#+TITLE: DCM - Dynamical Multi-Body Model :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} #+LATEX_HEADER_EXTRA: \bibliography{ref} #+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 \clearpage * System Kinematics :PROPERTIES: :header-args:matlab+: :tangle matlab/dcm_kinematics.m :END: ** Introduction :ignore: ** Matlab Init :noexport:ignore: #+begin_src matlab %% dcm_kinematics.m % Computation of the DCM kinematics #+end_src #+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) <> #+end_src #+begin_src matlab :exports none :results silent :noweb yes <> #+end_src #+begin_src matlab :tangle no :noweb yes <> #+end_src #+begin_src matlab :eval no :noweb yes <> #+end_src #+begin_src matlab :noweb yes <> #+end_src ** Bragg Angle #+begin_src matlab %% Tested bragg angles bragg = linspace(5, 80, 1000); % Bragg angle [deg] d_off = 10.5e-3; % Wanted offset between x-rays [m] #+end_src #+begin_src matlab %% Vertical Jack motion as a function of Bragg angle dz = d_off./(2*cos(bragg*pi/180)); #+end_src #+begin_src matlab :exports none %% Jack motion as a function of Bragg angle figure; plot(bragg, 1e3*dz) xlabel('Bragg angle [deg]'); ylabel('Jack Motion [mm]'); #+end_src #+begin_src matlab :tangle no :exports results :results file replace exportFig('figs/jack_motion_bragg_angle.pdf', 'width', 'wide', 'height', 'normal'); #+end_src #+name: fig:jack_motion_bragg_angle #+caption: Jack motion as a function of Bragg angle #+RESULTS: [[file:figs/jack_motion_bragg_angle.png]] #+begin_src matlab :results value replace :exports both %% Required Jack stroke ans = 1e3*(dz(end) - dz(1)) #+end_src #+RESULTS: : 24.963 ** Kinematics (111 Crystal) *** Introduction :ignore: The reference frame is taken at the center of the 111 second crystal. *** Interferometers - 111 Crystal Three interferometers are pointed to the bottom surface of the 111 crystal. The position of the measurement points are shown in Figure [[fig:sensor_111_crystal_points]] as well as the origin where the motion of the crystal is computed. #+begin_src latex :file sensor_111_crystal_points.pdf \begin{tikzpicture} % Crystal \draw (-15/2, -3.5/2) rectangle (15/2, 3.5/2); % Measurement Points \node[branch] (a1) at (-7, 1.5){}; \node[branch] (a2) at ( 0, -1.5){}; \node[branch] (a3) at ( 7, 1.5){}; % Labels \node[right] at (a1) {$\mathcal{O}_1 = (-0.07, -0.015)$}; \node[right] at (a2) {$\mathcal{O}_2 = (0, 0.015)$}; \node[left] at (a3) {$\mathcal{O}_3 = ( 0.07, -0.015)$}; % Origin \draw[->] (0, 0) node[] -- ++(1, 0) node[right]{$x$}; \draw[->] (0, 0) -- ++(0, -1) node[below]{$y$}; \draw[fill, color=black] (0, 0) circle (0.05); \node[left] at (0,0) {$\mathcal{O}_{111}$}; \end{tikzpicture} #+end_src #+name: fig:sensor_111_crystal_points #+caption: Bottom view of the second crystal 111. Position of the measurement points. #+RESULTS: [[file:figs/sensor_111_crystal_points.png]] The inverse kinematics consisting of deriving the interferometer measurements from the motion of the crystal (see Figure [[fig:schematic_sensor_jacobian_inverse_kinematics]]): \begin{equation} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \bm{J}_{s,111} \begin{bmatrix} d_z \\ r_y \\ r_x \end{bmatrix} \end{equation} #+begin_src latex :file schematic_sensor_jacobian_inverse_kinematics.pdf \begin{tikzpicture} % Blocs \node[block] (Js) {$\bm{J}_{s,111}$}; % Connections and labels \draw[->] ($(Js.west)+(-1.5,0)$) node[above right]{$\begin{bmatrix} d_z \\ r_y \\ r_x \end{bmatrix}$} -- (Js.west); \draw[->] (Js.east) -- ++(1.5, 0) node[above left]{$\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}$}; \end{tikzpicture} #+end_src #+name: fig:schematic_sensor_jacobian_inverse_kinematics #+caption: Inverse Kinematics - Interferometers #+RESULTS: [[file:figs/schematic_sensor_jacobian_inverse_kinematics.png]] From the Figure [[fig:sensor_111_crystal_points]], the inverse kinematics can be solved as follow (for small motion): \begin{equation} \bm{J}_{s,111} = \begin{bmatrix} 1 & 0.07 & -0.015 \\ 1 & 0 & 0.015 \\ 1 & -0.07 & -0.015 \end{bmatrix} \end{equation} #+begin_src matlab %% Sensor Jacobian matrix for 111 crystal J_s_111 = [1, 0.07, -0.015 1, 0, 0.015 1, -0.07, -0.015]; #+end_src #+begin_src matlab :exports results :results value table replace :tangle no data2orgtable(J_s_111, {}, {}, ' %.3f '); #+end_src #+name: tab:jacobian_sensor_111 #+caption: Sensor Jacobian $\bm{J}_{s,111}$ #+attr_latex: :environment tabularx :width 0.3\linewidth :align ccc #+attr_latex: :center t :booktabs t #+RESULTS: | 1.0 | 0.07 | -0.015 | | 1.0 | 0.0 | 0.015 | | 1.0 | -0.07 | -0.015 | The forward kinematics is solved by inverting the Jacobian matrix (see Figure [[fig:schematic_sensor_jacobian_forward_kinematics]]). \begin{equation} \begin{bmatrix} d_z \\ r_y \\ r_x \end{bmatrix} = \bm{J}_{s,111}^{-1} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} \end{equation} #+begin_src latex :file schematic_sensor_jacobian_forward_kinematics.pdf \begin{tikzpicture} % Blocs \node[block] (Js_inv) {$\bm{J}_{s,111}^{-1}$}; % Connections and labels \draw[->] ($(Js_inv.west)+(-1.5,0)$) node[above right]{$\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}$} -- (Js_inv.west); \draw[->] (Js_inv.east) -- ++(1.5, 0) node[above left]{$\begin{bmatrix} d_z \\ r_y \\ r_x \end{bmatrix}$}; \end{tikzpicture} #+end_src #+name: fig:schematic_sensor_jacobian_forward_kinematics #+caption: Forward Kinematics - Interferometers #+RESULTS: [[file:figs/schematic_sensor_jacobian_forward_kinematics.png]] #+begin_src matlab :exports results :results value table replace :tangle no data2orgtable(inv(J_s_111), {}, {}, ' %.2f '); #+end_src #+name: tab:inverse_jacobian_sensor_111 #+caption: Inverse of the sensor Jacobian $\bm{J}_{s,111}^{-1}$ #+attr_latex: :environment tabularx :width 0.3\linewidth :align ccc #+attr_latex: :center t :booktabs t #+RESULTS: | 0.25 | 0.5 | 0.25 | | 7.14 | 0.0 | -7.14 | | -16.67 | 33.33 | -16.67 | *** Piezo - 111 Crystal The location of the actuators with respect with the center of the 111 second crystal are shown in Figure [[fig:actuator_jacobian_111_points]]. #+name: fig:actuator_jacobian_111_points #+caption: Location of actuators with respect to the center of the 111 second crystal (bottom view) #+attr_latex: :width \linewidth [[file:figs/actuator_jacobian_111_points.png]] Inverse Kinematics consist of deriving the axial (z) motion of the 3 actuators from the motion of the crystal's center. \begin{equation} \begin{bmatrix} d_{u_r} \\ d_{u_h} \\ d_{d} \end{bmatrix} = \bm{J}_{a,111} \begin{bmatrix} d_z \\ r_y \\ r_x \end{bmatrix} \end{equation} #+begin_src latex :file schematic_actuator_jacobian_inverse_kinematics.pdf \begin{tikzpicture} % Blocs \node[block] (Ja) {$\bm{J}_{a,111}$}; % Connections and labels \draw[->] ($(Ja.west)+(-1.5,0)$) node[above right]{$\begin{bmatrix} d_z \\ r_y \\ r_x \end{bmatrix}$} -- (Ja.west); \draw[->] (Ja.east) -- ++(1.5, 0) node[above left]{$\begin{bmatrix} d_{u_r} \\ d_{u_h} \\ d_d \end{bmatrix}$}; \end{tikzpicture} #+end_src #+name: fig:schematic_sensor_jacobian_inverse_kinematics #+caption: Inverse Kinematics - Actuators #+RESULTS: [[file:figs/schematic_actuator_jacobian_inverse_kinematics.png]] Based on the geometry in Figure [[fig:actuator_jacobian_111_points]], we obtain: \begin{equation} \bm{J}_{a,111} = \begin{bmatrix} 1 & 0.14 & -0.1525 \\ 1 & 0.14 & 0.0675 \\ 1 & -0.14 & -0.0425 \end{bmatrix} \end{equation} #+begin_src matlab %% Actuator Jacobian - 111 crystal J_a_111 = [1, 0.14, -0.1525 1, 0.14, 0.0675 1, -0.14, -0.0425]; #+end_src #+begin_src matlab :exports results :results value table replace :tangle no data2orgtable(J_a_111, {}, {}, ' %.4f '); #+end_src #+name: tab:jacobian_actuator_111 #+caption: Actuator Jacobian $\bm{J}_{a,111}$ #+attr_latex: :environment tabularx :width 0.3\linewidth :align ccc #+attr_latex: :center t :booktabs t #+RESULTS: | 1.0 | 0.14 | -0.1525 | | 1.0 | 0.14 | 0.0675 | | 1.0 | -0.14 | -0.0425 | The forward Kinematics is solved by inverting the Jacobian matrix: \begin{equation} \begin{bmatrix} d_z \\ r_y \\ r_x \end{bmatrix} = \bm{J}_{a,111}^{-1} \begin{bmatrix} d_{u_r} \\ d_{u_h} \\ d_{d} \end{bmatrix} \end{equation} #+begin_src latex :file schematic_actuator_jacobian_forward_kinematics.pdf \begin{tikzpicture} % Blocs \node[block] (Ja_inv) {$\bm{J}_{a,111}^{-1}$}; % Connections and labels \draw[->] ($(Ja_inv.west)+(-1.5,0)$) node[above right]{$\begin{bmatrix} d_{u_r} \\ d_{u_h} \\ d_d \end{bmatrix}$} -- (Ja_inv.west); \draw[->] (Ja_inv.east) -- ++(1.5, 0) node[above left]{$\begin{bmatrix} d_z \\ r_y \\ r_x \end{bmatrix}$}; \end{tikzpicture} #+end_src #+name: fig:schematic_actuator_jacobian_forward_kinematics #+caption: Forward Kinematics - Actuators for 111 crystal #+RESULTS: [[file:figs/schematic_actuator_jacobian_forward_kinematics.png]] #+begin_src matlab :exports results :results value table replace :tangle no data2orgtable(inv(J_a_111), {}, {}, ' %.4f '); #+end_src #+name: tab:inverse_jacobian_actuator_111 #+caption: Inverse of the actuator Jacobian $\bm{J}_{a,111}^{-1}$ #+attr_latex: :environment tabularx :width 0.3\linewidth :align ccc #+attr_latex: :center t :booktabs t #+RESULTS: | 0.0568 | 0.4432 | 0.5 | | 1.7857 | 1.7857 | -3.5714 | | -4.5455 | 4.5455 | 0.0 | ** Save Kinematics #+begin_src matlab :exports none :tangle no save('matlab/mat/dcm_kinematics.mat', 'J_a_111', 'J_s_111') #+end_src #+begin_src matlab :eval no save('mat/dcm_kinematics.mat', 'J_a_111', 'J_s_111') #+end_src * System Identification :PROPERTIES: :header-args:matlab+: :tangle matlab/dcm_identification.m :END: ** Introduction :ignore: ** Matlab Init :noexport:ignore: #+begin_src matlab %% dcm_identification.m % Extraction of system dynamics using Simscape model #+end_src #+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) <> #+end_src #+begin_src matlab :exports none :results silent :noweb yes <> #+end_src #+begin_src matlab :tangle no :noweb yes <> #+end_src #+begin_src matlab :eval no :noweb yes <> #+end_src #+begin_src matlab :noweb yes <> #+end_src #+begin_src matlab :noweb yes <> #+end_src ** Identification Let's considered the system $\bm{G}(s)$ with: - 3 inputs: force applied to the 3 fast jacks - 3 outputs: measured displacement by the 3 interferometers pointing at the 111 second crystal It is schematically shown in Figure [[fig:schematic_system_inputs_outputs]]. #+begin_src latex :file schematic_system_inputs_outputs.pdf \begin{tikzpicture} % Blocs \node[block] (G) {$\bm{G}(s)$}; % Connections and labels \draw[->] ($(G.west)+(-1.5,0)$) node[above right]{$\begin{bmatrix} u_{u_r} \\ u_{u_h} \\ u_d \end{bmatrix}$} -- (G.west); \draw[->] (G.east) -- ++(1.5, 0) node[above left]{$\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}$}; \end{tikzpicture} #+end_src #+name: fig:schematic_system_inputs_outputs #+caption: Dynamical system with inputs and outputs #+RESULTS: [[file:figs/schematic_system_inputs_outputs.png]] The system is identified from the Simscape model. #+begin_src matlab %% Input/Output definition clear io; io_i = 1; %% Inputs % Control Input {3x1} [N] io(io_i) = linio([mdl, '/control_system'], 1, 'openinput'); io_i = io_i + 1; %% Outputs % Interferometers {3x1} [m] io(io_i) = linio([mdl, '/DCM'], 1, 'openoutput'); io_i = io_i + 1; #+end_src #+begin_src matlab %% Extraction of the dynamics G = linearize(mdl, io); #+end_src #+begin_src matlab :exports none %% Input and Output names G.InputName = {'u_ur', 'u_uh', 'u_d'}; G.OutputName = {'int_111_1', 'int_111_2', 'int_111_3'}; #+end_src #+begin_src matlab :results output replace :exports both :tangle no size(G) #+end_src #+RESULTS: : size(G) : State-space model with 3 outputs, 3 inputs, and 24 states. ** Plant in the frame of the fastjacks #+begin_src matlab load('mat/dcm_kinematics.mat'); #+end_src Using the forward and inverse kinematics, we can computed the dynamics from piezo forces to axial motion of the 3 fastjacks (see Figure [[fig:schematic_jacobian_frame_fastjack]]). #+begin_src latex :file schematic_jacobian_frame_fastjack.pdf \begin{tikzpicture} % Blocs \node[block] (G) {$\bm{G}(s)$}; \node[block, right=1.5 of G] (Js) {$\bm{J}_{s}^{-1}$}; \node[block, right=1.5 of Js] (Ja) {$\bm{J}_{a}$}; % Connections and labels \draw[->] ($(G.west)+(-1.5,0)$) node[above right]{$\begin{bmatrix} u_{u_r} \\ u_{u_h} \\ u_d \end{bmatrix}$} -- (G.west); \draw[->] (G.east) -- node[midway, above]{$\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}$} (Js.west); \draw[->] (Js.east) -- node[midway, above]{$\begin{bmatrix} d_z \\ r_y \\ r_x \end{bmatrix}$} (Ja.west); \draw[->] (Ja.east) -- ++(1.5, 0) node[above left]{$\begin{bmatrix} d_{u_r} \\ d_{u_h} \\ d_{d} \end{bmatrix}$}; \begin{scope}[on background layer] \node[fit={(G.south west) ($(Ja.east)+(0, 1.4)$)}, fill=black!20!white, draw, inner sep=6pt] (system) {}; \node[above] at (system.north) {$\bm{G}_{\text{fj}}(s)$}; \end{scope} \end{tikzpicture} #+end_src #+name: fig:schematic_jacobian_frame_fastjack #+caption: Use of Jacobian matrices to obtain the system in the frame of the fastjacks #+RESULTS: [[file:figs/schematic_jacobian_frame_fastjack.png]] #+begin_src matlab %% Compute the system in the frame of the fastjacks G_pz = J_a_111*inv(J_s_111)*G; #+end_src The DC gain of the new system shows that the system is well decoupled at low frequency. #+begin_src matlab :results value replace :exports both :tangle no dcgain(G_pz) #+end_src #+name: tab:dc_gain_plan_fj #+caption: DC gain of the plant in the frame of the fast jacks $\bm{G}_{\text{fj}}$ #+attr_latex: :environment tabularx :width 0.5\linewidth :align ccc #+attr_latex: :center t :booktabs t #+RESULTS: | 4.4407e-09 | 2.7656e-12 | 1.0132e-12 | | 2.7656e-12 | 4.4407e-09 | 1.0132e-12 | | 1.0109e-12 | 1.0109e-12 | 4.4424e-09 | The bode plot of $\bm{G}_{\text{fj}}(s)$ is shown in Figure [[fig:bode_plot_plant_fj]]. #+begin_src matlab :exports none %% Bode plot for the plant figure; tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); ax1 = nexttile([2,1]); hold on; plot(freqs, abs(squeeze(freqresp(G_pz(1,1), freqs, 'Hz'))), ... 'DisplayName', 'd'); plot(freqs, abs(squeeze(freqresp(G_pz(2,2), freqs, 'Hz'))), ... 'DisplayName', 'uh'); plot(freqs, abs(squeeze(freqresp(G_pz(3,3), freqs, 'Hz'))), ... 'DisplayName', 'ur'); for i = 1:2 for j = i+1:3 plot(freqs, abs(squeeze(freqresp(G_pz(i,j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ... 'HandleVisibility', 'off'); end end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]); legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 3); ylim([1e-13, 1e-6]); ax2 = nexttile; hold on; plot(freqs, 180/pi*angle(squeeze(freqresp(G_pz(1,1), freqs, 'Hz')))); plot(freqs, 180/pi*angle(squeeze(freqresp(G_pz(2,2), freqs, 'Hz')))); plot(freqs, 180/pi*angle(squeeze(freqresp(G_pz(3,3), freqs, 'Hz')))); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); hold off; yticks(-360:90:360); ylim([-180, 180]); linkaxes([ax1,ax2],'x'); xlim([freqs(1), freqs(end)]); #+end_src #+begin_src matlab :tangle no :exports results :results file replace exportFig('figs/bode_plot_plant_fj.pdf', 'width', 'wide', 'height', 'tall'); #+end_src #+name: fig:bode_plot_plant_fj #+caption: Bode plot of the diagonal and off-diagonal elements of the plant in the frame of the fast jacks #+RESULTS: [[file:figs/bode_plot_plant_fj.png]] #+begin_important Computing the system in the frame of the fastjack gives good decoupling at low frequency (until the first resonance of the system). #+end_important ** Plant in the frame of the crystal #+begin_src latex :file schematic_jacobian_frame_crystal.pdf \begin{tikzpicture} % Blocs \node[block] (G) {$\bm{G}(s)$}; \node[block, left=1.5 of G] (Ja) {$\bm{J}_{a}^{-T}$}; \node[block, right=1.5 of G] (Js) {$\bm{J}_{s}^{-1}$}; % Connections and labels \draw[->] ($(Ja.west)+(-1.5,0)$) node[above right]{$\begin{bmatrix} \mathcal{F}_{z} \\ \mathcal{M}_{y} \\ \mathcal{M}_{x} \end{bmatrix}$} -- (Ja.west); \draw[->] (Ja.east) -- node[midway, above]{$\begin{bmatrix} u_{u_r} \\ u_{u_h} \\ u_d \end{bmatrix}$} (G.west); \draw[->] (G.east) -- node[midway, above]{$\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}$} (Js.west); \draw[->] (Js.east) -- ++(1.5, 0) node[above left]{$\begin{bmatrix} d_z \\ r_y \\ r_x \end{bmatrix}$}; \begin{scope}[on background layer] \node[fit={(Ja.south west) ($(Js.east)+(0, 1.4)$)}, fill=black!20!white, draw, inner sep=6pt] (system) {}; \node[above] at (system.north) {$\bm{G}_{\text{cr}}(s)$}; \end{scope} \end{tikzpicture} #+end_src #+name: fig:schematic_jacobian_frame_crystal #+caption: Use of Jacobian matrices to obtain the system in the frame of the crystal #+RESULTS: [[file:figs/schematic_jacobian_frame_crystal.png]] #+begin_src matlab G_mr = inv(J_s_111)*G*inv(J_a_111'); #+end_src #+begin_src matlab :results value replace :exports both :tangle no dcgain(G_mr) #+end_src #+RESULTS: | 1.9978e-09 | 3.9657e-09 | 7.7944e-09 | | 3.9656e-09 | 8.4979e-08 | -1.5135e-17 | | 7.7944e-09 | -3.9252e-17 | 1.834e-07 | #+begin_src matlab :exports none %% Bode plot for the plant figure; tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); ax1 = nexttile([2,1]); hold on; plot(freqs, abs(squeeze(freqresp(G_mr(1,1), freqs, 'Hz'))), ... 'DisplayName', 'd'); plot(freqs, abs(squeeze(freqresp(G_mr(2,2), freqs, 'Hz'))), ... 'DisplayName', 'uh'); plot(freqs, abs(squeeze(freqresp(G_mr(3,3), freqs, 'Hz'))), ... 'DisplayName', 'ur'); for i = 1:2 for j = i+1:3 plot(freqs, abs(squeeze(freqresp(G_mr(i,j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ... 'HandleVisibility', 'off'); end end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]); legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 2); ax2 = nexttile; hold on; plot(freqs, 180/pi*angle(squeeze(freqresp(G_mr(1,1), freqs, 'Hz')))); plot(freqs, 180/pi*angle(squeeze(freqresp(G_mr(2,2), freqs, 'Hz')))); plot(freqs, 180/pi*angle(squeeze(freqresp(G_mr(3,3), freqs, 'Hz')))); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); hold off; yticks(-360:90:360); ylim([-180, 180]); linkaxes([ax1,ax2],'x'); xlim([freqs(1), freqs(end)]); #+end_src #+begin_src matlab :exports none %% Bode plot for the plant fig = figure; tiledlayout(3, 3, 'TileSpacing', 'Compact', 'Padding', 'None'); for i_out = 1:3 for i_in = 1:3 ax = nexttile; plot(freqs, abs(squeeze(freqresp(G_mr(i_out, i_in), freqs, 'Hz')))); set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); end end linkaxes(findall(fig, 'type', 'axes'),'xy'); xlim([freqs(1), freqs(end)]); #+end_src This results in a coupled system. The main reason is that, as we map forces to the center of the 111 crystal and not at the center of mass/stiffness of the moving part, vertical forces will induce rotation and torques will induce vertical motion. ** Plant at the center of stiffness :noexport: Here, we map the piezo forces at the center of stiffness. Let's first compute the Jacobian: * Active Damping Plant (Strain gauge) :PROPERTIES: :header-args:matlab+: :tangle matlab/dcm_active_damping_strain_gauges.m :END: ** Introduction :ignore: In this section, we wish to see whether if strain gauges fixed to the piezoelectric actuator can be used for active damping. ** Matlab Init :noexport:ignore: #+begin_src matlab %% dcm_active_damping_strain_gauges.m % Active Damping using relative motion sensors (strain gauges) #+end_src #+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) <> #+end_src #+begin_src matlab :exports none :results silent :noweb yes <> #+end_src #+begin_src matlab :tangle no :noweb yes <> #+end_src #+begin_src matlab :eval no :noweb yes <> #+end_src #+begin_src matlab :noweb yes <> #+end_src #+begin_src matlab :noweb yes <> #+end_src ** Identification #+begin_src matlab %% Input/Output definition clear io; io_i = 1; %% Inputs % Control Input {3x1} [N] io(io_i) = linio([mdl, '/u'], 1, 'openinput'); io_i = io_i + 1; % % Stepper Displacement {3x1} [m] % io(io_i) = linio([mdl, '/d'], 1, 'openinput'); io_i = io_i + 1; %% Outputs % Strain Gauges {3x1} [m] io(io_i) = linio([mdl, '/sg'], 1, 'openoutput'); io_i = io_i + 1; #+end_src #+begin_src matlab %% Extraction of the dynamics G_sg = linearize(mdl, io); #+end_src #+begin_src matlab :exports none G_sg.InputName = {'u_ur', 'u_uh', 'u_d'}; G_sg.OutputName = {'sg_ur', 'sg_uh', 'sg_d'}; #+end_src #+begin_src matlab :results value replace :exports both :tangle no dcgain(G_sg) #+end_src #+RESULTS: | -1.4113e-13 | 1.0339e-13 | 3.774e-14 | | 1.0339e-13 | -1.4113e-13 | 3.774e-14 | | 3.7792e-14 | 3.7792e-14 | -7.5585e-14 | #+begin_src matlab :exports none %% Bode plot for the plant figure; tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); ax1 = nexttile([2,1]); hold on; plot(freqs, abs(squeeze(freqresp(G_sg(1,1), freqs, 'Hz'))), ... 'DisplayName', 'd'); plot(freqs, abs(squeeze(freqresp(G_sg(2,2), freqs, 'Hz'))), ... 'DisplayName', 'uh'); plot(freqs, abs(squeeze(freqresp(G_sg(3,3), freqs, 'Hz'))), ... 'DisplayName', 'ur'); for i = 1:2 for j = i+1:3 plot(freqs, abs(squeeze(freqresp(G_sg(i,j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ... 'HandleVisibility', 'off'); end end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]); legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 2); ax2 = nexttile; hold on; plot(freqs, 180/pi*angle(squeeze(freqresp(G_sg(1,1), freqs, 'Hz')))); plot(freqs, 180/pi*angle(squeeze(freqresp(G_sg(2,2), freqs, 'Hz')))); plot(freqs, 180/pi*angle(squeeze(freqresp(G_sg(3,3), freqs, 'Hz')))); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); hold off; yticks(-360:90:360); ylim([-180, 180]); linkaxes([ax1,ax2],'x'); xlim([freqs(1), freqs(end)]); #+end_src * Active Damping Plant (Force Sensors) :PROPERTIES: :header-args:matlab+: :tangle matlab/dcm_active_damping_iff.m :END: ** Introduction :ignore: Force sensors are added above the piezoelectric actuators. They can consists of a simple piezoelectric ceramic stack. See for instance cite:fleming10_integ_strain_force_feedb_high. ** Matlab Init :noexport:ignore: #+begin_src matlab %% dcm_active_damping_iff.m % Test of Integral Force Feedback Strategy #+end_src #+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) <> #+end_src #+begin_src matlab :exports none :results silent :noweb yes <> #+end_src #+begin_src matlab :tangle no :noweb yes <> #+end_src #+begin_src matlab :eval no :noweb yes <> #+end_src #+begin_src matlab :noweb yes <> #+end_src #+begin_src matlab :noweb yes <> #+end_src ** Identification #+begin_src matlab %% Input/Output definition clear io; io_i = 1; %% Inputs % Control Input {3x1} [N] io(io_i) = linio([mdl, '/control_system'], 1, 'openinput'); io_i = io_i + 1; %% Outputs % Force Sensor {3x1} [m] io(io_i) = linio([mdl, '/DCM'], 3, 'openoutput'); io_i = io_i + 1; #+end_src #+begin_src matlab %% Extraction of the dynamics G_fs = linearize(mdl, io); #+end_src #+begin_src matlab :exports none G_fs.InputName = {'u_ur', 'u_uh', 'u_d'}; G_fs.OutputName = {'fs_ur', 'fs_uh', 'fs_d'}; #+end_src #+begin_src matlab :results value replace :exports both :tangle no dcgain(G_fs) #+end_src #+RESULTS: | -1.4113e-13 | 1.0339e-13 | 3.774e-14 | | 1.0339e-13 | -1.4113e-13 | 3.774e-14 | | 3.7792e-14 | 3.7792e-14 | -7.5585e-14 | #+begin_src matlab :exports none %% Bode plot for the plant figure; tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); ax1 = nexttile([2,1]); hold on; plot(freqs, abs(squeeze(freqresp(G_fs(1,1), freqs, 'Hz'))), ... 