#+TITLE: Nano-Hexapod - Test Bench :DRAWER: #+LANGUAGE: en #+EMAIL: dehaeze.thomas@gmail.com #+AUTHOR: Dehaeze Thomas #+HTML_LINK_HOME: ../index.html #+HTML_LINK_UP: ../index.html #+HTML_HEAD: #+HTML_HEAD: #+BIND: org-latex-image-default-option "scale=1" #+BIND: org-latex-image-default-width "" #+LaTeX_CLASS: scrreprt #+LaTeX_CLASS_OPTIONS: [a4paper, 10pt, DIV=12, parskip=full] #+LaTeX_HEADER_EXTRA: \input{preamble.tex} #+PROPERTY: header-args:matlab :session *MATLAB* #+PROPERTY: header-args:matlab+ :comments org #+PROPERTY: header-args:matlab+ :exports both #+PROPERTY: header-args:matlab+ :results none #+PROPERTY: header-args:matlab+ :eval no-export #+PROPERTY: header-args:matlab+ :noweb yes #+PROPERTY: header-args:matlab+ :mkdirp yes #+PROPERTY: header-args:matlab+ :output-dir figs #+PROPERTY: header-args:latex :headers '("\\usepackage{tikz}" "\\usepackage{import}" "\\import{$HOME/Cloud/tikz/org/}{config.tex}") #+PROPERTY: header-args:latex+ :imagemagick t :fit yes #+PROPERTY: header-args:latex+ :iminoptions -scale 100% -density 150 #+PROPERTY: header-args:latex+ :imoutoptions -quality 100 #+PROPERTY: header-args:latex+ :results file raw replace #+PROPERTY: header-args:latex+ :buffer no #+PROPERTY: header-args:latex+ :tangle no #+PROPERTY: header-args:latex+ :eval no-export #+PROPERTY: header-args:latex+ :exports results #+PROPERTY: header-args:latex+ :mkdirp yes #+PROPERTY: header-args:latex+ :output-dir figs #+PROPERTY: header-args:latex+ :post pdf2svg(file=*this*, ext="png") :END: #+begin_export html

This report is also available as a pdf.

#+end_export #+latex: \clearpage * Introduction :ignore: This document is dedicated to the experimental study of the nano-hexapod shown in Figure [[fig:picture_bench_granite_nano_hexapod]]. #+name: fig:picture_bench_granite_nano_hexapod #+caption: Nano-Hexapod #+attr_latex: :width \linewidth [[file:figs/IMG_20210608_152917.jpg]] #+begin_note Here are the documentation of the equipment used for this test bench (lots of them are shwon in Figure [[fig:picture_bench_granite_overview]]): - Voltage Amplifier: PiezoDrive [[file:doc/PD200-V7-R1.pdf][PD200]] - Amplified Piezoelectric Actuator: Cedrat [[file:doc/APA300ML.pdf][APA300ML]] - DAC/ADC: Speedgoat [[file:doc/IO131-OEM-Datasheet.pdf][IO313]] - Encoder: Renishaw [[file:doc/L-9517-9678-05-A_Data_sheet_VIONiC_series_en.pdf][Vionic]] and used [[file:doc/L-9517-9862-01-C_Data_sheet_RKLC_EN.pdf][Ruler]] - Interferometers: Attocube #+end_note #+name: fig:picture_bench_granite_overview #+caption: Nano-Hexapod and the control electronics #+attr_latex: :width \linewidth [[file:figs/IMG_20210608_154722.jpg]] In Figure [[fig:nano_hexapod_signals]] is shown a block diagram of the experimental setup. When possible, the notations are consistent with this diagram and summarized in Table [[tab:list_signals]]. #+begin_src latex :file nano_hexapod_signals.pdf \definecolor{instrumentation}{rgb}{0, 0.447, 0.741} \definecolor{mechanics}{rgb}{0.8500, 0.325, 0.098} \begin{tikzpicture} % Blocs \node[block={4.0cm}{3.0cm}, fill=mechanics!20!white] (nano_hexapod) {Mechanics}; \coordinate[] (inputF) at (nano_hexapod.west); \coordinate[] (outputL) at ($(nano_hexapod.south east)!0.8!(nano_hexapod.north east)$); \coordinate[] (outputF) at ($(nano_hexapod.south east)!0.2!(nano_hexapod.north east)$); \node[block, left= 0.8 of inputF, fill=instrumentation!20!white, align=center] (F_stack) {\tiny Actuator \\ \tiny stacks}; \node[block, left= 0.8 of F_stack, fill=instrumentation!20!white] (PD200) {PD200}; \node[DAC, left= 0.8 of PD200, fill=instrumentation!20!white] (F_DAC) {DAC}; \node[block, right=0.8 of outputF, fill=instrumentation!20!white, align=center] (Fm_stack){\tiny Sensor \\ \tiny stack}; \node[ADC, right=0.8 of Fm_stack,fill=instrumentation!20!white] (Fm_ADC) {ADC}; \node[block, right=0.8 of outputL, fill=instrumentation!20!white] (encoder) {\tiny Encoder}; % Connections and labels \draw[->] ($(F_DAC.west)+(-0.8,0)$) node[above right]{$\bm{u}$} node[below right]{$[V]$} -- node[sloped]{$/$} (F_DAC.west); \draw[->] (F_DAC.east) -- node[midway, above]{$\tilde{\bm{u}}$}node[midway, below]{$[V]$} (PD200.west); \draw[->] (PD200.east) -- node[midway, above]{$\bm{u}_a$}node[midway, below]{$[V]$} (F_stack.west); \draw[->] (F_stack.east) -- (inputF) node[above left]{$\bm{\tau}$}node[below left]{$[N]$}; \draw[->] (outputF) -- (Fm_stack.west) node[above left]{$\bm{\epsilon}$} node[below left]{$[m]$}; \draw[->] (Fm_stack.east) -- node[midway, above]{$\tilde{\bm{\tau}}_m$}node[midway, below]{$[V]$} (Fm_ADC.west); \draw[->] (Fm_ADC.east) -- node[sloped]{$/$} ++(0.8, 0)coordinate(end) node[above left]{$\bm{\tau}_m$}node[below left]{$[V]$}; \draw[->] (outputL) -- (encoder.west) node[above left]{$d\bm{\mathcal{L}}$} node[below left]{$[m]$}; \draw[->] (encoder.east) -- node[sloped]{$/$} (encoder-|end) node[above left]{$d\bm{\mathcal{L}}_m$}node[below left]{$[m]$}; % Nano-Hexapod \begin{scope}[on background layer] \node[fit={(F_stack.west|-nano_hexapod.south) (Fm_stack.east|-nano_hexapod.north)}, fill=black!20!white, draw, inner sep=2pt] (system) {}; \node[above] at (system.north) {Nano-Hexapod}; \end{scope} \end{tikzpicture} #+end_src #+name: fig:nano_hexapod_signals #+caption: Block diagram of the system with named signals #+attr_latex: :scale 1 [[file:figs/nano_hexapod_signals.png]] #+name: tab:list_signals #+caption: List of signals #+attr_latex: :environment tabularx :width \linewidth :align Xllll #+attr_latex: :center t :booktabs t :float t | | *Unit* | *Matlab* | *Vector* | *Elements* | |------------------------------------+-----------+-----------+-----------------------+----------------------| | Control Input (wanted DAC voltage) | =[V]= | =u= | $\bm{u}$ | $u_i$ | | DAC Output Voltage | =[V]= | =u= | $\tilde{\bm{u}}$ | $\tilde{u}_i$ | | PD200 Output Voltage | =[V]= | =ua= | $\bm{u}_a$ | $u_{a,i}$ | | Actuator applied force | =[N]= | =tau= | $\bm{\tau}$ | $\tau_i$ | |------------------------------------+-----------+-----------+-----------------------+----------------------| | Strut motion | =[m]= | =dL= | $d\bm{\mathcal{L}}$ | $d\mathcal{L}_i$ | | Encoder measured displacement | =[m]= | =dLm= | $d\bm{\mathcal{L}}_m$ | $d\mathcal{L}_{m,i}$ | |------------------------------------+-----------+-----------+-----------------------+----------------------| | Force Sensor strain | =[m]= | =epsilon= | $\bm{\epsilon}$ | $\epsilon_i$ | | Force Sensor Generated Voltage | =[V]= | =taum= | $\tilde{\bm{\tau}}_m$ | $\tilde{\tau}_{m,i}$ | | Measured Generated Voltage | =[V]= | =taum= | $\bm{\tau}_m$ | $\tau_{m,i}$ | |------------------------------------+-----------+-----------+-----------------------+----------------------| | Motion of the top platform | =[m,rad]= | =dX= | $d\bm{\mathcal{X}}$ | $d\mathcal{X}_i$ | | Metrology measured displacement | =[m,rad]= | =dXm= | $d\bm{\mathcal{X}}_m$ | $d\mathcal{X}_{m,i}$ | This document is divided in the following sections: - Section [[sec:encoders_struts]]: the dynamics of the nano-hexapod when the encoders are fixed to the struts is studied. - Section [[sec:encoders_plates]]: the same is done when the encoders are fixed to the plates. - Section [[sec:decentralized_hac_iff]]: a decentralized HAC-LAC strategy is studied and implemented. * Encoders fixed to the Struts - Dynamics <> ** Introduction :ignore: In this section, the encoders are fixed to the struts. It is divided in the following sections: - Section [[sec:enc_struts_plant_id]]: the transfer function matrix from the actuators to the force sensors and to the encoders is experimentally identified. - Section [[sec:enc_struts_comp_simscape]]: the obtained FRF matrix is compared with the dynamics of the simscape model - Section [[sec:enc_struts_iff]]: decentralized Integral Force Feedback (IFF) is applied and its performances are evaluated. - Section [[sec:enc_struts_modal_analysis]]: a modal analysis of the nano-hexapod is performed ** Identification of the dynamics <> *** Introduction :ignore: *** Matlab Init :noexport:ignore: #+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) <> #+end_src #+begin_src matlab :exports none :results silent :noweb yes <> #+end_src #+begin_src matlab :tangle no addpath('./matlab/mat/'); addpath('./matlab/src/'); addpath('./matlab/'); #+end_src #+begin_src matlab :eval no addpath('./mat/'); addpath('./src/'); #+end_src *** Load Measurement Data #+begin_src matlab %% Load Identification Data meas_data_lf = {}; for i = 1:6 meas_data_lf(i) = {load(sprintf('mat/frf_data_exc_strut_%i_noise_lf.mat', i), 't', 'Va', 'Vs', 'de')}; meas_data_hf(i) = {load(sprintf('mat/frf_data_exc_strut_%i_noise_hf.mat', i), 't', 'Va', 'Vs', 'de')}; end #+end_src *** Spectral Analysis - Setup #+begin_src matlab %% Setup useful variables % Sampling Time [s] Ts = (meas_data_lf{1}.t(end) - (meas_data_lf{1}.t(1)))/(length(meas_data_lf{1}.t)-1); % Sampling Frequency [Hz] Fs = 1/Ts; % Hannning Windows win = hanning(ceil(1*Fs)); % And we get the frequency vector [~, f] = tfestimate(meas_data_lf{1}.Va, meas_data_lf{1}.de, win, [], [], 1/Ts); i_lf = f < 250; % Points for low frequency excitation i_hf = f > 250; % Points for high frequency excitation #+end_src *** Transfer function from Actuator to Encoder First, let's compute the coherence from the excitation voltage and the displacement as measured by the encoders (Figure [[fig:enc_struts_dvf_coh]]). #+begin_src matlab %% Coherence coh_dvf = zeros(length(f), 6, 6); for i = 1:6 coh_dvf_lf = mscohere(meas_data_lf{i}.Va, meas_data_lf{i}.de, win, [], [], 1/Ts); coh_dvf_hf = mscohere(meas_data_hf{i}.Va, meas_data_hf{i}.de, win, [], [], 1/Ts); coh_dvf(:,:,i) = [coh_dvf_lf(i_lf, :); coh_dvf_hf(i_hf, :)]; end #+end_src #+begin_src matlab :exports none %% Coherence for the transfer function from u to dLm figure; hold on; for i = 1:5 for j = i+1:6 plot(f, coh_dvf(:, i, j), 'color', [0, 0, 0, 0.2], ... 'HandleVisibility', 'off'); end end for i =1:6 set(gca,'ColorOrderIndex',i) plot(f, coh_dvf(:, i, i), ... 'DisplayName', sprintf('$G_{dvf}(%i,%i)$', i, i)); end plot(f, coh_dvf(:, 1, 2), 'color', [0, 0, 0, 0.2], ... 'DisplayName', '$G_{dvf}(i,j)$'); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); xlabel('Frequency [Hz]'); ylabel('Coherence [-]'); xlim([20, 2e3]); ylim([0, 1]); legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 3); #+end_src #+begin_src matlab :tangle no :exports results :results file replace exportFig('figs/enc_struts_dvf_coh.pdf', 'width', 'wide', 'height', 'normal'); #+end_src #+name: fig:enc_struts_dvf_coh #+caption: Obtained coherence for the DVF plant #+RESULTS: [[file:figs/enc_struts_dvf_coh.png]] Then the 6x6 transfer function matrix is estimated (Figure [[fig:enc_struts_dvf_frf]]). #+begin_src matlab %% DVF Plant (transfer function from u to dLm) G_dvf = zeros(length(f), 6, 6); for i = 1:6 G_dvf_lf = tfestimate(meas_data_lf{i}.Va, meas_data_lf{i}.de, win, [], [], 1/Ts); G_dvf_hf = tfestimate(meas_data_hf{i}.Va, meas_data_hf{i}.de, win, [], [], 1/Ts); G_dvf(:,:,i) = [G_dvf_lf(i_lf, :); G_dvf_hf(i_hf, :)]; end #+end_src #+begin_src matlab :exports none %% Bode plot for the transfer function from u to dLm figure; tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); ax1 = nexttile([2,1]); hold on; for i = 1:5 for j = i+1:6 plot(f, abs(G_dvf(:, i, j)), 'color', [0, 0, 0, 0.2], ... 'HandleVisibility', 'off'); end end for i =1:6 set(gca,'ColorOrderIndex',i) plot(f, abs(G_dvf(:,i, i)), ... 'DisplayName', sprintf('$G_{dvf}(%i,%i)$', i, i)); set(gca,'ColorOrderIndex',i) plot(f, abs(G_dvf(:,i, i)), ... 'HandleVisibility', 'off'); end plot(f, abs(G_dvf(:, 1, 2)), 'color', [0, 0, 0, 0.2], ... 'DisplayName', '$G_{dvf}(i,j)$'); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); ylabel('Amplitude $d_e/V_a$ [m/V]'); set(gca, 'XTickLabel',[]); ylim([1e-9, 1e-3]); legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 3); ax2 = nexttile; hold on; for i =1:6 set(gca,'ColorOrderIndex',i) plot(f, 180/pi*angle(G_dvf(:,i, i))); end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); hold off; yticks(-360:90:360); linkaxes([ax1,ax2],'x'); xlim([20, 2e3]); #+end_src #+begin_src matlab :tangle no :exports results :results file replace exportFig('figs/enc_struts_dvf_frf.pdf', 'width', 'wide', 'height', 'tall'); #+end_src #+name: fig:enc_struts_dvf_frf #+caption: Measured FRF for the DVF plant #+RESULTS: [[file:figs/enc_struts_dvf_frf.png]] *** Transfer function from Actuator to Force Sensor First, let's compute the coherence from the excitation voltage and the displacement as measured by the encoders (Figure [[fig:enc_struts_iff_coh]]). #+begin_src matlab %% Coherence for the IFF plant coh_iff = zeros(length(f), 6, 6); for i = 1:6 coh_iff_lf = mscohere(meas_data_lf{i}.Va, meas_data_lf{i}.Vs, win, [], [], 1/Ts); coh_iff_hf = mscohere(meas_data_hf{i}.Va, meas_data_hf{i}.Vs, win, [], [], 1/Ts); coh_iff(:,:,i) = [coh_iff_lf(i_lf, :); coh_iff_hf(i_hf, :)]; end #+end_src #+begin_src matlab :exports none %% Coherence of the IFF Plant (transfer function from u to taum) figure; hold on; for i = 1:5 for j = i+1:6 plot(f, coh_iff(:, i, j), 'color', [0, 0, 0, 0.2], ... 'HandleVisibility', 'off'); end end for i =1:6 set(gca,'ColorOrderIndex',i) plot(f, coh_iff(:,i, i), ... 'DisplayName', sprintf('$G_{iff}(%i,%i)$', i, i)); end plot(f, coh_iff(:, 1, 2), 'color', [0, 0, 0, 0.2], ... 'DisplayName', '$G_{iff}(i,j)$'); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); xlabel('Frequency [Hz]'); ylabel('Coherence [-]'); xlim([20, 2e3]); ylim([0, 1]); legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 3); #+end_src #+begin_src matlab :tangle no :exports results :results file replace exportFig('figs/enc_struts_iff_coh.pdf', 'width', 'wide', 'height', 'normal'); #+end_src #+name: fig:enc_struts_iff_coh #+caption: Obtained coherence for the IFF plant #+RESULTS: [[file:figs/enc_struts_iff_coh.png]] Then the 6x6 transfer function matrix is estimated (Figure [[fig:enc_struts_iff_frf]]). #+begin_src matlab %% IFF Plant G_iff = zeros(length(f), 6, 6); for i = 1:6 G_iff_lf = tfestimate(meas_data_lf{i}.Va, meas_data_lf{i}.Vs, win, [], [], 1/Ts); G_iff_hf = tfestimate(meas_data_hf{i}.Va, meas_data_hf{i}.Vs, win, [], [], 1/Ts); G_iff(:,:,i) = [G_iff_lf(i_lf, :); G_iff_hf(i_hf, :)]; end #+end_src #+begin_src matlab :exports none %% Bode plot of the IFF Plant (transfer function from u to taum) figure; tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); ax1 = nexttile([2,1]); hold on; for i = 1:5 for j = i+1:6 plot(f, abs(G_iff(:, i, j)), 'color', [0, 0, 0, 0.2], ... 'HandleVisibility', 'off'); end end for i =1:6 set(gca,'ColorOrderIndex',i) plot(f, abs(G_iff(:,i , i)), ... 'DisplayName', sprintf('$G_{iff}(%i,%i)$', i, i)); end plot(f, abs(G_iff(:, 1, 2)), 'color', [0, 0, 0, 0.2], ... 'DisplayName', '$G_{iff}(i,j)$'); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); ylabel('Amplitude $V_s/V_a$ [V/V]'); set(gca, 'XTickLabel',[]); legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 3); ylim([1e-3, 1e2]); ax2 = nexttile; hold on; for i =1:6 set(gca,'ColorOrderIndex',i) plot(f, 180/pi*angle(G_iff(:,i, i))); end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); hold off; yticks(-360:90:360); linkaxes([ax1,ax2],'x'); xlim([20, 2e3]); #+end_src #+begin_src matlab :tangle no :exports results :results file replace exportFig('figs/enc_struts_iff_frf.pdf', 'width', 'wide', 'height', 'tall'); #+end_src #+name: fig:enc_struts_iff_frf #+caption: Measured FRF for the IFF plant #+RESULTS: [[file:figs/enc_struts_iff_frf.png]] *** Save Identified Plants #+begin_src matlab :tangle no save('matlab/mat/identified_plants_enc_struts.mat', 'f', 'Ts', 'G_iff', 'G_dvf') #+end_src #+begin_src matlab :exports none :eval no save('mat/identified_plants_enc_struts.mat', 'f', 'Ts', 'G_iff', 'G_dvf') #+end_src ** Jacobian :noexport: *** Introduction :ignore: The Jacobian is used to transform the excitation force in the cartesian frame as well as the displacements. Consider the plant shown in Figure [[fig:schematic_jacobian_in_out]] with: - $\tau$ the 6 input voltages (going to the PD200 amplifier and then to the APA) - $d\mathcal{L}$ the relative motion sensor outputs (encoders) - $\bm{\tau}_m$ the generated voltage of the force sensor stacks - $J_a$ and $J_s$ the Jacobians for the actuators and sensors #+begin_src latex :file schematic_jacobian_in_out.pdf \begin{tikzpicture} % Blocs \node[block={2.0cm}{2.0cm}] (P) {Plant}; \coordinate[] (inputF) at (P.west); \coordinate[] (outputL) at ($(P.south east)!0.8!(P.north east)$); \coordinate[] (outputF) at ($(P.south east)!0.2!(P.north east)$); \node[block, left= of inputF] (Ja) {$\bm{J}^{-T}_a$}; \node[block, right= of outputL] (Js) {$\bm{J}^{-1}_s$}; \node[block, right= of outputF] (Jf) {$\bm{J}^{-1}_s$}; % Connections and labels \draw[->] ($(Ja.west)+(-1,0)$) -- (Ja.west) node[above left]{$\bm{\mathcal{F}}$}; \draw[->] (Ja.east) -- (inputF) node[above left]{$\bm{\tau}$}; \draw[->] (outputL) -- (Js.west) node[above left]{$d\bm{\mathcal{L}}$}; \draw[->] (Js.east) -- ++(1, 0) node[above left]{$d\bm{\mathcal{X}}$}; \draw[->] (outputF) -- (Jf.west) node[above left]{$\bm{\tau}_m$}; \draw[->] (Jf.east) -- ++(1, 0) node[above left]{$\bm{\mathcal{F}}_m$}; \end{tikzpicture} #+end_src #+name: fig:schematic_jacobian_in_out #+caption: Plant in the cartesian Frame #+RESULTS: [[file:figs/schematic_jacobian_in_out.png]] First, we load the Jacobian matrix (same for the actuators and sensors). #+begin_src matlab load('jacobian.mat', 'J'); #+end_src *** Matlab Init :noexport:ignore: #+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) <> #+end_src #+begin_src matlab :exports none :results silent :noweb yes <> #+end_src #+begin_src matlab :tangle no addpath('./matlab/mat/'); addpath('./matlab/src/'); addpath('./matlab/'); #+end_src #+begin_src matlab :eval no addpath('./mat/'); addpath('./src/'); #+end_src #+begin_src matlab load('identified_plants_enc_struts.mat', 'f', 'Ts', 'G_iff', 'G_dvf') load('jacobian.mat', 'J'); #+end_src *** DVF Plant The transfer function from $\bm{\mathcal{F}}$ to $d\bm{\mathcal{X}}$ is computed and shown in Figure [[fig:enc_struts_dvf_cart_frf]]. #+begin_src matlab G_dvf_J = permute(pagemtimes(inv(J), pagemtimes(permute(G_dvf, [2 3 1]), inv(J'))), [3 1 2]); #+end_src #+begin_src matlab :exports none labels = {'$D_x/F_{x}$', '$D_y/F_{y}$', '$D_z/F_{z}$', '$R_{x}/M_{x}$', '$R_{y}/M_{y}$', '$R_{R}/M_{z}$'}; figure; tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); ax1 = nexttile([2,1]); hold on; for i = 1:5 for j = i+1:6 plot(f, abs(G_dvf_J(:, i, j)), 'color', [0, 0, 0, 0.2], ... 'HandleVisibility', 'off'); end end for i =1:6 set(gca,'ColorOrderIndex',i) plot(f, abs(G_dvf_J(:,i , i)), ... 'DisplayName', labels{i}); end plot(f, abs(G_dvf_J(:, 1, 2)), 'color', [0, 0, 0, 0.2], ... 'DisplayName', '$D_i/F_j$'); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); ylabel('Amplitude $d_e/V_a$ [m/V]'); set(gca, 'XTickLabel',[]); ylim([1e-7, 1e-1]); legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 3); ax2 = nexttile; hold on; for i =1:6 set(gca,'ColorOrderIndex',i) plot(f, 180/pi*angle(G_dvf_J(:,i , i))); end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); hold off; yticks(-360:90:360); linkaxes([ax1,ax2],'x'); xlim([20, 2e3]); #+end_src #+begin_src matlab :tangle no :exports results :results file replace exportFig('figs/enc_struts_dvf_cart_frf.pdf', 'width', 'wide', 'height', 'tall'); #+end_src #+name: fig:enc_struts_dvf_cart_frf #+caption: Measured FRF for the DVF plant in the cartesian frame #+RESULTS: [[file:figs/enc_struts_dvf_cart_frf.png]] *** IFF Plant The transfer function from $\bm{\mathcal{F}}$ to $\bm{\mathcal{F}}_m$ is computed and shown in Figure [[fig:enc_struts_iff_cart_frf]]. #+begin_src matlab G_iff_J = permute(pagemtimes(inv(J), pagemtimes(permute(G_iff, [2 3 1]), inv(J'))), [3 1 2]); #+end_src #+begin_src matlab :exports none labels = {'$F_{m,x}/F_{x}$', '$F_{m,y}/F_{y}$', '$F_{m,z}/F_{z}$', '$M_{m,x}/M_{x}$', '$M_{m,y}/M_{y}$', '$M_{m,z}/M_{z}$'}; figure; tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); ax1 = nexttile([2,1]); hold on; for i = 1:5 for j = i+1:6 plot(f, abs(G_iff_J(:, i, j)), 'color', [0, 0, 0, 0.2], ... 'HandleVisibility', 'off'); end end for i =1:6 set(gca,'ColorOrderIndex',i) plot(f, abs(G_iff_J(:,i, i)), ... 'DisplayName', labels{i}); end plot(f, abs(G_iff_J(:, 1, 2)), 'color', [0, 0, 0, 0.2], ... 'DisplayName', '$D_i/F_j$'); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); ylabel('Amplitude $d_e/V_a$ [m/V]'); set(gca, 'XTickLabel',[]); ylim([1e-3, 1e4]); legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 3); ax2 = nexttile; hold on; for i =1:6 set(gca,'ColorOrderIndex',i) plot(f, 180/pi*angle(G_iff_J(:,i, i))); end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); hold off; yticks(-360:90:360); linkaxes([ax1,ax2],'x'); xlim([20, 2e3]); #+end_src #+begin_src matlab :tangle no :exports results :results file replace exportFig('figs/enc_struts_iff_cart_frf.pdf', 'width', 'wide', 'height', 'tall'); #+end_src #+name: fig:enc_struts_iff_cart_frf #+caption: Measured FRF for the IFF plant in the cartesian frame #+RESULTS: [[file:figs/enc_struts_iff_cart_frf.png]] ** Comparison with the Simscape Model <> *** Introduction :ignore: In this section, the measured dynamics is compared with the dynamics estimated from the Simscape model. *** Matlab Init :noexport:ignore: #+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) <> #+end_src #+begin_src matlab :exports none :results silent :noweb yes <> #+end_src #+begin_src matlab :tangle no addpath('./matlab/mat/'); addpath('./matlab/src/'); addpath('./matlab/'); #+end_src #+begin_src matlab :eval no addpath('./mat/'); addpath('./src/'); #+end_src #+begin_src matlab :tangle no %% Add all useful folders to the path addpath('matlab/nass-simscape/matlab/nano_hexapod/') addpath('matlab/nass-simscape/STEPS/nano_hexapod/') addpath('matlab/nass-simscape/STEPS/png/') addpath('matlab/nass-simscape/src/') addpath('matlab/nass-simscape/mat/') #+end_src #+begin_src matlab :eval no %% Add all useful folders to the path addpath('nass-simscape/matlab/nano_hexapod/') addpath('nass-simscape/STEPS/nano_hexapod/') addpath('nass-simscape/STEPS/png/') addpath('nass-simscape/src/') addpath('nass-simscape/mat/') #+end_src #+begin_src matlab %% Open Simulink Model mdl = 'nano_hexapod_simscape'; options = linearizeOptions; options.SampleTime = 0; Rx = zeros(1, 7); open(mdl) #+end_src *** Load measured FRF #+begin_src matlab %% Load data load('identified_plants_enc_struts.mat', 'f', 'Ts', 'G_iff', 'G_dvf') #+end_src *** Dynamics from Actuator to Force Sensors #+begin_src matlab %% Initialize Nano-Hexapod n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... 