#+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: In this document, the dynamics of the nano-hexapod shown in Figure [[fig:picture_bench_granite_nano_hexapod]] is identified. #+begin_note Here are the documentation of the equipment used for this test bench: - Voltage Amplifier: PiezoDrive [[file:doc/PD200-V7-R1.pdf][PD200]] - 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_nano_hexapod #+caption: Nano-Hexapod #+attr_latex: :width \linewidth [[file:figs/IMG_20210608_152917.jpg]] #+name: fig:picture_bench_granite_overview #+caption: Nano-Hexapod and the control electronics #+attr_latex: :width \linewidth [[file:figs/IMG_20210608_154722.jpg]] #+begin_src latex :file nano_hexapod_signals.pdf \definecolor{instrumentation}{rgb}{0, 0.447, 0.741} \definecolor{mechanics}{rgb}{0.8500, 0.325, 0.098} \begin{tikzpicture} % Blocs \node[block={4.0cm}{3.0cm}, fill=mechanics!20!white] (nano_hexapod) {Mechanics}; \coordinate[] (inputF) at (nano_hexapod.west); \coordinate[] (outputL) at ($(nano_hexapod.south east)!0.8!(nano_hexapod.north east)$); \coordinate[] (outputF) at ($(nano_hexapod.south east)!0.2!(nano_hexapod.north east)$); \node[block, left= 0.8 of inputF, fill=instrumentation!20!white, align=center] (F_stack) {\tiny Actuator \\ \tiny stacks}; \node[block, left= 0.8 of F_stack, fill=instrumentation!20!white] (PD200) {PD200}; \node[DAC, left= 0.8 of PD200, fill=instrumentation!20!white] (F_DAC) {DAC}; \node[block, right=0.8 of outputF, fill=instrumentation!20!white, align=center] (Fm_stack){\tiny Sensor \\ \tiny stack}; \node[ADC, right=0.8 of Fm_stack,fill=instrumentation!20!white] (Fm_ADC) {ADC}; \node[block, right=0.8 of outputL, fill=instrumentation!20!white] (encoder) {\tiny Encoder}; % Connections and labels \draw[->] ($(F_DAC.west)+(-0.8,0)$) node[above right]{$\bm{u}$} node[below right]{$[V]$} -- node[sloped]{$/$} (F_DAC.west); \draw[->] (F_DAC.east) -- node[midway, above]{$\tilde{\bm{u}}$}node[midway, below]{$[V]$} (PD200.west); \draw[->] (PD200.east) -- node[midway, above]{$\bm{u}_a$}node[midway, below]{$[V]$} (F_stack.west); \draw[->] (F_stack.east) -- (inputF) node[above left]{$\bm{\tau}$}node[below left]{$[N]$}; \draw[->] (outputF) -- (Fm_stack.west) node[above left]{$\bm{\epsilon}$} node[below left]{$[m]$}; \draw[->] (Fm_stack.east) -- node[midway, above]{$\tilde{\bm{\tau}}_m$}node[midway, below]{$[V]$} (Fm_ADC.west); \draw[->] (Fm_ADC.east) -- node[sloped]{$/$} ++(0.8, 0)coordinate(end) node[above left]{$\bm{\tau}_m$}node[below left]{$[V]$}; \draw[->] (outputL) -- (encoder.west) node[above left]{$d\bm{\mathcal{L}}$} node[below left]{$[m]$}; \draw[->] (encoder.east) -- node[sloped]{$/$} (encoder-|end) node[above left]{$d\bm{\mathcal{L}}_m$}node[below left]{$[m]$}; % Nano-Hexapod \begin{scope}[on background layer] \node[fit={(F_stack.west|-nano_hexapod.south) (Fm_stack.east|-nano_hexapod.north)}, fill=black!20!white, draw, inner sep=2pt] (system) {}; \node[above] at (system.north) {Nano-Hexapod}; \end{scope} \end{tikzpicture} #+end_src #+name: fig:nano_hexapod_signals #+caption: Block diagram of the system with named signals #+attr_latex: :scale 1 #+RESULTS: [[file:figs/nano_hexapod_signals.png]] #+name: tab:list_signals #+caption: List of signals #+attr_latex: :environment tabularx :width \linewidth :align Xllll #+attr_latex: :center t :booktabs t :float t | | *Unit* | *Matlab* | *Vector* | *Elements* | |------------------------------------+-----------+-----------+-----------------------+----------------------| | Control Input (wanted DAC voltage) | =[V]= | =u= | $\bm{u}$ | $u_i$ | | DAC Output Voltage | =[V]= | =u= | $\tilde{\bm{u}}$ | $\tilde{u}_i$ | | PD200 Output Voltage | =[V]= | =ua= | $\bm{u}_a$ | $u_{a,i}$ | | Actuator