diff --git a/test-bench-nano-hexapod.org b/test-bench-nano-hexapod.org index 1bae01c..4cac7a1 100644 --- a/test-bench-nano-hexapod.org +++ b/test-bench-nano-hexapod.org @@ -101,15 +101,35 @@ Prefix is =test_nhexa= Add these documents: -- [ ] [[file:~/Cloud/work-projects/ID31-NASS/matlab/nass-simscape/org/nano_hexapod.org][nano_hexapod]] - [X] [[file:~/Cloud/work-projects/ID31-NASS/matlab/nass-nano-hexapod-assembly/nass-nano-hexapod-assembly.org][nass-nano-hexapod-assembly]] -- [ ] [[file:~/Cloud/work-projects/ID31-NASS/matlab/test-bench-vibration-table/vibration-table.org][test-bench-vibration-table]] +- [X] [[file:~/Cloud/work-projects/ID31-NASS/matlab/test-bench-vibration-table/vibration-table.org][test-bench-vibration-table]] - [ ] *Use corrected APA parameters in the initialization script* -- *Goal*: Tune model of nano-hexapod, validation of dynamics -- modal analysis soft table (first mode at xxx Hz => rigid body in Simscape) -- FRF + comp model (multiple masses) -- IFF and robustness to change of mass +*Goal of this test bench*: +- Properly mount the nano-hexapod +- Verify all is working +- Tune the complete nano-hexapod model + +*Basic outline*: +1. [ ] Mounting procedure + - [ ] Goal + - [ ] Procedure + - [ ] Results +2. [ ] Suspended table: + - [ ] Goal: identify dynamics of the nano-hexapod not coupled with the outside world + - [ ] Experimental modal analysis (first mode at 700 Hz => rigid body in Simscape)) + - [ ] Simscape model of the table, comparison of the obtained modes +3. [ ] Simscape model of the Nano-Hexapod? (Maybe already presented in second chapter, maybe this [[file:~/Cloud/work-projects/ID31-NASS/phd-thesis-chapters/B6-nass-design/nass-design.org::+TITLE: Nano Hexapod - Obtained Design][document]]) + Yes, but now the model is updated with the tuned models of the APA, Flexible joints, etc.. +4. [ ] Nano-Hexapod Dynamics + - Identified dynamics + - Comparison with the simscape model + - Effect of the payload mass + +Maybe the rest is not so interesting here as it will be presented again in the next sections. +- Robust Integral Force Feedback (LAC) +- High Authority Controller HAC + - Decoupling Strategy ** TODO [#A] Update the default APA parameters to have good match @@ -162,6 +182,3379 @@ end - git submodule? - Maybe just copy paste the directory as it will not change a lot now +** Analysis backup of HAC - Decoupling analysis +<> + +*** Introduction :ignore: + +In this section is studied the HAC-IFF architecture for the Nano-Hexapod. +More precisely: +- The LAC control is a decentralized integral force feedback as studied in Section ref:sec:test_nhexa_enc_plates_iff +- The HAC control is a decentralized controller working in the frame of the struts + +The corresponding control architecture is shown in Figure ref:fig:test_nhexa_control_architecture_hac_iff_struts with: +- $\bm{r}_{\mathcal{X}_n}$: the $6 \times 1$ reference signal in the cartesian frame +- $\bm{r}_{d\mathcal{L}}$: the $6 \times 1$ reference signal transformed in the frame of the struts thanks to the inverse kinematic +- $\bm{\epsilon}_{d\mathcal{L}}$: the $6 \times 1$ length error of the 6 struts +- $\bm{u}^\prime$: input of the damped plant +- $\bm{u}$: generated DAC voltages +- $\bm{\tau}_m$: measured force sensors +- $d\bm{\mathcal{L}}_m$: measured displacement of the struts by the encoders + +#+begin_src latex :file control_architecture_hac_iff_struts.pdf +\definecolor{instrumentation}{rgb}{0, 0.447, 0.741} +\definecolor{mechanics}{rgb}{0.8500, 0.325, 0.098} +\definecolor{control}{rgb}{0.4660, 0.6740, 0.1880} + +\begin{tikzpicture} + % Blocs + \node[block={3.0cm}{2.0cm}, fill=black!20!white] (P) {Plant}; + \coordinate[] (inputF) at ($(P.south west)!0.5!(P.north west)$); + \coordinate[] (outputF) at ($(P.south east)!0.2!(P.north east)$); + \coordinate[] (outputL) at ($(P.south east)!0.8!(P.north east)$); + + \node[block, below=0.4 of P, fill=control!20!white] (Kiff) {$\bm{K}_\text{IFF}$}; + \node[block, left=0.8 of inputF, fill=instrumentation!20!white] (pd200) {\tiny PD200}; + \node[addb, left=0.8 of pd200, fill=control!20!white] (addF) {}; + \node[block, left=0.8 of addF, fill=control!20!white] (K) {$\bm{K}_\mathcal{L}$}; + \node[addb={+}{}{-}{}{}, left=0.8 of K, fill=control!20!white] (subr) {}; + \node[block, align=center, left= of subr, fill=control!20!white] (J) {\tiny Inverse\\\tiny Kinematics}; + + % Connections and labels + \draw[->] (outputF) -- ++(1.0, 0) node[above left]{$\bm{\tau}_m$}; + \draw[->] ($(outputF) + (0.6, 0)$)node[branch]{} |- (Kiff.east); + \draw[->] (Kiff.west) -| (addF.south); + \draw[->] (addF.east) -- (pd200.west) node[above left]{$\bm{u}$}; + \draw[->] (pd200.east) -- (inputF) node[above left]{$\bm{u}_a$}; + + \draw[->] (outputL) -- ++(1.0, 0) node[below left]{$d\bm{\mathcal{L}_m}$}; + \draw[->] ($(outputL) + (0.6, 0)$)node[branch]{} -- ++(0, 1) -| (subr.north); + \draw[->] (subr.east) -- (K.west) node[above left]{$\bm{\epsilon}_{d\mathcal{L}}$}; + \draw[->] (K.east) -- (addF.west) node[above left]{$\bm{u}^\prime$}; + + \draw[->] (J.east) -- (subr.west) node[above left]{$\bm{r}_{d\mathcal{L}}$}; + \draw[<-] (J.west)node[above left]{$\bm{r}_{\mathcal{X}_n}$} -- ++(-1, 0); +\end{tikzpicture} +#+end_src + +#+name: fig:test_nhexa_control_architecture_hac_iff_struts +#+caption: HAC-LAC: IFF + Control in the frame of the legs +#+RESULTS: +[[file:figs/test_nhexa_control_architecture_hac_iff_struts.png]] + +This part is structured as follow: +- Section ref:sec:test_nhexa_hac_iff_struts_ref_track: some reference tracking tests are performed +- Section ref:sec:test_nhexa_hac_iff_struts_controller: the decentralized high authority controller is tuned using the Simscape model and is implemented and tested experimentally +- Section ref:sec:test_nhexa_interaction_analysis: an interaction analysis is performed, from which the best decoupling strategy can be determined +- Section ref:sec:test_nhexa_robust_hac_design: Robust High Authority Controller are designed + +*** Reference Tracking - Trajectories +:PROPERTIES: +:header-args:matlab+: :tangle matlab/scripts/reference_tracking_paths.m +:END: +<> +**** Introduction :ignore: +In this section, several trajectories representing the wanted pose (position and orientation) of the top platform with respect to the bottom platform are defined. + +These trajectories will be used to test the HAC-LAC architecture. + +In order to transform the wanted pose to the wanted displacement of the 6 struts, the inverse kinematic is required. +As a first approximation, the Jacobian matrix $\bm{J}$ can be used instead of using the full inverse kinematic equations. + +Therefore, the control architecture with the input trajectory $\bm{r}_{\mathcal{X}_n}$ is shown in Figure ref:fig:test_nhexa_control_architecture_hac_iff_L. + +#+begin_src latex :file control_architecture_hac_iff_struts_L.pdf +\definecolor{instrumentation}{rgb}{0, 0.447, 0.741} +\definecolor{mechanics}{rgb}{0.8500, 0.325, 0.098} +\definecolor{control}{rgb}{0.4660, 0.6740, 0.1880} + +\begin{tikzpicture} + % Blocs + \node[block={3.0cm}{2.0cm}, fill=black!20!white] (P) {Plant}; + \coordinate[] (inputF) at ($(P.south west)!0.5!(P.north west)$); + \coordinate[] (outputF) at ($(P.south east)!0.2!(P.north east)$); + \coordinate[] (outputL) at ($(P.south east)!0.8!(P.north east)$); + + \node[block, below=0.4 of P, fill=control!20!white] (Kiff) {$\bm{K}_\text{IFF}$}; + \node[block, left=0.8 of inputF, fill=instrumentation!20!white] (pd200) {\tiny PD200}; + \node[addb, left=0.8 of pd200, fill=control!20!white] (addF) {}; + \node[block, left=0.8 of addF, fill=control!20!white] (K) {$\bm{K}_\mathcal{L}$}; + \node[addb={+}{}{-}{}{}, left=0.8 of K, fill=control!20!white] (subr) {}; + \node[block, align=center, left= of subr, fill=control!20!white] (J) {$\bm{J}$}; + + % Connections and labels + \draw[->] (outputF) -- ++(1.0, 0) node[above left]{$\bm{\tau}_m$}; + \draw[->] ($(outputF) + (0.6, 0)$)node[branch]{} |- (Kiff.east); + \draw[->] (Kiff.west) -| (addF.south); + \draw[->] (addF.east) -- (pd200.west) node[above left]{$\bm{u}$}; + \draw[->] (pd200.east) -- (inputF) node[above left]{$\bm{u}_a$}; + + \draw[->] (outputL) -- ++(1.0, 0) node[below left]{$d\bm{\mathcal{L}_m}$}; + \draw[->] ($(outputL) + (0.6, 0)$)node[branch]{} -- ++(0, 1) -| (subr.north); + \draw[->] (subr.east) -- (K.west) node[above left]{$\bm{\epsilon}_{d\mathcal{L}}$}; + \draw[->] (K.east) -- (addF.west) node[above left]{$\bm{u}^\prime$}; + + \draw[->] (J.east) -- (subr.west) node[above left]{$\bm{r}_{d\mathcal{L}}$}; + \draw[<-] (J.west)node[above left]{$\bm{r}_{\mathcal{X}_n}$} -- ++(-1, 0); +\end{tikzpicture} +#+end_src + +#+name: fig:test_nhexa_control_architecture_hac_iff_L +#+caption: HAC-LAC: IFF + Control in the frame of the legs +#+RESULTS: +[[file:figs/test_nhexa_control_architecture_hac_iff_struts_L.png]] + +In the following sections, several reference trajectories are defined: +- Section ref:sec:test_nhexa_yz_scans: simple scans in the Y-Z plane +- Section ref:sec:test_nhexa_tilt_scans: scans in tilt are performed +- Section ref:sec:test_nhexa_nass_scans: scans with X-Y-Z translations in order to draw the word "NASS" + +**** Matlab Init :noexport:ignore: +#+begin_src matlab +%% reference_tracking_paths.m +% Computation of several reference paths +#+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 + +**** Y-Z Scans +<> +A function =generateYZScanTrajectory= has been developed in order to easily generate scans in the Y-Z plane. + +For instance, the following generated trajectory is represented in Figure ref:fig:test_nhexa_yz_scan_example_trajectory_yz_plane. +#+begin_src matlab +%% Generate the Y-Z trajectory scan +Rx_yz = generateYZScanTrajectory(... + 'y_tot', 4e-6, ... % Length of Y scans [m] + 'z_tot', 4e-6, ... % Total Z distance [m] + 'n', 5, ... % Number of Y scans + 'Ts', 1e-3, ... % Sampling Time [s] + 'ti', 1, ... % Time to go to initial position [s] + 'tw', 0, ... % Waiting time between each points [s] + 'ty', 0.6, ... % Time for a scan in Y [s] + 'tz', 0.2); % Time for a scan in Z [s] +#+end_src + +#+begin_src matlab :exports none +%% Plot the trajectory in the Y-Z plane +figure; +plot(Rx_yz(:,3), Rx_yz(:,4)); +xlabel('y [m]'); ylabel('z [m]'); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/yz_scan_example_trajectory_yz_plane.pdf', 'width', 'normal', 'height', 'normal'); +#+end_src + +#+name: fig:test_nhexa_yz_scan_example_trajectory_yz_plane +#+caption: Generated scan in the Y-Z plane +#+RESULTS: +[[file:figs/test_nhexa_yz_scan_example_trajectory_yz_plane.png]] + +The Y and Z positions as a function of time are shown in Figure ref:fig:test_nhexa_yz_scan_example_trajectory. + +#+begin_src matlab :exports none +%% Plot the Y-Z trajectory as a function of time +figure; +hold on; +plot(Rx_yz(:,1), Rx_yz(:,3), ... + 'DisplayName', 'Y motion') +plot(Rx_yz(:,1), Rx_yz(:,4), ... + 'DisplayName', 'Z motion') +hold off; +xlabel('Time [s]'); +ylabel('Displacement [m]'); +legend('location', 'northeast'); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/yz_scan_example_trajectory.pdf', 'width', 'wide', 'height', 'normal'); +#+end_src + +#+name: fig:test_nhexa_yz_scan_example_trajectory +#+caption: Y and Z trajectories as a function of time +#+RESULTS: +[[file:figs/test_nhexa_yz_scan_example_trajectory.png]] + +Using the Jacobian matrix, it is possible to compute the wanted struts lengths as a function of time: +\begin{equation} + \bm{r}_{d\mathcal{L}} = \bm{J} \bm{r}_{\mathcal{X}_n} +\end{equation} + +#+begin_src matlab :exports none +load('jacobian.mat', 'J'); +#+end_src + +#+begin_src matlab +%% Compute the reference in the frame of the legs +dL_ref = [J*Rx_yz(:, 2:7)']'; +#+end_src + +The reference signal for the strut length is shown in Figure ref:fig:test_nhexa_yz_scan_example_trajectory_struts. +#+begin_src matlab :exports none +%% Plot the reference in the frame of the legs +figure; +hold on; +for i=1:6 + plot(Rx_yz(:,1), dL_ref(:, i), ... + 'DisplayName', sprintf('$r_{d\\mathcal{L}_%i}$', i)) +end +xlabel('Time [s]'); ylabel('Strut Motion [m]'); +legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 2); +yticks(1e-6*[-5:5]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/yz_scan_example_trajectory_struts.pdf', 'width', 'wide', 'height', 'normal'); +#+end_src + +#+name: fig:test_nhexa_yz_scan_example_trajectory_struts +#+caption: Trajectories for the 6 individual struts +#+RESULTS: +[[file:figs/test_nhexa_yz_scan_example_trajectory_struts.png]] + +**** Tilt Scans +<> + +A function =generalSpiralAngleTrajectory= has been developed in order to easily generate $R_x,R_y$ tilt scans. + +For instance, the following generated trajectory is represented in Figure ref:fig:test_nhexa_tilt_scan_example_trajectory. +#+begin_src matlab +%% Generate the "tilt-spiral" trajectory scan +R_tilt = generateSpiralAngleTrajectory(... + 'R_tot', 20e-6, ... % Total Tilt [ad] + 'n_turn', 5, ... % Number of scans + 'Ts', 1e-3, ... % Sampling Time [s] + 't_turn', 1, ... % Turn time [s] + 't_end', 1); % End time to go back to zero [s] +#+end_src + +#+begin_src matlab :exports none +%% Plot the trajectory +figure; +plot(1e6*R_tilt(:,5), 1e6*R_tilt(:,6)); +xlabel('$R_x$ [$\mu$rad]'); ylabel('$R_y$ [$\mu$rad]'); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/tilt_scan_example_trajectory.pdf', 'width', 'normal', 'height', 'normal'); +#+end_src + +#+name: fig:test_nhexa_tilt_scan_example_trajectory +#+caption: Generated "spiral" scan +#+RESULTS: +[[file:figs/test_nhexa_tilt_scan_example_trajectory.png]] + +#+begin_src matlab :exports none +%% Compute the reference in the frame of the legs +load('jacobian.mat', 'J'); +dL_ref = [J*R_tilt(:, 2:7)']'; +#+end_src + +The reference signal for the strut length is shown in Figure ref:fig:test_nhexa_tilt_scan_example_trajectory_struts. +#+begin_src matlab :exports none +%% Plot the reference in the frame of the legs +figure; +hold on; +for i=1:6 + plot(R_tilt(:,1), dL_ref(:, i), ... + 'DisplayName', sprintf('$r_{d\\mathcal{L}_%i}$', i)) +end +xlabel('Time [s]'); ylabel('Strut Motion [m]'); +legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 2); +yticks(1e-6*[-5:5]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/tilt_scan_example_trajectory_struts.pdf', 'width', 'wide', 'height', 'normal'); +#+end_src + +#+name: fig:test_nhexa_tilt_scan_example_trajectory_struts +#+caption: Trajectories for the 6 individual struts - Tilt scan +#+RESULTS: +[[file:figs/test_nhexa_tilt_scan_example_trajectory_struts.png]] + +**** "NASS" reference path +<> +In this section, a reference path that "draws" the work "NASS" is developed. + +First, a series of points representing each letter are defined. +Between each letter, a negative Z motion is performed. +#+begin_src matlab +%% List of points that draws "NASS" +ref_path = [ ... + 0, 0,0; % Initial Position + 0,0,1; 0,4,1; 3,0,1; 3,4,1; % N + 3,4,0; 4,0,0; % Transition + 4,0,1; 4,3,1; 5,4,1; 6,4,1; 7,3,1; 7,2,1; 4,2,1; 4,3,1; 5,4,1; 6,4,1; 7,3,1; 7,0,1; % A + 7,0,0; 8,0,0; % Transition + 8,0,1; 11,0,1; 11,2,1; 8,2,1; 8,4,1; 11,4,1; % S + 11,4,0; 12,0,0; % Transition + 12,0,1; 15,0,1; 15,2,1; 12,2,1; 12,4,1; 15,4,1; % S + 15,4,0; + ]; + +%% Center the trajectory arround zero +ref_path = ref_path - (max(ref_path) - min(ref_path))/2; + +%% Define the X-Y-Z cuboid dimensions containing the trajectory +X_max = 10e-6; +Y_max = 4e-6; +Z_max = 2e-6; + +ref_path = ([X_max, Y_max, Z_max]./max(ref_path)).*ref_path; % [m] +#+end_src + +Then, using the =generateXYZTrajectory= function, the $6 \times 1$ trajectory signal is computed. +#+begin_src matlab +%% Generating the trajectory +Rx_nass = generateXYZTrajectory('points', ref_path); +#+end_src + +The trajectory in the X-Y plane is shown in Figure ref:fig:test_nhexa_ref_track_test_nass (the transitions between the letters are removed). +#+begin_src matlab :exports none +%% "NASS" trajectory in the X-Y plane +figure; +plot(1e6*Rx_nass(Rx_nass(:,4)>0, 2), 1e6*Rx_nass(Rx_nass(:,4)>0, 3), 'k.') +xlabel('X [$\mu m$]'); +ylabel('Y [$\mu m$]'); +axis equal; +xlim(1e6*[min(Rx_nass(:,2)), max(Rx_nass(:,2))]); +ylim(1e6*[min(Rx_nass(:,3)), max(Rx_nass(:,3))]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/ref_track_test_nass.pdf', 'width', 'wide', 'height', 'normal'); +#+end_src + +#+name: fig:test_nhexa_ref_track_test_nass +#+caption: Reference path corresponding to the "NASS" acronym +#+RESULTS: +[[file:figs/test_nhexa_ref_track_test_nass.png]] + +It can also be better viewed in a 3D representation as in Figure ref:fig:test_nhexa_ref_track_test_nass_3d. + +#+begin_src matlab :exports none +figure; +plot3(1e6*Rx_nass(:,2), 1e6*Rx_nass(:,3), 1e6*Rx_nass(:,4), 'k-'); +xlabel('x [$\mu m$]'); ylabel('y [$\mu m$]'); zlabel('z [$\mu m$]'); +view(-13, 41) +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/ref_track_test_nass_3d.pdf', 'width', 'normal', 'height', 'normal'); +#+end_src + +#+name: fig:test_nhexa_ref_track_test_nass_3d +#+caption: Reference path that draws "NASS" - 3D view +#+RESULTS: +[[file:figs/test_nhexa_ref_track_test_nass_3d.png]] + +*** First Basic High Authority Controller +:PROPERTIES: +:header-args:matlab+: :tangle matlab/scripts/hac_lac_first_try.m +:END: +<> +**** Introduction :ignore: +In this section, a simple decentralized high authority controller $\bm{K}_{\mathcal{L}}$ is developed to work without any payload. + +The diagonal controller is tuned using classical Loop Shaping in Section ref:sec:test_nhexa_hac_iff_no_payload_tuning. +The stability is verified in Section ref:sec:test_nhexa_hac_iff_no_payload_stability using the Simscape model. + +**** Matlab Init :noexport:ignore: +#+begin_src matlab +%% hac_lac_first_try.m +% Development and analysis of a first basic High Authority Controller +#+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 +%% Load the identified FRF and Simscape model +frf_iff = load('frf_iff_vib_table_m.mat', 'f', 'Ts', 'G_dL'); +sim_iff = load('sim_iff_vib_table_m.mat', 'G_dL'); +#+end_src + +**** HAC Controller +<> + +Let's first try to design a first decentralized controller with: +- a bandwidth of 100Hz +- sufficient phase margin +- simple and understandable components + +After some very basic and manual loop shaping, A diagonal controller is developed. +Each diagonal terms are identical and are composed of: +- A lead around 100Hz +- A first order low pass filter starting at 200Hz to add some robustness to high frequency modes +- A notch at 700Hz to cancel the flexible modes of the top plate +- A pure integrator + +#+begin_src matlab +%% Lead to increase phase margin +a = 2; % Amount of phase lead / width of the phase lead / high frequency gain +wc = 2*pi*100; % Frequency with the maximum phase lead [rad/s] + +H_lead = (1 + s/(wc/sqrt(a)))/(1 + s/(wc*sqrt(a))); + +%% Low Pass filter to increase robustness +H_lpf = 1/(1 + s/2/pi/200); + +%% Notch at the top-plate resonance +gm = 0.02; +xi = 0.3; +wn = 2*pi*700; + +H_notch = (s^2 + 2*gm*xi*wn*s + wn^2)/(s^2 + 2*xi*wn*s + wn^2); + +%% Decentralized HAC +Khac_iff_struts = -(1/(2.87e-5)) * ... % Gain + H_lead * ... % Lead + H_notch * ... % Notch + (2*pi*100/s) * ... % Integrator + eye(6); % 6x6 Diagonal +#+end_src + +This controller is saved for further use. +#+begin_src matlab :exports none :tangle no +save('matlab/data_sim/Khac_iff_struts.mat', 'Khac_iff_struts') +#+end_src + +#+begin_src matlab :eval no +save('data_sim/Khac_iff_struts.mat', 'Khac_iff_struts') +#+end_src + +The experimental loop gain is computed and shown in Figure ref:fig:test_nhexa_loop_gain_hac_iff_struts. +#+begin_src matlab +L_hac_iff_struts = pagemtimes(permute(frf_iff.G_dL{1}, [2 3 1]), squeeze(freqresp(Khac_iff_struts, frf_iff.f, 'Hz'))); +#+end_src + +#+begin_src matlab :exports none +%% Bode plot of the Loop Gain +figure; +tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); + + +ax1 = nexttile([2,1]); +hold on; +% Diagonal Elements Model +plot(frf_iff.f, abs(squeeze(L_hac_iff_struts(1,1,:))), 'color', colors(1,:), ... + 'DisplayName', 'Diagonal'); +for i = 2:6 + plot(frf_iff.f, abs(squeeze(L_hac_iff_struts(i,i,:))), 'color', colors(1,:), ... + 'HandleVisibility', 'off'); +end +plot(frf_iff.f, abs(squeeze(L_hac_iff_struts(1,2,:))), 'color', [colors(2,:), 0.2], ... + 'DisplayName', 'Off-Diag'); +for i = 1:5 + for j = i+1:6 + plot(frf_iff.f, abs(squeeze(L_hac_iff_struts(i,j,:))), 'color', [colors(2,:), 0.2], ... + 'HandleVisibility', 'off'); + end +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Loop Gain [-]'); set(gca, 'XTickLabel',[]); +ylim([1e-3, 1e2]); +legend('location', 'northeast'); + +ax2 = nexttile; +hold on; +for i =1:6 + plot(frf_iff.f, 180/pi*angle(squeeze(L_hac_iff_struts(i,i,:))), 'color', colors(1,:)); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); +hold off; +yticks(-360:90:360); +ylim([-180, 180]); + +linkaxes([ax1,ax2],'x'); +xlim([2, 2e3]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/loop_gain_hac_iff_struts.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:test_nhexa_loop_gain_hac_iff_struts +#+caption: Diagonal and off-diagonal elements of the Loop gain for "HAC-IFF-Struts" +#+RESULTS: +[[file:figs/test_nhexa_loop_gain_hac_iff_struts.png]] + +**** Verification of the Stability using the Simscape model +<> + +The HAC-IFF control strategy is implemented using Simscape. +#+begin_src matlab +%% Initialize the Simscape model in closed loop +n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... + 'flex_top_type', '4dof', ... + 'motion_sensor_type', 'plates', ... + 'actuator_type', 'flexible', ... + 'controller_type', 'hac-iff-struts'); +#+end_src + +#+begin_src matlab :exports none +support.type = 1; % On top of vibration table +payload.type = 3; % Payload / 1 "mass layer" + +load('Kiff_opt.mat', 'Kiff'); +#+end_src + +#+begin_src matlab +%% Identify the (damped) transfer function from u to dLm +clear io; io_i = 1; +io(io_i) = linio([mdl, '/du'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs +io(io_i) = linio([mdl, '/dL'], 1, 'openoutput'); io_i = io_i + 1; % Plate Displacement (encoder) +#+end_src + +We identify the closed-loop system. +#+begin_src matlab +%% Identification +Gd_iff_hac_opt = linearize(mdl, io, 0.0, options); +#+end_src + +And verify that it is indeed stable. +#+begin_src matlab :results value replace :exports both +%% Verify the stability +isstable(Gd_iff_hac_opt) +#+end_src + +#+RESULTS: +: 1 + +**** Experimental Validation +Both the Integral Force Feedback controller (developed in Section ref:sec:test_nhexa_enc_plates_iff) and the high authority controller working in the frame of the struts (developed in Section ref:sec:test_nhexa_hac_iff_struts_controller) are implemented experimentally. + +Two reference tracking experiments are performed to evaluate the stability and performances of the implemented control. + +#+begin_src matlab +%% Load the experimental data +load('hac_iff_struts_yz_scans.mat', 't', 'de') +#+end_src + +#+begin_src matlab :exports none +%% Reset initial time +t = t - t(1); +#+end_src + +The position of the top-platform is estimated using the Jacobian matrix: +#+begin_src matlab +%% Pose of the top platform from the encoder values +load('jacobian.mat', 'J'); +Xe = [inv(J)*de']'; +#+end_src + +#+begin_src matlab +%% Generate the Y-Z trajectory scan +Rx_yz = generateYZScanTrajectory(... + 'y_tot', 4e-6, ... % Length of Y scans [m] + 'z_tot', 8e-6, ... % Total Z distance [m] + 'n', 5, ... % Number of Y scans + 'Ts', 1e-3, ... % Sampling Time [s] + 'ti', 1, ... % Time to go to initial position [s] + 'tw', 0, ... % Waiting time between each points [s] + 'ty', 0.6, ... % Time for a scan in Y [s] + 'tz', 0.2); % Time for a scan in Z [s] +#+end_src + +The reference path as well as the measured position are partially shown in the Y-Z plane in Figure ref:fig:test_nhexa_yz_scans_exp_results_first_K. +#+begin_src matlab :exports none +%% Position and reference signal in the Y-Z plane +figure; +tiledlayout(1, 3, 'TileSpacing', 'Compact', 'Padding', 'None'); + +ax1 = nexttile; +hold on; +plot(1e6*Xe(t>2,2), 1e6*Xe(t>2,3)); +plot(1e6*Rx_yz(:,3), 1e6*Rx_yz(:,4), '--'); +hold off; +xlabel('Y [$\mu m$]'); ylabel('Z [$\mu m$]'); +xlim([-2.05, 2.05]); ylim([-4.1, 4.1]); +axis equal; + +ax2 = nexttile([1,2]); +hold on; +plot(1e6*Xe(:,2), 1e6*Xe(:,3), ... + 'DisplayName', '$\mathcal{X}_n$'); +plot(1e6*Rx_yz(:,3), 1e6*Rx_yz(:,4), '--', ... + 'DisplayName', '$r_{\mathcal{X}_n}$'); +hold off; +legend('location', 'northwest'); +xlabel('Y [$\mu m$]'); ylabel('Z [$\mu m$]'); +axis equal; +xlim([1.6, 2.1]); ylim([-4.1, -3.6]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/yz_scans_exp_results_first_K.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:test_nhexa_yz_scans_exp_results_first_K +#+caption: Measured position $\bm{\mathcal{X}}_n$ and reference signal $\bm{r}_{\mathcal{X}_n}$ in the Y-Z plane - Zoom on a change of direction +#+RESULTS: +[[file:figs/test_nhexa_yz_scans_exp_results_first_K.png]] + +#+begin_important +It is clear from Figure ref:fig:test_nhexa_yz_scans_exp_results_first_K that the position of the nano-hexapod effectively tracks to reference signal. +However, oscillations with amplitudes as large as 50nm can be observe. + +It turns out that the frequency of these oscillations is 100Hz which is corresponding to the crossover frequency of the High Authority Control loop. +This clearly indicates poor stability margins. +In the next section, the controller is re-designed to improve the stability margins. +#+end_important + +**** Controller with increased stability margins +The High Authority Controller is re-designed in order to improve the stability margins. +#+begin_src matlab +%% Lead +a = 5; % Amount of phase lead / width of the phase lead / high frequency gain +wc = 2*pi*110; % Frequency with the maximum phase lead [rad/s] + +H_lead = (1 + s/(wc/sqrt(a)))/(1 + s/(wc*sqrt(a))); + +%% Low Pass Filter +H_lpf = 1/(1 + s/2/pi/300); + +%% Notch +gm = 0.02; +xi = 0.5; +wn = 2*pi*700; + +H_notch = (s^2 + 2*gm*xi*wn*s + wn^2)/(s^2 + 2*xi*wn*s + wn^2); + +%% HAC Controller +Khac_iff_struts = -2.2e4 * ... % Gain + H_lead * ... % Lead + H_lpf * ... % Lead + H_notch * ... % Notch + (2*pi*100/s) * ... % Integrator + eye(6); % 6x6 Diagonal +#+end_src + +#+begin_src matlab :exports none +%% Load the FRF of the transfer function from u to dL with IFF +frf_iff = load('frf_iff_vib_table_m.mat', 'f', 'Ts', 'G_dL'); +#+end_src + +#+begin_src matlab :exports none +%% Compute the Loop Gain +L_frf = pagemtimes(permute(frf_iff.G_dL{1}, [2 3 1]), squeeze(freqresp(Khac_iff_struts, frf_iff.f, 'Hz'))); +#+end_src + +The bode plot of the new loop gain is shown in Figure ref:fig:test_nhexa_hac_iff_plates_exp_loop_gain_redesigned_K. +#+begin_src matlab :exports none +%% Bode plot for the transfer function from u to dLm +freqs = 2*logspace(1, 3, 1000); + +figure; +tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +% Diagonal Elements FRF +plot(frf_iff.f, abs(squeeze(L_frf(1,1,:))), 'color', colors(1,:), ... + 'DisplayName', 'Diagonal'); +for i = 2:6 + plot(frf_iff.f, abs(squeeze(L_frf(i,i,:))), 'color', colors(1,:), ... + 'HandleVisibility', 'off'); +end +plot(frf_iff.f, abs(squeeze(L_frf(1,2,:))), 'color', [colors(2,:), 0.2], ... + 'DisplayName', 'Off-Diag'); +for i = 1:5 + for j = i+1:6 + plot(frf_iff.f, abs(squeeze(L_frf(i,j,:))), 'color', [colors(2,:), 0.2], ... + 'HandleVisibility', 'off'); + end +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Loop Gain [-]'); set(gca, 'XTickLabel',[]); +ylim([1e-3, 1e2]); +legend('location', 'northeast'); + +ax2 = nexttile; +hold on; +for i =1:6 + plot(frf_iff.f, 180/pi*angle(squeeze(L_frf(i,i,:))), 'color', colors(1,:)); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); +hold off; +yticks(-360:90:360); +ylim([-180, 180]); + +linkaxes([ax1,ax2],'x'); +xlim([1, 2e3]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/hac_iff_plates_exp_loop_gain_redesigned_K.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:test_nhexa_hac_iff_plates_exp_loop_gain_redesigned_K +#+caption: Loop Gain for the updated decentralized HAC controller +#+RESULTS: +[[file:figs/test_nhexa_hac_iff_plates_exp_loop_gain_redesigned_K.png]] + +This new controller is implemented experimentally and several tracking tests are performed. +#+begin_src matlab +%% Load Measurements +load('hac_iff_more_lead_nass_scan.mat', 't', 'de') +#+end_src + +#+begin_src matlab :exports none +%% Reset Time +t = t - t(1); +#+end_src + +The pose of the top platform is estimated from the encoder position using the Jacobian matrix. +#+begin_src matlab +%% Compute the pose of the top platform +load('jacobian.mat', 'J'); +Xe = [inv(J)*de']'; +#+end_src + +#+begin_src matlab :exports none +%% Load the reference path +load('reference_path.mat', 'Rx_nass') +#+end_src + +The measured motion as well as the trajectory are shown in Figure ref:fig:test_nhexa_nass_scans_first_test_exp. +#+begin_src matlab :exports none +%% Plot the X-Y-Z "NASS" trajectory +figure; +hold on; +plot3(Xe(1:100:end,1), Xe(1:100:end,2), Xe(1:100:end,3)) +plot3(Rx_nass(1:100:end,2), Rx_nass(1:100:end,3), Rx_nass(1:100:end,4)) +hold off; +xlabel('x [$\mu m$]'); ylabel('y [$\mu m$]'); zlabel('z [$\mu m$]'); +view(-13, 41) +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/nass_scans_first_test_exp.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:test_nhexa_nass_scans_first_test_exp +#+caption: Measured position $\bm{\mathcal{X}}_n$ and reference signal $\bm{r}_{\mathcal{X}_n}$ for the "NASS" trajectory +#+RESULTS: +[[file:figs/test_nhexa_nass_scans_first_test_exp.png]] + +The trajectory and measured motion are also shown in the X-Y plane in Figure ref:fig:test_nhexa_ref_track_nass_exp_hac_iff_struts. +#+begin_src matlab :exports none +%% Estimate when the hexpod is on top position and drawing the letters +i_top = Xe(:,3) > 1.9e-6; +i_rx = Rx_nass(:,4) > 0; +#+end_src + +#+begin_src matlab :exports none +%% Plot the reference as well as the measurement in the X-Y plane +figure; +tiledlayout(1, 3, 'TileSpacing', 'Compact', 'Padding', 'None'); + +ax1 = nexttile([1,2]); +hold on; +scatter(1e6*Xe(i_top,1), 1e6*Xe(i_top,2),'.'); +plot(1e6*Rx_nass(i_rx,2), 1e6*Rx_nass(i_rx,3), '--'); +hold off; +xlabel('X [$\mu m$]'); ylabel('Y [$\mu m$]'); +axis equal; +xlim([-10.5, 10.5]); ylim([-4.5, 4.5]); + +ax2 = nexttile; +hold on; +scatter(1e6*Xe(i_top,1), 1e6*Xe(i_top,2),'.'); +plot(1e6*Rx_nass(i_rx,2), 1e6*Rx_nass(i_rx,3), '--'); +hold off; +xlabel('X [$\mu m$]'); ylabel('Y [$\mu m$]'); +axis equal; +xlim([4.5, 4.7]); ylim([-0.15, 0.05]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/ref_track_nass_exp_hac_iff_struts.pdf', 'width', 'full', 'height', 'tall'); +#+end_src + +#+name: fig:test_nhexa_ref_track_nass_exp_hac_iff_struts +#+caption: Reference path and measured motion in the X-Y plane +#+RESULTS: +[[file:figs/test_nhexa_ref_track_nass_exp_hac_iff_struts.png]] + +The orientation errors during all the scans are shown in Figure ref:fig:test_nhexa_nass_ref_rx_ry. +#+begin_src matlab :exports none +%% Orientation Errors +figure; +hold on; +plot(t(t>20&t<20.1), 1e6*Xe(t>20&t<20.1,4), '-', 'DisplayName', '$\epsilon_{\theta_x}$'); +plot(t(t>20&t<20.1), 1e6*Xe(t>20&t<20.1,5), '-', 'DisplayName', '$\epsilon_{\theta_y}$'); +plot(t(t>20&t<20.1), 1e6*Xe(t>20&t<20.1,6), '-', 'DisplayName', '$\epsilon_{\theta_z}$'); +hold off; +xlabel('Time [s]'); ylabel('Orientation Error [$\mu$ rad]'); +legend('location', 'northeast'); +#+end_src + +#+begin_src matlab :exports none +%% Orientation Errors +figure; +hold on; +plot(1e9*Xe(100000:100:end,4), 1e9*Xe(100000:100:end,5), '.'); +th = 0:pi/50:2*pi; +xunit = 90 * cos(th); +yunit = 90 * sin(th); +plot(xunit, yunit, '--'); +hold off; +xlabel('$R_x$ [nrad]'); ylabel('$R_y$ [nrad]'); +xlim([-100, 100]); +ylim([-100, 100]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/nass_ref_rx_ry.pdf', 'width', 500, 'height', 500); +#+end_src + +#+name: fig:test_nhexa_nass_ref_rx_ry +#+caption: Orientation errors during the scan +#+RESULTS: +[[file:figs/test_nhexa_nass_ref_rx_ry.png]] + +#+begin_important +Using the updated High Authority Controller, the nano-hexapod can follow trajectories with high accuracy (the position errors are in the order of 50nm peak to peak, and the orientation errors 300nrad peak to peak). +#+end_important + +*** Interaction Analysis and Decoupling +:PROPERTIES: +:header-args:matlab+: :tangle matlab/scripts/interaction_analysis_enc_plates.m +:END: +<> +**** Introduction :ignore: + +In this section, the interaction in the identified plant is estimated using the Relative Gain Array (RGA) [[cite:skogestad07_multiv_feedb_contr][Chap. 3.4]]. + +Then, several decoupling strategies are compared for the nano-hexapod. + +The RGA Matrix is defined as follow: +\begin{equation} + \text{RGA}(G(f)) = G(f) \times (G(f)^{-1})^T +\end{equation} + +Then, the RGA number is defined: +\begin{equation} +\text{RGA-num}(f) = \| \text{I - RGA(G(f))} \|_{\text{sum}} +\end{equation} + + +In this section, the plant with 2 added mass is studied. + +**** Matlab Init :noexport:ignore: +#+begin_src matlab +%% interaction_analysis_enc_plates.m +% Interaction analysis of several decoupling strategies +#+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 +%% Load the identified FRF and Simscape model +frf_iff = load('frf_iff_vib_table_m.mat', 'f', 'Ts', 'G_dL'); +sim_iff = load('sim_iff_vib_table_m.mat', 'G_dL'); +#+end_src + +**** Parameters +#+begin_src matlab +wc = 100; % Wanted crossover frequency [Hz] +[~, i_wc] = min(abs(frf_iff.f - wc)); % Indice corresponding to wc +#+end_src + +#+begin_src matlab +%% Plant to be decoupled +frf_coupled = frf_iff.G_dL{2}; +G_coupled = sim_iff.G_dL{2}; +#+end_src + +**** No Decoupling (Decentralized) +<> + +#+begin_src latex :file decoupling_arch_decentralized.pdf +\begin{tikzpicture} + \node[block] (G) {$\bm{G}$}; + + % Connections and labels + \draw[<-] (G.west) -- ++(-1.8, 0) node[above right]{$\bm{\tau}$}; + \draw[->] (G.east) -- ++( 1.8, 0) node[above left]{$d\bm{\mathcal{L}}$}; + + \begin{scope}[on background layer] + \node[fit={(G.south west) (G.north east)}, fill=black!10!white, draw, dashed, inner sep=16pt] (Gdec) {}; + \node[below right] at (Gdec.north west) {$\bm{G}_{\text{dec}}$}; + \end{scope} +\end{tikzpicture} +#+end_src + +#+name: fig:test_nhexa_decoupling_arch_decentralized +#+caption: Block diagram representing the plant. +#+RESULTS: +[[file:figs/test_nhexa_decoupling_arch_decentralized.png]] + +#+begin_src matlab :exports none +%% Decentralized Plant +figure; +tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +for i = 1:5 + for j = i+1:6 + plot(frf_iff.f, abs(frf_coupled(:,i,j)), 'color', [0,0,0,0.2], ... + 'HandleVisibility', 'off'); + end +end +set(gca,'ColorOrderIndex',1) +for i = 1:6 + plot(frf_iff.f, abs(frf_coupled(:,i,i)), ... + 'DisplayName', sprintf('$y_%i/u_%i$', i, i)); +end +plot(frf_iff.f, abs(frf_coupled(:,1,2)), 'color', [0,0,0,0.2], ... + 'DisplayName', 'Coupling'); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude'); set(gca, 'XTickLabel',[]); +ylim([1e-9, 1e-4]); +legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 3); + +ax2 = nexttile; +hold on; +for i =1:6 + plot(frf_iff.f, 180/pi*angle(frf_coupled(:,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); +ylim([-180, 180]); + +linkaxes([ax1,ax2],'x'); +xlim([10, 1e3]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/interaction_decentralized_plant.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:test_nhexa_interaction_decentralized_plant +#+caption: Bode Plot of the decentralized plant (diagonal and off-diagonal terms) +#+RESULTS: +[[file:figs/test_nhexa_interaction_decentralized_plant.png]] + +#+begin_src matlab :exports none +%% Decentralized RGA +RGA_dec = zeros(size(frf_coupled)); +for i = 1:length(frf_iff.f) + RGA_dec(i,:,:) = squeeze(frf_coupled(i,:,:)).*inv(squeeze(frf_coupled(i,:,:))).'; +end + +RGA_dec_sum = zeros(length(frf_iff), 1); +for i = 1:length(frf_iff.f) + RGA_dec_sum(i) = sum(sum(abs(eye(6) - squeeze(RGA_dec(i,:,:))))); +end +#+end_src + +#+begin_src matlab :exports none +%% RGA for Decentralized plant +figure; +plot(frf_iff.f, RGA_dec_sum, 'k-'); +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +xlabel('Frequency [Hz]'); ylabel('RGA Number'); +xlim([10, 1e3]); ylim([1e-2, 1e2]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/interaction_rga_decentralized.pdf', 'width', 'wide', 'height', 'normal'); +#+end_src + +#+name: fig:test_nhexa_interaction_rga_decentralized +#+caption: RGA number for the decentralized plant +#+RESULTS: +[[file:figs/test_nhexa_interaction_rga_decentralized.png]] + +**** Static Decoupling +<> + +#+begin_src latex :file decoupling_arch_static.pdf +\begin{tikzpicture} + \node[block] (G) {$\bm{G}$}; + \node[block, left=0.8 of G] (Ginv) {$\bm{\hat{G}}(j0)^{-1}$}; + + % Connections and labels + \draw[<-] (Ginv.west) -- ++(-1.8, 0) node[above right]{$\bm{u}$}; + \draw[->] (Ginv.east) -- (G.west) node[above left]{$\bm{\tau}$}; + \draw[->] (G.east) -- ++( 1.8, 0) node[above left]{$d\bm{\mathcal{L}}$}; + + \begin{scope}[on background layer] + \node[fit={(Ginv.south west) (G.north east)}, fill=black!10!white, draw, dashed, inner sep=16pt] (Gx) {}; + \node[below right] at (Gx.north west) {$\bm{G}_{\text{static}}$}; + \end{scope} +\end{tikzpicture} +#+end_src + +#+name: fig:test_nhexa_decoupling_arch_static +#+caption: Decoupling using the inverse of the DC gain of the plant +#+RESULTS: +[[file:figs/test_nhexa_decoupling_arch_static.png]] + +The DC gain is evaluated from the model as be have bad low frequency identification. + +#+begin_src matlab :exports none +%% Compute the inverse of the DC gain +G_model = G_coupled; +G_model.outputdelay = 0; % necessary for further inversion +dc_inv = inv(dcgain(G_model)); + +%% Compute the inversed plant +G_dL_sta = zeros(size(frf_coupled)); +for i = 1:length(frf_iff.f) + G_dL_sta(i,:,:) = squeeze(frf_coupled(i,:,:))*dc_inv; +end +#+end_src + +#+begin_src matlab :exports results :results value table replace :tangle no +data2orgtable(dc_inv, {}, {}, ' %.1f '); +#+end_src + +#+RESULTS: +| -62011.5 | 3910.6 | 4299.3 | 660.7 | -4016.5 | -4373.6 | +| 3914.4 | -61991.2 | -4356.8 | -4019.2 | 640.2 | 4281.6 | +| -4020.0 | -4370.5 | -62004.5 | 3914.6 | 4295.8 | 653.8 | +| 660.9 | 4292.4 | 3903.3 | -62012.2 | -4366.5 | -4008.9 | +| 4302.8 | 655.6 | -4025.8 | -4377.8 | -62006.0 | 3919.7 | +| -4377.9 | -4013.2 | 668.6 | 4303.7 | 3906.8 | -62019.3 | + +#+begin_src matlab :exports none +%% Bode plot of the static decoupled plant +figure; +tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +for i = 1:5 + for j = i+1:6 + plot(frf_iff.f, abs(G_dL_sta(:,i,j)), 'color', [0,0,0,0.2], ... + 'HandleVisibility', 'off'); + end +end +set(gca,'ColorOrderIndex',1) +for i = 1:6 + plot(frf_iff.f, abs(G_dL_sta(:,i,i)), ... + 'DisplayName', sprintf('$y_%i/u_%i$', i, i)); +end +plot(frf_iff.f, abs(G_dL_sta(:,1,2)), 'color', [0,0,0,0.2], ... + 'DisplayName', 'Coupling'); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude'); set(gca, 'XTickLabel',[]); +ylim([1e-3, 1e1]); +legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 2); + +ax2 = nexttile; +hold on; +for i =1:6 + plot(frf_iff.f, 180/pi*angle(G_dL_sta(:,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); +ylim([-180, 180]); + +linkaxes([ax1,ax2],'x'); +xlim([10, 1e3]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/interaction_static_dec_plant.