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< title > Stewart Platform - Vibration Isolation< / title >
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< meta name = "author" content = "Dehaeze Thomas" / >
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< h1 class = "title" > Stewart Platform - Vibration Isolation< / h1 >
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< div id = "table-of-contents" >
< h2 > Table of Contents< / h2 >
< div id = "text-table-of-contents" >
< ul >
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< li > < a href = "#org4b4dce5" > 1. HAC-LAC (Cascade) Control - Integral Control< / a >
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< ul >
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< li > < a href = "#org9a6463e" > 1.1. Introduction< / a > < / li >
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< li > < a href = "#org2308492" > 1.2. Initialization< / a > < / li >
< li > < a href = "#orgb5e88a6" > 1.3. Identification< / a >
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< ul >
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< li > < a href = "#org7e602f6" > 1.3.1. HAC - Without LAC< / a > < / li >
< li > < a href = "#org49dd47c" > 1.3.2. HAC - IFF< / a > < / li >
< li > < a href = "#orgc4bf514" > 1.3.3. HAC - DVF< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#org68ac3ce" > 1.4. Control Architecture< / a > < / li >
< li > < a href = "#org668a952" > 1.5. 6x6 Plant Comparison< / a > < / li >
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< li > < a href = "#orgc2cdba3" > 1.6. HAC - DVF< / a >
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< ul >
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< li > < a href = "#orgff953e4" > 1.6.1. Plant< / a > < / li >
< li > < a href = "#orgbd635c1" > 1.6.2. Controller Design< / a > < / li >
< li > < a href = "#org83c15a9" > 1.6.3. Obtained Performance< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#org5dfa0fd" > 1.7. HAC - IFF< / a >
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< ul >
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< li > < a href = "#orgeedcb6b" > 1.7.1. Plant< / a > < / li >
< li > < a href = "#orgd723b75" > 1.7.2. Controller Design< / a > < / li >
< li > < a href = "#org54813bf" > 1.7.3. Obtained Performance< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#org8e15485" > 1.8. Comparison< / a > < / li >
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< li > < a href = "#orgdc1bcf2" > 2. MIMO Analysis< / a >
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< ul >
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< li > < a href = "#org7fd66be" > 2.1. Initialization< / a > < / li >
< li > < a href = "#org06a70cd" > 2.2. Identification< / a >
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< ul >
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< li > < a href = "#org88e17f4" > 2.2.1. HAC - Without LAC< / a > < / li >
< li > < a href = "#orgb75c6b7" > 2.2.2. HAC - DVF< / a > < / li >
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< li > < a href = "#orgba8c7bf" > 2.2.3. Cartesian Frame< / a > < / li >
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< li > < a href = "#orgf9d0420" > 2.3. Singular Value Decomposition< / a > < / li >
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< li > < a href = "#orga095fa8" > 3. Diagonal Control based on the damped plant< / a >
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< ul >
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< li > < a href = "#orge901603" > 3.1. Initialization< / a > < / li >
< li > < a href = "#orgce73f0c" > 3.2. Identification< / a > < / li >
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< li > < a href = "#orgb5a063b" > 3.3. Steady State Decoupling< / a >
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< ul >
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< li > < a href = "#orgd0ce552" > 3.3.1. Pre-Compensator Design< / a > < / li >
< li > < a href = "#org41b76c6" > 3.3.2. Diagonal Control Design< / a > < / li >
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< li > < a href = "#org3228759" > 3.3.3. Results< / a > < / li >
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< / li >
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< li > < a href = "#orgb53dd48" > 3.4. Decoupling at Crossover< / a > < / li >
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< / li >
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< li > < a href = "#org639412c" > 4. Time Domain Simulation< / a >
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< ul >
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< li > < a href = "#orgc332811" > 4.1. Initialization< / a > < / li >
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< li > < a href = "#orgc72e6b5" > 4.2. HAC IFF< / a > < / li >
< li > < a href = "#org757f9e9" > 4.3. HAC-DVF< / a > < / li >
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< li > < a href = "#org620278d" > 4.4. Results< / a > < / li >
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< li > < a href = "#org6a9c87c" > 5. Functions< / a >
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< ul >
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< li > < a href = "#orgd1492e7" > 5.1. < code > initializeController< / code > : Initialize the Controller< / a >
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< ul >
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< li > < a href = "#orgae5eed0" > Function description< / a > < / li >
< li > < a href = "#orgda07b57" > Optional Parameters< / a > < / li >
< li > < a href = "#orgdb009ab" > Structure initialization< / a > < / li >
< li > < a href = "#org056c578" > Add Type< / a > < / li >
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< h2 id = "org4b4dce5" > < span class = "section-number-2" > 1< / span > HAC-LAC (Cascade) Control - Integral Control< / h2 >
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< h3 id = "org9a6463e" > < span class = "section-number-3" > 1.1< / span > Introduction< / h3 >
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< div class = "outline-text-3" id = "text-1-1" >
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< p >
In this section, we wish to study the use of the High Authority Control - Low Authority Control (HAC-LAC) architecture on the Stewart platform.
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< p >
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The control architectures are shown in Figures < a href = "#org000f34d" > 1< / a > and < a href = "#orgeaed076" > 2< / a > .
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< / p >
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< p >
First, the LAC loop is closed (the LAC control is described < a href = "active-damping.html" > here< / a > ), and then the HAC controller is designed and the outer loop is closed.
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< / p >
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< div id = "org000f34d" class = "figure" >
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< p > < img src = "figs/control_arch_hac_iff.png" alt = "control_arch_hac_iff.png" / >
< / p >
< p > < span class = "figure-number" > Figure 1: < / span > HAC-LAC architecture with IFF< / p >
< / div >
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< div id = "orgeaed076" class = "figure" >
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< p > < img src = "figs/control_arch_hac_dvf.png" alt = "control_arch_hac_dvf.png" / >
< / p >
< p > < span class = "figure-number" > Figure 2: < / span > HAC-LAC architecture with DVF< / p >
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< h3 id = "org2308492" > < span class = "section-number-3" > 1.2< / span > Initialization< / h3 >
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< div class = "outline-text-3" id = "text-1-2" >
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< p >
We first initialize the Stewart platform.
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > stewart = initializeStewartPlatform();
stewart = initializeFramesPositions(stewart, < span class = "org-string" > 'H'< / span > , 90e< span class = "org-type" > -< / span > 3, < span class = "org-string" > 'MO_B'< / span > , 45e< span class = "org-type" > -< / span > 3);
stewart = generateGeneralConfiguration(stewart);
stewart = computeJointsPose(stewart);
stewart = initializeStrutDynamics(stewart);
stewart = initializeJointDynamics(stewart, < span class = "org-string" > 'type_F'< / span > , < span class = "org-string" > 'universal'< / span > , < span class = "org-string" > 'type_M'< / span > , < span class = "org-string" > 'spherical'< / span > );
stewart = initializeCylindricalPlatforms(stewart);
stewart = initializeCylindricalStruts(stewart);
stewart = computeJacobian(stewart);
stewart = initializeStewartPose(stewart);
stewart = initializeInertialSensor(stewart, < span class = "org-string" > 'type'< / span > , < span class = "org-string" > 'none'< / span > );
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< p >
The rotation point of the ground is located at the origin of frame \(\{A\}\).
