tdehaeze/svd-control
1
0
Fork 0
Code Issues Pull Requests Releases Wiki Activity

Change gravimeter axis

Browse Source
This commit is contained in:
Thomas Dehaeze
2020-12-10 13:26:01 +01:00
parent 754716e4ad
commit 404c78505a
7 changed files with 496 additions and 72 deletions
Download Patch File Download Diff File Expand all files Collapse all files

90
index.org
View File

@@ -138,7 +138,7 @@ The parameters used for the simulation are the following:
G = linearize(mdl, io);
G.InputName = {'F1', 'F2', 'F3'};
G.OutputName = {'Ax1', 'Az1', 'Ax2', 'Az2'};
G.OutputName = {'Ax1', 'Ay1', 'Ax2', 'Ay2'};
#+end_src
The inputs and outputs of the plant are shown in Figure [[fig:gravimeter_plant_schematic]].
@@ -149,7 +149,7 @@ More precisely there are three inputs (the three actuator forces):
\end{equation}
And 4 outputs (the two 2-DoF accelerometers):
\begin{equation}
\bm{a} = \begin{bmatrix} a_{1x} \\ a_{1z} \\ a_{2x} \\ a_{2z} \end{bmatrix}
\bm{a} = \begin{bmatrix} a_{1x} \\ a_{1y} \\ a_{2x} \\ a_{2y} \end{bmatrix}
\end{equation}
#+begin_src latex :file gravimeter_plant_schematic.pdf :tangle no :exports results
@@ -158,7 +158,7 @@ And 4 outputs (the two 2-DoF accelerometers):
% Connections and labels
\draw[<-] (G.west) -- ++(-2.0, 0) node[above right]{$\bm{\tau} = \begin{bmatrix}\tau_1 \\ \tau_2 \\ \tau_2 \end{bmatrix}$};
\draw[->] (G.east) -- ++( 2.0, 0) node[above left]{$\bm{a} = \begin{bmatrix} a_{1x} \\ a_{1z} \\ a_{2x} \\ a_{2z} \end{bmatrix}$};
\draw[->] (G.east) -- ++( 2.0, 0) node[above left]{$\bm{a} = \begin{bmatrix} a_{1x} \\ a_{1y} \\ a_{2x} \\ a_{2y} \end{bmatrix}$};
\end{tikzpicture}
#+end_src
@@ -232,12 +232,12 @@ Consider the control architecture shown in Figure [[fig:gravimeter_decouple_jaco
The Jacobian matrix $J_{\tau}$ is used to transform forces applied by the three actuators into forces/torques applied on the gravimeter at its center of mass:
\begin{equation}
\begin{bmatrix} \tau_1 \\ \tau_2 \\ \tau_3 \end{bmatrix} = J_{\tau}^{-T} \begin{bmatrix} F_x \\ F_z \\ M_y \end{bmatrix}
\begin{bmatrix} \tau_1 \\ \tau_2 \\ \tau_3 \end{bmatrix} = J_{\tau}^{-T} \begin{bmatrix} F_x \\ F_y \\ M_z \end{bmatrix}
\end{equation}
The Jacobian matrix $J_{a}$ is used to compute the vertical acceleration, horizontal acceleration and rotational acceleration of the mass with respect to its center of mass:
\begin{equation}
\begin{bmatrix} a_x \\ a_z \\ a_{R_y} \end{bmatrix} = J_{a}^{-1} \begin{bmatrix} a_{x1} \\ a_{z1} \\ a_{x2} \\ a_{z2} \end{bmatrix}
\begin{bmatrix} a_x \\ a_y \\ a_{R_z} \end{bmatrix} = J_{a}^{-1} \begin{bmatrix} a_{x1} \\ a_{y1} \\ a_{x2} \\ a_{y2} \end{bmatrix}
\end{equation}
We thus define a new plant as defined in Figure [[fig:gravimeter_decouple_jacobian]].