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:test_nhexa_interaction_static_dec_plant +#+caption: Bode Plot of the static decoupled plant +#+RESULTS: +[[file:figs/test_nhexa_interaction_static_dec_plant.png]] + +#+begin_src matlab :exports none +%% Compute RGA Matrix +RGA_sta = zeros(size(frf_coupled)); +for i = 1:length(frf_iff.f) + RGA_sta(i,:,:) = squeeze(G_dL_sta(i,:,:)).*inv(squeeze(G_dL_sta(i,:,:))).'; +end + +%% Compute RGA-number +RGA_sta_sum = zeros(length(frf_iff), 1); +for i = 1:size(RGA_sta, 1) + RGA_sta_sum(i) = sum(sum(abs(eye(6) - squeeze(RGA_sta(i,:,:))))); +end +#+end_src + +#+begin_src matlab :exports none +%% Plot the RGA-number for statically decoupled plant +figure; +plot(frf_iff.f, RGA_sta_sum, 'k-'); +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +xlabel('Frequency [Hz]'); ylabel('RGA Number'); +xlim([10, 1e3]); ylim([1e-2, 1e2]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/interaction_rga_static_dec.pdf', 'width', 'wide', 'height', 'normal'); +#+end_src + +#+name: fig:test_nhexa_interaction_rga_static_dec +#+caption: RGA number for the statically decoupled plant +#+RESULTS: +[[file:figs/test_nhexa_interaction_rga_static_dec.png]] + +**** Decoupling at the Crossover +<> + +#+begin_src latex :file decoupling_arch_crossover.pdf +\begin{tikzpicture} + \node[block] (G) {$\bm{G}$}; + \node[block, left=0.8 of G] (Ginv) {$\bm{\hat{G}}(j\omega_c)^{-1}$}; + + % Connections and labels + \draw[<-] (Ginv.west) -- ++(-1.8, 0) node[above right]{$\bm{u}$}; + \draw[->] (Ginv.east) -- (G.west) node[above left]{$\bm{\tau}$}; + \draw[->] (G.east) -- ++( 1.8, 0) node[above left]{$d\bm{\mathcal{L}}$}; + + \begin{scope}[on background layer] + \node[fit={(Ginv.south west) (G.north east)}, fill=black!10!white, draw, dashed, inner sep=16pt] (Gx) {}; + \node[below right] at (Gx.north west) {$\bm{G}_{\omega_c}$}; + \end{scope} +\end{tikzpicture} +#+end_src + +#+name: fig:test_nhexa_decoupling_arch_crossover +#+caption: Decoupling using the inverse of a dynamical model $\bm{\hat{G}}$ of the plant dynamics $\bm{G}$ +#+RESULTS: +[[file:figs/test_nhexa_decoupling_arch_crossover.png]] + +#+begin_src matlab :exports none +%% Take complex matrix corresponding to the plant at 100Hz +V = squeeze(frf_coupled(i_wc,:,:)); + +%% Real approximation of inv(G(100Hz)) +D = pinv(real(V'*V)); +H1 = D*real(V'*diag(exp(1j*angle(diag(V*D*V.'))/2))); + +%% Compute the decoupled plant +G_dL_wc = zeros(size(frf_coupled)); +for i = 1:length(frf_iff.f) + G_dL_wc(i,:,:) = squeeze(frf_coupled(i,:,:))*H1; +end +#+end_src + +#+begin_src matlab :exports results :results value table replace :tangle no +data2orgtable(H1, {}, {}, ' %.1f '); +#+end_src + +#+RESULTS: +| 67229.8 | 3769.3 | -13704.6 | -23084.8 | -6318.2 | 23378.7 | +| 3486.2 | 67708.9 | 23220.0 | -6314.5 | -22699.8 | -14060.6 | +| -5731.7 | 22471.7 | 66701.4 | 3070.2 | -13205.6 | -21944.6 | +| -23305.5 | -14542.6 | 2743.2 | 70097.6 | 24846.8 | -5295.0 | +| -14882.9 | -22957.8 | -5344.4 | 25786.2 | 70484.6 | 2979.9 | +| 24353.3 | -5195.2 | -22449.0 | -14459.2 | 2203.6 | 69484.2 | + +#+begin_src matlab :exports none +%% Bode plot of the plant decoupled at the crossover +figure; +tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +for i = 1:5 + for j = i+1:6 + plot(frf_iff.f, abs(G_dL_wc(:,i,j)), 'color', [0,0,0,0.2], ... + 'HandleVisibility', 'off'); + end +end +for i = 1:6 + plot(frf_iff.f, abs(G_dL_wc(:,i,i)), ... + 'DisplayName', sprintf('$y_%i/u_%i$', i, i)); +end +plot(frf_iff.f, abs(G_dL_wc(:,1,2)), 'color', [0,0,0,0.2], ... + 'DisplayName', 'Coupling'); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude $d_L/V_a$ [m/V]'); set(gca, 'XTickLabel',[]); +ylim([1e-3, 1e1]); +legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 2); + +ax2 = nexttile; +hold on; +for i =1:6 + plot(frf_iff.f, 180/pi*angle(G_dL_wc(:,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); +ylim([-180, 180]); + +linkaxes([ax1,ax2],'x'); +xlim([10, 1e3]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/interaction_wc_plant.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:test_nhexa_interaction_wc_plant +#+caption: Bode Plot of the plant decoupled at the crossover +#+RESULTS: +[[file:figs/test_nhexa_interaction_wc_plant.png]] + +#+begin_src matlab +%% Compute RGA Matrix +RGA_wc = zeros(size(frf_coupled)); +for i = 1:length(frf_iff.f) + RGA_wc(i,:,:) = squeeze(G_dL_wc(i,:,:)).*inv(squeeze(G_dL_wc(i,:,:))).'; +end + +%% Compute RGA-number +RGA_wc_sum = zeros(size(RGA_wc, 1), 1); +for i = 1:size(RGA_wc, 1) + RGA_wc_sum(i) = sum(sum(abs(eye(6) - squeeze(RGA_wc(i,:,:))))); +end +#+end_src + +#+begin_src matlab :exports none +%% Plot the RGA-Number for the plant decoupled at crossover +figure; +plot(frf_iff.f, RGA_wc_sum, 'k-'); +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +xlabel('Frequency [Hz]'); ylabel('RGA Number'); +xlim([10, 1e3]); ylim([1e-2, 1e2]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/interaction_rga_wc.pdf', 'width', 'wide', 'height', 'normal'); +#+end_src + +#+name: fig:test_nhexa_interaction_rga_wc +#+caption: RGA number for the plant decoupled at the crossover +#+RESULTS: +[[file:figs/test_nhexa_interaction_rga_wc.png]] + +**** SVD Decoupling +<> + +#+begin_src latex :file decoupling_arch_svd.pdf +\begin{tikzpicture} + \node[block] (G) {$\bm{G}$}; + + \node[block, left=0.8 of G.west] (V) {$V^{-T}$}; + \node[block, right=0.8 of G.east] (U) {$U^{-1}$}; + + % Connections and labels + \draw[<-] (V.west) -- ++(-1.0, 0) node[above right]{$u$}; + \draw[->] (V.east) -- (G.west) node[above left]{$\bm{\tau}$}; + \draw[->] (G.east) -- (U.west) node[above left]{$d\bm{\mathcal{L}}$}; + \draw[->] (U.east) -- ++( 1.0, 0) node[above left]{$y$}; + + \begin{scope}[on background layer] + \node[fit={(V.south west) (G.north-|U.east)}, fill=black!10!white, draw, dashed, inner sep=14pt] (Gsvd) {}; + \node[below right] at (Gsvd.north west) {$\bm{G}_{SVD}$}; + \end{scope} +\end{tikzpicture} +#+end_src + +#+name: fig:test_nhexa_decoupling_arch_svd +#+caption: Decoupling using the Singular Value Decomposition +#+RESULTS: +[[file:figs/test_nhexa_decoupling_arch_svd.png]] + +#+begin_src matlab :exports none +%% Take complex matrix corresponding to the plant at 100Hz +V = squeeze(frf_coupled(i_wc,:,:)); + +%% Real approximation of G(100Hz) +D = pinv(real(V'*V)); +H1 = pinv(D*real(V'*diag(exp(1j*angle(diag(V*D*V.'))/2)))); + +%% Singular Value Decomposition +[U,S,V] = svd(H1); + +%% Compute the decoupled plant using SVD +G_dL_svd = zeros(size(frf_coupled)); +for i = 1:length(frf_iff.f) + G_dL_svd(i,:,:) = inv(U)*squeeze(frf_coupled(i,:,:))*inv(V'); +end +#+end_src + +#+begin_src matlab :exports none +%% Bode Plot of the SVD decoupled plant +figure; +tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +for i = 1:5 + for j = i+1:6 + plot(frf_iff.f, abs(G_dL_svd(:,i,j)), 'color', [0,0,0,0.2], ... + 'HandleVisibility', 'off'); + end +end +set(gca,'ColorOrderIndex',1) +for i = 1:6 + plot(frf_iff.f, abs(G_dL_svd(:,i,i)), ... + 'DisplayName', sprintf('$y_%i/u_%i$', i, i)); +end +plot(frf_iff.f, abs(G_dL_svd(:,1,2)), 'color', [0,0,0,0.2], ... + 'DisplayName', 'Coupling'); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude'); set(gca, 'XTickLabel',[]); +ylim([1e-9, 1e-4]); +legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 3); + +ax2 = nexttile; +hold on; +for i =1:6 + plot(frf_iff.f, 180/pi*angle(G_dL_svd(:,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); +ylim([-180, 180]); + +linkaxes([ax1,ax2],'x'); +xlim([10, 1e3]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/interaction_svd_plant.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:test_nhexa_interaction_svd_plant +#+caption: Bode Plot of the plant decoupled using the Singular Value Decomposition +#+RESULTS: +[[file:figs/test_nhexa_interaction_svd_plant.png]] + +#+begin_src matlab +%% Compute the RGA matrix for the SVD decoupled plant +RGA_svd = zeros(size(frf_coupled)); +for i = 1:length(frf_iff.f) + RGA_svd(i,:,:) = squeeze(G_dL_svd(i,:,:)).*inv(squeeze(G_dL_svd(i,:,:))).'; +end + +%% Compute the RGA-number +RGA_svd_sum = zeros(size(RGA_svd, 1), 1); +for i = 1:length(frf_iff.f) + RGA_svd_sum(i) = sum(sum(abs(eye(6) - squeeze(RGA_svd(i,:,:))))); +end +#+end_src + +#+begin_src matlab +%% RGA Number for the SVD decoupled plant +figure; +plot(frf_iff.f, RGA_svd_sum, 'k-'); +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +xlabel('Frequency [Hz]'); ylabel('RGA Number'); +xlim([10, 1e3]); ylim([1e-2, 1e2]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/interaction_rga_svd.pdf', 'width', 'wide', 'height', 'normal'); +#+end_src + +#+name: fig:test_nhexa_interaction_rga_svd +#+caption: RGA number for the plant decoupled using the SVD +#+RESULTS: +[[file:figs/test_nhexa_interaction_rga_svd.png]] + +**** Dynamic decoupling +<> + +#+begin_src latex :file decoupling_arch_dynamic.pdf +\begin{tikzpicture} + \node[block] (G) {$\bm{G}$}; + \node[block, left=0.8 of G] (Ginv) {$\bm{\hat{G}}^{-1}$}; + + % Connections and labels + \draw[<-] (Ginv.west) -- ++(-1.8, 0) node[above right]{$\bm{u}$}; + \draw[->] (Ginv.east) -- (G.west) node[above left]{$\bm{\tau}$}; + \draw[->] (G.east) -- ++( 1.8, 0) node[above left]{$d\bm{\mathcal{L}}$}; + + \begin{scope}[on background layer] + \node[fit={(Ginv.south west) (G.north east)}, fill=black!10!white, draw, dashed, inner sep=16pt] (Gx) {}; + \node[below right] at (Gx.north west) {$\bm{G}_{\text{inv}}$}; + \end{scope} +\end{tikzpicture} +#+end_src + +#+name: fig:test_nhexa_decoupling_arch_dynamic +#+caption: Decoupling using the inverse of a dynamical model $\bm{\hat{G}}$ of the plant dynamics $\bm{G}$ +#+RESULTS: +[[file:figs/test_nhexa_decoupling_arch_dynamic.png]] + +#+begin_src matlab :exports none +%% Compute the plant inverse from the model +G_model = G_coupled; +G_model.outputdelay = 0; % necessary for further inversion +G_inv = inv(G_model); + +%% Compute the decoupled plant +G_dL_inv = zeros(size(frf_coupled)); +for i = 1:length(frf_iff.f) + G_dL_inv(i,:,:) = squeeze(frf_coupled(i,:,:))*squeeze(evalfr(G_inv, 1j*2*pi*frf_iff.f(i))); +end +#+end_src + +#+begin_src matlab :exports none +%% Bode plot of the decoupled plant by full inversion +figure; +tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +for i = 1:5 + for j = i+1:6 + plot(frf_iff.f, abs(G_dL_inv(:,i,j)), 'color', [0,0,0,0.2], ... + 'HandleVisibility', 'off'); + end +end +set(gca,'ColorOrderIndex',1) +for i = 1:6 + plot(frf_iff.f, abs(G_dL_inv(:,i,i)), ... + 'DisplayName', sprintf('$y_%i/u_%i$', i, i)); +end +plot(frf_iff.f, abs(G_dL_inv(:,1,2)), 'color', [0,0,0,0.2], ... + 'DisplayName', 'Coupling'); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude'); set(gca, 'XTickLabel',[]); +ylim([1e-4, 1e1]); +legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 2); + +ax2 = nexttile; +hold on; +for i =1:6 + plot(frf_iff.f, 180/pi*angle(G_dL_inv(:,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); +ylim([-180, 180]); + +linkaxes([ax1,ax2],'x'); +xlim([10, 1e3]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/interaction_dynamic_dec_plant.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:test_nhexa_interaction_dynamic_dec_plant +#+caption: Bode Plot of the dynamically decoupled plant +#+RESULTS: +[[file:figs/test_nhexa_interaction_dynamic_dec_plant.png]] + +#+begin_src matlab :exports none +%% Compute the RGA matrix for the inverse based decoupled plant +RGA_inv = zeros(size(frf_coupled)); +for i = 1:length(frf_iff.f) + RGA_inv(i,:,:) = squeeze(G_dL_inv(i,:,:)).*inv(squeeze(G_dL_inv(i,:,:))).'; +end + +%% Compute the RGA-number +RGA_inv_sum = zeros(size(RGA_inv, 1), 1); +for i = 1:size(RGA_inv, 1) + RGA_inv_sum(i) = sum(sum(abs(eye(6) - squeeze(RGA_inv(i,:,:))))); +end +#+end_src + +#+begin_src matlab :exports none +%% RGA Number for the decoupled plant using full inversion +figure; +plot(frf_iff.f, RGA_inv_sum, 'k-'); +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +xlabel('Frequency [Hz]'); ylabel('RGA Number'); +xlim([10, 1e3]); ylim([1e-2, 1e2]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/interaction_rga_dynamic_dec.pdf', 'width', 'wide', 'height', 'normal'); +#+end_src + +#+name: fig:test_nhexa_interaction_rga_dynamic_dec +#+caption: RGA number for the dynamically decoupled plant +#+RESULTS: +[[file:figs/test_nhexa_interaction_rga_dynamic_dec.png]] + +**** Jacobian Decoupling - Center of Stiffness +<> + +#+begin_src latex :file decoupling_arch_jacobian_cok.pdf +\begin{tikzpicture} + \node[block] (G) {$\bm{G}$}; + \node[block, left=0.8 of G] (Jt) {$J_{s,\{K\}}^{-T}$}; + \node[block, right=0.8 of G] (Ja) {$J_{a,\{K\}}^{-1}$}; + + % Connections and labels + \draw[<-] (Jt.west) -- ++(-1.8, 0) node[above right]{$\bm{\mathcal{F}}_{\{K\}}$}; + \draw[->] (Jt.east) -- (G.west) node[above left]{$\bm{\tau}$}; + \draw[->] (G.east) -- (Ja.west) node[above left]{$d\bm{\mathcal{L}}$}; + \draw[->] (Ja.east) -- ++( 1.8, 0) node[above left]{$\bm{\mathcal{X}}_{\{K\}}$}; + + \begin{scope}[on background layer] + \node[fit={(Jt.south west) (Ja.north east)}, fill=black!10!white, draw, dashed, inner sep=16pt] (Gx) {}; + \node[below right] at (Gx.north west) {$\bm{G}_{\{K\}}$}; + \end{scope} +\end{tikzpicture} +#+end_src + +#+name: fig:test_nhexa_decoupling_arch_jacobian_cok +#+caption: Decoupling using Jacobian matrices evaluated at the Center of Stiffness +#+RESULTS: +[[file:figs/test_nhexa_decoupling_arch_jacobian_cok.png]] + +#+begin_src matlab :exports none +%% Initialize the Nano-Hexapod +n_hexapod = initializeNanoHexapodFinal('MO_B', -42e-3, ... + 'motion_sensor_type', 'plates'); + +%% Get the Jacobians +J_cok = n_hexapod.geometry.J; +Js_cok = n_hexapod.geometry.Js; + +%% Decouple plant using Jacobian (CoM) +G_dL_J_cok = zeros(size(frf_coupled)); +for i = 1:length(frf_iff.f) + G_dL_J_cok(i,:,:) = inv(Js_cok)*squeeze(frf_coupled(i,:,:))*inv(J_cok'); +end +#+end_src + +The obtained plant is shown in Figure ref:fig:test_nhexa_interaction_J_cok_plant_not_normalized. +We can see that the stiffness in the $x$, $y$ and $z$ directions are equal, which is due to the cubic architecture of the Stewart platform. + +#+begin_src matlab :exports none +%% Bode Plot of the SVD decoupled plant +figure; +tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +for i = 1:5 + for j = i+1:6 + plot(frf_iff.f, abs(G_dL_J_cok(:,i,j)), 'color', [0,0,0,0.2], ... + 'HandleVisibility', 'off'); + end +end +set(gca,'ColorOrderIndex',1) +plot(frf_iff.f, abs(G_dL_J_cok(:,1,1)), ... + 'DisplayName', '$D_x/\tilde{\mathcal{F}}_x$'); +plot(frf_iff.f, abs(G_dL_J_cok(:,2,2)), ... + 'DisplayName', '$D_y/\tilde{\mathcal{F}}_y$'); +plot(frf_iff.f, abs(G_dL_J_cok(:,3,3)), ... + 'DisplayName', '$D_z/\tilde{\mathcal{F}}_z$'); +plot(frf_iff.f, abs(G_dL_J_cok(:,4,4)), ... + 'DisplayName', '$R_x/\tilde{\mathcal{M}}_x$'); +plot(frf_iff.f, abs(G_dL_J_cok(:,5,5)), ... + 'DisplayName', '$R_y/\tilde{\mathcal{M}}_y$'); +plot(frf_iff.f, abs(G_dL_J_cok(:,6,6)), ... + 'DisplayName', '$R_z/\tilde{\mathcal{M}}_z$'); +plot(frf_iff.f, abs(G_dL_J_cok(:,1,2)), 'color', [0,0,0,0.2], ... + 'DisplayName', 'Coupling'); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude'); set(gca, 'XTickLabel',[]); +ylim([1e-8, 2e-2]); +legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 3); + +ax2 = nexttile; +hold on; +for i =1:6 + plot(frf_iff.f, 180/pi*angle(G_dL_J_cok(:,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); +ylim([-180, 180]); + +linkaxes([ax1,ax2],'x'); +xlim([10, 1e3]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/interaction_J_cok_plant_not_normalized.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:test_nhexa_interaction_J_cok_plant_not_normalized +#+caption: Bode Plot of the plant decoupled using the Jacobian evaluated at the "center of stiffness" +#+RESULTS: +[[file:figs/test_nhexa_interaction_J_cok_plant_not_normalized.png]] + +Because the plant in translation and rotation has very different gains, we choose to normalize the plant inputs such that the gain of the diagonal term is equal to $1$ at 100Hz. + +The results is shown in Figure ref:fig:test_nhexa_interaction_J_cok_plant. +#+begin_src matlab :exports none +%% Normalize the plant input +[~, i_100] = min(abs(frf_iff.f - 100)); +input_normalize = diag(1./diag(abs(squeeze(G_dL_J_cok(i_100,:,:))))); + +for i = 1:length(frf_iff.f) + G_dL_J_cok(i,:,:) = squeeze(G_dL_J_cok(i,:,:))*input_normalize; +end +#+end_src + +#+begin_src matlab :exports none +%% Bode Plot of the SVD decoupled plant +figure; +tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +for i = 1:5 + for j = i+1:6 + plot(frf_iff.f, abs(G_dL_J_cok(:,i,j)), 'color', [0,0,0,0.2], ... + 'HandleVisibility', 'off'); + end +end +set(gca,'ColorOrderIndex',1) +plot(frf_iff.f, abs(G_dL_J_cok(:,1,1)), ... + 'DisplayName', '$D_x/\tilde{\mathcal{F}}_x$'); +plot(frf_iff.f, abs(G_dL_J_cok(:,2,2)), ... + 'DisplayName', '$D_y/\tilde{\mathcal{F}}_y$'); +plot(frf_iff.f, abs(G_dL_J_cok(:,3,3)), ... + 'DisplayName', '$D_z/\tilde{\mathcal{F}}_z$'); +plot(frf_iff.f, abs(G_dL_J_cok(:,4,4)), ... + 'DisplayName', '$R_x/\tilde{\mathcal{M}}_x$'); +plot(frf_iff.f, abs(G_dL_J_cok(:,5,5)), ... + 'DisplayName', '$R_y/\tilde{\mathcal{M}}_y$'); +plot(frf_iff.f, abs(G_dL_J_cok(:,6,6)), ... + 'DisplayName', '$R_z/\tilde{\mathcal{M}}_z$'); +plot(frf_iff.f, abs(G_dL_J_cok(:,1,2)), 'color', [0,0,0,0.2], ... + 'DisplayName', 'Coupling'); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude'); set(gca, 'XTickLabel',[]); +ylim([1e-4, 1e1]); +legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 2); + +ax2 = nexttile; +hold on; +for i =1:6 + plot(frf_iff.f, 180/pi*angle(G_dL_J_cok(:,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); +ylim([-180, 180]); + +linkaxes([ax1,ax2],'x'); +xlim([10, 1e3]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/interaction_J_cok_plant.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:test_nhexa_interaction_J_cok_plant +#+caption: Bode Plot of the plant decoupled using the Jacobian evaluated at the "center of stiffness" +#+RESULTS: +[[file:figs/test_nhexa_interaction_J_cok_plant.png]] + +#+begin_src matlab :exports none +%% Compute RGA Matrix +RGA_cok = zeros(size(frf_coupled)); +for i = 1:length(frf_iff.f) + RGA_cok(i,:,:) = squeeze(G_dL_J_cok(i,:,:)).*inv(squeeze(G_dL_J_cok(i,:,:))).'; +end + +%% Compute RGA-number +RGA_cok_sum = zeros(length(frf_iff.f), 1); +for i = 1:length(frf_iff.f) + RGA_cok_sum(i) = sum(sum(abs(eye(6) - squeeze(RGA_cok(i,:,:))))); +end +#+end_src + +#+begin_src matlab :exports none +%% Plot the RGA-Number for the Jacobian (CoK) decoupled plant +figure; +plot(frf_iff.f, RGA_cok_sum, 'k-'); +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +xlabel('Frequency [Hz]'); ylabel('RGA Number'); +xlim([10, 1e3]); ylim([1e-2, 1e2]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/interaction_rga_J_cok.pdf', 'width', 'wide', 'height', 'normal'); +#+end_src + +#+name: fig:test_nhexa_interaction_rga_J_cok +#+caption: RGA number for the plant decoupled using the Jacobian evaluted at the Center of Stiffness +#+RESULTS: +[[file:figs/test_nhexa_interaction_rga_J_cok.png]] + +**** Jacobian Decoupling - Center of Mass +<> + +#+begin_src latex :file decoupling_arch_jacobian_com.pdf +\begin{tikzpicture} + \node[block] (G) {$\bm{G}$}; + \node[block, left=0.8 of G] (Jt) {$J_{s,\{M\}}^{-T}$}; + \node[block, right=0.8 of G] (Ja) {$J_{a,\{M\}}^{-1}$}; + + % Connections and labels + \draw[<-] (Jt.west) -- ++(-1.8, 0) node[above right]{$\bm{\mathcal{F}}_{\{M\}}$}; + \draw[->] (Jt.east) -- (G.west) node[above left]{$\bm{\tau}$}; + \draw[->] (G.east) -- (Ja.west) node[above left]{$d\bm{\mathcal{L}}$}; + \draw[->] (Ja.east) -- ++( 1.8, 0) node[above left]{$\bm{\mathcal{X}}_{\{M\}}$}; + + \begin{scope}[on background layer] + \node[fit={(Jt.south west) (Ja.north east)}, fill=black!10!white, draw, dashed, inner sep=16pt] (Gx) {}; + \node[below right] at (Gx.north west) {$\bm{G}_{\{M\}}$}; + \end{scope} +\end{tikzpicture} +#+end_src + +#+name: fig:test_nhexa_decoupling_arch_jacobian_com +#+caption: Decoupling using Jacobian matrices evaluated at the Center of Mass +#+RESULTS: +[[file:figs/test_nhexa_decoupling_arch_jacobian_com.png]] + +#+begin_src matlab :exports none +%% Initialize the Nano-Hexapod +n_hexapod = initializeNanoHexapodFinal('MO_B', 25e-3, ... + 'motion_sensor_type', 'plates'); + +%% Get the Jacobians +J_com = n_hexapod.geometry.J; +Js_com = n_hexapod.geometry.Js; + +%% Decouple plant using Jacobian (CoM) +G_dL_J_com = zeros(size(frf_coupled)); +for i = 1:length(frf_iff.f) + G_dL_J_com(i,:,:) = inv(Js_com)*squeeze(frf_coupled(i,:,:))*inv(J_com'); +end + +%% Normalize the plant input +[~, i_100] = min(abs(frf_iff.f - 100)); +input_normalize = diag(1./diag(abs(squeeze(G_dL_J_com(i_100,:,:))))); + +for i = 1:length(frf_iff.f) + G_dL_J_com(i,:,:) = squeeze(G_dL_J_com(i,:,:))*input_normalize; +end +#+end_src + +#+begin_src matlab :exports none +%% Bode Plot of the SVD decoupled plant +figure; +tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +for i = 1:5 + for j = i+1:6 + plot(frf_iff.f, abs(G_dL_J_com(:,i,j)), 'color', [0,0,0,0.2], ... + 'HandleVisibility', 'off'); + end +end +set(gca,'ColorOrderIndex',1) +plot(frf_iff.f, abs(G_dL_J_com(:,1,1)), ... + 'DisplayName', '$D_x/\tilde{\mathcal{F}}_x$'); +plot(frf_iff.f, abs(G_dL_J_com(:,2,2)), ... + 'DisplayName', '$D_y/\tilde{\mathcal{F}}_y$'); +plot(frf_iff.f, abs(G_dL_J_com(:,3,3)), ... + 'DisplayName', '$D_z/\tilde{\mathcal{F}}_z$'); +plot(frf_iff.f, abs(G_dL_J_com(:,4,4)), ... + 'DisplayName', '$R_x/\tilde{\mathcal{M}}_x$'); +plot(frf_iff.f, abs(G_dL_J_com(:,5,5)), ... + 'DisplayName', '$R_y/\tilde{\mathcal{M}}_y$'); +plot(frf_iff.f, abs(G_dL_J_com(:,6,6)), ... + 'DisplayName', '$R_z/\tilde{\mathcal{M}}_z$'); +plot(frf_iff.f, abs(G_dL_J_com(:,1,2)), 'color', [0,0,0,0.2], ... + 'DisplayName', 'Coupling'); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude'); set(gca, 'XTickLabel',[]); +ylim([1e-3, 1e1]); +legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 2); + +ax2 = nexttile; +hold on; +for i =1:6 + plot(frf_iff.f, 180/pi*angle(G_dL_J_com(:,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); +ylim([-180, 180]); + +linkaxes([ax1,ax2],'x'); +xlim([10, 1e3]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/interaction_J_com_plant.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:test_nhexa_interaction_J_com_plant +#+caption: Bode Plot of the plant decoupled using the Jacobian evaluated at the Center of Mass +#+RESULTS: +[[file:figs/test_nhexa_interaction_J_com_plant.png]] + +#+begin_src matlab :exports none +%% Compute RGA Matrix +RGA_com = zeros(size(frf_coupled)); +for i = 1:length(frf_iff.f) + RGA_com(i,:,:) = squeeze(G_dL_J_com(i,:,:)).*inv(squeeze(G_dL_J_com(i,:,:))).'; +end + +%% Compute RGA-number +RGA_com_sum = zeros(size(RGA_com, 1), 1); +for i = 1:size(RGA_com, 1) + RGA_com_sum(i) = sum(sum(abs(eye(6) - squeeze(RGA_com(i,:,:))))); +end +#+end_src + +#+begin_src matlab :exports none +%% Plot the RGA-Number for the Jacobian (CoM) decoupled plant +figure; +plot(frf_iff.f, RGA_com_sum, 'k-'); +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +xlabel('Frequency [Hz]'); ylabel('RGA Number'); +xlim([10, 1e3]); ylim([1e-2, 1e2]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/interaction_rga_J_com.pdf', 'width', 'wide', 'height', 'normal'); +#+end_src + +#+name: fig:test_nhexa_interaction_rga_J_com +#+caption: RGA number for the plant decoupled using the Jacobian evaluted at the Center of Mass +#+RESULTS: +[[file:figs/test_nhexa_interaction_rga_J_com.png]] + +**** Decoupling Comparison +<> + +Let's now compare all of the decoupling methods (Figure ref:fig:test_nhexa_interaction_compare_rga_numbers). + +#+begin_important +From Figure ref:fig:test_nhexa_interaction_compare_rga_numbers, the following remarks are made: +- *Decentralized plant*: well decoupled below suspension modes +- *Static inversion*: similar to the decentralized plant as the decentralized plant has already a good decoupling at low frequency +- *Crossover inversion*: the decoupling is improved around the crossover frequency as compared to the decentralized plant. However, the decoupling is increased at lower frequency. +- *SVD decoupling*: Very good decoupling up to 235Hz. Especially between 100Hz and 200Hz. +- *Dynamic Inversion*: the plant is very well decoupled at frequencies where the model is accurate (below 235Hz where flexible modes are not modelled). +- *Jacobian - Stiffness*: good decoupling at low frequency. The decoupling increases at the frequency of the suspension modes, but is acceptable up to the strut flexible modes (235Hz). +- *Jacobian - Mass*: bad decoupling at low frequency. Better decoupling above the frequency of the suspension modes, and acceptable decoupling up to the strut flexible modes (235Hz). +#+end_important + +#+begin_src matlab :exports none +%% Comparison of the RGA-Numbers +figure; +hold on; +plot(frf_iff.f, RGA_dec_sum, 'DisplayName', 'Decentralized'); +plot(frf_iff.f, RGA_sta_sum, 'DisplayName', 'Static inv.'); +plot(frf_iff.f, RGA_wc_sum, 'DisplayName', 'Crossover inv.'); +plot(frf_iff.f, RGA_svd_sum, 'DisplayName', 'SVD'); +plot(frf_iff.f, RGA_inv_sum, 'DisplayName', 'Dynamic inv.'); +plot(frf_iff.f, RGA_cok_sum, 'DisplayName', 'Jacobian - CoK'); +plot(frf_iff.f, RGA_com_sum, 'DisplayName', 'Jacobian - CoM'); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +xlabel('Frequency [Hz]'); ylabel('RGA Number'); +xlim([10, 1e3]); ylim([1e-2, 1e2]); +legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 2); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/interaction_compare_rga_numbers.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:test_nhexa_interaction_compare_rga_numbers +#+caption: Comparison of the obtained RGA-numbers for all the decoupling methods +#+RESULTS: +[[file:figs/test_nhexa_interaction_compare_rga_numbers.png]] + +**** Decoupling Robustness +<> + +Let's now see how the decoupling is changing when changing the payload's mass. +#+begin_src matlab +frf_new = frf_iff.G_dL{3}; +#+end_src + +#+begin_src matlab :exports none +%% Decentralized RGA +RGA_dec_b = zeros(size(frf_new)); +for i = 1:length(frf_iff.f) + RGA_dec_b(i,:,:) = squeeze(frf_new(i,:,:)).*inv(squeeze(frf_new(i,:,:))).'; +end + +RGA_dec_sum_b = zeros(length(frf_iff), 1); +for i = 1:length(frf_iff.f) + RGA_dec_sum_b(i) = sum(sum(abs(eye(6) - squeeze(RGA_dec_b(i,:,:))))); +end +#+end_src + +#+begin_src matlab :exports none +%% Static Decoupling +G_dL_sta_b = zeros(size(frf_new)); +for i = 1:length(frf_iff.f) + G_dL_sta_b(i,:,:) = squeeze(frf_new(i,:,:))*dc_inv; +end + +RGA_sta_b = zeros(size(frf_new)); +for i = 1:length(frf_iff.f) + RGA_sta_b(i,:,:) = squeeze(G_dL_sta_b(i,:,:)).*inv(squeeze(G_dL_sta_b(i,:,:))).'; +end + +RGA_sta_sum_b = zeros(size(RGA_sta_b, 1), 1); +for i = 1:size(RGA_sta_b, 1) + RGA_sta_sum_b(i) = sum(sum(abs(eye(6) - squeeze(RGA_sta_b(i,:,:))))); +end +#+end_src + +#+begin_src matlab :exports none +%% Crossover Decoupling +V = squeeze(frf_coupled(i_wc,:,:)); +D = pinv(real(V'*V)); +H1 = D*real(V'*diag(exp(1j*angle(diag(V*D*V.'))/2))); + +G_dL_wc_b = zeros(size(frf_new)); +for i = 1:length(frf_iff.f) + G_dL_wc_b(i,:,:) = squeeze(frf_new(i,:,:))*H1; +end + +RGA_wc_b = zeros(size(frf_new)); +for i = 1:length(frf_iff.f) + RGA_wc_b(i,:,:) = squeeze(G_dL_wc_b(i,:,:)).*inv(squeeze(G_dL_wc_b(i,:,:))).'; +end + +RGA_wc_sum_b = zeros(size(RGA_wc_b, 1), 1); +for i = 1:size(RGA_wc_b, 1) + RGA_wc_sum_b(i) = sum(sum(abs(eye(6) - squeeze(RGA_wc_b(i,:,:))))); +end +#+end_src + +#+begin_src matlab :exports none +%% SVD +V = squeeze(frf_coupled(i_wc,:,:)); +D = pinv(real(V'*V)); +H1 = pinv(D*real(V'*diag(exp(1j*angle(diag(V*D*V.'))/2)))); +[U,S,V] = svd(H1); + +G_dL_svd_b = zeros(size(frf_new)); +for i = 1:length(frf_iff.f) + G_dL_svd_b(i,:,:) = inv(U)*squeeze(frf_new(i,:,:))*inv(V'); +end + +RGA_svd_b = zeros(size(frf_new)); +for i = 1:length(frf_iff.f) + RGA_svd_b(i,:,:) = squeeze(G_dL_svd_b(i,:,:)).*inv(squeeze(G_dL_svd_b(i,:,:))).'; +end + +RGA_svd_sum_b = zeros(size(RGA_svd_b, 1), 1); +for i = 1:size(RGA_svd, 1) + RGA_svd_sum_b(i) = sum(sum(abs(eye(6) - squeeze(RGA_svd_b(i,:,:))))); +end +#+end_src + +#+begin_src matlab :exports none +%% Dynamic Decoupling +G_model = G_coupled; +G_model.outputdelay = 0; % necessary for further inversion +G_inv = inv(G_model); + +G_dL_inv_b = zeros(size(frf_new)); +for i = 1:length(frf_iff.f) + G_dL_inv_b(i,:,:) = squeeze(frf_new(i,:,:))*squeeze(evalfr(G_inv, 1j*2*pi*frf_iff.f(i))); +end + +RGA_inv_b = zeros(size(frf_new)); +for i = 1:length(frf_iff.f) + RGA_inv_b(i,:,:) = squeeze(G_dL_inv_b(i,:,:)).*inv(squeeze(G_dL_inv_b(i,:,:))).'; +end + +RGA_inv_sum_b = zeros(size(RGA_inv_b, 1), 1); +for i = 1:size(RGA_inv_b, 1) + RGA_inv_sum_b(i) = sum(sum(abs(eye(6) - squeeze(RGA_inv_b(i,:,:))))); +end +#+end_src + +#+begin_src matlab :exports none +%% Jacobian (CoK) +G_dL_J_cok_b = zeros(size(frf_new)); +for i = 1:length(frf_iff.f) + G_dL_J_cok_b(i,:,:) = inv(Js_cok)*squeeze(frf_new(i,:,:))*inv(J_cok'); +end + +RGA_cok_b = zeros(size(frf_new)); +for i = 1:length(frf_iff.f) + RGA_cok_b(i,:,:) = squeeze(G_dL_J_cok_b(i,:,:)).*inv(squeeze(G_dL_J_cok_b(i,:,:))).'; +end + +RGA_cok_sum_b = zeros(size(RGA_cok_b, 1), 1); +for i = 1:size(RGA_cok_b, 1) + RGA_cok_sum_b(i) = sum(sum(abs(eye(6) - squeeze(RGA_cok_b(i,:,:))))); +end +#+end_src + +#+begin_src matlab :exports none +%% Jacobian (CoM) +G_dL_J_com_b = zeros(size(frf_new)); +for i = 1:length(frf_iff.f) + G_dL_J_com_b(i,:,:) = inv(Js_com)*squeeze(frf_new(i,:,:))*inv(J_com'); +end + +RGA_com_b = zeros(size(frf_new)); +for i = 1:length(frf_iff.f) + RGA_com_b(i,:,:) = squeeze(G_dL_J_com_b(i,:,:)).*inv(squeeze(G_dL_J_com_b(i,:,:))).'; +end + +RGA_com_sum_b = zeros(size(RGA_com_b, 1), 1); +for i = 1:size(RGA_com_b, 1) + RGA_com_sum_b(i) = sum(sum(abs(eye(6) - squeeze(RGA_com_b(i,:,:))))); +end +#+end_src + +The obtained RGA-numbers are shown in Figure ref:fig:test_nhexa_interaction_compare_rga_numbers_rob. + +#+begin_important +From Figure ref:fig:test_nhexa_interaction_compare_rga_numbers_rob: +- The decoupling using the Jacobian evaluated at the "center of stiffness" seems to give the most robust results. +#+end_important + +#+begin_src matlab :exports none +%% Robustness of the Decoupling method +figure; +hold on; +plot(frf_iff.f, RGA_dec_sum, '-', 'DisplayName', 'Decentralized'); +plot(frf_iff.f, RGA_sta_sum, '-', 'DisplayName', 'Static inv.'); +plot(frf_iff.f, RGA_wc_sum, '-', 'DisplayName', 'Crossover inv.'); +plot(frf_iff.f, RGA_svd_sum, '-', 'DisplayName', 'SVD'); +plot(frf_iff.f, RGA_inv_sum, '-', 'DisplayName', 'Dynamic inv.'); +plot(frf_iff.f, RGA_cok_sum, '-', 'DisplayName', 'Jacobian - CoK'); +plot(frf_iff.f, RGA_com_sum, '-', 'DisplayName', 'Jacobian - CoM'); +set(gca,'ColorOrderIndex',1) +plot(frf_iff.f, RGA_dec_sum_b, '--', 'HandleVisibility', 'off'); +plot(frf_iff.f, RGA_sta_sum_b, '--', 'HandleVisibility', 'off'); +plot(frf_iff.f, RGA_wc_sum_b, '--', 'HandleVisibility', 'off'); +plot(frf_iff.f, RGA_svd_sum_b, '--', 'HandleVisibility', 'off'); +plot(frf_iff.f, RGA_inv_sum_b, '--', 'HandleVisibility', 'off'); +plot(frf_iff.f, RGA_cok_sum_b, '--', 'HandleVisibility', 'off'); +plot(frf_iff.f, RGA_com_sum_b, '--', 'HandleVisibility', 'off'); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +xlabel('Frequency [Hz]'); ylabel('RGA Number'); +xlim([10, 1e3]); ylim([1e-2, 1e2]); +legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 2); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/interaction_compare_rga_numbers_rob.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:test_nhexa_interaction_compare_rga_numbers_rob +#+caption: Change of the RGA-number with a change of the payload. Indication of the robustness of the inversion method. +#+RESULTS: +[[file:figs/test_nhexa_interaction_compare_rga_numbers_rob.png]] + +**** Conclusion + +#+begin_important +Several decoupling methods can be used: +- SVD +- Inverse +- Jacobian (CoK) +#+end_important + +#+name: tab:interaction_analysis_conclusion +#+caption: Summary of the interaction analysis and different decoupling strategies +#+attr_latex: :environment tabularx :width \linewidth :align lccc +#+attr_latex: :center t :booktabs t +| *Method* | *RGA* | *Diag Plant* | *Robustness* | +|----------------+-------+--------------+--------------| +| Decentralized | -- | Equal | ++ | +| Static dec. | -- | Equal | ++ | +| Crossover dec. | - | Equal | 0 | +| SVD | ++ | Diff | + | +| Dynamic dec. | ++ | Unity, equal | - | +| Jacobian - CoK | + | Diff | ++ | +| Jacobian - CoM | 0 | Diff | + | + +*** Robust High Authority Controller +:PROPERTIES: +:header-args:matlab+: :tangle matlab/scripts/hac_lac_enc_plates_suspended_table.m +:END: +<> +**** Introduction :ignore: +In this section we wish to develop a robust High Authority Controller (HAC) that is working for all payloads. + +cite:indri20_mechat_robot + +**** Matlab Init :noexport:ignore: +#+begin_src matlab +%% hac_lac_enc_plates_suspended_table.m +% Development and analysis of a robust High Authority Controller +#+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 +%% Load the identified FRF and Simscape model +frf_iff = load('frf_iff_vib_table_m.mat', 'f', 'Ts', 'G_dL'); +sim_iff = load('sim_iff_vib_table_m.mat', 'G_dL'); +#+end_src + +**** Using Jacobian evaluated at the center of stiffness +***** Decoupled Plant +#+begin_src matlab +G_nom = frf_iff.G_dL{2}; % Nominal Plant +#+end_src + +#+begin_src matlab :exports none +%% Initialize the Nano-Hexapod +n_hexapod = initializeNanoHexapodFinal('MO_B', -42e-3, ... + 'motion_sensor_type', 'plates'); + +%% Get the Jacobians +J_cok = n_hexapod.geometry.J; +Js_cok = n_hexapod.geometry.Js; + +%% Decouple plant using Jacobian (CoM) +G_dL_J_cok = zeros(size(G_nom)); +for i = 1:length(frf_iff.f) + G_dL_J_cok(i,:,:) = inv(Js_cok)*squeeze(G_nom(i,:,:))*inv(J_cok'); +end + +%% Normalize the plant input +[~, i_100] = min(abs(frf_iff.f - 10)); +input_normalize = diag(1./diag(abs(squeeze(G_dL_J_cok(i_100,:,:))))); + +for i = 1:length(frf_iff.f) + G_dL_J_cok(i,:,:) = squeeze(G_dL_J_cok(i,:,:))*input_normalize; +end +#+end_src + +#+begin_src matlab :exports none +%% Bode Plot of the decoupled plant +figure; +tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +for i = 1:5 + for j = i+1:6 + plot(frf_iff.f, abs(G_dL_J_cok(:,i,j)), 'color', [0,0,0,0.2], ... + 'HandleVisibility', 'off'); + end +end +set(gca,'ColorOrderIndex',1) +plot(frf_iff.f, abs(G_dL_J_cok(:,1,1)), ... + 'DisplayName', '$D_x/\tilde{\mathcal{F}}_x$'); +plot(frf_iff.f, abs(G_dL_J_cok(:,2,2)), ... + 'DisplayName', '$D_y/\tilde{\mathcal{F}}_y$'); +plot(frf_iff.f, abs(G_dL_J_cok(:,3,3)), ... + 'DisplayName', '$D_z/\tilde{\mathcal{F}}_z$'); +plot(frf_iff.f, abs(G_dL_J_cok(:,4,4)), ... + 'DisplayName', '$R_x/\tilde{\mathcal{M}}_x$'); +plot(frf_iff.f, abs(G_dL_J_cok(:,5,5)), ... + 'DisplayName', '$R_y/\tilde{\mathcal{M}}_y$'); +plot(frf_iff.f, abs(G_dL_J_cok(:,6,6)), ... + 'DisplayName', '$R_z/\tilde{\mathcal{M}}_z$'); +plot(frf_iff.f, abs(G_dL_J_cok(:,1,2)), 'color', [0,0,0,0.2], ... + 'DisplayName', 'Coupling'); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude'); set(gca, 'XTickLabel',[]); +ylim([1e-3, 1e1]); +legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 2); + +ax2 = nexttile; +hold on; +for i =1:6 + plot(frf_iff.f, 180/pi*angle(G_dL_J_cok(:,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); +ylim([-180, 180]); + +linkaxes([ax1,ax2],'x'); +xlim([10, 1e3]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/bode_plot_hac_iff_plant_jacobian_cok.