applied force | =[N]= | =tau= | $\bm{\tau}$ | $\tau_i$ | |------------------------------------+-----------+-----------+-----------------------+----------------------| | Strut motion | =[m]= | =dL= | $d\bm{\mathcal{L}}$ | $d\mathcal{L}_i$ | | Encoder measured displacement | =[m]= | =dLm= | $d\bm{\mathcal{L}}_m$ | $d\mathcal{L}_{m,i}$ | |------------------------------------+-----------+-----------+-----------------------+----------------------| | Force Sensor strain | =[m]= | =epsilon= | $\bm{\epsilon}$ | $\epsilon_i$ | | Force Sensor Generated Voltage | =[V]= | =taum= | $\tilde{\bm{\tau}}_m$ | $\tilde{\tau}_{m,i}$ | | Measured Generated Voltage | =[V]= | =taum= | $\bm{\tau}_m$ | $\tau_{m,i}$ | |------------------------------------+-----------+-----------+-----------------------+----------------------| | Motion of the top platform | =[m,rad]= | =dX= | $d\bm{\mathcal{X}}$ | $d\mathcal{X}_i$ | | Metrology measured displacement | =[m,rad]= | =dXm= | $d\bm{\mathcal{X}}_m$ | $d\mathcal{X}_{m,i}$ | * Encoders fixed to the Struts ** Introduction In this section, the encoders are fixed to the struts. ** 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 ** Identification of the dynamics *** Load 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 *** DVF Plant First, let's compute the coherence from the excitation voltage and the displacement as measured by the encoders (Figure [[fig:enc_struts_dvf_coh]]). #+begin_src matlab %% Coherence coh_dvf_lf = zeros(length(f), 6, 6); coh_dvf_hf = zeros(length(f), 6, 6); for i = 1:6 coh_dvf_lf(:, :, i) = mscohere(meas_data_lf{i}.Va, meas_data_lf{i}.de, win, [], [], 1/Ts); coh_dvf_hf(:, :, i) = mscohere(meas_data_hf{i}.Va, meas_data_hf{i}.de, win, [], [], 1/Ts); 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(i_lf), coh_dvf_lf(i_lf, i, j), 'color', [0, 0, 0, 0.2], ... 'HandleVisibility', 'off'); plot(f(i_hf), coh_dvf_hf(i_hf, i, j), 'color', [0, 0, 0, 0.2], ... 'HandleVisibility', 'off'); end end for i =1:6 set(gca,'ColorOrderIndex',i) plot(f(i_lf), coh_dvf_lf(i_lf,i, i), ... 'DisplayName', sprintf('$G_{dvf}(%i,%i)$', i, i)); set(gca,'ColorOrderIndex',i) plot(f(i_hf), coh_dvf_hf(i_hf,i, i), ... 'HandleVisibility', 'off'); end plot(f(i_lf), coh_dvf_lf(i_lf, 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_lf = zeros(length(f), 6, 6); G_dvf_hf = zeros(length(f), 6, 6); for i = 1:6 G_dvf_lf(:, :, i) = tfestimate(meas_data_lf{i}.Va, meas_data_lf{i}.de, win, [], [], 1/Ts); G_dvf_hf(:, :, i) = tfestimate(meas_data_hf{i}.Va, meas_data_hf{i}.de, win, [], [], 1/Ts); 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(i_lf), abs(G_dvf_lf(i_lf, i, j)), 'color', [0, 0, 0, 0.2], ... 'HandleVisibility', 'off'); plot(f(i_hf), abs(G_dvf_hf(i_hf, i, j)), 'color', [0, 0, 0, 0.2], ... 'HandleVisibility', 'off'); end end for i =1:6 set(gca,'ColorOrderIndex',i) plot(f(i_lf), abs(G_dvf_lf(i_lf,i, i)), ... 'DisplayName', sprintf('$G_{dvf}(%i,%i)$', i, i)); set(gca,'ColorOrderIndex',i) plot(f(i_hf), abs(G_dvf_hf(i_hf,i, i)), ... 'HandleVisibility', 'off'); end plot(f(i_lf), abs(G_dvf_lf(i_lf, 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(i_lf), 180/pi*angle(G_dvf_lf(i_lf,i, i))); set(gca,'ColorOrderIndex',i) plot(f(i_hf), 180/pi*angle(G_dvf_hf(i_hf,i, i))); end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); 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]] *** IFF Plant First, let's compute the coherence from the excitation voltage and the displacement as measured by the encoders (Figure [[fig:enc_struts_iff_coh]]). #+begin_src matlab %% Coherence for the IFF plant coh_iff_lf = zeros(length(f), 6, 6); coh_iff_hf = zeros(length(f), 6, 6); for i = 1:6 coh_iff_lf(:, :, i) = mscohere(meas_data_lf{i}.Va, meas_data_lf{i}.Vs, win, [], [], 1/Ts); coh_iff_hf(:, :, i) = mscohere(meas_data_hf{i}.Va, meas_data_hf{i}.Vs, win, [], [], 1/Ts); 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(i_lf), coh_iff_lf(i_lf, i, j), 'color', [0, 0, 0, 0.2], ... 