719 lines
27 KiB
Org Mode
719 lines
27 KiB
Org Mode
#+TITLE: Stewart Platform - Decentralized Active Damping
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:DRAWER:
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#+HTML_LINK_HOME: ./index.html
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#+HTML_LINK_UP: ./index.html
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#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="./css/htmlize.css"/>
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#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="./css/readtheorg.css"/>
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#+HTML_HEAD: <script src="./js/jquery.min.js"></script>
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#+HTML_HEAD: <script src="./js/bootstrap.min.js"></script>
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#+HTML_HEAD: <script src="./js/jquery.stickytableheaders.min.js"></script>
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#+HTML_HEAD: <script src="./js/readtheorg.js"></script>
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#+PROPERTY: header-args:matlab :session *MATLAB*
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#+PROPERTY: header-args:matlab+ :comments org
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#+PROPERTY: header-args:matlab+ :exports both
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#+PROPERTY: header-args:matlab+ :results none
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#+PROPERTY: header-args:matlab+ :eval no-export
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#+PROPERTY: header-args:matlab+ :noweb yes
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#+PROPERTY: header-args:matlab+ :mkdirp yes
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#+PROPERTY: header-args:matlab+ :output-dir figs
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:END:
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* Introduction :ignore:
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The following decentralized active damping techniques are briefly studied:
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- Inertial Control (proportional feedback of the absolute velocity): Section [[sec:active_damping_inertial]]
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- Integral Force Feedback: Section [[sec:active_damping_iff]]
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- Direct feedback of the relative velocity of each strut: Section [[sec:active_damping_dvf]]
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* Inertial Control
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:PROPERTIES:
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:header-args:matlab+: :tangle matlab/active_damping_inertial.m
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:header-args:matlab+: :comments org :mkdirp yes
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:END:
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<<sec:active_damping_inertial>>
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** Introduction :ignore:
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** Matlab Init :noexport:ignore:
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#+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
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#+begin_src matlab :exports none :results silent :noweb yes
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<<matlab-init>>
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#+end_src
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#+begin_src matlab
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simulinkproject('./');
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#+end_src
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#+begin_src matlab
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open('simulink/stewart_active_damping.slx')
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#+end_src
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** Identification of the Dynamics
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#+begin_src matlab
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stewart = initializeFramesPositions('H', 90e-3, 'MO_B', 45e-3);
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stewart = generateGeneralConfiguration(stewart);
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stewart = computeJointsPose(stewart);
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stewart = initializeStrutDynamics(stewart);
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stewart = initializeJointDynamics(stewart, 'disable', true);
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stewart = initializeCylindricalPlatforms(stewart);
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stewart = initializeCylindricalStruts(stewart);
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stewart = computeJacobian(stewart);
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stewart = initializeStewartPose(stewart);
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#+end_src
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#+begin_src matlab
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%% Options for Linearized
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options = linearizeOptions;
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options.SampleTime = 0;
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%% Name of the Simulink File
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mdl = 'stewart_active_damping';
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%% Input/Output definition
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clear io; io_i = 1;
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io(io_i) = linio([mdl, '/F'], 1, 'openinput'); io_i = io_i + 1; % Actuator Force Inputs [N]
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io(io_i) = linio([mdl, '/Vm'], 1, 'openoutput'); io_i = io_i + 1; % Absolute velocity of each leg [m/s]
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%% Run the linearization
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G = linearize(mdl, io, options);
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G.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
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G.OutputName = {'Vm1', 'Vm2', 'Vm3', 'Vm4', 'Vm5', 'Vm6'};
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#+end_src
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The transfer function from actuator forces to force sensors is shown in Figure [[fig:inertial_plant_coupling]].