'flex_top_type', '4dof', ... 'motion_sensor_type', 'struts', ... 'actuator_type', '2dof'); #+end_src #+begin_src matlab %% Identify the IFF Plant (transfer function from u to taum) 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, '/dum'], 1, 'openoutput'); io_i = io_i + 1; % Force Sensors Giff = exp(-s*Ts)*linearize(mdl, io, 0.0, options); #+end_src #+begin_src matlab :exports none %% Bode plot of the identified IFF Plant (Simscape) and measured FRF data freqs = 2*logspace(1, 3, 1000); figure; tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); ax1 = nexttile([2,1]); hold on; plot(f, abs(G_iff(:,1, 1)), 'color', [0,0,0,0.2], ... 'DisplayName', '$\tau_{m,i}/u_i$ - FRF') for i = 2:6 set(gca,'ColorOrderIndex',2) plot(f, abs(G_iff(:,i, i)), 'color', [0,0,0,0.2], ... 'HandleVisibility', 'off'); end set(gca,'ColorOrderIndex',2); plot(freqs, abs(squeeze(freqresp(Giff(1,1), freqs, 'Hz'))), '-', ... 'DisplayName', '$\tau_{m,i}/u_i$ - Model') for i = 2:6 set(gca,'ColorOrderIndex',2); plot(freqs, abs(squeeze(freqresp(Giff(i,i), freqs, 'Hz'))), '-', ... 'HandleVisibility', 'off'); end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); ylabel('Amplitude [V/V]'); set(gca, 'XTickLabel',[]); legend('location', 'southeast'); ax2 = nexttile; hold on; for i = 1:6 plot(f, 180/pi*angle(G_iff(:,i, i)), 'color', [0,0,0,0.2]); end for i = 1:6 set(gca,'ColorOrderIndex',2); plot(freqs, 180/pi*angle(squeeze(freqresp(Giff(i,i), freqs, 'Hz'))), '-'); end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); ylim([-180, 180]); yticks([-180, -90, 0, 90, 180]); linkaxes([ax1,ax2],'x'); xlim([freqs(1), freqs(end)]); #+end_src #+begin_src matlab :tangle no :exports results :results file replace exportFig('figs/enc_struts_iff_comp_simscape.pdf', 'width', 'wide', 'height', 'tall'); #+end_src #+name: fig:enc_struts_iff_comp_simscape #+caption: Diagonal elements of the IFF Plant #+RESULTS: [[file:figs/enc_struts_iff_comp_simscape.png]] #+begin_src matlab :exports none %% Bode plot of the identified IFF Plant (Simscape) and measured FRF data (off-diagonal elements) freqs = 2*logspace(1, 3, 1000); figure; hold on; % Off diagonal terms plot(f, abs(G_iff(:, 1, 2)), 'color', [0,0,0,0.2], ... 'DisplayName', '$\tau_{m,i}/u_j$ - FRF') for i = 1:5 for j = i+1:6 plot(f, abs(G_iff(:, i, j)), 'color', [0,0,0,0.2], ... 'HandleVisibility', 'off'); end end set(gca,'ColorOrderIndex',2); plot(freqs, abs(squeeze(freqresp(Giff(1, 2), freqs, 'Hz'))), ... 'DisplayName', '$\tau_{m,i}/u_j$ - Model') for i = 1:5 for j = i+1:6 set(gca,'ColorOrderIndex',2); plot(freqs, abs(squeeze(freqresp(Giff(i, j), freqs, 'Hz'))), ... 'HandleVisibility', 'off'); end end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); xlabel('Frequency [Hz]'); ylabel('Amplitude [V/V]'); xlim([freqs(1), freqs(end)]); ylim([1e-3, 1e2]); legend('location', 'northeast'); #+end_src #+begin_src matlab :tangle no :exports results :results file replace exportFig('figs/enc_struts_iff_comp_offdiag_simscape.pdf', 'width', 'wide', 'height', 'normal'); #+end_src #+name: fig:enc_struts_iff_comp_offdiag_simscape #+caption: Off diagonal elements of the IFF Plant #+RESULTS: [[file:figs/enc_struts_iff_comp_offdiag_simscape.png]] *** Dynamics from Actuator to Encoder #+begin_src matlab %% Initialization of the Nano-Hexapod n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... 'flex_top_type', '4dof', ... 'motion_sensor_type', 'struts', ... 'actuator_type', 'flexible'); #+end_src #+begin_src matlab %% Identify the DVF Plant (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, '/D'], 1, 'openoutput'); io_i = io_i + 1; % Encoders Gdvf = exp(-s*Ts)*linearize(mdl, io, 0.0, options); #+end_src #+begin_src matlab :exports none %% Diagonal elements of the DVF plant freqs = 2*logspace(1, 3, 1000); figure; tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); ax1 = nexttile([2,1]); hold on; plot(f, abs(G_dvf(:,1, 1)), 'color', [0,0,0,0.2], ... 'DisplayName', '$d\mathcal{L}_{m,i}/u_i$ - FRF') for i = 2:6 set(gca,'ColorOrderIndex',2) plot(f, abs(G_dvf(:,i, i)), 'color', [0,0,0,0.2], ... 'HandleVisibility', 'off'); end set(gca,'ColorOrderIndex',2); plot(freqs, abs(squeeze(freqresp(Gdvf(1,1), freqs, 'Hz'))), '-', ... 'DisplayName', '$d\mathcal{L}_{m,i}/u_i$ - Model') for i = 2:6 set(gca,'ColorOrderIndex',2); plot(freqs, abs(squeeze(freqresp(Gdvf(i,i), freqs, 'Hz'))), '-', ... 'HandleVisibility', 'off'); end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); ylabel('Amplitude [m/V]'); set(gca, 'XTickLabel',[]); ylim([1e-8, 1e-3]); legend('location', 'northeast'); ax2 = nexttile; hold on; for i = 1:6 plot(f, 180/pi*angle(G_dvf(:,i, i)), 'color', [0,0,0,0.2]); end for i = 1:6 set(gca,'ColorOrderIndex',2); plot(freqs, 180/pi*angle(squeeze(freqresp(Gdvf(i,i), freqs, 'Hz'))), '-'); end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); ylim([-180, 180]); yticks([-180, -90, 0, 90, 180]); linkaxes([ax1,ax2],'x'); xlim([freqs(1), freqs(end)]); #+end_src #+begin_src matlab :tangle no :exports results :results file replace exportFig('figs/enc_struts_dvf_comp_simscape.pdf', 'width', 'wide', 'height', 'tall'); #+end_src #+name: fig:enc_struts_dvf_comp_simscape #+caption: Diagonal elements of the DVF Plant #+RESULTS: [[file:figs/enc_struts_dvf_comp_simscape.png]] #+begin_src matlab :exports none %% Off-diagonal elements of the DVF plant freqs = 2*logspace(1, 3, 1000); figure; hold on; % Off diagonal terms plot(f, abs(G_dvf(:, 1, 2)), 'color', [0,0,0,0.2], ... 'DisplayName', '$d\mathcal{L}_{m,i}/u_j$ - FRF') for i = 1:5 for j = i+1:6 plot(f, abs(G_dvf(:, i, j)), 'color', [0,0,0,0.2], ... 'HandleVisibility', 'off'); end end set(gca,'ColorOrderIndex',2); plot(freqs, abs(squeeze(freqresp(Gdvf(1, 2), freqs, 'Hz'))), ... 'DisplayName', '$d\mathcal{L}_{m,i}/u_j$ - Model') for i = 1:5 for j = i+1:6 set(gca,'ColorOrderIndex',2); plot(freqs, abs(squeeze(freqresp(Gdvf(i, j), freqs, 'Hz'))), ... 'HandleVisibility', 'off'); end end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); xlabel('Frequency [Hz]'); ylabel('Amplitude [m/V]'); xlim([freqs(1), freqs(end)]); ylim([1e-8, 1e-3]); legend('location', 'northeast'); #+end_src #+begin_src matlab :tangle no :exports results :results file replace exportFig('figs/enc_struts_dvf_comp_offdiag_simscape.pdf', 'width', 'wide', 'height', 'normal'); #+end_src #+name: fig:enc_struts_dvf_comp_offdiag_simscape #+caption: Off diagonal elements of the DVF Plant #+RESULTS: [[file:figs/enc_struts_dvf_comp_offdiag_simscape.png]] *** Effect of a change in bending damping of the joints #+begin_src matlab %% Tested bending dampings [Nm/(rad/s)] cRs = [1e-3, 5e-3, 1e-2, 5e-2, 1e-1]; #+end_src #+begin_src matlab %% Identify the DVF Plant (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, '/D'], 1, 'openoutput'); io_i = io_i + 1; % Encoders #+end_src Then the identification is performed for all the values of the bending damping. #+begin_src matlab %% Idenfity the transfer function from actuator to encoder for all bending dampins Gs = {zeros(length(cRs), 1)}; for i = 1:length(cRs) n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... 'flex_top_type', '4dof', ... 'motion_sensor_type', 'struts', ... 'actuator_type', 'flexible', ... 'flex_bot_cRx', cRs(i), ... 'flex_bot_cRy', cRs(i), ... 'flex_top_cRx', cRs(i), ... 'flex_top_cRy', cRs(i)); G = exp(-s*Ts)*linearize(mdl, io, 0.0, options); G.InputName = {'Va1', 'Va2', 'Va3', 'Va4', 'Va5', 'Va6'}; G.OutputName = {'dL1', 'dL2', 'dL3', 'dL4', 'dL5', 'dL6'}; Gs(i) = {G}; end #+end_src #+begin_src matlab :exports none %% Plot the obtained direct transfer functions for all the bending stiffnesses freqs = 2*logspace(1, 3, 1000); figure; tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); ax1 = nexttile([2,1]); hold on; for i = 1:length(cRs) plot(freqs, abs(squeeze(freqresp(Gs{i}('dL1', 'Va1'), freqs, 'Hz'))), ... 'DisplayName', sprintf('$c_R = %.3f\\,[\\frac{Nm}{rad/s}]$', cRs(i))); end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); ylabel('Amplitude $d_L/V_a$ [m/V]'); set(gca, 'XTickLabel',[]); hold off; ylim([1e-8, 1e-3]); legend('location', 'southwest'); ax2 = nexttile; hold on; for i = 1:length(cRs) plot(freqs, 180/pi*angle(squeeze(freqresp(Gs{i}('dL1', 'Va1'), freqs, 'Hz')))); end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); hold off; yticks(-360:90:360); ylim([-180, 180]); linkaxes([ax1,ax2],'x'); xlim([20, 2e3]); #+end_src #+begin_src matlab :exports none %% Plot the obtained coupling transfer functions for all the bending stiffnesses freqs = 2*logspace(1, 3, 1000); figure; hold on; for i = 1:length(cRs) plot(freqs, abs(squeeze(freqresp(Gs{i}('dL2', 'Va1'), freqs, 'Hz'))), ... 'DisplayName', sprintf('$c_R = %.3f\\,[\\frac{Nm}{rad/s}]$', cRs(i))); end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); ylabel('Amplitude $d_L/V_a$ [m/V]'); set(gca, 'XTickLabel',[]); hold off; ylim([1e-8, 1e-3]); legend('location', 'southwest'); xlim([20, 2e3]); #+end_src - Could be nice - Actual damping is very small *** Effect of a change in damping factor of the APA #+begin_src matlab %% Tested bending dampings [Nm/(rad/s)] xis = [1e-3, 5e-3, 1e-2, 5e-2, 1e-1]; #+end_src #+begin_src matlab %% Identify the DVF Plant (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, '/D'], 1, 'openoutput'); io_i = io_i + 1; % Encoders #+end_src #+begin_src matlab %% Idenfity the transfer function from actuator to encoder for all bending dampins Gs = {zeros(length(xis), 1)}; for i = 1:length(xis) n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... 'flex_top_type', '4dof', ... 'motion_sensor_type', 'struts', ... 'actuator_type', 'flexible', ... 'actuator_xi', xis(i)); G = exp(-s*Ts)*linearize(mdl, io, 0.0, options); G.InputName = {'Va1', 'Va2', 'Va3', 'Va4', 'Va5', 'Va6'}; G.OutputName = {'dL1', 'dL2', 'dL3', 'dL4', 'dL5', 'dL6'}; Gs(i) = {G}; end #+end_src #+begin_src matlab :exports none %% Plot the obtained direct transfer functions for all the bending stiffnesses freqs = 2*logspace(1, 3, 1000); figure; tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); ax1 = nexttile([2,1]); hold on; for i = 1:length(xis) plot(freqs, abs(squeeze(freqresp(Gs{i}('dL1', 'Va1'), freqs, 'Hz'))), ... 'DisplayName', sprintf('$\\xi = %.3f$', xis(i))); end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); ylabel('Amplitude $d_L/V_a$ [m/V]'); set(gca, 'XTickLabel',[]); hold off; ylim([1e-8, 1e-3]); legend('location', 'southwest'); ax2 = nexttile; hold on; for i = 1:length(xis) plot(freqs, 180/pi*angle(squeeze(freqresp(Gs{i}('dL1', 'Va1'), freqs, 'Hz')))); end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); hold off; yticks(-360:90:360); ylim([-180, 180]); linkaxes([ax1,ax2],'x'); xlim([20, 2e3]); #+end_src #+begin_src matlab :tangle no :exports results :results file replace exportFig('figs/bode_Va_dL_effect_xi_damp.pdf', 'width', 'wide', 'height', 'tall'); #+end_src #+name: fig:bode_Va_dL_effect_xi_damp #+caption: Effect of the APA damping factor $\xi$ on the dynamics from $u$ to $d\mathcal{L}$ #+RESULTS: [[file:figs/bode_Va_dL_effect_xi_damp.png]] #+begin_src matlab :exports none %% Plot the obtained coupling transfer functions for all the bending stiffnesses freqs = 2*logspace(1, 3, 1000); figure; hold on; for i = 1:length(xis) plot(freqs, abs(squeeze(freqresp(Gs{i}('dL2', 'Va1'), freqs, 'Hz'))), ... 'DisplayName', sprintf('$c_R = %.3f\\,[\\frac{Nm}{rad/s}]$', xis(i))); end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); ylabel('Amplitude $d_L/V_a$ [m/V]'); set(gca, 'XTickLabel',[]); hold off; ylim([1e-8, 1e-3]); legend('location', 'southwest'); xlim([20, 2e3]); #+end_src #+begin_important Damping factor $\xi$ has a large impact on the damping of the "spurious resonances" at 200Hz and 300Hz. #+end_important #+begin_question Why is the damping factor does not change the damping of the first peak? #+end_question *** Effect of a change in stiffness damping coef of the APA #+begin_src matlab m_coef = 1e1; #+end_src #+begin_src matlab %% Tested bending dampings [Nm/(rad/s)] k_coefs = [1e-6, 5e-6, 1e-5, 5e-5, 1e-4]; #+end_src #+begin_src matlab %% Identify the DVF Plant (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, '/D'], 1, 'openoutput'); io_i = io_i + 1; % Encoders #+end_src #+begin_src matlab %% Idenfity the transfer function from actuator to encoder for all bending dampins Gs = {zeros(length(k_coefs), 1)}; n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... 'flex_top_type', '4dof', ... 'motion_sensor_type', 'struts', ... 'actuator_type', 'flexible'); for i = 1:length(k_coefs) k_coef = k_coefs(i); G = exp(-s*Ts)*linearize(mdl, io, 0.0, options); G.InputName = {'Va1', 'Va2', 'Va3', 'Va4', 'Va5', 'Va6'}; G.OutputName = {'dL1', 'dL2', 'dL3', 'dL4', 'dL5', 'dL6'}; Gs(i) = {G}; end #+end_src #+begin_src matlab :exports none %% Plot the obtained direct transfer functions for all the bending stiffnesses freqs = 2*logspace(1, 3, 1000); figure; tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); ax1 = nexttile([2,1]); hold on; for i = 1:length(k_coefs) plot(freqs, abs(squeeze(freqresp(Gs{i}('dL1', 'Va1'), freqs, 'Hz'))), ... 'DisplayName', sprintf('kcoef = %.0e', k_coefs(i))); end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); ylabel('Amplitude $d_L/V_a$ [m/V]'); set(gca, 'XTickLabel',[]); hold off; ylim([1e-8, 1e-3]); legend('location', 'southwest'); ax2 = nexttile; hold on; for i = 1:length(k_coefs) plot(freqs, 180/pi*angle(squeeze(freqresp(Gs{i}('dL1', 'Va1'), freqs, 'Hz')))); end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); hold off; yticks(-360:90:360); ylim([-180, 180]); linkaxes([ax1,ax2],'x'); xlim([20, 2e3]); #+end_src #+begin_src matlab :tangle no :exports results :results file replace exportFig('figs/bode_Va_dL_effect_k_coef.pdf', 'width', 'wide', 'height', 'tall'); #+end_src #+name: fig:bode_Va_dL_effect_k_coef #+caption: Effect of a change of the damping "stiffness coeficient" on the transfer function from $u$ to $d\mathcal{L}$ #+RESULTS: [[file:figs/bode_Va_dL_effect_k_coef.png]] #+begin_src matlab :exports none %% Plot the obtained coupling transfer functions for all the bending stiffnesses freqs = 2*logspace(1, 3, 1000); figure; hold on; for i = 1:length(xis) plot(freqs, abs(squeeze(freqresp(Gs{i}('dL2', 'Va1'), freqs, 'Hz'))), ... 'DisplayName', sprintf('$c_R = %.3f\\,[\\frac{Nm}{rad/s}]$', xis(i))); end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); ylabel('Amplitude $d_L/V_a$ [m/V]'); set(gca, 'XTickLabel',[]); hold off; ylim([1e-8, 1e-3]); legend('location', 'southwest'); xlim([20, 2e3]); #+end_src *** Effect of a change in mass damping coef of the APA #+begin_src matlab k_coef = 1e-6; #+end_src #+begin_src matlab %% Tested bending dampings [Nm/(rad/s)] m_coefs = [1e1, 5e1, 1e2, 5e2, 1e3]; #+end_src #+begin_src matlab %% Identify the DVF Plant (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, '/D'], 1, 'openoutput'); io_i = io_i + 1; % Encoders #+end_src #+begin_src matlab %% Idenfity the transfer function from actuator to encoder for all bending dampins Gs = {zeros(length(m_coefs), 1)}; n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... 'flex_top_type', '4dof', ... 'motion_sensor_type', 'struts', ... 'actuator_type', 'flexible'); for i = 1:length(m_coefs) m_coef = m_coefs(i); G = exp(-s*Ts)*linearize(mdl, io, 0.0, options); G.InputName = {'Va1', 'Va2', 'Va3', 'Va4', 'Va5', 'Va6'}; G.OutputName = {'dL1', 'dL2', 'dL3', 'dL4', 'dL5', 'dL6'}; Gs(i) = {G}; end #+end_src #+begin_src matlab :exports none %% Plot the obtained direct transfer functions for all the bending stiffnesses freqs = 2*logspace(1, 3, 1000); figure; tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); ax1 = nexttile([2,1]); hold on; for i = 1:length(m_coefs) plot(freqs, abs(squeeze(freqresp(Gs{i}('dL1', 'Va1'), freqs, 'Hz'))), ... 'DisplayName', sprintf('mcoef = %.0e', m_coefs(i))); end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); ylabel('Amplitude $d_L/V_a$ [m/V]'); set(gca, 'XTickLabel',[]); hold off; ylim([1e-8, 1e-3]); legend('location', 'southwest'); ax2 = nexttile; hold on; for i = 1:length(m_coefs) plot(freqs, 180/pi*angle(squeeze(freqresp(Gs{i}('dL1', 'Va1'), freqs, 'Hz')))); end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); hold off; yticks(-360:90:360); ylim([-180, 180]); linkaxes([ax1,ax2],'x'); xlim([20, 2e3]); #+end_src #+begin_src matlab :tangle no :exports results :results file replace exportFig('figs/bode_Va_dL_effect_m_coef.pdf', 'width', 'wide', 'height', 'tall'); #+end_src #+name: fig:bode_Va_dL_effect_m_coef #+caption: Effect of a change of the damping "mass coeficient" on the transfer function from $u$ to $d\mathcal{L}$ #+RESULTS: [[file:figs/bode_Va_dL_effect_m_coef.png]] #+begin_src matlab :exports none %% Plot the obtained coupling transfer functions for all the bending stiffnesses freqs = 2*logspace(1, 3, 1000); figure; hold on; for i = 1:length(xis) plot(freqs, abs(squeeze(freqresp(Gs{i}('dL2', 'Va1'), freqs, 'Hz'))), ... 'DisplayName', sprintf('$c_R = %.3f\\,[\\frac{Nm}{rad/s}]$', xis(i))); end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); ylabel('Amplitude $d_L/V_a$ [m/V]'); set(gca, 'XTickLabel',[]); hold off; ylim([1e-8, 1e-3]); legend('location', 'southwest'); xlim([20, 2e3]); #+end_src *** TODO Using Flexible model #+begin_src matlab d_aligns = [[-0.05, -0.3, 0]; [ 0, 0.5, 0]; [-0.1, -0.3, 0]; [ 0, 0.3, 0]; [-0.05, 0.05, 0]; [0, 0, 0]]*1e-3; #+end_src #+begin_src matlab d_aligns = zeros(6,3); % d_aligns(1,:) = [-0.05, -0.3, 0]*1e-3; d_aligns(2,:) = [ 0, 0.3, 0]*1e-3; #+end_src #+begin_src matlab %% Initialize Nano-Hexapod n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... 'flex_top_type', '4dof', ... 'motion_sensor_type', 'struts', ... 'actuator_type', 'flexible', ... 'actuator_d_align', d_aligns); #+end_src #+begin_question Why do we have smaller resonances when using flexible APA? On the test bench we have the same resonance as the 2DoF model. Could it be due to the compliance in other dof of the flexible model? #+end_question #+begin_src matlab %% Identify the DVF Plant (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, '/D'], 1, 'openoutput'); io_i = io_i + 1; % Encoders Gdvf = exp(-s*Ts)*linearize(mdl, io, 0.0, options); #+end_src #+begin_src matlab :exports none %% Comparison of the plants (encoder output) when tuning the misalignment freqs = 2*logspace(0, 3, 1000); figure; tiledlayout(2, 3, 'TileSpacing', 'Compact', 'Padding', 'None'); ax1 = nexttile(); hold on; plot(f, abs(G_dvf(:, 1, 1))); plot(freqs, abs(squeeze(freqresp(Gdvf(1,1), freqs, 'Hz')))); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); set(gca, 'XTickLabel',[]); ylabel('Amplitude [m/V]'); ax2 = nexttile(); hold on; plot(f, abs(G_dvf(:, 2, 2))); plot(freqs, abs(squeeze(freqresp(Gdvf(2,2), freqs, 'Hz')))); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]); ax3 = nexttile(); hold on; plot(f, abs(G_dvf(:, 3, 3)), 'DisplayName', 'Meas.'); plot(freqs, abs(squeeze(freqresp(Gdvf(3,3), freqs, 'Hz'))), ... 'DisplayName', 'Model'); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]); legend('location', 'southwest', 'FontSize', 8); ax4 = nexttile(); hold on; plot(f, abs(G_dvf(:, 4, 4))); plot(freqs, abs(squeeze(freqresp(Gdvf(4,4), freqs, 'Hz')))); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); xlabel('Frequency [Hz]'); ylabel('Amplitude [m/V]'); ax5 = nexttile(); hold on; plot(f, abs(G_dvf(:, 5, 5))); plot(freqs, abs(squeeze(freqresp(Gdvf(5,5), freqs, 'Hz')))); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]); ax6 = nexttile(); hold on; plot(f, abs(G_dvf(:, 6, 6))); plot(freqs, abs(squeeze(freqresp(Gdvf(6,6), freqs, 'Hz')))); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]); linkaxes([ax1,ax2,ax3,ax4,ax5,ax6],'xy'); % xlim([20, 2e3]); ylim([1e-8, 1e-3]); xlim([50, 5e2]); ylim([1e-6, 1e-3]); #+end_src #+begin_src matlab :exports none %% Diagonal elements of the DVF plant freqs = 6*logspace(1, 2, 2000); i_strut = 1; figure; tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); ax1 = nexttile([2,1]); hold on; plot(f, abs(G_dvf(:,i_strut, 2)), 'color', [0,0,0,0.2], ... 'DisplayName', '$d\mathcal{L}_{m,i}/u_i$ - FRF') plot(freqs, abs(squeeze(freqresp(Gdvf(2,2), freqs, 'Hz'))), '-', ... 'DisplayName', '$d\mathcal{L}_{m,i}/u_i$ - Model') hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); ylabel('Amplitude [m/V]'); set(gca, 'XTickLabel',[]); ylim([1e-8, 1e-3]); legend('location', 'northeast'); ax2 = nexttile; hold on; plot(f, 180/pi*angle(G_dvf(:,2, 2)), 'color', [0,0,0,0.2]); plot(freqs, 180/pi*angle(squeeze(freqresp(Gdvf(2,2), freqs, 'Hz'))), '-'); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); ylim([-180, 180]); yticks([-180, -90, 0, 90, 180]); linkaxes([ax1,ax2],'x'); xlim([freqs(1), freqs(end)]); #+end_src #+begin_src matlab :exports none %% Diagonal elements of the DVF plant freqs = 6*logspace(1, 2, 2000); figure; tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); ax1 = nexttile([2,1]); hold on; plot(f, abs(G_dvf(:,1, 1)), 'color', [0,0,0,0.2], ... 