'HandleVisibility', 'off'); plot(f(i_hf), coh_iff_hf(i_hf, i, j), 'color', [0, 0, 0, 0.2], ... 'HandleVisibility', 'off'); end end for i =1:6 set(gca,'ColorOrderIndex',i) plot(f(i_lf), coh_iff_lf(i_lf,i, i), ... 'DisplayName', sprintf('$G_{iff}(%i,%i)$', i, i)); set(gca,'ColorOrderIndex',i) plot(f(i_hf), coh_iff_hf(i_hf,i, i), ... 'HandleVisibility', 'off'); end plot(f(i_lf), coh_iff_lf(i_lf, 1, 2), 'color', [0, 0, 0, 0.2], ... 'DisplayName', '$G_{iff}(i,j)$'); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', '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_lf = zeros(length(f), 6, 6); G_iff_hf = zeros(length(f), 6, 6); for i = 1:6 G_iff_lf(:, :, i) = tfestimate(meas_data_lf{i}.Va, meas_data_lf{i}.Vs, win, [], [], 1/Ts); G_iff_hf(:, :, i) = tfestimate(meas_data_hf{i}.Va, meas_data_hf{i}.Vs, win, [], [], 1/Ts); 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(i_lf), abs(G_iff_lf(i_lf, i, j)), 'color', [0, 0, 0, 0.2], ... 'HandleVisibility', 'off'); plot(f(i_hf), abs(G_iff_hf(i_hf, i, j)), 'color', [0, 0, 0, 0.2], ... 'HandleVisibility', 'off'); end end for i =1:6 set(gca,'ColorOrderIndex',i) plot(f(i_lf), abs(G_iff_lf(i_lf,i, i)), ... 'DisplayName', sprintf('$G_{iff}(%i,%i)$', i, i)); set(gca,'ColorOrderIndex',i) plot(f(i_hf), abs(G_iff_hf(i_hf,i, i)), ... 'HandleVisibility', 'off'); end plot(f(i_lf), abs(G_iff_lf(i_lf, 1, 2)), 'color', [0, 0, 0, 0.2], ... 'DisplayName', '$G_{iff}(i,j)$'); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); ylabel('Amplitude $V_s/V_a$ [V/V]'); set(gca, 'XTickLabel',[]); legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 3); ylim([1e-3, 1e2]); ax2 = nexttile; hold on; for i =1:6 set(gca,'ColorOrderIndex',i) plot(f(i_lf), 180/pi*angle(G_iff_lf(i_lf,i, i))); set(gca,'ColorOrderIndex',i) plot(f(i_hf), 180/pi*angle(G_iff_hf(i_hf,i, i))); end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); hold off; yticks(-360:90:360); linkaxes([ax1,ax2],'x'); xlim([20, 2e3]); #+end_src #+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]] ** 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 *** 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_lf = permute(pagemtimes(inv(J), pagemtimes(permute(G_dvf_lf, [2 3 1]), inv(J'))), [3 1 2]); G_dvf_J_hf = permute(pagemtimes(inv(J), pagemtimes(permute(G_dvf_hf, [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(i_lf), abs(G_dvf_J_lf(i_lf, i, j)), 'color', [0, 0, 0, 0.2], ... 'HandleVisibility', 'off'); plot(f(i_hf), abs(G_dvf_J_hf(i_hf, i, j)), 'color', [0, 0, 0, 0.2], ... 'HandleVisibility', 'off'); end end for i =1:6 set(gca,'ColorOrderIndex',i) plot(f(i_lf), abs(G_dvf_J_lf(i_lf,i, i)), ... 'DisplayName', labels{i}); set(gca,'ColorOrderIndex',i) plot(f(i_hf), abs(G_dvf_J_hf(i_hf,i, i)), ... 'HandleVisibility', 'off'); end plot(f(i_lf), abs(G_dvf_J_lf(i_lf, 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(i_lf), 180/pi*angle(G_dvf_J_lf(i_lf,i, i))); set(gca,'ColorOrderIndex',i) plot(f(i_hf), 180/pi*angle(G_dvf_J_hf(i_hf,i, i))); end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); hold off; yticks(-360:90:360); linkaxes([ax1,ax2],'x'); xlim([20, 2e3]); #+end_src #+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_lf = permute(pagemtimes(inv(J), pagemtimes(permute(G_iff_lf, [2 3 1]), inv(J'))), [3 1 2]); G_iff_J_hf = permute(pagemtimes(inv(J), pagemtimes(permute(G_iff_hf, [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(i_lf), abs(G_iff_J_lf(i_lf, i, j)), 'color', [0, 0, 0, 0.2], ... 