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#+TITLE: Stewart Platform with Flexible Elements
:DRAWER:
#+STARTUP: overview
#+LANGUAGE: en
#+EMAIL: dehaeze.thomas@gmail.com
#+AUTHOR: Dehaeze Thomas
#+HTML_LINK_HOME: ./index.html
#+HTML_LINK_UP: ./index.html
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#+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>
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#+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:
* Simscape Model
** Matlab Init :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
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<<matlab-dir>>
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#+end_src
#+begin_src matlab :exports none :results silent :noweb yes
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<<matlab-init>>
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#+end_src
#+begin_src matlab :results none :exports none
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simulinkproject('../');
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#+end_src
** Flexible APA
#+begin_src matlab
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apa = load('./mat/APA300ML.mat', 'int_xyz', 'int_i', 'n_xyz', 'n_i', 'nodes', 'M', 'K');
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#+end_src
#+begin_src matlab :exports results :results value table replace :tangle no
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data2orgtable([length(apa.n_i); length(apa.int_i); size(apa.M,1) - 6*length(apa.int_i); size(apa.M,1)], {'Total number of Nodes', 'Number of interface Nodes', 'Number of Modes', 'Size of M and K matrices'}, {}, ' %.0f ');
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#+end_src
#+RESULTS:
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| Total number of Nodes | 7 |
| Number of interface Nodes | 7 |
| Number of Modes | 120 |
| Size of M and K matrices | 162 |
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#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
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data2orgtable([[1:length(apa.int_i)]', apa.int_i, apa.int_xyz], {}, {'Node i', 'Node Number', 'x [m]', 'y [m]', 'z [m]'}, ' %f ');
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#+end_src
#+caption: Coordinates of the interface nodes
#+RESULTS:
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| Node i | Node Number | x [m] | y [m] | z [m] |
|--------+-------------+---------+-------+--------|
| 1.0 | 697783.0 | 0.0 | 0.0 | -0.015 |
| 2.0 | 697784.0 | 0.0 | 0.0 | 0.015 |
| 3.0 | 697785.0 | -0.0325 | 0.0 | 0.0 |
| 4.0 | 697786.0 | -0.0125 | 0.0 | 0.0 |
| 5.0 | 697787.0 | -0.0075 | 0.0 | 0.0 |
| 6.0 | 697788.0 | 0.0125 | 0.0 | 0.0 |
| 7.0 | 697789.0 | 0.0325 | 0.0 | 0.0 |
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** Flexible Joint
#+begin_src matlab
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flex_joint = load('./mat/flexor_025.mat', 'int_xyz', 'int_i', 'n_xyz', 'n_i', 'nodes', 'M', 'K');
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#+end_src
#+begin_src matlab :exports results :results value table replace :tangle no
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data2orgtable([length(flex_joint.n_i); length(flex_joint.int_i); size(flex_joint.M,1) - 6*length(flex_joint.int_i); size(flex_joint.M,1)], {'Total number of Nodes', 'Number of interface Nodes', 'Number of Modes', 'Size of M and K matrices'}, {}, ' %.0f ');
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#+end_src
#+RESULTS:
| Total number of Nodes | 2 |
| Number of interface Nodes | 2 |
| Number of Modes | 6 |
| Size of M and K matrices | 18 |
#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
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data2orgtable([[1:length(flex_joint.int_i)]', flex_joint.int_i, flex_joint.int_xyz], {}, {'Node i', 'Node Number', 'x [m]', 'y [m]', 'z [m]'}, ' %f ');
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#+end_src
#+caption: Coordinates of the interface nodes
#+RESULTS:
| Node i | Node Number | x [m] | y [m] | z [m] |
|--------+-------------+-------+-------+-------|
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| 1.0 | 528875.0 | 0.0 | 0.0 | 0.0 |
| 2.0 | 528876.0 | 0.0 | 0.0 | -0.0 |
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#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
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data2orgtable([1e-6*flex_joint.K(3,3), flex_joint.K(4,4), flex_joint.K(5,5), flex_joint.K(6,6)]', {'Axial Stiffness [N/um]', 'Bending Stiffness [Nm/rad]', 'Bending Stiffness [Nm/rad]', 'Torsion Stiffness [Nm/rad]'}, {'*Caracteristic*', '*Value*'}, ' %0.f ');
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#+end_src
#+RESULTS:
| *Caracteristic* | *Value* |
|----------------------------+---------|
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| Axial Stiffness [N/um] | 94 |
| Bending Stiffness [Nm/rad] | 5 |
| Bending Stiffness [Nm/rad] | 5 |
| Torsion Stiffness [Nm/rad] | 260 |
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** Identification
And we identify the dynamics from force actuators to force sensors.
#+begin_src matlab
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%% Options for Linearized
options = linearizeOptions;
options.SampleTime = 0;
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%% Name of the Simulink File
mdl = 'stewart_platform_model';
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%% 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, '/Stewart Platform'], 1, 'openoutput', [], 'dLm'); io_i = io_i + 1; % Relative Displacement Outputs [m]
io(io_i) = linio([mdl, '/Stewart Platform'], 1, 'openoutput', [], 'Taum'); io_i = io_i + 1; % Force Sensors [N]
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#+end_src
#+begin_src matlab
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ground = initializeGround('type', 'none');
payload = initializePayload('type', 'rigid', 'm', 50);
controller = initializeController('type', 'open-loop');
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#+end_src
#+begin_src matlab
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disturbances = initializeDisturbances();
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#+end_src
** No Flexible Elements
#+begin_src matlab
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stewart = initializeStewartPlatform();
stewart = initializeFramesPositions(stewart);
stewart = generateGeneralConfiguration(stewart);
stewart = computeJointsPose(stewart);
stewart = initializeAmplifiedStrutDynamics(stewart, 'Ke', 1.5e6*ones(6,1), 'Ka', 40.5e6*ones(6,1), 'K1', 0.4e6*ones(6,1));
stewart = initializeJointDynamics(stewart, 'type_M', 'spherical_3dof', ...
'Kr_M', flex_joint.K(1,1)*ones(6,1), ...
'Ka_M', flex_joint.K(3,3)*ones(6,1), ...
'Kf_M', flex_joint.K(4,4)*ones(6,1), ...
'Kt_M', flex_joint.K(6,6)*ones(6,1), ...
'type_F', 'universal_3dof', ...
'Kr_F', flex_joint.K(1,1)*ones(6,1), ...
'Ka_F', flex_joint.K(3,3)*ones(6,1), ...
'Kf_F', flex_joint.K(4,4)*ones(6,1), ...
'Kt_F', flex_joint.K(6,6)*ones(6,1));
stewart = initializeCylindricalPlatforms(stewart);
stewart = initializeCylindricalStruts(stewart);
stewart = computeJacobian(stewart);
stewart = initializeStewartPose(stewart);
stewart = initializeInertialSensor(stewart);
references = initializeReferences(stewart);
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#+end_src
#+begin_src matlab
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%% Run the linearization
G = linearize(mdl, io, options);
G.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
G.OutputName = {'Dm1', 'Dm2', 'Dm3', 'Dm4', 'Dm5', 'Dm6', 'Fm1', 'Fm2', 'Fm3', 'Fm4', 'Fm5', 'Fm6'};
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#+end_src
** Flexible joints
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#+name: fig:simscape_flex_joints
#+caption: Figure caption
[[file:figs/simscape_flex_joints.png]]
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#+begin_src matlab
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stewart = initializeStewartPlatform();
stewart = initializeFramesPositions(stewart);
stewart = generateGeneralConfiguration(stewart);
stewart = computeJointsPose(stewart);
stewart = initializeAmplifiedStrutDynamics(stewart, 'Ke', 1.5e6*ones(6,1), 'Ka', 40.5e6*ones(6,1), 'K1', 0.4e6*ones(6,1));
stewart = initializeJointDynamics(stewart, 'type_F', 'flexible', 'K_F', flex_joint.K, 'M_F', flex_joint.M, 'n_xyz_F', flex_joint.n_xyz, 'xi_F', 0.1, 'step_file_F', 'mat/flexor_ID16.STEP', 'type_M', 'flexible', 'K_M', flex_joint.K, 'M_M', flex_joint.M, 'n_xyz_M', flex_joint.n_xyz, 'xi_M', 0.1, 'step_file_M', 'mat/flexor_ID16.STEP');
stewart = initializeCylindricalPlatforms(stewart);
stewart = initializeCylindricalStruts(stewart);
stewart = computeJacobian(stewart);
stewart = initializeStewartPose(stewart);
stewart = initializeInertialSensor(stewart);
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#+end_src
#+begin_src matlab
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%% Run the linearization
Gj = linearize(mdl, io, options);
Gj.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
Gj.OutputName = {'Dm1', 'Dm2', 'Dm3', 'Dm4', 'Dm5', 'Dm6', 'Fm1', 'Fm2', 'Fm3', 'Fm4', 'Fm5', 'Fm6'};
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#+end_src
** Flexible APA
Add picture of flexible platform
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#+name: fig:simscape_flex_apa
#+caption: Figure caption
[[file:figs/simscape_flex_apa.png]]
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#+begin_src matlab
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stewart = initializeStewartPlatform();
stewart = initializeFramesPositions(stewart);
stewart = generateGeneralConfiguration(stewart);
stewart = computeJointsPose(stewart);
stewart = initializeFlexibleStrutDynamics(stewart, 'H', 0.03, 'K', apa.K, 'M', apa.M, 'n_xyz', apa.n_xyz, 'xi', 0.1, 'Gf', -2.65e7, 'step_file', 'mat/APA300ML.STEP');
stewart = initializeJointDynamics(stewart, 'type_M', 'spherical_3dof', ...
'Kr_M', flex_joint.K(1,1)*ones(6,1), ...
'Ka_M', flex_joint.K(3,3)*ones(6,1), ...
'Kf_M', flex_joint.K(4,4)*ones(6,1), ...
'Kt_M', flex_joint.K(6,6)*ones(6,1), ...
'type_F', 'universal_3dof', ...
'Kr_F', flex_joint.K(1,1)*ones(6,1), ...
'Ka_F', flex_joint.K(3,3)*ones(6,1), ...
'Kf_F', flex_joint.K(4,4)*ones(6,1), ...
'Kt_F', flex_joint.K(6,6)*ones(6,1));
stewart = initializeCylindricalPlatforms(stewart);
stewart = initializeCylindricalStruts(stewart, 'type_F', 'none', 'type_M', 'none');
stewart = computeJacobian(stewart);
stewart = initializeStewartPose(stewart);
stewart = initializeInertialSensor(stewart);
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#+end_src
#+begin_src matlab
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%% Run the linearization
Ga = -linearize(mdl, io, options);
Ga.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
Ga.OutputName = {'Dm1', 'Dm2', 'Dm3', 'Dm4', 'Dm5', 'Dm6', 'Fm1', 'Fm2', 'Fm3', 'Fm4', 'Fm5', 'Fm6'};
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#+end_src
** Flexible Joints and APA
Add picture of flexible platform
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#+name: fig:simscape_flexible
#+caption: Figure caption
[[file:figs/simscape_flexible.png]]
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#+begin_src matlab
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stewart = initializeStewartPlatform();
stewart = initializeFramesPositions(stewart);
stewart = generateGeneralConfiguration(stewart);
stewart = computeJointsPose(stewart);
stewart = initializeFlexibleStrutDynamics(stewart, 'H', 0.03, 'K', apa.K, 'M', apa.M, 'n_xyz', apa.n_xyz, 'xi', 0.1, 'Gf', -2.65e7, 'step_file', 'mat/APA300ML.STEP');
stewart = initializeJointDynamics(stewart, 'type_F', 'flexible', 'K_F', flex_joint.K, 'M_F', flex_joint.M, 'n_xyz_F', flex_joint.n_xyz, 'xi_F', 0.1, 'step_file_F', 'mat/flexor_ID16.STEP', 'type_M', 'flexible', 'K_M', flex_joint.K, 'M_M', flex_joint.M, 'n_xyz_M', flex_joint.n_xyz, 'xi_M', 0.1, 'step_file_M', 'mat/flexor_ID16.STEP');
stewart = initializeCylindricalPlatforms(stewart);
stewart = initializeCylindricalStruts(stewart, 'type_F', 'none', 'type_M', 'none');
stewart = computeJacobian(stewart);
stewart = initializeStewartPose(stewart);
stewart = initializeInertialSensor(stewart);
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#+end_src
#+begin_src matlab
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Gf = -linearize(mdl, io, options);
Gf.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
Gf.OutputName = {'Dm1', 'Dm2', 'Dm3', 'Dm4', 'Dm5', 'Dm6', 'Fm1', 'Fm2', 'Fm3', 'Fm4', 'Fm5', 'Fm6'};
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#+end_src
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** Save
#+begin_src matlab
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save('./mat/flexible_stewart_platform.mat', 'G', 'Gj', 'Ga', 'Gf');
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#+end_src
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** Direct Velocity Feedback
#+begin_src matlab :exports none
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freqs = logspace(0, 3, 1000);
figure;
ax1 = subplot(2, 1, 1);
hold on;
for i = 1:6
plot(freqs, abs(squeeze(freqresp(G(i,i), freqs, 'Hz'))), 'color', [0 0.4470 0.7410 0.2]);
end
for i = 1:6
plot(freqs, abs(squeeze(freqresp(Gj(i,i), freqs, 'Hz'))), 'color', [0.8500 0.3250 0.0980 0.2]);
end
for i = 1:6
plot(freqs, abs(squeeze(freqresp(Ga(i,i), freqs, 'Hz'))), 'color', [0.9290 0.6940 0.1250 0.2]);
end
for i = 1:6
plot(freqs, abs(squeeze(freqresp(Gf(i,i), freqs, 'Hz'))), 'color', [0.4940 0.1840 0.5560 0.2]);
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
ax2 = subplot(2, 1, 2);
hold on;
for i = 1:6
plot(freqs, 180/pi*angle(squeeze(freqresp(G(i,i), freqs, 'Hz'))), 'color', [0 0.4470 0.7410 0.2]);
end
for i = 1:6
plot(freqs, 180/pi*angle(squeeze(freqresp(Gj(i,i), freqs, 'Hz'))), 'color', [0.8500 0.3250 0.0980 0.2]);
end
for i = 1:6
plot(freqs, 180/pi*angle(squeeze(freqresp(Ga(i,i), freqs, 'Hz'))), 'color', [0.9290 0.6940 0.1250 0.2]);
end
for i = 1:6
plot(freqs, 180/pi*angle(squeeze(freqresp(Gf(i,i), freqs, 'Hz'))), 'color', [0.4940 0.1840 0.5560 0.2]);
end
h = zeros(4, 1);
h(1) = plot(NaN, NaN, 'color', [0 0.4470 0.7410 0.2]);
h(2) = plot(NaN, NaN, 'color', [0.8500 0.3250 0.0980 0.2]);
h(3) = plot(NaN, NaN, 'color', [0.9290 0.6940 0.1250 0.2]);
h(4) = plot(NaN, NaN, 'color', [0.4940 0.1840 0.5560 0.2]);
legend(h, 'No flexible', 'Flexible Joints', 'Flexible APA', 'All Flexible');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-180, 180]);
yticks([-180, -90, 0, 90, 180]);
linkaxes([ax1,ax2],'x');
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#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
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exportFig('figs/flexible_elements_effect_dvf.pdf', 'width', 'full', 'height', 'full');
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#+end_src
#+name: fig:flexible_elements_effect_dvf
#+caption: Change of the DVF plant dynamics with the added flexible elements
#+RESULTS:
[[file:figs/flexible_elements_effect_dvf.png]]
** Integral Force Feedback
#+begin_src matlab :exports none
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freqs = logspace(0, 3, 1000);
figure;
ax1 = subplot(2, 1, 1);
hold on;
for i = 1:6
plot(freqs, abs(squeeze(freqresp(G(6+i,i), freqs, 'Hz'))), 'color', [0 0.4470 0.7410 0.2]);
end
for i = 1:6
plot(freqs, abs(squeeze(freqresp(Gj(6+i,i), freqs, 'Hz'))), 'color', [0.8500 0.3250 0.0980 0.2]);
end
for i = 1:6
plot(freqs, abs(squeeze(freqresp(Ga(6+i,i), freqs, 'Hz'))), 'color', [0.9290 0.6940 0.1250 0.2]);
end
for i = 1:6
plot(freqs, abs(squeeze(freqresp(Gf(6+i,i), freqs, 'Hz'))), 'color', [0.4940 0.1840 0.5560 0.2]);
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [N/N]'); set(gca, 'XTickLabel',[]);
ax2 = subplot(2, 1, 2);
hold on;
for i = 1:6
plot(freqs, 180/pi*angle(squeeze(freqresp(G(6+i,i), freqs, 'Hz'))), 'color', [0 0.4470 0.7410 0.2]);
end
for i = 1:6
plot(freqs, 180/pi*angle(squeeze(freqresp(Gj(6+i,i), freqs, 'Hz'))), 'color', [0.8500 0.3250 0.0980 0.2]);
end
for i = 1:6
plot(freqs, 180/pi*angle(squeeze(freqresp(Ga(6+i,i), freqs, 'Hz'))), 'color', [0.9290 0.6940 0.1250 0.2]);
end
for i = 1:6
plot(freqs, 180/pi*angle(squeeze(freqresp(Gf(6+i,i), freqs, 'Hz'))), 'color', [0.4940 0.1840 0.5560 0.2]);
end
h = zeros(4, 1);
h(1) = plot(NaN, NaN, 'color', [0 0.4470 0.7410 0.2]);
h(2) = plot(NaN, NaN, 'color', [0.8500 0.3250 0.0980 0.2]);
h(3) = plot(NaN, NaN, 'color', [0.9290 0.6940 0.1250 0.2]);
h(4) = plot(NaN, NaN, 'color', [0.4940 0.1840 0.5560 0.2]);
legend(h, 'No flexible', 'Flexible Joints', 'Flexible APA', 'All Flexible');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-180, 180]);
yticks([-180, -90, 0, 90, 180]);
linkaxes([ax1,ax2],'x');
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#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
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exportFig('figs/flexible_elements_effect_iff.pdf', 'width', 'full', 'height', 'full');
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#+end_src
#+name: fig:flexible_elements_effect_iff
#+caption: Change of the IFF plant dynamics with the added flexible elements
#+RESULTS:
[[file:figs/flexible_elements_effect_iff.png]]
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** Procedure to include flexible elements into Simscape
In order to model a flexible element with only few mass-spring-damper elements:
- FEM of the flexible element
- Super-Element extraction
- Parameters to extract
- For the flexible joint: axial, shear, bending and torsion stiffnesses
- For the APA: k1, ka, ke, c1
- This can be done directly from the Stiffness matrix or using identification from a simple Simscape model
** Conclusion
#+begin_important
The results seems to indicate that the model is well represented with only few degrees of freedom.
This permit to have a relatively sane number of states for the model.
#+end_important
* Control with flexible elements
** Matlab Init :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
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<<matlab-dir>>
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#+end_src
#+begin_src matlab :exports none :results silent :noweb yes
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<<matlab-init>>
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#+end_src
#+begin_src matlab :results none :exports none
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simulinkproject('../');
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#+end_src
** Flexible APA and Flexible Joint
#+begin_src matlab
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apa = load('./mat/APA300ML.mat', 'int_xyz', 'int_i', 'n_xyz', 'n_i', 'nodes', 'M', 'K');
flex_joint = load('./mat/flexor_ID16.mat', 'int_xyz', 'int_i', 'n_xyz', 'n_i', 'nodes', 'M', 'K');
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#+end_src
#+begin_src matlab
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stewart = initializeStewartPlatform();
stewart = initializeFramesPositions(stewart);
stewart = generateGeneralConfiguration(stewart);
stewart = computeJointsPose(stewart);
stewart = initializeFlexibleStrutDynamics(stewart, 'H', 0.03, 'K', apa.K, 'M', apa.M, 'n_xyz', apa.n_xyz, 'xi', 0.1, 'step_file', 'mat/APA300ML.STEP');
stewart = initializeJointDynamics(stewart, 'type_F', 'flexible', 'K_F', flex_joint.K, 'M_F', flex_joint.M, 'n_xyz_F', flex_joint.n_xyz, 'xi_F', 0.1, 'step_file_F', 'mat/flexor_ID16.STEP', 'type_M', 'flexible', 'K_M', flex_joint.K, 'M_M', flex_joint.M, 'n_xyz_M', flex_joint.n_xyz, 'xi_M', 0.1, 'step_file_M', 'mat/flexor_ID16.STEP');
stewart = initializeCylindricalPlatforms(stewart);
stewart = initializeCylindricalStruts(stewart, 'type_F', 'none', 'type_M', 'none');
stewart = computeJacobian(stewart);
stewart = initializeStewartPose(stewart);
stewart = initializeInertialSensor(stewart);
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#+end_src
#+begin_src matlab
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ground = initializeGround('type', 'none');
payload = initializePayload('type', 'rigid', 'm', 50);
controller = initializeController('type', 'open-loop');
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#+end_src
#+begin_src matlab
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disturbances = initializeDisturbances();
references = initializeReferences(stewart);
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#+end_src
** Identification
And we identify the dynamics from force actuators to force sensors.
#+begin_src matlab
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%% Options for Linearized
options = linearizeOptions;
options.SampleTime = 0;
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%% Name of the Simulink File
mdl = 'stewart_platform_model';
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%% 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, '/Stewart Platform'], 1, 'openoutput', [], 'dLm'); io_i = io_i + 1; % Relative Displacement Outputs [m]
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#+end_src
#+begin_src matlab
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G = -linearize(mdl, io, options);
G.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
G.OutputName = {'Dm1', 'Dm2', 'Dm3', 'Dm4', 'Dm5', 'Dm6'};
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#+end_src
** Decentralized Direct Velocity Feedback
#+begin_src matlab :exports none
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freqs = logspace(1, 4, 1000);
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figure;
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ax1 = subplot(2, 2, 1);
hold on;
for i = 1:6
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plot(freqs, abs(squeeze(freqresp(G(i, i), freqs, 'Hz'))));
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end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
title('Diagonal elements of the Plant');
ax2 = subplot(2, 2, 3);
hold on;
for i = 1:6
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plot(freqs, 180/pi*angle(squeeze(freqresp(G(i, i), freqs, 'Hz'))), 'DisplayName', sprintf('$\\epsilon_{\\mathcal{L}_%i}/\\tau_%i$', i, i));
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end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-180, 180]);
yticks([-180, -90, 0, 90, 180]);
legend('location', 'northwest');
ax3 = subplot(2, 2, 2);
hold on;
for i = 1:5
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for j = i+1:6
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plot(freqs, abs(squeeze(freqresp(G(i, j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2]);
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end
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end
set(gca,'ColorOrderIndex',1);
plot(freqs, abs(squeeze(freqresp(G(1, 1), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
title('Off-Diagonal elements of the Plant');
ax4 = subplot(2, 2, 4);
hold on;
for i = 1:5
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for j = i+1:6
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plot(freqs, 180/pi*angle(squeeze(freqresp(G(i, j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2]);
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end
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end
set(gca,'ColorOrderIndex',1);
plot(freqs, 180/pi*angle(squeeze(freqresp(G(1, 1), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-180, 180]);
yticks([-180, -90, 0, 90, 180]);
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linkaxes([ax1,ax2,ax3,ax4],'x');
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#+end_src
Controller Design
#+begin_src matlab
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Kl = -1e5*s/(1 + s/2/pi/2e2)/(1 + s/2/pi/5e2) * eye(6);
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#+end_src
#+begin_src matlab :exports none
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freqs = logspace(1, 4, 1000);
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figure;
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ax1 = subplot(2, 1, 1);
hold on;
plot(freqs, abs(squeeze(freqresp(G(1, 1)*Kl(1,1), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Loop Gain'); set(gca, 'XTickLabel',[]);
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ax2 = subplot(2, 1, 2);
hold on;
plot(freqs, 180/pi*angle(squeeze(freqresp(G(1, 1)*Kl(1,1), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-180, 180]);
yticks([-180, -90, 0, 90, 180]);
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linkaxes([ax1,ax2],'x');
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#+end_src
#+begin_src matlab :results replace value
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isstable(feedback(G(1:6,1:6)*Kl, eye(6), -1))
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#+end_src
#+RESULTS:
: 1
** HAC
#+begin_src matlab
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Kx = tf(zeros(6));
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controller = initializeController('type', 'ref-track-hac-dvf');
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#+end_src
#+begin_src matlab
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%% Name of the Simulink File
mdl = 'stewart_platform_model';
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%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Controller'], 1, 'input'); io_i = io_i + 1; % Actuator Force Inputs [N]
io(io_i) = linio([mdl, '/Relative Motion Sensor'], 1, 'openoutput'); io_i = io_i + 1; % Relative Displacement Outputs [m]
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%% Run the linearization
G = linearize(mdl, io);
G.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
G.OutputName = {'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'};
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#+end_src
#+begin_src matlab
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Gl = -stewart.kinematics.J*G;
Gl.OutputName = {'D1', 'D2', 'D3', 'D4', 'D5', 'D6'};
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#+end_src
#+begin_src matlab :exports none
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freqs = logspace(1, 4, 1000);
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figure;
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ax1 = subplot(2, 2, 1);
hold on;
for i = 1:6
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plot(freqs, abs(squeeze(freqresp(Gl(i, i), freqs, 'Hz'))));
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end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
title('Diagonal elements of the Plant');
ax2 = subplot(2, 2, 3);
hold on;
for i = 1:6
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plot(freqs, 180/pi*angle(squeeze(freqresp(Gl(i, i), freqs, 'Hz'))), 'DisplayName', sprintf('$\\epsilon_{\\mathcal{L}_%i}/\\tau_%i$', i, i));
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end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-180, 180]);
yticks([-180, -90, 0, 90, 180]);
legend('location', 'northwest');
ax3 = subplot(2, 2, 2);
hold on;
for i = 1:5
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for j = i+1:6
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plot(freqs, abs(squeeze(freqresp(Gl(i, j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2]);
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end
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end
set(gca,'ColorOrderIndex',1);
plot(freqs, abs(squeeze(freqresp(Gl(1, 1), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
title('Off-Diagonal elements of the Plant');
ax4 = subplot(2, 2, 4);
hold on;
for i = 1:5
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for j = i+1:6
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plot(freqs, 180/pi*angle(squeeze(freqresp(Gl(i, j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2]);
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end
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end
set(gca,'ColorOrderIndex',1);
plot(freqs, 180/pi*angle(squeeze(freqresp(Gl(1, 1), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-180, 180]);
yticks([-180, -90, 0, 90, 180]);
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linkaxes([ax1,ax2,ax3,ax4],'x');
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#+end_src
#+begin_src matlab
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wc = 2*pi*300;
Kl = diag(1./diag(abs(freqresp(Gl, wc)))) * wc/s * 1/(1 + s/3/wc);
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#+end_src
#+begin_src matlab :exports none
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freqs = logspace(1, 3, 1000);
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figure;
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ax1 = subplot(2, 1, 1);
hold on;
for i = 1:6
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plot(freqs, abs(squeeze(freqresp(Kl(i, i)*Gl(i, i), freqs, 'Hz'))));
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end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Loop Gain'); set(gca, 'XTickLabel',[]);
ax2 = subplot(2, 1, 2);
hold on;
for i = 1:6
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plot(freqs, 180/pi*angle(squeeze(freqresp(Kl(i, i)*Gl(i, i), freqs, 'Hz'))));
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end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-180, 180]);
yticks([-180, -90, 0, 90, 180]);
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linkaxes([ax1,ax2],'x');
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#+end_src
* Flexible Joint Specifications
** Matlab Init :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
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<<matlab-dir>>
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#+end_src
#+begin_src matlab :exports none :results silent :noweb yes
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<<matlab-init>>
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#+end_src
#+begin_src matlab :results none :exports none
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simulinkproject('../');
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#+end_src
** Stewart Platform Initialization
#+begin_src matlab
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stewart = initializeStewartPlatform();
stewart = initializeFramesPositions(stewart);
stewart = generateGeneralConfiguration(stewart);
stewart = computeJointsPose(stewart);
stewart = initializeAmplifiedStrutDynamics(stewart, 'Ke', 1.5e6*ones(6,1), 'Ka', 43e6*ones(6,1), 'K1', 0.4e6*ones(6,1), 'C1', 10*ones(6,1));
stewart = initializeCylindricalPlatforms(stewart);
stewart = initializeCylindricalStruts(stewart);
stewart = computeJacobian(stewart);
stewart = initializeStewartPose(stewart);
stewart = initializeInertialSensor(stewart);
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references = initializeReferences(stewart);
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#+end_src
#+begin_src matlab
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ground = initializeGround('type', 'none');
payload = initializePayload('type', 'rigid', 'm', 50);
controller = initializeController('type', 'open-loop');
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#+end_src
#+begin_src matlab
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disturbances = initializeDisturbances();
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#+end_src
#+begin_src matlab
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open('stewart_platform_model.slx')
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#+end_src
** Initialization :noexport:
#+begin_src matlab
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freqs = logspace(0, 3, 1000);
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c1 = [ 0 0.4470 0.7410 0.2]; % Blue
c2 = [0.8500 0.3250 0.0980 0.2]; % Orange
c3 = [0.9290 0.6940 0.1250 0.2]; % Yellow
c4 = [0.4940 0.1840 0.5560 0.2]; % Purple
c5 = [0.4660 0.6740 0.1880 0.2]; % Green
c6 = [0.3010 0.7450 0.9330 0.2]; % Light Blue
c7 = [0.6350 0.0780 0.1840 0.2]; % Red
colors = [c1; c2; c3; c4; c5; c6; c7];
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#+end_src
#+begin_src matlab
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%% Options for Linearized
options = linearizeOptions;
options.SampleTime = 0;
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%% Name of the Simulink File
mdl = 'stewart_platform_model';
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%% 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, '/Stewart Platform'], 1, 'openoutput', [], 'dLm'); io_i = io_i + 1; % Relative Motion - Legs [m]
io(io_i) = linio([mdl, '/Stewart Platform'], 1, 'openoutput', [], 'Taum'); io_i = io_i + 1; % Force Sensors [N]
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#+end_src
** Effect of the Bending Stiffness
#+begin_src matlab
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Kfs = [1, 10, 100, 1000]; % [Nm/rad]
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#+end_src
#+begin_src matlab :exports none
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Gs = {zeros(length(Kfs))};
for Kfs_i = 1:length(Kfs)
stewart = initializeJointDynamics(stewart, 'type_F', 'universal', ...
'type_M', 'spherical', ...
'Kf_M', Kfs(Kfs_i)*ones(6,1), ...
'Kt_M', 0*ones(6,1), ...
'Kf_F', Kfs(Kfs_i)*ones(6,1), ...
'Kt_F', 0*ones(6,1), ...
'Ka_M', 0*ones(6,1), ...
'Kr_M', 0*ones(6,1), ...
'Ka_F', 0*ones(6,1), ...
'Kr_F', 0*ones(6,1));
G = linearize(mdl, io, options);
G.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
G.OutputName = {'L1', 'L2', 'L3', 'L4', 'L5', 'L6', ...
'Fm1', 'Fm2', 'Fm3', 'Fm4', 'Fm5', 'Fm6'};
Gs(Kfs_i) = {G};
end
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#+end_src
#+begin_src matlab :exports none
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figure;
ax1 = subplot(2, 1, 1);
hold on;
for Kfs_i = 1:length(Kfs)
plot(freqs, abs(squeeze(freqresp(Gs{Kfs_i}('L1','F1'), freqs, 'Hz'))), 'color', colors(Kfs_i, :), 'DisplayName', sprintf('$k_f = %.0f [Nm/rad]$', Kfs(Kfs_i)));
for i = 2:6
plot(freqs, abs(squeeze(freqresp(Gs{Kfs_i}(sprintf('L%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'color', colors(Kfs_i, :), 'HandleVisibility', 'off');
end
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
legend('location', 'northeast');
ax2 = subplot(2, 1, 2);
hold on;
for Kfs_i = 1:length(Kfs)
for i = 1:6
plot(freqs, 180/pi*angle(squeeze(freqresp(Gs{Kfs_i}(sprintf('L%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'color', colors(Kfs_i, :));
end
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-180, 180]);
yticks([-180, -90, 0, 90, 180]);
linkaxes([ax1,ax2],'x');
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#+end_src
#+begin_src matlab :exports none
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figure;
ax1 = subplot(2, 1, 1);
hold on;
for Kfs_i = 1:length(Kfs)
plot(freqs, abs(squeeze(freqresp(Gs{Kfs_i}('Fm1','F1'), freqs, 'Hz'))), 'color', colors(Kfs_i, :), 'DisplayName', sprintf('$k_f = %.0f [Nm/rad]$', Kfs(Kfs_i)));
for i = 2:6
plot(freqs, abs(squeeze(freqresp(Gs{Kfs_i}(sprintf('Fm%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'color', colors(Kfs_i, :), 'HandleVisibility', 'off');
end
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
legend('location', 'northeast');
ax2 = subplot(2, 1, 2);
hold on;
for Kfs_i = 1:length(Kfs)
for i = 1:6
plot(freqs, 180/pi*angle(squeeze(freqresp(Gs{Kfs_i}(sprintf('Fm%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'color', colors(Kfs_i, :));
end
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-180, 180]);
yticks([-180, -90, 0, 90, 180]);
linkaxes([ax1,ax2],'x');
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#+end_src
** Effect of the Torsion Stiffness
#+begin_src matlab
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Kts = [10, 100, 500, 1000]; % [Nm/rad]
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#+end_src
#+begin_src matlab :exports none
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Gt = {zeros(length(Kts))};
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for Kts_i = 1:length(Kts)
stewart = initializeJointDynamics(stewart, 'type_F', 'universal', ...
'type_M', 'spherical', ...
'Kf_M', 0*ones(6,1), ...
'Kt_M', Kts(Kts_i)*ones(6,1), ...
'Kf_F', 0*ones(6,1), ...
'Kt_F', Kts(Kts_i)*ones(6,1), ...
'Ka_M', 0*ones(6,1), ...
'Kr_M', 0*ones(6,1), ...
'Ka_F', 0*ones(6,1), ...
'Kr_F', 0*ones(6,1));
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G = linearize(mdl, io, options);
G.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
G.OutputName = {'L1', 'L2', 'L3', 'L4', 'L5', 'L6', ...
'Fm1', 'Fm2', 'Fm3', 'Fm4', 'Fm5', 'Fm6'};
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Gt(Kts_i) = {G};
end
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#+end_src
#+begin_src matlab :exports none
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figure;
ax1 = subplot(2, 1, 1);
hold on;
for Kts_i = 1:length(Kts)
plot(freqs, abs(squeeze(freqresp(Gt{Kts_i}('L1','F1'), freqs, 'Hz'))), 'color', colors(Kts_i, :), 'DisplayName', sprintf('$k_t = %.0f [Nm/rad]$', Kts(Kts_i)));
for i = 2:6
plot(freqs, abs(squeeze(freqresp(Gt{Kts_i}(sprintf('L%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'color', colors(Kts_i, :), 'HandleVisibility', 'off');
end
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
legend('location', 'northeast');
ax2 = subplot(2, 1, 2);
hold on;
for Kts_i = 1:length(Kts)
for i = 1:6
plot(freqs, 180/pi*angle(squeeze(freqresp(Gt{Kts_i}(sprintf('L%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'color', colors(Kts_i, :));
end
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-180, 180]);
yticks([-180, -90, 0, 90, 180]);
linkaxes([ax1,ax2],'x');
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#+end_src
#+begin_src matlab :exports none
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figure;
ax1 = subplot(2, 1, 1);
hold on;
for Kts_i = 1:length(Kts)
plot(freqs, abs(squeeze(freqresp(Gs{Kts_i}('Fm1','F1'), freqs, 'Hz'))), 'color', colors(Kts_i, :), 'DisplayName', sprintf('$k_f = %.0f [Nm/rad]$', Kts(Kts_i)));
for i = 2:6
plot(freqs, abs(squeeze(freqresp(Gs{Kts_i}(sprintf('Fm%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'color', colors(Kts_i, :), 'HandleVisibility', 'off');
end
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
legend('location', 'northeast');
ax2 = subplot(2, 1, 2);
hold on;
for Kts_i = 1:length(Kts)
for i = 1:6
plot(freqs, 180/pi*angle(squeeze(freqresp(Gs{Kts_i}(sprintf('Fm%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'color', colors(Kts_i, :));
end
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-180, 180]);
yticks([-180, -90, 0, 90, 180]);
linkaxes([ax1,ax2],'x');
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#+end_src
** Effect of the Axial Stiffness
#+begin_src matlab
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Kas = [1e6, 1e7, 1e8, 5e8, 1e9]; % [N/m]
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#+end_src
#+begin_src matlab :exports none
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Ga = {zeros(length(Kas))};
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for Kas_i = 1:length(Kas)
stewart = initializeJointDynamics(stewart, 'type_F', 'universal_3dof', ...
'type_M', 'spherical_3dof', ...
'Kf_M', 0*ones(6,1), ...
'Kt_M', 0*ones(6,1), ...
'Kf_F', 0*ones(6,1), ...
'Kt_F', 0*ones(6,1), ...
'Ka_M', Kas(Kas_i)*ones(6,1), ...
'Kr_M', 0*ones(6,1), ...
'Ka_F', Kas(Kas_i)*ones(6,1), ...
'Kr_F', 0*ones(6,1));
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G = linearize(mdl, io, options);
G.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
G.OutputName = {'L1', 'L2', 'L3', 'L4', 'L5', 'L6', ...
'Fm1', 'Fm2', 'Fm3', 'Fm4', 'Fm5', 'Fm6'};
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Ga(Kas_i) = {G};
end
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#+end_src
#+begin_src matlab :exports none
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figure;
ax1 = subplot(2, 1, 1);
hold on;
for Kas_i = 1:length(Kas)
plot(freqs, abs(squeeze(freqresp(Ga{Kas_i}('L1','F1'), freqs, 'Hz'))), 'color', colors(Kas_i, :), 'DisplayName', sprintf('$k_a = %.0e [N/m]$', Kas(Kas_i)));
for i = 2:6
plot(freqs, abs(squeeze(freqresp(Ga{Kas_i}(sprintf('L%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'color', colors(Kas_i, :), 'HandleVisibility', 'off');
end
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
legend('location', 'northeast');
ax2 = subplot(2, 1, 2);
hold on;
for Kas_i = 1:length(Kas)
for i = 1:6
plot(freqs, 180/pi*angle(squeeze(freqresp(Ga{Kas_i}(sprintf('L%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'color', colors(Kas_i, :));
end
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-180, 180]);
yticks([-180, -90, 0, 90, 180]);
linkaxes([ax1,ax2],'x');
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#+end_src
#+begin_src matlab :exports none
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figure;
ax1 = subplot(2, 1, 1);
hold on;
for Kas_i = 1:length(Kas)
plot(freqs, abs(squeeze(freqresp(Ga{Kas_i}('Fm1','F1'), freqs, 'Hz'))), 'color', colors(Kas_i, :), 'DisplayName', sprintf('$k_a = %.0e [N/m]$', Kas(Kas_i)));
for i = 2:6
plot(freqs, abs(squeeze(freqresp(Ga{Kas_i}(sprintf('Fm%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'color', colors(Kas_i, :), 'HandleVisibility', 'off');
end
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
legend('location', 'northeast');
ax2 = subplot(2, 1, 2);
hold on;
for Kas_i = 1:length(Kas)
for i = 1:6
plot(freqs, 180/pi*angle(squeeze(freqresp(Ga{Kas_i}(sprintf('Fm%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'color', colors(Kas_i, :));
end
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-180, 180]);
yticks([-180, -90, 0, 90, 180]);
linkaxes([ax1,ax2],'x');
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#+end_src
** Effect of the Radial (Shear) Stiffness
#+begin_src matlab
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Krs = [1e4, 1e5, 1e6, 1e7]; % [N/m]
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#+end_src
#+begin_src matlab :exports none
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Gr = {zeros(length(Kas))};
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for Krs_i = 1:length(Krs)
stewart = initializeJointDynamics(stewart, 'type_F', 'universal_3dof', ...
'type_M', 'spherical_3dof', ...
'Kf_M', 0*ones(6,1), ...
'Kt_M', 0*ones(6,1), ...
'Kf_F', 0*ones(6,1), ...
'Kt_F', 0*ones(6,1), ...
'Ka_M', 1e15*ones(6,1), ...
'Kr_M', Krs(Krs_i)*ones(6,1), ...
'Ka_F', 1e15*ones(6,1), ...
'Kr_F', Krs(Krs_i)*ones(6,1));
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G = linearize(mdl, io, options);
G.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
G.OutputName = {'L1', 'L2', 'L3', 'L4', 'L5', 'L6', ...
'Fm1', 'Fm2', 'Fm3', 'Fm4', 'Fm5', 'Fm6'};
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Gr(Krs_i) = {G};
end
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#+end_src
#+begin_src matlab :exports none
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figure;
ax1 = subplot(2, 1, 1);
hold on;
for Krs_i = 1:length(Krs)
plot(freqs, abs(squeeze(freqresp(Gr{Krs_i}('L1','F1'), freqs, 'Hz'))), 'color', colors(Krs_i, :), 'DisplayName', sprintf('$k_r = %.0e [N/m]$', Krs(Krs_i)));
for i = 2:6
plot(freqs, abs(squeeze(freqresp(Gr{Krs_i}(sprintf('L%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'color', colors(Krs_i, :), 'HandleVisibility', 'off');
end
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
legend('location', 'northeast');
ax2 = subplot(2, 1, 2);
hold on;
for Krs_i = 1:length(Krs)
for i = 1:6
plot(freqs, 180/pi*angle(squeeze(freqresp(Gr{Krs_i}(sprintf('L%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'color', colors(Krs_i, :));
end
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-180, 180]);
yticks([-180, -90, 0, 90, 180]);
linkaxes([ax1,ax2],'x');
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#+end_src
#+begin_src matlab :exports none
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figure;
ax1 = subplot(2, 1, 1);
hold on;
for Krs_i = 1:length(Krs)
plot(freqs, abs(squeeze(freqresp(Gr{Krs_i}('Fm1','F1'), freqs, 'Hz'))), 'color', colors(Krs_i, :), 'DisplayName', sprintf('$k_r = %.0e [N/m]$', Krs(Krs_i)));
for i = 2:6
plot(freqs, abs(squeeze(freqresp(Gr{Krs_i}(sprintf('Fm%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'color', colors(Krs_i, :), 'HandleVisibility', 'off');
end
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
legend('location', 'northeast');
ax2 = subplot(2, 1, 2);
hold on;
for Krs_i = 1:length(Krs)
for i = 1:6
plot(freqs, 180/pi*angle(squeeze(freqresp(Gr{Krs_i}(sprintf('Fm%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'color', colors(Krs_i, :));
end
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-180, 180]);
yticks([-180, -90, 0, 90, 180]);
linkaxes([ax1,ax2],'x');
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#+end_src
** Comparison of perfect joint and worst specified joint
#+begin_src matlab :exports none
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stewart = initializeJointDynamics(stewart, 'type_F', 'universal_p', ...
'type_M', 'spherical_p');
Gp = linearize(mdl, io, options);
Gp.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
Gp.OutputName = {'L1', 'L2', 'L3', 'L4', 'L5', 'L6', ...
'Fm1', 'Fm2', 'Fm3', 'Fm4', 'Fm5', 'Fm6'};
stewart = initializeJointDynamics(stewart, 'type_F', 'universal_3dof', ...
'type_M', 'spherical_3dof', ...
'Kf_M', 100*ones(6,1), ...
'Kt_M', 500*ones(6,1), ...
'Kf_F', 100*ones(6,1), ...
'Kt_F', 500*ones(6,1), ...
'Ka_M', 200e6*ones(6,1), ...
'Kr_M', 1e6*ones(6,1), ...
'Ka_F', 200e6*ones(6,1), ...
'Kr_F', 1e6*ones(6,1));
G = linearize(mdl, io, options);
G.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
G.OutputName = {'L1', 'L2', 'L3', 'L4', 'L5', 'L6', ...
'Fm1', 'Fm2', 'Fm3', 'Fm4', 'Fm5', 'Fm6'};
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#+end_src
#+begin_src matlab :exports none
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figure;
ax1 = subplot(2, 1, 1);
hold on;
plot(freqs, abs(squeeze(freqresp(G('L1','F1'), freqs, 'Hz'))), 'k-', 'DisplayName', 'Real Joints');
for i = 2:6
plot(freqs, abs(squeeze(freqresp(G(sprintf('L%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'k-', 'HandleVisibility', 'off');
end
plot(freqs, abs(squeeze(freqresp(Gp('L1','F1'), freqs, 'Hz'))), 'b-', 'DisplayName', 'Perfect Joints');
for i = 2:6
plot(freqs, abs(squeeze(freqresp(Gp(sprintf('L%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'b-', 'HandleVisibility', 'off');
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
legend('location', 'northeast');
ax2 = subplot(2, 1, 2);
hold on;
for i = 1:6
plot(freqs, 180/pi*angle(squeeze(freqresp(G(sprintf('L%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'k-');
end
for i = 1:6
plot(freqs, 180/pi*angle(squeeze(freqresp(Gp(sprintf('L%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'b-');
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-180, 180]);
yticks([-180, -90, 0, 90, 180]);
linkaxes([ax1,ax2],'x');
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#+end_src
#+begin_src matlab :exports none
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figure;
ax1 = subplot(2, 1, 1);
hold on;
plot(freqs, abs(squeeze(freqresp(G('Fm1','F1'), freqs, 'Hz'))), 'k-', 'DisplayName', 'Real Joints');
for i = 2:6
plot(freqs, abs(squeeze(freqresp(G(sprintf('Fm%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'k-', 'HandleVisibility', 'off');
end
plot(freqs, abs(squeeze(freqresp(Gp('Fm1','F1'), freqs, 'Hz'))), 'b-', 'DisplayName', 'Perfect Joints');
for i = 2:6
plot(freqs, abs(squeeze(freqresp(Gp(sprintf('Fm%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'b-', 'HandleVisibility', 'off');
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
legend('location', 'northeast');
ax2 = subplot(2, 1, 2);
hold on;
for i = 1:6
plot(freqs, 180/pi*angle(squeeze(freqresp(G(sprintf('Fm%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'k-');
end
for i = 1:6
plot(freqs, 180/pi*angle(squeeze(freqresp(Gp(sprintf('Fm%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'b-');
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-180, 180]);
yticks([-180, -90, 0, 90, 180]);
linkaxes([ax1,ax2],'x');
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#+end_src
** Conclusion
Qualitatively:
| | *Specification* |
|-------------------+--------------------------------------------------|
| Axial Stiffness | Much larger than the Actuator axial stiffness |
| Shear Stiffness | |
| Bending Stiffness | Much smaller than the Actuator bending stiffness |
| Torsion Stiffness | |
Quantitatively:
| | *Specification* |
|-------------------+-----------------|
| Axial Stiffness | > 200 [N/um] |
| Shear Stiffness | > 1 [N/um] |
| Bending Stiffness | < 100 [Nm/rad] |
| Torsion Stiffness | < 500 [Nm/rad] |
* Flexible Joint Specifications with the APA300ML
** Matlab Init :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
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<<matlab-dir>>
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#+end_src
#+begin_src matlab :exports none :results silent :noweb yes
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<<matlab-init>>
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#+end_src
#+begin_src matlab :results none :exports none
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simulinkproject('../');
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#+end_src
** Stewart Platform Initialization
#+begin_src matlab
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apa = load('./mat/APA300ML.mat', 'int_xyz', 'int_i', 'n_xyz', 'n_i', 'nodes', 'M', 'K');
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#+end_src
#+begin_src matlab
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stewart = initializeStewartPlatform();
stewart = initializeFramesPositions(stewart);
stewart = generateGeneralConfiguration(stewart);
stewart = computeJointsPose(stewart);
stewart = initializeFlexibleStrutDynamics(stewart, 'H', 0.03, 'K', apa.K, 'M', apa.M, 'n_xyz', apa.n_xyz, 'xi', 0.1, 'step_file', 'mat/APA300ML.STEP');
stewart = initializeCylindricalPlatforms(stewart);
stewart = initializeCylindricalStruts(stewart, 'type_F', 'none', 'type_M', 'none');
stewart = computeJacobian(stewart);
stewart = initializeStewartPose(stewart);
stewart = initializeInertialSensor(stewart);
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references = initializeReferences(stewart);
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#+end_src
#+begin_src matlab
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ground = initializeGround('type', 'none');
payload = initializePayload('type', 'rigid', 'm', 50);
controller = initializeController('type', 'open-loop');
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#+end_src
#+begin_src matlab
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disturbances = initializeDisturbances();
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#+end_src
#+begin_src matlab
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open('stewart_platform_model.slx')
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#+end_src
** Initialization :noexport:
#+begin_src matlab
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freqs = logspace(0, 3, 1000);
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c1 = [ 0 0.4470 0.7410 0.2]; % Blue
c2 = [0.8500 0.3250 0.0980 0.2]; % Orange
c3 = [0.9290 0.6940 0.1250 0.2]; % Yellow
c4 = [0.4940 0.1840 0.5560 0.2]; % Purple
c5 = [0.4660 0.6740 0.1880 0.2]; % Green
c6 = [0.3010 0.7450 0.9330 0.2]; % Light Blue
c7 = [0.6350 0.0780 0.1840 0.2]; % Red
colors = [c1; c2; c3; c4; c5; c6; c7];
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#+end_src
#+begin_src matlab
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%% Options for Linearized
options = linearizeOptions;
options.SampleTime = 0;
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%% Name of the Simulink File
mdl = 'stewart_platform_model';
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%% 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, '/Stewart Platform'], 1, 'openoutput', [], 'dLm'); io_i = io_i + 1; % Relative Motion - Legs [m]
io(io_i) = linio([mdl, '/Stewart Platform'], 1, 'openoutput', [], 'Taum'); io_i = io_i + 1; % Force Sensors [N]
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#+end_src
** Comparison of perfect joint and worst specified joint
#+begin_src matlab :exports none
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stewart = initializeJointDynamics(stewart, 'type_F', 'universal_p', ...
'type_M', 'spherical_p');
Gp = linearize(mdl, io, options);
Gp.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
Gp.OutputName = {'L1', 'L2', 'L3', 'L4', 'L5', 'L6', ...
'Fm1', 'Fm2', 'Fm3', 'Fm4', 'Fm5', 'Fm6'};
stewart = initializeJointDynamics(stewart, 'type_F', 'universal_3dof', ...
'type_M', 'spherical_3dof', ...
'Kf_M', 100*ones(6,1), ...
'Kt_M', 500*ones(6,1), ...
'Kf_F', 100*ones(6,1), ...
'Kt_F', 500*ones(6,1), ...
'Ka_M', 200e6*ones(6,1), ...
'Kr_M', 1e6*ones(6,1), ...
'Ka_F', 200e6*ones(6,1), ...
'Kr_F', 1e6*ones(6,1));
G = linearize(mdl, io, options);
G.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
G.OutputName = {'L1', 'L2', 'L3', 'L4', 'L5', 'L6', ...
'Fm1', 'Fm2', 'Fm3', 'Fm4', 'Fm5', 'Fm6'};
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#+end_src
#+begin_src matlab :exports none
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figure;
ax1 = subplot(2, 1, 1);
hold on;
plot(freqs, abs(squeeze(freqresp(G('L1','F1'), freqs, 'Hz'))), 'k-', 'DisplayName', 'Real Joints');
for i = 2:6
plot(freqs, abs(squeeze(freqresp(G(sprintf('L%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'k-', 'HandleVisibility', 'off');
end
plot(freqs, abs(squeeze(freqresp(Gp('L1','F1'), freqs, 'Hz'))), 'b-', 'DisplayName', 'Perfect Joints');
for i = 2:6
plot(freqs, abs(squeeze(freqresp(Gp(sprintf('L%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'b-', 'HandleVisibility', 'off');
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
legend('location', 'northeast');
ax2 = subplot(2, 1, 2);
hold on;
for i = 1:6
plot(freqs, 180/pi*angle(squeeze(freqresp(G(sprintf('L%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'k-');
end
for i = 1:6
plot(freqs, 180/pi*angle(squeeze(freqresp(Gp(sprintf('L%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'b-');
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-180, 180]);
yticks([-180, -90, 0, 90, 180]);
linkaxes([ax1,ax2],'x');
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#+end_src
#+begin_src matlab :exports none
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figure;
ax1 = subplot(2, 1, 1);
hold on;
plot(freqs, abs(squeeze(freqresp(G('Fm1','F1'), freqs, 'Hz'))), 'k-', 'DisplayName', 'Real Joints');
for i = 2:6
plot(freqs, abs(squeeze(freqresp(G(sprintf('Fm%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'k-', 'HandleVisibility', 'off');
end
plot(freqs, abs(squeeze(freqresp(Gp('Fm1','F1'), freqs, 'Hz'))), 'b-', 'DisplayName', 'Perfect Joints');
for i = 2:6
plot(freqs, abs(squeeze(freqresp(Gp(sprintf('Fm%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'b-', 'HandleVisibility', 'off');
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
legend('location', 'northeast');
ax2 = subplot(2, 1, 2);
hold on;
for i = 1:6
plot(freqs, 180/pi*angle(squeeze(freqresp(G(sprintf('Fm%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'k-');
end
for i = 1:6
plot(freqs, 180/pi*angle(squeeze(freqresp(Gp(sprintf('Fm%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'b-');
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-180, 180]);
yticks([-180, -90, 0, 90, 180]);
linkaxes([ax1,ax2],'x');
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#+end_src
* Relative Motion Sensors
** Matlab Init :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
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<<matlab-dir>>
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#+end_src
#+begin_src matlab :exports none :results silent :noweb yes
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<<matlab-init>>
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#+end_src
#+begin_src matlab :results none :exports none
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simulinkproject('../');
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#+end_src
#+begin_src matlab
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open('stewart_platform_model.slx')
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#+end_src
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** Stewart Platform Initialization
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#+begin_src matlab
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stewart = initializeStewartPlatform();
stewart = initializeFramesPositions(stewart);
stewart = generateGeneralConfiguration(stewart);
stewart = computeJointsPose(stewart);
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#+end_src
#+begin_src matlab
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apa = load('./mat/APA300ML.mat', 'int_xyz', 'int_i', 'n_xyz', 'n_i', 'nodes', 'M', 'K');
stewart = initializeAmplifiedStrutDynamics(stewart, 'Ke', 1.5e6*ones(6,1), 'Ka', 40.5e6*ones(6,1), 'K1', 0.4e6*ones(6,1));
% stewart = initializeFlexibleStrutDynamics(stewart, 'H', 0.03, 'K', apa.K, 'M', apa.M, 'n_xyz', apa.n_xyz, 'xi', 0.1, 'step_file', 'mat/APA300ML.STEP');
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#+end_src
#+begin_src matlab
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flex_joint = load('./mat/flexor_025.mat', 'int_xyz', 'int_i', 'n_xyz', 'n_i', 'nodes', 'M', 'K');
stewart = initializeJointDynamics(stewart, 'type_M', 'spherical_3dof', ...
'Kr_M', flex_joint.K(1,1)*ones(6,1), ...
'Ka_M', flex_joint.K(3,3)*ones(6,1), ...
'Kf_M', flex_joint.K(4,4)*ones(6,1), ...
'Kt_M', flex_joint.K(6,6)*ones(6,1), ...
'type_F', 'universal_3dof', ...
'Kr_F', flex_joint.K(1,1)*ones(6,1), ...
'Ka_F', flex_joint.K(3,3)*ones(6,1), ...
'Kf_F', flex_joint.K(4,4)*ones(6,1), ...
'Kt_F', flex_joint.K(6,6)*ones(6,1));
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#+end_src
#+begin_src matlab
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stewart = initializeCylindricalPlatforms(stewart);
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#+end_src
#+begin_src matlab
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stewart = initializeCylindricalStruts(stewart);
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#+end_src
#+begin_src matlab
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stewart = computeJacobian(stewart);
stewart = initializeStewartPose(stewart);
stewart = initializeInertialSensor(stewart);
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references = initializeReferences(stewart);
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#+end_src
#+begin_src matlab
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ground = initializeGround('type', 'none');
payload = initializePayload('type', 'rigid', 'm', 50);
controller = initializeController('type', 'open-loop');
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#+end_src
#+begin_src matlab
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disturbances = initializeDisturbances();
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#+end_src
** Initialization :noexport:
#+begin_src matlab
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%% Options for Linearized
options = linearizeOptions;
options.SampleTime = 0;
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%% Name of the Simulink File
mdl = 'stewart_platform_model';
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%% 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, '/Stewart Platform'], 1, 'openoutput', [], 'dLm'); io_i = io_i + 1; % Relative Motion - Actuators [m]
io(io_i) = linio([mdl, '/Stewart Platform'], 1, 'openoutput', [], 'Lm'); io_i = io_i + 1; % Relative Motion - Legs [m]
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#+end_src
#+begin_src matlab :exports none
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G = linearize(mdl, io, options);
G.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
G.OutputName = {'L1', 'L2', 'L3', 'L4', 'L5', 'L6', ...
'D1', 'D2', 'D3', 'D4', 'D5', 'D6'};
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#+end_src
#+begin_src matlab :exports none
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freqs = logspace(1, 4, 5000);
figure;
ax1 = subplot(2, 1, 1);
hold on;
plot(freqs, abs(squeeze(freqresp(G('L1','F1'), freqs, 'Hz'))), 'k-', 'DisplayName', 'Actuator Stroke');
for i = 2:6
plot(freqs, abs(squeeze(freqresp(G(sprintf('L%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'k-', 'HandleVisibility', 'off');
end
plot(freqs, abs(squeeze(freqresp(G('D1','F1'), freqs, 'Hz'))), 'b-', 'DisplayName', 'Leg Stroke');
for i = 2:6
plot(freqs, abs(squeeze(freqresp(G(sprintf('D%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'b-', 'HandleVisibility', 'off');
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
legend('location', 'northeast');
ax2 = subplot(2, 1, 2);
hold on;
for i = 1:6
plot(freqs, 180/pi*angle(squeeze(freqresp(G(sprintf('L%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'k-');
end
for i = 1:6
plot(freqs, 180/pi*angle(squeeze(freqresp(G(sprintf('D%i', i),sprintf('F%i', i)), freqs, 'Hz'))), 'b-');
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-180, 180]);
yticks([-180, -90, 0, 90, 180]);
linkaxes([ax1,ax2],'x');
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#+end_src
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#+begin_src matlab :tangle no :exports results :results file replace
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exportFig('figs/comp_relative_motion_sensor_act_leg.pdf', 'width', 'full', 'height', 'full');
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#+end_src
#+name: fig:comp_relative_motion_sensor_act_leg
#+caption: Comparison of the dynamique from actuator to relative motion sensor located in parallel with the actuator or with the leg (flexible joints included)
#+RESULTS:
[[file:figs/comp_relative_motion_sensor_act_leg.png]]
* Struts with Encoders
** Matlab Init :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
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<<matlab-dir>>
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#+end_src
#+begin_src matlab :exports none :results silent :noweb yes
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<<matlab-init>>
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#+end_src
#+begin_src matlab :results none :exports none
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simulinkproject('../');
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#+end_src
** Flexible Strut
#+begin_src matlab
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apa = load('./mat/strut_encoder.mat', 'int_xyz', 'int_i', 'n_xyz', 'n_i', 'nodes', 'M', 'K');
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#+end_src
#+begin_src matlab :exports results :results value table replace :tangle no
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data2orgtable([length(apa.n_i); length(apa.int_i); size(apa.M,1) - 6*length(apa.int_i); size(apa.M,1)], {'Total number of Nodes', 'Number of interface Nodes', 'Number of Modes', 'Size of M and K matrices'}, {}, ' %.0f ');
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#+end_src
#+RESULTS:
| Total number of Nodes | 8 |
| Number of interface Nodes | 8 |
| Number of Modes | 6 |
| Size of M and K matrices | 54 |
#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
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data2orgtable([[1:length(apa.int_i)]', apa.int_i, apa.int_xyz], {}, {'Node i', 'Node Number', 'x [m]', 'y [m]', 'z [m]'}, ' %f ');
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#+end_src
#+caption: Coordinates of the interface nodes
#+RESULTS:
| Node i | Node Number | x [m] | y [m] | z [m] |
|--------+-------------+---------+--------+----------|
| 1.0 | 504411.0 | 0.0 | 0.0 | 0.0405 |
| 2.0 | 504412.0 | 0.0 | 0.0 | -0.0405 |
| 3.0 | 504413.0 | -0.0325 | 0.0 | 0.0 |
| 4.0 | 504414.0 | -0.0125 | 0.0 | 0.0 |
| 5.0 | 504415.0 | -0.0075 | 0.0 | 0.0 |
| 6.0 | 504416.0 | 0.0325 | 0.0 | 0.0 |
| 7.0 | 504417.0 | 0.004 | 0.0145 | -0.00175 |
| 8.0 | 504418.0 | 0.004 | 0.0166 | -0.00175 |
** Stewart Platform
#+begin_src matlab
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stewart = initializeStewartPlatform();
stewart = initializeFramesPositions(stewart, 'H', 95e-3, 'MO_B', 220e-3);
stewart = generateGeneralConfiguration(stewart, 'FH', 22.5e-3, 'FR', 114e-3, 'FTh', [ -11, 11, 120-11, 120+11, 240-11, 240+11]*(pi/180), ...
'MH', 22.5e-3, 'MR', 110e-3, 'MTh', [-60+15, 60-15, 60+15, 180-15, 180+15, -60-15]*(pi/180));
stewart = computeJointsPose(stewart);
stewart = initializeFlexibleStrutAndJointDynamics(stewart, 'H', (apa.int_xyz(1,3) - apa.int_xyz(2,3)), ...
'K', apa.K, ...
'M', apa.M, ...
'n_xyz', apa.n_xyz, ...
'xi', 0.1, ...
'Gf', -2.65e7, ...
'step_file', 'mat/APA300ML.STEP');
stewart = initializeSolidPlatforms(stewart);
stewart = initializeCylindricalStruts(stewart, 'type_F', 'none', 'type_M', 'none');
stewart = computeJacobian(stewart);
stewart = initializeStewartPose(stewart);
stewart = initializeInertialSensor(stewart);
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#+end_src
#+begin_src matlab
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disturbances = initializeDisturbances();
ground = initializeGround('type', 'none');
payload = initializePayload('type', 'rigid', 'm', 1);
controller = initializeController('type', 'open-loop');
references = initializeReferences(stewart);
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#+end_src
#+begin_src matlab
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%% Options for Linearized
options = linearizeOptions;
options.SampleTime = 0;
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%% Name of the Simulink File
mdl = 'stewart_platform_model';
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%% 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, '/Stewart Platform'], 1, 'openoutput', [], 'dLm'); io_i = io_i + 1; % Relative Displacement Outputs [m]
io(io_i) = linio([mdl, '/Stewart Platform'], 1, 'openoutput', [], 'Lm'); io_i = io_i + 1; % Force Sensors [N]
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#+end_src
#+begin_src matlab
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%% Run the linearization
G = linearize(mdl, io, options);
G.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
G.OutputName = {'Dm1', 'Dm2', 'Dm3', 'Dm4', 'Dm5', 'Dm6', ...
'D1', 'D2', 'D3', 'D4', 'D5', 'D6'};
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#+end_src
#+begin_src matlab :exports none
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freqs = logspace(0, 3, 1000);
figure;
ax1 = subplot(2, 1, 1);
hold on;
for i = 1:6
plot(freqs, abs(squeeze(freqresp(G(i,i), freqs, 'Hz'))), 'color', [0 0.4470 0.7410 0.2]);
end
for i = 1:6
plot(freqs, abs(squeeze(freqresp(G(6+i,i), freqs, 'Hz'))), 'color', [0.8500 0.3250 0.0980 0.2]);
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
ax2 = subplot(2, 1, 2);
hold on;
for i = 1:6
plot(freqs, 180/pi*angle(squeeze(freqresp(G(i,i), freqs, 'Hz'))), 'color', [0 0.4470 0.7410 0.2]);
end
for i = 1:6
plot(freqs, 180/pi*angle(squeeze(freqresp(G(6+i,i), freqs, 'Hz'))), 'color', [0.8500 0.3250 0.0980 0.2]);
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-180, 180]);
yticks([-180, -90, 0, 90, 180]);
linkaxes([ax1,ax2],'x');
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#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/comp_relative_motion_sensor_act_leg_encoder.pdf', 'width', 'full', 'height', 'full');
#+end_src
#+name: fig:comp_relative_motion_sensor_act_leg_encoder
#+caption:
#+RESULTS:
[[file:figs/comp_relative_motion_sensor_act_leg_encoder.png]]
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