Add flexible elements analysis
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docs/figs/flexible_elements_effect_dvf.pdf
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docs/figs/flexible_elements_effect_dvf.pdf
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docs/figs/flexible_elements_effect_dvf.png
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docs/figs/flexible_elements_effect_iff.pdf
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docs/figs/flexible_elements_effect_iff.pdf
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docs/figs/flexible_elements_effect_iff.png
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mat/APA300ML.STEP
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mat/APA300ML.STEP
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mat/APA300ML.mat
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mat/APA300ML.mat
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mat/flexor_ID16.STEP
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mat/flexor_ID16.STEP
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mat/flexor_ID16.mat
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mat/flexor_ID16.mat
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org/flexible-stewart-platform.org
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org/flexible-stewart-platform.org
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#+TITLE: Stewart Platform with Flexible Elements
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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/tikz/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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* Simscape Model
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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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** Flexible APA
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#+begin_src matlab
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apa = load('./mat/APA300ML.mat', 'int_xyz', 'int_i', 'n_xyz', 'n_i', 'nodes', 'M', 'K');
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#+end_src
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#+begin_src matlab :exports results :results value table replace :tangle no
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data2orgtable([length(apa.n_i); length(apa.int_i); size(apa.M,1) - 6*length(apa.int_i); size(apa.M,1)], {'Total number of Nodes', 'Number of interface Nodes', 'Number of Modes', 'Size of M and K matrices'}, {}, ' %.0f ');
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#+end_src
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#+RESULTS:
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| Total number of Nodes | 7 |
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| Number of interface Nodes | 7 |
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| Number of Modes | 6 |
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| Size of M and K matrices | 48 |
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#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
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data2orgtable([[1:length(apa.int_i)]', apa.int_i, apa.int_xyz], {}, {'Node i', 'Node Number', 'x [m]', 'y [m]', 'z [m]'}, ' %f ');
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#+end_src
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#+caption: Coordinates of the interface nodes
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#+RESULTS:
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| Node i | Node Number | x [m] | y [m] | z [m] |
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|--------+-------------+---------+--------+-------|
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| 1.0 | 53917.0 | 0.0 | -0.015 | 0.0 |
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| 2.0 | 53918.0 | 0.0 | 0.015 | 0.0 |
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| 3.0 | 53919.0 | -0.0325 | 0.0 | 0.0 |
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| 4.0 | 53920.0 | -0.0125 | 0.0 | 0.0 |
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| 5.0 | 53921.0 | -0.0075 | 0.0 | 0.0 |
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| 6.0 | 53922.0 | 0.0125 | 0.0 | 0.0 |
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| 7.0 | 53923.0 | 0.0325 | 0.0 | 0.0 |
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** Flexible Joint
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#+begin_src matlab
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flex_joint = load('./mat/flexor_ID16.mat', 'int_xyz', 'int_i', 'n_xyz', 'n_i', 'nodes', 'M', 'K');
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#+end_src
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#+begin_src matlab :exports results :results value table replace :tangle no
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data2orgtable([length(flex_joint.n_i); length(flex_joint.int_i); size(flex_joint.M,1) - 6*length(flex_joint.int_i); size(flex_joint.M,1)], {'Total number of Nodes', 'Number of interface Nodes', 'Number of Modes', 'Size of M and K matrices'}, {}, ' %.0f ');
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#+end_src
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#+RESULTS:
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| Total number of Nodes | 2 |
