attr_latex: :scale 1 +#+RESULTS: +[[file:figs/nano_hexapod_signals.png]] + +#+name: tab:list_signals +#+caption: List of signals +#+attr_latex: :environment tabularx :width \linewidth :align Xllll +#+attr_latex: :center t :booktabs t :float t +| | *Unit* | *Matlab* | *Vector* | *Elements* | +|------------------------------------+-----------+-----------+-----------------------+----------------------| +| Control Input (wanted DAC voltage) | =[V]= | =u= | $\bm{u}$ | $u_i$ | +| DAC Output Voltage | =[V]= | =u= | $\tilde{\bm{u}}$ | $\tilde{u}_i$ | +| PD200 Output Voltage | =[V]= | =ua= | $\bm{u}_a$ | $u_{a,i}$ | +| Actuator applied force | =[N]= | =tau= | $\bm{\tau}$ | $\tau_i$ | +|------------------------------------+-----------+-----------+-----------------------+----------------------| +| Strut motion | =[m]= | =dL= | $d\bm{\mathcal{L}}$ | $d\mathcal{L}_i$ | +| Encoder measured displacement | =[m]= | =dLm= | $d\bm{\mathcal{L}}_m$ | $d\mathcal{L}_{m,i}$ | +|------------------------------------+-----------+-----------+-----------------------+----------------------| +| Force Sensor strain | =[m]= | =epsilon= | $\bm{\epsilon}$ | $\epsilon_i$ | +| Force Sensor Generated Voltage | =[V]= | =taum= | $\tilde{\bm{\tau}}_m$ | $\tilde{\tau}_{m,i}$ | +| Measured Generated Voltage | =[V]= | =taum= | $\bm{\tau}_m$ | $\tau_{m,i}$ | +|------------------------------------+-----------+-----------+-----------------------+----------------------| +| Motion of the top platform | =[m,rad]= | =dX= | $d\bm{\mathcal{X}}$ | $d\mathcal{X}_i$ | +| Metrology measured displacement | =[m,rad]= | =dXm= | $d\bm{\mathcal{X}}_m$ | $d\mathcal{X}_{m,i}$ | + +* * Encoders fixed to the Struts ** Introduction In this section, the encoders are fixed to the struts. @@ -90,7 +161,8 @@ addpath('./mat/'); addpath('./src/'); #+end_src -** Load Data +** Identification of the dynamics +*** Load Data #+begin_src matlab meas_data_lf = {}; @@ -100,7 +172,7 @@ for i = 1:6 end #+end_src -** Spectral Analysis - Setup +*** Spectral Analysis - Setup #+begin_src matlab % Sampling Time [s] Ts = (meas_data_lf{1}.t(end) - (meas_data_lf{1}.t(1)))/(length(meas_data_lf{1}.t)-1); @@ -122,7 +194,7 @@ i_lf = f < 250; % Points for low frequency excitation i_hf = f > 250; % Points for high frequency excitation #+end_src -** DVF Plant +*** DVF Plant First, let's compute the coherence from the excitation voltage and the displacement as measured by the encoders (Figure [[fig:enc_struts_dvf_coh]]). #+begin_src matlab @@ -244,7 +316,7 @@ exportFig('figs/enc_struts_dvf_frf.pdf', 'width', 'wide', 'height', 'tall'); [[file:figs/enc_struts_dvf_frf.png]] -** IFF Plant +*** IFF Plant First, let's compute the coherence from the excitation voltage and the displacement as measured by the encoders (Figure [[fig:enc_struts_iff_coh]]). #+begin_src matlab @@ -365,11 +437,11 @@ exportFig('figs/enc_struts_iff_frf.pdf', 'width', 'wide', 'height', 'tall'); #+RESULTS: [[file:figs/enc_struts_iff_frf.png]] -** Jacobian +** Jacobian :noexport: *** Introduction :ignore: The Jacobian is used to transform the excitation force in the cartesian frame as well as the displacements. -Consider the plant shown in Figure [[fig:nano_hexapod_decentralized_schematic]] with: +Consider the plant shown in Figure [[fig:schematic_jacobian_in_out]] with: - $\tau$ the 6 input voltages (going to the PD200 amplifier and then to the APA) - $d\mathcal{L}$ the relative motion sensor outputs (encoders) - $\bm{\tau}_m$ the generated voltage of the force sensor stacks @@ -540,3 +612,337 @@ exportFig('figs/enc_struts_iff_cart_frf.pdf', 'width', 'wide', 'height', 'tall') #+caption: Measured FRF for the IFF plant in the cartesian frame #+RESULTS: [[file:figs/enc_struts_iff_cart_frf.png]] + +** Comparison with the Simscape Model +*** Introduction :ignore: +In this section, the measured dynamics is compared with the dynamics estimated from the Simscape model. + +*** Initialize :noexport: +#+begin_src matlab :tangle no +addpath('matlab/') +addpath('matlab/nass-simscape/matlab/nano_hexapod/') +addpath('matlab/nass-simscape/STEPS/nano_hexapod/') +addpath('matlab/nass-simscape/STEPS/png/') +addpath('matlab/nass-simscape/src/') +addpath('matlab/nass-simscape/mat/') +#+end_src + +#+begin_src matlab :eval no +addpath('nass-simscape/matlab/nano_hexapod/') +addpath('nass-simscape/STEPS/nano_hexapod/') +addpath('nass-simscape/STEPS/png/') +addpath('nass-simscape/src/') +addpath('nass-simscape/mat/') +#+end_src + +#+begin_src matlab +mdl = 'nano_hexapod_simscape'; + +options = linearizeOptions; +options.SampleTime = 0; + +open(mdl) +#+end_src + +*** Dynamics from Actuator to Force Sensors +#+begin_src matlab +n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '3dof', ... + 'flex_top_type', '2dof', ... + 'motion_sensor_type', 'struts', ... + 'actuator_type', '2dof'); +#+end_src + +#+begin_src matlab +%% Input/Output definition +clear io; io_i = 1; +io(io_i) = linio([mdl, '/F'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs +io(io_i) = linio([mdl, '/Fm'], 1, 'openoutput'); io_i = io_i + 1; % Force Sensors + +Giff = 20*exp(-s*Ts)*linearize(mdl, io, 0.0, options); +#+end_src + +#+begin_src matlab :exports none +freqs = 2*logspace(1, 3, 1000); + +figure; +tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +plot(freqs, abs(squeeze(freqresp(Giff(1,1), freqs, 'Hz'))), '-', ... + 'DisplayName', '$\tau_{m,i}/u_i$ - Model') +plot(f(i_lf), abs(G_iff_lf(i_lf,1, 1)), ... + 'DisplayName', '$\tau_{m,i}/u_i$ - FRF') +for i = 2:6 + set(gca,'ColorOrderIndex',1); + plot(freqs, abs(squeeze(freqresp(Giff(i,i), freqs, 'Hz'))), '-', ... + 'HandleVisibility', 'off'); +end +for i = 2:6 + set(gca,'ColorOrderIndex',2) + plot(f(i_lf), abs(G_iff_lf(i_lf,i, i)), ... + 'HandleVisibility', 'off'); + set(gca,'ColorOrderIndex',2) + plot(f(i_hf), abs(G_iff_hf(i_hf,i, i)), ... + 'HandleVisibility', 'off'); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude [V/V]'); set(gca, 'XTickLabel',[]); +legend('location', 'southeast'); + +ax2 = nexttile; +hold on; +for i = 1:6 + set(gca,'ColorOrderIndex',1); + plot(freqs, 180/pi*angle(squeeze(freqresp(Giff(i,i), freqs, 'Hz'))), '-'); +end +for i = 1:6 + set(gca,'ColorOrderIndex',2) + plot(f(i_lf), 180/pi*angle(G_iff_lf(i_lf,i, i))); + set(gca,'ColorOrderIndex',2) + plot(f(i_hf), 180/pi*angle(G_iff_hf(i_hf,i, i))); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); +ylim([-180, 180]); +yticks([-180, -90, 0, 90, 180]); + +linkaxes([ax1,ax2],'x'); +xlim([freqs(1), freqs(end)]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/enc_struts_iff_comp_simscape.