@@ -252,10 +252,10 @@ $\bm{G}_x(s)$ correspond to the $3 \times 3$transfer function matrix from forces
\node[block, right=0.6 of G] (Ja) {$J_{a}^{-1}$};
% Connections and labels
\draw[<-] (Jt.west) -- ++(-2.5, 0) node[above right]{$\bm{\mathcal{F}} = \begin{bmatrix}F_x \\ F_z \\ M_y \end{bmatrix}$};
\draw[<-] (Jt.west) -- ++(-2.5, 0) node[above right]{$\bm{\mathcal{F}} = \begin{bmatrix}F_x \\ F_y \\ M_z \end{bmatrix}$};
\draw[->] (Jt.east) -- (G.west) node[above left]{$\bm{\tau}$};
\draw[->] (G.east) -- (Ja.west) node[above left]{$\bm{a}$};
\draw[->] (Ja.east) -- ++( 2.6, 0) node[above left]{$\bm{\mathcal{A}} = \begin{bmatrix}a_x \\ a_z \\ a_{R_y} \end{bmatrix}$};
\draw[->] (Ja.east) -- ++( 2.6, 0) node[above left]{$\bm{\mathcal{A}} = \begin{bmatrix}a_x \\ a_y \\ a_{R_z} \end{bmatrix}$};
\begin{scope}[on background layer]
\node[fit={(Jt.south west) (Ja.north east)}, fill=black!10!white, draw, dashed, inner sep=14pt] (Gx) {};
@@ -284,8 +284,8 @@ The Jacobian corresponding to the sensors and actuators are defined below:
And the plant $\bm{G}_x$ is computed:
#+begin_src matlab
Gx = pinv(Ja)*G*pinv(Jt');
Gx.InputName = {'Fx', 'Fz', 'My'};
Gx.OutputName = {'Dx', 'Dz', 'Ry'};
Gx.InputName = {'Fx', 'Fy', 'Mz'};
Gx.OutputName = {'Dx', 'Dy', 'Rz'};
#+end_src
#+begin_src matlab :results output replace :exports results
@@ -736,7 +736,7 @@ Similarly, the bode plots of the diagonal elements and off-diagonal elements of
set(gca,'ColorOrderIndex',1)
plot(freqs, abs(squeeze(freqresp(Gx(1, 1), freqs, 'Hz'))), 'DisplayName', '$G_x(1,1) = A_x/F_x$');
plot(freqs, abs(squeeze(freqresp(Gx(2, 2), freqs, 'Hz'))), 'DisplayName', '$G_x(2,2) = A_y/F_y$');
plot(freqs, abs(squeeze(freqresp(Gx(3, 3), freqs, 'Hz'))), 'DisplayName', '$G_x(3,3) = R_y/M_y$');
plot(freqs, abs(squeeze(freqresp(Gx(3, 3), freqs, 'Hz'))), 'DisplayName', '$G_x(3,3) = R_z/M_z$');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Magnitude'); set(gca, 'XTickLabel',[]);
@@ -961,7 +961,7 @@ The obtained transmissibility in Open-loop, for the centralized control as well
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
set(gca, 'YTickLabel',[]); xlabel('Frequency [Hz]');
title('$D_z/D_{w,z}$');
title('$D_y/D_{w,y}$');
ax3 = nexttile;
hold on;
@@ -971,7 +971,7 @@ The obtained transmissibility in Open-loop, for the centralized control as well
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
set(gca, 'YTickLabel',[]); xlabel('Frequency [Hz]');
title('$R_y/R_{w,y}$');
title('$R_z/R_{w,z}$');
linkaxes([ax1,ax2,ax3],'xy');
xlim([freqs(1), freqs(end)]);
@@ -1040,7 +1040,7 @@ Let say we change the position of the actuators:
G = linearize(mdl, io);
G.InputName = {'F1', 'F2', 'F3'};
G.OutputName = {'Ax1', 'Az1', 'Ax2', 'Az2'};
G.OutputName = {'Ax1', 'Ay1', 'Ax2', 'Ay2'};
#+end_src
#+begin_src matlab :exports none
@@ -1077,7 +1077,7 @@ The closed-loop system are still stable, and their
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
set(gca, 'YTickLabel',[]); xlabel('Frequency [Hz]');
title('$D_z/D_{w,z}$');
title('$D_y/D_{w,y}$');
ax3 = nexttile;
hold on;
@@ -1087,7 +1087,7 @@ The closed-loop system are still stable, and their
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
set(gca, 'YTickLabel',[]); xlabel('Frequency [Hz]');
title('$R_y/R_{w,y}$');
title('$R_z/R_{w,z}$');
linkaxes([ax1,ax2,ax3],'xy');
xlim([freqs(1), freqs(end)]);
@@ -1145,7 +1145,7 @@ To do so, the actuators (springs) should be positioned such that the stiffness m
G = linearize(mdl, io);
G.InputName = {'F1', 'F2', 'F3'};
G.OutputName = {'Ax1', 'Az1', 'Ax2', 'Az2'};
G.OutputName = {'Ax1', 'Ay1', 'Ax2', 'Ay2'};
#+end_src
Decoupling at the CoM (Mass decoupled)
@@ -1162,8 +1162,8 @@ Decoupling at the CoM (Mass decoupled)
#+begin_src matlab
GM = pinv(JMa)*G*pinv(JMt');
GM.InputName = {'Fx', 'Fz', 'My'};
GM.OutputName = {'Dx', 'Dz', 'Ry'};
