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#+TITLE: Identification of the Stewart Platform using Simscape
:DRAWER:
#+STARTUP: overview
#+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>
#+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+ :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:
* Introduction :ignore:
In this document, we discuss the various methods to identify the behavior of the Stewart platform.
- [[sec:modal_analysis]]
- [[sec:transmissibility]]
- [[sec:compliance]]
* Modal Analysis of the Stewart Platform
<<sec:modal_analysis>>
** Introduction :ignore:
** Matlab Init :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
<<matlab-dir>>
#+end_src
#+begin_src matlab :exports none :results silent :noweb yes
<<matlab-init>>
#+end_src
#+begin_src matlab :results none :exports none
simulinkproject('../');
#+end_src
#+begin_src matlab
open('stewart_platform_model.slx')
#+end_src
** Initialize the Stewart Platform
#+begin_src matlab
stewart = initializeStewartPlatform();
stewart = initializeFramesPositions(stewart);
stewart = generateGeneralConfiguration(stewart);
stewart = computeJointsPose(stewart);
stewart = initializeStrutDynamics(stewart);
stewart = initializeJointDynamics(stewart, 'type_F', 'universal_p', 'type_M', 'spherical_p');
stewart = initializeCylindricalPlatforms(stewart);
stewart = initializeCylindricalStruts(stewart);
stewart = computeJacobian(stewart);
stewart = initializeStewartPose(stewart);
stewart = initializeInertialSensor(stewart);
#+end_src
#+begin_src matlab
ground = initializeGround('type', 'none');
payload = initializePayload('type', 'none');
controller = initializeController('type', 'open-loop');
#+end_src
** Identification
#+begin_src matlab
%% Options for Linearized
options = linearizeOptions;
options.SampleTime = 0;
%% Name of the Simulink File
mdl = 'stewart_platform_model';
%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Controller'], 1, 'openinput'); io_i = io_i + 1; % Actuator Force Inputs [N]
io(io_i) = linio([mdl, '/Relative Motion Sensor'], 1, 'openoutput'); io_i = io_i + 1; % Position/Orientation of {B} w.r.t. {A}
io(io_i) = linio([mdl, '/Relative Motion Sensor'], 2, 'openoutput'); io_i = io_i + 1; % Velocity of {B} w.r.t. {A}
%% Run the linearization
G = linearize(mdl, io);
% G.InputName = {'tau1', 'tau2', 'tau3', 'tau4', 'tau5', 'tau6'};
% G.OutputName = {'Xdx', 'Xdy', 'Xdz', 'Xrx', 'Xry', 'Xrz', 'Vdx', 'Vdy', 'Vdz', 'Vrx', 'Vry', 'Vrz'};
#+end_src
Let's check the size of =G=:
#+begin_src matlab :results replace output
size(G)
#+end_src
#+RESULTS:
: size(G)
: State-space model with 12 outputs, 6 inputs, and 18 states.
: 'org_babel_eoe'
: ans =
: 'org_babel_eoe'
We expect to have only 12 states (corresponding to the 6dof of the mobile platform).
#+begin_src matlab :results replace output
Gm = minreal(G);
#+end_src
#+RESULTS:
: Gm = minreal(G);
: 6 states removed.
And indeed, we obtain 12 states.
** Coordinate transformation
We can perform the following transformation using the =ss2ss= command.
#+begin_src matlab
Gt = ss2ss(Gm, Gm.C);
#+end_src
Then, the =C= matrix of =Gt= is the unity matrix which means that the states of the state space model are equal to the measurements $\bm{Y}$.
The measurements are the 6 displacement and 6 velocities of mobile platform with respect to $\{B\}$.