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#+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);
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hold on;
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for i = 2:6
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set(gca,'ColorOrderIndex',2);
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plot(freqs, abs(squeeze(freqresp(G(['Vm', num2str(i)], 'F1'), freqs, 'Hz'))));
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end
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set(gca,'ColorOrderIndex',1);
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plot(freqs, abs(squeeze(freqresp(G('Vm1', 'F1'), freqs, 'Hz'))));
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Amplitude [$\frac{m/s}{N}$]'); set(gca, 'XTickLabel',[]);
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ax2 = subplot(2, 1, 2);
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hold on;
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for i = 2:6
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set(gca,'ColorOrderIndex',2);
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p2 = plot(freqs, 180/pi*angle(squeeze(freqresp(G(['Vm', num2str(i)], 'F1'), freqs, 'Hz'))));
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end
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set(gca,'ColorOrderIndex',1);
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p1 = plot(freqs, 180/pi*angle(squeeze(freqresp(G('Vm1', 'F1'), freqs, 'Hz'))));
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
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ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
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ylim([-180, 180]);
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yticks([-180, -90, 0, 90, 180]);
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legend([p1, p2], {'$F_{m,i}/F_i$', '$F_{m,j}/F_i$'})
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linkaxes([ax1,ax2],'x');
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#+end_src
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#+header: :tangle no :exports results :results none :noweb yes
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#+begin_src matlab :var filepath="figs/inertial_plant_coupling.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
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<<plt-matlab>>
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#+end_src
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#+name: fig:inertial_plant_coupling
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#+caption: Transfer function from the Actuator force $F_{i}$ to the absolute velocity of the same leg $v_{m,i}$ and to the absolute velocity of the other legs $v_{m,j}$ with $i \neq j$ in grey ([[./figs/inertial_plant_coupling.png][png]], [[./figs/inertial_plant_coupling.pdf][pdf]])
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[[file:figs/inertial_plant_coupling.png]]
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** Effect of the Flexible Joint stiffness on the Dynamics
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We add some stiffness and damping in the flexible joints and we re-identify the dynamics.
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#+begin_src matlab
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stewart = initializeJointDynamics(stewart);
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Gf = linearize(mdl, io, options);
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Gf.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
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Gf.OutputName = {'Vm1', 'Vm2', 'Vm3', 'Vm4', 'Vm5', 'Vm6'};
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#+end_src
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The new dynamics from force actuator to force sensor is shown in Figure [[fig:inertial_plant_flexible_joint_decentralized]].
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#+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);
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hold on;
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plot(freqs, abs(squeeze(freqresp(G( 'Vm1', 'F1'), freqs, 'Hz'))));
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plot(freqs, abs(squeeze(freqresp(Gf('Vm1', 'F1'), freqs, 'Hz'))));
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Amplitude [$\frac{m/s}{N}$]'); set(gca, 'XTickLabel',[]);
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ax2 = subplot(2, 1, 2);
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hold on;
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plot(freqs, 180/pi*angle(squeeze(freqresp(G( 'Vm1', 'F1'), freqs, 'Hz'))), 'DisplayName', 'Perfect Joints');
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plot(freqs, 180/pi*angle(squeeze(freqresp(Gf('Vm1', 'F1'), freqs, 'Hz'))), 'DisplayName', 'Flexible Joints');
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
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ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
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ylim([-180, 180]);
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yticks([-180, -90, 0, 90, 180]);
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legend('location', 'southwest')
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linkaxes([ax1,ax2],'x');
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#+end_src
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#+header: :tangle no :exports results :results none :noweb yes
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#+begin_src matlab :var filepath="figs/inertial_plant_flexible_joint_decentralized.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
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<<plt-matlab>>
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#+end_src
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#+name: fig:inertial_plant_flexible_joint_decentralized
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#+caption: Transfer function from the Actuator force $F_{i}$ to the absolute velocity sensor $v_{m,i}$ ([[./figs/inertial_plant_flexible_joint_decentralized.png][png]], [[./figs/inertial_plant_flexible_joint_decentralized.pdf][pdf]])
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[[file:figs/inertial_plant_flexible_joint_decentralized.png]]
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** Obtained Damping
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The control is a performed in a decentralized manner.
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The $6 \times 6$ control is a diagonal matrix with pure proportional action on the diagonal:
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\[ K(s) = g
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\begin{bmatrix}
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1 & & 0 \\
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& \ddots & \\
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0 & & 1
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\end{bmatrix} \]
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The root locus is shown in figure [[fig:root_locus_inertial_rot_stiffness]] and the obtained pole damping function of the control gain is shown in figure [[fig:pole_damping_gain_inertial_rot_stiffness]].