'DisplayName', '$d\mathcal{L}_{m,i}/u_i$ - FRF') for i = 2:6 set(gca,'ColorOrderIndex',2) plot(f, abs(G_dvf(:,i, i)), 'color', [0,0,0,0.2], ... 'HandleVisibility', 'off'); end set(gca,'ColorOrderIndex',2); plot(freqs, abs(squeeze(freqresp(Gdvf(1,1), freqs, 'Hz'))), '-', ... 'DisplayName', '$d\mathcal{L}_{m,i}/u_i$ - Model') for i = 2:6 set(gca,'ColorOrderIndex',2); plot(freqs, abs(squeeze(freqresp(Gdvf(i,i), freqs, 'Hz'))), '-', ... 'HandleVisibility', 'off'); end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); ylabel('Amplitude [m/V]'); set(gca, 'XTickLabel',[]); ylim([1e-8, 1e-3]); legend('location', 'northeast'); ax2 = nexttile; hold on; for i = 1:6 plot(f, 180/pi*angle(G_dvf(:,i, i)), 'color', [0,0,0,0.2]); end for i = 1:6 set(gca,'ColorOrderIndex',2); plot(freqs, 180/pi*angle(squeeze(freqresp(Gdvf(i,i), freqs, 'Hz'))), '-'); end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); ylim([-180, 180]); yticks([-180, -90, 0, 90, 180]); linkaxes([ax1,ax2],'x'); xlim([freqs(1), freqs(end)]); #+end_src #+begin_src matlab %% Identify the IFF Plant (transfer function from u to taum) 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, '/dum'], 1, 'openoutput'); io_i = io_i + 1; % Force Sensors Giff = exp(-s*Ts)*linearize(mdl, io, 0.0, options); #+end_src #+begin_src matlab :exports none %% Bode plot of the identified IFF Plant (Simscape) and measured FRF data freqs = 2*logspace(1, 3, 1000); figure; tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); ax1 = nexttile([2,1]); hold on; plot(f, abs(G_iff(:,1, 1)), 'color', [0,0,0,0.2], ... 'DisplayName', '$\tau_{m,i}/u_i$ - FRF') for i = 2:6 set(gca,'ColorOrderIndex',2) plot(f, abs(G_iff(:,i, i)), 'color', [0,0,0,0.2], ... 'HandleVisibility', 'off'); end set(gca,'ColorOrderIndex',2); plot(freqs, abs(squeeze(freqresp(Giff(1,1), freqs, 'Hz'))), '-', ... 'DisplayName', '$\tau_{m,i}/u_i$ - Model') for i = 2:6 set(gca,'ColorOrderIndex',2); plot(freqs, abs(squeeze(freqresp(Giff(i,i), freqs, 'Hz'))), '-', ... 'HandleVisibility', 'off'); end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); ylabel('Amplitude [V/V]'); set(gca, 'XTickLabel',[]); legend('location', 'southeast'); ax2 = nexttile; hold on; for i = 1:6 plot(f, 180/pi*angle(G_iff(:,i, i)), 'color', [0,0,0,0.2]); end for i = 1:6 set(gca,'ColorOrderIndex',2); plot(freqs, 180/pi*angle(squeeze(freqresp(Giff(i,i), freqs, 'Hz'))), '-'); end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); ylim([-180, 180]); yticks([-180, -90, 0, 90, 180]); linkaxes([ax1,ax2],'x'); xlim([freqs(1), freqs(end)]); #+end_src *** Flexible model + encoders fixed to the plates #+begin_src matlab %% Identify the IFF Plant (transfer function from u to taum) 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, '/D'], 1, 'openoutput'); io_i = io_i + 1; % Force Sensors #+end_src #+begin_src matlab d_aligns = [[-0.05, -0.3, 0]; [ 0, 0.5, 0]; [-0.1, -0.3, 0]; [ 0, 0.3, 0]; [-0.05, 0.05, 0]; [0, 0, 0]]*1e-3; #+end_src #+begin_src matlab %% Initialize Nano-Hexapod n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... 'flex_top_type', '4dof', ... 'motion_sensor_type', 'struts', ... 'actuator_type', 'flexible', ... 'actuator_d_align', d_aligns); #+end_src #+begin_src matlab Gdvf_struts = exp(-s*Ts)*linearize(mdl, io, 0.0, options); #+end_src #+begin_src matlab %% Initialize Nano-Hexapod n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... 'flex_top_type', '4dof', ... 'motion_sensor_type', 'plates', ... 'actuator_type', 'flexible', ... 'actuator_d_align', d_aligns); #+end_src #+begin_src matlab Gdvf_plates = exp(-s*Ts)*linearize(mdl, io, 0.0, options); #+end_src #+begin_src matlab :exports none %% Plot the obtained direct transfer functions for all the bending stiffnesses freqs = 2*logspace(1, 3, 1000); figure; tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); ax1 = nexttile([2,1]); hold on; plot(freqs, abs(squeeze(freqresp(Gdvf_struts(1, 1), freqs, 'Hz'))), ... 'DisplayName', 'Struts'); plot(freqs, abs(squeeze(freqresp(Gdvf_plates(1, 1), freqs, 'Hz'))), ... 'DisplayName', 'Plates'); for i = 2:6 set(gca,'ColorOrderIndex',1); plot(freqs, abs(squeeze(freqresp(Gdvf_struts(i, i), freqs, 'Hz'))), ... 'HandleVisibility', 'off'); set(gca,'ColorOrderIndex',2); plot(freqs, abs(squeeze(freqresp(Gdvf_plates(i, i), freqs, 'Hz'))), ... 'HandleVisibility', 'off'); end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); ylabel('Amplitude $d_L/V_a$ [m/V]'); set(gca, 'XTickLabel',[]); hold off; ylim([1e-8, 1e-3]); legend('location', 'southwest'); ax2 = nexttile; hold on; for i = 1:6 set(gca,'ColorOrderIndex',1); plot(freqs, 180/pi*angle(squeeze(freqresp(Gdvf_struts(i, i), freqs, 'Hz')))); set(gca,'ColorOrderIndex',2); plot(freqs, 180/pi*angle(squeeze(freqresp(Gdvf_plates(i, i), freqs, 'Hz')))); end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); hold off; yticks(-360:90:360); ylim([-180, 180]); linkaxes([ax1,ax2],'x'); xlim([20, 2e3]); #+end_src #+begin_src matlab :tangle no :exports results :results file replace exportFig('figs/dvf_plant_comp_struts_plates.pdf', 'width', 'wide', 'height', 'tall'); #+end_src #+name: fig:dvf_plant_comp_struts_plates #+caption: Comparison of the dynamics from $V_a$ to $d_L$ when the encoders are fixed to the struts (blue) and to the plates (red). APA are modeled as a flexible element. #+RESULTS: [[file:figs/dvf_plant_comp_struts_plates.png]] ** Integral Force Feedback <> *** Introduction :ignore: In this section, the Integral Force Feedback (IFF) control strategy is applied to the nano-hexapod. The main goal of this to add damping to the nano-hexapod's modes. The control architecture is shown in Figure [[fig:control_architecture_iff_struts]] where $\bm{K}_\text{IFF}$ is a diagonal $6 \times 6$ controller. The system as then a new input $\bm{u}^\prime$, and the transfer function from $\bm{u}^\prime$ to $d\bm{\mathcal{L}}_m$ should be easier to control than the initial transfer function from $\bm{u}$ to $d\bm{\mathcal{L}}_m$. #+begin_src latex :file control_architecture_iff_struts.pdf \begin{tikzpicture} % Blocs \node[block={3.0cm}{2.0cm}] (P) {Plant}; \coordinate[] (inputF) at ($(P.south west)!0.5!(P.north west)$); \coordinate[] (outputF) at ($(P.south east)!0.7!(P.north east)$); \coordinate[] (outputL) at ($(P.south east)!0.3!(P.north east)$); \node[block, above=0.4 of P] (Kiff) {$\bm{K}_\text{IFF}$}; \node[addb, left= of inputF] (addF) {}; % Connections and labels \draw[->] (outputF) -- ++(1, 0) node[below left]{$\bm{\tau}_m$}; \draw[->] (outputL) -- ++(1, 0) node[below left]{$d\bm{\mathcal{L}}_m$}; \draw[->] ($(outputF) + (0.6, 0)$)node[branch]{} |- (Kiff.east); \draw[->] (Kiff.west) -| (addF.north); \draw[->] (addF.east) -- (inputF) node[above left]{$\bm{u}$}; \draw[<-] (addF.west) -- ++(-1, 0) node[above right]{$\bm{u}^\prime$}; \end{tikzpicture} #+end_src #+name: fig:control_architecture_iff_struts #+caption: Integral Force Feedback Strategy #+RESULTS: [[file:figs/control_architecture_iff_struts.png]] This section is structured as follow: - Section [[sec:iff_struts_plant_id]]: Using the Simscape model (APA taken as 2DoF model), the transfer function from $\bm{u}$ to $\bm{\tau}_m$ is identified. Based on the obtained dynamics, the control law is developed and the optimal gain is estimated using the Root Locus. - Section [[sec:iff_struts_effect_plant]]: Still using the Simscape model, the effect of the IFF gain on the the transfer function from $\bm{u}^\prime$ to $d\bm{\mathcal{L}}_m$ is studied. - Section [[sec:iff_struts_effect_plant_exp]]: The same is performed experimentally: several IFF gains are used and the damped plant is identified each time. - Section [[sec:iff_struts_opt_gain]]: The damped model and the identified damped system are compared for the optimal IFF gain. It is found that IFF indeed adds a lot of damping into the system. However it is not efficient in damping the spurious struts modes. - Section [[sec:iff_struts_comp_flex_model]]: Finally, a "flexible" model of the APA is used in the Simscape model and the optimally damped model is compared with the measurements. *** Matlab Init :noexport:ignore: #+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) <> #+end_src #+begin_src matlab :exports none :results silent :noweb yes <> #+end_src #+begin_src matlab :tangle no addpath('./matlab/mat/'); addpath('./matlab/src/'); addpath('./matlab/'); #+end_src #+begin_src matlab :eval no addpath('./mat/'); addpath('./src/'); #+end_src #+begin_src matlab load('identified_plants_enc_struts.mat', 'f', 'Ts', 'G_iff', 'G_dvf') #+end_src #+begin_src matlab :tangle no %% Add all useful folders to the path addpath('matlab/nass-simscape/matlab/nano_hexapod/') addpath('matlab/nass-simscape/STEPS/nano_hexapod/') addpath('matlab/nass-simscape/STEPS/png/') addpath('matlab/nass-simscape/src/') addpath('matlab/nass-simscape/mat/') #+end_src #+begin_src matlab :eval no %% Add all useful folders to the path addpath('nass-simscape/matlab/nano_hexapod/') addpath('nass-simscape/STEPS/nano_hexapod/') addpath('nass-simscape/STEPS/png/') addpath('nass-simscape/src/') addpath('nass-simscape/mat/') #+end_src #+begin_src matlab %% Open Simulink Model mdl = 'nano_hexapod_simscape'; options = linearizeOptions; options.SampleTime = 0; Rx = zeros(1, 7); open(mdl) #+end_src *** IFF Control Law and Optimal Gain <> Let's use a model of the Nano-Hexapod with the encoders fixed to the struts and the APA taken as 2DoF model. #+begin_src matlab %% Initialize Nano-Hexapod n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... 'flex_top_type', '4dof', ... 'motion_sensor_type', 'struts', ... 'actuator_type', '2dof'); #+end_src The transfer function from $\bm{u}$ to $\bm{\tau}_m$ is identified. #+begin_src matlab %% Identify the IFF Plant (transfer function from u to taum) 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, '/dum'], 1, 'openoutput'); io_i = io_i + 1; % Force Sensors Giff = exp(-s*Ts)*linearize(mdl, io, 0.0, options); #+end_src The IFF controller is defined as shown below: #+begin_src matlab %% IFF Controller Kiff_g1 = -(1/(s + 2*pi*40))*... % LPF: provides integral action above 40Hz (s/(s + 2*pi*30))*... % HPF: limit low frequency gain (1/(1 + s/2/pi/500))*... % LPF: more robust to high frequency resonances eye(6); % Diagonal 6x6 controller #+end_src Then, the poles of the system are shown in the complex plane as a function of the controller gain (i.e. Root Locus plot) in Figure [[fig:enc_struts_iff_root_locus]]. A gain of $400$ is chosen as the "optimal" gain as it visually seems to be the gain that adds the maximum damping to all the suspension modes simultaneously. #+begin_src matlab :exports none %% Root Locus for IFF gains = logspace(1, 4, 100); figure; hold on; % Pure Integrator set(gca,'ColorOrderIndex',1); plot(real(pole(Giff)), imag(pole(Giff)), 'x', 'DisplayName', '$g = 0$'); set(gca,'ColorOrderIndex',1); plot(real(tzero(Giff)), imag(tzero(Giff)), 'o', 'HandleVisibility', 'off'); for g = gains clpoles = pole(feedback(Giff, g*Kiff_g1*eye(6))); set(gca,'ColorOrderIndex',1); plot(real(clpoles), imag(clpoles), '.', 'HandleVisibility', 'off'); end g = 4e2; clpoles = pole(feedback(Giff, g*Kiff_g1*eye(6))); set(gca,'ColorOrderIndex',2); plot(real(clpoles), imag(clpoles), 'x', 'DisplayName', sprintf('$g=%.0f$', g)); hold off; axis square; xlim([-1250, 0]); ylim([0, 1250]); xlabel('Real Part'); ylabel('Imaginary Part'); legend('location', 'northwest'); #+end_src #+begin_src matlab :tangle no :exports results :results file replace exportFig('figs/enc_struts_iff_root_locus.pdf', 'width', 'wide', 'height', 'tall'); #+end_src #+name: fig:enc_struts_iff_root_locus #+caption: Root Locus for the IFF control strategy #+RESULTS: [[file:figs/enc_struts_iff_root_locus.png]] Then the "optimal" IFF controller is: #+begin_src matlab %% IFF controller with Optimal gain Kiff = 400*Kiff_g1; #+end_src And it is saved for further use. #+begin_src matlab :exports none :tangle no save('matlab/mat/Kiff.mat', 'Kiff') #+end_src #+begin_src matlab :eval no save('mat/Kiff.mat', 'Kiff') #+end_src The bode plots of the "diagonal" elements of the loop gain are shown in Figure [[fig:enc_struts_iff_opt_loop_gain]]. It is shown that the phase and gain margins are quite high and the loop gain is large arround the resonances. #+begin_src matlab :exports none %% Bode plot of the "decentralized loop gain" freqs = 2*logspace(1, 3, 1000); figure; tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); ax1 = nexttile([2,1]); hold on; plot(f, abs(squeeze(freqresp(Kiff(1,1), f, 'Hz')).*G_iff(:, 1, 1)), 'color', [0,0,0,0.2], ... 'DisplayName', '$\tau_{m,i}/u_i \cdot K_{iff}$ - FRF') for i = 2:6 set(gca,'ColorOrderIndex',2) plot(f, abs(squeeze(freqresp(Kiff(1,1), f, 'Hz')).*G_iff(:, i, i)), 'color', [0,0,0,0.2], ... 'HandleVisibility', 'off'); end set(gca,'ColorOrderIndex',2); plot(freqs, abs(squeeze(freqresp(Kiff(1,1)*Giff(1,1), freqs, 'Hz'))), '-', ... 'DisplayName', '$\tau_{m,i}/u_i \cdot K_{iff}$ - Model') for i = 2:6 set(gca,'ColorOrderIndex',2); plot(freqs, abs(squeeze(freqresp(Kiff(1,1)*Giff(i,i), freqs, 'Hz'))), '-', ... 'HandleVisibility', 'off'); end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); ylabel('Amplitude [V/V]'); set(gca, 'XTickLabel',[]); legend('location', 'northeast'); ax2 = nexttile; hold on; for i = 1:6 plot(f, 180/pi*angle(squeeze(freqresp(Kiff(1,1), f, 'Hz')).*G_iff(:, i, i)), 'color', [0,0,0,0.2]); end for i = 1:6 set(gca,'ColorOrderIndex',2); plot(freqs, 180/pi*angle(squeeze(freqresp(Kiff(1,1)*Giff(i,i), freqs, 'Hz'))), '-'); end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); ylim([-180, 180]); yticks([-180, -90, 0, 90, 180]); linkaxes([ax1,ax2],'x'); xlim([freqs(1), freqs(end)]); #+end_src #+begin_src matlab :tangle no :exports results :results file replace exportFig('figs/enc_struts_iff_opt_loop_gain.pdf', 'width', 'wide', 'height', 'tall'); #+end_src #+name: fig:enc_struts_iff_opt_loop_gain #+caption: Bode plot of the "decentralized loop gain" $G_\text{iff}(i,i) \times K_\text{iff}(i,i)$ #+RESULTS: [[file:figs/enc_struts_iff_opt_loop_gain.png]] *** Effect of IFF on the plant - Simulations <> Still using the Simscape model with encoders fixed to the struts and 2DoF APA, the IFF strategy is tested. #+begin_src matlab %% Initialize the Simscape model in closed loop n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... 'flex_top_type', '4dof', ... 'motion_sensor_type', 'struts', ... 'actuator_type', '2dof', ... 'controller_type', 'iff'); #+end_src The following IFF gains are tried: #+begin_src matlab %% Tested IFF gains iff_gains = [4, 10, 20, 40, 100, 200, 400]; #+end_src And the transfer functions from $\bm{u}^\prime$ to $d\bm{\mathcal{L}}_m$ are identified for all the IFF gains. #+begin_src matlab %% Identify the (damped) transfer function from u to dLm for different values of the IFF gain Gd_iff = {zeros(1, length(iff_gains))}; 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; % Strut Displacement (encoder) for i = 1:length(iff_gains) Kiff = iff_gains(i)*Kiff_g1*eye(6); % IFF Controller Gd_iff(i) = {exp(-s*Ts)*linearize(mdl, io, 0.0, options)}; isstable(Gd_iff{i}) end #+end_src The obtained dynamics are shown in Figure [[fig:enc_struts_iff_gains_effect_dvf_plant]]. #+begin_src matlab :exports none %% Bode plot of the transfer function from u to dLm for tested values of the IFF gain freqs = 2*logspace(1, 3, 1000); figure; tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); ax1 = nexttile([2,1]); hold on; for i = 1:length(iff_gains) plot(freqs, abs(squeeze(freqresp(Gd_iff{i}(1,1), freqs, 'Hz'))), '-', ... 'DisplayName', sprintf('$g = %.0f$', iff_gains(i))); end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); ylabel('Amplitude [m/V]'); set(gca, 'XTickLabel',[]); legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 2); ax2 = nexttile; hold on; for i = 1:length(iff_gains) plot(freqs, 180/pi*angle(squeeze(freqresp(Gd_iff{i}(1,1), freqs, 'Hz'))), '-'); end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); ylim([-180, 180]); yticks([-180, -90, 0, 90, 180]); linkaxes([ax1,ax2],'x'); xlim([freqs(1), freqs(end)]); #+end_src #+begin_src matlab :tangle no :exports results :results file replace exportFig('figs/enc_struts_iff_gains_effect_dvf_plant.pdf', 'width', 'wide', 'height', 'tall'); #+end_src #+name: fig:enc_struts_iff_gains_effect_dvf_plant #+caption: Effect of the IFF gain $g$ on the transfer function from $\bm{\tau}$ to $d\bm{\mathcal{L}}_m$ #+RESULTS: [[file:figs/enc_struts_iff_gains_effect_dvf_plant.png]] *** Effect of IFF on the plant - Experimental Results <> **** Introduction :ignore: The IFF strategy is applied experimentally and the transfer function from $\bm{u}^\prime$ to $d\bm{\mathcal{L}}_m$ is identified for all the defined values of the gain. **** Load Data First load the identification data. #+begin_src matlab %% Load Identification Data meas_iff_gains = {}; for i = 1:length(iff_gains) meas_iff_gains(i) = {load(sprintf('mat/iff_strut_1_noise_g_%i.mat', iff_gains(i)), 't', 'Vexc', 'Vs', 'de', 'u')}; end #+end_src **** Spectral Analysis - Setup And define the useful variables that will be used for the identification using the =tfestimate= function. #+begin_src matlab %% Setup useful variables % Sampling Time [s] Ts = (meas_iff_gains{1}.t(end) - (meas_iff_gains{1}.t(1)))/(length(meas_iff_gains{1}.t)-1); % Sampling Frequency [Hz] Fs = 1/Ts; % Hannning Windows win = hanning(ceil(1*Fs)); % And we get the frequency vector [~, f] = tfestimate(meas_iff_gains{1}.Vexc, meas_iff_gains{1}.de, win, [], [], 1/Ts); #+end_src **** DVF Plant The transfer functions are estimated for all the values of the gain. #+begin_src matlab %% DVF Plant (transfer function from u to dLm) G_iff_gains = {}; for i = 1:length(iff_gains) G_iff_gains{i} = tfestimate(meas_iff_gains{i}.Vexc, meas_iff_gains{i}.de(:,1), win, [], [], 1/Ts); end #+end_src The obtained dynamics as shown in the bode plot in Figure [[fig:comp_iff_gains_dvf_plant]]. The dashed curves are the results obtained using the model, and the solid curves the results from the experimental identification. #+begin_src matlab :exports none %% Bode plot of the transfer function from u to dLm for tested values of the IFF gain freqs = 2*logspace(1, 3, 1000); figure; tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); ax1 = nexttile([2,1]); hold on; for i = 1:length(iff_gains) plot(f, abs(G_iff_gains{i}), '-', ... 'DisplayName', sprintf('$g = %.0f$', iff_gains(i))); end set(gca,'ColorOrderIndex',1) for i = 1:length(iff_gains) plot(freqs, abs(squeeze(freqresp(Gd_iff{i}(1,1), freqs, 'Hz'))), '--', ... 'HandleVisibility', 'off'); end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); ylabel('Amplitude [m/V]'); set(gca, 'XTickLabel',[]); legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 2); ax2 = nexttile; hold on; for i =1:length(iff_gains) plot(f, 180/pi*angle(G_iff_gains{i}), '-'); end set(gca,'ColorOrderIndex',1) for i = 1:length(iff_gains) plot(freqs, 180/pi*angle(squeeze(freqresp(Gd_iff{i}(1,1), freqs, 'Hz'))), '--'); end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); ylim([-180, 180]); yticks([-180, -90, 0, 90, 180]); linkaxes([ax1,ax2],'x'); xlim([freqs(1), freqs(end)]); #+end_src #+begin_src matlab :tangle no :exports results :results file replace exportFig('figs/comp_iff_gains_dvf_plant.pdf', 'width', 'wide', 'height', 'tall'); #+end_src #+name: fig:comp_iff_gains_dvf_plant #+caption: Transfer function from $u$ to $d\mathcal{L}_m$ for multiple values of the IFF gain #+RESULTS: [[file:figs/comp_iff_gains_dvf_plant.png]] The bode plot is then zoomed on the suspension modes of the nano-hexapod in Figure [[fig:comp_iff_gains_dvf_plant_zoom]]. #+begin_src matlab :exports none xlim([20, 200]); #+end_src #+begin_src matlab :tangle no :exports results :results file replace exportFig('figs/comp_iff_gains_dvf_plant_zoom.pdf', 'width', 'wide', 'height', 'tall'); #+end_src #+name: fig:comp_iff_gains_dvf_plant_zoom #+caption: Transfer function from $u$ to $d\mathcal{L}_m$ for multiple values of the IFF gain (Zoom) #+RESULTS: [[file:figs/comp_iff_gains_dvf_plant_zoom.png]] #+begin_important The IFF control strategy is very effective for the damping of the suspension modes. It however does not damp the modes at 200Hz, 300Hz and 400Hz (flexible modes of the APA). Also, the experimental results and the models obtained from the Simscape model are in agreement concerning the damped system (up to the flexible modes). #+end_important **** Experimental Results - Comparison of the un-damped and fully damped system The un-damped and damped experimental plants are compared in Figure [[fig:comp_undamped_opt_iff_gain_diagonal]] (diagonal terms). It is very clear that all the suspension modes are very well damped thanks to IFF. However, there is little to no effect on the flexible modes of the struts and of the plate. #+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', 'None', 'Padding', 'None'); ax1 = nexttile([2,1]); hold on; % Un Damped measurement set(gca,'ColorOrderIndex',1) plot(f, abs(G_dvf(:, 1, 1)), ... 'DisplayName', 'Un-Damped') for i = 2:6 set(gca,'ColorOrderIndex',1) plot(f, abs(G_dvf(:,i , i)), ... 'HandleVisibility', 'off'); end % IFF Damped measurement set(gca,'ColorOrderIndex',2) plot(f, abs(G_iff_opt{1}(:,1)), ... 'DisplayName', 'Optimal gain') for i = 2:6 set(gca,'ColorOrderIndex',2) plot(f, abs(G_iff_opt{i}(:,i)), ... 