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > ground = initializeGround(< span class = "org-string" > 'type'< / span > , < span class = "org-string" > 'rigid'< / span > , < span class = "org-string" > 'rot_point'< / span > , stewart.platform_F.FO_A);
payload = initializePayload(< span class = "org-string" > 'type'< / span > , < span class = "org-string" > 'none'< / span > );
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< h3 id = "orgb5e88a6" > < span class = "section-number-3" > 1.3< / span > Identification< / h3 >
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< div class = "outline-text-3" id = "text-1-3" >
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< p >
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We identify the transfer function from the actuator forces \(\bm{\tau}\) to the absolute displacement of the mobile platform \(\bm{\mathcal{X}}\) in three different cases:
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< / p >
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< ul class = "org-ul" >
< li > Open Loop plant< / li >
< li > Already damped plant using Integral Force Feedback< / li >
< li > Already damped plant using Direct velocity feedback< / li >
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< h4 id = "org7e602f6" > < span class = "section-number-4" > 1.3.1< / span > HAC - Without LAC< / h4 >
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< pre class = "src src-matlab" > controller = initializeController(< span class = "org-string" > 'type'< / span > , < span class = "org-string" > 'open-loop'< / span > );
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< pre class = "src src-matlab" > < span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Name of the Simulink File< / span > < / span >
mdl = < span class = "org-string" > 'stewart_platform_model'< / span > ;
< span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Input/Output definition< / span > < / span >
clear io; io_i = 1;
io(io_i) = linio([mdl, < span class = "org-string" > '/Controller'< / span > ], 1, < span class = "org-string" > 'input'< / span > ); io_i = io_i < span class = "org-type" > +< / span > 1; < span class = "org-comment" > % Actuator Force Inputs [N]< / span >
io(io_i) = linio([mdl, < span class = "org-string" > '/Absolute Motion Sensor'< / span > ], 1, < span class = "org-string" > 'openoutput'< / span > ); io_i = io_i < span class = "org-type" > +< / span > 1; < span class = "org-comment" > % Absolute Sensor [m, rad]< / span >
< span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Run the linearization< / span > < / span >
G_ol = linearize(mdl, io);
G_ol.InputName = {< span class = "org-string" > 'F1'< / span > , < span class = "org-string" > 'F2'< / span > , < span class = "org-string" > 'F3'< / span > , < span class = "org-string" > 'F4'< / span > , < span class = "org-string" > 'F5'< / span > , < span class = "org-string" > 'F6'< / span > };
G_ol.OutputName = {< span class = "org-string" > 'Dx'< / span > , < span class = "org-string" > 'Dy'< / span > , < span class = "org-string" > 'Dz'< / span > , < span class = "org-string" > 'Rx'< / span > , < span class = "org-string" > 'Ry'< / span > , < span class = "org-string" > 'Rz'< / span > };
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< h4 id = "org49dd47c" > < span class = "section-number-4" > 1.3.2< / span > HAC - IFF< / h4 >
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< div class = "outline-text-4" id = "text-1-3-2" >
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< pre class = "src src-matlab" > controller = initializeController(< span class = "org-string" > 'type'< / span > , < span class = "org-string" > 'iff'< / span > );
K_iff = < span class = "org-type" > -< / span > (1e4< span class = "org-type" > /< / span > s)< span class = "org-type" > *< / span > eye(6);
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< / pre >
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< pre class = "src src-matlab" > < span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Name of the Simulink File< / span > < / span >
mdl = < span class = "org-string" > 'stewart_platform_model'< / span > ;
< span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Input/Output definition< / span > < / span >
clear io; io_i = 1;
io(io_i) = linio([mdl, < span class = "org-string" > '/Controller'< / span > ], 1, < span class = "org-string" > 'input'< / span > ); io_i = io_i < span class = "org-type" > +< / span > 1; < span class = "org-comment" > % Actuator Force Inputs [N]< / span >
io(io_i) = linio([mdl, < span class = "org-string" > '/Absolute Motion Sensor'< / span > ], 1, < span class = "org-string" > 'openoutput'< / span > ); io_i = io_i < span class = "org-type" > +< / span > 1; < span class = "org-comment" > % Absolute Sensor [m, rad]< / span >
< span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Run the linearization< / span > < / span >
G_iff = linearize(mdl, io);
G_iff.InputName = {< span class = "org-string" > 'F1'< / span > , < span class = "org-string" > 'F2'< / span > , < span class = "org-string" > 'F3'< / span > , < span class = "org-string" > 'F4'< / span > , < span class = "org-string" > 'F5'< / span > , < span class = "org-string" > 'F6'< / span > };
G_iff.OutputName = {< span class = "org-string" > 'Dx'< / span > , < span class = "org-string" > 'Dy'< / span > , < span class = "org-string" > 'Dz'< / span > , < span class = "org-string" > 'Rx'< / span > , < span class = "org-string" > 'Ry'< / span > , < span class = "org-string" > 'Rz'< / span > };
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< div id = "outline-container-orgc4bf514" class = "outline-4" >
< h4 id = "orgc4bf514" > < span class = "section-number-4" > 1.3.3< / span > HAC - DVF< / h4 >
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< pre class = "src src-matlab" > controller = initializeController(< span class = "org-string" > 'type'< / span > , < span class = "org-string" > 'dvf'< / span > );
K_dvf = < span class = "org-type" > -< / span > 1e4< span class = "org-type" > *< / span > s< span class = "org-type" > /< / span > (1< span class = "org-type" > +< / span > s< span class = "org-type" > /< / span > 2< span class = "org-type" > /< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > /< / span > 5000)< span class = "org-type" > *< / span > eye(6);
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< / pre >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > < span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Name of the Simulink File< / span > < / span >
mdl = < span class = "org-string" > 'stewart_platform_model'< / span > ;
< span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Input/Output definition< / span > < / span >
clear io; io_i = 1;
io(io_i) = linio([mdl, < span class = "org-string" > '/Controller'< / span > ], 1, < span class = "org-string" > 'input'< / span > ); io_i = io_i < span class = "org-type" > +< / span > 1; < span class = "org-comment" > % Actuator Force Inputs [N]< / span >
io(io_i) = linio([mdl, < span class = "org-string" > '/Absolute Motion Sensor'< / span > ], 1, < span class = "org-string" > 'openoutput'< / span > ); io_i = io_i < span class = "org-type" > +< / span > 1; < span class = "org-comment" > % Absolute Sensor [m, rad]< / span >
< span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Run the linearization< / span > < / span >
G_dvf = linearize(mdl, io);
G_dvf.InputName = {< span class = "org-string" > 'F1'< / span > , < span class = "org-string" > 'F2'< / span > , < span class = "org-string" > 'F3'< / span > , < span class = "org-string" > 'F4'< / span > , < span class = "org-string" > 'F5'< / span > , < span class = "org-string" > 'F6'< / span > };
G_dvf.OutputName = {< span class = "org-string" > 'Dx'< / span > , < span class = "org-string" > 'Dy'< / span > , < span class = "org-string" > 'Dz'< / span > , < span class = "org-string" > 'Rx'< / span > , < span class = "org-string" > 'Ry'< / span > , < span class = "org-string" > 'Rz'< / span > };
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< / pre >
< / div >
< / div >
< / div >
< / div >
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< div id = "outline-container-org68ac3ce" class = "outline-3" >
< h3 id = "org68ac3ce" > < span class = "section-number-3" > 1.4< / span > Control Architecture< / h3 >
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< div class = "outline-text-3" id = "text-1-4" >
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< p >
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We use the Jacobian to express the actuator forces in the cartesian frame, and thus we obtain the transfer functions from \(\bm{\mathcal{F}}\) to \(\bm{\mathcal{X}}\).