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#+begin_src matlab :exports none
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gains = logspace(0, 5, 1000);
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figure;
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hold on;
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plot(real(pole(G)), imag(pole(G)), 'x');
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plot(real(pole(Gf)), imag(pole(Gf)), 'x');
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set(gca,'ColorOrderIndex',1);
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plot(real(tzero(G)), imag(tzero(G)), 'o');
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plot(real(tzero(Gf)), imag(tzero(Gf)), 'o');
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for i = 1:length(gains)
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cl_poles = pole(feedback(G, gains(i)*eye(6)));
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set(gca,'ColorOrderIndex',1);
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plot(real(cl_poles), imag(cl_poles), '.');
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cl_poles = pole(feedback(Gf, gains(i)*eye(6)));
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set(gca,'ColorOrderIndex',2);
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plot(real(cl_poles), imag(cl_poles), '.');
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end
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ylim([0,2000]);
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xlim([-2000,0]);
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xlabel('Real Part')
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ylabel('Imaginary Part')
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axis square
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#+end_src
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#+header: :tangle no :exports results :results none :noweb yes
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#+begin_src matlab :var filepath="figs/root_locus_inertial_rot_stiffness.pdf" :var figsize="wide-tall" :post pdf2svg(file=*this*, ext="png")
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<<plt-matlab>>
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#+end_src
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#+name: fig:root_locus_inertial_rot_stiffness
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#+caption: Root Locus plot with Decentralized Inertial Control when considering the stiffness of flexible joints ([[./figs/root_locus_inertial_rot_stiffness.png][png]], [[./figs/root_locus_inertial_rot_stiffness.pdf][pdf]])
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[[file:figs/root_locus_inertial_rot_stiffness.png]]
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#+begin_src matlab :exports none
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gains = logspace(0, 5, 1000);
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figure;
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hold on;
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for i = 1:length(gains)
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set(gca,'ColorOrderIndex',1);
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cl_poles = pole(feedback(G, gains(i)*eye(6)));
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poles_damp = phase(cl_poles(imag(cl_poles)>0)) - pi/2;
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plot(gains(i)*ones(size(poles_damp)), poles_damp, '.');
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set(gca,'ColorOrderIndex',2);
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cl_poles = pole(feedback(Gf, gains(i)*eye(6)));
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poles_damp = phase(cl_poles(imag(cl_poles)>0)) - pi/2;
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plot(gains(i)*ones(size(poles_damp)), poles_damp, '.');
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end
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xlabel('Control Gain');
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ylabel('Damping of the Poles');
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set(gca, 'XScale', 'log');
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ylim([0,pi/2]);
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#+end_src
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#+header: :tangle no :exports results :results none :noweb yes
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#+begin_src matlab :var filepath="figs/pole_damping_gain_inertial_rot_stiffness.pdf" :var figsize="wide-tall" :post pdf2svg(file=*this*, ext="png")
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<<plt-matlab>>
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#+end_src
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#+name: fig:pole_damping_gain_inertial_rot_stiffness
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#+caption: Damping of the poles with respect to the gain of the Decentralized Inertial Control when considering the stiffness of flexible joints ([[./figs/pole_damping_gain_inertial_rot_stiffness.png][png]], [[./figs/pole_damping_gain_inertial_rot_stiffness.pdf][pdf]])
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[[file:figs/pole_damping_gain_inertial_rot_stiffness.png]]
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** Conclusion
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#+begin_important
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Joint stiffness does increase the resonance frequencies of the system but does not change the attainable damping when using relative motion sensors.
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#+end_important
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* Integral Force Feedback
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:PROPERTIES:
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:header-args:matlab+: :tangle matlab/active_damping_iff.m
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:header-args:matlab+: :comments org :mkdirp yes
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:END:
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<<sec:active_damping_iff>>
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** Introduction :ignore:
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** Matlab Init :noexport:ignore:
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#+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
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#+begin_src matlab :exports none :results silent :noweb yes
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<<matlab-init>>
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#+end_src
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#+begin_src matlab
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simulinkproject('./');
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#+end_src
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#+begin_src matlab
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open('simulink/stewart_active_damping.slx')
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#+end_src
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** Identification of the Dynamics with perfect Joints
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We first initialize the Stewart platform without joint stiffness.
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#+begin_src matlab
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stewart = initializeFramesPositions('H', 90e-3, 'MO_B', 45e-3);
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stewart = generateGeneralConfiguration(stewart);
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stewart = computeJointsPose(stewart);
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stewart = initializeStrutDynamics(stewart);
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stewart = initializeJointDynamics(stewart, 'disable', true);
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stewart = initializeCylindricalPlatforms(stewart);
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stewart = initializeCylindricalStruts(stewart);
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stewart = computeJacobian(stewart);
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stewart = initializeStewartPose(stewart);
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#+end_src
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And we identify the dynamics from force actuators to force sensors.
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#+begin_src matlab
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%% Options for Linearized
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options = linearizeOptions;
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options.SampleTime = 0;
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%% Name of the Simulink File
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mdl = 'stewart_active_damping';
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%% Input/Output definition
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clear io; io_i = 1;
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io(io_i) = linio([mdl, '/F'], 1, 'openinput'); io_i = io_i + 1; % Actuator Force Inputs [N]
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io(io_i) = linio([mdl, '/Fm'], 1, 'openoutput'); io_i = io_i + 1; % Force Sensor Outputs [N]
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%% Run the linearization
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G = linearize(mdl, io, options);
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G.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
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G.OutputName = {'Fm1', 'Fm2', 'Fm3', 'Fm4', 'Fm5', 'Fm6'};
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#+end_src
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The transfer function from actuator forces to force sensors is shown in Figure [[fig:iff_plant_coupling]].
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#+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);
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hold on;
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for i = 2:6
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set(gca,'ColorOrderIndex',2);
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plot(freqs, abs(squeeze(freqresp(G(['Fm', num2str(i)], 'F1'), freqs, 'Hz'))));
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end
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set(gca,'ColorOrderIndex',1);
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plot(freqs, abs(squeeze(freqresp(G('Fm1', 'F1'), freqs, 'Hz'))));
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Amplitude [N/N]'); set(gca, 'XTickLabel',[]);
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ax2 = subplot(2, 1, 2);
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hold on;
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for i = 2:6
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set(gca,'ColorOrderIndex',2);
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p2 = plot(freqs, 180/pi*angle(squeeze(freqresp(G(['Fm', num2str(i)], 'F1'), freqs, 'Hz'))));
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end
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set(gca,'ColorOrderIndex',1);
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p1 = plot(freqs, 180/pi*angle(squeeze(freqresp(G('Fm1', 'F1'), freqs, 'Hz'))));
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
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ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
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ylim([-180, 180]);
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yticks([-180, -90, 0, 90, 180]);
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legend([p1, p2], {'$F_{m,i}/F_i$', '$F_{m,j}/F_i$'})
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linkaxes([ax1,ax2],'x');
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#+end_src
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#+header: :tangle no :exports results :results none :noweb yes
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#+begin_src matlab :var filepath="figs/iff_plant_coupling.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
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<<plt-matlab>>
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#+end_src
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#+name: fig:iff_plant_coupling
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#+caption: Transfer function from the Actuator force $F_{i}$ to the Force sensor of the same leg $F_{m,i}$ and to the force sensor of the other legs $F_{m,j}$ with $i \neq j$ in grey ([[./figs/iff_plant_coupling.png][png]], [[./figs/iff_plant_coupling.pdf][pdf]])
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[[file:figs/iff_plant_coupling.png]]
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** Effect of the Flexible Joint stiffness on the Dynamics
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We add some stiffness and damping in the flexible joints and we re-identify the dynamics.
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#+begin_src matlab
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stewart = initializeJointDynamics(stewart);
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Gf = linearize(mdl, io, options);
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Gf.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
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Gf.OutputName = {'Fm1', 'Fm2', 'Fm3', 'Fm4', 'Fm5', 'Fm6'};
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#+end_src
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The new dynamics from force actuator to force sensor is shown in Figure [[fig:iff_plant_flexible_joint_decentralized]].
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#+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);
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hold on;
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plot(freqs, abs(squeeze(freqresp(G( 'Fm1', 'F1'), freqs, 'Hz'))));
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plot(freqs, abs(squeeze(freqresp(Gf('Fm1', 'F1'), freqs, 'Hz'))));
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Amplitude [N/N]'); set(gca, 'XTickLabel',[]);
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ax2 = subplot(2, 1, 2);
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hold on;
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plot(freqs, 180/pi*angle(squeeze(freqresp(G( 'Fm1', 'F1'), freqs, 'Hz'))), 'DisplayName', 'Perfect Joints');
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plot(freqs, 180/pi*angle(squeeze(freqresp(Gf('Fm1', 'F1'), freqs, 'Hz'))), 'DisplayName', 'Flexible Joints');
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|
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', 'southwest')
|
|
|
|
linkaxes([ax1,ax2],'x');
|
|
#+end_src
|
|
|
|
#+header: :tangle no :exports results :results none :noweb yes
|
|
#+begin_src matlab :var filepath="figs/iff_plant_flexible_joint_decentralized.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
|
<<plt-matlab>>
|
|
#+end_src
|
|
|
|
#+name: fig:iff_plant_flexible_joint_decentralized
|
|
#+caption: Transfer function from the Actuator force $F_{i}$ to the force sensor $F_{m,i}$ ([[./figs/iff_plant_flexible_joint_decentralized.png][png]], [[./figs/iff_plant_flexible_joint_decentralized.pdf][pdf]])
|
|
[[file:figs/iff_plant_flexible_joint_decentralized.png]]
|
|
|
|
** Obtained Damping
|
|
The control is a performed in a decentralized manner.
|
|
The $6 \times 6$ control is a diagonal matrix with pure integration action on the diagonal:
|
|
\[ K(s) = g
|
|
\begin{bmatrix}
|
|
\frac{1}{s} & & 0 \\
|
|
& \ddots & \\
|
|
0 & & \frac{1}{s}
|
|
\end{bmatrix} \]
|
|
|
|
The root locus is shown in figure [[fig:root_locus_iff_rot_stiffness]] and the obtained pole damping function of the control gain is shown in figure [[fig:pole_damping_gain_iff_rot_stiffness]].
|
|
#+begin_src matlab :exports none
|
|
gains = logspace(0, 5, 1000);
|
|
|
|
figure;
|
|
hold on;
|
|
plot(real(pole(G)), imag(pole(G)), 'x');
|
|
plot(real(pole(Gf)), imag(pole(Gf)), 'x');
|
|
set(gca,'ColorOrderIndex',1);
|
|
plot(real(tzero(G)), imag(tzero(G)), 'o');
|
|
plot(real(tzero(Gf)), imag(tzero(Gf)), 'o');
|
|
for i = 1:length(gains)
|
|
cl_poles = pole(feedback(G, (gains(i)/s)*eye(6)));
|
|
set(gca,'ColorOrderIndex',1);
|
|
plot(real(cl_poles), imag(cl_poles), '.');
|
|
cl_poles = pole(feedback(Gf, (gains(i)/s)*eye(6)));
|
|
set(gca,'ColorOrderIndex',2);
|
|
plot(real(cl_poles), imag(cl_poles), '.');
|
|
end
|
|
ylim([0,inf]);
|
|
xlim([-3000,0]);
|
|
xlabel('Real Part')
|
|
ylabel('Imaginary Part')
|
|
axis square
|
|
#+end_src
|
|
|
|
#+header: :tangle no :exports results :results none :noweb yes
|
|
#+begin_src matlab :var filepath="figs/root_locus_iff_rot_stiffness.pdf" :var figsize="wide-tall" :post pdf2svg(file=*this*, ext="png")
|
|
<<plt-matlab>>
|
|
#+end_src
|
|
|
|
#+name: fig:root_locus_iff_rot_stiffness
|
|
#+caption: Root Locus plot with Decentralized Integral Force Feedback when considering the stiffness of flexible joints ([[./figs/root_locus_iff_rot_stiffness.png][png]], [[./figs/root_locus_iff_rot_stiffness.pdf][pdf]])
|
|
[[file:figs/root_locus_iff_rot_stiffness.png]]
|
|
|
|
#+begin_src matlab :exports none
|
|
gains = logspace(0, 5, 1000);
|
|
|
|
figure;
|
|
hold on;
|
|
for i = 1:length(gains)
|
|
set(gca,'ColorOrderIndex',1);
|
|
cl_poles = pole(feedback(G, (gains(i)/s)*eye(6)));
|
|
poles_damp = phase(cl_poles(imag(cl_poles)>0)) - pi/2;
|
|
plot(gains(i)*ones(size(poles_damp)), poles_damp, '.');
|
|
set(gca,'ColorOrderIndex',2);
|
|
cl_poles = pole(feedback(Gf, (gains(i)/s)*eye(6)));
|
|
poles_damp = phase(cl_poles(imag(cl_poles)>0)) - pi/2;
|
|
plot(gains(i)*ones(size(poles_damp)), poles_damp, '.');
|
|
end
|
|
xlabel('Control Gain');
|
|
ylabel('Damping of the Poles');
|
|
set(gca, 'XScale', 'log');
|
|
ylim([0,pi/2]);
|
|
#+end_src
|
|
|
|
#+header: :tangle no :exports results :results none :noweb yes
|
|
#+begin_src matlab :var filepath="figs/pole_damping_gain_iff_rot_stiffness.pdf" :var figsize="wide-tall" :post pdf2svg(file=*this*, ext="png")
|
|
<<plt-matlab>>
|
|
#+end_src
|
|
|
|
#+name: fig:pole_damping_gain_iff_rot_stiffness
|
|
#+caption: Damping of the poles with respect to the gain of the Decentralized Integral Force Feedback when considering the stiffness of flexible joints ([[./figs/pole_damping_gain_iff_rot_stiffness.png][png]], [[./figs/pole_damping_gain_iff_rot_stiffness.pdf][pdf]])
|
|
[[file:figs/pole_damping_gain_iff_rot_stiffness.png]]
|
|
|
|
** Conclusion
|
|
#+begin_important
|
|
The joint stiffness has a huge impact on the attainable active damping performance when using force sensors.
|
|
Thus, if Integral Force Feedback is to be used in a Stewart platform with flexible joints, the rotational stiffness of the joints should be minimized.
|
|
#+end_important
|
|
|
|
* Direct Velocity Feedback
|
|
:PROPERTIES:
|
|
:header-args:matlab+: :tangle matlab/active_damping_dvf.m
|
|
:header-args:matlab+: :comments org :mkdirp yes
|
|
:END:
|
|
<<sec:active_damping_dvf>>
|
|
|
|
** 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
|
|
simulinkproject('./');
|
|
#+end_src
|
|
|
|
#+begin_src matlab
|
|
open('simulink/stewart_active_damping.slx')
|
|
#+end_src
|
|
|
|
** Identification of the Dynamics with perfect Joints
|
|
We first initialize the Stewart platform without joint stiffness.
|
|
#+begin_src matlab
|
|
stewart = initializeFramesPositions('H', 90e-3, 'MO_B', 45e-3);
|
|
stewart = generateGeneralConfiguration(stewart);
|
|
stewart = computeJointsPose(stewart);
|
|
stewart = initializeStrutDynamics(stewart);
|
|
stewart = initializeJointDynamics(stewart, 'disable', true);
|
|
stewart = initializeCylindricalPlatforms(stewart);
|
|
stewart = initializeCylindricalStruts(stewart);
|
|
stewart = computeJacobian(stewart);
|
|
stewart = initializeStewartPose(stewart);
|
|
#+end_src
|
|
|
|
And we identify the dynamics from force actuators to force sensors.
|
|
#+begin_src matlab
|
|
%% Options for Linearized
|
|
options = linearizeOptions;
|
|
options.SampleTime = 0;
|
|
|
|
%% Name of the Simulink File
|
|
mdl = 'stewart_active_damping';
|
|
|
|
%% Input/Output definition
|
|
clear io; io_i = 1;
|
|
io(io_i) = linio([mdl, '/F'], 1, 'openinput'); io_i = io_i + 1; % Actuator Force Inputs [N]
|
|
io(io_i) = linio([mdl, '/Dm'], 1, 'openoutput'); io_i = io_i + 1; % Relative Displacement Outputs [N]
|
|
|
|
%% Run the linearization
|
|
G = linearize(mdl, io, options);
|
|
G.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
|
|
G.OutputName = {'Dm1', 'Dm2', 'Dm3', 'Dm4', 'Dm5', 'Dm6'};
|
|
#+end_src
|
|
|
|
The transfer function from actuator forces to relative motion sensors is shown in Figure [[fig:dvf_plant_coupling]].
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(1, 3, 1000);
|
|
|
|
figure;
|
|
|
|
ax1 = subplot(2, 1, 1);
|
|
hold on;
|
|
for i = 2:6
|
|
set(gca,'ColorOrderIndex',2);
|
|
plot(freqs, abs(squeeze(freqresp(G(['Dm', num2str(i)], 'F1'), freqs, 'Hz'))));
|
|
end
|
|
set(gca,'ColorOrderIndex',1);
|
|
plot(freqs, abs(squeeze(freqresp(G('Dm1', 'F1'), freqs, 'Hz'))));
|
|
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 = 2:6
|
|
set(gca,'ColorOrderIndex',2);
|
|
p2 = plot(freqs, 180/pi*angle(squeeze(freqresp(G(['Dm', num2str(i)], 'F1'), freqs, 'Hz'))));
|
|
end
|
|
set(gca,'ColorOrderIndex',1);
|
|
p1 = plot(freqs, 180/pi*angle(squeeze(freqresp(G('Dm1', 'F1'), 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]);
|
|
legend([p1, p2], {'$D_{m,i}/F_i$', '$D_{m,j}/F_i$'})
|
|
|
|
linkaxes([ax1,ax2],'x');
|
|
#+end_src
|
|
|
|
#+header: :tangle no :exports results :results none :noweb yes
|
|
#+begin_src matlab :var filepath="figs/dvf_plant_coupling.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
|
<<plt-matlab>>
|
|
#+end_src
|
|
|
|
#+name: fig:dvf_plant_coupling
|
|
#+caption: Transfer function from the Actuator force $F_{i}$ to the Relative Motion Sensor $D_{m,j}$ with $i \neq j$ ([[./figs/dvf_plant_coupling.png][png]], [[./figs/dvf_plant_coupling.pdf][pdf]])
|
|
[[file:figs/dvf_plant_coupling.png]]
|
|
|
|
|
|
** Effect of the Flexible Joint stiffness on the Dynamics
|
|
We add some stiffness and damping in the flexible joints and we re-identify the dynamics.
|
|
#+begin_src matlab
|
|
stewart = initializeJointDynamics(stewart);
|
|
Gf = linearize(mdl, io, options);
|
|
Gf.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
|
|
Gf.OutputName = {'Dm1', 'Dm2', 'Dm3', 'Dm4', 'Dm5', 'Dm6'};
|
|
#+end_src
|
|
|
|
The new dynamics from force actuator to relative motion sensor is shown in Figure [[fig:dvf_plant_flexible_joint_decentralized]].
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(1, 3, 1000);
|
|
|
|
figure;
|
|
|
|
ax1 = subplot(2, 1, 1);
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G( 'Dm1', 'F1'), freqs, 'Hz'))));
|
|
plot(freqs, abs(squeeze(freqresp(Gf('Dm1', 'F1'), freqs, 'Hz'))));
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
|
|
|
|
ax2 = subplot(2, 1, 2);
|
|
hold on;
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(G( 'Dm1', 'F1'), freqs, 'Hz'))), 'DisplayName', 'Perfect Joints');
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(Gf('Dm1', 'F1'), freqs, 'Hz'))), 'DisplayName', 'Flexible Joints');
|
|
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', 'northeast');
|
|
|
|
linkaxes([ax1,ax2],'x');
|
|
#+end_src
|
|
|
|
#+header: :tangle no :exports results :results none :noweb yes
|
|
#+begin_src matlab :var filepath="figs/dvf_plant_flexible_joint_decentralized.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
|
<<plt-matlab>>
|
|
#+end_src
|
|
|
|
#+name: fig:dvf_plant_flexible_joint_decentralized
|
|
#+caption: Transfer function from the Actuator force $F_{i}$ to the relative displacement sensor $D_{m,i}$ ([[./figs/dvf_plant_flexible_joint_decentralized.png][png]], [[./figs/dvf_plant_flexible_joint_decentralized.pdf][pdf]])
|
|
[[file:figs/dvf_plant_flexible_joint_decentralized.png]]
|
|
|
|
** Obtained Damping
|
|
The control is a performed in a decentralized manner.
|
|
The $6 \times 6$ control is a diagonal matrix with pure derivative action on the diagonal:
|
|
\[ K(s) = g
|
|
\begin{bmatrix}
|
|
s & & \\
|
|
& \ddots & \\
|
|
& & s
|
|
\end{bmatrix} \]
|
|
|
|
The root locus is shown in figure [[fig:root_locus_dvf_rot_stiffness]] and the obtained pole damping function of the control gain is shown in figure [[fig:pole_damping_gain_dvf_rot_stiffness]].
