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