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > Gc_ol = minreal(G_ol)< span class = "org-type" > /< / span > stewart.kinematics.J< span class = "org-type" > '< / span > ;
Gc_ol.InputName = {< span class = "org-string" > 'Fx'< / span > , < span class = "org-string" > 'Fy'< / span > , < span class = "org-string" > 'Fz'< / span > , < span class = "org-string" > 'Mx'< / span > , < span class = "org-string" > 'My'< / span > , < span class = "org-string" > 'Mz'< / span > };
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Gc_iff = minreal(G_iff)< span class = "org-type" > /< / span > stewart.kinematics.J< span class = "org-type" > '< / span > ;
Gc_iff.InputName = {< span class = "org-string" > 'Fx'< / span > , < span class = "org-string" > 'Fy'< / span > , < span class = "org-string" > 'Fz'< / span > , < span class = "org-string" > 'Mx'< / span > , < span class = "org-string" > 'My'< / span > , < span class = "org-string" > 'Mz'< / span > };
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Gc_dvf = minreal(G_dvf)< span class = "org-type" > /< / span > stewart.kinematics.J< span class = "org-type" > '< / span > ;
Gc_dvf.InputName = {< span class = "org-string" > 'Fx'< / span > , < span class = "org-string" > 'Fy'< / span > , < span class = "org-string" > 'Fz'< / span > , < span class = "org-string" > 'Mx'< / span > , < span class = "org-string" > 'My'< / span > , < span class = "org-string" > 'Mz'< / span > };
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< / pre >
< / div >
< p >
We then design a controller based on the transfer functions from \(\bm{\mathcal{F}}\) to \(\bm{\mathcal{X}}\), finally, we will pre-multiply the controller by \(\bm{J}^{-T}\).
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< / p >
< / div >
< / div >
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< div id = "outline-container-org668a952" class = "outline-3" >
< h3 id = "org668a952" > < span class = "section-number-3" > 1.5< / span > 6x6 Plant Comparison< / h3 >
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< div class = "outline-text-3" id = "text-1-5" >
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< div id = "orgcf3a190" class = "figure" >
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< p > < img src = "figs/hac_lac_coupling_jacobian.png" alt = "hac_lac_coupling_jacobian.png" / >
< / p >
< p > < span class = "figure-number" > Figure 3: < / span > Norm of the transfer functions from \(\bm{\mathcal{F}}\) to \(\bm{\mathcal{X}}\) (< a href = "./figs/hac_lac_coupling_jacobian.png" > png< / a > , < a href = "./figs/hac_lac_coupling_jacobian.pdf" > pdf< / a > )< / p >
< / div >
< / div >
< / div >
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< div id = "outline-container-orgc2cdba3" class = "outline-3" >
< h3 id = "orgc2cdba3" > < span class = "section-number-3" > 1.6< / span > HAC - DVF< / h3 >
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< div class = "outline-text-3" id = "text-1-6" >
< / div >
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< div id = "outline-container-orgff953e4" class = "outline-4" >
< h4 id = "orgff953e4" > < span class = "section-number-4" > 1.6.1< / span > Plant< / h4 >
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< div class = "outline-text-4" id = "text-1-6-1" >
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< div id = "orgc08547a" class = "figure" >
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< p > < img src = "figs/hac_lac_plant_dvf.png" alt = "hac_lac_plant_dvf.png" / >
< / p >
< p > < span class = "figure-number" > Figure 4: < / span > Diagonal elements of the plant for HAC control when DVF is previously applied (< a href = "./figs/hac_lac_plant_dvf.png" > png< / a > , < a href = "./figs/hac_lac_plant_dvf.pdf" > pdf< / a > )< / p >
< / div >
< / div >
< / div >
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< div id = "outline-container-orgbd635c1" class = "outline-4" >
< h4 id = "orgbd635c1" > < span class = "section-number-4" > 1.6.2< / span > Controller Design< / h4 >
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< div class = "outline-text-4" id = "text-1-6-2" >
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< p >
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We design a diagonal controller with equal bandwidth for the 6 terms.
The controller is a pure integrator with a small lead near the crossover.
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< / p >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > wc = 2< span class = "org-type" > *< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > *< / span > 300; < span class = "org-comment" > % Wanted Bandwidth [rad/s]< / span >
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h = 1.2;
H_lead = 1< span class = "org-type" > /< / span > h< span class = "org-type" > *< / span > (1 < span class = "org-type" > +< / span > s< span class = "org-type" > /< / span > (wc< span class = "org-type" > /< / span > h))< span class = "org-type" > /< / span > (1 < span class = "org-type" > +< / span > s< span class = "org-type" > /< / span > (wc< span class = "org-type" > *< / span > h));
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Kd_dvf = diag(1< span class = "org-type" > ./< / span > abs(diag(freqresp(1< span class = "org-type" > /< / span > s< span class = "org-type" > *< / span > Gc_dvf, wc)))) < span class = "org-type" > .*< / span > H_lead < span class = "org-type" > .*< / span > 1< span class = "org-type" > /< / span > s;
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< / pre >
< / div >
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< div id = "org8108913" class = "figure" >
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< p > < img src = "figs/hac_lac_loop_gain_dvf.png" alt = "hac_lac_loop_gain_dvf.png" / >
< / p >
< p > < span class = "figure-number" > Figure 5: < / span > Diagonal elements of the Loop Gain for the HAC control (< a href = "./figs/hac_lac_loop_gain_dvf.png" > png< / a > , < a href = "./figs/hac_lac_loop_gain_dvf.pdf" > pdf< / a > )< / p >
< / div >
< p >
Finally, we pre-multiply the diagonal controller by \(\bm{J}^{-T}\) prior implementation.
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > K_hac_dvf = inv(stewart.kinematics.J< span class = "org-type" > '< / span > )< span class = "org-type" > *< / span > Kd_dvf;
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< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-org83c15a9" class = "outline-4" >
< h4 id = "org83c15a9" > < span class = "section-number-4" > 1.6.3< / span > Obtained Performance< / h4 >
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< div class = "outline-text-4" id = "text-1-6-3" >
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< p >
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We identify the transmissibility and compliance of the system.
