** 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_de'); sim_iff = load('sim_iff_vib_table_m.mat', 'G_de'); #+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_de{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_de'); #+end_src #+begin_src matlab :exports none %% Compute the Loop Gain L_frf = pagemtimes(permute(frf_iff.G_de{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_de'); sim_iff = load('sim_iff_vib_table_m.mat', 'G_de'); #+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_de{2}; G_coupled = sim_iff.G_de{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_de_sta = zeros(size(frf_coupled)); for i = 1:length(frf_iff.f) G_de_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_de_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_de_sta(:,i,i)), ... 'DisplayName', sprintf('$y_%i/u_%i$', i, i)); end plot(frf_iff.f, abs(G_de_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_de_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_de_sta(i,:,:)).*inv(squeeze(G_de_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_de_wc = zeros(size(frf_coupled)); for i = 1:length(frf_iff.f) G_de_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_de_wc(:,i,j)), 'color', [0,0,0,0.2], ... 'HandleVisibility', 'off'); end end for i = 1:6 plot(frf_iff.f, abs(G_de_wc(:,i,i)), ... 'DisplayName', sprintf('$y_%i/u_%i$', i, i)); end plot(frf_iff.f, abs(G_de_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_de_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_de_wc(i,:,:)).*inv(squeeze(G_de_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_de_svd = zeros(size(frf_coupled)); for i = 1:length(frf_iff.f) G_de_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_de_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_de_svd(:,i,i)), ... 'DisplayName', sprintf('$y_%i/u_%i$', i, i)); end plot(frf_iff.f, abs(G_de_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_de_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_de_svd(i,:,:)).*inv(squeeze(G_de_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_de_inv = zeros(size(frf_coupled)); for i = 1:length(frf_iff.f) G_de_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_de_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_de_inv(:,i,i)), ... 'DisplayName', sprintf('$y_%i/u_%i$', i, i)); end plot(frf_iff.f, abs(G_de_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_de_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_de_inv(i,:,:)).*inv(squeeze(G_de_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_de_J_cok = zeros(size(frf_coupled)); for i = 1:length(frf_iff.f) G_de_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_de_J_cok(:,i,j)), 'color', [0,0,0,0.2], ... 'HandleVisibility', 'off'); end end set(gca,'ColorOrderIndex',1) plot(frf_iff.f, abs(G_de_J_cok(:,1,1)), ... 'DisplayName', '$D_x/\tilde{\mathcal{F}}_x$'); plot(frf_iff.f, abs(G_de_J_cok(:,2,2)), ... 'DisplayName', '$D_y/\tilde{\mathcal{F}}_y$'); plot(frf_iff.f, abs(G_de_J_cok(:,3,3)), ... 'DisplayName', '$D_z/\tilde{\mathcal{F}}_z$'); plot(frf_iff.f, abs(G_de_J_cok(:,4,4)), ... 'DisplayName', '$R_x/\tilde{\mathcal{M}}_x$'); plot(frf_iff.f, abs(G_de_J_cok(:,5,5)), ... 'DisplayName', '$R_y/\tilde{\mathcal{M}}_y$'); plot(frf_iff.f, abs(G_de_J_cok(:,6,6)), ... 'DisplayName', '$R_z/\tilde{\mathcal{M}}_z$'); plot(frf_iff.f, abs(G_de_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_de_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_de_J_cok(i_100,:,:))))); for i = 1:length(frf_iff.f) G_de_J_cok(i,:,:) = squeeze(G_de_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_de_J_cok(:,i,j)), 'color', [0,0,0,0.2], ... 