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:test_nhexa_bode_plot_hac_iff_plant_jacobian_cok +#+caption: Bode plot of the decoupled plant using the Jacobian evaluated at the Center of Stiffness +#+RESULTS: +[[file:figs/test_nhexa_bode_plot_hac_iff_plant_jacobian_cok.png]] + +***** SISO Controller Design +As the diagonal elements of the plant are not equal, several SISO controllers are designed and then combined to form a diagonal controller. +All the diagonal terms of the controller consists of: +- A double integrator to have high gain at low frequency +- A lead around the crossover frequency to increase stability margins +- Two second order low pass filters above the crossover frequency to increase the robustness to high frequency modes + +#+begin_src matlab :exports none +%% Controller Ry,Rz + +% Wanted crossover frequency +wc_Rxy = 2*pi*80; + +% Lead +a = 8.0; % Amount of phase lead / width of the phase lead / high frequency gain +wc = wc_Rxy; % Frequency with the maximum phase lead [rad/s] +Kd_lead = (1 + s/(wc/sqrt(a)))/(1 + s/(wc*sqrt(a)))/sqrt(a); + +% Integrator +w0_int = wc_Rxy/2; % [rad/s] +xi_int = 0.3; + +Kd_int = (1 + 2*xi_int/w0_int*s + s^2/w0_int^2)/(s^2/w0_int^2); + +% Low Pass Filter (High frequency robustness) +w0_lpf = wc_Rxy*2; % Cut-off frequency [rad/s] +xi_lpf = 0.6; % Damping Ratio + +Kd_lpf = 1/(1 + 2*xi_lpf/w0_lpf*s + s^2/w0_lpf^2); + +w0_lpf_b = wc_Rxy*4; % Cut-off frequency [rad/s] +xi_lpf_b = 0.7; % Damping Ratio + +Kd_lpf_b = 1/(1 + 2*xi_lpf_b/w0_lpf_b*s + s^2/w0_lpf_b^2); + +% Unity Gain frequency +[~, i_80] = min(abs(frf_iff.f - wc_Rxy/2/pi)); + +% Combination of all the elements +Kd_Rxy = ... + -1/abs(G_dL_J_cok(i_80,4,4)) * ... + Kd_lead/abs(evalfr(Kd_lead, 1j*wc_Rxy)) * ... % Lead (gain of 1 at wc) + Kd_int /abs(evalfr(Kd_int, 1j*wc_Rxy)) * ... + Kd_lpf_b/abs(evalfr(Kd_lpf_b, 1j*wc_Rxy)) * ... + Kd_lpf /abs(evalfr(Kd_lpf, 1j*wc_Rxy)); % Low Pass Filter +#+end_src + +#+begin_src matlab :exports none +%% Controller Dx,Dy,Rz + +% Wanted crossover frequency +wc_Dxy = 2*pi*100; + +% Lead +a = 8.0; % Amount of phase lead / width of the phase lead / high frequency gain +wc = wc_Dxy; % Frequency with the maximum phase lead [rad/s] +Kd_lead = (1 + s/(wc/sqrt(a)))/(1 + s/(wc*sqrt(a)))/sqrt(a); + +% Integrator +w0_int = wc_Dxy/2; % [rad/s] +xi_int = 0.3; + +Kd_int = (1 + 2*xi_int/w0_int*s + s^2/w0_int^2)/(s^2/w0_int^2); + +% Low Pass Filter (High frequency robustness) +w0_lpf = wc_Dxy*2; % Cut-off frequency [rad/s] +xi_lpf = 0.6; % Damping Ratio + +Kd_lpf = 1/(1 + 2*xi_lpf/w0_lpf*s + s^2/w0_lpf^2); + +w0_lpf_b = wc_Dxy*4; % Cut-off frequency [rad/s] +xi_lpf_b = 0.7; % Damping Ratio + +Kd_lpf_b = 1/(1 + 2*xi_lpf_b/w0_lpf_b*s + s^2/w0_lpf_b^2); + +% Unity Gain frequency +[~, i_100] = min(abs(frf_iff.f - wc_Dxy/2/pi)); + +% Combination of all the elements +Kd_Dyx_Rz = ... + -1/abs(G_dL_J_cok(i_100,1,1)) * ... + Kd_int /abs(evalfr(Kd_int, 1j*wc_Dxy)) * ... % Integrator + Kd_lead/abs(evalfr(Kd_lead, 1j*wc_Dxy)) * ... % Lead (gain of 1 at wc) + Kd_lpf_b/abs(evalfr(Kd_lpf_b, 1j*wc_Dxy)) * ... % Lead (gain of 1 at wc) + Kd_lpf /abs(evalfr(Kd_lpf, 1j*wc_Dxy)); % Low Pass Filter +#+end_src + +#+begin_src matlab :exports none +%% Controller Dz + +% Wanted crossover frequency +wc_Dz = 2*pi*100; + +% Lead +a = 8.0; % Amount of phase lead / width of the phase lead / high frequency gain +wc = wc_Dz; % Frequency with the maximum phase lead [rad/s] +Kd_lead = (1 + s/(wc/sqrt(a)))/(1 + s/(wc*sqrt(a)))/sqrt(a); + +% Integrator +w0_int = wc_Dz/2; % [rad/s] +xi_int = 0.3; + +Kd_int = (1 + 2*xi_int/w0_int*s + s^2/w0_int^2)/(s^2/w0_int^2); + +% Low Pass Filter (High frequency robustness) +w0_lpf = wc_Dz*2; % Cut-off frequency [rad/s] +xi_lpf = 0.6; % Damping Ratio + +Kd_lpf = 1/(1 + 2*xi_lpf/w0_lpf*s + s^2/w0_lpf^2); + +w0_lpf_b = wc_Dz*4; % Cut-off frequency [rad/s] +xi_lpf_b = 0.7; % Damping Ratio + +Kd_lpf_b = 1/(1 + 2*xi_lpf_b/w0_lpf_b*s + s^2/w0_lpf_b^2); + +% Unity Gain frequency +[~, i_100] = min(abs(frf_iff.f - wc_Dz/2/pi)); + +% Combination of all the elements +Kd_Dz = ... + -1/abs(G_dL_J_cok(i_100,3,3)) * ... + Kd_int /abs(evalfr(Kd_int, 1j*wc_Dz)) * ... % Integrator + Kd_lead/abs(evalfr(Kd_lead, 1j*wc_Dz)) * ... % Lead (gain of 1 at wc) + Kd_lpf_b/abs(evalfr(Kd_lpf_b, 1j*wc_Dz)) * ... % Lead (gain of 1 at wc) + Kd_lpf /abs(evalfr(Kd_lpf, 1j*wc_Dz)); % Low Pass Filter +#+end_src + +#+begin_src matlab :exports none +%% Diagonal Controller +Kd_diag = blkdiag(Kd_Dyx_Rz, Kd_Dyx_Rz, Kd_Dz, Kd_Rxy, Kd_Rxy, Kd_Dyx_Rz); +#+end_src + +***** Obtained Loop Gain +#+begin_src matlab :exports none +%% Experimental Loop Gain +Lmimo = permute(pagemtimes(permute(G_dL_J_cok, [2,3,1]), squeeze(freqresp(Kd_diag, frf_iff.f, 'Hz'))), [3,1,2]); +#+end_src + +#+begin_src matlab :exports none +%% Bode plot of the experimental Loop Gain +figure; +tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +for i = 1:6 + plot(frf_iff.f, abs(Lmimo(:,i,i)), '-'); +end +for i = 1:5 + for j = i+1:6 + plot(frf_iff.f, abs(squeeze(Lmimo(:,i,j))), 'color', [0,0,0,0.2]); + end +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Loop Gain'); set(gca, 'XTickLabel',[]); +ylim([1e-3, 1e+3]); + +ax2 = nexttile; +hold on; +for i = 1:6 + plot(frf_iff.f, 180/pi*angle(Lmimo(:,i,i)), '-'); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); +hold off; +yticks(-360:45:360); +ylim([-180, 180]); + +linkaxes([ax1,ax2],'x'); +xlim([1, 2e3]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/bode_plot_hac_iff_loop_gain_jacobian_cok.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:test_nhexa_bode_plot_hac_iff_loop_gain_jacobian_cok +#+caption: Bode plot of the Loop Gain when using the Jacobian evaluated at the Center of Stiffness to decouple the system +#+RESULTS: +[[file:figs/test_nhexa_bode_plot_hac_iff_loop_gain_jacobian_cok.png]] + +#+begin_src matlab +%% Controller to be implemented +Kd = inv(J_cok')*input_normalize*ss(Kd_diag)*inv(Js_cok); +#+end_src + +***** Verification of the Stability +Now the stability of the feedback loop is verified using the generalized Nyquist criteria. + +#+begin_src matlab :exports none +%% Compute the Eigenvalues of the loop gain +Ldet = zeros(3, 6, length(frf_iff.f)); + +for i_mass = 1:3 + % Loop gain + Lmimo = pagemtimes(permute(frf_iff.G_dL{i_mass}, [2,3,1]),squeeze(freqresp(Kd, frf_iff.f, 'Hz'))); + for i_f = 2:length(frf_iff.f) + Ldet(i,:, i_f) = eig(squeeze(Lmimo(:,:,i_f))); + end +end +#+end_src + +#+begin_src matlab :exports none +%% Plot of the eigenvalues of L in the complex plane +figure; +hold on; +for i_mass = 2:3 + plot(real(squeeze(Ldet(i_mass, 1,:))), imag(squeeze(Ldet(i_mass, 1,:))), ... + '.', 'color', colors(i_mass+1, :), ... + 'DisplayName', sprintf('%i masses', i_mass)); + plot(real(squeeze(Ldet(i_mass, 1,:))), -imag(squeeze(Ldet(i_mass, 1,:))), ... + '.', 'color', colors(i_mass+1, :), ... + 'HandleVisibility', 'off'); + for i = 1:6 + plot(real(squeeze(Ldet(i_mass, i,:))), imag(squeeze(Ldet(i_mass, i,:))), ... + '.', 'color', colors(i_mass+1, :), ... + 'HandleVisibility', 'off'); + plot(real(squeeze(Ldet(i_mass, i,:))), -imag(squeeze(Ldet(i_mass, i,:))), ... + '.', 'color', colors(i_mass+1, :), ... + 'HandleVisibility', 'off'); + end +end +plot(-1, 0, 'kx', 'HandleVisibility', 'off'); +hold off; +set(gca, 'XScale', 'lin'); set(gca, 'YScale', 'lin'); +xlabel('Real'); ylabel('Imag'); +legend('location', 'southeast'); +xlim([-3, 1]); ylim([-2, 2]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/loci_hac_iff_loop_gain_jacobian_cok.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:test_nhexa_loci_hac_iff_loop_gain_jacobian_cok +#+caption: Loci of $L(j\omega)$ in the complex plane. +#+RESULTS: +[[file:figs/test_nhexa_loci_hac_iff_loop_gain_jacobian_cok.png]] + +***** Save for further analysis +#+begin_src matlab :exports none :tangle no +save('matlab/data_sim/Khac_iff_struts_jacobian_cok.mat', 'Kd') +#+end_src + +#+begin_src matlab :eval no +save('data_sim/Khac_iff_struts_jacobian_cok.mat', 'Kd') +#+end_src + +***** Sensitivity transfer function from the model +#+begin_src matlab :exports none +%% Open Simulink Model +mdl = 'nano_hexapod_simscape'; + +options = linearizeOptions; +options.SampleTime = 0; + +open(mdl) + +Rx = zeros(1, 7); + +colors = colororder; +#+end_src + +#+begin_src matlab :exports none +%% Initialize the Simscape model in closed loop +n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... + 'flex_top_type', '4dof', ... + 'motion_sensor_type', 'plates', ... + 'actuator_type', '2dof', ... + 'controller_type', 'hac-iff-struts'); + +support.type = 1; % On top of vibration table +payload.type = 2; % Payload +#+end_src + +#+begin_src matlab :exports none +%% Load controllers +load('Kiff_opt.mat', 'Kiff'); +Kiff = c2d(Kiff, Ts, 'Tustin'); +load('Khac_iff_struts_jacobian_cok.mat', 'Kd') +Khac_iff_struts = c2d(Kd, Ts, 'Tustin'); +#+end_src + +#+begin_src matlab :exports none +%% Identify the (damped) transfer function from u to dLm +clear io; io_i = 1; +io(io_i) = linio([mdl, '/Rx'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs +io(io_i) = linio([mdl, '/dL'], 1, 'output'); io_i = io_i + 1; % Plate Displacement (encoder) +#+end_src + +#+begin_src matlab :exports none +%% Identification of the dynamics +Gcl = linearize(mdl, io, 0.0, options); +#+end_src + +#+begin_src matlab :exports none +%% Computation of the sensitivity transfer function +S = eye(6) - inv(n_hexapod.geometry.J)*Gcl; +#+end_src + +The results are shown in Figure ref:fig:test_nhexa_sensitivity_hac_jacobian_cok_3m_comp_model. + +#+begin_src matlab :exports none +%% Bode plot for the transfer function from u to dLm +freqs = logspace(0, 3, 1000); + +figure; +hold on; +for i =1:6 + set(gca,'ColorOrderIndex',i); + plot(freqs, abs(squeeze(freqresp(S(i,i), freqs, 'Hz'))), '--', ... + 'DisplayName', sprintf('$S_{%s}$ - Model', labels{i})); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +xlabel('Frequency [Hz]'); ylabel('Sensitivity [-]'); +ylim([1e-4, 1e1]); +legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 3); +xlim([1, 1e3]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/sensitivity_hac_jacobian_cok_3m_comp_model.pdf', 'width', 'wide', 'height', 'normal'); +#+end_src + +#+name: fig:test_nhexa_sensitivity_hac_jacobian_cok_3m_comp_model +#+caption: Estimated sensitivity transfer functions for the HAC controller using the Jacobian estimated at the Center of Stiffness +#+RESULTS: +[[file:figs/test_nhexa_sensitivity_hac_jacobian_cok_3m_comp_model.png]] + +**** Using Singular Value Decomposition +***** Decoupled Plant +#+begin_src matlab +G_nom = frf_iff.G_dL{2}; % Nominal Plant +#+end_src + +#+begin_src matlab :exports none +%% Take complex matrix corresponding to the plant at 100Hz +wc = 100; % Wanted crossover frequency [Hz] +[~, i_wc] = min(abs(frf_iff.f - wc)); % Indice corresponding to wc + +V = squeeze(G_nom(i_wc,:,:)); + +%% Real approximation of G(100Hz) +D = pinv(real(V'*V)); +H1 = pinv(D*real(V'*diag(exp(1j*angle(diag(V*D*V.'))/2)))); + +%% Singular Value Decomposition +[U,S,V] = svd(H1); + +%% Compute the decoupled plant using SVD +G_dL_svd = zeros(size(G_nom)); +for i = 1:length(frf_iff.f) + G_dL_svd(i,:,:) = inv(U)*squeeze(G_nom(i,:,:))*inv(V'); +end +#+end_src + +#+begin_src matlab :exports none +%% Bode plot of the decoupled plant using SVD +figure; +tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +for i = 1:5 + for j = i+1:6 + plot(frf_iff.f, abs(G_dL_svd(:,i,j)), 'color', [0,0,0,0.2], ... + 'HandleVisibility', 'off'); + end +end +set(gca,'ColorOrderIndex',1); +for i = 1:6 + plot(frf_iff.f, abs(G_dL_svd(:,i,i)), ... + 'DisplayName', sprintf('$y_%i/u_%i$', i, i)); +end +plot(frf_iff.f, abs(G_dL_svd(:,1,2)), 'color', [0,0,0,0.2], ... + 'DisplayName', 'Coupling'); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude'); set(gca, 'XTickLabel',[]); +ylim([1e-9, 1e-4]); +legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 2); + +ax2 = nexttile; +hold on; +for i =1:6 + plot(frf_iff.f, 180/pi*angle(G_dL_svd(:,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); +ylim([-180, 180]); + +linkaxes([ax1,ax2],'x'); +xlim([10, 1e3]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/bode_plot_hac_iff_plant_svd.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:test_nhexa_bode_plot_hac_iff_plant_svd +#+caption: Bode plot of the decoupled plant using the SVD +#+RESULTS: +[[file:figs/test_nhexa_bode_plot_hac_iff_plant_svd.png]] + +***** Controller Design +#+begin_src matlab :exports none +%% Lead +a = 6.0; % Amount of phase lead / width of the phase lead / high frequency gain +wc = 2*pi*100; % Frequency with the maximum phase lead [rad/s] +Kd_lead = (1 + s/(wc/sqrt(a)))/(1 + s/(wc*sqrt(a)))/sqrt(a); + +%% Integrator +Kd_int = ((2*pi*50 + s)/(2*pi*0.1 + s))^2; + +%% Low Pass Filter (High frequency robustness) +w0_lpf = 2*pi*200; % Cut-off frequency [rad/s] +xi_lpf = 0.3; % Damping Ratio + +Kd_lpf = 1/(1 + 2*xi_lpf/w0_lpf*s + s^2/w0_lpf^2); + +%% Normalize Gain +Kd_norm = diag(1./abs(diag(squeeze(G_dL_svd(i_wc,:,:))))); + +%% Diagonal Control +Kd_diag = ... + Kd_norm * ... % Normalize gain at 100Hz + Kd_int /abs(evalfr(Kd_int, 1j*2*pi*100)) * ... % Integrator + Kd_lead/abs(evalfr(Kd_lead, 1j*2*pi*100)) * ... % Lead (gain of 1 at wc) + Kd_lpf /abs(evalfr(Kd_lpf, 1j*2*pi*100)); % Low Pass Filter +#+end_src + +#+begin_src matlab :exports none +%% MIMO Controller +Kd = -inv(V') * ... % Output decoupling + ss(Kd_diag) * ... + inv(U); % Input decoupling +#+end_src + +***** Loop Gain +#+begin_src matlab :exports none +%% Experimental Loop Gain +Lmimo = permute(pagemtimes(permute(G_nom, [2,3,1]),squeeze(freqresp(Kd, frf_iff.f, 'Hz'))), [3,1,2]); +#+end_src + +#+begin_src matlab :exports none +%% Loop gain when using SVD +figure; +tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +for i = 1:6 + plot(frf_iff.f, abs(Lmimo(:,i,i)), '-'); +end +for i = 1:5 + for j = i+1:6 + plot(frf_iff.f, abs(squeeze(Lmimo(:,i,j))), 'color', [0,0,0,0.2]); + end +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Loop Gain'); set(gca, 'XTickLabel',[]); +ylim([1e-3, 1e+3]); + +ax2 = nexttile; +hold on; +for i = 1:6 + plot(frf_iff.f, 180/pi*angle(Lmimo(:,i,i)), '-'); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); +hold off; +yticks(-360:30:360); +ylim([-180, 0]); + +linkaxes([ax1,ax2],'x'); +xlim([1, 2e3]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/bode_plot_hac_iff_loop_gain_svd.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:test_nhexa_bode_plot_hac_iff_loop_gain_svd +#+caption: Bode plot of Loop Gain when using the SVD +#+RESULTS: +[[file:figs/test_nhexa_bode_plot_hac_iff_loop_gain_svd.png]] + +***** Stability Verification +#+begin_src matlab +%% Compute the Eigenvalues of the loop gain +Ldet = zeros(3, 6, length(frf_iff.f)); + +for i = 1:3 + Lmimo = pagemtimes(permute(frf_iff.G_dL{i}, [2,3,1]),squeeze(freqresp(Kd, frf_iff.f, 'Hz'))); + for i_f = 2:length(frf_iff.f) + Ldet(i,:, i_f) = eig(squeeze(Lmimo(:,:,i_f))); + end +end +#+end_src + +#+begin_src matlab :exports none +%% Plot of the eigenvalues of L in the complex plane +figure; +hold on; +for i_mass = 2:3 + plot(real(squeeze(Ldet(i_mass, 1,:))), imag(squeeze(Ldet(i_mass, 1,:))), ... + '.', 'color', colors(i_mass+1, :), ... + 'DisplayName', sprintf('%i masses', i_mass)); + plot(real(squeeze(Ldet(i_mass, 1,:))), -imag(squeeze(Ldet(i_mass, 1,:))), ... + '.', 'color', colors(i_mass+1, :), ... + 'HandleVisibility', 'off'); + for i = 1:6 + plot(real(squeeze(Ldet(i_mass, i,:))), imag(squeeze(Ldet(i_mass, i,:))), ... + '.', 'color', colors(i_mass+1, :), ... + 'HandleVisibility', 'off'); + plot(real(squeeze(Ldet(i_mass, i,:))), -imag(squeeze(Ldet(i_mass, i,:))), ... + '.', 'color', colors(i_mass+1, :), ... + 'HandleVisibility', 'off'); + end +end +plot(-1, 0, 'kx', 'HandleVisibility', 'off'); +hold off; +set(gca, 'XScale', 'lin'); set(gca, 'YScale', 'lin'); +xlabel('Real'); ylabel('Imag'); +legend('location', 'southeast'); +xlim([-3, 1]); ylim([-2, 2]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/loci_hac_iff_loop_gain_svd.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:test_nhexa_loci_hac_iff_loop_gain_svd +#+caption: Locis of $L(j\omega)$ in the complex plane. +#+RESULTS: +[[file:figs/test_nhexa_loci_hac_iff_loop_gain_svd.png]] + +***** Save for further analysis +#+begin_src matlab :exports none :tangle no +save('matlab/data_sim/Khac_iff_struts_svd.mat', 'Kd') +#+end_src + +#+begin_src matlab :eval no +save('data_sim/Khac_iff_struts_svd.mat', 'Kd') +#+end_src + +***** Measured Sensitivity Transfer Function +The sensitivity transfer function is estimated by adding a reference signal $R_x$ consisting of a low pass filtered white noise, and measuring the position error $E_x$ at the same time. + +The transfer function from $R_x$ to $E_x$ is the sensitivity transfer function. + +In order to identify the sensitivity transfer function for all directions, six reference signals are used, one for each direction. + +#+begin_src matlab :exports none +%% Tested directions +labels = {'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'}; +#+end_src + +#+begin_src matlab :exports none +%% Load Identification Data +meas_hac_svd_3m = {}; + +for i = 1:6 + meas_hac_svd_3m(i) = {load(sprintf('T_S_meas_%s_3m_hac_svd_iff.mat', labels{i}), 't', 'Va', 'Vs', 'de', 'Rx')}; +end +#+end_src + +#+begin_src matlab :exports none +%% Setup useful variables +% Sampling Time [s] +Ts = (meas_hac_svd_3m{1}.t(end) - (meas_hac_svd_3m{1}.t(1)))/(length(meas_hac_svd_3m{1}.t)-1); + +% Sampling Frequency [Hz] +Fs = 1/Ts; + +% Hannning Windows +win = hanning(ceil(5*Fs)); + +% And we get the frequency vector +[~, f] = tfestimate(meas_hac_svd_3m{1}.Va, meas_hac_svd_3m{1}.de, win, Noverlap, Nfft, 1/Ts); +#+end_src + +#+begin_src matlab :exports none +%% Load Jacobian matrix +load('jacobian.mat', 'J'); + +%% Compute position error +for i = 1:6 + meas_hac_svd_3m{i}.Xm = [inv(J)*meas_hac_svd_3m{i}.de']'; + meas_hac_svd_3m{i}.Ex = meas_hac_svd_3m{i}.Rx - meas_hac_svd_3m{i}.Xm; +end +#+end_src + +An example is shown in Figure ref:fig:test_nhexa_ref_track_hac_svd_3m where both the reference signal and the measured position are shown for translations in the $x$ direction. + +#+begin_src matlab :exports none +figure; +hold on; +plot(meas_hac_svd_3m{1}.t, meas_hac_svd_3m{1}.Xm(:,1), 'DisplayName', 'Pos.') +plot(meas_hac_svd_3m{1}.t, meas_hac_svd_3m{1}.Rx(:,1), 'DisplayName', 'Ref.') +hold off; +xlabel('Time [s]'); ylabel('Dx motion [m]'); +xlim([20, 22]); +legend('location', 'northeast'); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/ref_track_hac_svd_3m.pdf', 'width', 'wide', 'height', 'normal'); +#+end_src + +#+name: fig:test_nhexa_ref_track_hac_svd_3m +#+caption: Reference position and measured position +#+RESULTS: +[[file:figs/test_nhexa_ref_track_hac_svd_3m.png]] + +#+begin_src matlab :exports none +%% Transfer function estimate of S +S_hac_svd_3m = zeros(length(f), 6, 6); + +for i = 1:6 + S_hac_svd_3m(:,:,i) = tfestimate(meas_hac_svd_3m{i}.Rx, meas_hac_svd_3m{i}.Ex, win, Noverlap, Nfft, 1/Ts); +end +#+end_src + +The sensitivity transfer functions estimated for all directions are shown in Figure ref:fig:test_nhexa_sensitivity_hac_svd_3m. + +#+begin_src matlab :exports none +%% Bode plot for the transfer function from u to dLm +figure; +hold on; +for i =1:6 + plot(f, abs(S_hac_svd_3m(:,i,i)), ... + 'DisplayName', sprintf('$S_{%s}$', labels{i})); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +xlabel('Frequency [Hz]'); ylabel('Sensitivity [-]'); +ylim([1e-4, 1e1]); +legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 3); +xlim([0.5, 1e3]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/sensitivity_hac_svd_3m.pdf', 'width', 'wide', 'height', 'normal'); +#+end_src + +#+name: fig:test_nhexa_sensitivity_hac_svd_3m +#+caption: Measured diagonal elements of the sensitivity transfer function matrix. +#+RESULTS: +[[file:figs/test_nhexa_sensitivity_hac_svd_3m.png]] + +#+begin_important +From Figure ref:fig:test_nhexa_sensitivity_hac_svd_3m: +- The sensitivity transfer functions are similar for all directions +- The disturbance attenuation at 1Hz is almost a factor 1000 as wanted +- The sensitivity transfer functions for $R_x$ and $R_y$ have high peak values which indicate poor stability margins. +#+end_important + +***** Sensitivity transfer function from the model +The sensitivity transfer function is now estimated using the model and compared with the one measured. + +#+begin_src matlab :exports none +%% Open Simulink Model +mdl = 'nano_hexapod_simscape'; + +options = linearizeOptions; +options.SampleTime = 0; + +open(mdl) + +Rx = zeros(1, 7); + +colors = colororder; +#+end_src + +#+begin_src matlab :exports none +%% Initialize the Simscape model in closed loop +n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... + 'flex_top_type', '4dof', ... + 'motion_sensor_type', 'plates', ... + 'actuator_type', '2dof', ... + 'controller_type', 'hac-iff-struts'); + +support.type = 1; % On top of vibration table +payload.type = 2; % Payload +#+end_src + +#+begin_src matlab :exports none +%% Load controllers +load('Kiff_opt.mat', 'Kiff'); +Kiff = c2d(Kiff, Ts, 'Tustin'); +load('Khac_iff_struts_svd.mat', 'Kd') +Khac_iff_struts = c2d(Kd, Ts, 'Tustin'); +#+end_src + +#+begin_src matlab :exports none +%% Identify the (damped) transfer function from u to dLm +clear io; io_i = 1; +io(io_i) = linio([mdl, '/Rx'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs +io(io_i) = linio([mdl, '/dL'], 1, 'output'); io_i = io_i + 1; % Plate Displacement (encoder) +#+end_src + +#+begin_src matlab :exports none +%% Identification of the dynamics +Gcl = linearize(mdl, io, 0.0, options); +#+end_src + +#+begin_src matlab :exports none +%% Computation of the sensitivity transfer function +S = eye(6) - inv(n_hexapod.geometry.J)*Gcl; +#+end_src + +The results are shown in Figure ref:fig:test_nhexa_sensitivity_hac_svd_3m_comp_model. +The model is quite effective in estimating the sensitivity transfer functions except around 60Hz were there is a peak for the measurement. + +#+begin_src matlab :exports none +%% Bode plot for the transfer function from u to dLm +freqs = logspace(0,3,1000); + +figure; +hold on; +for i =1:6 + set(gca,'ColorOrderIndex',i); + plot(f, abs(S_hac_svd_3m(:,i,i)), ... + 'DisplayName', sprintf('$S_{%s}$', labels{i})); + set(gca,'ColorOrderIndex',i); + plot(freqs, abs(squeeze(freqresp(S(i,i), freqs, 'Hz'))), '--', ... + 'DisplayName', sprintf('$S_{%s}$ - Model', labels{i})); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +xlabel('Frequency [Hz]'); ylabel('Sensitivity [-]'); +ylim([1e-4, 1e1]); +legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 3); +xlim([0.5, 1e3]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/sensitivity_hac_svd_3m_comp_model.pdf', 'width', 'wide', 'height', 'normal'); +#+end_src + +#+name: fig:test_nhexa_sensitivity_hac_svd_3m_comp_model +#+caption: Comparison of the measured sensitivity transfer functions with the model +#+RESULTS: +[[file:figs/test_nhexa_sensitivity_hac_svd_3m_comp_model.png]] + +**** Using (diagonal) Dynamical Inverse :noexport: +***** Decoupled Plant +#+begin_src matlab +G_nom = frf_iff.G_dL{2}; % Nominal Plant +G_model = sim_iff.G_dL{2}; % Model of the Plant +#+end_src + +#+begin_src matlab :exports none +%% Simplified model of the diagonal term +balred_opts = balredOptions('FreqIntervals', 2*pi*[0, 1000], 'StateElimMethod', 'Truncate'); + +G_red = balred(G_model(1,1), 8, balred_opts); +G_red.outputdelay = 0; % necessary for further inversion +#+end_src + +#+begin_src matlab +%% Inverse +G_inv = inv(G_red); +[G_z, G_p, G_g] = zpkdata(G_inv); +p_uns = real(G_p{1}) > 0; +G_p{1}(p_uns) = -G_p{1}(p_uns); +G_inv_stable = zpk(G_z, G_p, G_g); +#+end_src + +#+begin_src matlab :exports none +%% "Uncertainty" of inversed plant +freqs = logspace(0,3,1000); + +figure; +tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +for i_mass = i_masses + for i = 1 + plot(freqs, abs(squeeze(freqresp(G_inv_stable*sim_iff.G_dL{i_mass+1}(i,i), freqs, 'Hz'))), '-', 'color', colors(i_mass+1, :), ... + 'DisplayName', sprintf('$d\\mathcal{L}_i/u^\\prime_i$ - %i', i_mass)); + end +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude'); set(gca, 'XTickLabel',[]); +ylim([1e-1, 1e1]); +legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 4); + +ax2 = nexttile; +hold on; +for i_mass = i_masses + for i = 1 + plot(freqs, 180/pi*angle(squeeze(freqresp(G_inv_stable*sim_iff.G_dL{i_mass+1}(1,1), freqs, 'Hz'))), '-', 'color', colors(i_mass+1, :)); + end +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); +hold off; +yticks(-360:15:360); +ylim([-45, 45]); + +linkaxes([ax1,ax2],'x'); +xlim([freqs(1), freqs(end)]); +#+end_src + +***** Controller Design +#+begin_src matlab :exports none +% Wanted crossover frequency +wc = 2*pi*80; +[~, i_wc] = min(abs(frf_iff.f - wc/2/pi)); + +%% Lead +a = 20.0; % Amount of phase lead / width of the phase lead / high frequency gain +Kd_lead = (1 + s/(wc/sqrt(a)))/(1 + s/(wc*sqrt(a)))/sqrt(a); + +%% Integrator +Kd_int = ((wc)/(2*pi*0.2 + s))^2; + +%% Low Pass Filter (High frequency robustness) +w0_lpf = 2*wc; % Cut-off frequency [rad/s] +xi_lpf = 0.3; % Damping Ratio + +Kd_lpf = 1/(1 + 2*xi_lpf/w0_lpf*s + s^2/w0_lpf^2); + +w0_lpf_b = wc*4; % Cut-off frequency [rad/s] +xi_lpf_b = 0.7; % Damping Ratio + +Kd_lpf_b = 1/(1 + 2*xi_lpf_b/w0_lpf_b*s + s^2/w0_lpf_b^2); + +%% Normalize Gain +Kd_norm = diag(1./abs(diag(squeeze(G_dL_svd(i_wc,:,:))))); + +%% Diagonal Control +Kd_diag = ... + G_inv_stable * ... % Normalize gain at 100Hz + Kd_int /abs(evalfr(Kd_int, 1j*wc)) * ... % Integrator + Kd_lead/abs(evalfr(Kd_lead, 1j*wc)) * ... % Lead (gain of 1 at wc) + Kd_lpf /abs(evalfr(Kd_lpf, 1j*wc)); % Low Pass Filter +#+end_src + +#+begin_src matlab :exports none +Kd = ss(Kd_diag)*eye(6); +#+end_src + +***** Loop Gain +#+begin_src matlab :exports none +%% Experimental Loop Gain +Lmimo = permute(pagemtimes(permute(G_nom, [2,3,1]),squeeze(freqresp(Kd, frf_iff.f, 'Hz'))), [3,1,2]); +#+end_src + +#+begin_src matlab :exports none +%% Loop gain when using SVD +figure; +tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +for i = 1:6 + plot(frf_iff.f, abs(Lmimo(:,i,i)), '-'); +end +for i = 1:5 + for j = i+1:6 + plot(frf_iff.f, abs(squeeze(Lmimo(:,i,j))), 'color', [0,0,0,0.2]); + end +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Loop Gain'); set(gca, 'XTickLabel',[]); +ylim([1e-3, 1e+3]); + +ax2 = nexttile; +hold on; +for i = 1:6 + plot(frf_iff.f, 180/pi*angle(Lmimo(:,i,i)), '-'); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); +hold off; +yticks(-360:30:360); +ylim([-180, 0]); + +linkaxes([ax1,ax2],'x'); +xlim([1, 2e3]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/bode_plot_hac_iff_loop_gain_diag_inverse.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:test_nhexa_bode_plot_hac_iff_loop_gain_diag_inverse +#+caption: Bode plot of Loop Gain when using the Diagonal inversion +#+RESULTS: +[[file:figs/test_nhexa_bode_plot_hac_iff_loop_gain_diag_inverse.png]] + +***** Stability Verification +MIMO Nyquist with eigenvalues +#+begin_src matlab +%% Compute the Eigenvalues of the loop gain +Ldet = zeros(3, 6, length(frf_iff.f)); + +for i = 1:3 + Lmimo = pagemtimes(permute(frf_iff.G_dL{i}, [2,3,1]),squeeze(freqresp(Kd, frf_iff.f, 'Hz'))); + for i_f = 2:length(frf_iff.f) + Ldet(i,:, i_f) = eig(squeeze(Lmimo(:,:,i_f))); + end +end +#+end_src + +#+begin_src matlab :exports none +%% Plot of the eigenvalues of L in the complex plane +figure; +hold on; +for i_mass = 2:3 + plot(real(squeeze(Ldet(i_mass, 1,:))), imag(squeeze(Ldet(i_mass, 1,:))), ... + '.', 'color', colors(i_mass+1, :), ... + 'DisplayName', sprintf('%i masses', i_mass)); + plot(real(squeeze(Ldet(i_mass, 1,:))), -imag(squeeze(Ldet(i_mass, 1,:))), ... + '.', 'color', colors(i_mass+1, :), ... + 'HandleVisibility', 'off'); + for i = 1:6 + plot(real(squeeze(Ldet(i_mass, i,:))), imag(squeeze(Ldet(i_mass, i,:))), ... + '.', 'color', colors(i_mass+1, :), ... + 'HandleVisibility', 'off'); + plot(real(squeeze(Ldet(i_mass, i,:))), -imag(squeeze(Ldet(i_mass, i,:))), ... + '.', 'color', colors(i_mass+1, :), ... + 'HandleVisibility', 'off'); + end +end +plot(-1, 0, 'kx', 'HandleVisibility', 'off'); +hold off; +set(gca, 'XScale', 'lin'); set(gca, 'YScale', 'lin'); +xlabel('Real'); ylabel('Imag'); +legend('location', 'southeast'); +xlim([-3, 1]); ylim([-2, 2]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/loci_hac_iff_loop_gain_diag_inverse.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:test_nhexa_loci_hac_iff_loop_gain_diag_inverse +#+caption: Locis of $L(j\omega)$ in the complex plane. +#+RESULTS: +[[file:figs/test_nhexa_loci_hac_iff_loop_gain_diag_inverse.png]] + +#+begin_important +Even though the loop gain seems to be fine, the closed-loop system is unstable. +This might be due to the fact that there is large interaction in the plant. +We could look at the RGA-number to verify that. +#+end_important + +***** Save for further use +#+begin_src matlab :exports none :tangle no +save('matlab/data_sim/Khac_iff_struts_diag_inverse.mat', 'Kd') +#+end_src + +#+begin_src matlab :eval no +save('data_sim/Khac_iff_struts_diag_inverse.mat', 'Kd') +#+end_src + +**** Closed Loop Stability (Model) :noexport: +Verify stability using Simscape model +#+begin_src matlab +%% Initialize the Simscape model in closed loop +n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '2dof', ... + 'flex_top_type', '3dof', ... + 'motion_sensor_type', 'plates', ... + 'actuator_type', '2dof', ... + 'controller_type', 'hac-iff-struts'); +#+end_src + +#+begin_src matlab +%% IFF Controller +Kiff = -g_opt*Kiff_g1*eye(6); +Khac_iff_struts = Kd*eye(6); +#+end_src + +#+begin_src matlab +%% Identify the (damped) transfer function from u to dLm for different values of the IFF gain +clear io; io_i = 1; +io(io_i) = linio([mdl, '/du'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs +io(io_i) = linio([mdl, '/dL'], 1, 'openoutput'); io_i = io_i + 1; % Plate Displacement (encoder) +#+end_src + +#+begin_src matlab +GG_cl = {}; + +for i = i_masses + payload.type = i; + GG_cl(i+1) = {exp(-s*Ts)*linearize(mdl, io, 0.0, options)}; +end +#+end_src + +#+begin_src matlab +for i = i_masses + isstable(GG_cl{i+1}) +end +#+end_src + +MIMO Nyquist +#+begin_src matlab +Kdm = Kd*eye(6); + +Ldet = zeros(3, length(fb(i_lim))); + +for i = 1:3 + Lmimo = pagemtimes(permute(G_damp_m{i}(i_lim,:,:), [2,3,1]),squeeze(freqresp(Kdm, fb(i_lim), 'Hz'))); + Ldet(i,:) = arrayfun(@(t) det(eye(6) + squeeze(Lmimo(:,:,t))), 1:size(Lmimo,3)); +end +#+end_src + +#+begin_src matlab :exports none +%% Bode plot for the transfer function from u to dLm +figure; +hold on; +for i_mass = 3 + for i = 1 + plot(real(Ldet(i_mass,:)), imag(Ldet(i_mass,:)), ... + '-', 'color', colors(i_mass+1, :)); + end +end +hold off; +set(gca, 'XScale', 'lin'); set(gca, 'YScale', 'lin'); +xlabel('Real'); ylabel('Imag'); +xlim([-10, 1]); ylim([-4, 4]); +#+end_src + +MIMO Nyquist with eigenvalues +#+begin_src matlab +Kdm = Kd*eye(6); + +Ldet = zeros(3, 6, length(fb(i_lim))); + +for i = 1:3 + Lmimo = pagemtimes(permute(G_damp_m{i}(i_lim,:,:), [2,3,1]),squeeze(freqresp(Kdm, fb(i_lim), 'Hz'))); + for i_f = 1:length(fb(i_lim)) + Ldet(i,:, i_f) = eig(squeeze(Lmimo(:,:,i_f))); + end +end +#+end_src + +#+begin_src matlab :exports none +%% Bode plot for the transfer function from u to dLm +figure; +hold on; +for i_mass = 1 + for i = 1:6 + plot(real(squeeze(Ldet(i_mass, i,:))), imag(squeeze(Ldet(i_mass, i,:))), ... + '-', 'color', colors(i_mass+1, :)); + end +end +hold off; +set(gca, 'XScale', 'lin'); set(gca, 'YScale', 'lin'); +xlabel('Real'); ylabel('Imag'); +xlim([-10, 1]); ylim([-4, 2]); +#+end_src * Introduction :ignore: This document is dedicated to the experimental study of the nano-hexapod shown in Figure ref:fig:test_nhexa_picture_bench_granite_nano_hexapod. @@ -373,7 +3766,962 @@ This is probably better than that, but we reached the limit of the FARO arm's pr #+attr_latex: :width \linewidth [[file:figs/test_nhexa_nano_hexapod_mounted.jpg]] -* Encoders fixed to the plates +* Suspended Table +** Introduction + +This document is divided as follows: +- Section ref:sec:experimental_setup: the experimental setup and all the instrumentation are described +- Section ref:sec:meas_transformation: the mathematics used to compute the 6DoF motion of a solid body from several inertial sensor is derived +- Section ref:sec:simscape_model: a Simscape model of the vibration table is developed +- Section ref:sec:table_dynamics: the table dynamics is identified and compared with the Simscape model + +** Experimental Setup +<> +*** Introduction :ignore: +*** CAD Model + +#+name: fig:vibration-table-cad-view +#+caption: CAD View of the vibration table +#+attr_latex: :width 0.8\linewidth +[[file:figs/vibration-table-cad-view.png]] + +*** Instrumentation + +#+begin_note +Here are the documentation of the equipment used for this vibration table: +- Modal Shaker: Watson and Gearing +- Inertial Shaker: [[file:doc/inertial_shakers.pdf][IS20]] +- Viscoelastic supports: [[file:doc/810002_doc.pdf][810002]] +- Spring supports: [[file:doc/9129fcb6ec46bb52925bb16155a850f3be01c479.pdf][MV803-12CC]] +- Optical Table: [[https://www.thorlabs.com/thorproduct.cfm?partnumber=B4545A][B4545A]] +- Triaxial Accelerometer: [[https://www.pcb.com/products?model=356b18][356B18]] +- OROS +#+end_note + +*** Suspended table + +- Dimensions :: 450 mm x 450 mm x 60 mm +- Mass :: 21.30 kg + +#+name: fig:compliance_optical_table +#+caption: Compliance of the B4545A optical table +#+attr_latex: :width 0.8\linewidth +[[file:figs/B4545A_Compliance_inLb-780.png]] + +If we include including the bottom interface plate: +- Total mass: 30.7 kg +- CoM: 42mm below Center of optical table +- Ix = 0.54, Iy = 0.54, Iz = 1.07 (with respect to CoM) + +*** Inertial Sensors + +| Equipment | +|----------------------------------| +| (2x) 1D accelerometer [[https://www.pcbpiezotronics.fr/produit/accelerometre/393b05/][PCB 393B05]] | +| (4x) 3D accelerometer [[https://www.pcbpiezotronics.fr/produit/accelerometres/356b18/][PCB 356B18]] | + +** Compute the 6DoF solid body motion from several inertial sensors +:PROPERTIES: +:header-args:matlab+: :tangle matlab/meas_transformation.m +:END: +<> +*** Introduction :ignore: +Let's consider a solid body with several accelerometers attached to it (Figure ref:fig:local_to_global_coordinates). + +#+begin_src latex :file local_to_global_coordinates.pdf :post pdf2svg(file=*this*, ext="png") :exports results +\newcommand\irregularcircle[2]{% radius, irregularity + \pgfextra {\pgfmathsetmacro\len{(#1)+rand*(#2)}} + +(0:\len pt) + \foreach \a in {10,20,...,350}{ + \pgfextra {\pgfmathsetmacro\len{(#1)+rand*(#2)}} + -- +(\a:\len pt) + } -- cycle +} +\begin{tikzpicture} + \draw[rounded corners=1mm, fill=blue!30!white] (0, 0) \irregularcircle{3cm}{1mm}; + + \node[] (origin) at (0, 0) {$\bullet$}; + \begin{scope}[shift={(origin)}] + \def\axissize{0.8cm} + \draw[->] (0, 0) -- ++(\axissize, 0) node[above left]{$x$}; + \draw[->] (0, 0) -- ++(0, \axissize) node[below right]{$y$}; + \draw[fill, color=black] (0, 0) circle (0.05*\axissize); + \node[draw, circle, inner sep=0pt, minimum size=0.4*\axissize, label=left:$z$] (yaxis) at (0, 0){}; + \node[below right] at (0, 0){$\{O\}$}; + \end{scope} + + \coordinate[] (p1) at (-1.5, -1.5); + \coordinate[] (p2) at (-1.5, -1.5); + \coordinate[] (p3) at ( 1.5, -1.5); + \coordinate[] (p4) at ( 1.5, -1.5); + \coordinate[] (p5) at ( 1.5, 1.5); + \coordinate[] (p6) at ( 1.5, 1.5); + + \draw[->] (p1)node[]{$\bullet$} -- ++(0, 1)node[right]{$a_1$}; + \node[draw, circle, inner sep=0pt, minimum size=0.3cm, label=left:$a_2$] at (p2){}; + \draw[fill, color=black] (p2) circle (0.05); + \draw[->] (p3)node[]{$\bullet$} -- ++(1, 0)node[right]{$a_3$}; + \node[draw, circle, inner sep=0pt, minimum size=0.3cm, label=left:$a_4$] at (p4){}; + \draw[fill, color=black] (p4) circle (0.05); + \draw[->] (p5)node[]{$\bullet$} -- ++(1, 0)node[right]{$a_5$}; + \node[draw, circle, inner sep=0pt, minimum size=0.3cm, label=left:$a_6$] at (p6){}; + \draw[fill, color=black] (p6) circle (0.04); +\end{tikzpicture} +#+end_src + +#+name: fig:local_to_global_coordinates +#+caption: Schematic of the measured motions of a solid body +#+RESULTS: +[[file:figs/local_to_global_coordinates.png]] + +The goal of this section is to see how to compute the acceleration/angular acceleration of the solid body from the accelerations $\vec{a}_i$ measured by the accelerometers. + +The acceleration/angular acceleration of the solid body is defined as the vector ${}^O\vec{x}$: +\begin{equation} +{}^O\vec{x} = \begin{bmatrix} + \dot{v}_x \\ \dot{v}_y \\ \dot{v}_z \\ \dot{\omega}_x \\ \dot{\omega}_y \\ \dot{\omega}_z +\end{bmatrix} +\end{equation} + +As we want to measure 6dof, we suppose that we have 6 uniaxial acceleremoters (we could use more, but 6 is enough). +The measurement of the individual vectors is defined as the vector $\vec{a}$: +\begin{equation} +\vec{a} = \begin{bmatrix} + a_1 \\ a_2 \\ a_3 \\ a_4 \\ a_5 \\ a_6 +\end{bmatrix} +\end{equation} + +From the positions and orientations of the acceleremoters (defined in Section ref:sec:accelerometer_pos), it is quite straightforward to compute the accelerations measured by the sensors from the acceleration/angular acceleration of the solid body (Section [[sec:transformation_motion_to_acc]]). +From this, we can easily build a transformation matrix $M$, such that: +\begin{equation} +\vec{a} = M \cdot {}^O\vec{x} +\end{equation} + +If the matrix is invertible, we can just take the inverse in order to obtain the transformation matrix giving the 6dof acceleration of the solid body from the accelerometer measurements (Section ref:sec:transformation_acc_to_motion): +\begin{equation} +{}^O\vec{x} = M^{-1} \cdot \vec{a} +\end{equation} + +If it is not invertible, then it means that it is not possible to compute all 6dof of the solid body from the measurements. +The solution is then to change the location/orientation of some of the accelerometers. + +*** 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 +addpath('matlab/') +#+end_src + +*** Define accelerometers positions/orientations +<> +Let's first define the position and orientation of all measured accelerations with respect to a defined frame $\{O\}$. + +#+begin_src matlab +Opm = [-0.1875, -0.1875, -0.245; + -0.1875, -0.1875, -0.245; + 0.1875, -0.1875, -0.245; + 0.1875, -0.1875, -0.245; + 0.1875, 0.1875, -0.245; + 0.1875, 0.1875, -0.245]'; +#+end_src + +There are summarized in Table ref:tab:accelerometers_table_positions. + +#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*) +data2orgtable(Opm, {'x', 'y', 'z'}, {'$a_1$', '$a_2$', '$a_3$', '$a_4$', '$a_5$', '$a_6$'}, ' %.3f '); +#+end_src + +#+name: tab:accelerometers_table_positions +#+caption: Positions of the accelerometers fixed to the vibration table with respect to $\{O\}$ +#+attr_latex: :environment tabularx :width 0.55\linewidth :align Xcccccc +#+attr_latex: :center t :booktabs t :float t +#+RESULTS: +| | $a_1$ | $a_2$ | $a_3$ | $a_4$ | $a_5$ | $a_6$ | +|---+--------+--------+--------+--------+--------+--------| +| x | -0.188 | -0.188 | 0.188 | 0.188 | 0.188 | 0.188 | +| y | -0.188 | -0.188 | -0.188 | -0.188 | 0.188 | 0.188 | +| z | -0.245 | -0.245 | -0.245 | -0.245 | -0.245 | -0.245 | + +We then define the direction of the measured accelerations (unit vectors): +#+begin_src matlab +Osm = [0, 1, 0; + 0, 0, 1; + 1, 0, 0; + 0, 0, 1; + 1, 0, 0; + 0, 1, 0]'; +#+end_src + +They are summarized in Table ref:tab:accelerometers_table_orientations. + +#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*) +data2orgtable(Osm, {'x', 'y', 'z'}, {'$\hat{s}_1$', '$\hat{s}_2$', '$\hat{s}_3$', '$\hat{s}_4$', '$\hat{s}_5$', '$\hat{s}_6$'}, ' %.0f '); +#+end_src + +#+name: tab:accelerometers_table_orientations +#+caption: Orientations of the accelerometers fixed to the vibration table expressed in $\{O\}$ +#+attr_latex: :environment tabularx :width 0.35\linewidth :align Xcccccc +#+attr_latex: :center t :booktabs t :float t +#+RESULTS: +| | $\hat{s}_1$ | $\hat{s}_2$ | $\hat{s}_3$ | $\hat{s}_4$ | $\hat{s}_5$ | $\hat{s}_6$ | +|---+-------------+-------------+-------------+-------------+-------------+-------------| +| x | 0 | 0 | 1 | 0 | 1 | 0 | +| y | 1 | 0 | 0 | 0 | 0 | 0 | +| z | 0 | 1 | 0 | 1 | 0 | 1 | + +*** Transformation matrix from motion of the solid body to accelerometer measurements +<> + +Let's try to estimate the x-y-z acceleration of any point of the solid body from the acceleration/angular acceleration of the solid body expressed in $\{O\}$. +For any point $p_i$ of the solid body (corresponding to an accelerometer), we can write: +\begin{equation} +\begin{bmatrix} +a_{i,x} \\ a_{i,y} \\ a_{i,z} +\end{bmatrix} = \begin{bmatrix} +\dot{v}_x \\ \dot{v}_y \\ \dot{v}_z +\end{bmatrix} + p_i \times \begin{bmatrix} +\dot{\omega}_x \\ \dot{\omega}_y \\ \dot{\omega}_z +\end{bmatrix} +\end{equation} + +We can write the cross product as a matrix product using the skew-symmetric transformation: +\begin{equation} +\begin{bmatrix} +a_{i,x} \\ a_{i,y} \\ a_{i,z} +\end{bmatrix} = \begin{bmatrix} +\dot{v}_x \\ \dot{v}_y \\ \dot{v}_z +\end{bmatrix} + \underbrace{\begin{bmatrix} + 0 & p_{i,z} & -p_{i,y} \\ + -p_{i,z} & 0 & p_{i,x} \\ + p_{i,y} & -p_{i,x} & 0 +\end{bmatrix}}_{P_{i,[\times]}} \cdot \begin{bmatrix} +\dot{\omega}_x \\ \dot{\omega}_y \\ \dot{\omega}_z +\end{bmatrix} +\end{equation} + +If we now want to know the (scalar) acceleration $a_i$ of the point $p_i$ in the direction of the accelerometer direction $\hat{s}_i$, we can just project the 3d acceleration on $\hat{s}_i$: +\begin{equation} +a_i = \hat{s}_i^T \cdot \begin{bmatrix} +a_{i,x} \\ a_{i,y} \\ a_{i,z} +\end{bmatrix} = \hat{s}_i^T \cdot \begin{bmatrix} +\dot{v}_x \\ \dot{v}_y \\ \dot{v}_z +\end{bmatrix} + \left( \hat{s}_i^T \cdot P_{i,[\times]} \right) \cdot \begin{bmatrix} +\dot{\omega}_x \\ \dot{\omega}_y \\ \dot{\omega}_z +\end{bmatrix} +\end{equation} + +Which is equivalent as a simple vector multiplication: +\begin{equation} +a_i = \begin{bmatrix} +\hat{s}_i^T & \hat{s}_i^T \cdot P_{i,[\times]} +\end{bmatrix} +\begin{bmatrix} +\dot{v}_x \\ \dot{v}_y \\ \dot{v}_z \\ \dot{\omega}_x \\ \dot{\omega}_y \\ \dot{\omega}_z +\end{bmatrix} = \begin{bmatrix} +\hat{s}_i^T & \hat{s}_i^T \cdot P_{i,[\times]} +\end{bmatrix} {}^O\vec{x} +\end{equation} + +And finally we can combine the 6 (line) vectors for the 6 accelerometers to write that in a matrix form. +We obtain Eq. eqref:eq:M_matrix. +#+begin_important +The transformation from solid body acceleration ${}^O\vec{x}$ from sensor measured acceleration $\vec{a}$ is: +\begin{equation} \label{eq:M_matrix} +\vec{a} = \underbrace{\begin{bmatrix} +\hat{s}_1^T & \hat{s}_1^T \cdot P_{1,[\times]} \\ +\vdots & \vdots \\ +\hat{s}_6^T & \hat{s}_6^T \cdot P_{6,[\times]} +\end{bmatrix}}_{M} {}^O\vec{x} +\end{equation} + +with $\hat{s}_i$ the unit vector representing the measured direction of the i'th accelerometer expressed in frame $\{O\}$ and $P_{i,[\times]}$ the skew-symmetric matrix representing the cross product of the position of the i'th accelerometer expressed in frame $\{O\}$. +#+end_important + +Let's define such matrix using matlab: +#+begin_src matlab +M = zeros(length(Opm), 6); + +for i = 1:length(Opm) + Ri = [0, Opm(3,i), -Opm(2,i); + -Opm(3,i), 0, Opm(1,i); + Opm(2,i), -Opm(1,i), 0]; + M(i, 1:3) = Osm(:,i)'; + M(i, 4:6) = Osm(:,i)'*Ri; +end +#+end_src + +The obtained matrix is shown in Table ref:tab:effect_motion_on_meas. + +#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*) +data2orgtable(M, {'$a_1$', '$a_2$', '$a_3$', '$a_4$', '$a_5$', '$a_6$'}, {'$\dot{x}_x$', '$\dot{x}_y$', '$\dot{x}_z$', '$\dot{\omega}_x$', '$\dot{\omega}_y$', '$\dot{\omega}_z$'}, ' %.2f '); +#+end_src + +#+name: tab:effect_motion_on_meas +#+caption: Effect of a displacement/rotation on the 6 measurements +#+attr_latex: :environment tabularx :width 0.45\linewidth :align Xcccccc +#+attr_latex: :center t :booktabs t :float t +#+RESULTS: +| | $\dot{x}_x$ | $\dot{x}_y$ | $\dot{x}_z$ | $\dot{\omega}_x$ | $\dot{\omega}_y$ | $\dot{\omega}_z$ | +|-------+-------------+-------------+-------------+------------------+------------------+------------------| +| $a_1$ | 0.0 | 1.0 | 0.0 | 0.24 | 0.0 | -0.19 | +| $a_2$ | 0.0 | 0.0 | 1.0 | -0.19 | 0.19 | 0.0 | +| $a_3$ | 1.0 | 0.0 | 0.0 | 0.0 | -0.24 | 0.19 | +| $a_4$ | 0.0 | 0.0 | 1.0 | -0.19 | -0.19 | 0.0 | +| $a_5$ | 1.0 | 0.0 | 0.0 | 0.0 | -0.24 | -0.19 | +| $a_6$ | 0.0 | 0.0 | 1.0 | 0.19 | -0.19 | 0.0 | + +*** Compute the transformation matrix from accelerometer measurement to motion of the solid body +<> + +In order to compute the motion of the solid body ${}^O\vec{x}$ with respect to frame $\{O\}$ from the accelerometer measurements $\vec{a}$, we have to inverse the transformation matrix $M$. +\begin{equation} +{}^O\vec{x} = M^{-1} \vec{a} +\end{equation} + +We therefore need the determinant of $M$ to be non zero: +#+begin_src matlab :results value replace :exports both :tangle no +det(M) +#+end_src + +The obtained inverse of the matrix is shown in Table ref:tab:compute_motion_from_meas. + +#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*) +data2orgtable(inv(M), {'$\dot{x}_x$', '$\dot{x}_y$', '$\dot{x}_z$', '$\dot{\omega}_x$', '$\dot{\omega}_y$', '$\dot{\omega}_z$'}, {'$a_1$', '$a_2$', '$a_3$', '$a_4$', '$a_5$', '$a_6$'}, ' %.1f '); +#+end_src + +#+name: tab:compute_motion_from_meas +#+caption: Compute the displacement/rotation from the 6 measurements +#+attr_latex: :environment tabularx :width 0.45\linewidth :align Xcccccc +#+attr_latex: :center t :booktabs t :float t +#+RESULTS: +| | $a_1$ | $a_2$ | $a_3$ | $a_4$ | $a_5$ | $a_6$ | +|------------------+-------+-------+-------+-------+-------+-------| +| $\dot{x}_x$ | 0.0 | 0.7 | 0.5 | -0.7 | 0.5 | 0.0 | +| $\dot{x}_y$ | 1.0 | 0.0 | 0.5 | 0.7 | -0.5 | -0.7 | +| $\dot{x}_z$ | 0.0 | 0.5 | 0.0 | 0.0 | 0.0 | 0.5 | +| $\dot{\omega}_x$ | 0.0 | 0.0 | 0.0 | -2.7 | 0.0 | 2.7 | +| $\dot{\omega}_y$ | 0.0 | 2.7 | 0.0 | -2.7 | 0.0 | 0.0 | +| $\dot{\omega}_z$ | 0.0 | 0.0 | 2.7 | 0.0 | -2.7 | 0.0 | + +** Simscape Model +:PROPERTIES: +:header-args:matlab+: :tangle matlab/simscape_model.m +:END: +<> +*** Introduction :ignore: +In this section, the Simscape model of the vibration table is described. + +#+name: fig:simscape_vibration_table +#+caption: 3D representation of the simscape model +#+attr_latex: :width 0.8\linewidth +[[file:figs/simscape_vibration_table.png]] + +*** 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 +addpath('matlab/') +#+end_src + +#+begin_src matlab +% Run meas_transformation.m script in order to get the tranformation (jacobian) matrix +run('meas_transformation.m') +#+end_src + +#+begin_src matlab +% Open the Simulink File +open('vibration_table') +#+end_src + +*** Simscape Sub-systems +<> + +Parameters for sub-components of the simscape model are defined below. + +**** Springs +<> + +The 4 springs supporting the suspended optical table are modelled with "bushing joints" having stiffness and damping in the x-y-z directions: + +#+begin_src matlab +spring.kx = 1e4; % X- Stiffness [N/m] +spring.cx = 1e1; % X- Damping [N/(m/s)] + +spring.ky = 1e4; % Y- Stiffness [N/m] +spring.cy = 1e1; % Y- Damping [N/(m/s)] + +spring.kz = 1e4; % Z- Stiffness [N/m] +spring.cz = 1e1; % Z- Damping [N/(m/s)] + +spring.z0 = 32e-3; % Equilibrium z-length [m] +#+end_src + +And we can increase the "equilibrium position" of the vertical springs to take into account the gravity forces and start closer to equilibrium: +#+begin_src matlab +spring.dl = (30.5918/4)*9.80665/spring.kz; +#+end_src + +**** Inertial Shaker (IS20) +<> + +The inertial shaker is defined as two solid bodies: +- the "housing" that is fixed to the element that we want to excite +- the "inertial mass" that is suspended inside the housing + +The inertial mass is guided inside the housing and an actuator (coil and magnet) can be used to apply a force between the inertial mass and the support. +The "reacting" force on the support is then used as an excitation. + +#+name: tab:is20_characteristics +#+caption: Summary of the IS20 datasheet +#+attr_latex: :environment tabularx :width 0.4\linewidth :align lX +#+attr_latex: :center t :booktabs t :float t +| Characteristic | Value | +|-----------------+------------| +| Output Force | 20 N | +| Frequency Range | 10-3000 Hz | +| Moving Mass | 0.1 kg | +| Total Mass | 0.3 kg | + +From the datasheet in Table ref:tab:is20_characteristics, we can estimate the parameters of the physical shaker. + +These parameters are defined below +#+begin_src matlab +shaker.w0 = 2*pi*10; % Resonance frequency of moving mass [rad/s] +shaker.m = 0.1; % Moving mass [m] +shaker.m_tot = 0.3; % Total mass [m] +shaker.k = shaker.m*shaker.w0^2; % Spring constant [N/m] +shaker.c = 0.2*sqrt(shaker.k*shaker.m); % Damping [N/(m/s)] +#+end_src + +**** 3D accelerometer (356B18) +<> + +An accelerometer consists of 2 solids: +- a "housing" rigidly fixed to the measured body +- an "inertial mass" suspended inside the housing by springs and guided in the measured direction + +The relative motion between the housing and the inertial mass gives a measurement of the acceleration of the measured body (up to the suspension mode of the inertial mass). + +#+name: tab:356b18_characteristics +#+caption: Summary of the 356B18 datasheet +#+attr_latex: :environment tabularx :width 0.5\linewidth :align lX +#+attr_latex: :center t :booktabs t :float t +| Characteristic | Value | +|---------------------+---------------------| +| Sensitivity | 0.102 V/(m/s2) | +| Frequency Range | 0.5 to 3000 Hz | +| Resonance Frequency | > 20 kHz | +| Resolution | 0.0005 m/s2 rms | +| Weight | 0.025 kg | +| Size | 20.3x26.1x20.3 [mm] | + +Here are defined the parameters for the triaxial accelerometer: +#+begin_src matlab +acc_3d.m = 0.005; % Inertial mass [kg] +acc_3d.m_tot = 0.025; % Total mass [m] + +acc_3d.w0 = 2*pi*20e3; % Resonance frequency [rad/s] + +acc_3d.kx = acc_3d.m*acc_3d.w0^2; % Spring constant [N/m] +acc_3d.ky = acc_3d.m*acc_3d.w0^2; % Spring constant [N/m] +acc_3d.kz = acc_3d.m*acc_3d.w0^2; % Spring constant [N/m] + +acc_3d.cx = 1e2; % Damping [N/(m/s)] +acc_3d.cy = 1e2; % Damping [N/(m/s)] +acc_3d.cz = 1e2; % Damping [N/(m/s)] +#+end_src + +DC gain between support acceleration and inertial mass displacement is $-m/k$: +#+begin_src matlab +acc_3d.g_x = 1/(-acc_3d.m/acc_3d.kx); % [m/s^2/m] +acc_3d.g_y = 1/(-acc_3d.m/acc_3d.ky); % [m/s^2/m] +acc_3d.g_z = 1/(-acc_3d.m/acc_3d.kz); % [m/s^2/m] +#+end_src + +We also define the sensitivity in order to have the outputs in volts. +#+begin_src matlab +acc_3d.gV_x = 0.102; % [V/(m/s^2)] +acc_3d.gV_y = 0.102; % [V/(m/s^2)] +acc_3d.gV_z = 0.102; % [V/(m/s^2)] +#+end_src + +The problem with using such model for accelerometers is that this adds states to the identified models (2x3 states for each triaxial accelerometer). +These states represents the dynamics of the suspended inertial mass. +In the frequency band of interest (few Hz up to ~1 kHz), the dynamics of the inertial mass can be ignore (its resonance is way above 1kHz). +Therefore, we might as well use idealized "transform sensors" blocks as they will give the same result up to ~20kHz while allowing to reduce the number of identified states. + +The accelerometer model can be chosen by setting the =type= property: +#+begin_src matlab +acc_3d.type = 2; % 1: inertial mass, 2: perfect +#+end_src + +*** Identification +<> + +**** Number of states +Let's first use perfect 3d accelerometers: +#+begin_src matlab +acc_3d.type = 2; % 1: inertial mass, 2: perfect +#+end_src + +And identify the dynamics from the shaker force to the measured accelerations: +#+begin_src matlab +%% Name of the Simulink File +mdl = 'vibration_table'; + +%% Input/Output definition +clear io; io_i = 1; +io(io_i) = linio([mdl, '/F'], 1, 'openinput'); io_i = io_i + 1; +io(io_i) = linio([mdl, '/acc'], 1, 'openoutput'); io_i = io_i + 1; + +%% Run the linearization +Gp = linearize(mdl, io); +Gp.InputName = {'F'}; +Gp.OutputName = {'a1', 'a2', 'a3', 'a4', 'a5', 'a6'}; +#+end_src + +#+begin_src matlab :results output replace :exports results :tangle no +size(Gp) +#+end_src + +#+RESULTS: +: size(Gp) +: State-space model with 6 outputs, 1 inputs, and 12 states. + +We indeed have the 12 states corresponding to the 6 DoF of the suspended optical table. + +Let's now consider the inertial masses for the triaxial accelerometers: +#+begin_src matlab +acc_3d.type = 1; % 1: inertial mass, 2: perfect +#+end_src + +#+begin_src matlab +%% Name of the Simulink File +mdl = 'vibration_table'; + +%% Input/Output definition +clear io; io_i = 1; +io(io_i) = linio([mdl, '/F'], 1, 'openinput'); io_i = io_i + 1; +io(io_i) = linio([mdl, '/acc'], 1, 'openoutput'); io_i = io_i + 1; + +%% Run the linearization +Ga = linearize(mdl, io); +Ga.InputName = {'F'}; +Ga.OutputName = {'a1', 'a2', 'a3', 'a4', 'a5', 'a6'}; +#+end_src + +#+begin_src matlab :results output replace :exports results :tangle no +size(Ga) +#+end_src + +#+RESULTS: +: size(Ga) +: State-space model with 6 outputs, 1 inputs, and 30 states. + +And we can see that 18 states have been added. +This corresponds to 6 states for each triaxial accelerometers. + +**** Resonance frequencies and mode shapes +Let's now identify the resonance frequency and mode shapes associated with the suspension modes of the optical table. + +#+begin_src matlab +acc_3d.type = 2; % 1: inertial mass, 2: perfect + +%% Name of the Simulink File +mdl = 'vibration_table'; + +%% Input/Output definition +clear io; io_i = 1; +io(io_i) = linio([mdl, '/F'], 1, 'openinput'); io_i = io_i + 1; +io(io_i) = linio([mdl, '/a1,a2'], 1, 'openoutput'); io_i = io_i + 1; + +%% Run the linearization +G = linearize(mdl, io); +G.InputName = {'F'}; +G.OutputName = {'ax'}; +#+end_src + +Compute the resonance frequencies +#+begin_src matlab +ws = eig(G.A); +ws = ws(imag(ws) > 0); +#+end_src + +And the associated response of the optical table +#+begin_src matlab +x_mod = zeros(6, 6); + +for i = 1:length(ws) + xi = evalfr(G, ws(i)); + x_mod(:,i) = xi./norm(xi); +end +#+end_src + +The results are shown in Table ref:tab:mode_shapes. +The motion associated to the mode shapes are just indicative. + +#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*) +data2orgtable([flip(imag(ws)/2/pi)'; flip(abs(x_mod),2)], {'$\omega_0$ [Hz]', 'x', 'y', 'z', 'Rx', 'Ry', 'Rz'}, {}, ' %.1f '); +#+end_src + +#+name: tab:mode_shapes +#+caption: Resonance frequency and approximation of the mode shapes +#+attr_latex: :environment tabularx :width 0.4\linewidth :align Xcccccc +#+attr_latex: :center t :booktabs t :float t +#+RESULTS: +| $\omega_0$ [Hz] | 5.6 | 5.6 | 5.7 | 8.2 | 8.2 | 8.2 | +|-----------------+-----+-----+-----+-----+-----+-----| +| x | 0.1 | 0.7 | 0.0 | 0.0 | 0.2 | 0.0 | +| y | 0.7 | 0.1 | 0.0 | 0.0 | 0.0 | 0.2 | +| z | 0.0 | 0.0 | 1.0 | 0.0 | 0.0 | 0.0 | +| Rx | 0.7 | 0.1 | 0.0 | 0.0 | 0.1 | 1.0 | +| Ry | 0.1 | 0.7 | 0.0 | 0.0 | 1.0 | 0.1 | +| Rz | 0.0 | 0.0 | 0.0 | 1.0 | 0.0 | 0.0 | + +*** Verify transformation +#+begin_src matlab +%% Options for Linearized +options = linearizeOptions; +options.SampleTime = 0; + +%% Name of the Simulink File +mdl = 'vibration_table'; + +%% Input/Output definition +clear io; io_i = 1; +io(io_i) = linio([mdl, '/F'], 1, 'openinput'); io_i = io_i + 1; +io(io_i) = linio([mdl, '/acc'], 1, 'openoutput'); io_i = io_i + 1; +io(io_i) = linio([mdl, '/acc_O'], 1, 'openoutput'); io_i = io_i + 1; + +%% Run the linearization +G = linearize(mdl, io, 0.0, options); +G.InputName = {'F'}; +G.OutputName = {'a1', 'a2', 'a3', 'a4', 'a5', 'a6', ... + 'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'}; +#+end_src + +#+begin_src matlab +G_acc = inv(M)*G(1:6, 1); +G_id = G(7:12, 1); +#+end_src + +#+begin_src matlab +bodeFig({G_acc(1), G_id(1)}) +bodeFig({G_acc(2), G_id(2)}) +bodeFig({G_acc(3), G_id(3)}) +bodeFig({G_acc(4), G_id(4)}) +bodeFig({G_acc(5), G_id(5)}) +bodeFig({G_acc(6), G_id(6)}) +#+end_src + +** Nano-Hexapod +:PROPERTIES: +:header-args:matlab+: :tangle matlab/nano_hexapod.m +:END: +<> +*** Introduction :ignore: + +A configuration is added to be able to put the nano-hexapod on top of the vibration table as shown in Figure ref:fig:simscape_vibration_table. + +#+name: fig:simscape_vibration_table +#+caption: 3D representation of the simscape model with the nano-hexapod +#+attr_latex: :width 0.8\linewidth +[[file:figs/vibration_table_nano_hexapod_simscape.png]] + +The nano-hexapod's simscape model is taken from another [[https://git.tdehaeze.xyz/tdehaeze/nass-simscape][git repository]]. + +*** 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 +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 +% Open the Simulink File +open('vibration_table') +#+end_src + +*** Nano-Hexapod +#+begin_src matlab +n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... + 'flex_top_type', '4dof', ... + 'motion_sensor_type', 'plates', ... + 'actuator_type', 'flexible'); +#+end_src + +*** Computation of the transmissibility from accelerometer data +**** Introduction :ignore: + +The goal is to compute the $6 \times 6$ transfer function matrix corresponding to the transmissibility of the Nano-Hexapod. + +To do so, several accelerometers are located both on the vibration table and on the top of the nano-hexapod. + +The vibration table is then excited using a Shaker and all the accelerometers signals are recorded. + +Using transformation (jacobian) matrices, it is then possible to compute both the motion of the top and bottom platform of the nano-hexapod. + +Finally, it is possible to compute the $6 \times 6$ transmissibility matrix. + +Such procedure is explained in cite:marneffe04_stewar_platf_activ_vibrat_isolat. + +**** Jacobian matrices + +How to compute the Jacobian matrices is explained in Section ref:sec:meas_transformation. + +#+begin_src matlab +%% Bottom Accelerometers +Opb = [-0.1875, -0.1875, -0.245; + -0.1875, -0.1875, -0.245; + 0.1875, -0.1875, -0.245; + 0.1875, -0.1875, -0.245; + 0.1875, 0.1875, -0.245; + 0.1875, 0.1875, -0.245]'; + +Osb = [0, 1, 0; + 0, 0, 1; + 1, 0, 0; + 0, 0, 1; + 1, 0, 0; + 0, 0, 1;]'; + +Jb = zeros(length(Opb), 6); + +for i = 1:length(Opb) + Ri = [0, Opb(3,i), -Opb(2,i); + -Opb(3,i), 0, Opb(1,i); + Opb(2,i), -Opb(1,i), 0]; + Jb(i, 1:3) = Osb(:,i)'; + Jb(i, 4:6) = Osb(:,i)'*Ri; +end + +Jbinv = inv(Jb); +#+end_src + +#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*) +data2orgtable(Jbinv, {'$\dot{x}_x$', '$\dot{x}_y$', '$\dot{x}_z$', '$\dot{\omega}_x$', '$\dot{\omega}_y$', '$\dot{\omega}_z$'}, {'$a_1$', '$a_2$', '$a_3$', '$a_4$', '$a_5$', '$a_6$'}, ' %.1f '); +#+end_src + +#+RESULTS: +| | $a_1$ | $a_2$ | $a_3$ | $a_4$ | $a_5$ | $a_6$ | +|------------------+-------+-------+-------+-------+-------+-------| +| $\dot{x}_x$ | 0.0 | 0.7 | 0.5 | -0.7 | 0.5 | 0.0 | +| $\dot{x}_y$ | 1.0 | 0.0 | 0.5 | 0.7 | -0.5 | -0.7 | +| $\dot{x}_z$ | 0.0 | 0.5 | 0.0 | 0.0 | 0.0 | 0.5 | +| $\dot{\omega}_x$ | 0.0 | 0.0 | 0.0 | -2.7 | 0.0 | 2.7 | +| $\dot{\omega}_y$ | 0.0 | 2.7 | 0.0 | -2.7 | 0.0 | 0.0 | +| $\dot{\omega}_z$ | 0.0 | 0.0 | 2.7 | 0.0 | -2.7 | 0.0 | + +#+begin_src matlab +%% Top Accelerometers +Opt = [-0.1, 0, -0.150; + -0.1, 0, -0.150; + 0.05, 0.075, -0.150; + 0.05, 0.075, -0.150; + 0.05, -0.075, -0.150; + 0.05, -0.075, -0.150]'; + +Ost = [0, 1, 0; + 0, 0, 1; + 1, 0, 0; + 0, 0, 1; + 1, 0, 0; + 0, 0, 1;]'; + +Jt = zeros(length(Opt), 6); + +for i = 1:length(Opt) + Ri = [0, Opt(3,i), -Opt(2,i); + -Opt(3,i), 0, Opt(1,i); + Opt(2,i), -Opt(1,i), 0]; + Jt(i, 1:3) = Ost(:,i)'; + Jt(i, 4:6) = Ost(:,i)'*Ri; +end + +Jtinv = inv(Jt); +#+end_src + +#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*) +data2orgtable(Jtinv, {'$\dot{x}_x$', '$\dot{x}_y$', '$\dot{x}_z$', '$\dot{\omega}_x$', '$\dot{\omega}_y$', '$\dot{\omega}_z$'}, {'$b_1$', '$b_2$', '$b_3$', '$b_4$', '$b_5$', '$b_6$'}, ' %.1f '); +#+end_src + +#+RESULTS: +| | $b_1$ | $b_2$ | $b_3$ | $b_4$ | $b_5$ | $b_6$ | +|------------------+-------+-------+-------+-------+-------+-------| +| $\dot{x}_x$ | 0.0 | 1.0 | 0.5 | -0.5 | 0.5 | -0.5 | +| $\dot{x}_y$ | 1.0 | 0.0 | -0.7 | -1.0 | 0.7 | 1.0 | +| $\dot{x}_z$ | 0.0 | 0.3 | 0.0 | 0.3 | 0.0 | 0.3 | +| $\dot{\omega}_x$ | 0.0 | 0.0 | 0.0 | 6.7 | 0.0 | -6.7 | +| $\dot{\omega}_y$ | 0.0 | 6.7 | 0.0 | -3.3 | 0.0 | -3.3 | +| $\dot{\omega}_z$ | 0.0 | 0.0 | -6.7 | 0.0 | 6.7 | 0.0 | + +**** Using =linearize= function + +#+begin_src matlab +acc_3d.type = 2; % 1: inertial mass, 2: perfect + +%% Name of the Simulink File +mdl = 'vibration_table'; + +%% Input/Output definition +clear io; io_i = 1; +io(io_i) = linio([mdl, '/F_shaker'], 1, 'openinput'); io_i = io_i + 1; +io(io_i) = linio([mdl, '/acc'], 1, 'openoutput'); io_i = io_i + 1; +io(io_i) = linio([mdl, '/acc_top'], 1, 'openoutput'); io_i = io_i + 1; + +%% Run the linearization +G = linearize(mdl, io); +G.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'}; +G.OutputName = {'a1', 'a2', 'a3', 'a4', 'a5', 'a6', ... + 'b1', 'b2', 'b3', 'b4', 'b5', 'b6'}; +#+end_src + +#+begin_src matlab +Gb = Jbinv*G({'a1', 'a2', 'a3', 'a4', 'a5', 'a6'}, :); +Gt = Jtinv*G({'b1', 'b2', 'b3', 'b4', 'b5', 'b6'}, :); +#+end_src + +#+begin_src matlab +T = inv(Gb)*Gt; +T = minreal(T); +T = prescale(T, {2*pi*0.1, 2*pi*1e3}); +#+end_src + +#+begin_src matlab :exports none +freqs = logspace(0, 3, 1000); + +figure; +hold on; +for i = 1:6 + plot(freqs, abs(squeeze(freqresp(T(i, i), freqs, 'Hz')))); +end +for i = 1:5 + for j = i+1:6 + plot(freqs, abs(squeeze(freqresp(T(i, j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2]); + end +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +xlabel('Frequency [Hz]'); ylabel('Transmissibility'); +ylim([1e-4, 1e2]); +xlim([freqs(1), freqs(end)]); +#+end_src + +*** Comparison with "true" transmissibility + +#+begin_src matlab +%% Name of the Simulink File +mdl = 'test_transmissibility'; + +%% Input/Output definition +clear io; io_i = 1; +io(io_i) = linio([mdl, '/d'], 1, 'openinput'); io_i = io_i + 1; +io(io_i) = linio([mdl, '/acc'], 1, 'openoutput'); io_i = io_i + 1; + +%% Run the linearization +G = linearize(mdl, io); +G.InputName = {'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'}; +G.OutputName = {'Ax', 'Ay', 'Az', 'Bx', 'By', 'Bz'}; +#+end_src + +#+begin_src matlab +Tp = G/s^2; +#+end_src + +#+begin_src matlab :exports none +freqs = logspace(0, 3, 1000); + +figure; +hold on; +for i = 1:6 + plot(freqs, abs(squeeze(freqresp(Tp(i, i), freqs, 'Hz')))); +end +for i = 1:5 + for j = i+1:6 + plot(freqs, abs(squeeze(freqresp(Tp(i, j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2]); + end +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +xlabel('Frequency [Hz]'); ylabel('Transmissibility'); +ylim([1e-4, 1e2]); +xlim([freqs(1), freqs(end)]); +#+end_src + +** Identification of the table's dynamics +<> +*** 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 +addpath('matlab/') +#+end_src + +*** Mode Shapes + +#+name: tab:list_modes +#+caption: List of the identified modes +#+attr_latex: :environment tabularx :width 0.5\linewidth :align ccX +#+attr_latex: :center t :booktabs t :float t +| | Freq. [Hz] | Description | +|---+------------+---------------| +| 1 | 1.3 | X-translation | +| 2 | 1.3 | Y-translation | +| 3 | 1.95 | Z-rotation | +| 4 | 6.85 | Z-translation | +| 5 | 8.9 | Tilt | +| 6 | 8.9 | Tilt | +| 7 | 700 | Flexible Mode | + +#+name: fig:mode_shapes_rigid_table +#+caption: Mode shapes of the 6 suspension modes (from 1Hz to 9Hz) +#+attr_latex: :width \linewidth +[[file:figs/mode_shapes_rigid_table.gif]] + +#+name: fig:ModeShapeHF1_crop +#+caption: First flexible mode of the table at 700Hz +#+attr_latex: :width 0.3\linewidth +[[file:figs/ModeShapeHF1_crop.gif]] + + +* Nano-Hexapod Dynamics <> ** Introduction :ignore: @@ -389,6 +4737,230 @@ It is structured as follow: - Section ref:sec:test_nhexa_enc_plates_comp_simscape: The identified dynamics is compared with the Simscape model. - Section ref:sec:test_nhexa_enc_plates_iff: The Integral Force Feedback (IFF) control strategy is applied and the dynamics of the damped nano-hexapod is identified and compare with the Simscape model. +** Modal Analysis :noexport: +:PROPERTIES: +:header-args:matlab+: :tangle matlab/scripts/enc_struts_compliance_iff.m +:END: +<> + +- [ ] *This test was made using encoder fixed to the struts, is it relevant to put it here?* + +*** Introduction :ignore: +Several 3-axis accelerometers are fixed on the top platform of the nano-hexapod as shown in Figure ref:fig:test_nhexa_compliance_vertical_comp_iff. + +#+name: fig:test_nhexa_accelerometers_nano_hexapod +#+caption: Location of the accelerometers on top of the nano-hexapod +#+attr_latex: :width \linewidth +[[file:figs/test_nhexa_accelerometers_nano_hexapod.jpg]] + +The top platform is then excited using an instrumented hammer as shown in Figure ref:fig:test_nhexa_hammer_excitation_compliance_meas. + +#+name: fig:test_nhexa_hammer_excitation_compliance_meas +#+caption: Example of an excitation using an instrumented hammer +#+attr_latex: :width \linewidth +[[file:figs/test_nhexa_hammer_excitation_compliance_meas.jpg]] + +From this experiment, the resonance frequencies and the associated mode shapes can be computed (Section ref:sec:test_nhexa_modal_analysis_mode_shapes). +Then, in Section ref:sec:test_nhexa_compliance_effect_iff, the vertical compliance of the nano-hexapod is experimentally estimated. +Finally, in Section ref:sec:test_nhexa_compliance_effect_iff_comp_model, the measured compliance is compare with the estimated one from the Simscape model. + +*** Matlab Init :noexport:ignore: +#+begin_src matlab +%% enc_struts_compliance_iff.m +% Compare measured compliance and estimated compliance from the Simscape model +#+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 + +*** Obtained Mode Shapes +<> + +We can observe the mode shapes of the first 6 modes that are the suspension modes (the plate is behaving as a solid body) in Figure ref:fig:test_nhexa_mode_shapes_annotated. + +#+name: fig:test_nhexa_mode_shapes_annotated +#+caption: Measured mode shapes for the first six modes +#+attr_latex: :width \linewidth +[[file:figs/test_nhexa_mode_shapes_annotated.gif]] + +Then, there is a mode at 692Hz which corresponds to a flexible mode of the top plate (Figure ref:fig:test_nhexa_mode_shapes_flexible_annotated). + +#+name: fig:test_nhexa_mode_shapes_flexible_annotated +#+caption: First flexible mode at 692Hz +#+attr_latex: :width 0.3\linewidth +[[file:figs/test_nhexa_ModeShapeFlex1_crop.gif]] + +The obtained modes are summarized in Table ref:tab:description_modes. + +#+name: tab:description_modes +#+caption: Description of the identified modes +#+attr_latex: :environment tabularx :width 0.7\linewidth :align ccX +#+attr_latex: :center t :booktabs t +| *Mode* | *Freq. [Hz]* | *Description* | +|--------+--------------+----------------------------------------------| +| 1 | 105 | Suspension Mode: Y-translation | +| 2 | 107 | Suspension Mode: X-translation | +| 3 | 131 | Suspension Mode: Z-translation | +| 4 | 161 | Suspension Mode: Y-tilt | +| 5 | 162 | Suspension Mode: X-tilt | +| 6 | 180 | Suspension Mode: Z-rotation | +| 7 | 692 | (flexible) Membrane mode of the top platform | + +*** Nano-Hexapod Compliance - Effect of IFF +<> + +In this section, we wish to estimate the effectiveness of the IFF strategy regarding the compliance. + +The top plate is excited vertically using the instrumented hammer two times: +1. no control loop is used +2. decentralized IFF is used + +The data are loaded. +#+begin_src matlab +frf_ol = load('Measurement_Z_axis.mat'); % Open-Loop +frf_iff = load('Measurement_Z_axis_damped.mat'); % IFF +#+end_src + +The mean vertical motion of the top platform is computed by averaging all 5 vertical accelerometers. +#+begin_src matlab +%% Multiply by 10 (gain in m/s^2/V) and divide by 5 (number of accelerometers) +d_frf_ol = 10/5*(frf_ol.FFT1_H1_4_1_RMS_Y_Mod + frf_ol.FFT1_H1_7_1_RMS_Y_Mod + frf_ol.FFT1_H1_10_1_RMS_Y_Mod + frf_ol.FFT1_H1_13_1_RMS_Y_Mod + frf_ol.FFT1_H1_16_1_RMS_Y_Mod)./(2*pi*frf_ol.FFT1_H1_16_1_RMS_X_Val).^2; +d_frf_iff = 10/5*(frf_iff.FFT1_H1_4_1_RMS_Y_Mod + frf_iff.FFT1_H1_7_1_RMS_Y_Mod + frf_iff.FFT1_H1_10_1_RMS_Y_Mod + frf_iff.FFT1_H1_13_1_RMS_Y_Mod + frf_iff.FFT1_H1_16_1_RMS_Y_Mod)./(2*pi*frf_iff.FFT1_H1_16_1_RMS_X_Val).^2; +#+end_src + +The vertical compliance (magnitude of the transfer function from a vertical force applied on the top plate to the vertical motion of the top plate) is shown in Figure ref:fig:test_nhexa_compliance_vertical_comp_iff. +#+begin_src matlab :exports none +%% Comparison of the vertical compliances (OL and IFF) +figure; +hold on; +plot(frf_ol.FFT1_H1_16_1_RMS_X_Val, d_frf_ol, 'DisplayName', 'OL'); +plot(frf_iff.FFT1_H1_16_1_RMS_X_Val, d_frf_iff, 'DisplayName', 'IFF'); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +xlabel('Frequency [Hz]'); ylabel('Vertical Compliance [$m/N$]'); +xlim([20, 2e3]); ylim([2e-9, 2e-5]); +legend('location', 'northeast'); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/compliance_vertical_comp_iff.pdf', 'width', 'wide', 'height', 'normal'); +#+end_src + +#+name: fig:test_nhexa_compliance_vertical_comp_iff +#+caption: Measured vertical compliance with and without IFF +#+RESULTS: +[[file:figs/test_nhexa_compliance_vertical_comp_iff.png]] + +#+begin_important +From Figure ref:fig:test_nhexa_compliance_vertical_comp_iff, it is clear that the IFF control strategy is very effective in damping the suspensions modes of the nano-hexapod. +It also has the effect of (slightly) degrading the vertical compliance at low frequency. + +It also seems some damping can be added to the modes at around 205Hz which are flexible modes of the struts. +#+end_important + +*** Comparison with the Simscape Model +<> + +Let's initialize the Simscape model such that it corresponds to the experiment. +#+begin_src matlab +%% Nano-Hexapod is fixed on a rigid granite +support.type = 0; + +%% No Payload on top of the Nano-Hexapod +payload.type = 0; + +%% Initialize Nano-Hexapod in Open Loop +n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... + 'flex_top_type', '4dof', ... + 'motion_sensor_type', 'struts', ... + 'actuator_type', '2dof'); + +#+end_src + +And let's compare the measured vertical compliance with the vertical compliance as estimated from the Simscape model. + +The transfer function from a vertical external force to the absolute motion of the top platform is identified (with and without IFF) using the Simscape model. +#+begin_src matlab :exports none +%% Identify the IFF Plant (transfer function from u to taum) +clear io; io_i = 1; +io(io_i) = linio([mdl, '/Fz_ext'], 1, 'openinput'); io_i = io_i + 1; % External - Vertical force +io(io_i) = linio([mdl, '/Z_top_plat'], 1, 'openoutput'); io_i = io_i + 1; % Absolute vertical motion of top platform +#+end_src + +#+begin_src matlab :exports none +%% Perform the identifications +G_compl_z_ol = linearize(mdl, io, 0.0, options); +#+end_src + +#+begin_src matlab :exports none +%% Initialize Nano-Hexapod with IFF +Kiff = 400*(1/(s + 2*pi*40))*... % Low pass filter (provides integral action above 40Hz) + (s/(s + 2*pi*30))*... % High pass filter to limit low frequency gain + (1/(1 + s/2/pi/500))*... % Low pass filter to be more robust to high frequency resonances + eye(6); % Diagonal 6x6 controller + +%% Initialize the Nano-Hexapod with IFF +n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... + 'flex_top_type', '4dof', ... + 'motion_sensor_type', 'struts', ... + 'actuator_type', '2dof', ... + 'controller_type', 'iff'); + +%% Perform the identification +G_compl_z_iff = linearize(mdl, io, 0.0, options); +#+end_src + +The comparison is done in Figure ref:fig:test_nhexa_compliance_vertical_comp_model_iff. +Again, the model is quite accurate in predicting the (closed-loop) behavior of the system. + +#+begin_src matlab :exports none +%% Comparison of the measured compliance and the one obtained from the model +freqs = 2*logspace(1,3,1000); + +figure; +hold on; +plot(frf_ol.FFT1_H1_16_1_RMS_X_Val, d_frf_ol, '-', 'DisplayName', 'OL - Meas.'); +plot(frf_iff.FFT1_H1_16_1_RMS_X_Val, d_frf_iff, '-', 'DisplayName', 'IFF - Meas.'); +set(gca,'ColorOrderIndex',1) +plot(freqs, abs(squeeze(freqresp(G_compl_z_ol, freqs, 'Hz'))), '--', 'DisplayName', 'OL - Model') +plot(freqs, abs(squeeze(freqresp(G_compl_z_iff, freqs, 'Hz'))), '--', 'DisplayName', 'IFF - Model') +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +xlabel('Frequency [Hz]'); ylabel('Vertical Compliance [$m/N$]'); +xlim([20, 2e3]); ylim([2e-9, 2e-5]); +legend('location', 'northeast', 'FontSize', 8); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/compliance_vertical_comp_model_iff.pdf', 'width', 'wide', 'height', 'normal'); +#+end_src + +#+name: fig:test_nhexa_compliance_vertical_comp_model_iff +#+caption: Measured vertical compliance with and without IFF +#+RESULTS: +[[file:figs/test_nhexa_compliance_vertical_comp_model_iff.png]] + ** Identification of the dynamics :PROPERTIES: :header-args:matlab+: :tangle matlab/scripts/id_frf_enc_plates.m @@ -1058,537 +5630,6 @@ The Simscape model is quite accurate for the transfer function matrices from $\b The Simscape model can therefore be used to develop the control strategies. #+end_important -** Integral Force Feedback -:PROPERTIES: -:header-args:matlab+: :tangle matlab/scripts/iff_enc_plates.m -:END: -<> -*** Introduction :ignore: - -In this section, the Integral Force Feedback (IFF) control strategy is applied to the nano-hexapod in order to add damping to the suspension modes. - -The control architecture is shown in Figure ref:fig:test_nhexa_control_architecture_iff: -- $\bm{\tau}_m$ is the measured voltage of the 6 force sensors -- $\bm{K}_{\text{IFF}}$ is the $6 \times 6$ diagonal controller -- $\bm{u}$ is the plant input (voltage generated by the 6 DACs) -- $\bm{u}^\prime$ is the new plant inputs with added damping - -#+begin_src latex :file control_architecture_iff.pdf -\begin{tikzpicture} - % Blocs - \node[block={3.0cm}{2.0cm}] (P) {Plant}; - \coordinate[] (inputF) at ($(P.south west)!0.5!(P.north west)$); - \coordinate[] (outputF) at ($(P.south east)!0.7!(P.north east)$); - \coordinate[] (outputL) at ($(P.south east)!0.3!