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< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > controller = initializeController(< span class = "org-string" > 'type'< / span > , < span class = "org-string" > 'open-loop'< / span > );
[T_ol, T_norm_ol, freqs] = computeTransmissibility();
[C_ol, C_norm_ol, < span class = "org-type" > ~< / span > ] = computeCompliance();
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< / pre >
< / div >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > controller = initializeController(< span class = "org-string" > 'type'< / span > , < span class = "org-string" > 'dvf'< / span > );
[T_dvf, T_norm_dvf, < span class = "org-type" > ~< / span > ] = computeTransmissibility();
[C_dvf, C_norm_dvf, < span class = "org-type" > ~< / span > ] = computeCompliance();
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< / pre >
< / div >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > controller = initializeController(< span class = "org-string" > 'type'< / span > , < span class = "org-string" > 'hac-dvf'< / span > );
[T_hac_dvf, T_norm_hac_dvf, < span class = "org-type" > ~< / span > ] = computeTransmissibility();
[C_hac_dvf, C_norm_hac_dvf, < span class = "org-type" > ~< / span > ] = computeCompliance();
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< / pre >
< / div >
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< div id = "orgf16f5f2" class = "figure" >
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< p > < img src = "figs/hac_lac_C_T_dvf.png" alt = "hac_lac_C_T_dvf.png" / >
< / p >
< p > < span class = "figure-number" > Figure 6: < / span > Obtained Compliance and Transmissibility (< a href = "./figs/hac_lac_C_T_dvf.png" > png< / a > , < a href = "./figs/hac_lac_C_T_dvf.pdf" > pdf< / a > )< / p >
< / div >
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< / div >
< / div >
< / div >
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< div id = "outline-container-org5dfa0fd" class = "outline-3" >
< h3 id = "org5dfa0fd" > < span class = "section-number-3" > 1.7< / span > HAC - IFF< / h3 >
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< div class = "outline-text-3" id = "text-1-7" >
< / div >
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< div id = "outline-container-orgeedcb6b" class = "outline-4" >
< h4 id = "orgeedcb6b" > < span class = "section-number-4" > 1.7.1< / span > Plant< / h4 >
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< div class = "outline-text-4" id = "text-1-7-1" >
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< div id = "org66710a7" class = "figure" >
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< p > < img src = "figs/hac_lac_plant_iff.png" alt = "hac_lac_plant_iff.png" / >
< / p >
< p > < span class = "figure-number" > Figure 7: < / span > Diagonal elements of the plant for HAC control when IFF is previously applied (< a href = "./figs/hac_lac_plant_iff.png" > png< / a > , < a href = "./figs/hac_lac_plant_iff.pdf" > pdf< / a > )< / p >
< / div >
< / div >
< / div >
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< div id = "outline-container-orgd723b75" class = "outline-4" >
< h4 id = "orgd723b75" > < span class = "section-number-4" > 1.7.2< / span > Controller Design< / h4 >
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< div class = "outline-text-4" id = "text-1-7-2" >
< p >
We design a diagonal controller with equal bandwidth for the 6 terms.
The controller is a pure integrator with a small lead near the crossover.
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > wc = 2< span class = "org-type" > *< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > *< / span > 300; < span class = "org-comment" > % Wanted Bandwidth [rad/s]< / span >
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h = 1.2;
H_lead = 1< span class = "org-type" > /< / span > h< span class = "org-type" > *< / span > (1 < span class = "org-type" > +< / span > s< span class = "org-type" > /< / span > (wc< span class = "org-type" > /< / span > h))< span class = "org-type" > /< / span > (1 < span class = "org-type" > +< / span > s< span class = "org-type" > /< / span > (wc< span class = "org-type" > *< / span > h));
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Kd_iff = diag(1< span class = "org-type" > ./< / span > abs(diag(freqresp(1< span class = "org-type" > /< / span > s< span class = "org-type" > *< / span > Gc_iff, wc)))) < span class = "org-type" > .*< / span > H_lead < span class = "org-type" > .*< / span > 1< span class = "org-type" > /< / span > s;
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< / pre >
< / div >
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< div id = "org0b7f4f2" class = "figure" >
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< p > < img src = "figs/hac_lac_loop_gain_iff.png" alt = "hac_lac_loop_gain_iff.png" / >
< / p >
< p > < span class = "figure-number" > Figure 8: < / span > Diagonal elements of the Loop Gain for the HAC control (< a href = "./figs/hac_lac_loop_gain_iff.png" > png< / a > , < a href = "./figs/hac_lac_loop_gain_iff.pdf" > pdf< / a > )< / p >
< / div >
< p >
Finally, we pre-multiply the diagonal controller by \(\bm{J}^{-T}\) prior implementation.
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > K_hac_iff = inv(stewart.kinematics.J< span class = "org-type" > '< / span > )< span class = "org-type" > *< / span > Kd_iff;
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< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-org54813bf" class = "outline-4" >
< h4 id = "org54813bf" > < span class = "section-number-4" > 1.7.3< / span > Obtained Performance< / h4 >
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< div class = "outline-text-4" id = "text-1-7-3" >
< p >
We identify the transmissibility and compliance of the system.
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > controller = initializeController(< span class = "org-string" > 'type'< / span > , < span class = "org-string" > 'open-loop'< / span > );
[T_ol, T_norm_ol, freqs] = computeTransmissibility();
[C_ol, C_norm_ol, < span class = "org-type" > ~< / span > ] = computeCompliance();
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< / pre >
< / div >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > controller = initializeController(< span class = "org-string" > 'type'< / span > , < span class = "org-string" > 'iff'< / span > );
[T_iff, T_norm_iff, < span class = "org-type" > ~< / span > ] = computeTransmissibility();
[C_iff, C_norm_iff, < span class = "org-type" > ~< / span > ] = computeCompliance();
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< / pre >
< / div >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > controller = initializeController(< span class = "org-string" > 'type'< / span > , < span class = "org-string" > 'hac-iff'< / span > );
[T_hac_iff, T_norm_hac_iff, < span class = "org-type" > ~< / span > ] = computeTransmissibility();
[C_hac_iff, C_norm_hac_iff, < span class = "org-type" > ~< / span > ] = computeCompliance();
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< / pre >
< / div >
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< div id = "org6c57f46" class = "figure" >
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< p > < img src = "figs/hac_lac_C_T_iff.png" alt = "hac_lac_C_T_iff.png" / >
< / p >
< p > < span class = "figure-number" > Figure 9: < / span > Obtained Compliance and Transmissibility (< a href = "./figs/hac_lac_C_T_iff.png" > png< / a > , < a href = "./figs/hac_lac_C_T_iff.pdf" > pdf< / a > )< / p >
< / div >
< / div >
< / div >
< / div >
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< div id = "outline-container-org8e15485" class = "outline-3" >
< h3 id = "org8e15485" > < span class = "section-number-3" > 1.8< / span > Comparison< / h3 >
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< div class = "outline-text-3" id = "text-1-8" >
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< div id = "org34035be" class = "figure" >
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< p > < img src = "figs/hac_lac_C_full_comparison.png" alt = "hac_lac_C_full_comparison.png" / >
< / p >
< p > < span class = "figure-number" > Figure 10: < / span > Comparison of the norm of the Compliance matrices for the HAC-LAC architecture (< a href = "./figs/hac_lac_C_full_comparison.png" > png< / a > , < a href = "./figs/hac_lac_C_full_comparison.pdf" > pdf< / a > )< / p >
< / div >
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< div id = "org6db2ceb" class = "figure" >
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< p > < img src = "figs/hac_lac_T_full_comparison.png" alt = "hac_lac_T_full_comparison.png" / >
< / p >
< p > < span class = "figure-number" > Figure 11: < / span > Comparison of the norm of the Transmissibility matrices for the HAC-LAC architecture (< a href = "./figs/hac_lac_T_full_comparison.png" > png< / a > , < a href = "./figs/hac_lac_T_full_comparison.pdf" > pdf< / a > )< / p >
< / div >
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< div id = "orga195621" class = "figure" >
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< p > < img src = "figs/hac_lac_C_T_comparison.png" alt = "hac_lac_C_T_comparison.png" / >
< / p >
< p > < span class = "figure-number" > Figure 12: < / span > Comparison of the Frobenius norm of the Compliance and Transmissibility for the HAC-LAC architecture with both IFF and DVF (< a href = "./figs/hac_lac_C_T_comparison.png" > png< / a > , < a href = "./figs/hac_lac_C_T_comparison.pdf" > pdf< / a > )< / p >
< / div >
< / div >
< / div >
< / div >
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< div id = "outline-container-orgdc1bcf2" class = "outline-2" >
< h2 id = "orgdc1bcf2" > < span class = "section-number-2" > 2< / span > MIMO Analysis< / h2 >
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< div class = "outline-text-2" id = "text-2" >
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< p >
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Let’ s define the system as shown in figure < a href = "#org6f95566" > 13< / a > .