We could perform the transformation by hand:
#+begin_src matlab
At = Gm.C*Gm.A*pinv(Gm.C);
Bt = Gm.C*Gm.B;
Ct = eye(12);
Dt = zeros(12, 6);
Gt = ss(At, Bt, Ct, Dt);
#+end_src
** Analysis
#+begin_src matlab
[V,D] = eig(Gt.A);
#+end_src
#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
ws = imag(diag(D))/2/pi;
[ws,I] = sort(ws)
xi = 100*real(diag(D))./imag(diag(D));
xi = xi(I);
data2orgtable([[1:length(ws(ws>0))]', ws(ws>0), xi(xi>0)], {}, {'Mode Number', 'Resonance Frequency [Hz]', 'Damping Ratio [%]'}, ' %.1f ');
#+end_src
#+RESULTS:
| Mode Number | Resonance Frequency [Hz] | Damping Ratio [%] |
|-------------+--------------------------+-------------------|
| 1.0 | 780.6 | 0.4 |
| 2.0 | 780.6 | 0.3 |
| 3.0 | 903.9 | 0.3 |
| 4.0 | 1061.4 | 0.3 |
| 5.0 | 1061.4 | 0.2 |
| 6.0 | 1269.6 | 0.2 |
** Visualizing the modes
To visualize the i'th mode, we may excite the system using the inputs $U_i$ such that $B U_i$ is co-linear to $\xi_i$ (the mode we want to excite).
\[ U(t) = e^{\alpha t} ( ) \]
Let's first sort the modes and just take the modes corresponding to a eigenvalue with a positive imaginary part.
#+begin_src matlab
ws = imag(diag(D));
[ws,I] = sort(ws)
ws = ws(7:end); I = I(7:end);
#+end_src
#+begin_src matlab
for i = 1:length(ws)
#+end_src
#+begin_src matlab
i_mode = I(i); % the argument is the i'th mode
#+end_src
#+begin_src matlab
lambda_i = D(i_mode, i_mode);
xi_i = V(:,i_mode);
a_i = real(lambda_i);
b_i = imag(lambda_i);
#+end_src
Let do 10 periods of the mode.
#+begin_src matlab
t = linspace(0, 10/(imag(lambda_i)/2/pi), 1000);
U_i = pinv(Gt.B) * real(xi_i * lambda_i * (cos(b_i * t) + 1i*sin(b_i * t)));
#+end_src
#+begin_src matlab
U = timeseries(U_i, t);
#+end_src
Simulation:
#+begin_src matlab
load('mat/conf_simscape.mat');
set_param(conf_simscape, 'StopTime', num2str(t(end)));
sim(mdl);
#+end_src
Save the movie of the mode shape.
#+begin_src matlab
smwritevideo(mdl, sprintf('figs/mode%i', i), ...
'PlaybackSpeedRatio', 1/(b_i/2/pi), ...
'FrameRate', 30, ...
'FrameSize', [800, 400]);
#+end_src
#+begin_src matlab
end
#+end_src
#+name: fig:mode1
#+caption: Identified mode - 1
[[file:figs/mode1.gif]]
#+name: fig:mode3
#+caption: Identified mode - 3
[[file:figs/mode3.gif]]
#+name: fig:mode5
#+caption: Identified mode - 5
[[file:figs/mode5.gif]]
* Transmissibility Analysis
<<sec:transmissibility>>
** Introduction :ignore:
** Matlab Init :noexport:
#+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
simulinkproject('../');
#+end_src
#+begin_src matlab
open('stewart_platform_model.slx')
#+end_src
** Initialize the Stewart platform
#+begin_src matlab
stewart = initializeStewartPlatform();
stewart = initializeFramesPositions(stewart, 'H', 90e-3, 'MO_B', 45e-3);
stewart = generateGeneralConfiguration(stewart);
stewart = computeJointsPose(stewart);
stewart = initializeStrutDynamics(stewart);
stewart = initializeJointDynamics(stewart, 'type_F', 'universal_p', 'type_M', 'spherical_p');
stewart = initializeCylindricalPlatforms(stewart);
stewart = initializeCylindricalStruts(stewart);
stewart = computeJacobian(stewart);
stewart = initializeStewartPose(stewart);
stewart = initializeInertialSensor(stewart, 'type', 'accelerometer', 'freq', 5e3);
#+end_src
We set the rotation point of the ground to be at the same point at frames $\{A\}$ and $\{B\}$.