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:enc_struts_iff_comp_simscape +#+caption: Diagonal elements of the IFF Plant +#+RESULTS: +[[file:figs/enc_struts_iff_comp_simscape.png]] + +#+begin_src matlab :exports none +freqs = 2*logspace(1, 3, 1000); + +figure; +hold on; +% Off diagonal terms +set(gca,'ColorOrderIndex',1); +plot(freqs, abs(squeeze(freqresp(Giff(1, 2), freqs, 'Hz'))), ... + 'DisplayName', '$\tau_{m,i}/u_j$ - Model') +for i = 1:5 + for j = i+1:6 + set(gca,'ColorOrderIndex',1); + plot(freqs, abs(squeeze(freqresp(Giff(i, j), freqs, 'Hz'))), ... + 'HandleVisibility', 'off'); + end +end +set(gca,'ColorOrderIndex',2); +plot(f(i_lf), abs(G_iff_lf(i_lf, 1, 2)), ... + 'DisplayName', '$\tau_{m,i}/u_j$ - FRF') +for i = 1:5 + for j = i+1:6 + set(gca,'ColorOrderIndex',2); + plot(f(i_lf), abs(G_iff_lf(i_lf, i, j)), ... + 'HandleVisibility', 'off'); + set(gca,'ColorOrderIndex',2); + plot(f(i_hf), abs(G_iff_hf(i_hf, i, j)), ... + 'HandleVisibility', 'off'); + end +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +xlabel('Frequency [Hz]'); ylabel('Amplitude [V/V]'); +xlim([freqs(1), freqs(end)]); ylim([1e-3, 1e2]); +legend('location', 'northeast'); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/enc_struts_iff_comp_offdiag_simscape.pdf', 'width', 'wide', 'height', 'normal'); +#+end_src + +#+name: fig:enc_struts_iff_comp_offdiag_simscape +#+caption: Off diagonal elements of the IFF Plant +#+RESULTS: +[[file:figs/enc_struts_iff_comp_offdiag_simscape.png]] + +*** Dynamics from Actuator to Encoder +#+begin_src matlab +n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '3dof', ... + 'flex_top_type', '2dof', ... + 'motion_sensor_type', 'struts', ... + 'actuator_type', '2dof'); +#+end_src + +#+begin_src matlab +%% Input/Output definition +clear io; io_i = 1; +io(io_i) = linio([mdl, '/F'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs +io(io_i) = linio([mdl, '/D'], 1, 'openoutput'); io_i = io_i + 1; % Encoders + +Gdvf = 20*exp(-s*Ts)*linearize(mdl, io, 0.0, options); +#+end_src + +#+begin_src matlab :exports none +freqs = 2*logspace(1, 3, 1000); + +figure; +tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +plot(freqs, abs(squeeze(freqresp(Gdvf(1,1), freqs, 'Hz'))), '-', ... + 'DisplayName', '$d\mathcal{L}_{m,i}/u_i$ - Model') +plot(f(i_lf), abs(G_dvf_lf(i_lf,1, 1)), ... + 'DisplayName', '$d\mathcal{L}_{m,i}/u_i$ - FRF') +for i = 2:6 + set(gca,'ColorOrderIndex',1); + plot(freqs, abs(squeeze(freqresp(Gdvf(i,i), freqs, 'Hz'))), '-', ... + 'HandleVisibility', 'off'); +end +for i = 2:6 + set(gca,'ColorOrderIndex',2) + plot(f(i_lf), abs(G_dvf_lf(i_lf,i, i)), ... + 'HandleVisibility', 'off'); + set(gca,'ColorOrderIndex',2) + plot(f(i_hf), abs(G_dvf_hf(i_hf,i, i)), ... + 'HandleVisibility', 'off'); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude [m/V]'); set(gca, 'XTickLabel',[]); +legend('location', 'northeast'); +ylim([1e-8, 1e-3]); + +ax2 = nexttile; +hold on; +for i = 1:6 + set(gca,'ColorOrderIndex',1); + plot(freqs, 180/pi*angle(squeeze(freqresp(Gdvf(i,i), freqs, 'Hz'))), '-'); +end +for i = 1:6 + set(gca,'ColorOrderIndex',2) + plot(f(i_lf), 180/pi*angle(G_dvf_lf(i_lf,i, i))); + set(gca,'ColorOrderIndex',2) + plot(f(i_hf), 180/pi*angle(G_dvf_hf(i_hf,i, i))); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); +ylim([-180, 180]); +yticks([-180, -90, 0, 90, 180]); + +linkaxes([ax1,ax2],'x'); +xlim([freqs(1), freqs(end)]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/enc_struts_dvf_comp_simscape.