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< / p >
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< div id = "org6f95566" class = "figure" >
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< p > < img src = "figs/general_control_names.png" alt = "general_control_names.png" / >
< / p >
< p > < span class = "figure-number" > Figure 13: < / span > General Control Architecture< / p >
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< / div >
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< table id = "org8568d41" border = "2" cellspacing = "0" cellpadding = "6" rules = "groups" frame = "hsides" >
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< caption class = "t-above" > < span class = "table-number" > Table 1:< / span > Signals definition for the generalized plant< / caption >
< colgroup >
< col class = "org-left" / >
< col class = "org-left" / >
< col class = "org-left" / >
< / colgroup >
< thead >
< tr >
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< th scope = "col" class = "org-left" >   < / th >
< th scope = "col" class = "org-left" > < b > Symbol< / b > < / th >
< th scope = "col" class = "org-left" > < b > Meaning< / b > < / th >
< / tr >
< / thead >
< tbody >
< tr >
< td class = "org-left" > < b > Exogenous Inputs< / b > < / td >
< td class = "org-left" > \(\bm{\mathcal{X}}_w\)< / td >
< td class = "org-left" > Ground motion< / td >
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< / tr >
< tr >
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< td class = "org-left" >   < / td >
< td class = "org-left" > \(\bm{\mathcal{F}}_d\)< / td >
< td class = "org-left" > External Forces applied to the Payload< / td >
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< / tr >
< tr >
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< td class = "org-left" >   < / td >
< td class = "org-left" > \(\bm{r}\)< / td >
< td class = "org-left" > Reference signal for tracking< / td >
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< / tr >
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< / tbody >
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< tbody >
< tr >
< td class = "org-left" > < b > Exogenous Outputs< / b > < / td >
< td class = "org-left" > \(\bm{\mathcal{X}}\)< / td >
< td class = "org-left" > Absolute Motion of the Payload< / td >
< / tr >
< tr >
< td class = "org-left" >   < / td >
< td class = "org-left" > \(\bm{\tau}\)< / td >
< td class = "org-left" > Actuator Rate< / td >
< / tr >
< / tbody >
< tbody >
< tr >
< td class = "org-left" > < b > Sensed Outputs< / b > < / td >
< td class = "org-left" > \(\bm{\tau}_m\)< / td >
< td class = "org-left" > Force Sensors in each leg< / td >
< / tr >
< tr >
< td class = "org-left" >   < / td >
< td class = "org-left" > \(\delta \bm{\mathcal{L}}_m\)< / td >
< td class = "org-left" > Measured displacement of each leg< / td >
< / tr >
< tr >
< td class = "org-left" >   < / td >
< td class = "org-left" > \(\bm{\mathcal{X}}\)< / td >
< td class = "org-left" > Absolute Motion of the Payload< / td >
< / tr >
< / tbody >
< tbody >
< tr >
< td class = "org-left" > < b > Control Signals< / b > < / td >
< td class = "org-left" > \(\bm{\tau}\)< / td >
< td class = "org-left" > Actuator Inputs< / td >
< / tr >
< / tbody >
< / table >
< / div >
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< div id = "outline-container-org7fd66be" class = "outline-3" >
< h3 id = "org7fd66be" > < span class = "section-number-3" > 2.1< / span > Initialization< / h3 >
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< div class = "outline-text-3" id = "text-2-1" >
< p >
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We first initialize the Stewart platform.
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > stewart = initializeStewartPlatform();
stewart = initializeFramesPositions(stewart, < span class = "org-string" > 'H'< / span > , 90e< span class = "org-type" > -< / span > 3, < span class = "org-string" > 'MO_B'< / span > , 45e< span class = "org-type" > -< / span > 3);
stewart = generateGeneralConfiguration(stewart);
stewart = computeJointsPose(stewart);
stewart = initializeStrutDynamics(stewart);
stewart = initializeJointDynamics(stewart, < span class = "org-string" > 'type_F'< / span > , < span class = "org-string" > 'universal'< / span > , < span class = "org-string" > 'type_M'< / span > , < span class = "org-string" > 'spherical'< / span > );
stewart = initializeCylindricalPlatforms(stewart);
stewart = initializeCylindricalStruts(stewart);
stewart = computeJacobian(stewart);
stewart = initializeStewartPose(stewart);
stewart = initializeInertialSensor(stewart, < span class = "org-string" > 'type'< / span > , < span class = "org-string" > 'none'< / span > );
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< / pre >
< / div >
< p >
The rotation point of the ground is located at the origin of frame \(\{A\}\).
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > ground = initializeGround(< span class = "org-string" > 'type'< / span > , < span class = "org-string" > 'rigid'< / span > , < span class = "org-string" > 'rot_point'< / span > , stewart.platform_F.FO_A);
payload = initializePayload(< span class = "org-string" > 'type'< / span > , < span class = "org-string" > 'none'< / span > );
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< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-org06a70cd" class = "outline-3" >
< h3 id = "org06a70cd" > < span class = "section-number-3" > 2.2< / span > Identification< / h3 >
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< div class = "outline-text-3" id = "text-2-2" >
< / div >
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< div id = "outline-container-org88e17f4" class = "outline-4" >
< h4 id = "org88e17f4" > < span class = "section-number-4" > 2.2.1< / span > HAC - Without LAC< / h4 >
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< div class = "outline-text-4" id = "text-2-2-1" >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > controller = initializeController(< span class = "org-string" > 'type'< / span > , < span class = "org-string" > 'open-loop'< / span > );
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< / pre >
< / div >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > < span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Name of the Simulink File< / span > < / span >
mdl = < span class = "org-string" > 'stewart_platform_model'< / span > ;
< span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Input/Output definition< / span > < / span >
clear io; io_i = 1;
io(io_i) = linio([mdl, < span class = "org-string" > '/Controller'< / span > ], 1, < span class = "org-string" > 'input'< / span > ); io_i = io_i < span class = "org-type" > +< / span > 1; < span class = "org-comment" > % Actuator Force Inputs [N]< / span >
io(io_i) = linio([mdl, < span class = "org-string" > '/Absolute Motion Sensor'< / span > ], 1, < span class = "org-string" > 'openoutput'< / span > ); io_i = io_i < span class = "org-type" > +< / span > 1; < span class = "org-comment" > % Absolute Sensor [m, rad]< / span >
< span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Run the linearization< / span > < / span >
G_ol = linearize(mdl, io);
G_ol.InputName = {< span class = "org-string" > 'F1'< / span > , < span class = "org-string" > 'F2'< / span > , < span class = "org-string" > 'F3'< / span > , < span class = "org-string" > 'F4'< / span > , < span class = "org-string" > 'F5'< / span > , < span class = "org-string" > 'F6'< / span > };
G_ol.OutputName = {< span class = "org-string" > 'Dx'< / span > , < span class = "org-string" > 'Dy'< / span > , < span class = "org-string" > 'Dz'< / span > , < span class = "org-string" > 'Rx'< / span > , < span class = "org-string" > 'Ry'< / span > , < span class = "org-string" > 'Rz'< / span > };
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< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-orgb75c6b7" class = "outline-4" >
< h4 id = "orgb75c6b7" > < span class = "section-number-4" > 2.2.2< / span > HAC - DVF< / h4 >