'HandleVisibility', 'off'); end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); ylabel('Amplitude $d_e/V_{exc}$ [m/V]'); set(gca, 'XTickLabel',[]); ylim([1e-9, 1e-3]); legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 3); ax2 = nexttile; hold on; for i =1:6 set(gca,'ColorOrderIndex',1) plot(f, 180/pi*angle(G_dvf(i,i, i))); set(gca,'ColorOrderIndex',2) plot(f, 180/pi*angle(G_iff_opt{i}(:,i))); end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); hold off; yticks(-360:90:360); linkaxes([ax1,ax2],'x'); xlim([20, 2e3]); #+end_src #+begin_src matlab :tangle no :exports results :results file replace exportFig('figs/comp_undamped_opt_iff_gain_diagonal.pdf', 'width', 'wide', 'height', 'tall'); #+end_src #+name: fig:comp_undamped_opt_iff_gain_diagonal #+caption: Comparison of the diagonal elements of the tranfer function from $\bm{u}$ to $d\bm{\mathcal{L}}_m$ without active damping and with optimal IFF gain #+RESULTS: [[file:figs/comp_undamped_opt_iff_gain_diagonal.png]] *** Experimental Results - Damped Plant with Optimal gain <> **** Introduction :ignore: Let's now look at the $6 \times 6$ damped plant with the optimal gain $g = 400$. **** Load Data The experimental data are loaded. #+begin_src matlab %% Load Identification Data meas_iff_struts = {}; for i = 1:6 meas_iff_struts(i) = {load(sprintf('mat/iff_strut_%i_noise_g_400.mat', i), 't', 'Vexc', 'Vs', 'de', 'u')}; end #+end_src **** Spectral Analysis - Setup And the parameters useful for the spectral analysis are defined. #+begin_src matlab %% Setup useful variables % Sampling Time [s] Ts = (meas_iff_struts{1}.t(end) - (meas_iff_struts{1}.t(1)))/(length(meas_iff_struts{1}.t)-1); % Sampling Frequency [Hz] Fs = 1/Ts; % Hannning Windows win = hanning(ceil(1*Fs)); % And we get the frequency vector [~, f] = tfestimate(meas_iff_struts{1}.Vexc, meas_iff_struts{1}.de, win, [], [], 1/Ts); #+end_src **** DVF Plant Finally, the $6 \times 6$ plant is identified using the =tfestimate= function. #+begin_src matlab %% DVF Plant (transfer function from u to dLm) G_iff_opt = {}; for i = 1:6 G_iff_opt{i} = tfestimate(meas_iff_struts{i}.Vexc, meas_iff_struts{i}.de, win, [], [], 1/Ts); end #+end_src The obtained diagonal elements are compared with the model in Figure [[fig:damped_iff_plant_comp_diagonal]]. #+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', 'None', 'Padding', 'None'); ax1 = nexttile([2,1]); hold on; % Diagonal Elements FRF plot(f, abs(G_iff_opt{1}(:,1)), 'color', [0,0,0,0.2], ... 'DisplayName', '$d\mathcal{L}_{m,i}/u_i$ - FRF') for i = 2:6 plot(f, abs(G_iff_opt{i}(:,i)), 'color', [0,0,0,0.2], ... 'HandleVisibility', 'off'); end % Diagonal Elements Model set(gca,'ColorOrderIndex',2) plot(freqs, abs(squeeze(freqresp(Gd_iff{end}(1,1), freqs, 'Hz'))), '-', ... 'DisplayName', '$d\mathcal{L}_{m,i}/u_i$ - Model') for i = 2:6 set(gca,'ColorOrderIndex',2) plot(freqs, abs(squeeze(freqresp(Gd_iff{end}(i,i), freqs, 'Hz'))), '-', ... 'HandleVisibility', 'off'); end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); ylabel('Amplitude $d_e/V_{exc}$ [m/V]'); set(gca, 'XTickLabel',[]); ylim([1e-9, 1e-3]); legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 3); ax2 = nexttile; hold on; for i =1:6 plot(f, 180/pi*angle(G_iff_opt{i}(:,i)), 'color', [0,0,0,0.2]); set(gca,'ColorOrderIndex',2) plot(freqs, 180/pi*angle(squeeze(freqresp(Gd_iff{end}(i,i), freqs, 'Hz')))); end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); hold off; yticks(-360:90:360); linkaxes([ax1,ax2],'x'); xlim([20, 2e3]); #+end_src #+begin_src matlab :tangle no :exports results :results file replace exportFig('figs/damped_iff_plant_comp_diagonal.pdf', 'width', 'wide', 'height', 'tall'); #+end_src #+name: fig:damped_iff_plant_comp_diagonal #+caption: Comparison of the diagonal elements of the transfer functions from $\bm{u}$ to $d\bm{\mathcal{L}}_m$ with active damping (IFF) applied with an optimal gain $g = 400$ #+RESULTS: [[file:figs/damped_iff_plant_comp_diagonal.png]] And all the off-diagonal elements are compared with the model in Figure [[fig:damped_iff_plant_comp_off_diagonal]]. #+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', 'None', 'Padding', 'None'); ax1 = nexttile([2,1]); hold on; % Off diagonal FRF plot(f, abs(G_iff_opt{1}(:,2)), 'color', [0,0,0,0.2], ... 'DisplayName', '$d\mathcal{L}_{m,i}/u_j$ - FRF') for i = 1:5 for j = i+1:6 plot(f, abs(G_iff_opt{i}(:,j)), 'color', [0, 0, 0, 0.2], ... 'HandleVisibility', 'off'); end end % Off diagonal Model set(gca,'ColorOrderIndex',2) plot(freqs, abs(squeeze(freqresp(Gd_iff{end}(1,2), freqs, 'Hz'))), '-', ... 'DisplayName', '$d\mathcal{L}_{m,i}/u_j$ - Model') for i = 1:5 for j = i+1:6 set(gca,'ColorOrderIndex',2) plot(freqs, abs(squeeze(freqresp(Gd_iff{end}(i,j), freqs, 'Hz'))), ... 'HandleVisibility', 'off'); end end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); ylabel('Amplitude $d_e/V_{exc}$ [m/V]'); set(gca, 'XTickLabel',[]); ylim([1e-9, 1e-3]); legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 3); ax2 = nexttile; hold on; % Off diagonal FRF for i = 1:5 for j = i+1:6 plot(f, 180/pi*angle(G_iff_opt{i}(:,j)), 'color', [0, 0, 0, 0.2]); end end % Off diagonal Model for i = 1:5 for j = i+1:6 set(gca,'ColorOrderIndex',2) plot(freqs, 180/pi*angle(squeeze(freqresp(Gd_iff{end}(i,j), freqs, 'Hz')))); end end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); hold off; yticks(-360:90:360); linkaxes([ax1,ax2],'x'); xlim([20, 2e3]); #+end_src #+begin_src matlab :tangle no :exports results :results file replace exportFig('figs/damped_iff_plant_comp_off_diagonal.pdf', 'width', 'wide', 'height', 'tall'); #+end_src #+name: fig:damped_iff_plant_comp_off_diagonal #+caption: Comparison of the off-diagonal elements of the transfer functions from $\bm{u}$ to $d\bm{\mathcal{L}}_m$ with active damping (IFF) applied with an optimal gain $g = 400$ #+RESULTS: [[file:figs/damped_iff_plant_comp_off_diagonal.png]] #+begin_important With the IFF control strategy applied and the optimal gain used, the suspension modes are very well damped. Remains the un-damped flexible modes of the APA (200Hz, 300Hz, 400Hz), and the modes of the plates (700Hz). The Simscape model and the experimental results are in very good agreement. #+end_important *** Comparison with the Flexible model <> When using the 2-DoF model for the APA, the flexible modes of the struts were not modelled, and it was the main limitation of the model. Now, let's use a flexible model for the APA, and see if the obtained damped plant using the model is similar to the measured dynamics. First, the nano-hexapod is initialized. #+begin_src matlab %% Estimated misalignement of the struts d_aligns = [[-0.05, -0.3, 0]; [ 0, 0.5, 0]; [-0.1, -0.3, 0]; [ 0, 0.3, 0]; [-0.05, 0.05, 0]; [0, 0, 0]]*1e-3; %% Initialize Nano-Hexapod n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... 'flex_top_type', '4dof', ... 'motion_sensor_type', 'struts', ... 'actuator_type', 'flexible', ... 'actuator_d_align', d_aligns, ... 'controller_type', 'iff'); #+end_src And the "optimal" controller is loaded. #+begin_src matlab %% Optimal IFF controller load('Kiff.mat', 'Kiff'); #+end_src The transfer function from $\bm{u}^\prime$ to $d\bm{\mathcal{L}}_m$ is identified using the Simscape model. #+begin_src matlab %% Linearized inputs/outputs 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; % Strut Displacement (encoder) %% Identification of the plant Gd_iff = exp(-s*Ts)*linearize(mdl, io, 0.0, options); #+end_src The obtained diagonal elements are shown in Figure [[fig:enc_struts_iff_opt_damp_comp_flex_model_diag]] while the off-diagonal elements are shown in Figure [[fig:enc_struts_iff_opt_damp_comp_flex_model_off_diag]]. #+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', 'None', 'Padding', 'None'); ax1 = nexttile([2,1]); hold on; % Diagonal Elements FRF plot(f, abs(G_iff_opt{1}(:,1)), 'color', [0,0,0,0.2], ... 'DisplayName', '$d\mathcal{L}_{m,i}/u^\prime_i$ - FRF') for i = 2:6 plot(f, abs(G_iff_opt{i}(:,i)), 'color', [0,0,0,0.2], ... 'HandleVisibility', 'off'); end % Diagonal Elements Model set(gca,'ColorOrderIndex',2) plot(freqs, abs(squeeze(freqresp(Gd_iff(1,1), freqs, 'Hz'))), '-', ... 'DisplayName', '$d\mathcal{L}_{m,i}/u^\prime_i$ - Model') for i = 2:6 set(gca,'ColorOrderIndex',2) plot(freqs, abs(squeeze(freqresp(Gd_iff(i,i), freqs, 'Hz'))), '-', ... 'HandleVisibility', 'off'); end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); ylabel('Amplitude $d\mathcal{L}_m/u^\prime$ [m/V]'); set(gca, 'XTickLabel',[]); ylim([1e-9, 1e-3]); legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 3); ax2 = nexttile; hold on; for i =1:6 plot(f, 180/pi*angle(G_iff_opt{i}(:,i)), 'color', [0,0,0,0.2]); set(gca,'ColorOrderIndex',2) plot(freqs, 180/pi*angle(squeeze(freqresp(Gd_iff(i,i), freqs, 'Hz')))); end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); hold off; yticks(-360:90:360); linkaxes([ax1,ax2],'x'); xlim([20, 2e3]); #+end_src #+begin_src matlab :tangle no :exports results :results file replace exportFig('figs/enc_struts_iff_opt_damp_comp_flex_model_diag.pdf', 'width', 'wide', 'height', 'tall'); #+end_src #+name: fig:enc_struts_iff_opt_damp_comp_flex_model_diag #+caption: Diagonal elements of the transfer function from $\bm{u}^\prime$ to $d\bm{\mathcal{L}}_m$ - comparison of the measured FRF and the identified dynamics using the flexible model #+RESULTS: [[file:figs/enc_struts_iff_opt_damp_comp_flex_model_diag.png]] #+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', 'None', 'Padding', 'None'); ax1 = nexttile([2,1]); hold on; % Off diagonal FRF plot(f, abs(G_iff_opt{1}(:,2)), 'color', [0,0,0,0.2], ... 'DisplayName', '$d\mathcal{L}_{m,i}/u^\prime_j$ - FRF') for i = 1:5 for j = i+1:6 plot(f, abs(G_iff_opt{i}(:,j)), 'color', [0, 0, 0, 0.2], ... 'HandleVisibility', 'off'); end end % Off diagonal Model set(gca,'ColorOrderIndex',2) plot(freqs, abs(squeeze(freqresp(Gd_iff(1,2), freqs, 'Hz'))), '-', ... 'DisplayName', '$d\mathcal{L}_{m,i}/u^\prime_j$ - Model') for i = 1:5 for j = i+1:6 set(gca,'ColorOrderIndex',2) plot(freqs, abs(squeeze(freqresp(Gd_iff(i,j), freqs, 'Hz'))), ... 'HandleVisibility', 'off'); end end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); ylabel('Amplitude $d\mathcal{L}_m/u^\prime$ [m/V]'); set(gca, 'XTickLabel',[]); ylim([1e-9, 1e-3]); legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 3); ax2 = nexttile; hold on; % Off diagonal FRF for i = 1:5 for j = i+1:6 plot(f, 180/pi*angle(G_iff_opt{i}(:,j)), 'color', [0, 0, 0, 0.2]); end end % Off diagonal Model for i = 1:5 for j = i+1:6 set(gca,'ColorOrderIndex',2) plot(freqs, 180/pi*angle(squeeze(freqresp(Gd_iff(i,j), freqs, 'Hz')))); end end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); hold off; yticks(-360:90:360); linkaxes([ax1,ax2],'x'); xlim([20, 2e3]); #+end_src #+begin_src matlab :tangle no :exports results :results file replace exportFig('figs/enc_struts_iff_opt_damp_comp_flex_model_off_diag.pdf', 'width', 'wide', 'height', 'tall'); #+end_src #+name: fig:enc_struts_iff_opt_damp_comp_flex_model_off_diag #+caption: Off-diagonal elements of the transfer function from $\bm{u}^\prime$ to $d\bm{\mathcal{L}}_m$ - comparison of the measured FRF and the identified dynamics using the flexible model #+RESULTS: [[file:figs/enc_struts_iff_opt_damp_comp_flex_model_off_diag.png]] #+begin_important Using flexible models for the APA, the agreement between the Simscape model of the nano-hexapod and the measured FRF is very good. Only the flexible mode of the top-plate is not appearing in the model which is very logical as the top plate is taken as a solid body. #+end_important *** Conclusion #+begin_important The decentralized Integral Force Feedback strategy applied on the nano-hexapod is very effective in damping all the suspension modes. The Simscape model (especially when using a flexible model for the APA) is shown to be very accurate, even when IFF is applied. #+end_important ** Modal Analysis <> *** Introduction :ignore: Several 3-axis accelerometers are fixed on the top platform of the nano-hexapod as shown in Figure [[fig:compliance_vertical_comp_iff]]. #+name: fig:accelerometers_nano_hexapod #+caption: Location of the accelerometers on top of the nano-hexapod #+attr_latex: :width \linewidth [[file:figs/accelerometers_nano_hexapod.jpg]] The top platform is then excited using an instrumented hammer as shown in Figure [[fig:hammer_excitation_compliance_meas]]. #+name: fig:hammer_excitation_compliance_meas #+caption: Example of an excitation using an instrumented hammer #+attr_latex: :width \linewidth [[file:figs/hammer_excitation_compliance_meas.jpg]] From this experiment, the resonance frequencies and the associated mode shapes can be computed (Section [[sec:modal_analysis_mode_shapes]]). Then, in Section [[sec:compliance_effect_iff]], the vertical compliance of the nano-hexapod is experimentally estimated. Finally, in Section [[sec:compliance_effect_iff_comp_model]], the measured compliance is compare with the estimated one from the Simscape model. *** Matlab Init :noexport:ignore: #+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) <> #+end_src #+begin_src matlab :exports none :results silent :noweb yes <> #+end_src #+begin_src matlab :tangle no addpath('./matlab/mat/'); addpath('./matlab/src/'); addpath('./matlab/'); #+end_src #+begin_src matlab :eval no addpath('./mat/'); addpath('./src/'); #+end_src #+begin_src matlab :tangle no %% Add all useful folders to the path addpath('matlab/nass-simscape/matlab/nano_hexapod/') addpath('matlab/nass-simscape/STEPS/nano_hexapod/') addpath('matlab/nass-simscape/STEPS/png/') addpath('matlab/nass-simscape/src/') addpath('matlab/nass-simscape/mat/') #+end_src #+begin_src matlab :eval no %% Add all useful folders to the path addpath('nass-simscape/matlab/nano_hexapod/') addpath('nass-simscape/STEPS/nano_hexapod/') addpath('nass-simscape/STEPS/png/') addpath('nass-simscape/src/') addpath('nass-simscape/mat/') #+end_src #+begin_src matlab %% Open Simulink Model mdl = 'nano_hexapod_simscape'; options = linearizeOptions; options.SampleTime = 0; Rx = zeros(1, 7); open(mdl) #+end_src *** Obtained Mode Shapes <> We can observe the mode shapes of the first 6 modes that are the suspension modes (the plate is behaving as a solid body) in Figure [[fig:mode_shapes_annotated]]. #+name: fig:mode_shapes_annotated #+caption: Measured mode shapes for the first six modes #+attr_latex: :width \linewidth [[file:figs/mode_shapes_annotated.gif]] Then, there is a mode at 692Hz which corresponds to a flexible mode of the top plate (Figure [[fig:mode_shapes_flexible_annotated]]). #+name: fig:mode_shapes_flexible_annotated #+caption: First flexible mode at 692Hz #+attr_latex: :width 0.3\linewidth [[file:figs/ModeShapeFlex1_crop.gif]] The obtained modes are summarized in Table [[tab:description_modes]]. #+name: tab:description_modes #+caption: Description of the identified modes #+attr_latex: :environment tabularx :width 0.7\linewidth :align ccX #+attr_latex: :center t :booktabs t :float t | Mode | Freq. [Hz] | Description | |------+------------+----------------------------------------------| | 1 | 105 | Suspension Mode: Y-translation | | 2 | 107 | Suspension Mode: X-translation | | 3 | 131 | Suspension Mode: Z-translation | | 4 | 161 | Suspension Mode: Y-tilt | | 5 | 162 | Suspension Mode: X-tilt | | 6 | 180 | Suspension Mode: Z-rotation | | 7 | 692 | (flexible) Membrane mode of the top platform | *** Nano-Hexapod Compliance - Effect of IFF <> In this section, we wish to estimated the effectiveness of the IFF strategy concerning 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 is 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 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 [[fig:compliance_vertical_comp_iff]]. #+begin_src matlab :exports none 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:compliance_vertical_comp_iff #+caption: Measured vertical compliance with and without IFF #+RESULTS: [[file:figs/compliance_vertical_comp_iff.png]] #+begin_important From Figure [[fig: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 now 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, '/duz_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 %% Initialize Nano-Hexapod in Open Loop n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... 'flex_top_type', '4dof', ... 'motion_sensor_type', 'struts', ... 'actuator_type', '2dof'); G_compl_z_ol = linearize(mdl, io, 0.0, options); %% 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 n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... 'flex_top_type', '4dof', ... 'motion_sensor_type', 'struts', ... 'actuator_type', '2dof', ... 'controller_type', 'iff'); G_compl_z_iff = linearize(mdl, io, 0.0, options); #+end_src The comparison is done in Figure [[fig:compliance_vertical_comp_model_iff]]. Again, the model is quite accurate! #+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:compliance_vertical_comp_model_iff #+caption: Measured vertical compliance with and without IFF #+RESULTS: [[file:figs/compliance_vertical_comp_model_iff.png]] ** TODO Accelerometers fixed on the top platform :noexport: *** Introduction :ignore: #+name: fig:acc_top_plat_top_view #+caption: Accelerometers fixed on the top platform #+attr_latex: :width \linewidth [[file:figs/acc_top_plat_top_view.jpg]] *** Matlab Init :noexport:ignore: #+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) <> #+end_src #+begin_src matlab :exports none :results silent :noweb yes <> #+end_src #+begin_src matlab :tangle no addpath('./matlab/mat/'); addpath('./matlab/src/'); addpath('./matlab/'); #+end_src #+begin_src matlab :eval no addpath('./mat/'); addpath('./src/'); #+end_src #+begin_src matlab :tangle no %% Add all useful folders to the path addpath('matlab/nass-simscape/matlab/nano_hexapod/') addpath('matlab/nass-simscape/STEPS/nano_hexapod/') addpath('matlab/nass-simscape/STEPS/png/') addpath('matlab/nass-simscape/src/') addpath('matlab/nass-simscape/mat/') #+end_src #+begin_src matlab :eval no %% Add all useful folders to the path addpath('nass-simscape/matlab/nano_hexapod/') addpath('nass-simscape/STEPS/nano_hexapod/') addpath('nass-simscape/STEPS/png/') addpath('nass-simscape/src/') addpath('nass-simscape/mat/') #+end_src #+begin_src matlab %% Open Simulink Model mdl = 'nano_hexapod_simscape'; options = linearizeOptions; options.SampleTime = 0; Rx = zeros(1, 7); open(mdl) #+end_src *** Experimental Identification #+begin_src matlab %% Load Identification Data meas_acc = {}; for i = 1:6 meas_acc(i) = {load(sprintf('mat/meas_acc_top_plat_strut_%i.mat', i), 't', 'Va', 'de', 'Am')}; end #+end_src #+begin_src matlab %% Setup useful variables % Sampling Time [s] Ts = (meas_acc{1}.t(end) - (meas_acc{1}.t(1)))/(length(meas_acc{1}.t)-1); % Sampling Frequency [Hz] Fs = 1/Ts; % Hannning Windows win = hanning(ceil(1*Fs)); % And we get the frequency vector [~, f] = tfestimate(meas_acc{1}.Va, meas_acc{1}.de, win, [], [], 1/Ts); #+end_src The sensibility of the accelerometers are $0.1 V/g \approx 0.01 V/(m/s^2)$. #+begin_src matlab %% Compute the 6x6 transfer function matrix G_acc = zeros(length(f), 6, 6); for i = 1:6 G_acc(:,:,i) = tfestimate(meas_acc{i}.Va, 1/0.01*meas_acc{i}.Am, win, [], [], 1/Ts); end #+end_src *** Location and orientation of accelerometers #+begin_src matlab Opm = [ 0.047, -0.112, 10e-3; 0.047, -0.112, 10e-3; -0.113, 0.011, 10e-3; -0.113, 0.011, 10e-3; 0.040, 0.113, 10e-3; 0.040, 0.113, 10e-3]'; Osm = [-1, 0, 0; 0, 0, 1; 0, -1, 0; 0, 0, 1; -1, 0, 0; 0, 0, 1]'; #+end_src *** COM #+begin_src matlab Hbm = -15e-3; M = getTransformationMatrixAcc(Opm-[0;0;Hbm], Osm); J = getJacobianNanoHexapod(Hbm); #+end_src #+begin_src matlab G_acc_CoM = zeros(size(G_acc)); for i = 1:length(f) G_acc_CoM(i, :, :) = inv(M)*squeeze(G_acc(i, :, :))*inv(J'); end #+end_src #+begin_src matlab :exports none labels = {'$D_x/F_{x}$', '$D_y/F_{y}$', '$D_z/F_{z}$', '$R_{x}/M_{x}$', '$R_{y}/M_{y}$', '$R_{R}/M_{z}$'}; figure; tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); ax1 = nexttile([2,1]); hold on; for i = 1:2 for j = i+1:3 plot(f, abs(G_acc_CoM(:, i, j)./(-(2*pi*f).^2)), 'color', [0, 0, 0, 0.2], ... 'HandleVisibility', 'off'); end end for i =1:3 set(gca,'ColorOrderIndex',i) plot(f, abs(G_acc_CoM(:,i , i)./(-(2*pi*f).^2)), ... 'DisplayName', labels{i}); end plot(f, abs(G_acc_CoM(:, 1, 2)./(-(2*pi*f).^2)), 'color', [0, 0, 0, 0.2], ... 'DisplayName', '$D_i/F_j$'); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); ylabel('Amplitude $A_m/V_a$ [m/V]'); set(gca, 'XTickLabel',[]); ylim([1e-9, 1e-5]); legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 3); ax2 = nexttile; hold on; for i =1:3 set(gca,'ColorOrderIndex',i) plot(f, 180/pi*angle(G_acc_CoM(:,i , i)./(-(2*pi*f).^2))); end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); hold off; yticks(-360:90:360); linkaxes([ax1,ax2],'x'); xlim([20, 2e3]); #+end_src #+begin_src matlab :exports none labels = {'$D_x/F_{x}$', '$D_y/F_{y}$', '$D_z/F_{z}$', '$R_{x}/M_{x}$', '$R_{y}/M_{y}$', '$R_{R}/M_{z}$'}; figure; tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); ax1 = nexttile([2,1]); hold on; for i =1:6 set(gca,'ColorOrderIndex',i) plot(f, abs(G_acc_CoM(:,i , i)./(-(2*pi*f).^2)), ... 'DisplayName', labels{i}); end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); ylabel('Amplitude $A_m/V_a$ [m/V]'); set(gca, 'XTickLabel',[]); ylim([1e-9, 1e-3]); legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 3); ax2 = nexttile; hold on; for i =1:6 set(gca,'ColorOrderIndex',i) plot(f, 180/pi*angle(G_acc_CoM(:,i , i)./(-(2*pi*f).^2))); end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); hold off; yticks(-360:90:360); linkaxes([ax1,ax2],'x'); xlim([50, 5e2]); #+end_src *** COK #+begin_src matlab Hbm = -42.3e-3; M = getTransformationMatrixAcc(Opm-[0;0;Hbm], Osm); J = getJacobianNanoHexapod(Hbm); #+end_src #+begin_src matlab G_acc_CoK = zeros(size(G_acc)); for i = 1:length(f) G_acc_CoK(i, :, :) = inv(M)*squeeze(G_acc(i, :, :))*inv(J'); end #+end_src #+begin_src matlab :exports none labels = {'$D_x/F_{x}$', '$D_y/F_{y}$', '$D_z/F_{z}$', '$R_{x}/M_{x}$', '$R_{y}/M_{y}$', '$R_{R}/M_{z}$'}; figure; tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); ax1 = nexttile([2,1]); hold on; for i = 1:2 for j = i+1:3 plot(f, abs(G_acc_CoK(:, i, j)./(-(2*pi*f).^2)), 'color', [0, 0, 0, 0.2], ... 