(P.north east)$); - - \node[block, above=0.4 of P] (Kiff) {$\bm{K}_\text{IFF}$}; - \node[addb, left= of inputF] (addF) {}; - - % Connections and labels - \draw[->] (outputF) -- ++(1, 0) node[below left]{$\bm{\tau}_m$}; - \draw[->] (outputL) -- ++(1, 0) node[below left]{$d\bm{\mathcal{L}}_m$}; - - \draw[->] ($(outputF) + (0.6, 0)$)node[branch]{} |- (Kiff.east); - \draw[->] (Kiff.west) -| (addF.north); - \draw[->] (addF.east) -- (inputF) node[above left]{$\bm{u}$}; - \draw[<-] (addF.west) -- ++(-1, 0) node[above right]{$\bm{u}^\prime$}; -\end{tikzpicture} -#+end_src - -#+name: fig:test_nhexa_control_architecture_iff -#+caption: Integral Force Feedback Strategy -#+RESULTS: -[[file:figs/test_nhexa_control_architecture_iff.png]] - -- Section ref:sec:test_nhexa_enc_struts_effect_iff_plant - -*** Matlab Init :noexport:ignore: -#+begin_src matlab -%% iff_enc_plates.m -% Apply Integral Force Feedback for the nano-hexapod with encoders -% fixed to the plates -#+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 - -#+begin_src matlab -%% Load measured FRF and extracted Simscape model -frf_ol = load('identified_plants_enc_plates.mat', 'f', 'Ts', 'G_tau', 'G_dL'); -sim_ol = load('simscape_plants_enc_plates.mat', 'G_tau', 'G_dL'); -#+end_src - -*** Effect of IFF on the plant - Simscape Model -<> - -The nano-hexapod is initialized with 2DoF APA and the encoders fixed to the struts. -#+begin_src matlab -%% Initialize the Simscape model in closed loop -n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... - 'flex_top_type', '4dof', ... - 'motion_sensor_type', 'plates', ... - 'actuator_type', '2dof', ... - 'controller_type', 'iff'); - -support.type = 1; % On top of vibration table -payload.type = 0; % No Payload -#+end_src - -The same controller as the one developed when the encoder were fixed to the struts is used. -#+begin_src matlab -%% Optimal IFF controller -load('Kiff_struts_no_payload.mat', 'Kiff'); -#+end_src - -The transfer function from $\bm{u}^\prime$ to $d\bm{\mathcal{L}}_m$ is identified. -#+begin_src matlab -%% Identify the (damped) transfer function from u to dLm for different values of the IFF gain -clear io; io_i = 1; -io(io_i) = linio([mdl, '/du'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs -io(io_i) = linio([mdl, '/dL'], 1, 'openoutput'); io_i = io_i + 1; % Plate Displacement (encoder) - -%% Transfer function from u to dL (IFF) -G_dL = exp(-s*frf_ol.Ts)*linearize(mdl, io, 0.0, options); -#+end_src - -It is first verified that the system is stable: -#+begin_src matlab :results value replace :exports both :tangle no -isstable(G_dL) -#+end_src - -#+RESULTS: -: 1 - -The identified dynamics is saved for further use. -#+begin_src matlab :exports none :tangle no -%% Save Identified Plants -save('matlab/data_frf/simscape_plants_enc_plates_iff.mat', 'G_dL'); -#+end_src - -#+begin_src matlab :eval no -save('data_frf/simscape_plants_enc_plates_iff.mat', 'G_dL'); -#+end_src - -#+begin_src matlab :exports none -%% Load the Simscape model -sim_iff = load('simscape_plants_enc_plates_iff.mat', 'G_dL'); -#+end_src - -The diagonal and off-diagonal terms of the $6 \times 6$ transfer function matrices identified are compared in Figure ref:fig:test_nhexa_enc_plates_iff_gains_effect_dvf_plant. -It is shown, as was the case when the encoders were fixed to the struts, that the IFF control strategy is very effective in damping the suspension modes of the nano-hexapod. -#+begin_src matlab :exports none -%% Bode plot of the transfer function from u to dLm with and without IFF -figure; -tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); - -ax1 = nexttile([2,1]); -hold on; -plot(freqs, abs(squeeze(freqresp(sim_ol.G_dL(1,1), freqs, 'Hz'))), 'color', colors(1,:), ... - 'DisplayName', 'OL - Diag'); -plot(freqs, abs(squeeze(freqresp(sim_iff.G_dL(1,1), freqs, 'Hz'))), 'color', colors(2,:), ... - 'DisplayName', 'IFF - Diag'); -for i = 2:6 - plot(freqs, abs(squeeze(freqresp(sim_ol.G_dL(1,1), freqs, 'Hz'))), 'color', colors(1,:), ... - 'HandleVisibility', 'off'); -end -for i = 2:6 - plot(freqs, abs(squeeze(freqresp(sim_iff.G_dL(i,i), freqs, 'Hz'))), 'color', colors(2,:), ... - 'HandleVisibility', 'off'); -end - -plot(freqs, abs(squeeze(freqresp(sim_ol.G_dL(1,2), freqs, 'Hz'))), 'color', [colors(1,:), 0.2], ... - 'DisplayName', 'OL - Off-diag') -for i = 1:5 - for j = i+1:6 - plot(freqs, abs(squeeze(freqresp(sim_ol.G_dL(i,j), freqs, 'Hz'))), 'color', [colors(1,:), 0.2], ... - 'HandleVisibility', 'off'); - end -end -plot(freqs, abs(squeeze(freqresp(sim_iff.G_dL(1,2), freqs, 'Hz'))), 'color', [colors(2,:), 0.2], ... - 'DisplayName', 'IFF - Off-diag') -for i = 1:5 - for j = i+1:6 - plot(freqs, abs(squeeze(freqresp(sim_iff.G_dL(i,j), freqs, 'Hz'))), 'color', [colors(2,:), 0.2], ... - 'HandleVisibility', 'off'); - end -end -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Amplitude [m/V]'); set(gca, 'XTickLabel',[]); -legend('location', 'northeast', 'FontSize', 8); - -ax2 = nexttile; -hold on; -for i = 1:6 - plot(freqs, 180/pi*angle(squeeze(freqresp(sim_ol.G_dL(1,1), freqs, 'Hz'))), 'color', colors(1,:), ... - 'HandleVisibility', 'off'); - plot(freqs, 180/pi*angle(squeeze(freqresp(sim_iff.G_dL(i,i), freqs, 'Hz'))), 'color', colors(2,:), ... - 'HandleVisibility', 'off'); -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_plates_iff_gains_effect_dvf_plant.pdf', 'width', 'wide', 'height', 'tall'); -#+end_src - -#+name: fig:test_nhexa_enc_plates_iff_gains_effect_dvf_plant -#+caption: Effect of the IFF control strategy on the transfer function from $\bm{\tau}$ to $d\bm{\mathcal{L}}_m$ -#+RESULTS: -[[file:figs/test_nhexa_enc_plates_iff_gains_effect_dvf_plant.png]] - -*** Effect of IFF on the plant - FRF -The IFF control strategy is experimentally implemented. -The (damped) transfer function from $\bm{u}^\prime$ to $d\bm{\mathcal{L}}_m$ is experimentally identified. - -The identification data are loaded: -#+begin_src matlab -%% Load Identification Data -meas_iff_plates = {}; - -for i = 1:6 - meas_iff_plates(i) = {load(sprintf('frf_exc_iff_strut_%i_enc_plates_noise.mat', i), 't', 'Va', 'Vs', 'de', 'u')}; -end -#+end_src - -And the parameters used for the transfer function estimation are defined below. -#+begin_src matlab -% Sampling Time [s] -Ts = (meas_iff_plates{1}.t(end) - (meas_iff_plates{1}.t(1)))/(length(meas_iff_plates{1}.t)-1); - -% Hannning Windows -win = hanning(ceil(1/Ts)); - -% And we get the frequency vector -[~, f] = tfestimate(meas_iff_plates{1}.Va, meas_iff_plates{1}.de, win, Noverlap, Nfft, 1/Ts); -#+end_src - -The estimation is performed using the =tfestimate= command. -#+begin_src matlab -%% Estimation of the transfer function matrix from u to dL when IFF is applied -G_dL = zeros(length(f), 6, 6); - -for i = 1:6 - G_dL(:,:,i) = tfestimate(meas_iff_plates{i}.Va, meas_iff_plates{i}.de, win, Noverlap, Nfft, 1/Ts); -end -#+end_src - -The experimentally identified plant is saved for further use. -#+begin_src matlab :exports none:tangle no -save('matlab/data_frf/damped_plant_enc_plates.mat', 'f', 'Ts', 'G_dL') -#+end_src - -#+begin_src matlab :eval no -save('data_frf/damped_plant_enc_plates.mat', 'f', 'Ts', 'G_dL') -#+end_src - -#+begin_src matlab :exports none -frf_iff = load('damped_plant_enc_plates.mat', 'f', 'Ts', 'G_dL'); -#+end_src - -The obtained diagonal and off-diagonal elements of the transfer function from $\bm{u}^\prime$ to $d\bm{\mathcal{L}}_m$ are shown in Figure ref:fig:test_nhexa_enc_plant_plates_effect_iff both without and with IFF. -#+begin_src matlab :exports none -%% Bode plot of the transfer function from u to dLm with and without IFF (experimental results) -figure; -tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); - -ax1 = nexttile([2,1]); -hold on; -plot(frf_ol.f, abs(frf_ol.G_dL(:,1,1)), 'color', colors(1,:), ... - 'DisplayName', 'OL - Diag'); -plot(frf_iff.f, abs(frf_iff.G_dL(:,1,1)), 'color', colors(2,:), ... - 'DisplayName', 'IFF - Diag'); -for i = 2:6 - plot(frf_ol.f, abs(frf_ol.G_dL(:,1,1)), 'color', colors(1,:), ... - 'HandleVisibility', 'off'); - plot(frf_iff.f, abs(frf_iff.G_dL(:,i,i)), 'color', colors(2,:), ... - 'HandleVisibility', 'off'); -end - -plot(frf_ol.f, abs(frf_ol.G_dL(:,1,2)), 'color', [colors(1,:), 0.2], ... - 'DisplayName', 'OL - Off-diag') -for i = 1:5 - for j = i+1:6 - plot(frf_ol.f, abs(frf_ol.G_dL(:,i,j)), 'color', [colors(1,:), 0.2], ... - 'HandleVisibility', 'off'); - end -end -plot(frf_iff.f, abs(frf_iff.G_dL(:,1,2)), 'color', [colors(2,:), 0.2], ... - 'DisplayName', 'IFF - Off-diag') -for i = 1:5 - for j = i+1:6 - plot(frf_iff.f, abs(frf_iff.G_dL(:,i,j)), 'color', [colors(2,:), 0.2], ... - 'HandleVisibility', 'off'); - end -end -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Amplitude [m/V]'); set(gca, 'XTickLabel',[]); -legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2); - -ax2 = nexttile; -hold on; -for i = 1:6 - plot(frf_ol.f, 180/pi*angle(frf_ol.G_dL( :,i,i)), 'color', colors(1,:)) - plot(frf_iff.f, 180/pi*angle(frf_iff.G_dL(:,i,i)), 'color', colors(2,:)) -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_plant_plates_effect_iff.pdf', 'width', 'wide', 'height', 'tall'); -#+end_src - -#+name: fig:test_nhexa_enc_plant_plates_effect_iff -#+caption: Effect of the IFF control strategy on the transfer function from $\bm{\tau}$ to $d\bm{\mathcal{L}}_m$ -#+RESULTS: -[[file:figs/test_nhexa_enc_plant_plates_effect_iff.png]] - -#+begin_important -As was predicted with the Simscape model, the IFF control strategy is very effective in damping the suspension modes of the nano-hexapod. -Little damping is also applied on the first flexible mode of the strut at 235Hz. -However, no damping is applied on other modes, such as the flexible modes of the top plate. -#+end_important - -*** Comparison of the measured FRF and the Simscape model -Let's now compare the obtained damped plants obtained experimentally with the one extracted from Simscape: -- Figure ref:fig:test_nhexa_enc_plates_opt_iff_comp_simscape_all: the individual transfer function from $u_1^\prime$ to the six encoders are comapred -- Figure ref:fig:test_nhexa_damped_iff_plates_plant_comp_diagonal: all the diagonal elements are compared -- Figure ref:fig:test_nhexa_damped_iff_plates_plant_comp_off_diagonal: all the off-diagonal elements are compared - -#+begin_src matlab :exports none -%% Comparison of the plants (encoder output) when tuning the misalignment -i_input = 1; - -figure; -tiledlayout(2, 3, 'TileSpacing', 'Compact', 'Padding', 'None'); - -ax1 = nexttile(); -hold on; -plot(frf_iff.f, abs(frf_iff.G_dL(:, 1, i_input))); -plot(freqs, abs(squeeze(freqresp(sim_iff.G_dL(1, i_input), freqs, 'Hz')))); -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -set(gca, 'XTickLabel',[]); ylabel('Amplitude [m/V]'); -title(sprintf('$d\\tau_{m1}/u_{%i}$', i_input)); - -ax2 = nexttile(); -hold on; -plot(frf_iff.f, abs(frf_iff.G_dL(:, 2, i_input))); -plot(freqs, abs(squeeze(freqresp(sim_iff.G_dL(2, i_input), freqs, 'Hz')))); -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]); -title(sprintf('$d\\tau_{m2}/u_{%i}$', i_input)); - -ax3 = nexttile(); -hold on; -plot(frf_iff.f, abs(frf_iff.G_dL(:, 3, i_input)), ... - 'DisplayName', 'Meas.'); -plot(freqs, abs(squeeze(freqresp(sim_iff.G_dL(3, i_input), freqs, 'Hz'))), ... - 'DisplayName', 'Model'); -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]); -legend('location', 'southeast', 'FontSize', 8); -title(sprintf('$d\\tau_{m3}/u_{%i}$', i_input)); - -ax4 = nexttile(); -hold on; -plot(frf_iff.f, abs(frf_iff.G_dL(:, 4, i_input))); -plot(freqs, abs(squeeze(freqresp(sim_iff.G_dL(4, i_input), freqs, 'Hz')))); -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -xlabel('Frequency [Hz]'); ylabel('Amplitude [m/V]'); -title(sprintf('$d\\tau_{m4}/u_{%i}$', i_input)); - -ax5 = nexttile(); -hold on; -plot(frf_iff.f, abs(frf_iff.G_dL(:, 5, i_input))); -plot(freqs, abs(squeeze(freqresp(sim_iff.G_dL(5, i_input), freqs, 'Hz')))); -hold off; -xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]); -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -title(sprintf('$d\\tau_{m5}/u_{%i}$', i_input)); - -ax6 = nexttile(); -hold on; -plot(frf_iff.f, abs(frf_iff.G_dL(:, 6, i_input))); -plot(freqs, abs(squeeze(freqresp(sim_iff.G_dL(6, i_input), freqs, 'Hz')))); -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]); -title(sprintf('$d\\tau_{m6}/u_{%i}$', i_input)); - -linkaxes([ax1,ax2,ax3,ax4,ax5,ax6],'xy'); -xlim([freqs(1), 1e3]); ylim([1e-8, 1e-4]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/enc_plates_opt_iff_comp_simscape_all.pdf', 'width', 'full', 'height', 'tall'); -#+end_src - -#+name: fig:test_nhexa_enc_plates_opt_iff_comp_simscape_all -#+caption: FRF from one actuator to all the encoders when the plant is damped using IFF -#+RESULTS: -[[file:figs/test_nhexa_enc_plates_opt_iff_comp_simscape_all.png]] - -#+begin_src matlab :exports none -%% Bode plot for the transfer function from u to dLm -figure; -tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); - -ax1 = nexttile([2,1]); -hold on; -% Diagonal Elements FRF -plot(frf_iff.f, abs(frf_iff.G_dL(:,1,1)), 'color', [colors(1,:), 0.2], ... - 'DisplayName', '$d\mathcal{L}_{m,i}/u_i$ - FRF') -for i = 2:6 - plot(frf_iff.f, abs(frf_iff.G_dL(:,i,i)), 'color', [colors(1,:), 0.2], ... - 'HandleVisibility', 'off'); -end - -% Diagonal Elements Model -plot(freqs, abs(squeeze(freqresp(sim_iff.G_dL(1,1), freqs, 'Hz'))), 'color', [colors(2,:),0.2], ... - 'DisplayName', '$d\mathcal{L}_{m,i}/u_i$ - Model') -for i = 2:6 - plot(freqs, abs(squeeze(freqresp(sim_iff.G_dL(i,i), freqs, 'Hz'))), 'color', [colors(2,:),0.2], ... - 'HandleVisibility', 'off'); -end - -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Amplitude $d_e/V_{exc}$ [m/V]'); set(gca, 'XTickLabel',[]); -ylim([1e-7, 1e-4]); -legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 3); - -ax2 = nexttile; -hold on; -for i =1:6 - plot(frf_iff.f, 180/pi*angle(frf_iff.G_dL(:,i,i)), 'color', [colors(1,:), 0.2]); - plot(freqs, 180/pi*angle(squeeze(freqresp(sim_iff.G_dL(i,i), freqs, 'Hz'))), 'color', [colors(2,:),0.2]); -end -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); -xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); -hold off; -yticks(-360:90:360); -ylim([-180, 180]); - -linkaxes([ax1,ax2],'x'); -xlim([20, 2e3]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/damped_iff_plates_plant_comp_diagonal.pdf', 'width', 'wide', 'height', 'tall'); -#+end_src - -#+name: fig:test_nhexa_damped_iff_plates_plant_comp_diagonal -#+caption: Comparison of the diagonal elements of the transfer functions from $\bm{u}$ to $d\bm{\mathcal{L}}_m$ with active damping (IFF) applied with an optimal gain $g = 400$ -#+RESULTS: -[[file:figs/test_nhexa_damped_iff_plates_plant_comp_diagonal.png]] - -#+begin_src matlab :exports none -%% Bode plot for the transfer function from u to dLm -figure; -tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); - -ax1 = nexttile([2,1]); -hold on; -% Off diagonal FRF -plot(frf_iff.f, abs(frf_iff.G_dL(:,1,2)), 'color', [colors(1,:), 0.2], ... - 'DisplayName', '$d\mathcal{L}_{m,i}/u_j$ - FRF') -for i = 1:5 - for j = i+1:6 - plot(frf_iff.f, abs(frf_iff.G_dL(:,i,j)), 'color', [colors(1,:), 0.2], ... - 'HandleVisibility', 'off'); - end -end - -% Off diagonal Model -plot(freqs, abs(squeeze(freqresp(sim_iff.G_dL(1,2), freqs, 'Hz'))), 'color', [colors(2,:), 0.2], ... - 'DisplayName', '$d\mathcal{L}_{m,i}/u_j$ - Model') -for i = 1:5 - for j = i+1:6 - plot(freqs, abs(squeeze(freqresp(sim_iff.G_dL(i,j), freqs, 'Hz'))), 'color', [colors(2,:), 0.2], ... - 'HandleVisibility', 'off'); - end -end - -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Amplitude [m/V]'); set(gca, 'XTickLabel',[]); -ylim([1e-8, 1e-4]); -legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 3); - -ax2 = nexttile; -hold on; -for i = 1:5 - for j = i+1:6 - plot(frf_iff.f, 180/pi*angle(frf_iff.G_dL(:,i,j)), 'color', [colors(1,:), 0.2]); - plot(freqs, 180/pi*angle(squeeze(freqresp(sim_iff.G_dL(i,j), freqs, 'Hz'))), 'color', [colors(2,:),0.2]); - end -end -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); -xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); -hold off; -yticks(-360:90:360); -ylim([-180, 180]); - -linkaxes([ax1,ax2],'x'); -xlim([20, 2e3]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/damped_iff_plates_plant_comp_off_diagonal.pdf', 'width', 'wide', 'height', 'tall'); -#+end_src - -#+name: fig:test_nhexa_damped_iff_plates_plant_comp_off_diagonal -#+caption: Comparison of the off-diagonal elements of the transfer functions from $\bm{u}$ to $d\bm{\mathcal{L}}_m$ with active damping (IFF) applied with an optimal gain $g = 400$ -#+RESULTS: -[[file:figs/test_nhexa_damped_iff_plates_plant_comp_off_diagonal.png]] - -#+begin_important -From Figures ref:fig:test_nhexa_damped_iff_plates_plant_comp_diagonal and ref:fig:test_nhexa_damped_iff_plates_plant_comp_off_diagonal, it is clear that the Simscape model very well represents the dynamics of the nano-hexapod. -This is true to around 400Hz, then the dynamics depends on the flexible modes of the top plate which are not modelled. -#+end_important - ** Effect of Payload mass on the Dynamics :PROPERTIES: :header-args:matlab+: :tangle matlab/scripts/id_frf_enc_plates_effect_payload.m @@ -2255,938 +6296,6 @@ exportFig('figs/comp_masses_model_exp_iff_zoom.pdf', 'width', 'wide', 'height', #+RESULTS: [[file:figs/test_nhexa_comp_masses_model_exp_iff_zoom.png]] -** Integral Force Feedback Controller -:PROPERTIES: -:header-args:matlab+: :tangle matlab/scripts/iff_robust_enc_plates.m -:END: -<> -*** Introduction :ignore: -In this section, we wish to develop the Integral Force Feedback controller that is robust with respect to the added payload. - -*** Matlab Init :noexport:ignore: -#+begin_src matlab -%% iff_robust_enc_plates.m -% Development of IFF controller that works for all payloads -% Estimation of performances using the Simscape model and validation with measurements -#+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 -%% Load the identified FRF and Simscape model -frf_ol = load('frf_vib_table_m.mat', 'f', 'Ts', 'G_tau', 'G_dL'); -sim_ol = load('sim_vib_table_m.mat', 'G_tau', 'G_dL'); -#+end_src - -*** Robust IFF Controller -Based on the measured FRF from $\bm{u}$ to $\bm{\tau}_{m}$, the following IFF controller is developed: -#+begin_src matlab -%% IFF Controller -Kiff_g1 = -(1/(s + 2*pi*20))*... % LPF: provides integral action above 20[Hz] - (s/(s + 2*pi*20))*... % HPF: limit low frequency gain - (1/(1 + s/2/pi/400)); % LPF: more robust to high frequency resonances -#+end_src - -Then, the Root Locus plot of Figure ref:fig:test_nhexa_iff_root_locus_masses is used to estimate the optimal gain. -This Root Locus plot is computed from the Simscape model. -#+begin_src matlab :exports none -%% Root Locus for IFF -gains = logspace(1, 3, 100); - -figure; - -hold on; -% Pure Integrator -for i_mass = 0:3 - plot(real(pole(sim_ol.G_tau{i_mass+1})), imag(pole(sim_ol.G_tau{i_mass+1})), 'x', 'color', colors(i_mass+1, :), ... - 'DisplayName', sprintf('OL Poles - %i', i_mass)); - plot(real(tzero(sim_ol.G_tau{i_mass+1})), imag(tzero(sim_ol.G_tau{i_mass+1})), 'o', 'color', colors(i_mass+1, :), ... - 'HandleVisibility', 'off'); -end - -for i_mass = 0:3 - for g = gains - clpoles = pole(feedback(sim_ol.G_tau{i_mass+1}, g*Kiff_g1*eye(6), +1)); - plot(real(clpoles), imag(clpoles), '.', 'color', colors(i_mass+1, :), ... - 'HandleVisibility', 'off'); - end -end - -g_opt = 2e2; - -clpoles = pole(feedback(sim_ol.G_tau{1}, g_opt*Kiff_g1*eye(6), +1)); -plot(real(clpoles), imag(clpoles), 'kx', ... - 'DisplayName', sprintf('$g = %.0f$', g_opt)); -for i_mass = 1:3 - clpoles = pole(feedback(sim_ol.G_tau{i_mass+1}, g_opt*Kiff_g1*eye(6), +1)); - plot(real(clpoles), imag(clpoles), 'kx', ... - 'HandleVisibility', 'off'); -end -axis square; -xlim([-700, 0]); ylim([0, 1400]); -xlabel('Real Part'); ylabel('Imaginary Part'); -legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 3); -rootLocusOverlay('xis', [0.01, 0.1, 0.2, 0.3, 0.4, 0.5], 'R', 1e3); -hold off; -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/iff_root_locus_masses.pdf', 'width', 'wide', 'height', 'tall'); -#+end_src - -#+name: fig:test_nhexa_iff_root_locus_masses -#+caption: Root Locus for the IFF control strategy (for all payload conditions). -#+RESULTS: -[[file:figs/test_nhexa_iff_root_locus_masses.png]] - -#+begin_src matlab :exports none :tangle no -%% Verify close-loop stability for all payloads -for i_mass = 0:3 - clpoles = isstable(feedback(sim_ol.G_tau{i_mass+1}, g_opt*Kiff_g1*eye(6), +1)); - sum(real(clpoles)>0) -end -#+end_src - -The found optimal IFF controller is: -#+begin_src matlab -%% Optimal controller -g_opt = 2e2; -Kiff = g_opt*Kiff_g1*eye(6); -#+end_src - -It is saved for further use. -#+begin_src matlab :exports none :tangle no -save('matlab/data_sim/Kiff_robust_opt.mat', 'Kiff') -#+end_src - -#+begin_src matlab :eval no -save('data_sim/Kiff_robust_opt.mat', 'Kiff') -#+end_src - -The corresponding experimental loop gains are shown in Figure ref:fig:test_nhexa_iff_loop_gain_masses. -#+begin_src matlab :exports none -%% Bode plot for the transfer function from u to dLm -figure; -tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); - -ax1 = nexttile([2,1]); -hold on; -for i_mass = 0:3 - for i = 1:6 - plot(frf_ol.f, abs(squeeze(freqresp(Kiff(i,i), frf_ol.f, 'Hz')).*frf_ol.G_tau{i_mass+1}(:,i,i)), '-', 'color', [colors(i_mass+1,:), 0.2]); - end -end -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Loop Gain [-]'); set(gca, 'XTickLabel',[]); -ylim([1e-2, 1e2]); - -ax2 = nexttile; -hold on; -for i_mass = 0:3 - for i = 1:6 - plot(frf_ol.f, 180/pi*angle(squeeze(freqresp(-Kiff(i,i), frf_ol.f, 'Hz')).*frf_ol.G_tau{i_mass+1}(:,i,i)), '-', 'color', [colors(i_mass+1,:), 0.2]); - end -end -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); -xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); -hold off; -yticks(-360:90:360); -ylim([-180, 180]); - -linkaxes([ax1,ax2],'x'); -xlim([10, 1e3]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/iff_loop_gain_masses.pdf', 'width', 'wide', 'height', 'tall'); -#+end_src - -#+name: fig:test_nhexa_iff_loop_gain_masses -#+caption: Loop gain for the Integral Force Feedback controller -#+RESULTS: -[[file:figs/test_nhexa_iff_loop_gain_masses.png]] - -#+begin_important -Based on the above analysis: -- The same IFF controller can be used to damp the suspension modes for all payload conditions -- The IFF controller should be robust -#+end_important - -*** Estimated Damped Plant from the Simscape model -Let's initialize the simscape model with the nano-hexapod fixed on top of the vibration table. -#+begin_src matlab -%% Initialize the Simscape model in closed loop -n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... - 'flex_top_type', '4dof', ... - 'motion_sensor_type', 'plates', ... - 'actuator_type', '2dof', ... - 'controller_type', 'iff'); - -support.type = 1; % On top of vibration table -#+end_src - -And Load the IFF controller. -#+begin_src matlab :exports none -%% Make sure IFF controller is loaded -load('Kiff_robust_opt.mat', 'Kiff') -#+end_src - -Finally, let's identify the damped plant from $\bm{u}^\prime$ to $d\bm{\mathcal{L}}_m$ for all payloads. -#+begin_src matlab -%% Identify the (damped) transfer function from u to dLm -clear io; io_i = 1; -io(io_i) = linio([mdl, '/du'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs -io(io_i) = linio([mdl, '/dL'], 1, 'openoutput'); io_i = io_i + 1; % Plate Displacement (encoder) - -%% Identify for all add masses -G_dL = {}; - -for i = i_masses - payload.type = i; - G_dL(i+1) = {exp(-s*frf_ol.Ts)*linearize(mdl, io, 0.0, options)}; -end -#+end_src - -The identified dynamics are then saved for further use. -#+begin_src matlab :exports none :tangle no -%% Save the identified dynamics -save('matlab/data_frf/sim_iff_vib_table_m.mat', 'G_dL'); -#+end_src - -#+begin_src matlab :eval no -save('data_frf/sim_iff_vib_table_m.mat', 'G_dL'); -#+end_src - -#+begin_src matlab -%% Load the identified dynamics -sim_iff = load('sim_iff_vib_table_m.mat', 'G_dL'); -#+end_src - -#+begin_src matlab :exports none :tangle no -%% Verify Stability -for i = i_masses - isstable(sim_iff.G_dL{i+1}) -end -#+end_src - -#+begin_src matlab :exports none -%% Bode plot for the transfer function from u to dLm - With and without IFF -figure; -tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); - -freqs = logspace(1,3,1000); - -ax1 = nexttile([2,1]); -hold on; -for i_mass = i_masses - for i = 1 - plot(freqs, abs(squeeze(freqresp(sim_ol.G_dL{i_mass+1}(i,i), freqs, 'Hz'))), '-', 'color', [colors(i_mass+1, :), 0.5], ... - 'DisplayName', sprintf('$d\\mathcal{L}_i/u_i$ - %i', i_mass)); - plot(freqs, abs(squeeze(freqresp(sim_iff.G_dL{i_mass+1}(i,i), freqs, 'Hz'))), '-', 'color', colors(i_mass+1, :), ... - 'DisplayName', sprintf('$d\\mathcal{L}_i/u^\\prime_i$ - %i', i_mass)); - end -end -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Amplitude [m/V]'); set(gca, 'XTickLabel',[]); -ylim([1e-7, 1e-3]); -legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 4); - -ax2 = nexttile; -hold on; -for i_mass = i_masses - for i = 1 - plot(freqs, 180/pi*angle(squeeze(freqresp(sim_ol.G_dL{i_mass+1}(1,1), freqs, 'Hz'))), '-', 'color', [colors(i_mass+1, :), 0.5]); - plot(freqs, 180/pi*angle(squeeze(freqresp(sim_iff.G_dL{i_mass+1}(1,1), freqs, 'Hz'))), '-', 'color', colors(i_mass+1, :)); - end -end -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); -xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); -hold off; -yticks(-360:90:360); -ylim([-180, 180]); - -linkaxes([ax1,ax2],'x'); -xlim([freqs(1), freqs(end)]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/damped_plant_model_masses.pdf', 'width', 'wide', 'height', 'tall'); -#+end_src - -#+name: fig:test_nhexa_damped_plant_model_masses -#+caption: Transfer function from $u_i$ to $d\mathcal{L}_{m,i}$ (without active damping) and from $u^\prime_i$ to $d\mathcal{L}_{m,i}$ (with IFF) -#+RESULTS: -[[file:figs/test_nhexa_damped_plant_model_masses.png]] - -*** Compute the identified FRF with IFF -Several experimental identifications are done in order to identify the dynamics from $\bm{u}^\prime$ to $d\bm{\mathcal{L}}_m$ with the robust IFF controller implemented and with various payloads. - -The following data are loaded: -- =Va=: the excitation voltage for the damped plant (corresponding to $u^\prime_i$) -- =de=: the measured motion by the 6 encoders (corresponding to $d\bm{\mathcal{L}}_m$) -#+begin_src matlab -%% Load Identification Data -meas_added_mass = {}; - -for i_mass = i_masses - for i_strut = 1:6 - meas_iff_mass(i_strut, i_mass+1) = {load(sprintf('frf_data_exc_strut_%i_iff_vib_table_%im.mat', i_strut, i_mass), 't', 'Va', 'de')}; - end -end -#+end_src - -The window =win= and the frequency vector =f= are defined. -#+begin_src matlab -% Sampling Time [s] -Ts = (meas_iff_mass{1,1}.t(end) - (meas_iff_mass{1,1}.t(1)))/(length(meas_iff_mass{1,1}.t)-1); - -% Hannning Windows -win = hanning(ceil(1/Ts)); - -% And we get the frequency vector -[~, f] = tfestimate(meas_iff_mass{1,1}.Va, meas_iff_mass{1,1}.de, win, Noverlap, Nfft, 1/Ts); -#+end_src - -Finally the $6 \times 6$ transfer function matrix from $\bm{u}^\prime$ to $d\bm{\mathcal{L}}_m$ is estimated: -#+begin_src matlab -%% DVF Plant (transfer function from u to dLm) -G_dL = {}; - -for i_mass = i_masses - G_dL(i_mass+1) = {zeros(length(f), 6, 6)}; - for i_strut = 1:6 - G_dL{i_mass+1}(:,:,i_strut) = tfestimate(meas_iff_mass{i_strut, i_mass+1}.Va, meas_iff_mass{i_strut, i_mass+1}.de, win, Noverlap, Nfft, 1/Ts); - end -end -#+end_src - -The identified dynamics are then saved for further use. -#+begin_src matlab :exports none :tangle no -save('matlab/data_frf/frf_iff_vib_table_m.mat', 'f', 'Ts', 'G_dL'); -#+end_src - -#+begin_src matlab :eval no -save('mat/frf_iff_vib_table_m.mat', 'f', 'Ts', 'G_dL'); -#+end_src - -*** Comparison of the measured FRF and the Simscape model -#+begin_src matlab :exports none -%% Load the Measured FRF of the damped plant -frf_iff = load('frf_iff_vib_table_m.mat', 'f', 'Ts', 'G_dL'); -#+end_src - -The following figures are computed: -- Figure ref:fig:test_nhexa_damped_iff_plant_meas_frf: the measured damped FRF are displayed -- Figure ref:fig:test_nhexa_comp_undamped_damped_plant_meas_frf: the open-loop and damped FRF are compared (diagonal elements) -- Figure ref:fig:test_nhexa_comp_iff_plant_frf_sim: the obtained damped FRF is compared with the identified damped from using the Simscape model - -#+begin_src matlab :exports none -%% Diagonal and Off Diagonal elements of the damped plants -figure; -tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); - -ax1 = nexttile([2,1]); -hold on; -for i_mass = i_masses - plot(frf_iff.f, abs(frf_iff.G_dL{i_mass+1}(:,1,1)), 'color', colors(i_mass+1,:), ... - 'DisplayName', sprintf('$d\\mathcal{L}_{m,i}/u^\\prime_i$ - %i', i_mass)); - for i = 2:6 - plot(frf_iff.f, abs(frf_iff.G_dL{i_mass+1}(:,i,i)), 'color', colors(i_mass+1,:), ... - 'HandleVisibility', 'off'); - end - plot(frf_iff.f, abs(frf_iff.G_dL{i_mass+1}(:,1,2)), 'color', [colors(i_mass+1,:), 0.2], ... - 'DisplayName', sprintf('$d\\mathcal{L}_{m,i}/u^\\prime_j$ - %i', i_mass)); - for i = 1:5 - for j = i+1:6 - plot(frf_iff.f, abs(frf_iff.G_dL{i_mass+1}(:,i,j)), 'color', [colors(i_mass+1,:), 0.2], ... - 'HandleVisibility', 'off'); - end - end -end -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Amplitude [m/V]'); set(gca, 'XTickLabel',[]); -ylim([1e-9, 1e-4]); - -legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 2); - -ax2 = nexttile; -hold on; -for i_mass = i_masses - for i =1:6 - plot(frf_iff.f, 180/pi*angle(frf_iff.G_dL{i_mass+1}(:,i, i)), 'color', colors(i_mass+1,:)); - end -end -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); -xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); -hold off; -yticks(-360:90:360); -ylim([-180, 180]); - -linkaxes([ax1,ax2],'x'); -xlim([10, 1e3]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/damped_iff_plant_meas_frf.pdf', 'width', 'wide', 'height', 'tall'); -#+end_src - -#+name: fig:test_nhexa_damped_iff_plant_meas_frf -#+caption: Diagonal and off-diagonal of the measured FRF matrix for the damped plant -#+RESULTS: -[[file:figs/test_nhexa_damped_iff_plant_meas_frf.png]] - -#+begin_src matlab :exports none -%% Comparison of the OL and IFF identified FRF -figure; -tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); - -ax1 = nexttile([2,1]); -hold on; -for i_mass = i_masses - plot(frf_ol.f, abs(frf_ol.G_dL{i_mass+1}(:,1,1)), '-', 'color', [colors(i_mass+1, :), 0.5], ... - 'DisplayName', sprintf('$d\\mathcal{L}_i/u_i$ - %i', i_mass)); - plot(frf_iff.f, abs(frf_iff.G_dL{i_mass+1}(:,1,1)), '-', 'color', colors(i_mass+1, :), ... - 'DisplayName', sprintf('$d\\mathcal{L}_i/u^\\prime_i$ - %i', i_mass)); -end -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Amplitude $d_L/V_a$ [m/V]'); set(gca, 'XTickLabel',[]); -ylim([1e-7, 1e-3]); -legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 2); - -ax2 = nexttile; -hold on; -for i_mass = i_masses - plot(frf_ol.f, 180/pi*angle(frf_ol.G_dL{i_mass+1}(:,1,1)), '-', 'color', [colors(i_mass+1, :), 0.5]); - plot(frf_iff.f, 180/pi*angle(frf_iff.G_dL{i_mass+1}(:,1,1)), '-', 'color', colors(i_mass+1, :)); -end -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); -xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); -hold off; -yticks(-360:90:360); -ylim([-180, 180]); - -linkaxes([ax1,ax2],'x'); -xlim([10, 1e3]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/comp_undamped_damped_plant_meas_frf.pdf', 'width', 'wide', 'height', 'tall'); -#+end_src - -#+name: fig:test_nhexa_comp_undamped_damped_plant_meas_frf -#+caption: Damped and Undamped measured FRF (diagonal elements) -#+RESULTS: -[[file:figs/test_nhexa_comp_undamped_damped_plant_meas_frf.png]] - -#+begin_src matlab :exports none -%% Comparison of the measured FRF and identified TF of the damped plant -figure; -tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); - -freqs = logspace(1,3,1000); - -ax1 = nexttile([2,1]); -hold on; -for i_mass = i_masses - plot(frf_iff.f, abs(frf_iff.G_dL{i_mass+1}(:,1, 1)), 'color', [colors(i_mass+1,:), 0.2], ... - 'DisplayName', sprintf('$d\\mathcal{L}_{m,i}/u^\\prime_i$ - FRF %i', i_mass)); - for i = 2:6 - plot(frf_iff.f, abs(frf_iff.G_dL{i_mass+1}(:,i, i)), 'color', [colors(i_mass+1,:), 0.2], ... - 'HandleVisibility', 'off'); - end - set(gca, 'ColorOrderIndex', i_mass+1) - plot(freqs, abs(squeeze(freqresp(sim_iff.G_dL{i_mass+1}(1,1), freqs, 'Hz'))), '--', ... - 'DisplayName', sprintf('$d\\mathcal{L}_{m,i}/u^\\prime_i$ - Sim %i', i_mass)); -end -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Amplitude [m/V]'); set(gca, 'XTickLabel',[]); -ylim([1e-8, 1e-4]); -legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 2); - -ax2 = nexttile; -hold on; -for i_mass = i_masses - for i =1:6 - plot(frf_iff.f, 180/pi*angle(frf_iff.G_dL{i_mass+1}(:,i, i)), 'color', [colors(i_mass+1,:), 0.2]); - end - set(gca, 'ColorOrderIndex', i_mass+1) - plot(freqs, 180/pi*angle(squeeze(freqresp(sim_iff.G_dL{i_mass+1}(1,1), freqs, 'Hz'))), '--'); -end -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); -xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); -hold off; -yticks(-360:90:360); -ylim([-180, 180]); - -linkaxes([ax1,ax2],'x'); -xlim([10, 1e3]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/comp_iff_plant_frf_sim.pdf', 'width', 'wide', 'height', 'tall'); -#+end_src - -#+name: fig:test_nhexa_comp_iff_plant_frf_sim -#+caption: Comparison of the measured FRF and the identified dynamics from the Simscape model -#+RESULTS: -[[file:figs/test_nhexa_comp_iff_plant_frf_sim.png]] - -#+begin_important -The IFF control strategy effectively damps all the suspensions modes of the nano-hexapod whatever the payload is. -The obtained plant is easier to control (provided the flexible modes of the top platform are well damped). -#+end_important - -*** Change of coupling with IFF -Let's see how the Integral Force Feedback is changing the interaction in the system. - -To study that, the RGA-number are computed both for the undamped plant and for the damped plant using IFF. - -#+begin_src matlab :exports none -%% Decentralized RGA - Undamped Plant -RGA_num_ol = zeros(length(frf_ol.f), length(i_masses)); -for i_mass = i_masses - for i = 1:length(frf_ol.f) - RGA_num_ol(i, i_mass+1) = sum(sum(abs(eye(6) - squeeze(frf_ol.G_dL{i_mass+1}(i,:,:)).*inv(squeeze(frf_ol.G_dL{i_mass+1}(i,:,:))).'))); - end -end -#+end_src - -#+begin_src matlab :exports none -%% Decentralized RGA - Damped Plant using IFF -RGA_num_iff = zeros(length(frf_iff.f), length(i_masses)); -for i_mass = i_masses - for i = 1:length(frf_iff.f) - RGA_num_iff(i, i_mass+1) = sum(sum(abs(eye(6) - squeeze(frf_iff.G_dL{i_mass+1}(i,:,:)).*inv(squeeze(frf_iff.G_dL{i_mass+1}(i,:,:))).'))); - end -end -#+end_src - -Both are compared in Figure ref:fig:test_nhexa_rga_num_ol_iff_masses. - -#+begin_src matlab :exports none -%% RGA for Decentralized plant - With and Without IFF -figure; -hold on; -for i_mass = i_masses - plot(frf_ol.f, RGA_num_ol(:,i_mass+1), '-', 'color', colors(i_mass+1,:), ... - 'DisplayName', sprintf('RGA-num OL - %im', i_mass)); - plot(frf_iff.f, RGA_num_iff(:,i_mass+1), '--', 'color', colors(i_mass+1,:), ... - 'DisplayName', sprintf('RGA-num IFF - %im', i_mass)); -end -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -xlabel('Frequency [Hz]'); ylabel('RGA Number'); -xlim([10, 1e3]); ylim([1e-2, 1e2]); -legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/rga_num_ol_iff_masses.pdf', 'width', 'full', 'height', 'tall'); -#+end_src - -#+name: fig:test_nhexa_rga_num_ol_iff_masses -#+caption: Comparison of the RGA-Number (interaction estimate) without and without IFF -#+RESULTS: -[[file:figs/test_nhexa_rga_num_ol_iff_masses.png]] - -#+begin_important -From Figure ref:fig:test_nhexa_picture_unbalanced_payload, it is clear that the interaction in the system is largest near the resonances. -The Integral Force Feedback controller, by reducing the amplitude at the resonances, also reduces the coupling near these resonances. -It however increases the coupling bellow the frequency of the suspension modes. -#+end_important - -** Un-Balanced mass -:PROPERTIES: -:header-args:matlab+: :tangle matlab/scripts/unbalanced_mass_enc_plates.m -:END: -*** Introduction - -In this section, we wish to see if a payload with a center of mass not aligned with the symmetry axis of the nano-hexapod could cause any issue. - -To study that, the payload shown in Figure ref:fig:test_nhexa_picture_unbalanced_payload is used. - -#+name: fig:test_nhexa_picture_unbalanced_payload -#+caption: Nano-Hexapod with unbalanced payload -#+attr_latex: :width \linewidth -[[file:figs/test_nhexa_picture_unbalanced_payload.jpg]] - -*** Matlab Init :noexport:ignore: -#+begin_src matlab -%% unbalanced_mass_enc_plates.m -% Study of the effect of an un-balanced payload on the plant dynamics -#+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 - -*** Compute the identified FRF with IFF -The following data are loaded: -- =Va=: the excitation voltage for the damped plant (corresponding to $u^\prime_i$) -- =de=: the measured motion by the 6 encoders (corresponding to $d\bm{\mathcal{L}}_m$) -#+begin_src matlab -%% Load Identification Data -meas_added_mass = {zeros(6,1)}; - -for i_strut = 1:6 - meas_iff_mass(i_strut) = {load(sprintf('frf_data_exc_strut_%i_iff_vib_table_1m_unbalanced.mat', i_strut), 't', 'Va', 'de')}; -end -#+end_src - -The window =win= and the frequency vector =f= are defined. -#+begin_src matlab -% Sampling Time [s] -Ts = (meas_iff_mass{1}.t(end) - (meas_iff_mass{1}.t(1)))/(length(meas_iff_mass{1}.t)-1); - -% Hannning Windows -win = hanning(ceil(1/Ts)); - -% And we get the frequency vector -[~, f] = tfestimate(meas_iff_mass{1}.Va, meas_iff_mass{1}.de, win, Noverlap, Nfft, 1/Ts); -#+end_src - -Finally the $6 \times 6$ transfer function matrix from $\bm{u}^\prime$ to $d\bm{\mathcal{L}}_m$ is estimated: -#+begin_src matlab -%% DVF Plant (transfer function from u to dLm) -G_dL = zeros(length(f), 6, 6); -for i_strut = 1:6 - G_dL(:,:,i_strut) = tfestimate(meas_iff_mass{i_strut}.Va, meas_iff_mass{i_strut}.de, win, Noverlap, Nfft, 1/Ts); -end -#+end_src - -The identified dynamics are then saved for further use. -#+begin_src matlab :exports none :tangle no -save('matlab/data_frf/frf_iff_unbalanced_vib_table_m.mat', 'f', 'Ts', 'G_dL'); -#+end_src - -#+begin_src matlab :eval no -save('data_frf/frf_iff_unbalanced_vib_table_m.mat', 'f', 'Ts', 'G_dL'); -#+end_src - -*** Effect of an unbalanced payload -#+begin_src matlab :exports none -%% Load the Measured FRF of the damped plant -frf_unb_iff = load('frf_iff_unbalanced_vib_table_m.mat', 'f', 'Ts', 'G_dL'); -#+end_src - -The transfer functions from $u_i$ to $d\mathcal{L}_i$ are shown in Figure ref:fig:test_nhexa_frf_damp_unbalanced_mass. -Due to the unbalanced payload, the system is not symmetrical anymore, and therefore each of the diagonal elements are not equal. -This is due to the fact that each strut is not affected by the same inertia. - -#+begin_src matlab :exports none -%% Diagonal and Off Diagonal elements of the damped plants -figure; -tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); - -ax1 = nexttile([2,1]); -hold on; -for i = 1:6 - plot(frf_unb_iff.f, abs(frf_unb_iff.G_dL(:,i,i)), 'color', colors(i,:), ... - 'DisplayName', sprintf('$d\\mathcal{L}_{m,%i}/u^\\prime_%i$', i, i)); -end -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Amplitude $d_L/V_a$ [m/V]'); set(gca, 'XTickLabel',[]); -ylim([5e-8, 3e-5]); -legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 2); - -ax2 = nexttile; -hold on; -for i =1:6 - plot(frf_unb_iff.f, 180/pi*angle(frf_unb_iff.G_dL(:,i, i)), 'color', colors(i,:)); -end -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); -xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); -hold off; -yticks(-360:90:360); -ylim([-180, 180]); - -linkaxes([ax1,ax2],'x'); -xlim([10, 1e3]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/frf_damp_unbalanced_mass.pdf', 'width', 'wide', 'height', 'tall'); -#+end_src - -#+name: fig:test_nhexa_frf_damp_unbalanced_mass -#+caption: Transfer function from $u_i$ to $d\mathcal{L}_i$ for the nano-hexapod with an unbalanced payload -#+RESULTS: -[[file:figs/test_nhexa_frf_damp_unbalanced_mass.png]] - - - -** Modal Analysis :noexport: -:PROPERTIES: -:header-args:matlab+: :tangle matlab/scripts/enc_struts_compliance_iff.m -:END: -<> - -- [ ] *This test was made using encoder fixed to the struts, is it relevant to put it here?* - -*** Introduction :ignore: -Several 3-axis accelerometers are fixed on the top platform of the nano-hexapod as shown in Figure ref:fig:test_nhexa_compliance_vertical_comp_iff. - -#+name: fig:test_nhexa_accelerometers_nano_hexapod -#+caption: Location of the accelerometers on top of the nano-hexapod -#+attr_latex: :width \linewidth -[[file:figs/test_nhexa_accelerometers_nano_hexapod.jpg]] - -The top platform is then excited using an instrumented hammer as shown in Figure ref:fig:test_nhexa_hammer_excitation_compliance_meas. - -#+name: fig:test_nhexa_hammer_excitation_compliance_meas -#+caption: Example of an excitation using an instrumented hammer -#+attr_latex: :width \linewidth -[[file:figs/test_nhexa_hammer_excitation_compliance_meas.jpg]] - -From this experiment, the resonance frequencies and the associated mode shapes can be computed (Section ref:sec:test_nhexa_modal_analysis_mode_shapes). -Then, in Section ref:sec:test_nhexa_compliance_effect_iff, the vertical compliance of the nano-hexapod is experimentally estimated. -Finally, in Section ref:sec:test_nhexa_compliance_effect_iff_comp_model, the measured compliance is compare with the estimated one from the Simscape model. - -*** Matlab Init :noexport:ignore: -#+begin_src matlab -%% enc_struts_compliance_iff.m -% Compare measured compliance and estimated compliance from the Simscape model -#+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 - -*** Obtained Mode Shapes -<> - -We can observe the mode shapes of the first 6 modes that are the suspension modes (the plate is behaving as a solid body) in Figure ref:fig:test_nhexa_mode_shapes_annotated. - -#+name: fig:test_nhexa_mode_shapes_annotated -#+caption: Measured mode shapes for the first six modes -#+attr_latex: :width \linewidth -[[file:figs/test_nhexa_mode_shapes_annotated.gif]] - -Then, there is a mode at 692Hz which corresponds to a flexible mode of the top plate (Figure ref:fig:test_nhexa_mode_shapes_flexible_annotated). - -#+name: fig:test_nhexa_mode_shapes_flexible_annotated -#+caption: First flexible mode at 692Hz -#+attr_latex: :width 0.3\linewidth -[[file:figs/test_nhexa_ModeShapeFlex1_crop.gif]] - -The obtained modes are summarized in Table ref:tab:description_modes. - -#+name: tab:description_modes -#+caption: Description of the identified modes -#+attr_latex: :environment tabularx :width 0.7\linewidth :align ccX -#+attr_latex: :center t :booktabs t -| *Mode* | *Freq. [Hz]* | *Description* | -|--------+--------------+----------------------------------------------| -| 1 | 105 | Suspension Mode: Y-translation | -| 2 | 107 | Suspension Mode: X-translation | -| 3 | 131 | Suspension Mode: Z-translation | -| 4 | 161 | Suspension Mode: Y-tilt | -| 5 | 162 | Suspension Mode: X-tilt | -| 6 | 180 | Suspension Mode: Z-rotation | -| 7 | 692 | (flexible) Membrane mode of the top platform | - -*** Nano-Hexapod Compliance - Effect of IFF -<> - -In this section, we wish to estimate the effectiveness of the IFF strategy regarding the compliance. - -The top plate is excited vertically using the instrumented hammer two times: -1. no control loop is used -2. decentralized IFF is used - -The data are loaded. -#+begin_src matlab -frf_ol = load('Measurement_Z_axis.mat'); % Open-Loop -frf_iff = load('Measurement_Z_axis_damped.mat'); % IFF -#+end_src - -The mean vertical motion of the top platform is computed by averaging all 5 vertical accelerometers. -#+begin_src matlab -%% Multiply by 10 (gain in m/s^2/V) and divide by 5 (number of accelerometers) -d_frf_ol = 10/5*(frf_ol.FFT1_H1_4_1_RMS_Y_Mod + frf_ol.FFT1_H1_7_1_RMS_Y_Mod + frf_ol.FFT1_H1_10_1_RMS_Y_Mod + frf_ol.FFT1_H1_13_1_RMS_Y_Mod + frf_ol.FFT1_H1_16_1_RMS_Y_Mod)./(2*pi*frf_ol.FFT1_H1_16_1_RMS_X_Val).