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< div class = "outline-text-4" id = "text-2-2-2" >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > controller = initializeController(< span class = "org-string" > 'type'< / span > , < span class = "org-string" > 'dvf'< / span > );
K_dvf = < span class = "org-type" > -< / span > 1e4< span class = "org-type" > *< / span > s< span class = "org-type" > /< / span > (1< span class = "org-type" > +< / span > s< span class = "org-type" > /< / span > 2< span class = "org-type" > /< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > /< / span > 5000)< span class = "org-type" > *< / span > eye(6);
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< / pre >
< / div >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > < span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Name of the Simulink File< / span > < / span >
mdl = < span class = "org-string" > 'stewart_platform_model'< / span > ;
< span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Input/Output definition< / span > < / span >
clear io; io_i = 1;
io(io_i) = linio([mdl, < span class = "org-string" > '/Controller'< / span > ], 1, < span class = "org-string" > 'input'< / span > ); io_i = io_i < span class = "org-type" > +< / span > 1; < span class = "org-comment" > % Actuator Force Inputs [N]< / span >
io(io_i) = linio([mdl, < span class = "org-string" > '/Absolute Motion Sensor'< / span > ], 1, < span class = "org-string" > 'openoutput'< / span > ); io_i = io_i < span class = "org-type" > +< / span > 1; < span class = "org-comment" > % Absolute Sensor [m, rad]< / span >
< span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Run the linearization< / span > < / span >
G_dvf = linearize(mdl, io);
G_dvf.InputName = {< span class = "org-string" > 'F1'< / span > , < span class = "org-string" > 'F2'< / span > , < span class = "org-string" > 'F3'< / span > , < span class = "org-string" > 'F4'< / span > , < span class = "org-string" > 'F5'< / span > , < span class = "org-string" > 'F6'< / span > };
G_dvf.OutputName = {< span class = "org-string" > 'Dx'< / span > , < span class = "org-string" > 'Dy'< / span > , < span class = "org-string" > 'Dz'< / span > , < span class = "org-string" > 'Rx'< / span > , < span class = "org-string" > 'Ry'< / span > , < span class = "org-string" > 'Rz'< / span > };
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< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-orgba8c7bf" class = "outline-4" >
< h4 id = "orgba8c7bf" > < span class = "section-number-4" > 2.2.3< / span > Cartesian Frame< / h4 >
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< div class = "outline-text-4" id = "text-2-2-3" >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > Gc_ol = minreal(G_ol)< span class = "org-type" > /< / span > stewart.kinematics.J< span class = "org-type" > '< / span > ;
Gc_ol.InputName = {< span class = "org-string" > 'Fx'< / span > , < span class = "org-string" > 'Fy'< / span > , < span class = "org-string" > 'Fz'< / span > , < span class = "org-string" > 'Mx'< / span > , < span class = "org-string" > 'My'< / span > , < span class = "org-string" > 'Mz'< / span > };
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Gc_dvf = minreal(G_dvf)< span class = "org-type" > /< / span > stewart.kinematics.J< span class = "org-type" > '< / span > ;
Gc_dvf.InputName = {< span class = "org-string" > 'Fx'< / span > , < span class = "org-string" > 'Fy'< / span > , < span class = "org-string" > 'Fz'< / span > , < span class = "org-string" > 'Mx'< / span > , < span class = "org-string" > 'My'< / span > , < span class = "org-string" > 'Mz'< / span > };
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< / pre >
< / div >
< / div >
< / div >
< / div >
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< div id = "outline-container-orgf9d0420" class = "outline-3" >
< h3 id = "orgf9d0420" > < span class = "section-number-3" > 2.3< / span > Singular Value Decomposition< / h3 >
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< div class = "outline-text-3" id = "text-2-3" >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > freqs = logspace(1, 4, 1000);
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U_ol = zeros(6,6,length(freqs));
S_ol = zeros(6,length(freqs));
V_ol = zeros(6,6,length(freqs));
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U_dvf = zeros(6,6,length(freqs));
S_dvf = zeros(6,length(freqs));
V_dvf = zeros(6,6,length(freqs));
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< span class = "org-keyword" > for< / span > < span class = "org-variable-name" > < span class = "org-constant" > i< / span > < / span > = < span class = "org-constant" > 1:length(freqs)< / span >
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[U,S,V] = svd(freqresp(Gc_ol, freqs(< span class = "org-constant" > i< / span > ), < span class = "org-string" > 'Hz'< / span > ));
U_ol(< span class = "org-type" > :< / span > ,< span class = "org-type" > :< / span > ,< span class = "org-constant" > i< / span > ) = U;
S_ol(< span class = "org-type" > :< / span > ,< span class = "org-constant" > i< / span > ) = diag(S);
V_ol(< span class = "org-type" > :< / span > ,< span class = "org-type" > :< / span > ,< span class = "org-constant" > i< / span > ) = V;
[U,S,V] = svd(freqresp(Gc_dvf, freqs(< span class = "org-constant" > i< / span > ), < span class = "org-string" > 'Hz'< / span > ));
U_dvf(< span class = "org-type" > :< / span > ,< span class = "org-type" > :< / span > ,< span class = "org-constant" > i< / span > ) = U;
S_dvf(< span class = "org-type" > :< / span > ,< span class = "org-constant" > i< / span > ) = diag(S);
V_dvf(< span class = "org-type" > :< / span > ,< span class = "org-type" > :< / span > ,< span class = "org-constant" > i< / span > ) = V;
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< span class = "org-keyword" > end< / span >
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< / pre >
< / div >
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< / div >
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< div id = "outline-container-orga095fa8" class = "outline-2" >
< h2 id = "orga095fa8" > < span class = "section-number-2" > 3< / span > Diagonal Control based on the damped plant< / h2 >
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< div class = "outline-text-2" id = "text-3" >
< p >
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From (< a href = "#citeproc_bib_item_1" > Skogestad and Postlethwaite 2007< / a > ), a simple approach to multivariable control is the following two-step procedure:
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< / p >
< ol class = "org-ol" >
< li > < b > Design a pre-compensator< / b > \(W_1\), which counteracts the interactions in the plant and results in a new < b > shaped plant< / b > \(G_S(s) = G(s) W_1(s)\) which is < b > more diagonal and easier to control< / b > than the original plant \(G(s)\).< / li >
< li > < b > Design a diagonal controller< / b > \(K_S(s)\) for the shaped plant using methods similar to those for SISO systems.< / li >
< / ol >
< p >
The overall controller is then:
\[ K(s) = W_1(s)K_s(s) \]
< / p >
< p >
There are mainly three different cases:
< / p >
< ol class = "org-ol" >
< li > < b > Dynamic decoupling< / b > : \(G_S(s)\) is diagonal at all frequencies. For that we can choose \(W_1(s) = G^{-1}(s)\) and this is an inverse-based controller.< / li >
< li > < b > Steady-state decoupling< / b > : \(G_S(0)\) is diagonal. This can be obtained by selecting \(W_1(s) = G^{-1}(0)\).< / li >
< li > < b > Approximate decoupling at frequency \(\w_0\)< / b > : \(G_S(j\w_0)\) is as diagonal as possible. Decoupling the system at \(\w_0\) is a good choice because the effect on performance of reducing interaction is normally greatest at this frequency.< / li >
< / ol >
< / div >
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< div id = "outline-container-orge901603" class = "outline-3" >
< h3 id = "orge901603" > < span class = "section-number-3" > 3.1< / span > Initialization< / h3 >
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< div class = "outline-text-3" id = "text-3-1" >
< p >
We first initialize the Stewart platform.
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > stewart = initializeStewartPlatform();
stewart = initializeFramesPositions(stewart, < span class = "org-string" > 'H'< / span > , 90e< span class = "org-type" > -< / span > 3, < span class = "org-string" > 'MO_B'< / span > , 45e< span class = "org-type" > -< / span > 3);
stewart = generateGeneralConfiguration(stewart);
stewart = computeJointsPose(stewart);
stewart = initializeStrutDynamics(stewart);
stewart = initializeJointDynamics(stewart, < span class = "org-string" > 'type_F'< / span > , < span class = "org-string" > 'universal'< / span > , < span class = "org-string" > 'type_M'< / span > , < span class = "org-string" > 'spherical'< / span > );
stewart = initializeCylindricalPlatforms(stewart);
stewart = initializeCylindricalStruts(stewart);
stewart = computeJacobian(stewart);
stewart = initializeStewartPose(stewart);
stewart = initializeInertialSensor(stewart, < span class = "org-string" > 'type'< / span > , < span class = "org-string" > 'none'< / span > );
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< / pre >
< / div >
< p >
The rotation point of the ground is located at the origin of frame \(\{A\}\).
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > ground = initializeGround(< span class = "org-string" > 'type'< / span > , < span class = "org-string" > 'rigid'< / span > , < span class = "org-string" > 'rot_point'< / span > , stewart.platform_F.FO_A);
payload = initializePayload(< span class = "org-string" > 'type'< / span > , < span class = "org-string" > 'none'< / span > );
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< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-orgce73f0c" class = "outline-3" >
< h3 id = "orgce73f0c" > < span class = "section-number-3" > 3.2< / span > Identification< / h3 >
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< div class = "outline-text-3" id = "text-3-2" >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > controller = initializeController(< span class = "org-string" > 'type'< / span > , < span class = "org-string" > 'dvf'< / span > );
K_dvf = < span class = "org-type" > -< / span > 1e4< span class = "org-type" > *< / span > s< span class = "org-type" > /< / span > (1< span class = "org-type" > +< / span > s< span class = "org-type" > /< / span > 2< span class = "org-type" > /< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > /< / span > 5000)< span class = "org-type" > *< / span > eye(6);
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< / pre >
< / div >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > < span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Name of the Simulink File< / span > < / span >
mdl = < span class = "org-string" > 'stewart_platform_model'< / span > ;
< span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Input/Output definition< / span > < / span >
clear io; io_i = 1;
io(io_i) = linio([mdl, < span class = "org-string" > '/Controller'< / span > ], 1, < span class = "org-string" > 'input'< / span > ); io_i = io_i < span class = "org-type" > +< / span > 1; < span class = "org-comment" > % Actuator Force Inputs [N]< / span >
io(io_i) = linio([mdl, < span class = "org-string" > '/Absolute Motion Sensor'< / span > ], 1, < span class = "org-string" > 'openoutput'< / span > ); io_i = io_i < span class = "org-type" > +< / span > 1; < span class = "org-comment" > % Absolute Sensor [m, rad]< / span >
< span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Run the linearization< / span > < / span >
G_dvf = linearize(mdl, io);
G_dvf.InputName = {< span class = "org-string" > 'F1'< / span > , < span class = "org-string" > 'F2'< / span > , < span class = "org-string" > 'F3'< / span > , < span class = "org-string" > 'F4'< / span > , < span class = "org-string" > 'F5'< / span > , < span class = "org-string" > 'F6'< / span > };
G_dvf.OutputName = {< span class = "org-string" > 'Dx'< / span > , < span class = "org-string" > 'Dy'< / span > , < span class = "org-string" > 'Dz'< / span > , < span class = "org-string" > 'Rx'< / span > , < span class = "org-string" > 'Ry'< / span > , < span class = "org-string" > 'Rz'< / span > };
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< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-orgb5a063b" class = "outline-3" >
< h3 id = "orgb5a063b" > < span class = "section-number-3" > 3.3< / span > Steady State Decoupling< / h3 >
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< div class = "outline-text-3" id = "text-3-3" >
< / div >
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< div id = "outline-container-orgd0ce552" class = "outline-4" >
< h4 id = "orgd0ce552" > < span class = "section-number-4" > 3.3.1< / span > Pre-Compensator Design< / h4 >
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< div class = "outline-text-4" id = "text-3-3-1" >
< p >
We choose \(W_1 = G^{-1}(0)\).
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > W1 = inv(freqresp(G_dvf, 0));
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< / pre >
< / div >
< p >
The (static) decoupled plant is \(G_s(s) = G(s) W_1\).
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > Gs = G_dvf< span class = "org-type" > *< / span > W1;
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< / pre >
< / div >
< p >
In the case of the Stewart platform, the pre-compensator for static decoupling is equal to \(\mathcal{K} \bm{J}\):
< / p >
\begin{align*}
W_1 & = \left( \frac{\bm{\mathcal{X}}}{\bm{\tau}}(s=0) \right)^{-1}\\
& = \left( \frac{\bm{\mathcal{X}}}{\bm{\tau}}(s=0) \bm{J}^T \right)^{-1}\\
& = \left( \bm{C} \bm{J}^T \right)^{-1}\\
& = \left( \bm{J}^{-1} \mathcal{K}^{-1} \right)^{-1}\\
& = \mathcal{K} \bm{J}
\end{align*}
< p >
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The static decoupled plant is schematic shown in Figure < a href = "#org6f0c40d" > 14< / a > and the bode plots of its diagonal elements are shown in Figure < a href = "#orgd6f1b1d" > 15< / a > .
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< / p >
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< div id = "org6f0c40d" class = "figure" >
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< p > < img src = "figs/control_arch_static_decoupling.png" alt = "control_arch_static_decoupling.png" / >
< / p >
< p > < span class = "figure-number" > Figure 14: < / span > Static Decoupling of the Stewart platform< / p >
< / div >
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< div id = "orgd6f1b1d" class = "figure" >
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< p > < img src = "figs/static_decoupling_diagonal_plant.png" alt = "static_decoupling_diagonal_plant.png" / >
< / p >
< p > < span class = "figure-number" > Figure 15: < / span > Bode plot of the diagonal elements of \(G_s(s)\) (< a href = "./figs/static_decoupling_diagonal_plant.png" > png< / a > , < a href = "./figs/static_decoupling_diagonal_plant.pdf" > pdf< / a > )< / p >
< / div >
< / div >
< / div >
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< div id = "outline-container-org41b76c6" class = "outline-4" >
< h4 id = "org41b76c6" > < span class = "section-number-4" > 3.3.2< / span > Diagonal Control Design< / h4 >
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< div class = "outline-text-4" id = "text-3-3-2" >
< p >
We design a diagonal controller \(K_s(s)\) that consist of a pure integrator and a lead around the crossover.