#+begin_src matlab
ground = initializeGround('type', 'rigid', 'rot_point', stewart.platform_F.FO_A);
payload = initializePayload('type', 'rigid');
controller = initializeController('type', 'open-loop');
#+end_src
** Transmissibility
#+begin_src matlab
%% Options for Linearized
options = linearizeOptions;
options.SampleTime = 0;
%% Name of the Simulink File
mdl = 'stewart_platform_model';
%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Disturbances/D_w'], 1, 'openinput'); io_i = io_i + 1; % Base Motion [m, rad]
io(io_i) = linio([mdl, '/Absolute Motion Sensor'], 1, 'openoutput'); io_i = io_i + 1; % Absolute Motion [m, rad]
%% Run the linearization
T = linearize(mdl, io, options);
T.InputName = {'Wdx', 'Wdy', 'Wdz', 'Wrx', 'Wry', 'Wrz'};
T.OutputName = {'Edx', 'Edy', 'Edz', 'Erx', 'Ery', 'Erz'};
#+end_src
#+begin_src matlab
freqs = logspace(1, 4, 1000);
figure;
for ix = 1:6
for iy = 1:6
subplot(6, 6, (ix-1)*6 + iy);
hold on;
plot(freqs, abs(squeeze(freqresp(T(ix, iy), freqs, 'Hz'))), 'k-');
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylim([1e-5, 10]);
xlim([freqs(1), freqs(end)]);
if ix < 6
xticklabels({});
end
if iy > 1
yticklabels({});
end
end
end
#+end_src
From cite:preumont07_six_axis_singl_stage_activ, one can use the Frobenius norm of the transmissibility matrix to obtain a scalar indicator of the transmissibility performance of the system:
\begin{align*}
\| \bm{T}(\omega) \| &= \sqrt{\text{Trace}[\bm{T}(\omega) \bm{T}(\omega)^H]}\\
&= \sqrt{\Sigma_{i=1}^6 \Sigma_{j=1}^6 |T_{ij}|^2}
\end{align*}
#+begin_src matlab
freqs = logspace(1, 4, 1000);
T_norm = zeros(length(freqs), 1);
for i = 1:length(freqs)
T_norm(i) = sqrt(trace(freqresp(T, freqs(i), 'Hz')*freqresp(T, freqs(i), 'Hz')'));
end
#+end_src
And we normalize by a factor $\sqrt{6}$ to obtain a performance metric comparable to the transmissibility of a one-axis isolator:
\[ \Gamma(\omega) = \|\bm{T}(\omega)\| / \sqrt{6} \]
#+begin_src matlab
Gamma = T_norm/sqrt(6);
#+end_src
#+begin_src matlab
figure;
plot(freqs, Gamma)
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
#+end_src
* Compliance Analysis
<<sec:compliance>>
** Introduction :ignore:
** Matlab Init :noexport:
#+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
simulinkproject('../');
#+end_src
#+begin_src matlab
open('stewart_platform_model.slx')
#+end_src
** Initialize the Stewart platform
#+begin_src matlab
stewart = initializeStewartPlatform();
stewart = initializeFramesPositions(stewart, 'H', 90e-3, 'MO_B', 45e-3);
stewart = generateGeneralConfiguration(stewart);
stewart = computeJointsPose(stewart);
stewart = initializeStrutDynamics(stewart);
stewart = initializeJointDynamics(stewart, 'type_F', 'universal_p', 'type_M', 'spherical_p');
stewart = initializeCylindricalPlatforms(stewart);
stewart = initializeCylindricalStruts(stewart);
stewart = computeJacobian(stewart);
stewart = initializeStewartPose(stewart);
stewart = initializeInertialSensor(stewart, 'type', 'accelerometer', 'freq', 5e3);
#+end_src
We set the rotation point of the ground to be at the same point at frames $\{A\}$ and $\{B\}$.