'HandleVisibility', 'off'); end end set(gca,'ColorOrderIndex',1) plot(frf_iff.f, abs(G_de_J_cok(:,1,1)), ... 'DisplayName', '$D_x/\tilde{\mathcal{F}}_x$'); plot(frf_iff.f, abs(G_de_J_cok(:,2,2)), ... 'DisplayName', '$D_y/\tilde{\mathcal{F}}_y$'); plot(frf_iff.f, abs(G_de_J_cok(:,3,3)), ... 'DisplayName', '$D_z/\tilde{\mathcal{F}}_z$'); plot(frf_iff.f, abs(G_de_J_cok(:,4,4)), ... 'DisplayName', '$R_x/\tilde{\mathcal{M}}_x$'); plot(frf_iff.f, abs(G_de_J_cok(:,5,5)), ... 'DisplayName', '$R_y/\tilde{\mathcal{M}}_y$'); plot(frf_iff.f, abs(G_de_J_cok(:,6,6)), ... 'DisplayName', '$R_z/\tilde{\mathcal{M}}_z$'); plot(frf_iff.f, abs(G_de_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_de_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_de_J_cok(i,:,:)).*inv(squeeze(G_de_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_de_J_com = zeros(size(frf_coupled)); for i = 1:length(frf_iff.f) G_de_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_de_J_com(i_100,:,:))))); for i = 1:length(frf_iff.f) G_de_J_com(i,:,:) = squeeze(G_de_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_de_J_com(:,i,j)), 'color', [0,0,0,0.2], ... 'HandleVisibility', 'off'); end end set(gca,'ColorOrderIndex',1) plot(frf_iff.f, abs(G_de_J_com(:,1,1)), ... 'DisplayName', '$D_x/\tilde{\mathcal{F}}_x$'); plot(frf_iff.f, abs(G_de_J_com(:,2,2)), ... 'DisplayName', '$D_y/\tilde{\mathcal{F}}_y$'); plot(frf_iff.f, abs(G_de_J_com(:,3,3)), ... 'DisplayName', '$D_z/\tilde{\mathcal{F}}_z$'); plot(frf_iff.f, abs(G_de_J_com(:,4,4)), ... 'DisplayName', '$R_x/\tilde{\mathcal{M}}_x$'); plot(frf_iff.f, abs(G_de_J_com(:,5,5)), ... 'DisplayName', '$R_y/\tilde{\mathcal{M}}_y$'); plot(frf_iff.f, abs(G_de_J_com(:,6,6)), ... 'DisplayName', '$R_z/\tilde{\mathcal{M}}_z$'); plot(frf_iff.f, abs(G_de_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_de_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_de_J_com(i,:,:)).*inv(squeeze(G_de_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_de{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_de_sta_b = zeros(size(frf_new)); for i = 1:length(frf_iff.f) G_de_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_de_sta_b(i,:,:)).*inv(squeeze(G_de_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_de_wc_b = zeros(size(frf_new)); for i = 1:length(frf_iff.f) G_de_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_de_wc_b(i,:,:)).*inv(squeeze(G_de_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_de_svd_b = zeros(size(frf_new)); for i = 1:length(frf_iff.f) G_de_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_de_svd_b(i,:,:)).*inv(squeeze(G_de_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_de_inv_b = zeros(size(frf_new)); for i = 1:length(frf_iff.f) G_de_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_de_inv_b(i,:,:)).*inv(squeeze(G_de_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_de_J_cok_b = zeros(size(frf_new)); for i = 1:length(frf_iff.f) G_de_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_de_J_cok_b(i,:,:)).*inv(squeeze(G_de_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_de_J_com_b = zeros(size(frf_new)); for i = 1:length(frf_iff.f) G_de_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_de_J_com_b(i,:,:)).*inv(squeeze(G_de_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_de'); sim_iff = load('sim_iff_vib_table_m.mat', 'G_de'); #+end_src **** Using Jacobian evaluated at the center of stiffness ***** Decoupled Plant #+begin_src matlab G_nom = frf_iff.G_de{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_de_J_cok = zeros(size(G_nom)); for i = 1:length(frf_iff.f) G_de_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_de_J_cok(i_100,:,:))))); for i = 1:length(frf_iff.f) G_de_J_cok(i,:,:) = squeeze(G_de_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_de_J_cok(:,i,j)), 'color', [0,0,0,0.2], ... 'HandleVisibility', 'off'); end end set(gca,'ColorOrderIndex',1) plot(frf_iff.f, abs(G_de_J_cok(:,1,1)), ... 'DisplayName', '$D_x/\tilde{\mathcal{F}}_x$'); plot(frf_iff.f, abs(G_de_J_cok(:,2,2)), ... 'DisplayName', '$D_y/\tilde{\mathcal{F}}_y$'); plot(frf_iff.f, abs(G_de_J_cok(:,3,3)), ... 'DisplayName', '$D_z/\tilde{\mathcal{F}}_z$'); plot(frf_iff.f, abs(G_de_J_cok(:,4,4)), ... 'DisplayName', '$R_x/\tilde{\mathcal{M}}_x$'); plot(frf_iff.f, abs(G_de_J_cok(:,5,5)), ... 'DisplayName', '$R_y/\tilde{\mathcal{M}}_y$'); plot(frf_iff.f, abs(G_de_J_cok(:,6,6)), ... 'DisplayName', '$R_z/\tilde{\mathcal{M}}_z$'); plot(frf_iff.f, abs(G_de_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_de_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_de_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_de_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_de_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_de_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_de{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_de{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_de_svd = zeros(size(G_nom)); for i = 1:length(frf_iff.f) G_de_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_de_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_de_svd(:,i,i)), ... 'DisplayName', sprintf('$y_%i/u_%i$', i, i)); end plot(frf_iff.f, abs(G_de_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_de_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_de_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_de{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_de{2}; % Nominal Plant G_model = sim_iff.G_de{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_de{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_de{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_de_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_de{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 ** Other Backups *** 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]] *** 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 *** Rigidification of the added payloads - [ ] figure #+begin_src matlab %% Load Identification Data meas_added_mass = {}; for i_strut = 1:6 meas_added_mass(i_strut) = {load(sprintf('frf_data_exc_strut_%i_spindle_1m_solid.mat', i_strut), 't', 'Va', 'Vs', 'de')}; end #+end_src Finally the $6 \times 6$ transfer function matrices from $\bm{u}$ to $d\bm{\mathcal{L}}_m$ and from $\bm{u}$ to $\bm{\tau}_m$ are identified: #+begin_src matlab %% DVF Plant (transfer function from u to dLm) G_de = zeros(length(f), 6, 6); for i_strut = 1:6 G_de(:,:,i_strut) = tfestimate(meas_added_mass{i_strut}.Va, meas_added_mass{i_strut}.de, win, Noverlap, Nfft, 1/Ts); end %% IFF Plant (transfer function from u to taum) G_Vs = zeros(length(f), 6, 6); for i_strut = 1:6 G_Vs(:,:,i_strut) = tfestimate(meas_added_mass{i_strut}.Va, meas_added_mass{i_strut}.Vs, win, Noverlap, Nfft, 1/Ts); end #+end_src #+begin_src matlab :exports none %% Bode plot for the transfer function from u to dLm - Several payloads figure; tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); ax1 = nexttile([2,1]); hold on; % Diagonal terms for i = 1:6 plot(frf_ol.f, abs(frf_ol.G_de{1}(:,i, i)), 'color', colors(1,:)); plot(f, abs(G_de(:,i, i)), 'color', colors(2,:)); end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); xlabel('Frequency [Hz]'); ylabel('Amplitude [m/V]'); ylim([1e-8, 1e-3]); xlim([20, 2e3]); #+end_src #+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 = 1:6 plot(frf_ol.f, abs(frf_ol.G_de(:,i, i)), 'color', colors(1,:)); plot(f, abs(G_de(:,i, i)), 'color', colors(2,:)); end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); xlabel('Frequency [Hz]'); ylabel('Amplitude [m/V]'); ylim([1e-8, 1e-3]); xlim([10, 1e3]); #+end_src *** Table with signals #+name: tab:list_signals #+caption: List of signals #+attr_latex: :environment tabularx :width \linewidth :align Xllll #+attr_latex: :center t :booktabs t | | *Unit* | *Matlab* | *Vector* | *Elements* | |------------------------------------+-----------+-----------+-----------------------+----------------------| | Control Input (wanted DAC voltage) | =[V]= | =u= | $\bm{u}$ | $u_i$ | | DAC Output Voltage | =[V]= | =u= | $\tilde{\bm{u}}$ | $\tilde{u}_i$ | | PD200 Output Voltage | =[V]= | =ua= | $\bm{u}_a$ | $u_{a,i}$ | | Actuator applied force | =[N]= | =tau= | $\bm{\tau}$ | $\tau_i$ | |------------------------------------+-----------+-----------+-----------------------+----------------------| | Strut motion | =[m]= | =dL= | $d\bm{\mathcal{L}}$ | $d\mathcal{L}_i$ | | Encoder measured displacement | =[m]= | =dLm= | $d\bm{\mathcal{L}}_m$ | $d\mathcal{L}_{m,i}$ | |------------------------------------+-----------+-----------+-----------------------+----------------------| | Force Sensor strain | =[m]= | =epsilon= | $\bm{\epsilon}$ | $\epsilon_i$ | | Force Sensor Generated Voltage | =[V]= | =taum= | $\tilde{\bm{\tau}}_m$ | $\tilde{\tau}_{m,i}$ | | Measured Generated Voltage | =[V]= | =taum= | $\bm{\tau}_m$ | $\tau_{m,i}$ | |------------------------------------+-----------+-----------+-----------------------+----------------------| | Motion of the top platform | =[m,rad]= | =dX= | $d\bm{\mathcal{X}}$ | $d\mathcal{X}_i$ | | Metrology measured displacement | =[m,rad]= | =dXm= | $d\bm{\mathcal{X}}_m$ | $d\mathcal{X}_{m,i}$ | *** RGA The RGA-number, which is a measure of the interaction in the system, is computed for the transfer function matrix from $\mathbf{u}$ to $d\mathbf{\mathcal{L}}_m$ for all the payloads. The obtained numbers are compared in Figure ref:fig:test_nhexa_rga_num_ol_masses. #+begin_src matlab :exports none %% Decentralized RGA - Undamped Plant RGA_num = zeros(length(frf_ol.f), 4); for i_mass = [0:3] for i = 1:length(frf_ol.f) RGA_num(i, i_mass+1) = sum(sum(abs(eye(6) - squeeze(frf_ol.G_de{i_mass+1}(i,:,:)).*inv(squeeze(frf_ol.G_de{i_mass+1}(i,:,:))).'))); end end #+end_src #+begin_src matlab :exports none %% RGA for Decentralized plant figure; hold on; for i_mass = [0:3] plot(frf_ol.f, RGA_num(:,i_mass+1), '-', 'color', colors(i_mass+1,:), ... 'DisplayName', sprintf('RGA-num - %i mass', 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 :exports none %% Decentralized RGA - Undamped Plant RGA = zeros(length(frf_ol.f), 6, 6); for i = 1:length(frf_ol.f) RGA(i, :, :) = squeeze(frf_ol.G_Vs{1}(i,:,:)).*inv(squeeze(frf_ol.G_Vs{1}(i,:,:))).'; end #+end_src #+begin_src matlab :exports none %% RGA figure; hold on; for i = 1:6 plot(frf_ol.f, abs(RGA(:,i,i)), 'k-') end for i = 1:5 for j = i+1:6 plot(frf_ol.f, abs(RGA(:,i,j)), 'r-') end end hold off; 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/rga_num_ol_masses.pdf', 'width', 'wide', 'height', 'normal'); #+end_src #+name: fig:test_nhexa_rga_num_ol_masses #+caption: RGA-number for the open-loop transfer function from $\mathbf{u}$ to $d\mathbf{\mathcal{L}}_m$ #+RESULTS: [[file:figs/test_nhexa_rga_num_ol_masses.png]] #+begin_important From Figure ref:fig:test_nhexa_rga_num_ol_masses, it is clear that the coupling is quite large starting from the first suspension mode of the nano-hexapod. Therefore, is the payload's mass is increase, the coupling in the system start to become unacceptably large at lower frequencies. #+end_important