'HandleVisibility', 'off'); plot(f(i_hf), abs(G_iff_J_hf(i_hf, i, j)), 'color', [0, 0, 0, 0.2], ... 'HandleVisibility', 'off'); end end for i =1:6 set(gca,'ColorOrderIndex',i) plot(f(i_lf), abs(G_iff_J_lf(i_lf,i, i)), ... 'DisplayName', labels{i}); set(gca,'ColorOrderIndex',i) plot(f(i_hf), abs(G_iff_J_hf(i_hf,i, i)), ... 'HandleVisibility', 'off'); end plot(f(i_lf), abs(G_iff_J_lf(i_lf, 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(i_lf), 180/pi*angle(G_iff_J_lf(i_lf,i, i))); set(gca,'ColorOrderIndex',i) plot(f(i_hf), 180/pi*angle(G_iff_J_hf(i_hf,i, i))); end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); hold off; yticks(-360:90:360); linkaxes([ax1,ax2],'x'); xlim([20, 2e3]); #+end_src #+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. *** Initialize :noexport: #+begin_src matlab :tangle no %% Add all useful folders to the path addpath('matlab/') addpath('matlab/nass-simscape/matlab/nano_hexapod/') addpath('matlab/nass-simscape/STEPS/nano_hexapod/') addpath('matlab/nass-simscape/STEPS/png/') addpath('matlab/nass-simscape/src/') addpath('matlab/nass-simscape/mat/') #+end_src #+begin_src matlab :eval no %% 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) #+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, '/F'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs io(io_i) = linio([mdl, '/Fm'], 1, 'openoutput'); io_i = io_i + 1; % Force Sensors Giff = 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(i_lf), abs(G_iff_lf(i_lf,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(i_lf), abs(G_iff_lf(i_lf,i, i)), 'color', [0,0,0,0.2], ... 'HandleVisibility', 'off'); set(gca,'ColorOrderIndex',2) plot(f(i_hf), abs(G_iff_hf(i_hf,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(i_lf), 180/pi*angle(G_iff_lf(i_lf,i, i)), 'color', [0,0,0,0.2]); plot(f(i_hf), 180/pi*angle(G_iff_hf(i_hf,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(i_lf), abs(G_iff_lf(i_lf, 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(i_lf), abs(G_iff_lf(i_lf, i, j)), 'color', [0,0,0,0.2], ... 'HandleVisibility', 'off'); plot(f(i_hf), abs(G_iff_hf(i_hf, i, j)), 'color', [0,0,0,0.2], ... 'HandleVisibility', 'off'); end end 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', '2dof'); #+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, '/F'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs io(io_i) = linio([mdl, '/D'], 1, 'openoutput'); io_i = io_i + 1; % Encoders Gdvf = 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(i_lf), abs(G_dvf_lf(i_lf,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(i_lf), abs(G_dvf_lf(i_lf,i, i)), 'color', [0,0,0,0.2], ... 'HandleVisibility', 'off'); set(gca,'ColorOrderIndex',2) plot(f(i_hf), abs(G_dvf_hf(i_hf,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(i_lf), 180/pi*angle(G_dvf_lf(i_lf,i, i)), 'color', [0,0,0,0.2]); plot(f(i_hf), 180/pi*angle(G_dvf_hf(i_hf,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(i_lf), abs(G_dvf_lf(i_lf, 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(i_lf), abs(G_dvf_lf(i_lf, i, j)), 'color', [0,0,0,0.2], ... 'HandleVisibility', 'off'); plot(f(i_hf), abs(G_dvf_hf(i_hf, 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]] ** Integral Force Feedback *** Root Locus and Decentralized Loop gain #+begin_src matlab %% IFF Controller Kiff_g1 = (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 #+end_src #+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 = g*Kiff_g1; #+end_src #+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(i_lf), abs(squeeze(freqresp(Kiff(1,1), f(i_lf), 'Hz')).*G_iff_lf(i_lf,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(i_lf), abs(squeeze(freqresp(Kiff(1,1), f(i_lf), 'Hz')).