|
|
#+begin_src matlab :exports none
|
|
gains = logspace(0, 5, 1000);
|
|
|
|
figure;
|
|
hold on;
|
|
plot(real(pole(G)), imag(pole(G)), 'x');
|
|
plot(real(pole(Gf)), imag(pole(Gf)), 'x');
|
|
set(gca,'ColorOrderIndex',1);
|
|
plot(real(tzero(G)), imag(tzero(G)), 'o');
|
|
plot(real(tzero(Gf)), imag(tzero(Gf)), 'o');
|
|
for i = 1:length(gains)
|
|
cl_poles = pole(feedback(G, (gains(i)*s)*eye(6)));
|
|
set(gca,'ColorOrderIndex',1);
|
|
plot(real(cl_poles), imag(cl_poles), '.');
|
|
cl_poles = pole(feedback(Gf, (gains(i)*s)*eye(6)));
|
|
set(gca,'ColorOrderIndex',2);
|
|
plot(real(cl_poles), imag(cl_poles), '.');
|
|
end
|
|
ylim([0,inf]);
|
|
xlim([-3000,0]);
|
|
xlabel('Real Part')
|
|
ylabel('Imaginary Part')
|
|
axis square
|
|
#+end_src
|
|
|
|
#+header: :tangle no :exports results :results none :noweb yes
|
|
#+begin_src matlab :var filepath="figs/root_locus_dvf_rot_stiffness.pdf" :var figsize="wide-tall" :post pdf2svg(file=*this*, ext="png")
|
|
<<plt-matlab>>
|
|
#+end_src
|
|
|
|
#+name: fig:root_locus_dvf_rot_stiffness
|
|
#+caption: Root Locus plot with Direct Velocity Feedback when considering the Stiffness of flexible joints ([[./figs/root_locus_dvf_rot_stiffness.png][png]], [[./figs/root_locus_dvf_rot_stiffness.pdf][pdf]])
|
|
[[file:figs/root_locus_dvf_rot_stiffness.png]]
|
|
|
|
#+begin_src matlab :exports none
|
|
gains = logspace(0, 5, 1000);
|
|
|
|
figure;
|
|
hold on;
|
|
for i = 1:length(gains)
|
|
set(gca,'ColorOrderIndex',1);
|
|
cl_poles = pole(feedback(G, (gains(i)*s)*eye(6)));
|
|
poles_damp = phase(cl_poles(imag(cl_poles)>0)) - pi/2;
|
|
plot(gains(i)*ones(size(poles_damp)), poles_damp, '.');
|
|
set(gca,'ColorOrderIndex',2);
|
|
cl_poles = pole(feedback(Gf, (gains(i)*s)*eye(6)));
|
|
poles_damp = phase(cl_poles(imag(cl_poles)>0)) - pi/2;
|
|
plot(gains(i)*ones(size(poles_damp)), poles_damp, '.');
|
|
end
|
|
xlabel('Control Gain');
|
|
ylabel('Damping of the Poles');
|
|
set(gca, 'XScale', 'log');
|
|
ylim([0,pi/2]);
|
|
#+end_src
|
|
|
|
#+header: :tangle no :exports results :results none :noweb yes
|
|
#+begin_src matlab :var filepath="figs/pole_damping_gain_dvf_rot_stiffness.pdf" :var figsize="wide-tall" :post pdf2svg(file=*this*, ext="png")
|
|
<<plt-matlab>>
|
|
#+end_src
|
|
|
|
#+name: fig:pole_damping_gain_dvf_rot_stiffness
|
|
#+caption: Damping of the poles with respect to the gain of the Direct Velocity Feedback when considering the Stiffness of flexible joints ([[./figs/pole_damping_gain_dvf_rot_stiffness.png][png]], [[./figs/pole_damping_gain_dvf_rot_stiffness.pdf][pdf]])
|
|
[[file:figs/pole_damping_gain_dvf_rot_stiffness.png]]
|
|
|
|
** Conclusion
|
|
#+begin_important
|
|
Joint stiffness does increase the resonance frequencies of the system but does not change the attainable damping when using relative motion sensors.
|
|
#+end_important
|