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2021-11-30 11:16:48 +01:00
#+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: <link rel="stylesheet" type="text/css" href="https://research.tdehaeze.xyz/css/style.css"/>
#+HTML_HEAD: <script type="text/javascript" src="https://research.tdehaeze.xyz/js/script.js"></script>
#+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
<hr>
<p>This report is also available as a <a href="./dcm-simscape.pdf">pdf</a>.</p>
<hr>
#+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)
<<matlab-dir>>
#+end_src
#+begin_src matlab :exports none :results silent :noweb yes
<<matlab-init>>
#+end_src
#+begin_src matlab :tangle no :noweb yes
<<m-init-path>>
#+end_src
#+begin_src matlab :eval no :noweb yes
<<m-init-path-tangle>>
#+end_src
#+begin_src matlab :noweb yes
<<m-init-other>>
#+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)
<<matlab-dir>>
#+end_src
#+begin_src matlab :exports none :results silent :noweb yes
<<matlab-init>>
#+end_src
#+begin_src matlab :tangle no :noweb yes
<<m-init-path>>
#+end_src
#+begin_src matlab :eval no :noweb yes
<<m-init-path-tangle>>
#+end_src
#+begin_src matlab :noweb yes
<<m-init-simscape>>
#+end_src
#+begin_src matlab :noweb yes
<<m-init-other>>
#+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)
<<matlab-dir>>
#+end_src
#+begin_src matlab :exports none :results silent :noweb yes
<<matlab-init>>
#+end_src
#+begin_src matlab :tangle no :noweb yes
<<m-init-path>>
#+end_src
#+begin_src matlab :eval no :noweb yes
<<m-init-path-tangle>>
#+end_src
#+begin_src matlab :noweb yes
<<m-init-simscape>>
#+end_src
#+begin_src matlab :noweb yes
<<m-init-other>>
#+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)
<<matlab-dir>>
#+end_src
#+begin_src matlab :exports none :results silent :noweb yes
<<matlab-init>>
#+end_src
#+begin_src matlab :tangle no :noweb yes
<<m-init-path>>
#+end_src
#+begin_src matlab :eval no :noweb yes
<<m-init-path-tangle>>
#+end_src
#+begin_src matlab :noweb yes
<<m-init-simscape>>
#+end_src
#+begin_src matlab :noweb yes
<<m-init-other>>
#+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)
<<matlab-dir>>
#+end_src
#+begin_src matlab :exports none :results silent :noweb yes
<<matlab-init>>
#+end_src
#+begin_src matlab :tangle no :noweb yes
<<m-init-path>>
#+end_src
#+begin_src matlab :eval no :noweb yes
<<m-init-path-tangle>>
#+end_src
#+begin_src matlab :noweb yes
<<m-init-simscape>>
#+end_src
#+begin_src matlab :noweb yes
<<m-init-other>>
#+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
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