*G_iff_lf(i_lf,i, i)), 'color', [0,0,0,0.2], ... 'HandleVisibility', 'off'); set(gca,'ColorOrderIndex',2) plot(f(i_hf), abs(squeeze(freqresp(Kiff(1,1), f(i_hf), 'Hz')).*G_iff_hf(i_hf,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(i_lf), 180/pi*angle(squeeze(freqresp(Kiff(1,1), f(i_lf), 'Hz')).*G_iff_lf(i_lf,i, i)), 'color', [0,0,0,0.2]); plot(f(i_hf), 180/pi*angle(squeeze(freqresp(Kiff(1,1), f(i_hf), 'Hz')).*G_iff_hf(i_hf,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]] *** Multiple Gains - Simulation #+begin_src matlab %% Tested IFF gains iff_gains = [4, 10, 20, 40, 100, 200, 400]; #+end_src #+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 #+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, '/F'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs io(io_i) = linio([mdl, '/D'], 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 #+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]] *** Experimental Results - Gains **** Introduction :ignore: Let's look at the damping introduced by IFF as a function of the IFF gain and compare that with the results obtained using the Simscape model. **** Load 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 #+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 #+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 #+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('$g_{iff} = %.0f$', 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]] #+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). This is very logical. Also, the experimental results and the models obtained from the Simscape model are in agreement. #+end_important **** Experimental Results - Comparison of the un-damped and fully damped system #+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(i_lf), abs(G_dvf_lf(i_lf,1, 1)), ... 'DisplayName', 'Un-Damped') set(gca,'ColorOrderIndex',1) plot(f(i_hf), abs(G_dvf_hf(i_hf,1, 1)), ... 'HandleVisibility', 'off'); for i = 2:6 set(gca,'ColorOrderIndex',1) plot(f(i_lf), abs(G_dvf_lf(i_lf,i, i)), ... 'HandleVisibility', 'off'); set(gca,'ColorOrderIndex',1) plot(f(i_hf), abs(G_dvf_hf(i_hf,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(i_lf), 180/pi*angle(G_dvf_lf(i_lf,i, i))); set(gca,'ColorOrderIndex',1) plot(f(i_hf), 180/pi*angle(G_dvf_hf(i_hf,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 #+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 #+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 #+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 #+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]] #+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 dapmed. Remains the undamped flexible modes of the APA, and the modes of the plates. The Simscape model and the experimental results are in very good agreement. #+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]] *** Effectiveness of the IFF Strategy - Compliance 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-hexapode. It also has the effect of degrading (slightly) the vertical compliance at low frequency. #+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, '/Fz_ext'], 1, 'openinput'); io_i = io_i + 1; % External - Vertical force io(io_i) = linio([mdl, '/Z_top_plat'], 1, 'openoutput'); io_i = io_i + 1; % Absolute vertical motion of top platform %% 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 quire 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]] *** Obtained Mode Shapes Then, several excitation are performed using the instrumented Hammer and the mode shapes are extracted. 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_annotated]]). #+name: fig:mode_shapes_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 | * Encoders fixed to the plates ** Introduction :ignore: