258 lines
8.0 KiB
Org Mode
258 lines
8.0 KiB
Org Mode
#+TITLE: Stewart Platform - Tracking Control
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:DRAWER:
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#+STARTUP: overview
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#+LANGUAGE: en
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#+EMAIL: dehaeze.thomas@gmail.com
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#+AUTHOR: Dehaeze Thomas
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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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#+PROPERTY: header-args:latex :headers '("\\usepackage{tikz}" "\\usepackage{import}" "\\import{$HOME/Cloud/thesis/latex/}{config.tex}")
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#+PROPERTY: header-args:latex+ :imagemagick t :fit yes
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#+PROPERTY: header-args:latex+ :iminoptions -scale 100% -density 150
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#+PROPERTY: header-args:latex+ :imoutoptions -quality 100
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#+PROPERTY: header-args:latex+ :results file raw replace
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#+PROPERTY: header-args:latex+ :buffer no
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#+PROPERTY: header-args:latex+ :eval no-export
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#+PROPERTY: header-args:latex+ :exports results
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#+PROPERTY: header-args:latex+ :mkdirp yes
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#+PROPERTY: header-args:latex+ :output-dir figs
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#+PROPERTY: header-args:latex+ :post pdf2svg(file=*this*, ext="png")
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:END:
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* First Control Architecture
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** Matlab Init :noexport:
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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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** Control Schematic
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#+begin_src latex :file control_measure_rotating_2dof.pdf
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\begin{tikzpicture}
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% Blocs
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\node[block] (J) at (0, 0) {$J$};
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\node[addb={+}{}{}{}{-}, right=1 of J] (subr) {};
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\node[block, right=0.8 of subr] (K) {$K_{L}$};
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\node[block, right=1 of K] (G) {$G_{L}$};
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% Connections and labels
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\draw[<-] (J.west)node[above left]{$\bm{r}_{n}$} -- ++(-1, 0);
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\draw[->] (J.east) -- (subr.west) node[above left]{$\bm{r}_{L}$};
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\draw[->] (subr.east) -- (K.west) node[above left]{$\bm{\epsilon}_{L}$};
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\draw[->] (K.east) -- (G.west) node[above left]{$\bm{\tau}$};
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\draw[->] (G.east) node[above right]{$\bm{L}$} -| ($(G.east)+(1, -1)$) -| (subr.south);
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\end{tikzpicture}
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#+end_src
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#+RESULTS:
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[[file:figs/control_measure_rotating_2dof.png]]
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** Initialize the Stewart platform
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#+begin_src matlab
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stewart = initializeStewartPlatform();
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stewart = initializeFramesPositions(stewart, '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, 'type_F', 'universal_p', 'type_M', 'spherical_p');
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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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stewart = initializeInertialSensor(stewart, 'type', 'accelerometer', 'freq', 5e3);
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#+end_src
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#+begin_src matlab
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ground = initializeGround('type', 'none');
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payload = initializePayload('type', 'none');
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#+end_src
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** Identification of the plant
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Let's identify the transfer function from $\bm{\tau}$ to $\bm{L}$.
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#+begin_src matlab
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%% Name of the Simulink File
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mdl = 'stewart_platform_model';
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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, '/Controller'], 1, 'openinput'); io_i = io_i + 1; % Actuator Force Inputs [N]
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io(io_i) = linio([mdl, '/Stewart Platform'], 1, 'openoutput', [], 'dLm'); io_i = io_i + 1; % Relative Displacement Outputs [m]
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%% Run the linearization
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G = linearize(mdl, io);
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G.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
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G.OutputName = {'L1', 'L2', 'L3', 'L4', 'L5', 'L6'};
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#+end_src
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** Plant Analysis
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Diagonal terms
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#+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);
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hold on;
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for i = 1:6
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plot(freqs, abs(squeeze(freqresp(G(i, i), freqs, 'Hz'))));
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end
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Amplitude [m/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 = 1:6
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plot(freqs, 180/pi*angle(squeeze(freqresp(G(i, i), freqs, 'Hz'))));
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end
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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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linkaxes([ax1,ax2],'x');
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#+end_src
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Compare to off-diagonal terms
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#+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);
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hold on;
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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
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set(gca,'ColorOrderIndex',1);
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plot(freqs, abs(squeeze(freqresp(G(1, 1), 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 [m/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 = 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
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set(gca,'ColorOrderIndex',1);
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plot(freqs, 180/pi*angle(squeeze(freqresp(G(1, 1), 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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linkaxes([ax1,ax2],'x');
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#+end_src
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** Controller Design
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One integrator should be present in the controller.
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A lead is added around the crossover frequency which is set to be around 500Hz.
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#+begin_src matlab
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% wint = 2*pi*100; % Integrate until [rad]
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% wlead = 2*pi*500; % Location of the lead [rad]
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% hlead = 2; % Lead strengh
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% Kl = 1e6 * ... % Gain
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% (s + wint)/(s) * ... % Integrator until 100Hz
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% (1 + s/(wlead/hlead)/(1 + s/(wlead*hlead))); % Lead
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wc = 2*pi*100;
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Kl = 1/abs(freqresp(G(1,1), wc)) * wc/s * 1/(1 + s/(3*wc));
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Kl = Kl * eye(6);
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#+end_src
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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(Kl(1,1)*G(1, 1), 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 [m/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(Kl(1,1)*G(1, 1), 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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linkaxes([ax1,ax2],'x');
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#+end_src
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#+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);
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hold on;
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for i = 1:5
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for j = i+1:6
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plot(freqs, abs(squeeze(freqresp(Kl(i,i)*G(i, j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2]);
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end
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end
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set(gca,'ColorOrderIndex',1);
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plot(freqs, abs(squeeze(freqresp(Kl(1,1)*G(1, 1), 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 [m/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 = 1:5
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for j = i+1:6
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plot(freqs, 180/pi*angle(squeeze(freqresp(Kl(i, i)*G(i, j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2]);
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end
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end
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set(gca,'ColorOrderIndex',1);
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plot(freqs, 180/pi*angle(squeeze(freqresp(Kl(1,1)*G(1, 1), 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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linkaxes([ax1,ax2],'x');
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#+end_src
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