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| Number of interface Nodes | 2 |
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| Number of Modes | 6 |
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| Size of M and K matrices | 18 |
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#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
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data2orgtable([[1:length(flex_joint.int_i)]', flex_joint.int_i, flex_joint.int_xyz], {}, {'Node i', 'Node Number', 'x [m]', 'y [m]', 'z [m]'}, ' %f ');
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#+end_src
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#+caption: Coordinates of the interface nodes
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#+RESULTS:
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| Node i | Node Number | x [m] | y [m] | z [m] |
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|--------+-------------+-------+-------+-------|
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| 1.0 | 181278.0 | 0.0 | 0.0 | 0.0 |
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| 2.0 | 181279.0 | 0.0 | 0.0 | -0.0 |
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#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
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data2orgtable([1e-6*flex_joint.K(3,3), flex_joint.K(4,4), flex_joint.K(5,5), flex_joint.K(6,6)]', {'Axial Stiffness [N/um]', 'Bending Stiffness [Nm/rad]', 'Bending Stiffness [Nm/rad]', 'Torsion Stiffness [Nm/rad]'}, {'*Caracteristic*', '*Value*'}, ' %0.f ');
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#+end_src
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#+RESULTS:
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| *Caracteristic* | *Value* |
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|----------------------------+---------|
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| Axial Stiffness [N/um] | 119 |
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| Bending Stiffness [Nm/rad] | 33 |
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| Bending Stiffness [Nm/rad] | 33 |
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| Torsion Stiffness [Nm/rad] | 236 |
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** Identification
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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_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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io(io_i) = linio([mdl, '/Stewart Platform'], 1, 'openoutput', [], 'Taum'); io_i = io_i + 1; % Force Sensors [N]
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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', 'rigid', 'm', 50);
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controller = initializeController('type', 'open-loop');
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#+end_src
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#+begin_src matlab
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disturbances = initializeDisturbances();
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references = initializeReferences(stewart);
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#+end_src
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** No Flexible Elements
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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, 'K', 1.8e6*ones(6,1));
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stewart = initializeAmplifiedStrutDynamics(stewart, 'Kr', 0.9e6*ones(6,1), 'Ka', 0.9e6*ones(6,1));
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stewart = initializeJointDynamics(stewart, 'Kf_M', 33*ones(6,1), 'Kt_M', 235*ones(6,1), 'Kf_F', 33*ones(6,1), 'Kt_F', 235*ones(6,1));
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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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%% 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 = {'Dm1', 'Dm2', 'Dm3', 'Dm4', 'Dm5', 'Dm6', 'Fm1', 'Fm2', 'Fm3', 'Fm4', 'Fm5', 'Fm6'};
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#+end_src
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** Flexible joints
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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 = initializeAmplifiedStrutDynamics(stewart, 'Kr', 0.9e6*ones(6,1), 'Ka', 0.9e6*ones(6,1));
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stewart = initializeJointDynamics(stewart, 'type_F', 'flexible', 'K_F', flex_joint.K, 'M_F', flex_joint.M, 'n_xyz_F', flex_joint.n_xyz, 'xi_F', 0.1, 'step_file_F', 'mat/flexor_ID16.STEP', 'type_M', 'flexible', 'K_M', flex_joint.K, 'M_M', flex_joint.M, 'n_xyz_M', flex_joint.n_xyz, 'xi_M', 0.1, 'step_file_M', 'mat/flexor_ID16.STEP');
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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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%% Run the linearization
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Gj = linearize(mdl, io, options);
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Gj.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
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Gj.OutputName = {'Dm1', 'Dm2', 'Dm3', 'Dm4', 'Dm5', 'Dm6', 'Fm1', 'Fm2', 'Fm3', 'Fm4', 'Fm5', 'Fm6'};
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#+end_src