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:enc_struts_dvf_comp_simscape +#+caption: Diagonal elements of the DVF Plant +#+RESULTS: +[[file:figs/enc_struts_dvf_comp_simscape.png]] + +#+begin_src matlab :exports none +freqs = 2*logspace(1, 3, 1000); + +figure; +hold on; +% Off diagonal terms +set(gca,'ColorOrderIndex',1); +plot(freqs, abs(squeeze(freqresp(Gdvf(1, 2), freqs, 'Hz'))), ... + 'DisplayName', '$d\mathcal{L}_{m,i}/u_j$ - Model') +for i = 1:5 + for j = i+1:6 + set(gca,'ColorOrderIndex',1); + plot(freqs, abs(squeeze(freqresp(Gdvf(i, j), freqs, 'Hz'))), ... + 'HandleVisibility', 'off'); + end +end +set(gca,'ColorOrderIndex',2); +plot(f(i_lf), abs(G_dvf_lf(i_lf, 1, 2)), ... + 'DisplayName', '$d\mathcal{L}_{m,i}/u_j$ - FRF') +for i = 1:5 + for j = i+1:6 + set(gca,'ColorOrderIndex',2); + plot(f(i_lf), abs(G_dvf_lf(i_lf, i, j)), ... + 'HandleVisibility', 'off'); + set(gca,'ColorOrderIndex',2); + plot(f(i_hf), abs(G_dvf_hf(i_hf, i, j)), ... + 'HandleVisibility', 'off'); + end +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +xlabel('Frequency [Hz]'); ylabel('Amplitude [m/V]'); +xlim([freqs(1), freqs(end)]); ylim([1e-8, 1e-3]); +legend('location', 'northeast'); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/enc_struts_dvf_comp_offdiag_simscape.pdf', 'width', 'wide', 'height', 'normal'); +#+end_src + +#+name: fig:enc_struts_dvf_comp_offdiag_simscape +#+caption: Off diagonal elements of the DVF Plant +#+RESULTS: +[[file:figs/enc_struts_dvf_comp_offdiag_simscape.png]] + +** TODO Integral Force Feedback +*** Plant +#+begin_src matlab :exports none +figure; +tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +for i = 1:5 + for j = i+1:6 + plot(f(i_lf), abs(G_iff_lf(i_lf, i, j)), 'color', [0, 0, 0, 0.2], ... + 'HandleVisibility', 'off'); + plot(f(i_hf), abs(G_iff_hf(i_hf, i, j)), 'color', [0, 0, 0, 0.2], ... + 'HandleVisibility', 'off'); + end +end +for i =1:6 + set(gca,'ColorOrderIndex',i) + plot(f(i_lf), abs(G_iff_lf(i_lf,i, i)), ... + 'DisplayName', sprintf('$G_{iff}(%i,%i)$', i, i)); + set(gca,'ColorOrderIndex',i) + plot(f(i_hf), abs(G_iff_hf(i_hf,i, i)), ... + 'HandleVisibility', 'off'); +end +plot(f(i_lf), abs(G_iff_lf(i_lf, 1, 2)), 'color', [0, 0, 0, 0.2], ... + 'DisplayName', '$G_{iff}(i,j)$'); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude $V_s/V_a$ [V/V]'); set(gca, 'XTickLabel',[]); +legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 3); +ylim([1e-3, 1e2]); + +ax2 = nexttile; +hold on; +for i =1:6 + set(gca,'ColorOrderIndex',i) + plot(f(i_lf), 180/pi*angle(G_iff_lf(i_lf,i, i))); + set(gca,'ColorOrderIndex',i) + plot(f(i_hf), 180/pi*angle(G_iff_hf(i_hf,i, i))); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); +hold off; +yticks(-360:90:360); + +linkaxes([ax1,ax2],'x'); +xlim([20, 2e3]); +#+end_src + +*** Root Locus + +*** Gains + +*** Experimental Results diff --git a/test-bench-nano-hexapod.pdf b/test-bench-nano-hexapod.pdf index 8b00a26f7810f6abba08f4c13ab2493d54063b08..e60de65737f05db11b009ab7261ae9db91f1e6f8 100644 GIT binary patch delta 234592 zcmZsjcRbba|NpZ~Rx-0U899?<%N8jkvPWjNWIM96XSU3cP{>S?%w#6X-kBj0lHaBG zJFoZc_xb*D-EKY4>y*cJ&eQ#Np7-l@>c}m|iw`Kpd(K77UazdJITO3x`m%4b{y?gR z*wXQi+dWrHGl!EuIhorLqf=xxlrQ2Tks^r~aQU$7BOf&03R9ig6vryWPME!9sW9@< z-Vc8l6yJ8fl5?6^_6o)PUo(S7#mqdJ?EpdjFy67geB7M`-3;DkvZUgdmS^y4kw&oLoLuk2 z+y=sW+uQ!|fX7DEF}j6>Lt;m7+l;zk-f8J$0tiW+LN8{|eHDtbK3n}^KenMvo4MV` znip$j1yDa*rgmm~X(?@YF8}WvEQ(?kW<7bQB_xQis98E%-?L%GAYp&r?WSAmZewVQ zrtiAbyv%23GJ-YSpM7>IQWw-syq16`cNK2UIq;;IJJDo{Kqf9paYghdk44%U+Q^AL zn#OAs)mQN+6bcylB+ljWQtMGfQx)l*)wa*0#TWDFk@_Ve0Nn1v!ch$6MG}pL67_zqhDT_HYJec{%@3#gDLfI=^t1P+U49z%`V5 zCPj{pewz-LME0e*sd7Qua|`aOsC>dTwa2>ery|A=^w%Y>KS(POlVdnmt$&M8VpB~P 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