^2; -d_frf_iff = 10/5*(frf_iff.FFT1_H1_4_1_RMS_Y_Mod + frf_iff.FFT1_H1_7_1_RMS_Y_Mod + frf_iff.FFT1_H1_10_1_RMS_Y_Mod + frf_iff.FFT1_H1_13_1_RMS_Y_Mod + frf_iff.FFT1_H1_16_1_RMS_Y_Mod)./(2*pi*frf_iff.FFT1_H1_16_1_RMS_X_Val).^2; -#+end_src - -The vertical compliance (magnitude of the transfer function from a vertical force applied on the top plate to the vertical motion of the top plate) is shown in Figure ref:fig:test_nhexa_compliance_vertical_comp_iff. -#+begin_src matlab :exports none -%% Comparison of the vertical compliances (OL and IFF) -figure; -hold on; -plot(frf_ol.FFT1_H1_16_1_RMS_X_Val, d_frf_ol, 'DisplayName', 'OL'); -plot(frf_iff.FFT1_H1_16_1_RMS_X_Val, d_frf_iff, 'DisplayName', 'IFF'); -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -xlabel('Frequency [Hz]'); ylabel('Vertical Compliance [$m/N$]'); -xlim([20, 2e3]); ylim([2e-9, 2e-5]); -legend('location', 'northeast'); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/compliance_vertical_comp_iff.pdf', 'width', 'wide', 'height', 'normal'); -#+end_src - -#+name: fig:test_nhexa_compliance_vertical_comp_iff -#+caption: Measured vertical compliance with and without IFF -#+RESULTS: -[[file:figs/test_nhexa_compliance_vertical_comp_iff.png]] - -#+begin_important -From Figure ref:fig:test_nhexa_compliance_vertical_comp_iff, it is clear that the IFF control strategy is very effective in damping the suspensions modes of the nano-hexapod. -It also has the effect of (slightly) degrading the vertical compliance at low frequency. - -It also seems some damping can be added to the modes at around 205Hz which are flexible modes of the struts. -#+end_important - -*** Comparison with the Simscape Model -<> - -Let's initialize the Simscape model such that it corresponds to the experiment. -#+begin_src matlab -%% Nano-Hexapod is fixed on a rigid granite -support.type = 0; - -%% No Payload on top of the Nano-Hexapod -payload.type = 0; - -%% Initialize Nano-Hexapod in Open Loop -n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... - 'flex_top_type', '4dof', ... - 'motion_sensor_type', 'struts', ... - 'actuator_type', '2dof'); - -#+end_src - -And let's compare the measured vertical compliance with the vertical compliance as estimated from the Simscape model. - -The transfer function from a vertical external force to the absolute motion of the top platform is identified (with and without IFF) using the Simscape model. -#+begin_src matlab :exports none -%% Identify the IFF Plant (transfer function from u to taum) -clear io; io_i = 1; -io(io_i) = linio([mdl, '/Fz_ext'], 1, 'openinput'); io_i = io_i + 1; % External - Vertical force -io(io_i) = linio([mdl, '/Z_top_plat'], 1, 'openoutput'); io_i = io_i + 1; % Absolute vertical motion of top platform -#+end_src - -#+begin_src matlab :exports none -%% Perform the identifications -G_compl_z_ol = linearize(mdl, io, 0.0, options); -#+end_src - -#+begin_src matlab :exports none -%% Initialize Nano-Hexapod with IFF -Kiff = 400*(1/(s + 2*pi*40))*... % Low pass filter (provides integral action above 40Hz) - (s/(s + 2*pi*30))*... % High pass filter to limit low frequency gain - (1/(1 + s/2/pi/500))*... % Low pass filter to be more robust to high frequency resonances - eye(6); % Diagonal 6x6 controller - -%% Initialize the Nano-Hexapod with IFF -n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... - 'flex_top_type', '4dof', ... - 'motion_sensor_type', 'struts', ... - 'actuator_type', '2dof', ... - 'controller_type', 'iff'); - -%% Perform the identification -G_compl_z_iff = linearize(mdl, io, 0.0, options); -#+end_src - -The comparison is done in Figure ref:fig:test_nhexa_compliance_vertical_comp_model_iff. -Again, the model is quite accurate in predicting the (closed-loop) behavior of the system. - -#+begin_src matlab :exports none -%% Comparison of the measured compliance and the one obtained from the model -freqs = 2*logspace(1,3,1000); - -figure; -hold on; -plot(frf_ol.FFT1_H1_16_1_RMS_X_Val, d_frf_ol, '-', 'DisplayName', 'OL - Meas.'); -plot(frf_iff.FFT1_H1_16_1_RMS_X_Val, d_frf_iff, '-', 'DisplayName', 'IFF - Meas.'); -set(gca,'ColorOrderIndex',1) -plot(freqs, abs(squeeze(freqresp(G_compl_z_ol, freqs, 'Hz'))), '--', 'DisplayName', 'OL - Model') -plot(freqs, abs(squeeze(freqresp(G_compl_z_iff, freqs, 'Hz'))), '--', 'DisplayName', 'IFF - Model') -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -xlabel('Frequency [Hz]'); ylabel('Vertical Compliance [$m/N$]'); -xlim([20, 2e3]); ylim([2e-9, 2e-5]); -legend('location', 'northeast', 'FontSize', 8); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/compliance_vertical_comp_model_iff.pdf', 'width', 'wide', 'height', 'normal'); -#+end_src - -#+name: fig:test_nhexa_compliance_vertical_comp_model_iff -#+caption: Measured vertical compliance with and without IFF -#+RESULTS: -[[file:figs/test_nhexa_compliance_vertical_comp_model_iff.png]] - ** Conclusion #+begin_important In this section, the dynamics of the nano-hexapod with the encoders fixed to the plates is studied. @@ -3200,3453 +6309,17 @@ It has been found that: Only the flexible modes of the top plates seems to be problematic for control. #+end_important -* Decentralized High Authority Control with Integral Force Feedback -<> - -** Introduction :ignore: - -In this section is studied the HAC-IFF architecture for the Nano-Hexapod. -More precisely: -- The LAC control is a decentralized integral force feedback as studied in Section ref:sec:test_nhexa_enc_plates_iff -- The HAC control is a decentralized controller working in the frame of the struts - -The corresponding control architecture is shown in Figure ref:fig:test_nhexa_control_architecture_hac_iff_struts with: -- $\bm{r}_{\mathcal{X}_n}$: the $6 \times 1$ reference signal in the cartesian frame -- $\bm{r}_{d\mathcal{L}}$: the $6 \times 1$ reference signal transformed in the frame of the struts thanks to the inverse kinematic -- $\bm{\epsilon}_{d\mathcal{L}}$: the $6 \times 1$ length error of the 6 struts -- $\bm{u}^\prime$: input of the damped plant -- $\bm{u}$: generated DAC voltages -- $\bm{\tau}_m$: measured force sensors -- $d\bm{\mathcal{L}}_m$: measured displacement of the struts by the encoders - -#+begin_src latex :file control_architecture_hac_iff_struts.pdf -\definecolor{instrumentation}{rgb}{0, 0.447, 0.741} -\definecolor{mechanics}{rgb}{0.8500, 0.325, 0.098} -\definecolor{control}{rgb}{0.4660, 0.6740, 0.1880} - -\begin{tikzpicture} - % Blocs - \node[block={3.0cm}{2.0cm}, fill=black!20!white] (P) {Plant}; - \coordinate[] (inputF) at ($(P.south west)!0.5!(P.north west)$); - \coordinate[] (outputF) at ($(P.south east)!0.2!(P.north east)$); - \coordinate[] (outputL) at ($(P.south east)!0.8!(P.north east)$); - - \node[block, below=0.4 of P, fill=control!20!white] (Kiff) {$\bm{K}_\text{IFF}$}; - \node[block, left=0.8 of inputF, fill=instrumentation!20!white] (pd200) {\tiny PD200}; - \node[addb, left=0.8 of pd200, fill=control!20!white] (addF) {}; - \node[block, left=0.8 of addF, fill=control!20!white] (K) {$\bm{K}_\mathcal{L}$}; - \node[addb={+}{}{-}{}{}, left=0.8 of K, fill=control!20!white] (subr) {}; - \node[block, align=center, left= of subr, fill=control!20!white] (J) {\tiny Inverse\\\tiny Kinematics}; - - % Connections and labels - \draw[->] (outputF) -- ++(1.0, 0) node[above left]{$\bm{\tau}_m$}; - \draw[->] ($(outputF) + (0.6, 0)$)node[branch]{} |- (Kiff.east); - \draw[->] (Kiff.west) -| (addF.south); - \draw[->] (addF.east) -- (pd200.west) node[above left]{$\bm{u}$}; - \draw[->] (pd200.east) -- (inputF) node[above left]{$\bm{u}_a$}; - - \draw[->] (outputL) -- ++(1.0, 0) node[below left]{$d\bm{\mathcal{L}_m}$}; - \draw[->] ($(outputL) + (0.6, 0)$)node[branch]{} -- ++(0, 1) -| (subr.north); - \draw[->] (subr.east) -- (K.west) node[above left]{$\bm{\epsilon}_{d\mathcal{L}}$}; - \draw[->] (K.east) -- (addF.west) node[above left]{$\bm{u}^\prime$}; - - \draw[->] (J.east) -- (subr.west) node[above left]{$\bm{r}_{d\mathcal{L}}$}; - \draw[<-] (J.west)node[above left]{$\bm{r}_{\mathcal{X}_n}$} -- ++(-1, 0); -\end{tikzpicture} -#+end_src - -#+name: fig:test_nhexa_control_architecture_hac_iff_struts -#+caption: HAC-LAC: IFF + Control in the frame of the legs -#+RESULTS: -[[file:figs/test_nhexa_control_architecture_hac_iff_struts.png]] - -This part is structured as follow: -- Section ref:sec:test_nhexa_hac_iff_struts_ref_track: some reference tracking tests are performed -- Section ref:sec:test_nhexa_hac_iff_struts_controller: the decentralized high authority controller is tuned using the Simscape model and is implemented and tested experimentally -- Section ref:sec:test_nhexa_interaction_analysis: an interaction analysis is performed, from which the best decoupling strategy can be determined -- Section ref:sec:test_nhexa_robust_hac_design: Robust High Authority Controller are designed - -** Reference Tracking - Trajectories -:PROPERTIES: -:header-args:matlab+: :tangle matlab/scripts/reference_tracking_paths.m -:END: -<> -*** Introduction :ignore: -In this section, several trajectories representing the wanted pose (position and orientation) of the top platform with respect to the bottom platform are defined. - -These trajectories will be used to test the HAC-LAC architecture. - -In order to transform the wanted pose to the wanted displacement of the 6 struts, the inverse kinematic is required. -As a first approximation, the Jacobian matrix $\bm{J}$ can be used instead of using the full inverse kinematic equations. - -Therefore, the control architecture with the input trajectory $\bm{r}_{\mathcal{X}_n}$ is shown in Figure ref:fig:test_nhexa_control_architecture_hac_iff_L. - -#+begin_src latex :file control_architecture_hac_iff_struts_L.pdf -\definecolor{instrumentation}{rgb}{0, 0.447, 0.741} -\definecolor{mechanics}{rgb}{0.8500, 0.325, 0.098} -\definecolor{control}{rgb}{0.4660, 0.6740, 0.1880} - -\begin{tikzpicture} - % Blocs - \node[block={3.0cm}{2.0cm}, fill=black!20!white] (P) {Plant}; - \coordinate[] (inputF) at ($(P.south west)!0.5!(P.north west)$); - \coordinate[] (outputF) at ($(P.south east)!0.2!(P.north east)$); - \coordinate[] (outputL) at ($(P.south east)!0.8!(P.north east)$); - - \node[block, below=0.4 of P, fill=control!20!white] (Kiff) {$\bm{K}_\text{IFF}$}; - \node[block, left=0.8 of inputF, fill=instrumentation!20!white] (pd200) {\tiny PD200}; - \node[addb, left=0.8 of pd200, fill=control!20!white] (addF) {}; - \node[block, left=0.8 of addF, fill=control!20!white] (K) {$\bm{K}_\mathcal{L}$}; - \node[addb={+}{}{-}{}{}, left=0.8 of K, fill=control!20!white] (subr) {}; - \node[block, align=center, left= of subr, fill=control!20!white] (J) {$\bm{J}$}; - - % Connections and labels - \draw[->] (outputF) -- ++(1.0, 0) node[above left]{$\bm{\tau}_m$}; - \draw[->] ($(outputF) + (0.6, 0)$)node[branch]{} |- (Kiff.east); - \draw[->] (Kiff.west) -| (addF.south); - \draw[->] (addF.east) -- (pd200.west) node[above left]{$\bm{u}$}; - \draw[->] (pd200.east) -- (inputF) node[above left]{$\bm{u}_a$}; - - \draw[->] (outputL) -- ++(1.0, 0) node[below left]{$d\bm{\mathcal{L}_m}$}; - \draw[->] ($(outputL) + (0.6, 0)$)node[branch]{} -- ++(0, 1) -| (subr.north); - \draw[->] (subr.east) -- (K.west) node[above left]{$\bm{\epsilon}_{d\mathcal{L}}$}; - \draw[->] (K.east) -- (addF.west) node[above left]{$\bm{u}^\prime$}; - - \draw[->] (J.east) -- (subr.west) node[above left]{$\bm{r}_{d\mathcal{L}}$}; - \draw[<-] (J.west)node[above left]{$\bm{r}_{\mathcal{X}_n}$} -- ++(-1, 0); -\end{tikzpicture} -#+end_src - -#+name: fig:test_nhexa_control_architecture_hac_iff_L -#+caption: HAC-LAC: IFF + Control in the frame of the legs -#+RESULTS: -[[file:figs/test_nhexa_control_architecture_hac_iff_struts_L.png]] - -In the following sections, several reference trajectories are defined: -- Section ref:sec:test_nhexa_yz_scans: simple scans in the Y-Z plane -- Section ref:sec:test_nhexa_tilt_scans: scans in tilt are performed -- Section ref:sec:test_nhexa_nass_scans: scans with X-Y-Z translations in order to draw the word "NASS" - -*** Matlab Init :noexport:ignore: -#+begin_src matlab -%% reference_tracking_paths.m -% Computation of several reference paths -#+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 - -*** Y-Z Scans -<> -A function =generateYZScanTrajectory= has been developed in order to easily generate scans in the Y-Z plane. - -For instance, the following generated trajectory is represented in Figure ref:fig:test_nhexa_yz_scan_example_trajectory_yz_plane. -#+begin_src matlab -%% Generate the Y-Z trajectory scan -Rx_yz = generateYZScanTrajectory(... - 'y_tot', 4e-6, ... % Length of Y scans [m] - 'z_tot', 4e-6, ... % Total Z distance [m] - 'n', 5, ... % Number of Y scans - 'Ts', 1e-3, ... % Sampling Time [s] - 'ti', 1, ... % Time to go to initial position [s] - 'tw', 0, ... % Waiting time between each points [s] - 'ty', 0.6, ... % Time for a scan in Y [s] - 'tz', 0.2); % Time for a scan in Z [s] -#+end_src - -#+begin_src matlab :exports none -%% Plot the trajectory in the Y-Z plane -figure; -plot(Rx_yz(:,3), Rx_yz(:,4)); -xlabel('y [m]'); ylabel('z [m]'); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/yz_scan_example_trajectory_yz_plane.pdf', 'width', 'normal', 'height', 'normal'); -#+end_src - -#+name: fig:test_nhexa_yz_scan_example_trajectory_yz_plane -#+caption: Generated scan in the Y-Z plane -#+RESULTS: -[[file:figs/test_nhexa_yz_scan_example_trajectory_yz_plane.png]] - -The Y and Z positions as a function of time are shown in Figure ref:fig:test_nhexa_yz_scan_example_trajectory. - -#+begin_src matlab :exports none -%% Plot the Y-Z trajectory as a function of time -figure; -hold on; -plot(Rx_yz(:,1), Rx_yz(:,3), ... - 'DisplayName', 'Y motion') -plot(Rx_yz(:,1), Rx_yz(:,4), ... - 'DisplayName', 'Z motion') -hold off; -xlabel('Time [s]'); -ylabel('Displacement [m]'); -legend('location', 'northeast'); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/yz_scan_example_trajectory.pdf', 'width', 'wide', 'height', 'normal'); -#+end_src - -#+name: fig:test_nhexa_yz_scan_example_trajectory -#+caption: Y and Z trajectories as a function of time -#+RESULTS: -[[file:figs/test_nhexa_yz_scan_example_trajectory.png]] - -Using the Jacobian matrix, it is possible to compute the wanted struts lengths as a function of time: -\begin{equation} - \bm{r}_{d\mathcal{L}} = \bm{J} \bm{r}_{\mathcal{X}_n} -\end{equation} - -#+begin_src matlab :exports none -load('jacobian.mat', 'J'); -#+end_src - -#+begin_src matlab -%% Compute the reference in the frame of the legs -dL_ref = [J*Rx_yz(:, 2:7)']'; -#+end_src - -The reference signal for the strut length is shown in Figure ref:fig:test_nhexa_yz_scan_example_trajectory_struts. -#+begin_src matlab :exports none -%% Plot the reference in the frame of the legs -figure; -hold on; -for i=1:6 - plot(Rx_yz(:,1), dL_ref(:, i), ... - 'DisplayName', sprintf('$r_{d\\mathcal{L}_%i}$', i)) -end -xlabel('Time [s]'); ylabel('Strut Motion [m]'); -legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 2); -yticks(1e-6*[-5:5]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/yz_scan_example_trajectory_struts.pdf', 'width', 'wide', 'height', 'normal'); -#+end_src - -#+name: fig:test_nhexa_yz_scan_example_trajectory_struts -#+caption: Trajectories for the 6 individual struts -#+RESULTS: -[[file:figs/test_nhexa_yz_scan_example_trajectory_struts.png]] - -*** Tilt Scans -<> - -A function =generalSpiralAngleTrajectory= has been developed in order to easily generate $R_x,R_y$ tilt scans. - -For instance, the following generated trajectory is represented in Figure ref:fig:test_nhexa_tilt_scan_example_trajectory. -#+begin_src matlab -%% Generate the "tilt-spiral" trajectory scan -R_tilt = generateSpiralAngleTrajectory(... - 'R_tot', 20e-6, ... % Total Tilt [ad] - 'n_turn', 5, ... % Number of scans - 'Ts', 1e-3, ... % Sampling Time [s] - 't_turn', 1, ... % Turn time [s] - 't_end', 1); % End time to go back to zero [s] -#+end_src - -#+begin_src matlab :exports none -%% Plot the trajectory -figure; -plot(1e6*R_tilt(:,5), 1e6*R_tilt(:,6)); -xlabel('$R_x$ [$\mu$rad]'); ylabel('$R_y$ [$\mu$rad]'); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/tilt_scan_example_trajectory.pdf', 'width', 'normal', 'height', 'normal'); -#+end_src - -#+name: fig:test_nhexa_tilt_scan_example_trajectory -#+caption: Generated "spiral" scan -#+RESULTS: -[[file:figs/test_nhexa_tilt_scan_example_trajectory.png]] - -#+begin_src matlab :exports none -%% Compute the reference in the frame of the legs -load('jacobian.mat', 'J'); -dL_ref = [J*R_tilt(:, 2:7)']'; -#+end_src - -The reference signal for the strut length is shown in Figure ref:fig:test_nhexa_tilt_scan_example_trajectory_struts. -#+begin_src matlab :exports none -%% Plot the reference in the frame of the legs -figure; -hold on; -for i=1:6 - plot(R_tilt(:,1), dL_ref(:, i), ... - 'DisplayName', sprintf('$r_{d\\mathcal{L}_%i}$', i)) -end -xlabel('Time [s]'); ylabel('Strut Motion [m]'); -legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 2); -yticks(1e-6*[-5:5]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/tilt_scan_example_trajectory_struts.pdf', 'width', 'wide', 'height', 'normal'); -#+end_src - -#+name: fig:test_nhexa_tilt_scan_example_trajectory_struts -#+caption: Trajectories for the 6 individual struts - Tilt scan -#+RESULTS: -[[file:figs/test_nhexa_tilt_scan_example_trajectory_struts.png]] - -*** "NASS" reference path -<> -In this section, a reference path that "draws" the work "NASS" is developed. - -First, a series of points representing each letter are defined. -Between each letter, a negative Z motion is performed. -#+begin_src matlab -%% List of points that draws "NASS" -ref_path = [ ... - 0, 0,0; % Initial Position - 0,0,1; 0,4,1; 3,0,1; 3,4,1; % N - 3,4,0; 4,0,0; % Transition - 4,0,1; 4,3,1; 5,4,1; 6,4,1; 7,3,1; 7,2,1; 4,2,1; 4,3,1; 5,4,1; 6,4,1; 7,3,1; 7,0,1; % A - 7,0,0; 8,0,0; % Transition - 8,0,1; 11,0,1; 11,2,1; 8,2,1; 8,4,1; 11,4,1; % S - 11,4,0; 12,0,0; % Transition - 12,0,1; 15,0,1; 15,2,1; 12,2,1; 12,4,1; 15,4,1; % S - 15,4,0; - ]; - -%% Center the trajectory arround zero -ref_path = ref_path - (max(ref_path) - min(ref_path))/2; - -%% Define the X-Y-Z cuboid dimensions containing the trajectory -X_max = 10e-6; -Y_max = 4e-6; -Z_max = 2e-6; - -ref_path = ([X_max, Y_max, Z_max]./max(ref_path)).*ref_path; % [m] -#+end_src - -Then, using the =generateXYZTrajectory= function, the $6 \times 1$ trajectory signal is computed. -#+begin_src matlab -%% Generating the trajectory -Rx_nass = generateXYZTrajectory('points', ref_path); -#+end_src - -The trajectory in the X-Y plane is shown in Figure ref:fig:test_nhexa_ref_track_test_nass (the transitions between the letters are removed). -#+begin_src matlab :exports none -%% "NASS" trajectory in the X-Y plane -figure; -plot(1e6*Rx_nass(Rx_nass(:,4)>0, 2), 1e6*Rx_nass(Rx_nass(:,4)>0, 3), 'k.') -xlabel('X [$\mu m$]'); -ylabel('Y [$\mu m$]'); -axis equal; -xlim(1e6*[min(Rx_nass(:,2)), max(Rx_nass(:,2))]); -ylim(1e6*[min(Rx_nass(:,3)), max(Rx_nass(:,3))]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/ref_track_test_nass.pdf', 'width', 'wide', 'height', 'normal'); -#+end_src - -#+name: fig:test_nhexa_ref_track_test_nass -#+caption: Reference path corresponding to the "NASS" acronym -#+RESULTS: -[[file:figs/test_nhexa_ref_track_test_nass.png]] - -It can also be better viewed in a 3D representation as in Figure ref:fig:test_nhexa_ref_track_test_nass_3d. - -#+begin_src matlab :exports none -figure; -plot3(1e6*Rx_nass(:,2), 1e6*Rx_nass(:,3), 1e6*Rx_nass(:,4), 'k-'); -xlabel('x [$\mu m$]'); ylabel('y [$\mu m$]'); zlabel('z [$\mu m$]'); -view(-13, 41) -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/ref_track_test_nass_3d.pdf', 'width', 'normal', 'height', 'normal'); -#+end_src - -#+name: fig:test_nhexa_ref_track_test_nass_3d -#+caption: Reference path that draws "NASS" - 3D view -#+RESULTS: -[[file:figs/test_nhexa_ref_track_test_nass_3d.png]] - -** First Basic High Authority Controller -:PROPERTIES: -:header-args:matlab+: :tangle matlab/scripts/hac_lac_first_try.m -:END: -<> -*** Introduction :ignore: -In this section, a simple decentralized high authority controller $\bm{K}_{\mathcal{L}}$ is developed to work without any payload. - -The diagonal controller is tuned using classical Loop Shaping in Section ref:sec:test_nhexa_hac_iff_no_payload_tuning. -The stability is verified in Section ref:sec:test_nhexa_hac_iff_no_payload_stability using the Simscape model. - -*** Matlab Init :noexport:ignore: -#+begin_src matlab -%% hac_lac_first_try.m -% Development and analysis of a first basic High Authority Controller -#+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 -%% Load the identified FRF and Simscape model -frf_iff = load('frf_iff_vib_table_m.mat', 'f', 'Ts', 'G_dL'); -sim_iff = load('sim_iff_vib_table_m.mat', 'G_dL'); -#+end_src - -*** HAC Controller -<> - -Let's first try to design a first decentralized controller with: -- a bandwidth of 100Hz -- sufficient phase margin -- simple and understandable components - -After some very basic and manual loop shaping, A diagonal controller is developed. -Each diagonal terms are identical and are composed of: -- A lead around 100Hz -- A first order low pass filter starting at 200Hz to add some robustness to high frequency modes -- A notch at 700Hz to cancel the flexible modes of the top plate -- A pure integrator - -#+begin_src matlab -%% Lead to increase phase margin -a = 2; % Amount of phase lead / width of the phase lead / high frequency gain -wc = 2*pi*100; % Frequency with the maximum phase lead [rad/s] - -H_lead = (1 + s/(wc/sqrt(a)))/(1 + s/(wc*sqrt(a))); - -%% Low Pass filter to increase robustness -H_lpf = 1/(1 + s/2/pi/200); - -%% Notch at the top-plate resonance -gm = 0.02; -xi = 0.3; -wn = 2*pi*700; - -H_notch = (s^2 + 2*gm*xi*wn*s + wn^2)/(s^2 + 2*xi*wn*s + wn^2); - -%% Decentralized HAC -Khac_iff_struts = -(1/(2.87e-5)) * ... % Gain - H_lead * ... % Lead - H_notch * ... % Notch - (2*pi*100/s) * ... % Integrator - eye(6); % 6x6 Diagonal -#+end_src - -This controller is saved for further use. -#+begin_src matlab :exports none :tangle no -save('matlab/data_sim/Khac_iff_struts.mat', 'Khac_iff_struts') -#+end_src - -#+begin_src matlab :eval no -save('data_sim/Khac_iff_struts.mat', 'Khac_iff_struts') -#+end_src - -The experimental loop gain is computed and shown in Figure ref:fig:test_nhexa_loop_gain_hac_iff_struts. -#+begin_src matlab -L_hac_iff_struts = pagemtimes(permute(frf_iff.G_dL{1}, [2 3 1]), squeeze(freqresp(Khac_iff_struts, frf_iff.f, 'Hz'))); -#+end_src - -#+begin_src matlab :exports none -%% Bode plot of the Loop Gain -figure; -tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); - - -ax1 = nexttile([2,1]); -hold on; -% Diagonal Elements Model -plot(frf_iff.f, abs(squeeze(L_hac_iff_struts(1,1,:))), 'color', colors(1,:), ... - 'DisplayName', 'Diagonal'); -for i = 2:6 - plot(frf_iff.f, abs(squeeze(L_hac_iff_struts(i,i,:))), 'color', colors(1,:), ... - 'HandleVisibility', 'off'); -end -plot(frf_iff.f, abs(squeeze(L_hac_iff_struts(1,2,:))), 'color', [colors(2,:), 0.2], ... - 'DisplayName', 'Off-Diag'); -for i = 1:5 - for j = i+1:6 - plot(frf_iff.f, abs(squeeze(L_hac_iff_struts(i,j,:))), 'color', [colors(2,:), 0.2], ... - 'HandleVisibility', 'off'); - end -end -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Loop Gain [-]'); set(gca, 'XTickLabel',[]); -ylim([1e-3, 1e2]); -legend('location', 'northeast'); - -ax2 = nexttile; -hold on; -for i =1:6 - plot(frf_iff.f, 180/pi*angle(squeeze(L_hac_iff_struts(i,i,:))), 'color', colors(1,:)); -end -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); -xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); -hold off; -yticks(-360:90:360); -ylim([-180, 180]); - -linkaxes([ax1,ax2],'x'); -xlim([2, 2e3]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/loop_gain_hac_iff_struts.pdf', 'width', 'wide', 'height', 'tall'); -#+end_src - -#+name: fig:test_nhexa_loop_gain_hac_iff_struts -#+caption: Diagonal and off-diagonal elements of the Loop gain for "HAC-IFF-Struts" -#+RESULTS: -[[file:figs/test_nhexa_loop_gain_hac_iff_struts.png]] - -*** Verification of the Stability using the Simscape model -<> - -The HAC-IFF control strategy is implemented using Simscape. -#+begin_src matlab -%% Initialize the Simscape model in closed loop -n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... - 'flex_top_type', '4dof', ... - 'motion_sensor_type', 'plates', ... - 'actuator_type', 'flexible', ... - 'controller_type', 'hac-iff-struts'); -#+end_src - -#+begin_src matlab :exports none -support.type = 1; % On top of vibration table -payload.type = 3; % Payload / 1 "mass layer" - -load('Kiff_opt.mat', 'Kiff'); -#+end_src - -#+begin_src matlab -%% Identify the (damped) transfer function from u to dLm -clear io; io_i = 1; -io(io_i) = linio([mdl, '/du'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs -io(io_i) = linio([mdl, '/dL'], 1, 'openoutput'); io_i = io_i + 1; % Plate Displacement (encoder) -#+end_src - -We identify the closed-loop system. -#+begin_src matlab -%% Identification -Gd_iff_hac_opt = linearize(mdl, io, 0.0, options); -#+end_src - -And verify that it is indeed stable. -#+begin_src matlab :results value replace :exports both -%% Verify the stability -isstable(Gd_iff_hac_opt) -#+end_src - -#+RESULTS: -: 1 - -*** Experimental Validation -Both the Integral Force Feedback controller (developed in Section ref:sec:test_nhexa_enc_plates_iff) and the high authority controller working in the frame of the struts (developed in Section ref:sec:test_nhexa_hac_iff_struts_controller) are implemented experimentally. - -Two reference tracking experiments are performed to evaluate the stability and performances of the implemented control. - -#+begin_src matlab -%% Load the experimental data -load('hac_iff_struts_yz_scans.mat', 't', 'de') -#+end_src - -#+begin_src matlab :exports none -%% Reset initial time -t = t - t(1); -#+end_src - -The position of the top-platform is estimated using the Jacobian matrix: -#+begin_src matlab -%% Pose of the top platform from the encoder values -load('jacobian.mat', 'J'); -Xe = [inv(J)*de']'; -#+end_src - -#+begin_src matlab -%% Generate the Y-Z trajectory scan -Rx_yz = generateYZScanTrajectory(... - 'y_tot', 4e-6, ... % Length of Y scans [m] - 'z_tot', 8e-6, ... % Total Z distance [m] - 'n', 5, ... % Number of Y scans - 'Ts', 1e-3, ... % Sampling Time [s] - 'ti', 1, ... % Time to go to initial position [s] - 'tw', 0, ... % Waiting time between each points [s] - 'ty', 0.6, ... % Time for a scan in Y [s] - 'tz', 0.2); % Time for a scan in Z [s] -#+end_src - -The reference path as well as the measured position are partially shown in the Y-Z plane in Figure ref:fig:test_nhexa_yz_scans_exp_results_first_K. -#+begin_src matlab :exports none -%% Position and reference signal in the Y-Z plane -figure; -tiledlayout(1, 3, 'TileSpacing', 'Compact', 'Padding', 'None'); - -ax1 = nexttile; -hold on; -plot(1e6*Xe(t>2,2), 1e6*Xe(t>2,3)); -plot(1e6*Rx_yz(:,3), 1e6*Rx_yz(:,4), '--'); -hold off; -xlabel('Y [$\mu m$]'); ylabel('Z [$\mu m$]'); -xlim([-2.05, 2.05]); ylim([-4.1, 4.1]); -axis equal; - -ax2 = nexttile([1,2]); -hold on; -plot(1e6*Xe(:,2), 1e6*Xe(:,3), ... - 'DisplayName', '$\mathcal{X}_n$'); -plot(1e6*Rx_yz(:,3), 1e6*Rx_yz(:,4), '--', ... - 'DisplayName', '$r_{\mathcal{X}_n}$'); -hold off; -legend('location', 'northwest'); -xlabel('Y [$\mu m$]'); ylabel('Z [$\mu m$]'); -axis equal; -xlim([1.6, 2.1]); ylim([-4.1, -3.6]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/yz_scans_exp_results_first_K.pdf', 'width', 'wide', 'height', 'tall'); -#+end_src - -#+name: fig:test_nhexa_yz_scans_exp_results_first_K -#+caption: Measured position $\bm{\mathcal{X}}_n$ and reference signal $\bm{r}_{\mathcal{X}_n}$ in the Y-Z plane - Zoom on a change of direction -#+RESULTS: -[[file:figs/test_nhexa_yz_scans_exp_results_first_K.png]] - -#+begin_important -It is clear from Figure ref:fig:test_nhexa_yz_scans_exp_results_first_K that the position of the nano-hexapod effectively tracks to reference signal. -However, oscillations with amplitudes as large as 50nm can be observe. - -It turns out that the frequency of these oscillations is 100Hz which is corresponding to the crossover frequency of the High Authority Control loop. -This clearly indicates poor stability margins. -In the next section, the controller is re-designed to improve the stability margins. -#+end_important - -*** Controller with increased stability margins -The High Authority Controller is re-designed in order to improve the stability margins. -#+begin_src matlab -%% Lead -a = 5; % Amount of phase lead / width of the phase lead / high frequency gain -wc = 2*pi*110; % Frequency with the maximum phase lead [rad/s] - -H_lead = (1 + s/(wc/sqrt(a)))/(1 + s/(wc*sqrt(a))); - -%% Low Pass Filter -H_lpf = 1/(1 + s/2/pi/300); - -%% Notch -gm = 0.02; -xi = 0.5; -wn = 2*pi*700; - -H_notch = (s^2 + 2*gm*xi*wn*s + wn^2)/(s^2 + 2*xi*wn*s + wn^2); - -%% HAC Controller -Khac_iff_struts = -2.2e4 * ... % Gain - H_lead * ... % Lead - H_lpf * ... % Lead - H_notch * ... % Notch - (2*pi*100/s) * ... % Integrator - eye(6); % 6x6 Diagonal -#+end_src - -#+begin_src matlab :exports none -%% Load the FRF of the transfer function from u to dL with IFF -frf_iff = load('frf_iff_vib_table_m.mat', 'f', 'Ts', 'G_dL'); -#+end_src - -#+begin_src matlab :exports none -%% Compute the Loop Gain -L_frf = pagemtimes(permute(frf_iff.G_dL{1}, [2 3 1]), squeeze(freqresp(Khac_iff_struts, frf_iff.f, 'Hz'))); -#+end_src - -The bode plot of the new loop gain is shown in Figure ref:fig:test_nhexa_hac_iff_plates_exp_loop_gain_redesigned_K. -#+begin_src matlab :exports none -%% Bode plot for the transfer function from u to dLm -freqs = 2*logspace(1, 3, 1000); - -figure; -tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); - -ax1 = nexttile([2,1]); -hold on; -% Diagonal Elements FRF -plot(frf_iff.f, abs(squeeze(L_frf(1,1,:))), 'color', colors(1,:), ... - 'DisplayName', 'Diagonal'); -for i = 2:6 - plot(frf_iff.f, abs(squeeze(L_frf(i,i,:))), 'color', colors(1,:), ... - 'HandleVisibility', 'off'); -end -plot(frf_iff.f, abs(squeeze(L_frf(1,2,:))), 'color', [colors(2,:), 0.2], ... - 'DisplayName', 'Off-Diag'); -for i = 1:5 - for j = i+1:6 - plot(frf_iff.f, abs(squeeze(L_frf(i,j,:))), 'color', [colors(2,:), 0.2], ... - 'HandleVisibility', 'off'); - end -end -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Loop Gain [-]'); set(gca, 'XTickLabel',[]); -ylim([1e-3, 1e2]); -legend('location', 'northeast'); - -ax2 = nexttile; -hold on; -for i =1:6 - plot(frf_iff.f, 180/pi*angle(squeeze(L_frf(i,i,:))), 'color', colors(1,:)); -end -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); -xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); -hold off; -yticks(-360:90:360); -ylim([-180, 180]); - -linkaxes([ax1,ax2],'x'); -xlim([1, 2e3]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/hac_iff_plates_exp_loop_gain_redesigned_K.pdf', 'width', 'wide', 'height', 'tall'); -#+end_src - -#+name: fig:test_nhexa_hac_iff_plates_exp_loop_gain_redesigned_K -#+caption: Loop Gain for the updated decentralized HAC controller -#+RESULTS: -[[file:figs/test_nhexa_hac_iff_plates_exp_loop_gain_redesigned_K.png]] - -This new controller is implemented experimentally and several tracking tests are performed. -#+begin_src matlab -%% Load Measurements -load('hac_iff_more_lead_nass_scan.mat', 't', 'de') -#+end_src - -#+begin_src matlab :exports none -%% Reset Time -t = t - t(1); -#+end_src - -The pose of the top platform is estimated from the encoder position using the Jacobian matrix. -#+begin_src matlab -%% Compute the pose of the top platform -load('jacobian.mat', 'J'); -Xe = [inv(J)*de']'; -#+end_src - -#+begin_src matlab :exports none -%% Load the reference path -load('reference_path.mat', 'Rx_nass') -#+end_src - -The measured motion as well as the trajectory are shown in Figure ref:fig:test_nhexa_nass_scans_first_test_exp. -#+begin_src matlab :exports none -%% Plot the X-Y-Z "NASS" trajectory -figure; -hold on; -plot3(Xe(1:100:end,1), Xe(1:100:end,2), Xe(1:100:end,3)) -plot3(Rx_nass(1:100:end,2), Rx_nass(1:100:end,3), Rx_nass(1:100:end,4)) -hold off; -xlabel('x [$\mu m$]'); ylabel('y [$\mu m$]'); zlabel('z [$\mu m$]'); -view(-13, 41) -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/nass_scans_first_test_exp.pdf', 'width', 'wide', 'height', 'tall'); -#+end_src - -#+name: fig:test_nhexa_nass_scans_first_test_exp -#+caption: Measured position $\bm{\mathcal{X}}_n$ and reference signal $\bm{r}_{\mathcal{X}_n}$ for the "NASS" trajectory -#+RESULTS: -[[file:figs/test_nhexa_nass_scans_first_test_exp.png]] - -The trajectory and measured motion are also shown in the X-Y plane in Figure ref:fig:test_nhexa_ref_track_nass_exp_hac_iff_struts. -#+begin_src matlab :exports none -%% Estimate when the hexpod is on top position and drawing the letters -i_top = Xe(:,3) > 1.9e-6; -i_rx = Rx_nass(:,4) > 0; -#+end_src - -#+begin_src matlab :exports none -%% Plot the reference as well as the measurement in the X-Y plane -figure; -tiledlayout(1, 3, 'TileSpacing', 'Compact', 'Padding', 'None'); - -ax1 = nexttile([1,2]); -hold on; -scatter(1e6*Xe(i_top,1), 1e6*Xe(i_top,2),'.'); -plot(1e6*Rx_nass(i_rx,2), 1e6*Rx_nass(i_rx,3), '--'); -hold off; -xlabel('X [$\mu m$]'); ylabel('Y [$\mu m$]'); -axis equal; -xlim([-10.5, 10.5]); ylim([-4.5, 4.5]); - -ax2 = nexttile; -hold on; -scatter(1e6*Xe(i_top,1), 1e6*Xe(i_top,2),'.'); -plot(1e6*Rx_nass(i_rx,2), 1e6*Rx_nass(i_rx,3), '--'); -hold off; -xlabel('X [$\mu m$]'); ylabel('Y [$\mu m$]'); -axis equal; -xlim([4.5, 4.7]); ylim([-0.15, 0.05]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/ref_track_nass_exp_hac_iff_struts.pdf', 'width', 'full', 'height', 'tall'); -#+end_src - -#+name: fig:test_nhexa_ref_track_nass_exp_hac_iff_struts -#+caption: Reference path and measured motion in the X-Y plane -#+RESULTS: -[[file:figs/test_nhexa_ref_track_nass_exp_hac_iff_struts.png]] - -The orientation errors during all the scans are shown in Figure ref:fig:test_nhexa_nass_ref_rx_ry. -#+begin_src matlab :exports none -%% Orientation Errors -figure; -hold on; -plot(t(t>20&t<20.1), 1e6*Xe(t>20&t<20.1,4), '-', 'DisplayName', '$\epsilon_{\theta_x}$'); -plot(t(t>20&t<20.1), 1e6*Xe(t>20&t<20.1,5), '-', 'DisplayName', '$\epsilon_{\theta_y}$'); -plot(t(t>20&t<20.1), 1e6*Xe(t>20&t<20.1,6), '-', 'DisplayName', '$\epsilon_{\theta_z}$'); -hold off; -xlabel('Time [s]'); ylabel('Orientation Error [$\mu$ rad]'); -legend('location', 'northeast'); -#+end_src - -#+begin_src matlab :exports none -%% Orientation Errors -figure; -hold on; -plot(1e9*Xe(100000:100:end,4), 1e9*Xe(100000:100:end,5), '.'); -th = 0:pi/50:2*pi; -xunit = 90 * cos(th); -yunit = 90 * sin(th); -plot(xunit, yunit, '--'); -hold off; -xlabel('$R_x$ [nrad]'); ylabel('$R_y$ [nrad]'); -xlim([-100, 100]); -ylim([-100, 100]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/nass_ref_rx_ry.pdf', 'width', 500, 'height', 500); -#+end_src - -#+name: fig:test_nhexa_nass_ref_rx_ry -#+caption: Orientation errors during the scan -#+RESULTS: -[[file:figs/test_nhexa_nass_ref_rx_ry.png]] - -#+begin_important -Using the updated High Authority Controller, the nano-hexapod can follow trajectories with high accuracy (the position errors are in the order of 50nm peak to peak, and the orientation errors 300nrad peak to peak). -#+end_important - -** Interaction Analysis and Decoupling -:PROPERTIES: -:header-args:matlab+: :tangle matlab/scripts/interaction_analysis_enc_plates.m -:END: -<> -*** Introduction :ignore: - -In this section, the interaction in the identified plant is estimated using the Relative Gain Array (RGA) [[cite:skogestad07_multiv_feedb_contr][Chap. 3.4]]. - -Then, several decoupling strategies are compared for the nano-hexapod. - -The RGA Matrix is defined as follow: -\begin{equation} - \text{RGA}(G(f)) = G(f) \times (G(f)^{-1})^T -\end{equation} - -Then, the RGA number is defined: -\begin{equation} -\text{RGA-num}(f) = \| \text{I - RGA(G(f))} \|_{\text{sum}} -\end{equation} - - -In this section, the plant with 2 added mass is studied. - -*** Matlab Init :noexport:ignore: -#+begin_src matlab -%% interaction_analysis_enc_plates.m -% Interaction analysis of several decoupling strategies -#+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 -%% Load the identified FRF and Simscape model -frf_iff = load('frf_iff_vib_table_m.mat', 'f', 'Ts', 'G_dL'); -sim_iff = load('sim_iff_vib_table_m.mat', 'G_dL'); -#+end_src - -*** Parameters -#+begin_src matlab -wc = 100; % Wanted crossover frequency [Hz] -[~, i_wc] = min(abs(frf_iff.f - wc)); % Indice corresponding to wc -#+end_src - -#+begin_src matlab -%% Plant to be decoupled -frf_coupled = frf_iff.G_dL{2}; -G_coupled = sim_iff.G_dL{2}; -#+end_src - -*** No Decoupling (Decentralized) -<> - -#+begin_src latex :file decoupling_arch_decentralized.pdf -\begin{tikzpicture} - \node[block] (G) {$\bm{G}$}; - - % Connections and labels - \draw[<-] (G.west) -- ++(-1.8, 0) node[above right]{$\bm{\tau}$}; - \draw[->] (G.east) -- ++( 1.8, 0) node[above left]{$d\bm{\mathcal{L}}$}; - - \begin{scope}[on background layer] - \node[fit={(G.south west) (G.north east)}, fill=black!10!white, draw, dashed, inner sep=16pt] (Gdec) {}; - \node[below right] at (Gdec.north west) {$\bm{G}_{\text{dec}}$}; - \end{scope} -\end{tikzpicture} -#+end_src - -#+name: fig:test_nhexa_decoupling_arch_decentralized -#+caption: Block diagram representing the plant. -#+RESULTS: -[[file:figs/test_nhexa_decoupling_arch_decentralized.png]] - -#+begin_src matlab :exports none -%% Decentralized Plant -figure; -tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); - -ax1 = nexttile([2,1]); -hold on; -for i = 1:5 - for j = i+1:6 - plot(frf_iff.f, abs(frf_coupled(:,i,j)), 'color', [0,0,0,0.2], ... - 'HandleVisibility', 'off'); - end -end -set(gca,'ColorOrderIndex',1) -for i = 1:6 - plot(frf_iff.f, abs(frf_coupled(:,i,i)), ... - 'DisplayName', sprintf('$y_%i/u_%i$', i, i)); -end -plot(frf_iff.f, abs(frf_coupled(:,1,2)), 'color', [0,0,0,0.2], ... - 'DisplayName', 'Coupling'); -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Amplitude'); set(gca, 'XTickLabel',[]); -ylim([1e-9, 1e-4]); -legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 3); - -ax2 = nexttile; -hold on; -for i =1:6 - plot(frf_iff.f, 180/pi*angle(frf_coupled(:,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); -ylim([-180, 180]); - -linkaxes([ax1,ax2],'x'); -xlim([10, 1e3]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/interaction_decentralized_plant.pdf', 'width', 'wide', 'height', 'tall'); -#+end_src - -#+name: fig:test_nhexa_interaction_decentralized_plant -#+caption: Bode Plot of the decentralized plant (diagonal and off-diagonal terms) -#+RESULTS: -[[file:figs/test_nhexa_interaction_decentralized_plant.png]] - -#+begin_src matlab :exports none -%% Decentralized RGA -RGA_dec = zeros(size(frf_coupled)); -for i = 1:length(frf_iff.f) - RGA_dec(i,:,:) = squeeze(frf_coupled(i,:,:)).