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** Flexible APA
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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 = initializeFlexibleStrutDynamics(stewart, 'H', 0.03, 'K', apa.K, 'M', apa.M, 'n_xyz', apa.n_xyz, 'xi', 0.1, 'step_file', 'mat/APA300ML.STEP');
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stewart = initializeJointDynamics(stewart, 'Kf_M', 33*ones(6,1), 'Kt_M', 235, 'Kf_F', 33*ones(6,1), 'Kt_F', 235);
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stewart = initializeCylindricalPlatforms(stewart);
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stewart = initializeCylindricalStruts(stewart, 'type_F', 'none', 'type_M', 'none');
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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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%% Run the linearization
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Ga = -linearize(mdl, io, options);
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Ga.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
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Ga.OutputName = {'Dm1', 'Dm2', 'Dm3', 'Dm4', 'Dm5', 'Dm6', 'Fm1', 'Fm2', 'Fm3', 'Fm4', 'Fm5', 'Fm6'};
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#+end_src
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** Flexible Joints and APA
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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 = initializeFlexibleStrutDynamics(stewart, 'H', 0.03, 'K', apa.K, 'M', apa.M, 'n_xyz', apa.n_xyz, 'xi', 0.1, 'step_file', 'mat/APA300ML.STEP');
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stewart = initializeJointDynamics(stewart, 'type_F', 'flexible', 'K_F', flex_joint.K, 'M_F', flex_joint.M, 'n_xyz_F', flex_joint.n_xyz, 'xi_F', 0.1, 'step_file_F', 'mat/flexor_ID16.STEP', 'type_M', 'flexible', 'K_M', flex_joint.K, 'M_M', flex_joint.M, 'n_xyz_M', flex_joint.n_xyz, 'xi_M', 0.1, 'step_file_M', 'mat/flexor_ID16.STEP');
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stewart = initializeCylindricalPlatforms(stewart);
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stewart = initializeCylindricalStruts(stewart, 'type_F', 'none', 'type_M', 'none');
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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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Gf = -linearize(mdl, io, options);
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Gf.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
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Gf.OutputName = {'Dm1', 'Dm2', 'Dm3', 'Dm4', 'Dm5', 'Dm6', 'Fm1', 'Fm2', 'Fm3', 'Fm4', 'Fm5', 'Fm6'};
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#+end_src
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** Direct Velocity Feedback
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#+begin_src matlab :exports none
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freqs = logspace(0, 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 = 1:6
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plot(freqs, abs(squeeze(freqresp(G(i,i), freqs, 'Hz'))), 'color', [0 0.4470 0.7410 0.2]);
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end
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for i = 1:6
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plot(freqs, abs(squeeze(freqresp(Gj(i,i), freqs, 'Hz'))), 'color', [0.8500 0.3250 0.0980 0.2]);
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end
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for i = 1:6
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plot(freqs, abs(squeeze(freqresp(44*Ga(i,i), freqs, 'Hz'))), 'color', [0.9290 0.6940 0.1250 0.2]);
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end
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for i = 1:6
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plot(freqs, abs(squeeze(freqresp(44*Gf(i,i), freqs, 'Hz'))), 'color', [0.4940 0.1840 0.5560 0.2]);
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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'))), 'color', [0 0.4470 0.7410 0.2]);
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end
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for i = 1:6
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plot(freqs, 180/pi*angle(squeeze(freqresp(Gj(i,i), freqs, 'Hz'))), 'color', [0.8500 0.3250 0.0980 0.2]);
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end
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for i = 1:6
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plot(freqs, 180/pi*angle(squeeze(freqresp(Ga(i,i), freqs, 'Hz'))), 'color', [0.9290 0.6940 0.1250 0.2]);
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end
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for i = 1:6
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plot(freqs, 180/pi*angle(squeeze(freqresp(Gf(i,i), freqs, 'Hz'))), 'color', [0.4940 0.1840 0.5560 0.2]);
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end
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h = zeros(4, 1);
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h(1) = plot(NaN, NaN, 'color', [0 0.4470 0.7410 0.2]);
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h(2) = plot(NaN, NaN, 'color', [0.8500 0.3250 0.0980 0.2]);