#+begin_src matlab
ground = initializeGround('type', 'none');
payload = initializePayload('type', 'rigid');
controller = initializeController('type', 'open-loop');
#+end_src
** Compliance
#+begin_src matlab
%% Options for Linearized
options = linearizeOptions;
options.SampleTime = 0;
%% Name of the Simulink File
mdl = 'stewart_platform_model';
%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Disturbances/F_ext'], 1, 'openinput'); io_i = io_i + 1; % Base Motion [m, rad]
io(io_i) = linio([mdl, '/Absolute Motion Sensor'], 1, 'openoutput'); io_i = io_i + 1; % Absolute Motion [m, rad]
%% Run the linearization
C = linearize(mdl, io, options);
C.InputName = {'Fdx', 'Fdy', 'Fdz', 'Mdx', 'Mdy', 'Mdz'};
C.OutputName = {'Edx', 'Edy', 'Edz', 'Erx', 'Ery', 'Erz'};
#+end_src
#+begin_src matlab
freqs = logspace(1, 4, 1000);
figure;
for ix = 1:6
for iy = 1:6
subplot(6, 6, (ix-1)*6 + iy);
hold on;
plot(freqs, abs(squeeze(freqresp(C(ix, iy), freqs, 'Hz'))), 'k-');
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylim([1e-10, 1e-3]);
xlim([freqs(1), freqs(end)]);
if ix < 6
xticklabels({});
end
if iy > 1
yticklabels({});
end
end
end
#+end_src
We can try to use the Frobenius norm to obtain a scalar value representing the 6-dof compliance of the Stewart platform.
#+begin_src matlab
freqs = logspace(1, 4, 1000);
C_norm = zeros(length(freqs), 1);
for i = 1:length(freqs)
C_norm(i) = sqrt(trace(freqresp(C, freqs(i), 'Hz')*freqresp(C, freqs(i), 'Hz')'));
end
#+end_src
#+begin_src matlab
figure;
plot(freqs, C_norm)
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
#+end_src
* Functions
** Compute the Transmissibility
:PROPERTIES:
:header-args:matlab+: :tangle ../src/computeTransmissibility.m
:header-args:matlab+: :comments none :mkdirp yes :eval no
:END:
<<sec:computeTransmissibility>>
*** Function description
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
function [T, T_norm, freqs] = computeTransmissibility(args)
% computeTransmissibility -
%
% Syntax: [T, T_norm, freqs] = computeTransmissibility(args)
%
% Inputs:
% - args - Structure with the following fields:
% - plots [true/false] - Should plot the transmissilibty matrix and its Frobenius norm
% - freqs [] - Frequency vector to estimate the Frobenius norm
%
% Outputs:
% - T [6x6 ss] - Transmissibility matrix
% - T_norm [length(freqs)x1] - Frobenius norm of the Transmissibility matrix
% - freqs [length(freqs)x1] - Frequency vector in [Hz]
#+end_src
*** Optional Parameters
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
arguments
args.plots logical {mustBeNumericOrLogical} = false
args.freqs double {mustBeNumeric, mustBeNonnegative} = logspace(1,4,1000)
end
#+end_src
#+begin_src matlab
freqs = args.freqs;
#+end_src
*** Identification of the Transmissibility Matrix
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
%% Options for Linearized
options = linearizeOptions;
options.SampleTime = 0;
%% Name of the Simulink File
mdl = 'stewart_platform_model';
%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Disturbances/D_w'], 1, 'openinput'); io_i = io_i + 1; % Base Motion [m, rad]
io(io_i) = linio([mdl, '/Absolute Motion Sensor'], 1, 'output'); io_i = io_i + 1; % Absolute Motion [m, rad]
%% Run the linearization
T = linearize(mdl, io, options);
T.InputName = {'Wdx', 'Wdy', 'Wdz', 'Wrx', 'Wry', 'Wrz'};
T.OutputName = {'Edx', 'Edy', 'Edz', 'Erx', 'Ery', 'Erz'};
#+end_src
If wanted, the 6x6 transmissibility matrix is plotted.
#+begin_src matlab
p_handle = zeros(6*6,1);
if args.plots
fig = figure;
for ix = 1:6
for iy = 1:6
p_handle((ix-1)*6 + iy) = subplot(6, 6, (ix-1)*6 + iy);
hold on;
plot(freqs, abs(squeeze(freqresp(T(ix, iy), freqs, 'Hz'))), 'k-');
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
if ix < 6
xticklabels({});
end
if iy > 1
yticklabels({});
end
end
end
linkaxes(p_handle, 'xy')
xlim([freqs(1), freqs(end)]);
ylim([1e-5, 1e2]);
han = axes(fig, 'visible', 'off');
han.XLabel.Visible = 'on';
han.YLabel.Visible = 'on';
xlabel(han, 'Frequency [Hz]');
ylabel(han, 'Transmissibility [m/m]');
end
#+end_src
*** Computation of the Frobenius norm
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
T_norm = zeros(length(freqs), 1);
for i = 1:length(freqs)
T_norm(i) = sqrt(trace(freqresp(T, freqs(i), 'Hz')*freqresp(T, freqs(i), 'Hz')'));
end
#+end_src
#+begin_src matlab
T_norm = T_norm/sqrt(6);
#+end_src
#+begin_src matlab
if args.plots
figure;
plot(freqs, T_norm)
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]');
ylabel('Transmissibility - Frobenius Norm');
end
#+end_src
** Compute the Compliance
:PROPERTIES:
:header-args:matlab+: :tangle ../src/computeCompliance.m
:header-args:matlab+: :comments none :mkdirp yes :eval no
:END:
<<sec:computeCompliance>>
*** Function description
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
function [C, C_norm, freqs] = computeCompliance(args)
% computeCompliance -
%
% Syntax: [C, C_norm, freqs] = computeCompliance(args)
%
% Inputs:
% - args - Structure with the following fields:
% - plots [true/false] - Should plot the transmissilibty matrix and its Frobenius norm
% - freqs [] - Frequency vector to estimate the Frobenius norm
%
% Outputs:
% - C [6x6 ss] - Compliance matrix
% - C_norm [length(freqs)x1] - Frobenius norm of the Compliance matrix
% - freqs [length(freqs)x1] - Frequency vector in [Hz]
#+end_src
*** Optional Parameters
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
arguments
args.plots logical {mustBeNumericOrLogical} = false
args.freqs double {mustBeNumeric, mustBeNonnegative} = logspace(1,4,1000)
end
#+end_src
#+begin_src matlab
freqs = args.freqs;
#+end_src
*** Identification of the Compliance Matrix
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
%% Options for Linearized
options = linearizeOptions;
options.SampleTime = 0;
%% Name of the Simulink File
mdl = 'stewart_platform_model';
%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Disturbances/F_ext'], 1, 'openinput'); io_i = io_i + 1; % External forces [N, N*m]
io(io_i) = linio([mdl, '/Absolute Motion Sensor'], 1, 'output'); io_i = io_i + 1; % Absolute Motion [m, rad]
%% Run the linearization
C = linearize(mdl, io, options);
C.InputName = {'Fdx', 'Fdy', 'Fdz', 'Mdx', 'Mdy', 'Mdz'};
C.OutputName = {'Edx', 'Edy', 'Edz', 'Erx', 'Ery', 'Erz'};
#+end_src
If wanted, the 6x6 transmissibility matrix is plotted.
#+begin_src matlab
p_handle = zeros(6*6,1);
if args.plots
fig = figure;
for ix = 1:6
for iy = 1:6
p_handle((ix-1)*6 + iy) = subplot(6, 6, (ix-1)*6 + iy);
hold on;
plot(freqs, abs(squeeze(freqresp(C(ix, iy), freqs, 'Hz'))), 'k-');
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
if ix < 6
xticklabels({});
end
if iy > 1
yticklabels({});
end
end
end
linkaxes(p_handle, 'xy')
xlim([freqs(1), freqs(end)]);
han = axes(fig, 'visible', 'off');
han.XLabel.Visible = 'on';
han.YLabel.Visible = 'on';
xlabel(han, 'Frequency [Hz]');
ylabel(han, 'Compliance [m/N, rad/(N*m)]');
end
#+end_src
*** Computation of the Frobenius norm
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
freqs = args.freqs;
C_norm = zeros(length(freqs), 1);
for i = 1:length(freqs)
C_norm(i) = sqrt(trace(freqresp(C, freqs(i), 'Hz')*freqresp(C, freqs(i), 'Hz')'));
end
#+end_src
#+begin_src matlab
if args.plots
figure;
plot(freqs, C_norm)
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]');
ylabel('Compliance - Frobenius Norm');
end
#+end_src
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