'DisplayName', 'd'); plot(freqs, abs(squeeze(freqresp(G_fs(2,2), freqs, 'Hz'))), ... 'DisplayName', 'uh'); plot(freqs, abs(squeeze(freqresp(G_fs(3,3), freqs, 'Hz'))), ... 'DisplayName', 'ur'); plot(freqs, abs(squeeze(freqresp(G_fs(1,2), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ... 'DisplayName', 'off-diag'); for i = 1:2 for j = i+1:3 plot(freqs, abs(squeeze(freqresp(G_fs(i,j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ... 'HandleVisibility', 'off'); end end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]); legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 2); ylim([1e-13, 1e-7]); ax2 = nexttile; hold on; plot(freqs, 180/pi*angle(squeeze(freqresp(G_fs(1,1), freqs, 'Hz')))); plot(freqs, 180/pi*angle(squeeze(freqresp(G_fs(2,2), freqs, 'Hz')))); plot(freqs, 180/pi*angle(squeeze(freqresp(G_fs(3,3), freqs, 'Hz')))); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); hold off; yticks(-360:90:360); ylim([-180, 180]); linkaxes([ax1,ax2],'x'); xlim([freqs(1), freqs(end)]); #+end_src #+begin_src matlab :tangle no :exports results :results file replace exportFig('figs/iff_plant_bode_plot.pdf', 'width', 'wide', 'height', 'tall'); #+end_src #+name: fig:iff_plant_bode_plot #+caption: Bode plot of IFF Plant #+RESULTS: [[file:figs/iff_plant_bode_plot.png]] ** Controller - Root Locus #+begin_src matlab Kiff_g1 = eye(3)*1/(1 + s/2/pi/20); #+end_src #+begin_src matlab :exports none %% Root Locus for IFF gains = logspace(9, 12, 200); figure; hold on; plot(real(pole(G_fs)), imag(pole(G_fs)), 'x', 'color', colors(1,:), ... 'DisplayName', '$g = 0$'); plot(real(tzero(G_fs)), imag(tzero(G_fs)), 'o', 'color', colors(1,:), ... 'HandleVisibility', 'off'); for g = gains clpoles = pole(feedback(G_fs, g*Kiff_g1, +1)); plot(real(clpoles), imag(clpoles), '.', 'color', colors(1,:), ... 'HandleVisibility', 'off'); end % Optimal gain g = 8e10; clpoles = pole(feedback(G_fs, g*Kiff_g1, +1)); plot(real(clpoles), imag(clpoles), 'x', 'color', colors(2,:), ... 'DisplayName', sprintf('$g=%.0e$', g)); hold off; axis square; xlim([-2700, 0]); ylim([0, 2700]); xlabel('Real Part'); ylabel('Imaginary Part'); legend('location', 'northwest'); #+end_src #+begin_src matlab :tangle no :exports results :results file replace exportFig('figs/iff_root_locus.pdf', 'width', 'wide', 'height', 'tall'); #+end_src #+name: fig:iff_root_locus #+caption: Root Locus plot for the IFF Control strategy #+RESULTS: [[file:figs/iff_root_locus.png]] #+begin_src matlab %% Integral Force Feedback Controller Kiff = g*Kiff_g1; #+end_src ** Damped Plant #+begin_src matlab %% Input/Output definition clear io; io_i = 1; %% Inputs % Control Input {3x1} [N] io(io_i) = linio([mdl, '/control_system'], 1, 'input'); io_i = io_i + 1; %% Outputs % Force Sensor {3x1} [m] io(io_i) = linio([mdl, '/DCM'], 1, 'openoutput'); io_i = io_i + 1; #+end_src #+begin_src matlab %% DCM Kinematics load('mat/dcm_kinematics.mat'); #+end_src #+begin_src matlab %% Identification of the Open Loop plant controller.type = 0; % Open Loop G_ol = J_a_111*inv(J_s_111)*linearize(mdl, io); G_ol.InputName = {'u_ur', 'u_uh', 'u_d'}; G_ol.OutputName = {'d_ur', 'd_uh', 'd_d'}; #+end_src #+begin_src matlab %% Identification of the damped plant with IFF controller.type = 1; % IFF G_dp = J_a_111*inv(J_s_111)*linearize(mdl, io); G_dp.InputName = {'u_ur', 'u_uh', 'u_d'}; G_dp.OutputName = {'d_ur', 'd_uh', 'd_d'}; #+end_src #+begin_src matlab :exports none %% Comparison of the damped and undamped plant figure; tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); ax1 = nexttile([2,1]); hold on; plot(freqs, abs(squeeze(freqresp(G_ol(1,1), freqs, 'Hz'))), ... 'DisplayName', 'd - OL'); plot(freqs, abs(squeeze(freqresp(G_ol(2,2), freqs, 'Hz'))), ... 'DisplayName', 'uh - OL'); plot(freqs, abs(squeeze(freqresp(G_ol(3,3), freqs, 'Hz'))), ... 'DisplayName', 'ur - OL'); set(gca,'ColorOrderIndex',1) plot(freqs, abs(squeeze(freqresp(G_dp(1,1), freqs, 'Hz'))), '--', ... 'DisplayName', 'd - IFF'); plot(freqs, abs(squeeze(freqresp(G_dp(2,2), freqs, 'Hz'))), '--', ... 'DisplayName', 'uh - IFF'); plot(freqs, abs(squeeze(freqresp(G_dp(3,3), freqs, 'Hz'))), '--', ... 'DisplayName', 'ur - IFF'); for i = 1:2 for j = i+1:3 plot(freqs, abs(squeeze(freqresp(G_dp(i,j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ... 'HandleVisibility', 'off'); end end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]); legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2); ylim([1e-12, 1e-6]); ax2 = nexttile; hold on; plot(freqs, 180/pi*angle(squeeze(freqresp(G_ol(1,1), freqs, 'Hz')))); plot(freqs, 180/pi*angle(squeeze(freqresp(G_ol(2,2), freqs, 'Hz')))); plot(freqs, 180/pi*angle(squeeze(freqresp(G_ol(3,3), freqs, 'Hz')))); set(gca,'ColorOrderIndex',1) plot(freqs, 180/pi*angle(squeeze(freqresp(G_dp(1,1), freqs, 'Hz'))), '--'); plot(freqs, 180/pi*angle(squeeze(freqresp(G_dp(2,2), freqs, 'Hz'))), '--'); plot(freqs, 180/pi*angle(squeeze(freqresp(G_dp(3,3), freqs, 'Hz'))), '--'); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); hold off; yticks(-360:90:360); ylim([-180, 0]); linkaxes([ax1,ax2],'x'); xlim([freqs(1), freqs(end)]); #+end_src #+begin_src matlab :tangle no :exports results :results file replace exportFig('figs/comp_damped_undamped_plant_iff_bode_plot.pdf', 'width', 'wide', 'height', 'tall'); #+end_src #+name: fig:comp_damped_undamped_plant_iff_bode_plot #+caption: Bode plot of both the open-loop plant and the damped plant using IFF #+RESULTS: [[file:figs/comp_damped_undamped_plant_iff_bode_plot.png]] #+begin_important The Integral Force Feedback control strategy is very effective in damping the suspension modes of the DCM. #+end_important ** Save #+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 * HAC-LAC (IFF) architecture :PROPERTIES: :header-args:matlab+: :tangle matlab/dcm_hac_iff.m :END: ** Introduction :ignore: ** Matlab Init :noexport:ignore: #+begin_src matlab %% dcm_hac_iff.m % Development of the HAC-IFF control strategy #+end_src #+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) <> #+end_src #+begin_src matlab :exports none :results silent :noweb yes <> #+end_src #+begin_src matlab :tangle no :noweb yes <> #+end_src #+begin_src matlab :eval no :noweb yes <> #+end_src #+begin_src matlab :noweb yes <> #+end_src #+begin_src matlab :noweb yes <> #+end_src * Helping Functions :noexport: ** Initialize Path #+NAME: m-init-path #+BEGIN_SRC matlab %% Path for functions, data and scripts addpath('./matlab/mat/'); % Path for data addpath('./matlab/'); % Path for scripts %% Simscape Model - Nano Hexapod addpath('./matlab/STEPS/') #+END_SRC #+NAME: m-init-path-tangle #+BEGIN_SRC matlab %% Path for functions, data and scripts addpath('./mat/'); % Path for data %% Simscape Model - Nano Hexapod addpath('./STEPS/') #+END_SRC ** Initialize Simscape Model #+NAME: m-init-simscape #+begin_src matlab %% Initialize Parameters for Simscape model controller.type = 0; % Open Loop Control %% Options for Linearization options = linearizeOptions; options.SampleTime = 0; %% Open Simulink Model mdl = 'simscape_dcm'; open(mdl) #+end_src ** Initialize other elements #+NAME: m-init-other #+BEGIN_SRC matlab %% Colors for the figures colors = colororder; %% Frequency Vector freqs = logspace(1, 3, 1000); #+END_SRC * Bibliography :ignore: #+latex: \printbibliography