'HandleVisibility', 'off'); end end for i =1:3 set(gca,'ColorOrderIndex',i) plot(f, abs(G_acc_CoK(:,i , i)./(-(2*pi*f).^2)), ... 'DisplayName', labels{i}); end plot(f, abs(G_acc_CoK(:, 1, 2)./(-(2*pi*f).^2)), 'color', [0, 0, 0, 0.2], ... 'DisplayName', '$D_i/F_j$'); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); ylabel('Amplitude $A_m/V_a$ [m/V]'); set(gca, 'XTickLabel',[]); ylim([1e-9, 1e-5]); legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 3); ax2 = nexttile; hold on; for i =1:3 set(gca,'ColorOrderIndex',i) plot(f, 180/pi*angle(G_acc_CoK(:,i , i)./(-(2*pi*f).^2))); end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); hold off; yticks(-360:90:360); linkaxes([ax1,ax2],'x'); xlim([20, 2e3]); #+end_src #+begin_src matlab :exports none labels = {'$D_x/\mathcal{F}_x$', '$D_y/\mathcal{F}_y$', '$D_z/\mathcal{F}_z$', ... '$R_x/\mathcal{M}_x$', '$R_y/\mathcal{M}_y$', '$R_z/\mathcal{M}_z$'}; figure; hold on; for i =1:6 set(gca,'ColorOrderIndex',i) plot(f, abs(G_acc_CoK(:,i,i)./(-(2*pi*f).^2)), ... 'DisplayName', labels{i}); end plot(f, abs(G_acc_CoK(:,1,2)./(-(2*pi*f).^2)), 'color', [0, 0, 0, 0.2], ... 'DisplayName', 'Off-Diagonal'); for i = 1:5 for j = i+1:6 plot(f, abs(G_acc_CoK(:,i,j)./(-(2*pi*f).^2)), 'color', [0, 0, 0, 0.2], ... 'HandleVisibility', 'off'); end end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); xlabel('Frequency [Hz]'); ylabel('Amplitude $X_m/V_a$ [m/V]'); xlim([50, 5e2]); ylim([1e-7, 1e-1]); legend('location', 'southwest'); #+end_src *** Comp with the Simscape Model #+begin_src matlab n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... 'flex_top_type', '4dof', ... 'motion_sensor_type', 'struts', ... 'actuator_type', 'flexible', ... 'MO_B', -42.3e-3); #+end_src #+begin_src matlab %% Input/Output definition 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, '/D'], 1, 'openoutput'); io_i = io_i + 1; % Relative Motion Outputs G = linearize(mdl, io, 0.0, options); G.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'}; G.OutputName = {'D1', 'D2', 'D3', 'D4', 'D5', 'D6'}; #+end_src Then use the Jacobian matrices to obtain the "cartesian" centralized plant. #+begin_src matlab Gc = inv(n_hexapod.geometry.J)*... G*... inv(n_hexapod.geometry.J'); #+end_src #+begin_src matlab :exports none freqs = 2*logspace(1, 3, 1000); labels = {'$D_x/\mathcal{F}_x$', '$D_y/\mathcal{F}_y$', '$D_z/\mathcal{F}_z$', ... '$R_x/\mathcal{M}_x$', '$R_y/\mathcal{M}_y$', '$R_z/\mathcal{M}_z$'}; figure; hold on; for i = 1:6 plot(freqs, abs(squeeze(freqresp(Gc(i,i), freqs, 'Hz'))), '-', ... 'DisplayName', labels{i}); end plot(freqs, abs(squeeze(freqresp(Gc(1, 2), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ... 'DisplayName', 'Off-Diagonal'); for i = 1:5 for j = i+1:6 plot(freqs, abs(squeeze(freqresp(Gc(i, j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ... 'HandleVisibility', 'off'); end end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); xlabel('Frequency [Hz]'); ylabel('Amplitude [m/N,rad/N/m]'); xlim([50, 5e2]); ylim([1e-7, 1e-1]); legend('location', 'southwest'); #+end_src ** Conclusion #+begin_important From the previous analysis, several conclusions can be drawn: - Decentralized IFF is very effective in damping the "suspension" modes of the nano-hexapod (Figure [[fig:comp_undamped_opt_iff_gain_diagonal]]) - Decentralized IFF does not damp the "spurious" modes of the struts nor the flexible modes of the top plate (Figure [[fig:comp_undamped_opt_iff_gain_diagonal]]) - Even though the Simscape model and the experimentally measured FRF are in good agreement (Figures [[fig:enc_struts_iff_opt_damp_comp_flex_model_diag]] and [[fig:enc_struts_iff_opt_damp_comp_flex_model_off_diag]]), the obtain dynamics from the control inputs $\bm{u}$ and the encoders $d\bm{\mathcal{L}}_m$ is very difficult to control Therefore, in the following sections, the encoders will be fixed to the plates. The goal is to be less sensitive to the flexible modes of the struts. #+end_important * Encoders fixed to the plates - Dynamics <> ** Introduction :ignore: In this section, the encoders are fixed to the plates rather than to the struts as shown in Figure [[fig:enc_fixed_to_struts]]. #+name: fig:enc_fixed_to_struts #+caption: Nano-Hexapod with encoders fixed to the struts #+attr_latex: :width \linewidth [[file:figs/IMG_20210625_083801.jpg]] It is structured as follow: - Section [[sec:enc_plates_plant_id]]: The dynamics of the nano-hexapod is identified. - Section [[sec:enc_plates_comp_simscape]]: The identified dynamics is compared with the Simscape model. - Section [[sec:enc_plates_iff]]: The Integral Force Feedback (IFF) control strategy is applied and the dynamics of the damped nano-hexapod is identified and compare with the Simscape model. ** Identification of the dynamics <> *** Introduction :ignore: In this section, the dynamics of the nano-hexapod with the encoders fixed to the plates is identified. First, the measurement data are loaded in Section [[sec:enc_plates_plant_id_setup]], then the transfer function matrix from the actuators to the encoders are estimated in Section [[sec:enc_plates_plant_id_dvf]]. Finally, the transfer function matrix from the actuators to the force sensors is estimated in Section [[sec:enc_plates_plant_id_iff]]. *** Matlab Init :noexport:ignore: #+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) <> #+end_src #+begin_src matlab :exports none :results silent :noweb yes <> #+end_src #+begin_src matlab :tangle no addpath('./matlab/mat/'); addpath('./matlab/src/'); addpath('./matlab/'); #+end_src #+begin_src matlab :eval no addpath('./mat/'); addpath('./src/'); #+end_src *** Data Loading and Spectral Analysis Setup <> The actuators are excited one by one using a low pass filtered white noise. For each excitation, the 6 force sensors and 6 encoders are measured and saved. #+begin_src matlab %% Load Identification Data meas_data_lf = {}; for i = 1:6 meas_data_lf(i) = {load(sprintf('mat/frf_exc_strut_%i_enc_plates_noise.mat', i), 't', 'Va', 'Vs', 'de')}; end #+end_src #+begin_src matlab :exports none %% Setup useful variables % Sampling Time [s] Ts = (meas_data_lf{1}.t(end) - (meas_data_lf{1}.t(1)))/(length(meas_data_lf{1}.t)-1); % Sampling Frequency [Hz] Fs = 1/Ts; % Hannning Windows win = hanning(ceil(1*Fs)); % And we get the frequency vector [~, f] = tfestimate(meas_data_lf{1}.Va, meas_data_lf{1}.de, win, [], [], 1/Ts); #+end_src *** Transfer function from Actuator to Encoder <> Let's compute the coherence from the excitation voltage $\bm{u}$ and the displacement $d\bm{\mathcal{L}}_m$ as measured by the encoders. #+begin_src matlab %% Coherence coh_dvf = zeros(length(f), 6, 6); for i = 1:6 coh_dvf(:, :, i) = mscohere(meas_data_lf{i}.Va, meas_data_lf{i}.de, win, [], [], 1/Ts); end #+end_src The obtained coherence shown in Figure [[fig:enc_plates_dvf_coh]] is quite good up to 400Hz. #+begin_src matlab :exports none %% Coherence for the transfer function from u to dLm figure; hold on; for i = 1:5 for j = i+1:6 plot(f, coh_dvf(:, i, j), 'color', [0, 0, 0, 0.2], ... 'HandleVisibility', 'off'); end end for i =1:6 set(gca,'ColorOrderIndex',i) plot(f, coh_dvf(:, i, i), ... 'DisplayName', sprintf('$G_{dvf}(%i,%i)$', i, i)); end plot(f, coh_dvf(:, 1, 2), 'color', [0, 0, 0, 0.2], ... 'DisplayName', '$G_{dvf}(i,j)$'); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); xlabel('Frequency [Hz]'); ylabel('Coherence [-]'); xlim([20, 2e3]); ylim([0, 1]); legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 3); #+end_src #+begin_src matlab :tangle no :exports results :results file replace exportFig('figs/enc_plates_dvf_coh.pdf', 'width', 'wide', 'height', 'normal'); #+end_src #+name: fig:enc_plates_dvf_coh #+caption: Obtained coherence for the DVF plant #+RESULTS: [[file:figs/enc_plates_dvf_coh.png]] Then the 6x6 transfer function matrix is estimated. #+begin_src matlab %% DVF Plant (transfer function from u to dLm) G_dvf = zeros(length(f), 6, 6); for i = 1:6 G_dvf(:,:,i) = tfestimate(meas_data_lf{i}.Va, meas_data_lf{i}.de, win, [], [], 1/Ts); end #+end_src The diagonal and off-diagonal terms of this transfer function matrix are shown in Figure [[fig:enc_plates_dvf_frf]]. #+begin_src matlab :exports none %% Bode plot for the transfer function from u to dLm figure; tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); ax1 = nexttile([2,1]); hold on; for i = 1:5 for j = i+1:6 plot(f, abs(G_dvf(:, i, j)), 'color', [0, 0, 0, 0.2], ... 'HandleVisibility', 'off'); end end for i =1:6 set(gca,'ColorOrderIndex',i) plot(f, abs(G_dvf(:,i, i)), ... 'DisplayName', sprintf('$G_{dvf}(%i,%i)$', i, i)); set(gca,'ColorOrderIndex',i) plot(f, abs(G_dvf(:,i, i)), ... 'HandleVisibility', 'off'); end plot(f, abs(G_dvf(:, 1, 2)), 'color', [0, 0, 0, 0.2], ... 'DisplayName', '$G_{dvf}(i,j)$'); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); ylabel('Amplitude $d_e/V_a$ [m/V]'); set(gca, 'XTickLabel',[]); ylim([1e-9, 1e-3]); legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 3); ax2 = nexttile; hold on; for i =1:6 set(gca,'ColorOrderIndex',i) plot(f, 180/pi*angle(G_dvf(:,i, i))); end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); hold off; yticks(-360:90:360); linkaxes([ax1,ax2],'x'); xlim([20, 2e3]); #+end_src #+begin_src matlab :tangle no :exports results :results file replace exportFig('figs/enc_plates_dvf_frf.pdf', 'width', 'wide', 'height', 'tall'); #+end_src #+name: fig:enc_plates_dvf_frf #+caption: Measured FRF for the DVF plant #+RESULTS: [[file:figs/enc_plates_dvf_frf.png]] #+begin_important From Figure [[fig:enc_plates_dvf_frf]], we can draw few conclusions on the transfer functions from $\bm{u}$ to $d\bm{\mathcal{L}}_m$ when the encoders are fixed to the plates: - the decoupling is rather good at low frequency (below the first suspension mode). The low frequency gain is constant for the off diagonal terms, whereas when the encoders where fixed to the struts, the low frequency gain of the off-diagonal terms were going to zero (Figure [[fig:enc_struts_dvf_frf]]). - the flexible modes of the struts at 226Hz and 337Hz are indeed shown in the transfer functions, but their amplitudes are rather low. - the diagonal terms have alternating poles and zeros up to at least 600Hz: the flexible modes of the struts are not affecting the alternating pole/zero pattern. This what not the case when the encoders were fixed to the struts (Figure [[fig:enc_struts_dvf_frf]]). #+end_important *** Transfer function from Actuator to Force Sensor <> Let's now compute the coherence from the excitation voltage $\bm{u}$ and the voltage $\bm{\tau}_m$ generated by the Force senors. #+begin_src matlab %% Coherence for the IFF plant coh_iff = zeros(length(f), 6, 6); for i = 1:6 coh_iff(:,:,i) = mscohere(meas_data_lf{i}.Va, meas_data_lf{i}.Vs, win, [], [], 1/Ts); end #+end_src The coherence is shown in Figure [[fig:enc_plates_iff_coh]], and is very good for from 10Hz up to 2kHz. #+begin_src matlab :exports none %% Coherence of the IFF Plant (transfer function from u to taum) figure; hold on; for i = 1:5 for j = i+1:6 plot(f, coh_iff(:, i, j), 'color', [0, 0, 0, 0.2], ... 'HandleVisibility', 'off'); end end for i =1:6 set(gca,'ColorOrderIndex',i) plot(f, coh_iff(:,i, i), ... 'DisplayName', sprintf('$G_{iff}(%i,%i)$', i, i)); end plot(f, coh_iff(:, 1, 2), 'color', [0, 0, 0, 0.2], ... 'DisplayName', '$G_{iff}(i,j)$'); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); xlabel('Frequency [Hz]'); ylabel('Coherence [-]'); xlim([20, 2e3]); ylim([0, 1]); legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 3); #+end_src #+begin_src matlab :tangle no :exports results :results file replace exportFig('figs/enc_plates_iff_coh.pdf', 'width', 'wide', 'height', 'normal'); #+end_src #+name: fig:enc_plates_iff_coh #+caption: Obtained coherence for the IFF plant #+RESULTS: [[file:figs/enc_plates_iff_coh.png]] Then the 6x6 transfer function matrix is estimated. #+begin_src matlab %% IFF Plant G_iff = zeros(length(f), 6, 6); for i = 1:6 G_iff(:,:,i) = tfestimate(meas_data_lf{i}.Va, meas_data_lf{i}.Vs, win, [], [], 1/Ts); end #+end_src The bode plot of the diagonal and off-diagonal terms are shown in Figure [[fig:enc_plates_iff_frf]]. #+begin_src matlab :exports none %% Bode plot of the IFF Plant (transfer function from u to taum) figure; tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); ax1 = nexttile([2,1]); hold on; for i = 1:5 for j = i+1:6 plot(f, abs(G_iff(:, i, j)), 'color', [0, 0, 0, 0.2], ... 'HandleVisibility', 'off'); end end for i =1:6 set(gca,'ColorOrderIndex',i) plot(f, abs(G_iff(:,i , i)), ... 'DisplayName', sprintf('$G_{iff}(%i,%i)$', i, i)); end plot(f, abs(G_iff(:, 1, 2)), 'color', [0, 0, 0, 0.2], ... 'DisplayName', '$G_{iff}(i,j)$'); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); ylabel('Amplitude $V_s/V_a$ [V/V]'); set(gca, 'XTickLabel',[]); legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 3); ylim([1e-3, 1e2]); ax2 = nexttile; hold on; for i =1:6 set(gca,'ColorOrderIndex',i) plot(f, 180/pi*angle(G_iff(:,i, i))); end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); hold off; yticks(-360:90:360); linkaxes([ax1,ax2],'x'); xlim([20, 2e3]); #+end_src #+begin_src matlab :tangle no :exports results :results file replace exportFig('figs/enc_plates_iff_frf.pdf', 'width', 'wide', 'height', 'tall'); #+end_src #+name: fig:enc_plates_iff_frf #+caption: Measured FRF for the IFF plant #+RESULTS: [[file:figs/enc_plates_iff_frf.png]] #+begin_important It is shown in Figure [[fig:enc_plates_iff_comp_simscape_all]] that: - The IFF plant has alternating poles and zeros - The first flexible mode of the struts as 235Hz is appearing, and therefore is should be possible to add some damping to this mode using IFF - The decoupling is quite good at low frequency (below the first model) as well as high frequency (above the last suspension mode, except near the flexible modes of the top plate) #+end_important *** Save Identified Plants The identified dynamics is saved for further use. #+begin_src matlab :exports none :tangle no save('matlab/mat/identified_plants_enc_plates.mat', 'f', 'Ts', 'G_iff', 'G_dvf') #+end_src #+begin_src matlab :eval no save('mat/identified_plants_enc_plates.mat', 'f', 'Ts', 'G_iff', 'G_dvf') #+end_src ** Comparison with the Simscape Model <> *** Introduction :ignore: In this section, the measured dynamics done in Section [[sec:enc_plates_plant_id]] is compared with the dynamics estimated from the Simscape model. *** Matlab Init :noexport:ignore: #+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) <> #+end_src #+begin_src matlab :exports none :results silent :noweb yes <> #+end_src #+begin_src matlab :tangle no addpath('./matlab/mat/'); addpath('./matlab/src/'); addpath('./matlab/'); #+end_src #+begin_src matlab :eval no addpath('./mat/'); addpath('./src/'); #+end_src #+begin_src matlab :tangle no %% Add all useful folders to the path addpath('matlab/nass-simscape/matlab/nano_hexapod/') addpath('matlab/nass-simscape/STEPS/nano_hexapod/') addpath('matlab/nass-simscape/STEPS/png/') addpath('matlab/nass-simscape/src/') addpath('matlab/nass-simscape/mat/') #+end_src #+begin_src matlab :eval no %% Add all useful folders to the path addpath('nass-simscape/matlab/nano_hexapod/') addpath('nass-simscape/STEPS/nano_hexapod/') addpath('nass-simscape/STEPS/png/') addpath('nass-simscape/src/') addpath('nass-simscape/mat/') #+end_src #+begin_src matlab %% Load identification data load('identified_plants_enc_plates.mat', 'f', 'Ts', 'G_iff', 'G_dvf') #+end_src #+begin_src matlab %% Open Simulink Model mdl = 'nano_hexapod_simscape'; options = linearizeOptions; options.SampleTime = 0; Rx = zeros(1, 7); open(mdl) #+end_src *** Identification Setup The nano-hexapod is initialized with the APA taken as flexible models. #+begin_src matlab %% Initialize Nano-Hexapod n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... 'flex_top_type', '4dof', ... 'motion_sensor_type', 'plates', ... 'actuator_type', 'flexible'); #+end_src *** Dynamics from Actuator to Force Sensors Then the transfer function from $\bm{u}$ to $\bm{\tau}_m$ is identified using the Simscape model. #+begin_src matlab %% Identify the IFF Plant (transfer function from u to taum) 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, '/dum'], 1, 'openoutput'); io_i = io_i + 1; % Force Sensors Giff = exp(-s*Ts)*linearize(mdl, io, 0.0, options); #+end_src The identified dynamics is compared with the measured FRF: - Figure [[fig:enc_plates_iff_comp_simscape_all]]: the individual transfer function from $u_1$ (the DAC voltage for the first actuator) to the force sensors of all 6 struts are compared - Figure [[fig:enc_plates_iff_comp_simscape]]: all the diagonal elements are compared - Figure [[fig:enc_plates_iff_comp_offdiag_simscape]]: all the off-diagonal elements are compared #+begin_src matlab :exports none %% Comparison of the plants (encoder output) when tuning the misalignment freqs = 2*logspace(1, 3, 1000); i_input = 1; figure; tiledlayout(2, 3, 'TileSpacing', 'Compact', 'Padding', 'None'); ax1 = nexttile(); hold on; plot(f, abs(G_iff(:, 1, i_input))); plot(freqs, abs(squeeze(freqresp(Giff(1, i_input), freqs, 'Hz')))); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); set(gca, 'XTickLabel',[]); ylabel('Amplitude [m/V]'); title(sprintf('$d\\tau_{m1}/u_{%i}$', i_input)); ax2 = nexttile(); hold on; plot(f, abs(G_iff(:, 2, i_input))); plot(freqs, abs(squeeze(freqresp(Giff(2, i_input), freqs, 'Hz')))); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]); title(sprintf('$d\\tau_{m2}/u_{%i}$', i_input)); ax3 = nexttile(); hold on; plot(f, abs(G_iff(:, 3, i_input)), ... 'DisplayName', 'Meas.'); plot(freqs, abs(squeeze(freqresp(Giff(3, i_input), freqs, 'Hz'))), ... 'DisplayName', 'Model'); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]); legend('location', 'southeast', 'FontSize', 8); title(sprintf('$d\\tau_{m3}/u_{%i}$', i_input)); ax4 = nexttile(); hold on; plot(f, abs(G_iff(:, 4, i_input))); plot(freqs, abs(squeeze(freqresp(Giff(4, i_input), freqs, 'Hz')))); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); xlabel('Frequency [Hz]'); ylabel('Amplitude [m/V]'); title(sprintf('$d\\tau_{m4}/u_{%i}$', i_input)); ax5 = nexttile(); hold on; plot(f, abs(G_iff(:, 5, i_input))); plot(freqs, abs(squeeze(freqresp(Giff(5, i_input), freqs, 'Hz')))); hold off; xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]); set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); title(sprintf('$d\\tau_{m5}/u_{%i}$', i_input)); ax6 = nexttile(); hold on; plot(f, abs(G_iff(:, 6, i_input))); plot(freqs, abs(squeeze(freqresp(Giff(6, i_input), freqs, 'Hz')))); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]); title(sprintf('$d\\tau_{m6}/u_{%i}$', i_input)); linkaxes([ax1,ax2,ax3,ax4,ax5,ax6],'xy'); xlim([20, 2e3]); ylim([1e-2, 1e2]); #+end_src #+begin_src matlab :tangle no :exports results :results file replace exportFig('figs/enc_plates_iff_comp_simscape_all.pdf', 'width', 'full', 'height', 'tall'); #+end_src #+name: fig:enc_plates_iff_comp_simscape_all #+caption: IFF Plant for the first actuator input and all the force senosrs #+RESULTS: [[file:figs/enc_plates_iff_comp_simscape_all.png]] #+begin_src matlab :exports none %% Bode plot of the identified IFF Plant (Simscape) and measured FRF data freqs = 2*logspace(1, 3, 1000); figure; tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); ax1 = nexttile([2,1]); hold on; plot(f, abs(G_iff(:,1, 1)), 'color', [0,0,0,0.2], ... 'DisplayName', '$\tau_{m,i}/u_i$ - FRF') for i = 2:6 set(gca,'ColorOrderIndex',2) plot(f, abs(G_iff(:,i, i)), 'color', [0,0,0,0.2], ... 'HandleVisibility', 'off'); end set(gca,'ColorOrderIndex',2); plot(freqs, abs(squeeze(freqresp(Giff(1,1), freqs, 'Hz'))), '-', ... 'DisplayName', '$\tau_{m,i}/u_i$ - Model') for i = 2:6 set(gca,'ColorOrderIndex',2); plot(freqs, abs(squeeze(freqresp(Giff(i,i), freqs, 'Hz'))), '-', ... 'HandleVisibility', 'off'); end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); ylabel('Amplitude [V/V]'); set(gca, 'XTickLabel',[]); legend('location', 'southeast'); ax2 = nexttile; hold on; for i = 1:6 plot(f, 180/pi*angle(G_iff(:,i, i)), 'color', [0,0,0,0.2]); end for i = 1:6 set(gca,'ColorOrderIndex',2); plot(freqs, 180/pi*angle(squeeze(freqresp(Giff(i,i), freqs, 'Hz'))), '-'); end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); ylim([-180, 180]); yticks([-180, -90, 0, 90, 180]); linkaxes([ax1,ax2],'x'); xlim([freqs(1), freqs(end)]); #+end_src #+begin_src matlab :tangle no :exports results :results file replace exportFig('figs/enc_plates_iff_comp_simscape.pdf', 'width', 'wide', 'height', 'tall'); #+end_src #+name: fig:enc_plates_iff_comp_simscape #+caption: Diagonal elements of the IFF Plant #+RESULTS: [[file:figs/enc_plates_iff_comp_simscape.png]] #+begin_src matlab :exports none %% Bode plot of the identified IFF Plant (Simscape) and measured FRF data (off-diagonal elements) freqs = 2*logspace(1, 3, 1000); figure; hold on; % Off diagonal terms plot(f, abs(G_iff(:, 1, 2)), 'color', [0,0,0,0.2], ... 'DisplayName', '$\tau_{m,i}/u_j$ - FRF') for i = 1:5 for j = i+1:6 plot(f, abs(G_iff(:, i, j)), 'color', [0,0,0,0.2], ... 'HandleVisibility', 'off'); end end set(gca,'ColorOrderIndex',2); plot(freqs, abs(squeeze(freqresp(Giff(1, 2), freqs, 'Hz'))), ... 'DisplayName', '$\tau_{m,i}/u_j$ - Model') for i = 1:5 for j = i+1:6 set(gca,'ColorOrderIndex',2); plot(freqs, abs(squeeze(freqresp(Giff(i, j), freqs, 'Hz'))), ... 'HandleVisibility', 'off'); end end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); xlabel('Frequency [Hz]'); ylabel('Amplitude [V/V]'); xlim([freqs(1), freqs(end)]); ylim([1e-3, 1e2]); legend('location', 'northeast'); #+end_src #+begin_src matlab :tangle no :exports results :results file replace exportFig('figs/enc_plates_iff_comp_offdiag_simscape.pdf', 'width', 'wide', 'height', 'normal'); #+end_src #+name: fig:enc_plates_iff_comp_offdiag_simscape #+caption: Off diagonal elements of the IFF Plant #+RESULTS: [[file:figs/enc_plates_iff_comp_offdiag_simscape.png]] *** Dynamics from Actuator to Encoder Now, the dynamics from the DAC voltage $\bm{u}$ to the encoders $d\bm{\mathcal{L}}_m$ is estimated using the Simscape model. #+begin_src matlab %% Identify the DVF Plant (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; % Encoders Gdvf = exp(-s*Ts)*linearize(mdl, io, 0.0, options); #+end_src The identified dynamics is compared with the measured FRF: - Figure [[fig:enc_plates_dvf_comp_simscape_all]]: the individual transfer function from $u_3$ (the DAC voltage for the actuator number 3) to the six encoders - Figure [[fig:enc_plates_dvf_comp_simscape]]: all the diagonal elements are compared - Figure [[fig:enc_plates_dvf_comp_offdiag_simscape]]: all the off-diagonal elements are compared #+begin_src matlab :exports none %% Comparison of the plants (encoder output) when tuning the misalignment freqs = 2*logspace(1, 3, 1000); i_input = 3; figure; tiledlayout(2, 3, 'TileSpacing', 'Compact', 'Padding', 'None'); ax1 = nexttile(); hold on; plot(f, abs(G_dvf(:, 1, i_input))); plot(freqs, abs(squeeze(freqresp(Gdvf(1, i_input), freqs, 'Hz')))); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); set(gca, 'XTickLabel',[]); ylabel('Amplitude [m/V]'); title(sprintf('$d\\mathcal{L}_{m1}/u_{%i}$', i_input)); ax2 = nexttile(); hold on; plot(f, abs(G_dvf(:, 2, i_input))); plot(freqs, abs(squeeze(freqresp(Gdvf(2, i_input), freqs, 'Hz')))); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]); title(sprintf('$d\\mathcal{L}_{m2}/u_{%i}$', i_input)); ax3 = nexttile(); hold on; plot(f, abs(G_dvf(:, 3, i_input)), ... 'DisplayName', 'Meas.'); plot(freqs, abs(squeeze(freqresp(Gdvf(3, i_input), freqs, 'Hz'))), ... 'DisplayName', 'Model'); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]); legend('location', 'southeast', 'FontSize', 8); title(sprintf('$d\\mathcal{L}_{m3}/u_{%i}$', i_input)); ax4 = nexttile(); hold on; plot(f, abs(G_dvf(:, 4, i_input))); plot(freqs, abs(squeeze(freqresp(Gdvf(4, i_input), freqs, 'Hz')))); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); xlabel('Frequency [Hz]'); ylabel('Amplitude [m/V]'); title(sprintf('$d\\mathcal{L}_{m4}/u_{%i}$', i_input)); ax5 = nexttile(); hold on; plot(f, abs(G_dvf(:, 5, i_input))); plot(freqs, abs(squeeze(freqresp(Gdvf(5, i_input), freqs, 'Hz')))); hold off; xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]); set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); title(sprintf('$d\\mathcal{L}_{m5}/u_{%i}$', i_input)); ax6 = nexttile(); hold on; plot(f, abs(G_dvf(:, 6, i_input))); plot(freqs, abs(squeeze(freqresp(Gdvf(6, i_input), freqs, 'Hz')))); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]); title(sprintf('$d\\mathcal{L}_{m6}/u_{%i}$', i_input)); linkaxes([ax1,ax2,ax3,ax4,ax5,ax6],'xy'); xlim([40, 4e2]); ylim([1e-8, 1e-2]); #+end_src #+begin_src matlab :tangle no :exports results :results file replace exportFig('figs/enc_plates_dvf_comp_simscape_all.pdf', 'width', 'full', 'height', 'tall'); #+end_src #+name: fig:enc_plates_dvf_comp_simscape_all #+caption: DVF Plant for the first actuator input and all the encoders #+RESULTS: [[file:figs/enc_plates_dvf_comp_simscape_all.png]] #+begin_src matlab :exports none %% Diagonal elements of the DVF plant freqs = 2*logspace(1, 3, 1000); figure; tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); ax1 = nexttile([2,1]); hold on; plot(f, abs(G_dvf(:,1, 1)), 'color', [0,0,0,0.2], ... 'DisplayName', '$d\mathcal{L}_{m,i}/u_i$ - FRF') for i = 2:6 set(gca,'ColorOrderIndex',2) plot(f, abs(G_dvf(:,i, i)), 'color', [0,0,0,0.2], ... 'HandleVisibility', 'off'); end set(gca,'ColorOrderIndex',2); plot(freqs, abs(squeeze(freqresp(Gdvf(1,1), freqs, 'Hz'))), '-', ... 'DisplayName', '$d\mathcal{L}_{m,i}/u_i$ - Model') for i = 2:6 set(gca,'ColorOrderIndex',2); plot(freqs, abs(squeeze(freqresp(Gdvf(i,i), freqs, 'Hz'))), '-', ... 'HandleVisibility', 'off'); end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); ylabel('Amplitude [m/V]'); set(gca, 'XTickLabel',[]); ylim([1e-8, 1e-3]); legend('location', 'northeast'); ax2 = nexttile; hold on; for i = 1:6 plot(f, 180/pi*angle(G_dvf(:,i, i)), 'color', [0,0,0,0.2]); end for i = 1:6 set(gca,'ColorOrderIndex',2); plot(freqs, 180/pi*angle(squeeze(freqresp(Gdvf(i,i), freqs, 'Hz'))), '-'); end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); ylim([-180, 180]); yticks([-180, -90, 0, 90, 180]); linkaxes([ax1,ax2],'x'); xlim([freqs(1), freqs(end)]); #+end_src #+begin_src matlab :tangle no :exports results :results file replace exportFig('figs/enc_plates_dvf_comp_simscape.pdf', 'width', 'wide', 'height', 'tall'); #+end_src #+name: fig:enc_plates_dvf_comp_simscape #+caption: Diagonal elements of the DVF Plant #+RESULTS: [[file:figs/enc_plates_dvf_comp_simscape.png]] #+begin_src matlab :exports none %% Off-diagonal elements of the DVF plant freqs = 2*logspace(1, 3, 1000); figure; hold on; % Off diagonal terms plot(f, abs(G_dvf(:, 1, 2)), 'color', [0,0,0,0.2], ... 'DisplayName', '$d\mathcal{L}_{m,i}/u_j$ - FRF') for i = 1:5 for j = i+1:6 plot(f, abs(G_dvf(:, i, j)), 'color', [0,0,0,0.2], ... 'HandleVisibility', 'off'); end end set(gca,'ColorOrderIndex',2); plot(freqs, abs(squeeze(freqresp(Gdvf(1, 2), freqs, 'Hz'))), ... 'DisplayName', '$d\mathcal{L}_{m,i}/u_j$ - Model') for i = 1:5 for j = i+1:6 set(gca,'ColorOrderIndex',2); plot(freqs, abs(squeeze(freqresp(Gdvf(i, j), freqs, 'Hz'))), ... 'HandleVisibility', 'off'); end end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); xlabel('Frequency [Hz]'); ylabel('Amplitude [m/V]'); xlim([freqs(1), freqs(end)]); ylim([1e-8, 1e-3]); legend('location', 'northeast'); #+end_src #+begin_src matlab :tangle no :exports results :results file replace exportFig('figs/enc_plates_dvf_comp_offdiag_simscape.pdf', 'width', 'wide', 'height', 'normal'); #+end_src #+name: fig:enc_plates_dvf_comp_offdiag_simscape #+caption: Off diagonal elements of the DVF Plant #+RESULTS: [[file:figs/enc_plates_dvf_comp_offdiag_simscape.png]] *** Conclusion #+begin_important The Simscape model is quite accurate for the transfer function matrices from $\bm{u}$ to $\bm{\tau}_m$ and from $\bm{u}$ to $d\bm{\mathcal{L}}_m$ except at frequencies of the flexible modes of the top-plate. The Simscape model can therefore be used to develop the control strategies. #+end_important ** Integral Force Feedback <> *** Introduction :ignore: In this section, the Integral Force Feedback (IFF) control strategy is applied to the nano-hexapod in order to add damping to the suspension modes. The control architecture is shown in Figure [[fig:control_architecture_iff]]: - $\bm{\tau}_m$ is the measured voltage of the 6 force sensors - $\bm{K}_{\text{IFF}}$ is the $6 \times 6$ diagonal controller - $\bm{u}$ is the plant input (voltage generated by the 6 DACs) - $\bm{u}^\prime$ is the new plant inputs with added damping #+begin_src latex :file control_architecture_iff.pdf \begin{tikzpicture} % Blocs \node[block={3.0cm}{2.0cm}] (P) {Plant}; \coordinate[] (inputF) at ($(P.south west)!0.5!(P.north west)$); \coordinate[] (outputF) at ($(P.south east)!0.7!(P.north east)$); \coordinate[] (outputL) at ($(P.south east)!0.3!(P.north east)$); \node[block, above=0.4 of P] (Kiff) {$\bm{K}_\text{IFF}$}; \node[addb, left= of inputF] (addF) {}; % Connections and labels \draw[->] (outputF) -- ++(1, 0) node[below left]{$\bm{\tau}_m$}; \draw[->] (outputL) -- ++(1, 0) node[below left]{$d\bm{\mathcal{L}}_m$}; \draw[->] ($(outputF) + (0.6, 0)$)node[branch]{} |- (Kiff.east); \draw[->] (Kiff.west) -| (addF.north); \draw[->] (addF.east) -- (inputF) node[above left]{$\bm{u}$}; \draw[<-] (addF.west) -- ++(-1, 0) node[above right]{$\bm{u}^\prime$}; \end{tikzpicture} #+end_src #+name: fig:control_architecture_iff #+caption: Integral Force Feedback Strategy #+RESULTS: [[file:figs/control_architecture_iff.png]] - Section [[sec:enc_struts_effect_iff_plant]] *** Matlab Init :noexport:ignore: #+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) <> #+end_src #+begin_src matlab :exports none :results silent :noweb yes <> #+end_src #+begin_src matlab :tangle no addpath('./matlab/mat/'); addpath('./matlab/src/'); addpath('./matlab/'); #+end_src #+begin_src matlab :eval no addpath('./mat/'); addpath('./src/'); #+end_src #+begin_src matlab load('identified_plants_enc_plates.mat', 'f', 'Ts', 'G_iff', 'G_dvf') #+end_src #+begin_src matlab :tangle no %% Add all useful folders to the path addpath('matlab/nass-simscape/matlab/nano_hexapod/') addpath('matlab/nass-simscape/STEPS/nano_hexapod/') addpath('matlab/nass-simscape/STEPS/png/') addpath('matlab/nass-simscape/src/') addpath('matlab/nass-simscape/mat/') #+end_src #+begin_src matlab :eval no %% Add all useful folders to the path addpath('nass-simscape/matlab/nano_hexapod/') addpath('nass-simscape/STEPS/nano_hexapod/') addpath('nass-simscape/STEPS/png/') addpath('nass-simscape/src/') addpath('nass-simscape/mat/') #+end_src #+begin_src matlab %% Open Simulink Model mdl = 'nano_hexapod_simscape'; options = linearizeOptions; options.SampleTime = 0; open(mdl) Rx = zeros(1, 7); colors = colororder; #+end_src *** Effect of IFF on the plant - Simscape Model <> The nano-hexapod is initialized with flexible APA and the encoders fixed to the struts. #+begin_src matlab %% Initialize the Simscape model in closed loop n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... 'flex_top_type', '4dof', ... 'motion_sensor_type', 'plates', ... 'actuator_type', 'flexible'); #+end_src The same controller as the one developed when the encoder were fixed to the struts is used. #+begin_src matlab %% Optimal IFF controller load('Kiff.mat', 'Kiff') #+end_src The transfer function from $\bm{u}^\prime$ to $d\bm{\mathcal{L}}_m$ is identified. #+begin_src matlab %% Identify the (damped) transfer function from u to dLm for different values of the IFF gain clear io; io_i = 1; io(io_i) = linio([mdl, '/du'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs io(io_i) = linio([mdl, '/dL'], 1, 'openoutput'); io_i = io_i + 1; % Plate Displacement (encoder) #+end_src First in Open-Loop: #+begin_src matlab %% Transfer function from u to dL (open-loop) Gd_ol = exp(-s*Ts)*linearize(mdl, io, 0.0, options); #+end_src And then with the IFF controller: #+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', 'iff'); %% Transfer function from u to dL (IFF) Gd_iff = exp(-s*Ts)*linearize(mdl, io, 0.0, options); #+end_src It is first verified that the system is stable: #+begin_src matlab :results value replace :exports both :tangle no isstable(Gd_iff) #+end_src #+RESULTS: : 1 The diagonal and off-diagonal terms of the $6 \times 6$ transfer function matrices identified are compared in Figure [[fig:enc_plates_iff_gains_effect_dvf_plant]]. It is shown, as was the case when the encoders were fixed to the struts, that the IFF control strategy is very effective in damping the suspension modes of the nano-hexapod. #+begin_src matlab :exports none %% Bode plot of the transfer function from u to dLm for tested values of the IFF gain freqs = 2*logspace(1, 3, 1000); figure; tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); ax1 = nexttile([2,1]); hold on; plot(freqs, abs(squeeze(freqresp(Gd_ol(1,1), freqs, 'Hz'))), '-', ... 'DisplayName', 'OL - Diag'); plot(freqs, abs(squeeze(freqresp(Gd_iff(1,1), freqs, 'Hz'))), '-', ... 'DisplayName', 'IFF - Diag'); for i = 2:6 set(gca,'ColorOrderIndex',1); plot(freqs, abs(squeeze(freqresp(Gd_ol(1,1), freqs, 'Hz'))), '-', ... 'HandleVisibility', 'off'); end for i = 2:6 set(gca,'ColorOrderIndex',2); plot(freqs, abs(squeeze(freqresp(Gd_iff(i,i), freqs, 'Hz'))), '-', ... 'HandleVisibility', 'off'); end plot(freqs, abs(squeeze(freqresp(Gd_ol(1,2), freqs, 'Hz'))), 'color', [colors(1,:), 0.2], ... 'DisplayName', 'OL - Off-diag') for i = 1:5 for j = i+1:6 plot(freqs, abs(squeeze(freqresp(Gd_ol(i,j), freqs, 'Hz'))), 'color', [colors(1,:), 0.2], ... 'HandleVisibility', 'off'); end end plot(freqs, abs(squeeze(freqresp(Gd_iff(1,2), freqs, 'Hz'))), 'color', [colors(2,:), 0.2], ... 'DisplayName', 'IFF - Off-diag') for i = 1:5 for j = i+1:6 plot(freqs, abs(squeeze(freqresp(Gd_iff(i,j), freqs, 'Hz'))), 'color', [colors(2,:), 0.2], ... 'HandleVisibility', 'off'); end end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); ylabel('Amplitude [m/V]'); set(gca, 'XTickLabel',[]); legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 2); ax2 = nexttile; hold on; for i = 1:6 set(gca,'ColorOrderIndex',1); plot(freqs, 180/pi*angle(squeeze(freqresp(Gd_ol(1,1), freqs, 'Hz'))), '-', ... 'HandleVisibility', 'off'); set(gca,'ColorOrderIndex',2); plot(freqs, 180/pi*angle(squeeze(freqresp(Gd_iff(i,i), freqs, 'Hz'))), '-', ... 'HandleVisibility', 'off'); end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); ylim([-180, 180]); yticks([-180, -90, 0, 90, 180]); linkaxes([ax1,ax2],'x'); xlim([freqs(1), freqs(end)]); #+end_src #+begin_src matlab :tangle no :exports results :results file replace exportFig('figs/enc_plates_iff_gains_effect_dvf_plant.pdf', 'width', 'wide', 'height', 'tall'); #+end_src #+name: fig:enc_plates_iff_gains_effect_dvf_plant #+caption: Effect of the IFF control strategy on the transfer function from $\bm{\tau}$ to $d\bm{\mathcal{L}}_m$ #+RESULTS: [[file:figs/enc_plates_iff_gains_effect_dvf_plant.png]] *** Effect of IFF on the plant - FRF The IFF control strategy is experimentally implemented. The (damped) transfer function from $\bm{u}^\prime$ to $d\bm{\mathcal{L}}_m$ is experimentally identified. The identification data are loaded: #+begin_src matlab %% Load Identification Data meas_iff_plates = {}; for i = 1:6 meas_iff_plates(i) = {load(sprintf('mat/frf_exc_iff_strut_%i_enc_plates_noise.mat', i), 't', 'Va', 'Vs', 'de', 'u')}; end #+end_src And the parameters used for the transfer function estimation are defined below. #+begin_src matlab % Sampling Time [s] Ts = (meas_iff_plates{1}.t(end) - (meas_iff_plates{1}.t(1)))/(length(meas_iff_plates{1}.t)-1); % Hannning Windows win = hanning(ceil(1*Fs)); % And we get the frequency vector [~, f] = tfestimate(meas_iff_plates{1}.Va, meas_iff_plates{1}.de, win, [], [], 1/Ts); #+end_src The estimation is performed using the =tfestimate= command. #+begin_src matlab %% Estimation of the transfer function matrix from u to dL when IFF is applied G_enc_iff_opt = zeros(length(f), 6, 6); for i = 1:6 G_enc_iff_opt(:,:,i) = tfestimate(meas_iff_plates{i}.Va, meas_iff_plates{i}.de, win, [], [], 1/Ts); end #+end_src The obtained diagonal and off-diagonal elements of the transfer function from $\bm{u}^\prime$ to $d\bm{\mathcal{L}}_m$ are shown in Figure [[fig:enc_plant_plates_effect_iff]] both without and with IFF. #+begin_src matlab :exports none %% Bode plot of the transfer function from u to dLm for tested values of the IFF gain freqs = 2*logspace(1, 3, 1000); figure; tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); ax1 = nexttile([2,1]); hold on; plot(f, abs(G_dvf(:,1,1)), '-', ... 'DisplayName', 'OL - Diag'); plot(f, abs(G_enc_iff_opt(:,1,1)), '-', ... 'DisplayName', 'IFF - Diag'); for i = 2:6 set(gca,'ColorOrderIndex',1); plot(f, abs(G_dvf(:,1,1)), '-', ... 'HandleVisibility', 'off'); end for i = 2:6 set(gca,'ColorOrderIndex',2); plot(f, abs(G_enc_iff_opt(:,i,i)), '-', ... 'HandleVisibility', 'off'); end plot(f, abs(G_dvf(:,1,2)), 'color', [colors(1,:), 0.2], ... 'DisplayName', 'OL - Off-diag') for i = 1:5 for j = i+1:6 plot(f, abs(G_dvf(:,i,j)), 'color', [colors(1,:), 0.2], ... 'HandleVisibility', 'off'); end end plot(f, abs(G_enc_iff_opt(:,1,2)), 'color', [colors(2,:), 0.2], ... 'DisplayName', 'IFF - Off-diag') for i = 1:5 for j = i+1:6 plot(f, abs(G_enc_iff_opt(:,i,j)), 'color', [colors(2,:), 0.2], ... 'HandleVisibility', 'off'); end end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); ylabel('Amplitude [m/V]'); set(gca, 'XTickLabel',[]); legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2); ax2 = nexttile; hold on; for i = 1:6 set(gca,'ColorOrderIndex',1); plot(f, 180/pi*angle(G_dvf(:,1,1)), '-') set(gca,'ColorOrderIndex',2); plot(f, 180/pi*angle(G_enc_iff_opt(:,i,i)), '-') end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); ylim([-180, 180]); yticks([-180, -90, 0, 90, 180]); linkaxes([ax1,ax2],'x'); xlim([freqs(1), freqs(end)]); #+end_src #+begin_src matlab :tangle no :exports results :results file replace exportFig('figs/enc_plant_plates_effect_iff.pdf', 'width', 'wide', 'height', 'tall'); #+end_src #+name: fig:enc_plant_plates_effect_iff #+caption: Effect of the IFF control strategy on the transfer function from $\bm{\tau}$ to $d\bm{\mathcal{L}}_m$ #+RESULTS: [[file:figs/enc_plant_plates_effect_iff.png]] #+begin_important As was predicted with the Simscape model, the IFF control strategy is very effective in damping the suspension modes of the nano-hexapod. Little damping is also applied on the first flexible mode of the strut at 235Hz. However, no damping is applied on other modes, such as the flexible modes of the top plate. #+end_important *** Comparison of the measured FRF and the Simscape model Let's now compare the obtained damped plants obtained experimentally with the one extracted from Simscape: - Figure [[fig:enc_plates_opt_iff_comp_simscape_all]]: the individual transfer function from $u_1^\prime$ to the six encoders are comapred - Figure [[fig:damped_iff_plates_plant_comp_diagonal]]: all the diagonal elements are compared - Figure [[fig:damped_iff_plates_plant_comp_off_diagonal]]: all the off-diagonal elements are compared #+begin_src matlab :exports none %% Comparison of the plants (encoder output) when tuning the misalignment freqs = 2*logspace(1, 3, 1000); i_input = 1; figure; tiledlayout(2, 3, 'TileSpacing', 'Compact', 'Padding', 'None'); ax1 = nexttile(); hold on; plot(f, abs(G_enc_iff_opt(:, 1, i_input))); plot(freqs, abs(squeeze(freqresp(Gd_iff(1, i_input), freqs, 'Hz')))); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); set(gca, 'XTickLabel',[]); ylabel('Amplitude [m/V]'); title(sprintf('$d\\tau_{m1}/u_{%i}$', i_input)); ax2 = nexttile(); hold on; plot(f, abs(G_enc_iff_opt(:, 2, i_input))); plot(freqs, abs(squeeze(freqresp(Gd_iff(2, i_input), freqs, 'Hz')))); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]); title(sprintf('$d\\tau_{m2}/u_{%i}$', i_input)); ax3 = nexttile(); hold on; plot(f, abs(G_enc_iff_opt(:, 3, i_input)), ... 'DisplayName', 'Meas.'); plot(freqs, abs(squeeze(freqresp(Gd_iff(3, i_input), freqs, 'Hz'))), ... 'DisplayName', 'Model'); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]); legend('location', 'southeast', 'FontSize', 8); title(sprintf('$d\\tau_{m3}/u_{%i}$', i_input)); ax4 = nexttile(); hold on; plot(f, abs(G_enc_iff_opt(:, 4, i_input))); plot(freqs, abs(squeeze(freqresp(Gd_iff(4, i_input), freqs, 'Hz')))); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); xlabel('Frequency [Hz]'); ylabel('Amplitude [m/V]'); title(sprintf('$d\\tau_{m4}/u_{%i}$', i_input)); ax5 = nexttile(); hold on; plot(f, abs(G_enc_iff_opt(:, 5, i_input))); plot(freqs, abs(squeeze(freqresp(Gd_iff(5, i_input), freqs, 'Hz')))); hold off; xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]); set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); title(sprintf('$d\\tau_{m5}/u_{%i}$', i_input)); ax6 = nexttile(); hold on; plot(f, abs(G_enc_iff_opt(:, 6, i_input))); plot(freqs, abs(squeeze(freqresp(Gd_iff(6, i_input), freqs, 'Hz')))); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]); title(sprintf('$d\\tau_{m6}/u_{%i}$', i_input)); linkaxes([ax1,ax2,ax3,ax4,ax5,ax6],'xy'); xlim([20, 2e3]); ylim([1e-8, 1e-4]); #+end_src #+begin_src matlab :tangle no :exports results :results file replace exportFig('figs/enc_plates_opt_iff_comp_simscape_all.pdf', 'width', 'full', 'height', 'tall'); #+end_src #+name: fig:enc_plates_opt_iff_comp_simscape_all #+caption: FRF from one actuator to all the encoders when the plant is damped using IFF #+RESULTS: [[file:figs/enc_plates_opt_iff_comp_simscape_all.png]] #+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', 'None', 'Padding', 'None'); ax1 = nexttile([2,1]); hold on; % Diagonal Elements FRF plot(f, abs(G_enc_iff_opt(:,1,1)), 'color', [colors(1,:), 0.2], ... 'DisplayName', '$d\mathcal{L}_{m,i}/u_i$ - FRF') for i = 2:6 plot(f, abs(G_enc_iff_opt(:,i,i)), 'color', [colors(1,:), 0.2], ... 'HandleVisibility', 'off'); end % Diagonal Elements Model set(gca,'ColorOrderIndex',2) plot(freqs, abs(squeeze(freqresp(Gd_iff(1,1), freqs, 'Hz'))), '-', ... 'DisplayName', '$d\mathcal{L}_{m,i}/u_i$ - Model') for i = 2:6 set(gca,'ColorOrderIndex',2) plot(freqs, abs(squeeze(freqresp(Gd_iff(i,i), freqs, 'Hz'))), '-', ... 'HandleVisibility', 'off'); end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); ylabel('Amplitude $d_e/V_{exc}$ [m/V]'); set(gca, 'XTickLabel',[]); ylim([1e-8, 1e-4]); legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 3); ax2 = nexttile; hold on; for i =1:6 plot(f, 180/pi*angle(G_enc_iff_opt(:,i,i)), 'color', [colors(1,:), 0.2]); set(gca,'ColorOrderIndex',2) plot(freqs, 180/pi*angle(squeeze(freqresp(Gd_iff(i,i), freqs, 'Hz')))); end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); hold off; yticks(-360:90:360); ylim([-180, 180]); linkaxes([ax1,ax2],'x'); xlim([20, 2e3]); #+end_src #+begin_src matlab :tangle no :exports results :results file replace exportFig('figs/damped_iff_plates_plant_comp_diagonal.pdf', 'width', 'wide', 'height', 'tall'); #+end_src #+name: fig:damped_iff_plates_plant_comp_diagonal #+caption: Comparison of the diagonal elements of the transfer functions from $\bm{u}$ to $d\bm{\mathcal{L}}_m$ with active damping (IFF) applied with an optimal gain $g = 400$ #+RESULTS: [[file:figs/damped_iff_plates_plant_comp_diagonal.png]] #+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', 'None', 'Padding', 'None'); ax1 = nexttile([2,1]); hold on; % Off diagonal FRF plot(f, abs(G_enc_iff_opt(:,1,2)), 'color', [colors(1,:), 0.2], ... 'DisplayName', '$d\mathcal{L}_{m,i}/u_j$ - FRF') for i = 1:5 for j = i+1:6 plot(f, abs(G_enc_iff_opt(:,i,j)), 'color', [colors(1,:), 0.2], ... 'HandleVisibility', 'off'); end end % Off diagonal Model set(gca,'ColorOrderIndex',2) plot(freqs, abs(squeeze(freqresp(Gd_iff(1,2), freqs, 'Hz'))), '-', ... 'DisplayName', '$d\mathcal{L}_{m,i}/u_j$ - Model') for i = 1:5 for j = i+1:6 set(gca,'ColorOrderIndex',2) plot(freqs, abs(squeeze(freqresp(Gd_iff(i,j), freqs, 'Hz'))), ... 'HandleVisibility', 'off'); end end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); ylabel('Amplitude $d_e/V_{exc}$ [m/V]'); set(gca, 'XTickLabel',[]); ylim([1e-8, 1e-4]); legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 3); ax2 = nexttile; hold on; % Off diagonal FRF for i = 1:5 for j = i+1:6 plot(f, 180/pi*angle(G_enc_iff_opt(:,i,j)), 'color', [colors(1,:), 0.2]); end end % Off diagonal Model for i = 1:5 for j = i+1:6 set(gca,'ColorOrderIndex',2) plot(freqs, 180/pi*angle(squeeze(freqresp(Gd_iff(i,j), freqs, 'Hz')))); end end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); hold off; yticks(-360:90:360); ylim([-180, 180]); linkaxes([ax1,ax2],'x'); xlim([20, 2e3]); #+end_src #+begin_src matlab :tangle no :exports results :results file replace exportFig('figs/damped_iff_plates_plant_comp_off_diagonal.pdf', 'width', 'wide', 'height', 'tall'); #+end_src #+name: fig:damped_iff_plates_plant_comp_off_diagonal #+caption: Comparison of the off-diagonal elements of the transfer functions from $\bm{u}$ to $d\bm{\mathcal{L}}_m$ with active damping (IFF) applied with an optimal gain $g = 400$ #+RESULTS: [[file:figs/damped_iff_plates_plant_comp_off_diagonal.png]] #+begin_important From Figures [[fig:damped_iff_plates_plant_comp_diagonal]] and [[fig:damped_iff_plates_plant_comp_off_diagonal]], it is clear that the Simscape model very well represents the dynamics of the nano-hexapod. This is true to around 400Hz, then the dynamics depends on the flexible modes of the top plate which are not modelled. #+end_important *** Save Damped Plant The experimentally identified plant is saved for further use. #+begin_src matlab :exports none:tangle no save('matlab/mat/damped_plant_enc_plates.mat', 'f', 'Ts', 'G_enc_iff_opt') #+end_src #+begin_src matlab :eval no save('mat/damped_plant_enc_plates.mat', 'f', 'Ts', 'G_enc_iff_opt') #+end_src ** Conclusion #+begin_important In this section, the dynamics of the nano-hexapod with the encoders fixed to the plates is studied. It has been found that: - The measured dynamics is in agreement with the dynamics of the simscape model, up to the flexible modes of the top plate. See figures [[fig:enc_plates_iff_comp_simscape]] and [[fig:enc_plates_iff_comp_offdiag_simscape]] for the transfer function to the force sensors and Figures [[fig:enc_plates_dvf_comp_simscape]] and [[fig:enc_plates_dvf_comp_offdiag_simscape]]for the transfer functions to the encoders - The Integral Force Feedback strategy is very effective in damping the suspension modes of the nano-hexapod (Figure [[fig:enc_plant_plates_effect_iff]]). - The transfer function from $\bm{u}^\prime$ to $d\bm{\mathcal{L}}_m$ show nice dynamical properties and is a much better candidate for the high-authority-control than when the encoders were fixed to the struts. At least up to the flexible modes of the top plate, the diagonal elements of the transfer function matrix have alternating poles and zeros, and the phase is moving smoothly. Only the flexible modes of the top plates seems to be problematic for control. #+end_important * Decentralized High Authority Control with Integral Force Feedback <> ** Introduction :ignore: In this section is studied the HAC-LAC architecture for the Nano-Hexapod. More precisely: - The LAC control is a decentralized force feedback as studied in Section [[sec: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 [[fig: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 \begin{tikzpicture} % Blocs \node[block={3.0cm}{2.0cm}] (P) {Plant}; \coordinate[] (inputF) at ($(P.south west)!0.5!(P.north west)$); \coordinate[] (outputF) at ($(P.south east)!0.8!(P.north east)$); \coordinate[] (outputL) at ($(P.south east)!0.2!(P.north east)$); \node[block, above=0.4 of P] (Kiff) {$\bm{K}_\text{IFF}$}; \node[addb, left= of inputF] (addF) {}; \node[block, left= of addF] (K) {$\bm{K}_\mathcal{L}$}; \node[addb={+}{}{}{}{-}, left= of K] (subr) {}; \node[block, align=center, left= of subr] (J) {Inverse\\Kinematics}; % Connections and labels \draw[->] (outputF) -- ++(1, 0) node[below left]{$\bm{\tau}_m$}; \draw[->] ($(outputF) + (0.6, 0)$)node[branch]{} |- (Kiff.east); \draw[->] (Kiff.west) -| (addF.north); \draw[->] (addF.east) -- (inputF) node[above left]{$\bm{u}$}; \draw[->] (outputL) -- ++(1, 0) node[above left]{$d\bm{\mathcal{L}_m}$}; \draw[->] ($(outputL) + (0.6, 0)$)node[branch]{} -- ++(0, -1) -| (subr.south); \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:control_architecture_hac_iff_struts #+caption: HAC-LAC: IFF + Control in the frame of the legs #+RESULTS: [[file:figs/control_architecture_hac_iff_struts.png]] - Section [[sec:hac_iff_struts_controller]]: the decentralized high authority controller is tuned using the Simscape model - Section [[sec:hac_iff_struts_first_tests]]: the controller is implemented and tested experimentally - Section [[sec:hac_iff_struts_ref_track]]: some reference tracking tests are performed ** High Authority Controller <> *** Introduction :ignore: In this section, the decentralized high authority controller $\bm{K}_{\mathcal{L}}$ is first tuned using the Simscape model. *** Matlab Init :noexport:ignore: #+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) <> #+end_src #+begin_src matlab :exports none :results silent :noweb yes <> #+end_src #+begin_src matlab :tangle no addpath('./matlab/mat/'); addpath('./matlab/src/'); addpath('./matlab/'); #+end_src #+begin_src matlab :eval no %% Add useful folders to the path addpath('./mat/'); addpath('./src/'); #+end_src #+begin_src matlab :tangle no addpath('matlab/nass-simscape/matlab/nano_hexapod/') addpath('matlab/nass-simscape/STEPS/nano_hexapod/') addpath('matlab/nass-simscape/STEPS/png/') addpath('matlab/nass-simscape/src/') addpath('matlab/nass-simscape/mat/') #+end_src #+begin_src matlab :eval no %% Add other useful folders to the path related to the Simscape model addpath('nass-simscape/matlab/nano_hexapod/') addpath('nass-simscape/STEPS/nano_hexapod/') addpath('nass-simscape/STEPS/png/') addpath('nass-simscape/src/') addpath('nass-simscape/mat/') #+end_src #+begin_src matlab %% Load the FRF of the transfer function from u to dL with IFF load('damped_plant_enc_plates.mat', 'f', 'Ts', 'G_enc_iff_opt') #+end_src #+begin_src matlab %% Open Simulink Model mdl = 'nano_hexapod_simscape'; options = linearizeOptions; options.SampleTime = 0; open(mdl) %% Initialize the Rerference path to zero Rx = zeros(1, 7); %% Colors for the figures colors = colororder; #+end_src *** Simscape Model First initialized the nano-hexapod with a flexible APA model and with the IFF control strategy. #+begin_src matlab %% Initialize the Simscape model n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... 'flex_top_type', '4dof', ... 'motion_sensor_type', 'plates', ... 'actuator_type', 'flexible', ... 'controller_type', 'iff'); #+end_src Then the controller is loaded #+begin_src matlab %% Load the decentralized IFF controller load('Kiff.mat', 'Kiff') #+end_src The inputs and outputs for the transfer function estimation are defined. #+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 And the plant from $\bm{u}^\prime$ to $d\bm{\mathcal{L}}_m$ is identified and the bode plot of its diagonal terms are shown in Figure [[fig:hac_iff_struts_enc_plates_plant_bode]]. #+begin_src matlab %% Identified of the damped TF from u' to dL Gd_iff_opt = exp(-s*Ts)*linearize(mdl, io, 0.0, options); #+end_src #+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', 'None', 'Padding', 'None'); ax1 = nexttile([2,1]); hold on; % Diagonal Elements Model plot(freqs, abs(squeeze(freqresp(Gd_iff_opt(1,1), freqs, 'Hz'))), 'color', colors(1,:), ... 'DisplayName', 'Diagonal'); for i = 2:6 plot(freqs, abs(squeeze(freqresp(Gd_iff_opt(i,i), freqs, 'Hz'))), 'color', colors(1,:), ... 'HandleVisibility', 'off'); end plot(freqs, abs(squeeze(freqresp(Gd_iff_opt(1,2), freqs, 'Hz'))), 'color', [colors(2,:), 0.2], ... 'DisplayName', 'Off-Diag'); for i = 1:5 for j = i+1:6 plot(freqs, abs(squeeze(freqresp(Gd_iff_opt(i,j), freqs, 'Hz'))), 'color', [colors(2,:), 0.2], ... 'HandleVisibility', 'off'); end end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); ylabel('Amplitude $d\mathcal{L}_m/u^\prime$ [m/V]'); set(gca, 'XTickLabel',[]); ylim([1e-8, 1e-4]); legend('location', 'northeast') ax2 = nexttile; hold on; for i =1:6 plot(freqs, 180/pi*angle(squeeze(freqresp(Gd_iff_opt(i,i), freqs, 'Hz'))), '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([20, 2e3]); #+end_src #+begin_src matlab :tangle no :exports results :results file replace exportFig('figs/hac_iff_struts_enc_plates_plant_bode.pdf', 'width', 'wide', 'height', 'tall'); #+end_src #+name: fig:hac_iff_struts_enc_plates_plant_bode #+caption: Transfer functions from $u$ to $d\mathcal{L}_m$ with IFF (diagonal and off-diagonal elements) #+RESULTS: [[file:figs/hac_iff_struts_enc_plates_plant_bode.png]] *** 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, the following controller is developed: #+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/mat/Khac_iff_struts.mat', 'Khac_iff_struts') #+end_src #+begin_src matlab :eval no save('mat/Khac_iff_struts.mat', 'Khac_iff_struts') #+end_src The Loop Gain is computed and shown in Figure [[fig:loop_gain_hac_iff_struts]]. #+begin_src matlab Lhac_iff_struts = Khac_iff_struts*Gd_iff_opt; #+end_src #+begin_src matlab :exports none %% Bode plot for the transfer function from u to dLm freqs = 2*logspace(0, 3, 1000); figure; tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); ax1 = nexttile([2,1]); hold on; % Diagonal Elements Model plot(freqs, abs(squeeze(freqresp(Lhac_iff_struts(i,i), freqs, 'Hz'))), 'color', colors(1,:), ... 'DisplayName', 'Diagonal'); for i = 2:6 plot(freqs, abs(squeeze(freqresp(Lhac_iff_struts(i,i), freqs, 'Hz'))), 'color', colors(1,:), ... 'HandleVisibility', 'off'); end plot(freqs, abs(squeeze(freqresp(Lhac_iff_struts(1,2), freqs, 'Hz'))), 'color', [colors(2,:), 0.2], ... 'DisplayName', 'Off-Diag'); for i = 1:5 for j = i+1:6 plot(freqs, abs(squeeze(freqresp(Lhac_iff_struts(i,j), freqs, 'Hz'))), '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(freqs, 180/pi*angle(squeeze(freqresp(Lhac_iff_struts(i,i), freqs, 'Hz'))), '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:loop_gain_hac_iff_struts #+caption: Diagonal and off-diagonal elements of the Loop gain for "HAC-IFF-Struts" #+RESULTS: [[file:figs/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 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 Loop Gain Now, the loop gain is estimated from the measured FRF. #+begin_src matlab L_frf = zeros(size(G_enc_iff_opt)); for i = 1:size(G_enc_iff_opt, 1) L_frf(i, :, :) = squeeze(G_enc_iff_opt(i,:,:))*freqresp(Khac_iff_struts, f(i), 'Hz'); end #+end_src The bode plot of the loop gain is shown in Figure [[fig:hac_iff_plates_exp_loop_gain_diag]]. #+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', 'None', 'Padding', 'None'); ax1 = nexttile([2,1]); hold on; % Diagonal Elements FRF plot(f, abs(L_frf(:,1,1)), 'color', colors(1,:), ... 'DisplayName', 'Diagonal'); for i = 2:6 plot(f, abs(L_frf(:,i,i)), 'color', colors(1,:), ... 'HandleVisibility', 'off'); end plot(f, abs(L_frf(:,1,2)), 'color', [colors(2,:), 0.2], ... 'DisplayName', 'Off-Diag'); for i = 1:5 for j = i+1:6 plot(f, abs(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(f, 180/pi*angle(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_diag.pdf', 'width', 'wide', 'height', 'tall'); #+end_src #+name: fig:hac_iff_plates_exp_loop_gain_diag #+caption: Diagonal and Off-diagonal elements of the Loop gain (experimental data) #+RESULTS: [[file:figs/hac_iff_plates_exp_loop_gain_diag.png]] ** Reference Tracking - Trajectories <> *** 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 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 [[fig:control_architecture_hac_iff_L]]. #+begin_src latex :file control_architecture_hac_iff_L.pdf \begin{tikzpicture} % Blocs \node[block={3.0cm}{3.0cm}] (P) {Plant}; \coordinate[] (inputF) at ($(P.south west)!0.5!(P.north west)$); \coordinate[] (outputF) at ($(P.south east)!0.8!(P.north east)$); \coordinate[] (outputL) at ($(P.south east)!0.2!(P.north east)$); \node[block, above=0.4 of P] (Kiff) {$\bm{K}_\text{IFF}$}; \node[addb, left= of inputF] (addF) {}; \node[block, left= of addF] (K) {$\bm{K}_\mathcal{L}$}; \node[addb={+}{}{}{}{-}, left= of K] (subr) {}; \node[block, left= of subr] (J) {$J$}; % Connections and labels \draw[->] (outputF) -- ++(1, 0) node[below left]{$\bm{\tau}_m$}; \draw[->] ($(outputF) + (0.6, 0)$)node[branch]{} |- (Kiff.east); \draw[->] (Kiff.west) -| (addF.north); \draw[->] (addF.east) -- (inputF) node[above left]{$\bm{u}$}; \draw[->] (outputL) -- ++(1, 0) node[above left]{$d\bm{\mathcal{L}}$}; \draw[->] ($(outputL) + (0.6, 0)$)node[branch]{} -- ++(0, -1) -| (subr.south); \draw[->] (subr.east) -- (K.west) node[above left]{$\bm{\epsilon}_{d\mathcal{L}_m}$}; \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:control_architecture_hac_iff_L #+caption: HAC-LAC: IFF + Control in the frame of the legs #+RESULTS: [[file:figs/control_architecture_hac_iff_L.png]] In the following sections, several reference trajectories are defined: - Section [[sec:yz_scans]]: simple scans in the Y-Z plane - Section [[sec:tilt_scans]]: scans in tilt are performed - Section [[sec:nass_scans]]: scans with X-Y-Z translations in order to draw the word "NASS" *** Matlab Init :noexport:ignore: #+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) <> #+end_src #+begin_src matlab :exports none :results silent :noweb yes <> #+end_src #+begin_src matlab :tangle no addpath('./matlab/mat/'); addpath('./matlab/src/'); addpath('./matlab/'); #+end_src #+begin_src matlab :eval no addpath('./mat/'); addpath('./src/'); #+end_src #+begin_src matlab :tangle no %% Add all useful folders to the path addpath('matlab/nass-simscape/matlab/nano_hexapod/') addpath('matlab/nass-simscape/STEPS/nano_hexapod/') addpath('matlab/nass-simscape/STEPS/png/') addpath('matlab/nass-simscape/src/') addpath('matlab/nass-simscape/mat/') #+end_src #+begin_src matlab :eval no %% Add all useful folders to the path addpath('nass-simscape/matlab/nano_hexapod/') addpath('nass-simscape/STEPS/nano_hexapod/') addpath('nass-simscape/STEPS/png/') addpath('nass-simscape/src/') addpath('nass-simscape/mat/') #+end_src #+begin_src matlab %% Open Simulink Model mdl = 'nano_hexapod_simscape'; options = linearizeOptions; options.SampleTime = 0; open(mdl) colors = colororder; #+end_src *** Y-Z Scans <> **** Generate the Scan A function =generateYZScanTrajectory= has been developed (accessible [[sec:generateYZScanTrajectory][here]]) in order to easily generate scans in the Y-Z plane. For instance, the following generated trajectory is represented in Figure [[fig: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', 8e-6, ... % Total Z distance [m] 'n', 5, ... % Number of Y scans 'Ts', 1e-3, ... % Sampling Time [s] 'ti', 2, ... % Time to go to initial position [s] 'tw', 0.5, ... % Waiting time between each points [s] 'ty', 2, ... % Time for a scan in Y [s] 'tz', 1); % 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:yz_scan_example_trajectory_yz_plane #+caption: Generated scan in the Y-Z plane #+RESULTS: [[file:figs/yz_scan_example_trajectory_yz_plane.png]] The Y and Z positions as a function of time are shown in Figure [[fig: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:yz_scan_example_trajectory #+caption: Y and Z trajectories as a function of time #+RESULTS: [[file:figs/yz_scan_example_trajectory.png]] **** Reference Signal for the Strut lengths 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 dL_ref = [n_hexapod.geometry.J*Rx_yz(:, 2:7)']'; #+end_src The reference signal for the strut length is shown in Figure [[fig:yz_scan_example_trajectory_struts]]. #+begin_src matlab :exports none 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:yz_scan_example_trajectory_struts #+caption: Trajectories for the 6 individual struts #+RESULTS: [[file:figs/yz_scan_example_trajectory_struts.png]] **** Time domain simulation with 2DoF model Before trying to follow this reference with the nano-hexapod, let's try to do it using the Simscape model. The nano-hexapod is initialized with the APA modelled as 2DoF system (for the simulation to run quickly). #+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 :exports none %% Run the simulation set_param(mdl,'StopTime', num2str(Rx_yz(end,1))) set_param(mdl,'SimulationCommand','start') #+end_src #+begin_src matlab :exports none %% Reset motion at t=0 out.X.Data = out.X.Data - out.X.Data(1,:); #+end_src The reference path as well as the measured motion are compared in Figure [[fig:ref_track_hac_iff_struts_yz_plane]]. #+begin_src matlab :exports none %% Plot both the reference path and the measured motion figure; hold on; set(gca,'ColorOrderIndex',2) plot(1e6*out.X.Data(:,2), 1e6*out.X.Data(:,3), '-', ... 'DisplayName', 'Meas. Motion') plot(1e6*Rx_yz(:,3), 1e6*Rx_yz(:,4), 'k--', ... 'DisplayName', 'Reference Path') hold off; xlabel('X displacement [$\mu m$]'); ylabel('Y displacement [$\mu m$]'); legend('location', 'southwest'); #+end_src #+begin_src matlab :tangle no :exports results :results file replace exportFig('figs/ref_track_hac_iff_struts_yz_plane.pdf', 'width', 'wide', 'height', 'tall'); #+end_src #+name: fig:ref_track_hac_iff_struts_yz_plane #+caption: Simulated Y-Z motion #+RESULTS: [[file:figs/ref_track_hac_iff_struts_yz_plane.png]] The motion errors are computed and shown in Figure [[fig:ref_track_hac_iff_struts_pos_error]]. It is clear that the hexapod is indeed tracking the reference path. However, in this simulation, no disturbances are included nor sensor noises. #+begin_src matlab :exports none %% Motion errors figure; tiledlayout(1, 2, 'TileSpacing', 'None', 'Padding', 'None'); ax1 = nexttile; hold on; plot(out.X.Time, 1e9*(out.X.Data(:,1) - Rx_yz(:,2)), 'DisplayName', '$\epsilon_x$') plot(out.X.Time, 1e9*(out.X.Data(:,2) - Rx_yz(:,3)), 'DisplayName', '$\epsilon_y$') plot(out.X.Time, 1e9*(out.X.Data(:,3) - Rx_yz(:,4)), 'DisplayName', '$\epsilon_z$') hold off; xlabel('Time [s]'); ylabel('Position Errors [nm]'); legend('location', 'northeast'); ax2 = nexttile; hold on; plot(out.X.Time, 1e6*(out.X.Data(:,4) - Rx_yz(:,5)), 'DisplayName', '$\epsilon_{R_x}$') plot(out.X.Time, 1e6*(out.X.Data(:,5) - Rx_yz(:,6)), 'DisplayName', '$\epsilon_{R_y}$') plot(out.X.Time, 1e6*(out.X.Data(:,6) - Rx_yz(:,7)), 'DisplayName', '$\epsilon_{R_z}$') hold off; xlabel('Time [s]'); ylabel('Orientation Errors [$\mu rad$]'); legend('location', 'northeast'); #+end_src #+begin_src matlab :tangle no :exports results :results file replace exportFig('figs/ref_track_hac_iff_struts_pos_error.pdf', 'width', 'full', 'height', 'tall'); #+end_src #+name: fig:ref_track_hac_iff_struts_pos_error #+caption: Positioning errors as a function of time #+RESULTS: [[file:figs/ref_track_hac_iff_struts_pos_error.png]] *** TODO Tilt Scans <> *** "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 [[fig: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:ref_track_test_nass #+caption: Reference path corresponding to the "NASS" acronym #+RESULTS: [[file:figs/ref_track_test_nass.png]] It can also be better viewed in a 3D representation as in Figure [[fig: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:ref_track_test_nass_3d #+caption: Reference path that draws "NASS" - 3D view #+RESULTS: [[file:figs/ref_track_test_nass_3d.png]] ** First Experimental Tests with the HAC <> *** Introduction :ignore: Both the Integral Force Feedback controller (developed in Section [[sec:enc_plates_iff]]) and the high authority controller working in the frame of the struts (developed in Section [[sec:hac_iff_struts_controller]]) are implemented experimentally. *** Matlab Init :noexport:ignore: #+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) <> #+end_src #+begin_src matlab :exports none :results silent :noweb yes <> #+end_src #+begin_src matlab :tangle no addpath('./matlab/mat/'); addpath('./matlab/src/'); addpath('./matlab/'); #+end_src #+begin_src matlab :eval no %% Add useful folders to the path addpath('./mat/'); addpath('./src/'); #+end_src #+begin_src matlab :tangle no addpath('matlab/nass-simscape/matlab/nano_hexapod/') addpath('matlab/nass-simscape/STEPS/nano_hexapod/') addpath('matlab/nass-simscape/STEPS/png/') addpath('matlab/nass-simscape/src/') addpath('matlab/nass-simscape/mat/') #+end_src #+begin_src matlab :eval no %% Add other useful folders to the path related to the Simscape model addpath('nass-simscape/matlab/nano_hexapod/') addpath('nass-simscape/STEPS/nano_hexapod/') addpath('nass-simscape/STEPS/png/') addpath('nass-simscape/src/') addpath('nass-simscape/mat/') #+end_src #+begin_src matlab %% Open Simulink Model mdl = 'nano_hexapod_simscape'; options = linearizeOptions; options.SampleTime = 0; open(mdl) %% Initialize the Rerference path to zero Rx = zeros(1, 7); %% Colors for the figures colors = colororder; #+end_src *** Initial Controller The controller designed in Section [[sec:hac_iff_struts_controller]] is implemented experimentally and some reference tracking tests are performed. #+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 The reference path as well as the measured position are partially shown in the Y-Z plane in Figure [[fig: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', 'None', '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: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/yz_scans_exp_results_first_K.png]] #+begin_important It is clear from Figure [[fig: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 load('damped_plant_enc_plates.mat', 'f', 'Ts', 'G_enc_iff_opt') #+end_src #+begin_src matlab :exports none %% Compute the Loop Gain L_frf = zeros(size(G_enc_iff_opt)); for i = 1:size(G_enc_iff_opt, 1) L_frf(i, :, :) = squeeze(G_enc_iff_opt(i,:,:))*freqresp(Khac_iff_struts, f(i), 'Hz'); end #+end_src The bode plot of the new loop gain is shown in Figure [[fig: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', 'None', 'Padding', 'None'); ax1 = nexttile([2,1]); hold on; % Diagonal Elements FRF plot(f, abs(L_frf(:,1,1)), 'color', colors(1,:), ... 'DisplayName', 'Diagonal'); for i = 2:6 plot(f, abs(L_frf(:,i,i)), 'color', colors(1,:), ... 'HandleVisibility', 'off'); end plot(f, abs(L_frf(:,1,2)), 'color', [colors(2,:), 0.2], ... 'DisplayName', 'Off-Diag'); for i = 1:5 for j = i+1:6 plot(f, abs(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(f, 180/pi*angle(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:hac_iff_plates_exp_loop_gain_redesigned_K #+caption: Loop Gain for the updated decentralized HAC controller #+RESULTS: [[file:figs/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 [[fig:nass_scans_first_test_exp]]. #+begin_src matlab :exports none %% Plot the X-Y-Z "NASS" trajectory figure; hold on; plot3(Xe(:,1), Xe(:,2), Xe(:,3)) plot3(Rx_nass(:,2), Rx_nass(:,3), Rx_nass(:,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: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/nass_scans_first_test_exp.png]] The trajectory and measured motion are also shown in the X-Y plane in Figure [[fig: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', 'None', '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:ref_track_nass_exp_hac_iff_struts #+caption: Reference path and measured motion in the X-Y plane #+RESULTS: [[file:figs/ref_track_nass_exp_hac_iff_struts.png]] The orientations errors as a function of time are shown in Figure [[fig:ref_track_nass_exp_hac_iff_struts_errors_angle]]. #+begin_src matlab :exports none %% Orientation Errors figure; hold on; plot(t(t>10), 1e6*Xe(t>10,4), '-', 'DisplayName', '$\epsilon_{\theta_x}$'); plot(t(t>10), 1e6*Xe(t>10,5), '-', 'DisplayName', '$\epsilon_{\theta_y}$'); plot(t(t>10), 1e6*Xe(t>10,6), '-', 'DisplayName', '$\epsilon_{\theta_z}$'); hold off; xlabel('Time [s]'); ylabel('Orientation Error [$\mu$ rad]'); legend('location', 'northeast'); #+end_src #+begin_src matlab :tangle no :exports results :results file replace exportFig('figs/ref_track_nass_exp_hac_iff_struts_errors_angle.pdf', 'width', 'wide', 'height', 'normal'); #+end_src #+name: fig:ref_track_nass_exp_hac_iff_struts_errors_angle #+caption: Orientation errors as a function of time during the "NASS" trajectory #+RESULTS: [[file:figs/ref_track_nass_exp_hac_iff_struts_errors_angle.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 *** Noise Budgeting :noexport: The updated controller is implemented, and the encoder and force sensors signals are recorded while zero motion is imposed. #+begin_src matlab :exports none %% Load load('hac_iff_more_lead_huddle.mat', 't', 'Va', 'Vs', 'de') #+end_src #+begin_src matlab :exports none %% Remove first seconds Va = Va(t>15, :); de = de(t>15, :); Vs = Vs(t>15, :); t = t( t>15, :); %% Reset time vector t = t - t(1); #+end_src #+begin_src matlab % Sampling Time [s] Ts = (t(end) - (t(1)))/(length(t)-1); % Sampling Frequency [Hz] Fs = 1/Ts; % Hannning Windows win = hanning(ceil(1*Fs)); % And we get the frequency vector [~, f] = pwelch(de(:,1), win, 0, [], Fs); #+end_src #+begin_src matlab [pxx, f] = pwelch(de, win, 0, [], Fs); #+end_src #+begin_src matlab :exports none %% Amplitude spectral density of the residual motion figure; hold on; for i = 1:6 plot(f, sqrt(pxx(:, i)), '-'); end hold off; xlabel('Frequency [Hz]'); ylabel('Amplitude Spectral Density [$m/\sqrt{Hz}$]'); set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log'); xlim([1, Fs/2]); #+end_src #+begin_src matlab figure; plot(t, de) #+end_src #+begin_src matlab figure; plot(t, Vs) #+end_src #+begin_src matlab rms(de) #+end_src #+begin_src matlab Xe = [inv(n_hexapod.geometry.J)*de']'; #+end_src * Centralized High Authority Control with Integral Force Feedback :noexport: ** HAC Control - Cartesian Frame #+begin_src matlab load('damped_plant_enc_plates.mat', 'f', 'Ts', 'G_enc_iff_opt') load('jacobian.mat', 'J'); #+end_src From the transfer function from $\bm{\tau}$ to $d\bm{\mathcal{L}}_m$, let's compute the transfer function from $\bm{\mathcal{F}}$ to $d\bm{\mathcal{X}}_m$. #+begin_src matlab G_dvf_J = permute(pagemtimes(inv(J), pagemtimes(permute(G_enc_iff_opt, [2 3 1]), inv(J'))), [3 1 2]); #+end_src #+begin_src matlab :exports none labels = {'$D_x/F_{x}$', '$D_y/F_{y}$', '$D_z/F_{z}$', '$R_{x}/M_{x}$', '$R_{y}/M_{y}$', '$R_{R}/M_{z}$'}; figure; tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); ax1 = nexttile([2,1]); hold on; for i = 1:5 for j = i+1:6 plot(f, abs(G_dvf_J(:, i, j)), 'color', [0, 0, 0, 0.2], ... 'HandleVisibility', 'off'); end end for i =1:6 set(gca,'ColorOrderIndex',i) plot(f, abs(G_dvf_J(:,i , i)), ... 'DisplayName', labels{i}); end plot(f, abs(G_dvf_J(:, 1, 2)), 'color', [0, 0, 0, 0.2], ... 'DisplayName', '$D_i/F_j$'); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); ylabel('Amplitude $d_e/V_a$ [m/V]'); set(gca, 'XTickLabel',[]); ylim([1e-7, 1e-1]); legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 3); ax2 = nexttile; hold on; for i =1:6 set(gca,'ColorOrderIndex',i) plot(f, 180/pi*angle(G_dvf_J(:,i , i))); end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); hold off; yticks(-360:90:360); linkaxes([ax1,ax2],'x'); xlim([20, 2e3]); #+end_src * Feedforward Control :noexport: <> ** Introduction :ignore: #+begin_src latex :file control_architecture_iff_feedforward.pdf \begin{tikzpicture} % Blocs \node[block={3.0cm}{3.0cm}] (P) {Plant}; \coordinate[] (inputF) at ($(P.south west)!0.5!(P.north west)$); \coordinate[] (outputF) at ($(P.south east)!0.8!(P.north east)$); \coordinate[] (outputX) at ($(P.south east)!0.5!(P.north east)$); \coordinate[] (outputL) at ($(P.south east)!0.2!(P.north east)$); \node[block, above=0.4 of P] (Kiff) {$\bm{K}_\text{IFF}$}; \node[addb, left= of inputF] (addF) {}; \node[block, left= of addF] (Kff) {$\bm{K}_{\mathcal{L},\text{ff}}$}; \node[block, align=center, left= of Kff] (J) {Inverse\\Kinematics}; % Connections and labels \draw[->] (outputF) -- ++(1, 0) node[below left]{$\bm{\tau}_m$}; \draw[->] ($(outputF) + (0.6, 0)$)node[branch]{} |- (Kiff.east); \draw[->] (Kiff.west) -| (addF.north); \draw[->] (addF.east) -- (inputF) node[above left]{$\bm{u}$}; \draw[->] (outputL) -- ++(1, 0) node[above left]{$d\bm{\mathcal{L}}$}; \draw[->] (outputX) -- ++(1, 0) node[above left]{$\bm{\mathcal{X}}$}; \draw[->] (Kff.east) -- (addF.west) node[above left]{$\bm{u}_{\text{ff}}$}; \draw[->] (J.east) -- (Kff.west) node[above left]{$\bm{r}_{d\mathcal{L}}$}; \draw[<-] (J.west)node[above left]{$\bm{r}_{\mathcal{X}}$} -- ++(-1, 0); \end{tikzpicture} #+end_src #+name: fig:control_architecture_iff_feedforward #+caption: Feedforward control in the frame of the legs #+RESULTS: [[file:figs/control_architecture_iff_feedforward.png]] Main problems: - Non-linearity: Creep, Hysteresis - Variability of the plant ** Matlab Init :noexport:ignore: #+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) <> #+end_src #+begin_src matlab :exports none :results silent :noweb yes <> #+end_src #+begin_src matlab :tangle no addpath('./matlab/mat/'); addpath('./matlab/src/'); addpath('./matlab/'); #+end_src #+begin_src matlab :eval no addpath('./mat/'); addpath('./src/'); #+end_src #+begin_src matlab load('damped_plant_enc_plates.mat', 'f', 'Ts', 'G_enc_iff_opt') #+end_src #+begin_src matlab :tangle no %% Add all useful folders to the path addpath('matlab/nass-simscape/matlab/nano_hexapod/') addpath('matlab/nass-simscape/STEPS/nano_hexapod/') addpath('matlab/nass-simscape/STEPS/png/') addpath('matlab/nass-simscape/src/') addpath('matlab/nass-simscape/mat/') #+end_src #+begin_src matlab :eval no %% Add all useful folders to the path addpath('nass-simscape/matlab/nano_hexapod/') addpath('nass-simscape/STEPS/nano_hexapod/') addpath('nass-simscape/STEPS/png/') addpath('nass-simscape/src/') addpath('nass-simscape/mat/') #+end_src #+begin_src matlab %% Open Simulink Model mdl = 'nano_hexapod_simscape'; options = linearizeOptions; options.SampleTime = 0; open(mdl) Rx = zeros(1, 7); colors = colororder; #+end_src ** Simple Feedforward Controller Let's estimate the mean DC gain for the damped plant (diagonal elements:) #+begin_src matlab :results value replace :exports results :tangle no mean(diag(abs(squeeze(mean(G_enc_iff_opt(f>2 & f<4,:,:)))))) #+end_src #+RESULTS: : 1.773e-05 The feedforward controller is then taken as the inverse of this gain (the minus sign is there manually added as it is "removed" by the =abs= function): #+begin_src matlab Kff_iff_L = -1/mean(diag(abs(squeeze(mean(G_enc_iff_opt(f>2 & f<4,:,:)))))); #+end_src The open-loop gain (feedforward controller times the damped plant) is shown in Figure [[fig:open_loop_gain_feedforward_iff_struts]]. #+begin_src matlab :exports none %% Bode plot of the transfer function from u to dLm for tested values of the IFF gain figure; tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); ax1 = nexttile([2,1]); hold on; for i = 1:6 set(gca,'ColorOrderIndex',1); plot(f, abs(Kff_iff_L*G_enc_iff_opt(:,i,i)), 'k-'); end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); ylabel('Amplitude [-]'); set(gca, 'XTickLabel',[]); ylim([1e-2, 1e1]); ax2 = nexttile; hold on; for i = 1:6 set(gca,'ColorOrderIndex',1); plot(f, 180/pi*angle(Kff_iff_L*G_enc_iff_opt(:,i,i)), 'k-') end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); ylim([-180, 180]); yticks([-180, -90, 0, 90, 180]); linkaxes([ax1,ax2],'x'); xlim([1, 2e3]); #+end_src #+begin_src matlab :tangle no :exports results :results file replace exportFig('figs/open_loop_gain_feedforward_iff_struts.pdf', 'width', 'wide', 'height', 'tall'); #+end_src #+name: fig:open_loop_gain_feedforward_iff_struts #+caption: Diagonal elements of the "open loop gain" #+RESULTS: [[file:figs/open_loop_gain_feedforward_iff_struts.png]] And save the feedforward controller for further use: #+begin_src matlab Kff_iff_L = zpk(Kff_iff_L)*eye(6); #+end_src #+begin_src matlab :tangle no save('matlab/mat/feedforward_iff.mat', 'Kff_iff_L') #+end_src #+begin_src matlab :exports none :eval no save('mat/feedforward_iff.mat', 'Kff_iff_L') #+end_src ** Test with Simscape Model #+begin_src matlab load('reference_path.mat', 'Rx_yz'); #+end_src * Further work :noexport: ** Feedback/Feedforward control in the frame of the struts *** Introduction :ignore: #+begin_src latex :file control_architecture_hac_iff_L_feedforward.pdf \begin{tikzpicture} % Blocs \node[block={3.0cm}{3.0cm}] (P) {Plant}; \coordinate[] (inputF) at ($(P.south west)!0.5!(P.north west)$); \coordinate[] (outputF) at ($(P.south east)!0.8!(P.north east)$); \coordinate[] (outputX) at ($(P.south east)!0.5!(P.north east)$); \coordinate[] (outputL) at ($(P.south east)!0.2!(P.north east)$); \node[block, above=0.4 of P] (Kiff) {$\bm{K}_\text{IFF}$}; \node[addb, left= of inputF] (addF) {}; \node[block, left= of addF] (K) {$\bm{K}_\mathcal{L}$}; \node[block, above= of K] (Kff) {$\bm{K}_{\mathcal{L},\text{ff}}$}; \node[addb, left= of K] (subr) {}; \node[block, align=center, left= of subr] (J) {Inverse\\Kinematics}; % Connections and labels \draw[->] (outputF) -- ++(1, 0) node[below left]{$\bm{\tau}_m$}; \draw[->] ($(outputF) + (0.6, 0)$)node[branch]{} |- (Kiff.east); \draw[->] (Kiff.west) -| (addF.north); \draw[->] (addF.east) -- (inputF) node[above left]{$\bm{u}$}; \draw[->] (outputL) -- ++(1, 0) node[above left]{$d\bm{\mathcal{L}}$}; \draw[->] ($(outputL) + (0.6, 0)$)node[branch]{} -- ++(0, -1) -| (subr.south); \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[->] (outputX) -- ++(1, 0) node[above left]{$\bm{\mathcal{X}}_n$}; \draw[->] (J.east) -- (subr.west); \draw[->] ($(J.east) + (0.4, 0)$)node[branch]{} node[below]{$\bm{r}_{d\mathcal{L}}$} |- (Kff.west); \draw[->] (Kff.east) -- ++(0.5, 0) -- (addF.north west); \draw[<-] (J.west)node[above left]{$\bm{r}_{\mathcal{X}_n}$} -- ++(-1, 0); \end{tikzpicture} #+end_src #+name: fig:control_architecture_hac_iff_L_feedforward #+caption: Feedback/Feedforward control in the frame of the legs #+RESULTS: [[file:figs/control_architecture_hac_iff_L_feedforward.png]] * Functions ** =generateXYZTrajectory= :PROPERTIES: :header-args:matlab+: :tangle matlab/src/generateXYZTrajectory.m :header-args:matlab+: :comments none :mkdirp yes :eval no :END: <> *** Function description :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab function [ref] = generateXYZTrajectory(args) % generateXYZTrajectory - % % Syntax: [ref] = generateXYZTrajectory(args) % % Inputs: % - args % % Outputs: % - ref - Reference Signal #+end_src *** Optional Parameters :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab arguments args.points double {mustBeNumeric} = zeros(2, 3) % [m] args.ti (1,1) double {mustBeNumeric, mustBePositive} = 1 % Time to go to first point and after last point [s] args.tw (1,1) double {mustBeNumeric, mustBePositive} = 0.5 % Time wait between each point [s] args.tm (1,1) double {mustBeNumeric, mustBePositive} = 1 % Motion time between points [s] args.Ts (1,1) double {mustBeNumeric, mustBePositive} = 1e-3 % Sampling Time [s] end #+end_src *** Initialize Time Vectors :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab time_i = 0:args.Ts:args.ti; time_w = 0:args.Ts:args.tw; time_m = 0:args.Ts:args.tm; #+end_src *** XYZ Trajectory :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab % Go to initial position xyz = (args.points(1,:))'*(time_i/args.ti); % Wait xyz = [xyz, xyz(:,end).*ones(size(time_w))]; % Scans for i = 2:size(args.points, 1) % Go to next point xyz = [xyz, xyz(:,end) + (args.points(i,:)' - xyz(:,end))*(time_m/args.tm)]; % Wait a litle bit xyz = [xyz, xyz(:,end).*ones(size(time_w))]; end % End motion xyz = [xyz, xyz(:,end) - xyz(:,end)*(time_i/args.ti)]; #+end_src *** Reference Signal :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab t = 0:args.Ts:args.Ts*(length(xyz) - 1); #+end_src #+begin_src matlab ref = zeros(length(xyz), 7); ref(:, 1) = t; ref(:, 2:4) = xyz'; #+end_src ** =generateYZScanTrajectory= :PROPERTIES: :header-args:matlab+: :tangle matlab/src/generateYZScanTrajectory.m :header-args:matlab+: :comments none :mkdirp yes :eval no :END: <> *** Function description :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab function [ref] = generateYZScanTrajectory(args) % generateYZScanTrajectory - % % Syntax: [ref] = generateYZScanTrajectory(args) % % Inputs: % - args % % Outputs: % - ref - Reference Signal #+end_src *** Optional Parameters :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab arguments args.y_tot (1,1) double {mustBeNumeric} = 10e-6 % [m] args.z_tot (1,1) double {mustBeNumeric} = 10e-6 % [m] args.n (1,1) double {mustBeInteger, mustBePositive} = 10 % [-] args.Ts (1,1) double {mustBeNumeric, mustBePositive} = 1e-4 % [s] args.ti (1,1) double {mustBeNumeric, mustBePositive} = 1 % [s] args.tw (1,1) double {mustBeNumeric, mustBePositive} = 1 % [s] args.ty (1,1) double {mustBeNumeric, mustBePositive} = 1 % [s] args.tz (1,1) double {mustBeNumeric, mustBePositive} = 1 % [s] end #+end_src *** Initialize Time Vectors :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab time_i = 0:args.Ts:args.ti; time_w = 0:args.Ts:args.tw; time_y = 0:args.Ts:args.ty; time_z = 0:args.Ts:args.tz; #+end_src *** Y and Z vectors :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab % Go to initial position y = (time_i/args.ti)*(args.y_tot/2); % Wait y = [y, y(end)*ones(size(time_w))]; % Scans for i = 1:args.n if mod(i,2) == 0 y = [y, -(args.y_tot/2) + (time_y/args.ty)*args.y_tot]; else y = [y, (args.y_tot/2) - (time_y/args.ty)*args.y_tot]; end if i < args.n y = [y, y(end)*ones(size(time_z))]; end end % Wait a litle bit y = [y, y(end)*ones(size(time_w))]; % End motion y = [y, y(end) - y(end)*time_i/args.ti]; #+end_src #+begin_src matlab % Go to initial position z = (time_i/args.ti)*(args.z_tot/2); % Wait z = [z, z(end)*ones(size(time_w))]; % Scans for i = 1:args.n z = [z, z(end)*ones(size(time_y))]; if i < args.n z = [z, z(end) - (time_z/args.tz)*args.z_tot/(args.n-1)]; end end % Wait a litle bit z = [z, z(end)*ones(size(time_w))]; % End motion z = [z, z(end) - z(end)*time_i/args.ti]; #+end_src *** Reference Signal :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab t = 0:args.Ts:args.Ts*(length(y) - 1); #+end_src #+begin_src matlab ref = zeros(length(y), 7); ref(:, 1) = t; ref(:, 3) = y; ref(:, 4) = z; #+end_src ** =getTransformationMatrixAcc= :PROPERTIES: :header-args:matlab+: :tangle matlab/src/getTransformationMatrixAcc.m :header-args:matlab+: :comments none :mkdirp yes :eval no :END: <> *** Function description :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab function [M] = getTransformationMatrixAcc(Opm, Osm) % getTransformationMatrixAcc - % % Syntax: [M] = getTransformationMatrixAcc(Opm, Osm) % % Inputs: % - Opm - Nx3 (N = number of accelerometer measurements) X,Y,Z position of accelerometers % - Opm - Nx3 (N = number of accelerometer measurements) Unit vectors representing the accelerometer orientation % % Outputs: % - M - Transformation Matrix #+end_src *** Transformation matrix from motion of the solid body to accelerometer measurements :PROPERTIES: :UNNUMBERED: t :END: Let's try to estimate the x-y-z acceleration of any point of the solid body from the acceleration/angular acceleration of the solid body expressed in $\{O\}$. For any point $p_i$ of the solid body (corresponding to an accelerometer), we can write: \begin{equation} \begin{bmatrix} a_{i,x} \\ a_{i,y} \\ a_{i,z} \end{bmatrix} = \begin{bmatrix} \dot{v}_x \\ \dot{v}_y \\ \dot{v}_z \end{bmatrix} + p_i \times \begin{bmatrix} \dot{\omega}_x \\ \dot{\omega}_y \\ \dot{\omega}_z \end{bmatrix} \end{equation} We can write the cross product as a matrix product using the skew-symmetric transformation: \begin{equation} \begin{bmatrix} a_{i,x} \\ a_{i,y} \\ a_{i,z} \end{bmatrix} = \begin{bmatrix} \dot{v}_x \\ \dot{v}_y \\ \dot{v}_z \end{bmatrix} + \underbrace{\begin{bmatrix} 0 & p_{i,z} & -p_{i,y} \\ -p_{i,z} & 0 & p_{i,x} \\ p_{i,y} & -p_{i,x} & 0 \end{bmatrix}}_{P_{i,[\times]}} \cdot \begin{bmatrix} \dot{\omega}_x \\ \dot{\omega}_y \\ \dot{\omega}_z \end{bmatrix} \end{equation} If we now want to know the (scalar) acceleration $a_i$ of the point $p_i$ in the direction of the accelerometer direction $\hat{s}_i$, we can just project the 3d acceleration on $\hat{s}_i$: \begin{equation} a_i = \hat{s}_i^T \cdot \begin{bmatrix} a_{i,x} \\ a_{i,y} \\ a_{i,z} \end{bmatrix} = \hat{s}_i^T \cdot \begin{bmatrix} \dot{v}_x \\ \dot{v}_y \\ \dot{v}_z \end{bmatrix} + \left( \hat{s}_i^T \cdot P_{i,[\times]} \right) \cdot \begin{bmatrix} \dot{\omega}_x \\ \dot{\omega}_y \\ \dot{\omega}_z \end{bmatrix} \end{equation} Which is equivalent as a simple vector multiplication: \begin{equation} a_i = \begin{bmatrix} \hat{s}_i^T & \hat{s}_i^T \cdot P_{i,[\times]} \end{bmatrix} \begin{bmatrix} \dot{v}_x \\ \dot{v}_y \\ \dot{v}_z \\ \dot{\omega}_x \\ \dot{\omega}_y \\ \dot{\omega}_z \end{bmatrix} = \begin{bmatrix} \hat{s}_i^T & \hat{s}_i^T \cdot P_{i,[\times]} \end{bmatrix} {}^O\vec{x} \end{equation} And finally we can combine the 6 (line) vectors for the 6 accelerometers to write that in a matrix form. We obtain Eq. eqref:eq:M_matrix. #+begin_important The transformation from solid body acceleration ${}^O\vec{x}$ from sensor measured acceleration $\vec{a}$ is: \begin{equation} \label{eq:M_matrix} \vec{a} = \underbrace{\begin{bmatrix} \hat{s}_1^T & \hat{s}_1^T \cdot P_{1,[\times]} \\ \vdots & \vdots \\ \hat{s}_6^T & \hat{s}_6^T \cdot P_{6,[\times]} \end{bmatrix}}_{M} {}^O\vec{x} \end{equation} with $\hat{s}_i$ the unit vector representing the measured direction of the i'th accelerometer expressed in frame $\{O\}$ and $P_{i,[\times]}$ the skew-symmetric matrix representing the cross product of the position of the i'th accelerometer expressed in frame $\{O\}$. #+end_important Let's define such matrix using matlab: #+begin_src matlab M = zeros(length(Opm), 6); for i = 1:length(Opm) Ri = [0, Opm(3,i), -Opm(2,i); -Opm(3,i), 0, Opm(1,i); Opm(2,i), -Opm(1,i), 0]; M(i, 1:3) = Osm(:,i)'; M(i, 4:6) = Osm(:,i)'*Ri; end #+end_src #+begin_src matlab end #+end_src ** =getJacobianNanoHexapod= :PROPERTIES: :header-args:matlab+: :tangle matlab/src/getJacobianNanoHexapod.m :header-args:matlab+: :comments none :mkdirp yes :eval no :END: <> *** Function description :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab function [J] = getJacobianNanoHexapod(Hbm) % getJacobianNanoHexapod - % % Syntax: [J] = getJacobianNanoHexapod(Hbm) % % Inputs: % - Hbm - Height of {B} w.r.t. {M} [m] % % Outputs: % - J - Jacobian Matrix #+end_src *** Transformation matrix from motion of the solid body to accelerometer measurements :PROPERTIES: :UNNUMBERED: t :END: #+begin_src matlab Fa = [[-86.05, -74.78, 22.49], [ 86.05, -74.78, 22.49], [ 107.79, -37.13, 22.49], [ 21.74, 111.91, 22.49], [-21.74, 111.91, 22.49], [-107.79, -37.13, 22.49]]'*1e-3; % Ai w.r.t. {F} [m] Mb = [[-28.47, -106.25, -22.50], [ 28.47, -106.25, -22.50], [ 106.25, 28.47, -22.50], [ 77.78, 77.78, -22.50], [-77.78, 77.78, -22.50], [-106.25, 28.47, -22.50]]'*1e-3; % Bi w.r.t. {M} [m] H = 95e-3; % Stewart platform height [m] Fb = Mb + [0; 0; H]; % Bi w.r.t. {F} [m] si = Fb - Fa; si = si./vecnorm(si); % Normalize Bb = Mb - [0; 0; Hbm]; J = [si', cross(Bb, si)']; #+end_src