*inv(squeeze(frf_coupled(i,:,:))).'; -end - -RGA_dec_sum = zeros(length(frf_iff), 1); -for i = 1:length(frf_iff.f) - RGA_dec_sum(i) = sum(sum(abs(eye(6) - squeeze(RGA_dec(i,:,:))))); -end -#+end_src - -#+begin_src matlab :exports none -%% RGA for Decentralized plant -figure; -plot(frf_iff.f, RGA_dec_sum, 'k-'); -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -xlabel('Frequency [Hz]'); ylabel('RGA Number'); -xlim([10, 1e3]); ylim([1e-2, 1e2]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/interaction_rga_decentralized.pdf', 'width', 'wide', 'height', 'normal'); -#+end_src - -#+name: fig:test_nhexa_interaction_rga_decentralized -#+caption: RGA number for the decentralized plant -#+RESULTS: -[[file:figs/test_nhexa_interaction_rga_decentralized.png]] - -*** Static Decoupling -<> - -#+begin_src latex :file decoupling_arch_static.pdf -\begin{tikzpicture} - \node[block] (G) {$\bm{G}$}; - \node[block, left=0.8 of G] (Ginv) {$\bm{\hat{G}}(j0)^{-1}$}; - - % Connections and labels - \draw[<-] (Ginv.west) -- ++(-1.8, 0) node[above right]{$\bm{u}$}; - \draw[->] (Ginv.east) -- (G.west) node[above left]{$\bm{\tau}$}; - \draw[->] (G.east) -- ++( 1.8, 0) node[above left]{$d\bm{\mathcal{L}}$}; - - \begin{scope}[on background layer] - \node[fit={(Ginv.south west) (G.north east)}, fill=black!10!white, draw, dashed, inner sep=16pt] (Gx) {}; - \node[below right] at (Gx.north west) {$\bm{G}_{\text{static}}$}; - \end{scope} -\end{tikzpicture} -#+end_src - -#+name: fig:test_nhexa_decoupling_arch_static -#+caption: Decoupling using the inverse of the DC gain of the plant -#+RESULTS: -[[file:figs/test_nhexa_decoupling_arch_static.png]] - -The DC gain is evaluated from the model as be have bad low frequency identification. - -#+begin_src matlab :exports none -%% Compute the inverse of the DC gain -G_model = G_coupled; -G_model.outputdelay = 0; % necessary for further inversion -dc_inv = inv(dcgain(G_model)); - -%% Compute the inversed plant -G_dL_sta = zeros(size(frf_coupled)); -for i = 1:length(frf_iff.f) - G_dL_sta(i,:,:) = squeeze(frf_coupled(i,:,:))*dc_inv; -end -#+end_src - -#+begin_src matlab :exports results :results value table replace :tangle no -data2orgtable(dc_inv, {}, {}, ' %.1f '); -#+end_src - -#+RESULTS: -| -62011.5 | 3910.6 | 4299.3 | 660.7 | -4016.5 | -4373.6 | -| 3914.4 | -61991.2 | -4356.8 | -4019.2 | 640.2 | 4281.6 | -| -4020.0 | -4370.5 | -62004.5 | 3914.6 | 4295.8 | 653.8 | -| 660.9 | 4292.4 | 3903.3 | -62012.2 | -4366.5 | -4008.9 | -| 4302.8 | 655.6 | -4025.8 | -4377.8 | -62006.0 | 3919.7 | -| -4377.9 | -4013.2 | 668.6 | 4303.7 | 3906.8 | -62019.3 | - -#+begin_src matlab :exports none -%% Bode plot of the static decoupled plant -figure; -tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); - -ax1 = nexttile([2,1]); -hold on; -for i = 1:5 - for j = i+1:6 - plot(frf_iff.f, abs(G_dL_sta(:,i,j)), 'color', [0,0,0,0.2], ... - 'HandleVisibility', 'off'); - end -end -set(gca,'ColorOrderIndex',1) -for i = 1:6 - plot(frf_iff.f, abs(G_dL_sta(:,i,i)), ... - 'DisplayName', sprintf('$y_%i/u_%i$', i, i)); -end -plot(frf_iff.f, abs(G_dL_sta(:,1,2)), 'color', [0,0,0,0.2], ... - 'DisplayName', 'Coupling'); -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Amplitude'); set(gca, 'XTickLabel',[]); -ylim([1e-3, 1e1]); -legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 2); - -ax2 = nexttile; -hold on; -for i =1:6 - plot(frf_iff.f, 180/pi*angle(G_dL_sta(:,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); -ylim([-180, 180]); - -linkaxes([ax1,ax2],'x'); -xlim([10, 1e3]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/interaction_static_dec_plant.pdf', 'width', 'wide', 'height', 'tall'); -#+end_src - -#+name: fig:test_nhexa_interaction_static_dec_plant -#+caption: Bode Plot of the static decoupled plant -#+RESULTS: -[[file:figs/test_nhexa_interaction_static_dec_plant.png]] - -#+begin_src matlab :exports none -%% Compute RGA Matrix -RGA_sta = zeros(size(frf_coupled)); -for i = 1:length(frf_iff.f) - RGA_sta(i,:,:) = squeeze(G_dL_sta(i,:,:)).*inv(squeeze(G_dL_sta(i,:,:))).'; -end - -%% Compute RGA-number -RGA_sta_sum = zeros(length(frf_iff), 1); -for i = 1:size(RGA_sta, 1) - RGA_sta_sum(i) = sum(sum(abs(eye(6) - squeeze(RGA_sta(i,:,:))))); -end -#+end_src - -#+begin_src matlab :exports none -%% Plot the RGA-number for statically decoupled plant -figure; -plot(frf_iff.f, RGA_sta_sum, 'k-'); -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -xlabel('Frequency [Hz]'); ylabel('RGA Number'); -xlim([10, 1e3]); ylim([1e-2, 1e2]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/interaction_rga_static_dec.pdf', 'width', 'wide', 'height', 'normal'); -#+end_src - -#+name: fig:test_nhexa_interaction_rga_static_dec -#+caption: RGA number for the statically decoupled plant -#+RESULTS: -[[file:figs/test_nhexa_interaction_rga_static_dec.png]] - -*** Decoupling at the Crossover -<> - -#+begin_src latex :file decoupling_arch_crossover.pdf -\begin{tikzpicture} - \node[block] (G) {$\bm{G}$}; - \node[block, left=0.8 of G] (Ginv) {$\bm{\hat{G}}(j\omega_c)^{-1}$}; - - % Connections and labels - \draw[<-] (Ginv.west) -- ++(-1.8, 0) node[above right]{$\bm{u}$}; - \draw[->] (Ginv.east) -- (G.west) node[above left]{$\bm{\tau}$}; - \draw[->] (G.east) -- ++( 1.8, 0) node[above left]{$d\bm{\mathcal{L}}$}; - - \begin{scope}[on background layer] - \node[fit={(Ginv.south west) (G.north east)}, fill=black!10!white, draw, dashed, inner sep=16pt] (Gx) {}; - \node[below right] at (Gx.north west) {$\bm{G}_{\omega_c}$}; - \end{scope} -\end{tikzpicture} -#+end_src - -#+name: fig:test_nhexa_decoupling_arch_crossover -#+caption: Decoupling using the inverse of a dynamical model $\bm{\hat{G}}$ of the plant dynamics $\bm{G}$ -#+RESULTS: -[[file:figs/test_nhexa_decoupling_arch_crossover.png]] - -#+begin_src matlab :exports none -%% Take complex matrix corresponding to the plant at 100Hz -V = squeeze(frf_coupled(i_wc,:,:)); - -%% Real approximation of inv(G(100Hz)) -D = pinv(real(V'*V)); -H1 = D*real(V'*diag(exp(1j*angle(diag(V*D*V.'))/2))); - -%% Compute the decoupled plant -G_dL_wc = zeros(size(frf_coupled)); -for i = 1:length(frf_iff.f) - G_dL_wc(i,:,:) = squeeze(frf_coupled(i,:,:))*H1; -end -#+end_src - -#+begin_src matlab :exports results :results value table replace :tangle no -data2orgtable(H1, {}, {}, ' %.1f '); -#+end_src - -#+RESULTS: -| 67229.8 | 3769.3 | -13704.6 | -23084.8 | -6318.2 | 23378.7 | -| 3486.2 | 67708.9 | 23220.0 | -6314.5 | -22699.8 | -14060.6 | -| -5731.7 | 22471.7 | 66701.4 | 3070.2 | -13205.6 | -21944.6 | -| -23305.5 | -14542.6 | 2743.2 | 70097.6 | 24846.8 | -5295.0 | -| -14882.9 | -22957.8 | -5344.4 | 25786.2 | 70484.6 | 2979.9 | -| 24353.3 | -5195.2 | -22449.0 | -14459.2 | 2203.6 | 69484.2 | - -#+begin_src matlab :exports none -%% Bode plot of the plant decoupled at the crossover -figure; -tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); - -ax1 = nexttile([2,1]); -hold on; -for i = 1:5 - for j = i+1:6 - plot(frf_iff.f, abs(G_dL_wc(:,i,j)), 'color', [0,0,0,0.2], ... - 'HandleVisibility', 'off'); - end -end -for i = 1:6 - plot(frf_iff.f, abs(G_dL_wc(:,i,i)), ... - 'DisplayName', sprintf('$y_%i/u_%i$', i, i)); -end -plot(frf_iff.f, abs(G_dL_wc(:,1,2)), 'color', [0,0,0,0.2], ... - 'DisplayName', 'Coupling'); -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Amplitude $d_L/V_a$ [m/V]'); set(gca, 'XTickLabel',[]); -ylim([1e-3, 1e1]); -legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 2); - -ax2 = nexttile; -hold on; -for i =1:6 - plot(frf_iff.f, 180/pi*angle(G_dL_wc(:,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); -ylim([-180, 180]); - -linkaxes([ax1,ax2],'x'); -xlim([10, 1e3]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/interaction_wc_plant.pdf', 'width', 'wide', 'height', 'tall'); -#+end_src - -#+name: fig:test_nhexa_interaction_wc_plant -#+caption: Bode Plot of the plant decoupled at the crossover -#+RESULTS: -[[file:figs/test_nhexa_interaction_wc_plant.png]] - -#+begin_src matlab -%% Compute RGA Matrix -RGA_wc = zeros(size(frf_coupled)); -for i = 1:length(frf_iff.f) - RGA_wc(i,:,:) = squeeze(G_dL_wc(i,:,:)).*inv(squeeze(G_dL_wc(i,:,:))).'; -end - -%% Compute RGA-number -RGA_wc_sum = zeros(size(RGA_wc, 1), 1); -for i = 1:size(RGA_wc, 1) - RGA_wc_sum(i) = sum(sum(abs(eye(6) - squeeze(RGA_wc(i,:,:))))); -end -#+end_src - -#+begin_src matlab :exports none -%% Plot the RGA-Number for the plant decoupled at crossover -figure; -plot(frf_iff.f, RGA_wc_sum, 'k-'); -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -xlabel('Frequency [Hz]'); ylabel('RGA Number'); -xlim([10, 1e3]); ylim([1e-2, 1e2]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/interaction_rga_wc.pdf', 'width', 'wide', 'height', 'normal'); -#+end_src - -#+name: fig:test_nhexa_interaction_rga_wc -#+caption: RGA number for the plant decoupled at the crossover -#+RESULTS: -[[file:figs/test_nhexa_interaction_rga_wc.png]] - -*** SVD Decoupling -<> - -#+begin_src latex :file decoupling_arch_svd.pdf -\begin{tikzpicture} - \node[block] (G) {$\bm{G}$}; - - \node[block, left=0.8 of G.west] (V) {$V^{-T}$}; - \node[block, right=0.8 of G.east] (U) {$U^{-1}$}; - - % Connections and labels - \draw[<-] (V.west) -- ++(-1.0, 0) node[above right]{$u$}; - \draw[->] (V.east) -- (G.west) node[above left]{$\bm{\tau}$}; - \draw[->] (G.east) -- (U.west) node[above left]{$d\bm{\mathcal{L}}$}; - \draw[->] (U.east) -- ++( 1.0, 0) node[above left]{$y$}; - - \begin{scope}[on background layer] - \node[fit={(V.south west) (G.north-|U.east)}, fill=black!10!white, draw, dashed, inner sep=14pt] (Gsvd) {}; - \node[below right] at (Gsvd.north west) {$\bm{G}_{SVD}$}; - \end{scope} -\end{tikzpicture} -#+end_src - -#+name: fig:test_nhexa_decoupling_arch_svd -#+caption: Decoupling using the Singular Value Decomposition -#+RESULTS: -[[file:figs/test_nhexa_decoupling_arch_svd.png]] - -#+begin_src matlab :exports none -%% Take complex matrix corresponding to the plant at 100Hz -V = squeeze(frf_coupled(i_wc,:,:)); - -%% Real approximation of G(100Hz) -D = pinv(real(V'*V)); -H1 = pinv(D*real(V'*diag(exp(1j*angle(diag(V*D*V.'))/2)))); - -%% Singular Value Decomposition -[U,S,V] = svd(H1); - -%% Compute the decoupled plant using SVD -G_dL_svd = zeros(size(frf_coupled)); -for i = 1:length(frf_iff.f) - G_dL_svd(i,:,:) = inv(U)*squeeze(frf_coupled(i,:,:))*inv(V'); -end -#+end_src - -#+begin_src matlab :exports none -%% Bode Plot of the SVD decoupled plant -figure; -tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); - -ax1 = nexttile([2,1]); -hold on; -for i = 1:5 - for j = i+1:6 - plot(frf_iff.f, abs(G_dL_svd(:,i,j)), 'color', [0,0,0,0.2], ... - 'HandleVisibility', 'off'); - end -end -set(gca,'ColorOrderIndex',1) -for i = 1:6 - plot(frf_iff.f, abs(G_dL_svd(:,i,i)), ... - 'DisplayName', sprintf('$y_%i/u_%i$', i, i)); -end -plot(frf_iff.f, abs(G_dL_svd(:,1,2)), 'color', [0,0,0,0.2], ... - 'DisplayName', 'Coupling'); -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Amplitude'); set(gca, 'XTickLabel',[]); -ylim([1e-9, 1e-4]); -legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 3); - -ax2 = nexttile; -hold on; -for i =1:6 - plot(frf_iff.f, 180/pi*angle(G_dL_svd(:,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); -ylim([-180, 180]); - -linkaxes([ax1,ax2],'x'); -xlim([10, 1e3]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/interaction_svd_plant.pdf', 'width', 'wide', 'height', 'tall'); -#+end_src - -#+name: fig:test_nhexa_interaction_svd_plant -#+caption: Bode Plot of the plant decoupled using the Singular Value Decomposition -#+RESULTS: -[[file:figs/test_nhexa_interaction_svd_plant.png]] - -#+begin_src matlab -%% Compute the RGA matrix for the SVD decoupled plant -RGA_svd = zeros(size(frf_coupled)); -for i = 1:length(frf_iff.f) - RGA_svd(i,:,:) = squeeze(G_dL_svd(i,:,:)).*inv(squeeze(G_dL_svd(i,:,:))).'; -end - -%% Compute the RGA-number -RGA_svd_sum = zeros(size(RGA_svd, 1), 1); -for i = 1:length(frf_iff.f) - RGA_svd_sum(i) = sum(sum(abs(eye(6) - squeeze(RGA_svd(i,:,:))))); -end -#+end_src - -#+begin_src matlab -%% RGA Number for the SVD decoupled plant -figure; -plot(frf_iff.f, RGA_svd_sum, 'k-'); -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -xlabel('Frequency [Hz]'); ylabel('RGA Number'); -xlim([10, 1e3]); ylim([1e-2, 1e2]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/interaction_rga_svd.pdf', 'width', 'wide', 'height', 'normal'); -#+end_src - -#+name: fig:test_nhexa_interaction_rga_svd -#+caption: RGA number for the plant decoupled using the SVD -#+RESULTS: -[[file:figs/test_nhexa_interaction_rga_svd.png]] - -*** Dynamic decoupling -<> - -#+begin_src latex :file decoupling_arch_dynamic.pdf -\begin{tikzpicture} - \node[block] (G) {$\bm{G}$}; - \node[block, left=0.8 of G] (Ginv) {$\bm{\hat{G}}^{-1}$}; - - % Connections and labels - \draw[<-] (Ginv.west) -- ++(-1.8, 0) node[above right]{$\bm{u}$}; - \draw[->] (Ginv.east) -- (G.west) node[above left]{$\bm{\tau}$}; - \draw[->] (G.east) -- ++( 1.8, 0) node[above left]{$d\bm{\mathcal{L}}$}; - - \begin{scope}[on background layer] - \node[fit={(Ginv.south west) (G.north east)}, fill=black!10!white, draw, dashed, inner sep=16pt] (Gx) {}; - \node[below right] at (Gx.north west) {$\bm{G}_{\text{inv}}$}; - \end{scope} -\end{tikzpicture} -#+end_src - -#+name: fig:test_nhexa_decoupling_arch_dynamic -#+caption: Decoupling using the inverse of a dynamical model $\bm{\hat{G}}$ of the plant dynamics $\bm{G}$ -#+RESULTS: -[[file:figs/test_nhexa_decoupling_arch_dynamic.png]] - -#+begin_src matlab :exports none -%% Compute the plant inverse from the model -G_model = G_coupled; -G_model.outputdelay = 0; % necessary for further inversion -G_inv = inv(G_model); - -%% Compute the decoupled plant -G_dL_inv = zeros(size(frf_coupled)); -for i = 1:length(frf_iff.f) - G_dL_inv(i,:,:) = squeeze(frf_coupled(i,:,:))*squeeze(evalfr(G_inv, 1j*2*pi*frf_iff.f(i))); -end -#+end_src - -#+begin_src matlab :exports none -%% Bode plot of the decoupled plant by full inversion -figure; -tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); - -ax1 = nexttile([2,1]); -hold on; -for i = 1:5 - for j = i+1:6 - plot(frf_iff.f, abs(G_dL_inv(:,i,j)), 'color', [0,0,0,0.2], ... - 'HandleVisibility', 'off'); - end -end -set(gca,'ColorOrderIndex',1) -for i = 1:6 - plot(frf_iff.f, abs(G_dL_inv(:,i,i)), ... - 'DisplayName', sprintf('$y_%i/u_%i$', i, i)); -end -plot(frf_iff.f, abs(G_dL_inv(:,1,2)), 'color', [0,0,0,0.2], ... - 'DisplayName', 'Coupling'); -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Amplitude'); set(gca, 'XTickLabel',[]); -ylim([1e-4, 1e1]); -legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 2); - -ax2 = nexttile; -hold on; -for i =1:6 - plot(frf_iff.f, 180/pi*angle(G_dL_inv(:,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); -ylim([-180, 180]); - -linkaxes([ax1,ax2],'x'); -xlim([10, 1e3]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/interaction_dynamic_dec_plant.pdf', 'width', 'wide', 'height', 'tall'); -#+end_src - -#+name: fig:test_nhexa_interaction_dynamic_dec_plant -#+caption: Bode Plot of the dynamically decoupled plant -#+RESULTS: -[[file:figs/test_nhexa_interaction_dynamic_dec_plant.png]] - -#+begin_src matlab :exports none -%% Compute the RGA matrix for the inverse based decoupled plant -RGA_inv = zeros(size(frf_coupled)); -for i = 1:length(frf_iff.f) - RGA_inv(i,:,:) = squeeze(G_dL_inv(i,:,:)).*inv(squeeze(G_dL_inv(i,:,:))).'; -end - -%% Compute the RGA-number -RGA_inv_sum = zeros(size(RGA_inv, 1), 1); -for i = 1:size(RGA_inv, 1) - RGA_inv_sum(i) = sum(sum(abs(eye(6) - squeeze(RGA_inv(i,:,:))))); -end -#+end_src - -#+begin_src matlab :exports none -%% RGA Number for the decoupled plant using full inversion -figure; -plot(frf_iff.f, RGA_inv_sum, 'k-'); -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -xlabel('Frequency [Hz]'); ylabel('RGA Number'); -xlim([10, 1e3]); ylim([1e-2, 1e2]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/interaction_rga_dynamic_dec.pdf', 'width', 'wide', 'height', 'normal'); -#+end_src - -#+name: fig:test_nhexa_interaction_rga_dynamic_dec -#+caption: RGA number for the dynamically decoupled plant -#+RESULTS: -[[file:figs/test_nhexa_interaction_rga_dynamic_dec.png]] - -*** Jacobian Decoupling - Center of Stiffness -<> - -#+begin_src latex :file decoupling_arch_jacobian_cok.pdf -\begin{tikzpicture} - \node[block] (G) {$\bm{G}$}; - \node[block, left=0.8 of G] (Jt) {$J_{s,\{K\}}^{-T}$}; - \node[block, right=0.8 of G] (Ja) {$J_{a,\{K\}}^{-1}$}; - - % Connections and labels - \draw[<-] (Jt.west) -- ++(-1.8, 0) node[above right]{$\bm{\mathcal{F}}_{\{K\}}$}; - \draw[->] (Jt.east) -- (G.west) node[above left]{$\bm{\tau}$}; - \draw[->] (G.east) -- (Ja.west) node[above left]{$d\bm{\mathcal{L}}$}; - \draw[->] (Ja.east) -- ++( 1.8, 0) node[above left]{$\bm{\mathcal{X}}_{\{K\}}$}; - - \begin{scope}[on background layer] - \node[fit={(Jt.south west) (Ja.north east)}, fill=black!10!white, draw, dashed, inner sep=16pt] (Gx) {}; - \node[below right] at (Gx.north west) {$\bm{G}_{\{K\}}$}; - \end{scope} -\end{tikzpicture} -#+end_src - -#+name: fig:test_nhexa_decoupling_arch_jacobian_cok -#+caption: Decoupling using Jacobian matrices evaluated at the Center of Stiffness -#+RESULTS: -[[file:figs/test_nhexa_decoupling_arch_jacobian_cok.png]] - -#+begin_src matlab :exports none -%% Initialize the Nano-Hexapod -n_hexapod = initializeNanoHexapodFinal('MO_B', -42e-3, ... - 'motion_sensor_type', 'plates'); - -%% Get the Jacobians -J_cok = n_hexapod.geometry.J; -Js_cok = n_hexapod.geometry.Js; - -%% Decouple plant using Jacobian (CoM) -G_dL_J_cok = zeros(size(frf_coupled)); -for i = 1:length(frf_iff.f) - G_dL_J_cok(i,:,:) = inv(Js_cok)*squeeze(frf_coupled(i,:,:))*inv(J_cok'); -end -#+end_src - -The obtained plant is shown in Figure ref:fig:test_nhexa_interaction_J_cok_plant_not_normalized. -We can see that the stiffness in the $x$, $y$ and $z$ directions are equal, which is due to the cubic architecture of the Stewart platform. - -#+begin_src matlab :exports none -%% Bode Plot of the SVD decoupled plant -figure; -tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); - -ax1 = nexttile([2,1]); -hold on; -for i = 1:5 - for j = i+1:6 - plot(frf_iff.f, abs(G_dL_J_cok(:,i,j)), 'color', [0,0,0,0.2], ... - 'HandleVisibility', 'off'); - end -end -set(gca,'ColorOrderIndex',1) -plot(frf_iff.f, abs(G_dL_J_cok(:,1,1)), ... - 'DisplayName', '$D_x/\tilde{\mathcal{F}}_x$'); -plot(frf_iff.f, abs(G_dL_J_cok(:,2,2)), ... - 'DisplayName', '$D_y/\tilde{\mathcal{F}}_y$'); -plot(frf_iff.f, abs(G_dL_J_cok(:,3,3)), ... - 'DisplayName', '$D_z/\tilde{\mathcal{F}}_z$'); -plot(frf_iff.f, abs(G_dL_J_cok(:,4,4)), ... - 'DisplayName', '$R_x/\tilde{\mathcal{M}}_x$'); -plot(frf_iff.f, abs(G_dL_J_cok(:,5,5)), ... - 'DisplayName', '$R_y/\tilde{\mathcal{M}}_y$'); -plot(frf_iff.f, abs(G_dL_J_cok(:,6,6)), ... - 'DisplayName', '$R_z/\tilde{\mathcal{M}}_z$'); -plot(frf_iff.f, abs(G_dL_J_cok(:,1,2)), 'color', [0,0,0,0.2], ... - 'DisplayName', 'Coupling'); -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Amplitude'); set(gca, 'XTickLabel',[]); -ylim([1e-8, 2e-2]); -legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 3); - -ax2 = nexttile; -hold on; -for i =1:6 - plot(frf_iff.f, 180/pi*angle(G_dL_J_cok(:,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); -ylim([-180, 180]); - -linkaxes([ax1,ax2],'x'); -xlim([10, 1e3]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/interaction_J_cok_plant_not_normalized.pdf', 'width', 'wide', 'height', 'tall'); -#+end_src - -#+name: fig:test_nhexa_interaction_J_cok_plant_not_normalized -#+caption: Bode Plot of the plant decoupled using the Jacobian evaluated at the "center of stiffness" -#+RESULTS: -[[file:figs/test_nhexa_interaction_J_cok_plant_not_normalized.png]] - -Because the plant in translation and rotation has very different gains, we choose to normalize the plant inputs such that the gain of the diagonal term is equal to $1$ at 100Hz. - -The results is shown in Figure ref:fig:test_nhexa_interaction_J_cok_plant. -#+begin_src matlab :exports none -%% Normalize the plant input -[~, i_100] = min(abs(frf_iff.f - 100)); -input_normalize = diag(1./diag(abs(squeeze(G_dL_J_cok(i_100,:,:))))); - -for i = 1:length(frf_iff.f) - G_dL_J_cok(i,:,:) = squeeze(G_dL_J_cok(i,:,:))*input_normalize; -end -#+end_src - -#+begin_src matlab :exports none -%% Bode Plot of the SVD decoupled plant -figure; -tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); - -ax1 = nexttile([2,1]); -hold on; -for i = 1:5 - for j = i+1:6 - plot(frf_iff.f, abs(G_dL_J_cok(:,i,j)), 'color', [0,0,0,0.2], ... - 'HandleVisibility', 'off'); - end -end -set(gca,'ColorOrderIndex',1) -plot(frf_iff.f, abs(G_dL_J_cok(:,1,1)), ... - 'DisplayName', '$D_x/\tilde{\mathcal{F}}_x$'); -plot(frf_iff.f, abs(G_dL_J_cok(:,2,2)), ... - 'DisplayName', '$D_y/\tilde{\mathcal{F}}_y$'); -plot(frf_iff.f, abs(G_dL_J_cok(:,3,3)), ... - 'DisplayName', '$D_z/\tilde{\mathcal{F}}_z$'); -plot(frf_iff.f, abs(G_dL_J_cok(:,4,4)), ... - 'DisplayName', '$R_x/\tilde{\mathcal{M}}_x$'); -plot(frf_iff.f, abs(G_dL_J_cok(:,5,5)), ... - 'DisplayName', '$R_y/\tilde{\mathcal{M}}_y$'); -plot(frf_iff.f, abs(G_dL_J_cok(:,6,6)), ... - 'DisplayName', '$R_z/\tilde{\mathcal{M}}_z$'); -plot(frf_iff.f, abs(G_dL_J_cok(:,1,2)), 'color', [0,0,0,0.2], ... - 'DisplayName', 'Coupling'); -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Amplitude'); set(gca, 'XTickLabel',[]); -ylim([1e-4, 1e1]); -legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 2); - -ax2 = nexttile; -hold on; -for i =1:6 - plot(frf_iff.f, 180/pi*angle(G_dL_J_cok(:,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); -ylim([-180, 180]); - -linkaxes([ax1,ax2],'x'); -xlim([10, 1e3]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/interaction_J_cok_plant.pdf', 'width', 'wide', 'height', 'tall'); -#+end_src - -#+name: fig:test_nhexa_interaction_J_cok_plant -#+caption: Bode Plot of the plant decoupled using the Jacobian evaluated at the "center of stiffness" -#+RESULTS: -[[file:figs/test_nhexa_interaction_J_cok_plant.png]] - -#+begin_src matlab :exports none -%% Compute RGA Matrix -RGA_cok = zeros(size(frf_coupled)); -for i = 1:length(frf_iff.f) - RGA_cok(i,:,:) = squeeze(G_dL_J_cok(i,:,:)).*inv(squeeze(G_dL_J_cok(i,:,:))).'; -end - -%% Compute RGA-number -RGA_cok_sum = zeros(length(frf_iff.f), 1); -for i = 1:length(frf_iff.f) - RGA_cok_sum(i) = sum(sum(abs(eye(6) - squeeze(RGA_cok(i,:,:))))); -end -#+end_src - -#+begin_src matlab :exports none -%% Plot the RGA-Number for the Jacobian (CoK) decoupled plant -figure; -plot(frf_iff.f, RGA_cok_sum, 'k-'); -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -xlabel('Frequency [Hz]'); ylabel('RGA Number'); -xlim([10, 1e3]); ylim([1e-2, 1e2]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/interaction_rga_J_cok.pdf', 'width', 'wide', 'height', 'normal'); -#+end_src - -#+name: fig:test_nhexa_interaction_rga_J_cok -#+caption: RGA number for the plant decoupled using the Jacobian evaluted at the Center of Stiffness -#+RESULTS: -[[file:figs/test_nhexa_interaction_rga_J_cok.png]] - -*** Jacobian Decoupling - Center of Mass -<> - -#+begin_src latex :file decoupling_arch_jacobian_com.pdf -\begin{tikzpicture} - \node[block] (G) {$\bm{G}$}; - \node[block, left=0.8 of G] (Jt) {$J_{s,\{M\}}^{-T}$}; - \node[block, right=0.8 of G] (Ja) {$J_{a,\{M\}}^{-1}$}; - - % Connections and labels - \draw[<-] (Jt.west) -- ++(-1.8, 0) node[above right]{$\bm{\mathcal{F}}_{\{M\}}$}; - \draw[->] (Jt.east) -- (G.west) node[above left]{$\bm{\tau}$}; - \draw[->] (G.east) -- (Ja.west) node[above left]{$d\bm{\mathcal{L}}$}; - \draw[->] (Ja.east) -- ++( 1.8, 0) node[above left]{$\bm{\mathcal{X}}_{\{M\}}$}; - - \begin{scope}[on background layer] - \node[fit={(Jt.south west) (Ja.north east)}, fill=black!10!white, draw, dashed, inner sep=16pt] (Gx) {}; - \node[below right] at (Gx.north west) {$\bm{G}_{\{M\}}$}; - \end{scope} -\end{tikzpicture} -#+end_src - -#+name: fig:test_nhexa_decoupling_arch_jacobian_com -#+caption: Decoupling using Jacobian matrices evaluated at the Center of Mass -#+RESULTS: -[[file:figs/test_nhexa_decoupling_arch_jacobian_com.png]] - -#+begin_src matlab :exports none -%% Initialize the Nano-Hexapod -n_hexapod = initializeNanoHexapodFinal('MO_B', 25e-3, ... - 'motion_sensor_type', 'plates'); - -%% Get the Jacobians -J_com = n_hexapod.geometry.J; -Js_com = n_hexapod.geometry.Js; - -%% Decouple plant using Jacobian (CoM) -G_dL_J_com = zeros(size(frf_coupled)); -for i = 1:length(frf_iff.f) - G_dL_J_com(i,:,:) = inv(Js_com)*squeeze(frf_coupled(i,:,:))*inv(J_com'); -end - -%% Normalize the plant input -[~, i_100] = min(abs(frf_iff.f - 100)); -input_normalize = diag(1./diag(abs(squeeze(G_dL_J_com(i_100,:,:))))); - -for i = 1:length(frf_iff.f) - G_dL_J_com(i,:,:) = squeeze(G_dL_J_com(i,:,:))*input_normalize; -end -#+end_src - -#+begin_src matlab :exports none -%% Bode Plot of the SVD decoupled plant -figure; -tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); - -ax1 = nexttile([2,1]); -hold on; -for i = 1:5 - for j = i+1:6 - plot(frf_iff.f, abs(G_dL_J_com(:,i,j)), 'color', [0,0,0,0.2], ... - 'HandleVisibility', 'off'); - end -end -set(gca,'ColorOrderIndex',1) -plot(frf_iff.f, abs(G_dL_J_com(:,1,1)), ... - 'DisplayName', '$D_x/\tilde{\mathcal{F}}_x$'); -plot(frf_iff.f, abs(G_dL_J_com(:,2,2)), ... - 'DisplayName', '$D_y/\tilde{\mathcal{F}}_y$'); -plot(frf_iff.f, abs(G_dL_J_com(:,3,3)), ... - 'DisplayName', '$D_z/\tilde{\mathcal{F}}_z$'); -plot(frf_iff.f, abs(G_dL_J_com(:,4,4)), ... - 'DisplayName', '$R_x/\tilde{\mathcal{M}}_x$'); -plot(frf_iff.f, abs(G_dL_J_com(:,5,5)), ... - 'DisplayName', '$R_y/\tilde{\mathcal{M}}_y$'); -plot(frf_iff.f, abs(G_dL_J_com(:,6,6)), ... - 'DisplayName', '$R_z/\tilde{\mathcal{M}}_z$'); -plot(frf_iff.f, abs(G_dL_J_com(:,1,2)), 'color', [0,0,0,0.2], ... - 'DisplayName', 'Coupling'); -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Amplitude'); set(gca, 'XTickLabel',[]); -ylim([1e-3, 1e1]); -legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 2); - -ax2 = nexttile; -hold on; -for i =1:6 - plot(frf_iff.f, 180/pi*angle(G_dL_J_com(:,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); -ylim([-180, 180]); - -linkaxes([ax1,ax2],'x'); -xlim([10, 1e3]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/interaction_J_com_plant.pdf', 'width', 'wide', 'height', 'tall'); -#+end_src - -#+name: fig:test_nhexa_interaction_J_com_plant -#+caption: Bode Plot of the plant decoupled using the Jacobian evaluated at the Center of Mass -#+RESULTS: -[[file:figs/test_nhexa_interaction_J_com_plant.png]] - -#+begin_src matlab :exports none -%% Compute RGA Matrix -RGA_com = zeros(size(frf_coupled)); -for i = 1:length(frf_iff.f) - RGA_com(i,:,:) = squeeze(G_dL_J_com(i,:,:)).*inv(squeeze(G_dL_J_com(i,:,:))).'; -end - -%% Compute RGA-number -RGA_com_sum = zeros(size(RGA_com, 1), 1); -for i = 1:size(RGA_com, 1) - RGA_com_sum(i) = sum(sum(abs(eye(6) - squeeze(RGA_com(i,:,:))))); -end -#+end_src - -#+begin_src matlab :exports none -%% Plot the RGA-Number for the Jacobian (CoM) decoupled plant -figure; -plot(frf_iff.f, RGA_com_sum, 'k-'); -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -xlabel('Frequency [Hz]'); ylabel('RGA Number'); -xlim([10, 1e3]); ylim([1e-2, 1e2]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/interaction_rga_J_com.pdf', 'width', 'wide', 'height', 'normal'); -#+end_src - -#+name: fig:test_nhexa_interaction_rga_J_com -#+caption: RGA number for the plant decoupled using the Jacobian evaluted at the Center of Mass -#+RESULTS: -[[file:figs/test_nhexa_interaction_rga_J_com.png]] - -*** Decoupling Comparison -<> - -Let's now compare all of the decoupling methods (Figure ref:fig:test_nhexa_interaction_compare_rga_numbers). - -#+begin_important -From Figure ref:fig:test_nhexa_interaction_compare_rga_numbers, the following remarks are made: -- *Decentralized plant*: well decoupled below suspension modes -- *Static inversion*: similar to the decentralized plant as the decentralized plant has already a good decoupling at low frequency -- *Crossover inversion*: the decoupling is improved around the crossover frequency as compared to the decentralized plant. However, the decoupling is increased at lower frequency. -- *SVD decoupling*: Very good decoupling up to 235Hz. Especially between 100Hz and 200Hz. -- *Dynamic Inversion*: the plant is very well decoupled at frequencies where the model is accurate (below 235Hz where flexible modes are not modelled). -- *Jacobian - Stiffness*: good decoupling at low frequency. The decoupling increases at the frequency of the suspension modes, but is acceptable up to the strut flexible modes (235Hz). -- *Jacobian - Mass*: bad decoupling at low frequency. Better decoupling above the frequency of the suspension modes, and acceptable decoupling up to the strut flexible modes (235Hz). -#+end_important - -#+begin_src matlab :exports none -%% Comparison of the RGA-Numbers -figure; -hold on; -plot(frf_iff.f, RGA_dec_sum, 'DisplayName', 'Decentralized'); -plot(frf_iff.f, RGA_sta_sum, 'DisplayName', 'Static inv.'); -plot(frf_iff.f, RGA_wc_sum, 'DisplayName', 'Crossover inv.'); -plot(frf_iff.f, RGA_svd_sum, 'DisplayName', 'SVD'); -plot(frf_iff.f, RGA_inv_sum, 'DisplayName', 'Dynamic inv.'); -plot(frf_iff.f, RGA_cok_sum, 'DisplayName', 'Jacobian - CoK'); -plot(frf_iff.f, RGA_com_sum, 'DisplayName', 'Jacobian - CoM'); -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -xlabel('Frequency [Hz]'); ylabel('RGA Number'); -xlim([10, 1e3]); ylim([1e-2, 1e2]); -legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 2); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/interaction_compare_rga_numbers.pdf', 'width', 'wide', 'height', 'tall'); -#+end_src - -#+name: fig:test_nhexa_interaction_compare_rga_numbers -#+caption: Comparison of the obtained RGA-numbers for all the decoupling methods -#+RESULTS: -[[file:figs/test_nhexa_interaction_compare_rga_numbers.png]] - -*** Decoupling Robustness -<> - -Let's now see how the decoupling is changing when changing the payload's mass. -#+begin_src matlab -frf_new = frf_iff.G_dL{3}; -#+end_src - -#+begin_src matlab :exports none -%% Decentralized RGA -RGA_dec_b = zeros(size(frf_new)); -for i = 1:length(frf_iff.f) - RGA_dec_b(i,:,:) = squeeze(frf_new(i,:,:)).*inv(squeeze(frf_new(i,:,:))).'; -end - -RGA_dec_sum_b = zeros(length(frf_iff), 1); -for i = 1:length(frf_iff.f) - RGA_dec_sum_b(i) = sum(sum(abs(eye(6) - squeeze(RGA_dec_b(i,:,:))))); -end -#+end_src - -#+begin_src matlab :exports none -%% Static Decoupling -G_dL_sta_b = zeros(size(frf_new)); -for i = 1:length(frf_iff.f) - G_dL_sta_b(i,:,:) = squeeze(frf_new(i,:,:))*dc_inv; -end - -RGA_sta_b = zeros(size(frf_new)); -for i = 1:length(frf_iff.f) - RGA_sta_b(i,:,:) = squeeze(G_dL_sta_b(i,:,:)).*inv(squeeze(G_dL_sta_b(i,:,:))).'; -end - -RGA_sta_sum_b = zeros(size(RGA_sta_b, 1), 1); -for i = 1:size(RGA_sta_b, 1) - RGA_sta_sum_b(i) = sum(sum(abs(eye(6) - squeeze(RGA_sta_b(i,:,:))))); -end -#+end_src - -#+begin_src matlab :exports none -%% Crossover Decoupling -V = squeeze(frf_coupled(i_wc,:,:)); -D = pinv(real(V'*V)); -H1 = D*real(V'*diag(exp(1j*angle(diag(V*D*V.'))/2))); - -G_dL_wc_b = zeros(size(frf_new)); -for i = 1:length(frf_iff.f) - G_dL_wc_b(i,:,:) = squeeze(frf_new(i,:,:))*H1; -end - -RGA_wc_b = zeros(size(frf_new)); -for i = 1:length(frf_iff.f) - RGA_wc_b(i,:,:) = squeeze(G_dL_wc_b(i,:,:)).*inv(squeeze(G_dL_wc_b(i,:,:))).'; -end - -RGA_wc_sum_b = zeros(size(RGA_wc_b, 1), 1); -for i = 1:size(RGA_wc_b, 1) - RGA_wc_sum_b(i) = sum(sum(abs(eye(6) - squeeze(RGA_wc_b(i,:,:))))); -end -#+end_src - -#+begin_src matlab :exports none -%% SVD -V = squeeze(frf_coupled(i_wc,:,:)); -D = pinv(real(V'*V)); -H1 = pinv(D*real(V'*diag(exp(1j*angle(diag(V*D*V.'))/2)))); -[U,S,V] = svd(H1); - -G_dL_svd_b = zeros(size(frf_new)); -for i = 1:length(frf_iff.f) - G_dL_svd_b(i,:,:) = inv(U)*squeeze(frf_new(i,:,:))*inv(V'); -end - -RGA_svd_b = zeros(size(frf_new)); -for i = 1:length(frf_iff.f) - RGA_svd_b(i,:,:) = squeeze(G_dL_svd_b(i,:,:)).*inv(squeeze(G_dL_svd_b(i,:,:))).'; -end - -RGA_svd_sum_b = zeros(size(RGA_svd_b, 1), 1); -for i = 1:size(RGA_svd, 1) - RGA_svd_sum_b(i) = sum(sum(abs(eye(6) - squeeze(RGA_svd_b(i,:,:))))); -end -#+end_src - -#+begin_src matlab :exports none -%% Dynamic Decoupling -G_model = G_coupled; -G_model.outputdelay = 0; % necessary for further inversion -G_inv = inv(G_model); - -G_dL_inv_b = zeros(size(frf_new)); -for i = 1:length(frf_iff.f) - G_dL_inv_b(i,:,:) = squeeze(frf_new(i,:,:))*squeeze(evalfr(G_inv, 1j*2*pi*frf_iff.f(i))); -end - -RGA_inv_b = zeros(size(frf_new)); -for i = 1:length(frf_iff.f) - RGA_inv_b(i,:,:) = squeeze(G_dL_inv_b(i,:,:)).*inv(squeeze(G_dL_inv_b(i,:,:))).'; -end - -RGA_inv_sum_b = zeros(size(RGA_inv_b, 1), 1); -for i = 1:size(RGA_inv_b, 1) - RGA_inv_sum_b(i) = sum(sum(abs(eye(6) - squeeze(RGA_inv_b(i,:,:))))); -end -#+end_src - -#+begin_src matlab :exports none -%% Jacobian (CoK) -G_dL_J_cok_b = zeros(size(frf_new)); -for i = 1:length(frf_iff.f) - G_dL_J_cok_b(i,:,:) = inv(Js_cok)*squeeze(frf_new(i,:,:))*inv(J_cok'); -end - -RGA_cok_b = zeros(size(frf_new)); -for i = 1:length(frf_iff.f) - RGA_cok_b(i,:,:) = squeeze(G_dL_J_cok_b(i,:,:)).*inv(squeeze(G_dL_J_cok_b(i,:,:))).'; -end - -RGA_cok_sum_b = zeros(size(RGA_cok_b, 1), 1); -for i = 1:size(RGA_cok_b, 1) - RGA_cok_sum_b(i) = sum(sum(abs(eye(6) - squeeze(RGA_cok_b(i,:,:))))); -end -#+end_src - -#+begin_src matlab :exports none -%% Jacobian (CoM) -G_dL_J_com_b = zeros(size(frf_new)); -for i = 1:length(frf_iff.f) - G_dL_J_com_b(i,:,:) = inv(Js_com)*squeeze(frf_new(i,:,:))*inv(J_com'); -end - -RGA_com_b = zeros(size(frf_new)); -for i = 1:length(frf_iff.f) - RGA_com_b(i,:,:) = squeeze(G_dL_J_com_b(i,:,:)).*inv(squeeze(G_dL_J_com_b(i,:,:))).'; -end - -RGA_com_sum_b = zeros(size(RGA_com_b, 1), 1); -for i = 1:size(RGA_com_b, 1) - RGA_com_sum_b(i) = sum(sum(abs(eye(6) - squeeze(RGA_com_b(i,:,:))))); -end -#+end_src - -The obtained RGA-numbers are shown in Figure ref:fig:test_nhexa_interaction_compare_rga_numbers_rob. - -#+begin_important -From Figure ref:fig:test_nhexa_interaction_compare_rga_numbers_rob: -- The decoupling using the Jacobian evaluated at the "center of stiffness" seems to give the most robust results. -#+end_important - -#+begin_src matlab :exports none -%% Robustness of the Decoupling method -figure; -hold on; -plot(frf_iff.f, RGA_dec_sum, '-', 'DisplayName', 'Decentralized'); -plot(frf_iff.f, RGA_sta_sum, '-', 'DisplayName', 'Static inv.'); -plot(frf_iff.f, RGA_wc_sum, '-', 'DisplayName', 'Crossover inv.'); -plot(frf_iff.f, RGA_svd_sum, '-', 'DisplayName', 'SVD'); -plot(frf_iff.f, RGA_inv_sum, '-', 'DisplayName', 'Dynamic inv.'); -plot(frf_iff.f, RGA_cok_sum, '-', 'DisplayName', 'Jacobian - CoK'); -plot(frf_iff.f, RGA_com_sum, '-', 'DisplayName', 'Jacobian - CoM'); -set(gca,'ColorOrderIndex',1) -plot(frf_iff.f, RGA_dec_sum_b, '--', 'HandleVisibility', 'off'); -plot(frf_iff.f, RGA_sta_sum_b, '--', 'HandleVisibility', 'off'); -plot(frf_iff.f, RGA_wc_sum_b, '--', 'HandleVisibility', 'off'); -plot(frf_iff.f, RGA_svd_sum_b, '--', 'HandleVisibility', 'off'); -plot(frf_iff.f, RGA_inv_sum_b, '--', 'HandleVisibility', 'off'); -plot(frf_iff.f, RGA_cok_sum_b, '--', 'HandleVisibility', 'off'); -plot(frf_iff.f, RGA_com_sum_b, '--', 'HandleVisibility', 'off'); -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -xlabel('Frequency [Hz]'); ylabel('RGA Number'); -xlim([10, 1e3]); ylim([1e-2, 1e2]); -legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 2); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/interaction_compare_rga_numbers_rob.pdf', 'width', 'wide', 'height', 'tall'); -#+end_src - -#+name: fig:test_nhexa_interaction_compare_rga_numbers_rob -#+caption: Change of the RGA-number with a change of the payload. Indication of the robustness of the inversion method. -#+RESULTS: -[[file:figs/test_nhexa_interaction_compare_rga_numbers_rob.png]] - -*** Conclusion - -#+begin_important -Several decoupling methods can be used: -- SVD -- Inverse -- Jacobian (CoK) -#+end_important - -#+name: tab:interaction_analysis_conclusion -#+caption: Summary of the interaction analysis and different decoupling strategies -#+attr_latex: :environment tabularx :width \linewidth :align lccc -#+attr_latex: :center t :booktabs t -| *Method* | *RGA* | *Diag Plant* | *Robustness* | -|----------------+-------+--------------+--------------| -| Decentralized | -- | Equal | ++ | -| Static dec. | -- | Equal | ++ | -| Crossover dec. | - | Equal | 0 | -| SVD | ++ | Diff | + | -| Dynamic dec. | ++ | Unity, equal | - | -| Jacobian - CoK | + | Diff | ++ | -| Jacobian - CoM | 0 | Diff | + | - -** Robust High Authority Controller -:PROPERTIES: -:header-args:matlab+: :tangle matlab/scripts/hac_lac_enc_plates_suspended_table.m -:END: -<> -*** Introduction :ignore: -In this section we wish to develop a robust High Authority Controller (HAC) that is working for all payloads. - -cite:indri20_mechat_robot - -*** Matlab Init :noexport:ignore: -#+begin_src matlab -%% hac_lac_enc_plates_suspended_table.m -% Development and analysis of a robust High Authority Controller -#+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 -%% Load the identified FRF and Simscape model -frf_iff = load('frf_iff_vib_table_m.mat', 'f', 'Ts', 'G_dL'); -sim_iff = load('sim_iff_vib_table_m.mat', 'G_dL'); -#+end_src - -*** Using Jacobian evaluated at the center of stiffness -**** Decoupled Plant -#+begin_src matlab -G_nom = frf_iff.G_dL{2}; % Nominal Plant -#+end_src - -#+begin_src matlab :exports none -%% Initialize the Nano-Hexapod -n_hexapod = initializeNanoHexapodFinal('MO_B', -42e-3, ... - 'motion_sensor_type', 'plates'); - -%% Get the Jacobians -J_cok = n_hexapod.geometry.J; -Js_cok = n_hexapod.geometry.Js; - -%% Decouple plant using Jacobian (CoM) -G_dL_J_cok = zeros(size(G_nom)); -for i = 1:length(frf_iff.f) - G_dL_J_cok(i,:,:) = inv(Js_cok)*squeeze(G_nom(i,:,:))*inv(J_cok'); -end - -%% Normalize the plant input -[~, i_100] = min(abs(frf_iff.f - 10)); -input_normalize = diag(1./diag(abs(squeeze(G_dL_J_cok(i_100,:,:))))); - -for i = 1:length(frf_iff.f) - G_dL_J_cok(i,:,:) = squeeze(G_dL_J_cok(i,:,:))*input_normalize; -end -#+end_src - -#+begin_src matlab :exports none -%% Bode Plot of the decoupled plant -figure; -tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); - -ax1 = nexttile([2,1]); -hold on; -for i = 1:5 - for j = i+1:6 - plot(frf_iff.f, abs(G_dL_J_cok(:,i,j)), 'color', [0,0,0,0.2], ... - 'HandleVisibility', 'off'); - end -end -set(gca,'ColorOrderIndex',1) -plot(frf_iff.f, abs(G_dL_J_cok(:,1,1)), ... - 'DisplayName', '$D_x/\tilde{\mathcal{F}}_x$'); -plot(frf_iff.f, abs(G_dL_J_cok(:,2,2)), ... - 'DisplayName', '$D_y/\tilde{\mathcal{F}}_y$'); -plot(frf_iff.f, abs(G_dL_J_cok(:,3,3)), ... - 'DisplayName', '$D_z/\tilde{\mathcal{F}}_z$'); -plot(frf_iff.f, abs(G_dL_J_cok(:,4,4)), ... - 'DisplayName', '$R_x/\tilde{\mathcal{M}}_x$'); -plot(frf_iff.f, abs(G_dL_J_cok(:,5,5)), ... - 'DisplayName', '$R_y/\tilde{\mathcal{M}}_y$'); -plot(frf_iff.f, abs(G_dL_J_cok(:,6,6)), ... - 'DisplayName', '$R_z/\tilde{\mathcal{M}}_z$'); -plot(frf_iff.f, abs(G_dL_J_cok(:,1,2)), 'color', [0,0,0,0.2], ... - 'DisplayName', 'Coupling'); -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Amplitude'); set(gca, 'XTickLabel',[]); -ylim([1e-3, 1e1]); -legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 2); - -ax2 = nexttile; -hold on; -for i =1:6 - plot(frf_iff.f, 180/pi*angle(G_dL_J_cok(:,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); -ylim([-180, 180]); - -linkaxes([ax1,ax2],'x'); -xlim([10, 1e3]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/bode_plot_hac_iff_plant_jacobian_cok.pdf', 'width', 'wide', 'height', 'tall'); -#+end_src - -#+name: fig:test_nhexa_bode_plot_hac_iff_plant_jacobian_cok -#+caption: Bode plot of the decoupled plant using the Jacobian evaluated at the Center of Stiffness -#+RESULTS: -[[file:figs/test_nhexa_bode_plot_hac_iff_plant_jacobian_cok.png]] - -**** SISO Controller Design -As the diagonal elements of the plant are not equal, several SISO controllers are designed and then combined to form a diagonal controller. -All the diagonal terms of the controller consists of: -- A double integrator to have high gain at low frequency -- A lead around the crossover frequency to increase stability margins -- Two second order low pass filters above the crossover frequency to increase the robustness to high frequency modes - -#+begin_src matlab :exports none -%% Controller Ry,Rz - -% Wanted crossover frequency -wc_Rxy = 2*pi*80; - -% Lead -a = 8.0; % Amount of phase lead / width of the phase lead / high frequency gain -wc = wc_Rxy; % Frequency with the maximum phase lead [rad/s] -Kd_lead = (1 + s/(wc/sqrt(a)))/(1 + s/(wc*sqrt(a)))/sqrt(a); - -% Integrator -w0_int = wc_Rxy/2; % [rad/s] -xi_int = 0.3; - -Kd_int = (1 + 2*xi_int/w0_int*s + s^2/w0_int^2)/(s^2/w0_int^2); - -% Low Pass Filter (High frequency robustness) -w0_lpf = wc_Rxy*2; % Cut-off frequency [rad/s] -xi_lpf = 0.6; % Damping Ratio - -Kd_lpf = 1/(1 + 2*xi_lpf/w0_lpf*s + s^2/w0_lpf^2); - -w0_lpf_b = wc_Rxy*4; % Cut-off frequency [rad/s] -xi_lpf_b = 0.7; % Damping Ratio - -Kd_lpf_b = 1/(1 + 2*xi_lpf_b/w0_lpf_b*s + s^2/w0_lpf_b^2); - -% Unity Gain frequency -[~, i_80] = min(abs(frf_iff.f - wc_Rxy/2/pi)); - -% Combination of all the elements -Kd_Rxy = ... - -1/abs(G_dL_J_cok(i_80,4,4)) * ... - Kd_lead/abs(evalfr(Kd_lead, 1j*wc_Rxy)) * ... % Lead (gain of 1 at wc) - Kd_int /abs(evalfr(Kd_int, 1j*wc_Rxy)) * ... - Kd_lpf_b/abs(evalfr(Kd_lpf_b, 1j*wc_Rxy)) * ... - Kd_lpf /abs(evalfr(Kd_lpf, 1j*wc_Rxy)); % Low Pass Filter -#+end_src - -#+begin_src matlab :exports none -%% Controller Dx,Dy,Rz - -% Wanted crossover frequency -wc_Dxy = 2*pi*100; - -% Lead -a = 8.0; % Amount of phase lead / width of the phase lead / high frequency gain -wc = wc_Dxy; % Frequency with the maximum phase lead [rad/s] -Kd_lead = (1 + s/(wc/sqrt(a)))/(1 + s/(wc*sqrt(a)))/sqrt(a); - -% Integrator -w0_int = wc_Dxy/2; % [rad/s] -xi_int = 0.3; - -Kd_int = (1 + 2*xi_int/w0_int*s + s^2/w0_int^2)/(s^2/w0_int^2); - -% Low Pass Filter (High frequency robustness) -w0_lpf = wc_Dxy*2; % Cut-off frequency [rad/s] -xi_lpf = 0.6; % Damping Ratio - -Kd_lpf = 1/(1 + 2*xi_lpf/w0_lpf*s + s^2/w0_lpf^2); - -w0_lpf_b = wc_Dxy*4; % Cut-off frequency [rad/s] -xi_lpf_b = 0.7; % Damping Ratio - -Kd_lpf_b = 1/(1 + 2*xi_lpf_b/w0_lpf_b*s + s^2/w0_lpf_b^2); - -% Unity Gain frequency -[~, i_100] = min(abs(frf_iff.f - wc_Dxy/2/pi)); - -% Combination of all the elements -Kd_Dyx_Rz = ... - -1/abs(G_dL_J_cok(i_100,1,1)) * ... - Kd_int /abs(evalfr(Kd_int, 1j*wc_Dxy)) * ... % Integrator - Kd_lead/abs(evalfr(Kd_lead, 1j*wc_Dxy)) * ... % Lead (gain of 1 at wc) - Kd_lpf_b/abs(evalfr(Kd_lpf_b, 1j*wc_Dxy)) * ... % Lead (gain of 1 at wc) - Kd_lpf /abs(evalfr(Kd_lpf, 1j*wc_Dxy)); % Low Pass Filter -#+end_src - -#+begin_src matlab :exports none -%% Controller Dz - -% Wanted crossover frequency -wc_Dz = 2*pi*100; - -% Lead -a = 8.0; % Amount of phase lead / width of the phase lead / high frequency gain -wc = wc_Dz; % Frequency with the maximum phase lead [rad/s] -Kd_lead = (1 + s/(wc/sqrt(a)))/(1 + s/(wc*sqrt(a)))/sqrt(a); - -% Integrator -w0_int = wc_Dz/2; % [rad/s] -xi_int = 0.3; - -Kd_int = (1 + 2*xi_int/w0_int*s + s^2/w0_int^2)/(s^2/w0_int^2); - -% Low Pass Filter (High frequency robustness) -w0_lpf = wc_Dz*2; % Cut-off frequency [rad/s] -xi_lpf = 0.6; % Damping Ratio - -Kd_lpf = 1/(1 + 2*xi_lpf/w0_lpf*s + s^2/w0_lpf^2); - -w0_lpf_b = wc_Dz*4; % Cut-off frequency [rad/s] -xi_lpf_b = 0.7; % Damping Ratio - -Kd_lpf_b = 1/(1 + 2*xi_lpf_b/w0_lpf_b*s + s^2/w0_lpf_b^2); - -% Unity Gain frequency -[~, i_100] = min(abs(frf_iff.f - wc_Dz/2/pi)); - -% Combination of all the elements -Kd_Dz = ... - -1/abs(G_dL_J_cok(i_100,3,3)) * ... - Kd_int /abs(evalfr(Kd_int, 1j*wc_Dz)) * ... % Integrator - Kd_lead/abs(evalfr(Kd_lead, 1j*wc_Dz)) * ... % Lead (gain of 1 at wc) - Kd_lpf_b/abs(evalfr(Kd_lpf_b, 1j*wc_Dz)) * ... % Lead (gain of 1 at wc) - Kd_lpf /abs(evalfr(Kd_lpf, 1j*wc_Dz)); % Low Pass Filter -#+end_src - -#+begin_src matlab :exports none -%% Diagonal Controller -Kd_diag = blkdiag(Kd_Dyx_Rz, Kd_Dyx_Rz, Kd_Dz, Kd_Rxy, Kd_Rxy, Kd_Dyx_Rz); -#+end_src - -**** Obtained Loop Gain -#+begin_src matlab :exports none -%% Experimental Loop Gain -Lmimo = permute(pagemtimes(permute(G_dL_J_cok, [2,3,1]), squeeze(freqresp(Kd_diag, frf_iff.f, 'Hz'))), [3,1,2]); -#+end_src - -#+begin_src matlab :exports none -%% Bode plot of the experimental Loop Gain -figure; -tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); - -ax1 = nexttile([2,1]); -hold on; -for i = 1:6 - plot(frf_iff.f, abs(Lmimo(:,i,i)), '-'); -end -for i = 1:5 - for j = i+1:6 - plot(frf_iff.f, abs(squeeze(Lmimo(:,i,j))), 'color', [0,0,0,0.2]); - end -end -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Loop Gain'); set(gca, 'XTickLabel',[]); -ylim([1e-3, 1e+3]); - -ax2 = nexttile; -hold on; -for i = 1:6 - plot(frf_iff.f, 180/pi*angle(Lmimo(:,i,i)), '-'); -end -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); -xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); -hold off; -yticks(-360:45:360); -ylim([-180, 180]); - -linkaxes([ax1,ax2],'x'); -xlim([1, 2e3]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/bode_plot_hac_iff_loop_gain_jacobian_cok.pdf', 'width', 'wide', 'height', 'tall'); -#+end_src - -#+name: fig:test_nhexa_bode_plot_hac_iff_loop_gain_jacobian_cok -#+caption: Bode plot of the Loop Gain when using the Jacobian evaluated at the Center of Stiffness to decouple the system -#+RESULTS: -[[file:figs/test_nhexa_bode_plot_hac_iff_loop_gain_jacobian_cok.png]] - -#+begin_src matlab -%% Controller to be implemented -Kd = inv(J_cok')*input_normalize*ss(Kd_diag)*inv(Js_cok); -#+end_src - -**** Verification of the Stability -Now the stability of the feedback loop is verified using the generalized Nyquist criteria. - -#+begin_src matlab :exports none -%% Compute the Eigenvalues of the loop gain -Ldet = zeros(3, 6, length(frf_iff.f)); - -for i_mass = 1:3 - % Loop gain - Lmimo = pagemtimes(permute(frf_iff.G_dL{i_mass}, [2,3,1]),squeeze(freqresp(Kd, frf_iff.f, 'Hz'))); - for i_f = 2:length(frf_iff.f) - Ldet(i,:, i_f) = eig(squeeze(Lmimo(:,:,i_f))); - end -end -#+end_src - -#+begin_src matlab :exports none -%% Plot of the eigenvalues of L in the complex plane -figure; -hold on; -for i_mass = 2:3 - plot(real(squeeze(Ldet(i_mass, 1,:))), imag(squeeze(Ldet(i_mass, 1,:))), ... - '.', 'color', colors(i_mass+1, :), ... - 'DisplayName', sprintf('%i masses', i_mass)); - plot(real(squeeze(Ldet(i_mass, 1,:))), -imag(squeeze(Ldet(i_mass, 1,:))), ... - '.', 'color', colors(i_mass+1, :), ... - 'HandleVisibility', 'off'); - for i = 1:6 - plot(real(squeeze(Ldet(i_mass, i,:))), imag(squeeze(Ldet(i_mass, i,:))), ... - '.', 'color', colors(i_mass+1, :), ... - 'HandleVisibility', 'off'); - plot(real(squeeze(Ldet(i_mass, i,:))), -imag(squeeze(Ldet(i_mass, i,:))), ... - '.', 'color', colors(i_mass+1, :), ... - 'HandleVisibility', 'off'); - end -end -plot(-1, 0, 'kx', 'HandleVisibility', 'off'); -hold off; -set(gca, 'XScale', 'lin'); set(gca, 'YScale', 'lin'); -xlabel('Real'); ylabel('Imag'); -legend('location', 'southeast'); -xlim([-3, 1]); ylim([-2, 2]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/loci_hac_iff_loop_gain_jacobian_cok.pdf', 'width', 'wide', 'height', 'tall'); -#+end_src - -#+name: fig:test_nhexa_loci_hac_iff_loop_gain_jacobian_cok -#+caption: Loci of $L(j\omega)$ in the complex plane. -#+RESULTS: -[[file:figs/test_nhexa_loci_hac_iff_loop_gain_jacobian_cok.png]] - -**** Save for further analysis -#+begin_src matlab :exports none :tangle no -save('matlab/data_sim/Khac_iff_struts_jacobian_cok.mat', 'Kd') -#+end_src - -#+begin_src matlab :eval no -save('data_sim/Khac_iff_struts_jacobian_cok.mat', 'Kd') -#+end_src - -**** Sensitivity transfer function from the model -#+begin_src matlab :exports none -%% Open Simulink Model -mdl = 'nano_hexapod_simscape'; - -options = linearizeOptions; -options.SampleTime = 0; - -open(mdl) - -Rx = zeros(1, 7); - -colors = colororder; -#+end_src - -#+begin_src matlab :exports none -%% Initialize the Simscape model in closed loop -n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... - 'flex_top_type', '4dof', ... - 'motion_sensor_type', 'plates', ... - 'actuator_type', '2dof', ... - 'controller_type', 'hac-iff-struts'); - -support.type = 1; % On top of vibration table -payload.type = 2; % Payload -#+end_src - -#+begin_src matlab :exports none -%% Load controllers -load('Kiff_opt.mat', 'Kiff'); -Kiff = c2d(Kiff, Ts, 'Tustin'); -load('Khac_iff_struts_jacobian_cok.mat', 'Kd') -Khac_iff_struts = c2d(Kd, Ts, 'Tustin'); -#+end_src - -#+begin_src matlab :exports none -%% Identify the (damped) transfer function from u to dLm -clear io; io_i = 1; -io(io_i) = linio([mdl, '/Rx'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs -io(io_i) = linio([mdl, '/dL'], 1, 'output'); io_i = io_i + 1; % Plate Displacement (encoder) -#+end_src - -#+begin_src matlab :exports none -%% Identification of the dynamics -Gcl = linearize(mdl, io, 0.0, options); -#+end_src - -#+begin_src matlab :exports none -%% Computation of the sensitivity transfer function -S = eye(6) - inv(n_hexapod.geometry.J)*Gcl; -#+end_src - -The results are shown in Figure ref:fig:test_nhexa_sensitivity_hac_jacobian_cok_3m_comp_model. - -#+begin_src matlab :exports none -%% Bode plot for the transfer function from u to dLm -freqs = logspace(0, 3, 1000); - -figure; -hold on; -for i =1:6 - set(gca,'ColorOrderIndex',i); - plot(freqs, abs(squeeze(freqresp(S(i,i), freqs, 'Hz'))), '--', ... - 'DisplayName', sprintf('$S_{%s}$ - Model', labels{i})); -end -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -xlabel('Frequency [Hz]'); ylabel('Sensitivity [-]'); -ylim([1e-4, 1e1]); -legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 3); -xlim([1, 1e3]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/sensitivity_hac_jacobian_cok_3m_comp_model.pdf', 'width', 'wide', 'height', 'normal'); -#+end_src - -#+name: fig:test_nhexa_sensitivity_hac_jacobian_cok_3m_comp_model -#+caption: Estimated sensitivity transfer functions for the HAC controller using the Jacobian estimated at the Center of Stiffness -#+RESULTS: -[[file:figs/test_nhexa_sensitivity_hac_jacobian_cok_3m_comp_model.png]] - -*** Using Singular Value Decomposition -**** Decoupled Plant -#+begin_src matlab -G_nom = frf_iff.G_dL{2}; % Nominal Plant -#+end_src - -#+begin_src matlab :exports none -%% Take complex matrix corresponding to the plant at 100Hz -wc = 100; % Wanted crossover frequency [Hz] -[~, i_wc] = min(abs(frf_iff.f - wc)); % Indice corresponding to wc - -V = squeeze(G_nom(i_wc,:,:)); - -%% Real approximation of G(100Hz) -D = pinv(real(V'*V)); -H1 = pinv(D*real(V'*diag(exp(1j*angle(diag(V*D*V.'))/2)))); - -%% Singular Value Decomposition -[U,S,V] = svd(H1); - -%% Compute the decoupled plant using SVD -G_dL_svd = zeros(size(G_nom)); -for i = 1:length(frf_iff.f) - G_dL_svd(i,:,:) = inv(U)*squeeze(G_nom(i,:,:))*inv(V'); -end -#+end_src - -#+begin_src matlab :exports none -%% Bode plot of the decoupled plant using SVD -figure; -tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); - -ax1 = nexttile([2,1]); -hold on; -for i = 1:5 - for j = i+1:6 - plot(frf_iff.f, abs(G_dL_svd(:,i,j)), 'color', [0,0,0,0.2], ... - 'HandleVisibility', 'off'); - end -end -set(gca,'ColorOrderIndex',1); -for i = 1:6 - plot(frf_iff.f, abs(G_dL_svd(:,i,i)), ... - 'DisplayName', sprintf('$y_%i/u_%i$', i, i)); -end -plot(frf_iff.f, abs(G_dL_svd(:,1,2)), 'color', [0,0,0,0.2], ... - 'DisplayName', 'Coupling'); -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Amplitude'); set(gca, 'XTickLabel',[]); -ylim([1e-9, 1e-4]); -legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 2); - -ax2 = nexttile; -hold on; -for i =1:6 - plot(frf_iff.f, 180/pi*angle(G_dL_svd(:,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); -ylim([-180, 180]); - -linkaxes([ax1,ax2],'x'); -xlim([10, 1e3]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/bode_plot_hac_iff_plant_svd.pdf', 'width', 'wide', 'height', 'tall'); -#+end_src - -#+name: fig:test_nhexa_bode_plot_hac_iff_plant_svd -#+caption: Bode plot of the decoupled plant using the SVD -#+RESULTS: -[[file:figs/test_nhexa_bode_plot_hac_iff_plant_svd.png]] - -**** Controller Design -#+begin_src matlab :exports none -%% Lead -a = 6.0; % Amount of phase lead / width of the phase lead / high frequency gain -wc = 2*pi*100; % Frequency with the maximum phase lead [rad/s] -Kd_lead = (1 + s/(wc/sqrt(a)))/(1 + s/(wc*sqrt(a)))/sqrt(a); - -%% Integrator -Kd_int = ((2*pi*50 + s)/(2*pi*0.1 + s))^2; - -%% Low Pass Filter (High frequency robustness) -w0_lpf = 2*pi*200; % Cut-off frequency [rad/s] -xi_lpf = 0.3; % Damping Ratio - -Kd_lpf = 1/(1 + 2*xi_lpf/w0_lpf*s + s^2/w0_lpf^2); - -%% Normalize Gain -Kd_norm = diag(1./abs(diag(squeeze(G_dL_svd(i_wc,:,:))))); - -%% Diagonal Control -Kd_diag = ... - Kd_norm * ... % Normalize gain at 100Hz - Kd_int /abs(evalfr(Kd_int, 1j*2*pi*100)) * ... % Integrator - Kd_lead/abs(evalfr(Kd_lead, 1j*2*pi*100)) * ... % Lead (gain of 1 at wc) - Kd_lpf /abs(evalfr(Kd_lpf, 1j*2*pi*100)); % Low Pass Filter -#+end_src - -#+begin_src matlab :exports none -%% MIMO Controller -Kd = -inv(V') * ... % Output decoupling - ss(Kd_diag) * ... - inv(U); % Input decoupling -#+end_src - -**** Loop Gain -#+begin_src matlab :exports none -%% Experimental Loop Gain -Lmimo = permute(pagemtimes(permute(G_nom, [2,3,1]),squeeze(freqresp(Kd, frf_iff.f, 'Hz'))), [3,1,2]); -#+end_src - -#+begin_src matlab :exports none -%% Loop gain when using SVD -figure; -tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); - -ax1 = nexttile([2,1]); -hold on; -for i = 1:6 - plot(frf_iff.f, abs(Lmimo(:,i,i)), '-'); -end -for i = 1:5 - for j = i+1:6 - plot(frf_iff.f, abs(squeeze(Lmimo(:,i,j))), 'color', [0,0,0,0.2]); - end -end -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Loop Gain'); set(gca, 'XTickLabel',[]); -ylim([1e-3, 1e+3]); - -ax2 = nexttile; -hold on; -for i = 1:6 - plot(frf_iff.f, 180/pi*angle(Lmimo(:,i,i)), '-'); -end -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); -xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); -hold off; -yticks(-360:30:360); -ylim([-180, 0]); - -linkaxes([ax1,ax2],'x'); -xlim([1, 2e3]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/bode_plot_hac_iff_loop_gain_svd.pdf', 'width', 'wide', 'height', 'tall'); -#+end_src - -#+name: fig:test_nhexa_bode_plot_hac_iff_loop_gain_svd -#+caption: Bode plot of Loop Gain when using the SVD -#+RESULTS: -[[file:figs/test_nhexa_bode_plot_hac_iff_loop_gain_svd.png]] - -**** Stability Verification -#+begin_src matlab -%% Compute the Eigenvalues of the loop gain -Ldet = zeros(3, 6, length(frf_iff.f)); - -for i = 1:3 - Lmimo = pagemtimes(permute(frf_iff.G_dL{i}, [2,3,1]),squeeze(freqresp(Kd, frf_iff.f, 'Hz'))); - for i_f = 2:length(frf_iff.f) - Ldet(i,:, i_f) = eig(squeeze(Lmimo(:,:,i_f))); - end -end -#+end_src - -#+begin_src matlab :exports none -%% Plot of the eigenvalues of L in the complex plane -figure; -hold on; -for i_mass = 2:3 - plot(real(squeeze(Ldet(i_mass, 1,:))), imag(squeeze(Ldet(i_mass, 1,:))), ... - '.', 'color', colors(i_mass+1, :), ... - 'DisplayName', sprintf('%i masses', i_mass)); - plot(real(squeeze(Ldet(i_mass, 1,:))), -imag(squeeze(Ldet(i_mass, 1,:))), ... - '.', 'color', colors(i_mass+1, :), ... - 'HandleVisibility', 'off'); - for i = 1:6 - plot(real(squeeze(Ldet(i_mass, i,:))), imag(squeeze(Ldet(i_mass, i,:))), ... - '.', 'color', colors(i_mass+1, :), ... - 'HandleVisibility', 'off'); - plot(real(squeeze(Ldet(i_mass, i,:))), -imag(squeeze(Ldet(i_mass, i,:))), ... - '.', 'color', colors(i_mass+1, :), ... - 'HandleVisibility', 'off'); - end -end -plot(-1, 0, 'kx', 'HandleVisibility', 'off'); -hold off; -set(gca, 'XScale', 'lin'); set(gca, 'YScale', 'lin'); -xlabel('Real'); ylabel('Imag'); -legend('location', 'southeast'); -xlim([-3, 1]); ylim([-2, 2]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/loci_hac_iff_loop_gain_svd.pdf', 'width', 'wide', 'height', 'tall'); -#+end_src - -#+name: fig:test_nhexa_loci_hac_iff_loop_gain_svd -#+caption: Locis of $L(j\omega)$ in the complex plane. -#+RESULTS: -[[file:figs/test_nhexa_loci_hac_iff_loop_gain_svd.png]] - -**** Save for further analysis -#+begin_src matlab :exports none :tangle no -save('matlab/data_sim/Khac_iff_struts_svd.mat', 'Kd') -#+end_src - -#+begin_src matlab :eval no -save('data_sim/Khac_iff_struts_svd.mat', 'Kd') -#+end_src - -**** Measured Sensitivity Transfer Function -The sensitivity transfer function is estimated by adding a reference signal $R_x$ consisting of a low pass filtered white noise, and measuring the position error $E_x$ at the same time. - -The transfer function from $R_x$ to $E_x$ is the sensitivity transfer function. - -In order to identify the sensitivity transfer function for all directions, six reference signals are used, one for each direction. - -#+begin_src matlab :exports none -%% Tested directions -labels = {'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'}; -#+end_src - -#+begin_src matlab :exports none -%% Load Identification Data -meas_hac_svd_3m = {}; - -for i = 1:6 - meas_hac_svd_3m(i) = {load(sprintf('T_S_meas_%s_3m_hac_svd_iff.mat', labels{i}), 't', 'Va', 'Vs', 'de', 'Rx')}; -end -#+end_src - -#+begin_src matlab :exports none -%% Setup useful variables -% Sampling Time [s] -Ts = (meas_hac_svd_3m{1}.t(end) - (meas_hac_svd_3m{1}.t(1)))/(length(meas_hac_svd_3m{1}.t)-1); - -% Sampling Frequency [Hz] -Fs = 1/Ts; - -% Hannning Windows -win = hanning(ceil(5*Fs)); - -% And we get the frequency vector -[~, f] = tfestimate(meas_hac_svd_3m{1}.Va, meas_hac_svd_3m{1}.de, win, Noverlap, Nfft, 1/Ts); -#+end_src - -#+begin_src matlab :exports none -%% Load Jacobian matrix -load('jacobian.mat', 'J'); - -%% Compute position error -for i = 1:6 - meas_hac_svd_3m{i}.Xm = [inv(J)*meas_hac_svd_3m{i}.de']'; - meas_hac_svd_3m{i}.Ex = meas_hac_svd_3m{i}.Rx - meas_hac_svd_3m{i}.Xm; -end -#+end_src - -An example is shown in Figure ref:fig:test_nhexa_ref_track_hac_svd_3m where both the reference signal and the measured position are shown for translations in the $x$ direction. - -#+begin_src matlab :exports none -figure; -hold on; -plot(meas_hac_svd_3m{1}.t, meas_hac_svd_3m{1}.Xm(:,1), 'DisplayName', 'Pos.') -plot(meas_hac_svd_3m{1}.t, meas_hac_svd_3m{1}.Rx(:,1), 'DisplayName', 'Ref.') -hold off; -xlabel('Time [s]'); ylabel('Dx motion [m]'); -xlim([20, 22]); -legend('location', 'northeast'); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/ref_track_hac_svd_3m.pdf', 'width', 'wide', 'height', 'normal'); -#+end_src - -#+name: fig:test_nhexa_ref_track_hac_svd_3m -#+caption: Reference position and measured position -#+RESULTS: -[[file:figs/test_nhexa_ref_track_hac_svd_3m.png]] - -#+begin_src matlab :exports none -%% Transfer function estimate of S -S_hac_svd_3m = zeros(length(f), 6, 6); - -for i = 1:6 - S_hac_svd_3m(:,:,i) = tfestimate(meas_hac_svd_3m{i}.Rx, meas_hac_svd_3m{i}.Ex, win, Noverlap, Nfft, 1/Ts); -end -#+end_src - -The sensitivity transfer functions estimated for all directions are shown in Figure ref:fig:test_nhexa_sensitivity_hac_svd_3m. - -#+begin_src matlab :exports none -%% Bode plot for the transfer function from u to dLm -figure; -hold on; -for i =1:6 - plot(f, abs(S_hac_svd_3m(:,i,i)), ... - 'DisplayName', sprintf('$S_{%s}$', labels{i})); -end -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -xlabel('Frequency [Hz]'); ylabel('Sensitivity [-]'); -ylim([1e-4, 1e1]); -legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 3); -xlim([0.5, 1e3]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/sensitivity_hac_svd_3m.pdf', 'width', 'wide', 'height', 'normal'); -#+end_src - -#+name: fig:test_nhexa_sensitivity_hac_svd_3m -#+caption: Measured diagonal elements of the sensitivity transfer function matrix. -#+RESULTS: -[[file:figs/test_nhexa_sensitivity_hac_svd_3m.png]] - -#+begin_important -From Figure ref:fig:test_nhexa_sensitivity_hac_svd_3m: -- The sensitivity transfer functions are similar for all directions -- The disturbance attenuation at 1Hz is almost a factor 1000 as wanted -- The sensitivity transfer functions for $R_x$ and $R_y$ have high peak values which indicate poor stability margins. -#+end_important - -**** Sensitivity transfer function from the model -The sensitivity transfer function is now estimated using the model and compared with the one measured. - -#+begin_src matlab :exports none -%% Open Simulink Model -mdl = 'nano_hexapod_simscape'; - -options = linearizeOptions; -options.SampleTime = 0; - -open(mdl) - -Rx = zeros(1, 7); - -colors = colororder; -#+end_src - -#+begin_src matlab :exports none -%% Initialize the Simscape model in closed loop -n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... - 'flex_top_type', '4dof', ... - 'motion_sensor_type', 'plates', ... - 'actuator_type', '2dof', ... - 'controller_type', 'hac-iff-struts'); - -support.type = 1; % On top of vibration table -payload.type = 2; % Payload -#+end_src - -#+begin_src matlab :exports none -%% Load controllers -load('Kiff_opt.mat', 'Kiff'); -Kiff = c2d(Kiff, Ts, 'Tustin'); -load('Khac_iff_struts_svd.mat', 'Kd') -Khac_iff_struts = c2d(Kd, Ts, 'Tustin'); -#+end_src - -#+begin_src matlab :exports none -%% Identify the (damped) transfer function from u to dLm -clear io; io_i = 1; -io(io_i) = linio([mdl, '/Rx'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs -io(io_i) = linio([mdl, '/dL'], 1, 'output'); io_i = io_i + 1; % Plate Displacement (encoder) -#+end_src - -#+begin_src matlab :exports none -%% Identification of the dynamics -Gcl = linearize(mdl, io, 0.0, options); -#+end_src - -#+begin_src matlab :exports none -%% Computation of the sensitivity transfer function -S = eye(6) - inv(n_hexapod.geometry.J)*Gcl; -#+end_src - -The results are shown in Figure ref:fig:test_nhexa_sensitivity_hac_svd_3m_comp_model. -The model is quite effective in estimating the sensitivity transfer functions except around 60Hz were there is a peak for the measurement. - -#+begin_src matlab :exports none -%% Bode plot for the transfer function from u to dLm -freqs = logspace(0,3,1000); - -figure; -hold on; -for i =1:6 - set(gca,'ColorOrderIndex',i); - plot(f, abs(S_hac_svd_3m(:,i,i)), ... - 'DisplayName', sprintf('$S_{%s}$', labels{i})); - set(gca,'ColorOrderIndex',i); - plot(freqs, abs(squeeze(freqresp(S(i,i), freqs, 'Hz'))), '--', ... - 'DisplayName', sprintf('$S_{%s}$ - Model', labels{i})); -end -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -xlabel('Frequency [Hz]'); ylabel('Sensitivity [-]'); -ylim([1e-4, 1e1]); -legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 3); -xlim([0.5, 1e3]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/sensitivity_hac_svd_3m_comp_model.pdf', 'width', 'wide', 'height', 'normal'); -#+end_src - -#+name: fig:test_nhexa_sensitivity_hac_svd_3m_comp_model -#+caption: Comparison of the measured sensitivity transfer functions with the model -#+RESULTS: -[[file:figs/test_nhexa_sensitivity_hac_svd_3m_comp_model.png]] - -*** Using (diagonal) Dynamical Inverse :noexport: -**** Decoupled Plant -#+begin_src matlab -G_nom = frf_iff.G_dL{2}; % Nominal Plant -G_model = sim_iff.G_dL{2}; % Model of the Plant -#+end_src - -#+begin_src matlab :exports none -%% Simplified model of the diagonal term -balred_opts = balredOptions('FreqIntervals', 2*pi*[0, 1000], 'StateElimMethod', 'Truncate'); - -G_red = balred(G_model(1,1), 8, balred_opts); -G_red.outputdelay = 0; % necessary for further inversion -#+end_src - -#+begin_src matlab -%% Inverse -G_inv = inv(G_red); -[G_z, G_p, G_g] = zpkdata(G_inv); -p_uns = real(G_p{1}) > 0; -G_p{1}(p_uns) = -G_p{1}(p_uns); -G_inv_stable = zpk(G_z, G_p, G_g); -#+end_src - -#+begin_src matlab :exports none -%% "Uncertainty" of inversed plant -freqs = logspace(0,3,1000); - -figure; -tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); - -ax1 = nexttile([2,1]); -hold on; -for i_mass = i_masses - for i = 1 - plot(freqs, abs(squeeze(freqresp(G_inv_stable*sim_iff.G_dL{i_mass+1}(i,i), freqs, 'Hz'))), '-', 'color', colors(i_mass+1, :), ... - 'DisplayName', sprintf('$d\\mathcal{L}_i/u^\\prime_i$ - %i', i_mass)); - end -end -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Amplitude'); set(gca, 'XTickLabel',[]); -ylim([1e-1, 1e1]); -legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 4); - -ax2 = nexttile; -hold on; -for i_mass = i_masses - for i = 1 - plot(freqs, 180/pi*angle(squeeze(freqresp(G_inv_stable*sim_iff.G_dL{i_mass+1}(1,1), freqs, 'Hz'))), '-', 'color', colors(i_mass+1, :)); - end -end -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); -xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); -hold off; -yticks(-360:15:360); -ylim([-45, 45]); - -linkaxes([ax1,ax2],'x'); -xlim([freqs(1), freqs(end)]); -#+end_src - -**** Controller Design -#+begin_src matlab :exports none -% Wanted crossover frequency -wc = 2*pi*80; -[~, i_wc] = min(abs(frf_iff.f - wc/2/pi)); - -%% Lead -a = 20.0; % Amount of phase lead / width of the phase lead / high frequency gain -Kd_lead = (1 + s/(wc/sqrt(a)))/(1 + s/(wc*sqrt(a)))/sqrt(a); - -%% Integrator -Kd_int = ((wc)/(2*pi*0.2 + s))^2; - -%% Low Pass Filter (High frequency robustness) -w0_lpf = 2*wc; % Cut-off frequency [rad/s] -xi_lpf = 0.3; % Damping Ratio - -Kd_lpf = 1/(1 + 2*xi_lpf/w0_lpf*s + s^2/w0_lpf^2); - -w0_lpf_b = wc*4; % Cut-off frequency [rad/s] -xi_lpf_b = 0.7; % Damping Ratio - -Kd_lpf_b = 1/(1 + 2*xi_lpf_b/w0_lpf_b*s + s^2/w0_lpf_b^2); - -%% Normalize Gain -Kd_norm = diag(1./abs(diag(squeeze(G_dL_svd(i_wc,:,:))))); - -%% Diagonal Control -Kd_diag = ... - G_inv_stable * ... % Normalize gain at 100Hz - Kd_int /abs(evalfr(Kd_int, 1j*wc)) * ... % Integrator - Kd_lead/abs(evalfr(Kd_lead, 1j*wc)) * ... % Lead (gain of 1 at wc) - Kd_lpf /abs(evalfr(Kd_lpf, 1j*wc)); % Low Pass Filter -#+end_src - -#+begin_src matlab :exports none -Kd = ss(Kd_diag)*eye(6); -#+end_src - -**** Loop Gain -#+begin_src matlab :exports none -%% Experimental Loop Gain -Lmimo = permute(pagemtimes(permute(G_nom, [2,3,1]),squeeze(freqresp(Kd, frf_iff.f, 'Hz'))), [3,1,2]); -#+end_src - -#+begin_src matlab :exports none -%% Loop gain when using SVD -figure; -tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); - -ax1 = nexttile([2,1]); -hold on; -for i = 1:6 - plot(frf_iff.f, abs(Lmimo(:,i,i)), '-'); -end -for i = 1:5 - for j = i+1:6 - plot(frf_iff.f, abs(squeeze(Lmimo(:,i,j))), 'color', [0,0,0,0.2]); - end -end -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Loop Gain'); set(gca, 'XTickLabel',[]); -ylim([1e-3, 1e+3]); - -ax2 = nexttile; -hold on; -for i = 1:6 - plot(frf_iff.f, 180/pi*angle(Lmimo(:,i,i)), '-'); -end -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); -xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); -hold off; -yticks(-360:30:360); -ylim([-180, 0]); - -linkaxes([ax1,ax2],'x'); -xlim([1, 2e3]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/bode_plot_hac_iff_loop_gain_diag_inverse.pdf', 'width', 'wide', 'height', 'tall'); -#+end_src - -#+name: fig:test_nhexa_bode_plot_hac_iff_loop_gain_diag_inverse -#+caption: Bode plot of Loop Gain when using the Diagonal inversion -#+RESULTS: -[[file:figs/test_nhexa_bode_plot_hac_iff_loop_gain_diag_inverse.png]] - -**** Stability Verification -MIMO Nyquist with eigenvalues -#+begin_src matlab -%% Compute the Eigenvalues of the loop gain -Ldet = zeros(3, 6, length(frf_iff.f)); - -for i = 1:3 - Lmimo = pagemtimes(permute(frf_iff.G_dL{i}, [2,3,1]),squeeze(freqresp(Kd, frf_iff.f, 'Hz'))); - for i_f = 2:length(frf_iff.f) - Ldet(i,:, i_f) = eig(squeeze(Lmimo(:,:,i_f))); - end -end -#+end_src - -#+begin_src matlab :exports none -%% Plot of the eigenvalues of L in the complex plane -figure; -hold on; -for i_mass = 2:3 - plot(real(squeeze(Ldet(i_mass, 1,:))), imag(squeeze(Ldet(i_mass, 1,:))), ... - '.', 'color', colors(i_mass+1, :), ... - 'DisplayName', sprintf('%i masses', i_mass)); - plot(real(squeeze(Ldet(i_mass, 1,:))), -imag(squeeze(Ldet(i_mass, 1,:))), ... - '.', 'color', colors(i_mass+1, :), ... - 'HandleVisibility', 'off'); - for i = 1:6 - plot(real(squeeze(Ldet(i_mass, i,:))), imag(squeeze(Ldet(i_mass, i,:))), ... - '.', 'color', colors(i_mass+1, :), ... - 'HandleVisibility', 'off'); - plot(real(squeeze(Ldet(i_mass, i,:))), -imag(squeeze(Ldet(i_mass, i,:))), ... - '.', 'color', colors(i_mass+1, :), ... - 'HandleVisibility', 'off'); - end -end -plot(-1, 0, 'kx', 'HandleVisibility', 'off'); -hold off; -set(gca, 'XScale', 'lin'); set(gca, 'YScale', 'lin'); -xlabel('Real'); ylabel('Imag'); -legend('location', 'southeast'); -xlim([-3, 1]); ylim([-2, 2]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/loci_hac_iff_loop_gain_diag_inverse.pdf', 'width', 'wide', 'height', 'tall'); -#+end_src - -#+name: fig:test_nhexa_loci_hac_iff_loop_gain_diag_inverse -#+caption: Locis of $L(j\omega)$ in the complex plane. -#+RESULTS: -[[file:figs/test_nhexa_loci_hac_iff_loop_gain_diag_inverse.png]] - -#+begin_important -Even though the loop gain seems to be fine, the closed-loop system is unstable. -This might be due to the fact that there is large interaction in the plant. -We could look at the RGA-number to verify that. -#+end_important - -**** Save for further use -#+begin_src matlab :exports none :tangle no -save('matlab/data_sim/Khac_iff_struts_diag_inverse.mat', 'Kd') -#+end_src - -#+begin_src matlab :eval no -save('data_sim/Khac_iff_struts_diag_inverse.mat', 'Kd') -#+end_src - -*** Closed Loop Stability (Model) :noexport: -Verify stability using Simscape model -#+begin_src matlab -%% Initialize the Simscape model in closed loop -n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '2dof', ... - 'flex_top_type', '3dof', ... - 'motion_sensor_type', 'plates', ... - 'actuator_type', '2dof', ... - 'controller_type', 'hac-iff-struts'); -#+end_src - -#+begin_src matlab -%% IFF Controller -Kiff = -g_opt*Kiff_g1*eye(6); -Khac_iff_struts = Kd*eye(6); -#+end_src - -#+begin_src matlab -%% Identify the (damped) transfer function from u to dLm for different values of the IFF gain -clear io; io_i = 1; -io(io_i) = linio([mdl, '/du'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs -io(io_i) = linio([mdl, '/dL'], 1, 'openoutput'); io_i = io_i + 1; % Plate Displacement (encoder) -#+end_src - -#+begin_src matlab -GG_cl = {}; - -for i = i_masses - payload.type = i; - GG_cl(i+1) = {exp(-s*Ts)*linearize(mdl, io, 0.0, options)}; -end -#+end_src - -#+begin_src matlab -for i = i_masses - isstable(GG_cl{i+1}) -end -#+end_src - -MIMO Nyquist -#+begin_src matlab -Kdm = Kd*eye(6); - -Ldet = zeros(3, length(fb(i_lim))); - -for i = 1:3 - Lmimo = pagemtimes(permute(G_damp_m{i}(i_lim,:,:), [2,3,1]),squeeze(freqresp(Kdm, fb(i_lim), 'Hz'))); - Ldet(i,:) = arrayfun(@(t) det(eye(6) + squeeze(Lmimo(:,:,t))), 1:size(Lmimo,3)); -end -#+end_src - -#+begin_src matlab :exports none -%% Bode plot for the transfer function from u to dLm -figure; -hold on; -for i_mass = 3 - for i = 1 - plot(real(Ldet(i_mass,:)), imag(Ldet(i_mass,:)), ... - '-', 'color', colors(i_mass+1, :)); - end -end -hold off; -set(gca, 'XScale', 'lin'); set(gca, 'YScale', 'lin'); -xlabel('Real'); ylabel('Imag'); -xlim([-10, 1]); ylim([-4, 4]); -#+end_src - -MIMO Nyquist with eigenvalues -#+begin_src matlab -Kdm = Kd*eye(6); - -Ldet = zeros(3, 6, length(fb(i_lim))); - -for i = 1:3 - Lmimo = pagemtimes(permute(G_damp_m{i}(i_lim,:,:), [2,3,1]),squeeze(freqresp(Kdm, fb(i_lim), 'Hz'))); - for i_f = 1:length(fb(i_lim)) - Ldet(i,:, i_f) = eig(squeeze(Lmimo(:,:,i_f))); - end -end -#+end_src - -#+begin_src matlab :exports none -%% Bode plot for the transfer function from u to dLm -figure; -hold on; -for i_mass = 1 - for i = 1:6 - plot(real(squeeze(Ldet(i_mass, i,:))), imag(squeeze(Ldet(i_mass, i,:))), ... - '-', 'color', colors(i_mass+1, :)); - end -end -hold off; -set(gca, 'XScale', 'lin'); set(gca, 'YScale', 'lin'); -xlabel('Real'); ylabel('Imag'); -xlim([-10, 1]); ylim([-4, 2]); -#+end_src -* Noise Budgeting :noexport: -** Introduction :ignore: - -Noise sources: -- PD200 => plant -- DAC => plant x 20 -- Encoder => direct output -- ADC (Force Sensor) => added when closing the loop (controller + plant) - -Disturbances Sources: -- Ground motion - -** 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 - -** Measurements -#+begin_src matlab -noise_enc = load('noise_meas_100s_20kHz.mat', 't', 'x'); -noise_enc.Ts = (noise_enc.t(end) - (noise_enc.t(1)))/(length(noise_enc.t)-1); -noise_enc.win = hanning(ceil(1/noise_enc.Ts)); -noise_enc.x = noise_enc.x - noise_enc.x(1); -[noise_enc.pxx, noise_enc.f] = pwelch(noise_enc.x, noise_enc.win, Noverlap, Nfft, 1/noise_enc.Ts); -#+end_src - -#+begin_src matlab :exports none -noise_ol = load('noise_meas_2m_ol.mat', 't', 'Vs', 'de'); -noise_ol.Ts = (noise_ol.t(end) - (noise_ol.t(1)))/(length(noise_ol.t)-1); -noise_ol.win = hanning(ceil(1/noise_ol.Ts)); -[noise_ol.pxx, noise_ol.f] = pwelch(noise_ol.de(:,1), noise_ol.win, Noverlap, Nfft, 1/noise_ol.Ts); -#+end_src - -#+begin_src matlab :exports none -noise_iff = load('noise_meas_2m_iff.mat', 't', 'de'); -noise_iff.Ts = (noise_iff.t(end) - (noise_iff.t(1)))/(length(noise_iff.t)-1); -noise_iff.win = hanning(ceil(1/noise_iff.Ts)); -[noise_iff.pxx, noise_iff.f] = pwelch(noise_iff.de(:,1), noise_iff.win, Noverlap, Nfft, 1/noise_iff.Ts); -#+end_src - -#+begin_src matlab :exports none -figure; -hold on; -plot(noise_ol.f, sqrt(noise_ol.pxx), 'DisplayName', 'OL'); -plot(noise_iff.f, sqrt(noise_iff.pxx), 'DisplayName', 'IFF'); -plot(noise_enc.f, sqrt(noise_enc.pxx), 'DisplayName', 'Encoder'); -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -xlabel('Frequency [Hz]'); ylabel('ASD [$m/\sqrt{Hz}$]'); -legend('location', 'northeast'); -xlim([1, Fs/2]); ylim([1e-11, 1e-7]); -#+end_src - - * Bibliography :ignore: #+latex: \printbibliography[heading=bibintoc,title={Bibliography}] -* Functions :noexport: +* Glossary :ignore: +[[printglossaries:]] + +# #+latex: \printglossary[type=\acronymtype] +# #+latex: \printglossary[type=\glossarytype] +# #+latex: \printglossary + +* Matlab Functions :noexport: ** =generateXYZTrajectory= :PROPERTIES: :header-args:matlab+: :tangle matlab/src/generateXYZTrajectory.m @@ -7112,3 +6785,5 @@ i_masses = 0:3; %% Frequency Vector freqs = 2*logspace(1, 3, 1000); #+END_SRC + +* Footnotes