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h(3) = plot(NaN, NaN, 'color', [0.9290 0.6940 0.1250 0.2]);
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h(4) = plot(NaN, NaN, 'color', [0.4940 0.1840 0.5560 0.2]);
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legend(h, 'No flexible', 'Flexible Joints', 'Flexible APA', 'All Flexible');
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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 :tangle no :exports results :results file replace
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exportFig('figs/flexible_elements_effect_dvf.pdf', 'width', 'full', 'height', 'full');
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#+end_src
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#+name: fig:flexible_elements_effect_dvf
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#+caption: Change of the DVF plant dynamics with the added flexible elements
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#+RESULTS:
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[[file:figs/flexible_elements_effect_dvf.png]]
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** Integral Force Feedback
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#+begin_src matlab :exports none
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freqs = logspace(0, 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 = 1:6
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plot(freqs, abs(squeeze(freqresp(G(6+i,i), freqs, 'Hz'))), 'color', [0 0.4470 0.7410 0.2]);
|
||||
end
|
||||
for i = 1:6
|
||||
plot(freqs, abs(squeeze(freqresp(Gj(6+i,i), freqs, 'Hz'))), 'color', [0.8500 0.3250 0.0980 0.2]);
|
||||
end
|
||||
for i = 1:6
|
||||
plot(freqs, abs(squeeze(freqresp(1e9*Ga(6+i,i), freqs, 'Hz'))), 'color', [0.9290 0.6940 0.1250 0.2]);
|
||||
end
|
||||
for i = 1:6
|
||||
plot(freqs, abs(squeeze(freqresp(1e9*Gf(6+i,i), freqs, 'Hz'))), 'color', [0.4940 0.1840 0.5560 0.2]);
|
||||
end
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
|
||||
|
||||
ax2 = subplot(2, 1, 2);
|
||||
hold on;
|
||||
for i = 1:6
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(G(6+i,i), freqs, 'Hz'))), 'color', [0 0.4470 0.7410 0.2]);
|
||||
end
|
||||
for i = 1:6
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(Gj(6+i,i), freqs, 'Hz'))), 'color', [0.8500 0.3250 0.0980 0.2]);
|
||||
end
|
||||
for i = 1:6
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(Ga(6+i,i), freqs, 'Hz'))), 'color', [0.9290 0.6940 0.1250 0.2]);
|
||||
end
|
||||
for i = 1:6
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(Gf(6+i,i), freqs, 'Hz'))), 'color', [0.4940 0.1840 0.5560 0.2]);
|
||||
end
|
||||
h = zeros(4, 1);
|
||||
h(1) = plot(NaN, NaN, 'color', [0 0.4470 0.7410 0.2]);
|
||||
h(2) = plot(NaN, NaN, 'color', [0.8500 0.3250 0.0980 0.2]);
|
||||
h(3) = plot(NaN, NaN, 'color', [0.9290 0.6940 0.1250 0.2]);
|
||||
h(4) = plot(NaN, NaN, 'color', [0.4940 0.1840 0.5560 0.2]);
|
||||
legend(h, 'No flexible', 'Flexible Joints', 'Flexible APA', 'All Flexible');
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
||||
ylim([-180, 180]);
|
||||
yticks([-180, -90, 0, 90, 180]);
|
||||
|
||||
linkaxes([ax1,ax2],'x');
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :tangle no :exports results :results file replace
|
||||
exportFig('figs/flexible_elements_effect_iff.pdf', 'width', 'full', 'height', 'full');
|
||||
#+end_src
|
||||
|
||||
#+name: fig:flexible_elements_effect_iff
|
||||
#+caption: Change of the IFF plant dynamics with the added flexible elements
|
||||
#+RESULTS:
|
||||
[[file:figs/flexible_elements_effect_iff.png]]
|
||||
38
src/initializeFlexibleStrutDynamics.m
Normal file
38
src/initializeFlexibleStrutDynamics.m
Normal file
@ -0,0 +1,38 @@
|
||||
function [stewart] = initializeFlexibleStrutDynamics(stewart, args)
|
||||
% initializeFlexibleStrutDynamics - Add Stiffness and Damping properties of each strut
|
||||
%
|
||||
% Syntax: [stewart] = initializeFlexibleStrutDynamics(args)
|
||||
%
|
||||
% Inputs:
|
||||
% - args - Structure with the following fields:
|
||||
% - K [nxn] - Vertical stiffness contribution of the piezoelectric stack [N/m]
|
||||
% - M [nxn] - Vertical damping contribution of the piezoelectric stack [N/(m/s)]
|
||||
% - xi [1x1] - Vertical (residual) stiffness when the piezoelectric stack is removed [N/m]
|
||||
% - step_file [6x1] - Vertical (residual) damping when the piezoelectric stack is removed [N/(m/s)]
|
||||
%
|
||||
% Outputs:
|
||||
% - stewart - updated Stewart structure with the added fields:
|
||||
|
||||
arguments
|
||||
stewart
|
||||
args.K double {mustBeNumeric} = zeros(6,6)
|
||||
args.M double {mustBeNumeric} = zeros(6,6)
|
||||
args.H double {mustBeNumeric} = 0
|
||||
args.n_xyz double {mustBeNumeric} = zeros(2,3)
|
||||
args.xi double {mustBeNumeric} = 0.1
|
||||
args.step_file char {} = ''
|
||||
end
|
||||
|
||||
stewart.actuators.ax_off = (stewart.geometry.l(1) - args.H)/2; % Axial Offset at the ends of the actuator
|
||||
|
||||
stewart.actuators.type = 3;
|
||||
|
||||
stewart.actuators.Km = args.K;
|
||||
stewart.actuators.Mm = args.M;
|
||||
|
||||
stewart.actuators.n_xyz = args.n_xyz;
|
||||
stewart.actuators.xi = args.xi;
|
||||
|
||||
stewart.actuators.step_file = args.step_file;
|
||||
|
||||
stewart.actuators.K = args.K(3,3); % Axial Stiffness
|
||||
Loading…
Reference in New Issue
Block a user