GM.InputName = {'Fx', 'Fy', 'Mz'};
GM.OutputName = {'Dx', 'Dy', 'Rz'};
#+end_src
#+begin_src matlab :exports none
@@ -1195,7 +1195,7 @@ Decoupling at the CoM (Mass decoupled)
#+end_src
#+name: fig:jac_decoupling_M
#+caption:
#+caption: Diagonal and off-diagonal elements of the decoupled plant
#+RESULTS:
[[file:figs/jac_decoupling_M.png]]
@@ -1220,8 +1220,8 @@ Decoupling at the point where K is diagonal (x = 0, y = -h/2 from the schematic
And the plant $\bm{G}_x$ is computed:
#+begin_src matlab
GK = pinv(JKa)*G*pinv(JKt');
GK.InputName = {'Fx', 'Fz', 'My'};
GK.OutputName = {'Dx', 'Dz', 'Ry'};
GK.InputName = {'Fx', 'Fy', 'Mz'};
GK.OutputName = {'Dx', 'Dy', 'Rz'};
#+end_src
#+begin_src matlab :exports none
@@ -1253,7 +1253,7 @@ And the plant $\bm{G}_x$ is computed:
#+end_src
#+name: fig:jac_decoupling_K
#+caption:
#+caption: Diagonal and off-diagonal elements of the decoupled plant
#+RESULTS:
[[file:figs/jac_decoupling_K.png]]
@@ -1286,7 +1286,7 @@ To do so, the actuator position should be modified
G = linearize(mdl, io);
G.InputName = {'F1', 'F2', 'F3'};
G.OutputName = {'Ax1', 'Az1', 'Ax2', 'Az2'};
G.OutputName = {'Ax1', 'Ay1', 'Ax2', 'Ay2'};
#+end_src
#+begin_src matlab
@@ -1302,8 +1302,8 @@ To do so, the actuator position should be modified
#+begin_src matlab
GKM = pinv(JMa)*G*pinv(JMt');
GKM.InputName = {'Fx', 'Fz', 'My'};
GKM.OutputName = {'Dx', 'Dz', 'Ry'};
GKM.InputName = {'Fx', 'Fy', 'Mz'};
GKM.OutputName = {'Dx', 'Dy', 'Rz'};
#+end_src
#+begin_src matlab :exports none
@@ -1335,7 +1335,7 @@ To do so, the actuator position should be modified
#+end_src
#+name: fig:jac_decoupling_KM
#+caption:
#+caption: Diagonal and off-diagonal elements of the decoupled plant
#+RESULTS:
[[file:figs/jac_decoupling_KM.png]]
@@ -1373,7 +1373,7 @@ Or it can be decoupled at high frequency if the Jacobians are evaluated at the C
G = linearize(mdl, io);
G.InputName = {'F1', 'F2', 'F3'};
G.OutputName = {'Ax1', 'Az1', 'Ax2', 'Az2'};
G.OutputName = {'Ax1', 'Ay1', 'Ax2', 'Ay2'};
#+end_src
#+begin_src matlab
@@ -1405,7 +1405,7 @@ Or it can be decoupled at high frequency if the Jacobians are evaluated at the C
G = linearize(mdl, io);
G.InputName = {'F1', 'F2', 'F3'};
G.OutputName = {'Ax1', 'Az1', 'Ax2', 'Az2'};
G.OutputName = {'Ax1', 'Ay1', 'Ax2', 'Ay2'};
#+end_src
#+begin_src matlab
@@ -1430,8 +1430,8 @@ Or it can be decoupled at high frequency if the Jacobians are evaluated at the C
#+begin_src matlab
GM = pinv(JMa)*G*pinv(JMt');
GM.InputName = {'Fx', 'Fz', 'My'};
GM.OutputName = {'Dx', 'Dz', 'Ry'};
GM.InputName = {'Fx', 'Fy', 'Mz'};
GM.OutputName = {'Dx', 'Dy', 'Rz'};
#+end_src
#+begin_src matlab :exports none
@@ -1506,36 +1506,6 @@ Or it can be decoupled at high frequency if the Jacobians are evaluated at the C
ylim([1e-8, 1e0]);
#+end_src
** SVD U and V matrices :noexport:
#+begin_src matlab
la = l/2; % Position of Act. [m]
ha = 0; % Position of Act. [m]
#+end_src
#+begin_src matlab
c = 2e1; % Actuator Damping [N/(m/s)]
#+end_src
#+begin_src matlab
%% Name of the Simulink File
mdl = 'gravimeter';
%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/F1'], 1, 'openinput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/F2'], 1, 'openinput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/F3'], 1, 'openinput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Acc_side'], 1, 'openoutput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Acc_side'], 2, 'openoutput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Acc_top'], 1, 'openoutput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Acc_top'], 2, 'openoutput'); io_i = io_i + 1;
G = linearize(mdl, io);
G.InputName = {'F1', 'F2', 'F3'};
G.OutputName = {'Ax1', 'Az1', 'Ax2', 'Az2'};
#+end_src
* Stewart Platform - Simscape Model
:PROPERTIES:
:header-args:matlab+: :tangle stewart_platform/script.m