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > wc = 2< span class = "org-type" > *< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > *< / span > 300; < span class = "org-comment" > % Wanted Bandwidth [rad/s]< / span >
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h = 1.5;
H_lead = 1< span class = "org-type" > /< / span > h< span class = "org-type" > *< / span > (1 < span class = "org-type" > +< / span > s< span class = "org-type" > /< / span > (wc< span class = "org-type" > /< / span > h))< span class = "org-type" > /< / span > (1 < span class = "org-type" > +< / span > s< span class = "org-type" > /< / span > (wc< span class = "org-type" > *< / span > h));
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Ks_dvf = diag(1< span class = "org-type" > ./< / span > abs(diag(freqresp(1< span class = "org-type" > /< / span > s< span class = "org-type" > *< / span > Gs, wc)))) < span class = "org-type" > .*< / span > H_lead < span class = "org-type" > .*< / span > 1< span class = "org-type" > /< / span > s;
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< / pre >
< / div >
< p >
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The overall controller is then \(K(s) = W_1 K_s(s)\) as shown in Figure < a href = "#org2d2d9e8" > 16< / a > .
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< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > K_hac_dvf = W1 < span class = "org-type" > *< / span > Ks_dvf;
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< / pre >
< / div >
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< div id = "org2d2d9e8" class = "figure" >
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< p > < img src = "figs/control_arch_static_decoupling_K.png" alt = "control_arch_static_decoupling_K.png" / >
< / p >
< p > < span class = "figure-number" > Figure 16: < / span > Controller including the static decoupling matrix< / p >
< / div >
< / div >
< / div >
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< div id = "outline-container-org3228759" class = "outline-4" >
< h4 id = "org3228759" > < span class = "section-number-4" > 3.3.3< / span > Results< / h4 >
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< div class = "outline-text-4" id = "text-3-3-3" >
< p >
We identify the transmissibility and compliance of the Stewart platform under open-loop and closed-loop control.
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > controller = initializeController(< span class = "org-string" > 'type'< / span > , < span class = "org-string" > 'open-loop'< / span > );
[T_ol, T_norm_ol, freqs] = computeTransmissibility();
[C_ol, C_norm_ol, < span class = "org-type" > ~< / span > ] = computeCompliance();
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< / pre >
< / div >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > controller = initializeController(< span class = "org-string" > 'type'< / span > , < span class = "org-string" > 'hac-dvf'< / span > );
[T_hac_dvf, T_norm_hac_dvf, < span class = "org-type" > ~< / span > ] = computeTransmissibility();
[C_hac_dvf, C_norm_hac_dvf, < span class = "org-type" > ~< / span > ] = computeCompliance();
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< / pre >
< / div >
< p >
The results are shown in figure
< / p >
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< div id = "orgb6c4828" class = "figure" >
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< p > < img src = "figs/static_decoupling_C_T_frobenius_norm.png" alt = "static_decoupling_C_T_frobenius_norm.png" / >
< / p >
< p > < span class = "figure-number" > Figure 17: < / span > Frobenius norm of the Compliance and transmissibility matrices (< a href = "./figs/static_decoupling_C_T_frobenius_norm.png" > png< / a > , < a href = "./figs/static_decoupling_C_T_frobenius_norm.pdf" > pdf< / a > )< / p >
< / div >
< / div >
< / div >
< / div >
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< div id = "outline-container-orgb53dd48" class = "outline-3" >
< h3 id = "orgb53dd48" > < span class = "section-number-3" > 3.4< / span > Decoupling at Crossover< / h3 >
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< div class = "outline-text-3" id = "text-3-4" >
< ul class = "org-ul" >
< li class = "off" > < code > [  ]< / code > Find a method for real approximation of a complex matrix< / li >
< / ul >
< / div >
< / div >
< / div >
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< div id = "outline-container-org639412c" class = "outline-2" >
< h2 id = "org639412c" > < span class = "section-number-2" > 4< / span > Time Domain Simulation< / h2 >
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< div class = "outline-text-2" id = "text-4" >
< / div >
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< div id = "outline-container-orgc332811" class = "outline-3" >
< h3 id = "orgc332811" > < span class = "section-number-3" > 4.1< / span > Initialization< / h3 >
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< div class = "outline-text-3" id = "text-4-1" >
< p >
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We first initialize the Stewart platform.
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > stewart = initializeStewartPlatform();
stewart = initializeFramesPositions(stewart, < span class = "org-string" > 'H'< / span > , 90e< span class = "org-type" > -< / span > 3, < span class = "org-string" > 'MO_B'< / span > , 45e< span class = "org-type" > -< / span > 3);
stewart = generateGeneralConfiguration(stewart);
stewart = computeJointsPose(stewart);
stewart = initializeStrutDynamics(stewart);
stewart = initializeJointDynamics(stewart, < span class = "org-string" > 'type_F'< / span > , < span class = "org-string" > 'universal'< / span > , < span class = "org-string" > 'type_M'< / span > , < span class = "org-string" > 'spherical'< / span > );
stewart = initializeCylindricalPlatforms(stewart);
stewart = initializeCylindricalStruts(stewart);
stewart = computeJacobian(stewart);
stewart = initializeStewartPose(stewart);
stewart = initializeInertialSensor(stewart, < span class = "org-string" > 'type'< / span > , < span class = "org-string" > 'none'< / span > );
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< / pre >
< / div >
< p >
The rotation point of the ground is located at the origin of frame \(\{A\}\).
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > ground = initializeGround(< span class = "org-string" > 'type'< / span > , < span class = "org-string" > 'rigid'< / span > , < span class = "org-string" > 'rot_point'< / span > , stewart.platform_F.FO_A);
payload = initializePayload(< span class = "org-string" > 'type'< / span > , < span class = "org-string" > 'none'< / span > );
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< / pre >
< / div >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > load(< span class = "org-string" > './mat/motion_error_ol.mat'< / span > , < span class = "org-string" > 'Eg'< / span > )
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< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-orgc72e6b5" class = "outline-3" >
< h3 id = "orgc72e6b5" > < span class = "section-number-3" > 4.2< / span > HAC IFF< / h3 >
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< div class = "outline-text-3" id = "text-4-2" >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > controller = initializeController(< span class = "org-string" > 'type'< / span > , < span class = "org-string" > 'iff'< / span > );
K_iff = < span class = "org-type" > -< / span > (1e4< span class = "org-type" > /< / span > s)< span class = "org-type" > *< / span > eye(6);
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< span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Name of the Simulink File< / span > < / span >
mdl = < span class = "org-string" > 'stewart_platform_model'< / span > ;
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< span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Input/Output definition< / span > < / span >
clear io; io_i = 1;
io(io_i) = linio([mdl, < span class = "org-string" > '/Controller'< / span > ], 1, < span class = "org-string" > 'input'< / span > ); io_i = io_i < span class = "org-type" > +< / span > 1; < span class = "org-comment" > % Actuator Force Inputs [N]< / span >
io(io_i) = linio([mdl, < span class = "org-string" > '/Absolute Motion Sensor'< / span > ], 1, < span class = "org-string" > 'openoutput'< / span > ); io_i = io_i < span class = "org-type" > +< / span > 1; < span class = "org-comment" > % Absolute Sensor [m, rad]< / span >
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< span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Run the linearization< / span > < / span >
G_iff = linearize(mdl, io);
G_iff.InputName = {< span class = "org-string" > 'F1'< / span > , < span class = "org-string" > 'F2'< / span > , < span class = "org-string" > 'F3'< / span > , < span class = "org-string" > 'F4'< / span > , < span class = "org-string" > 'F5'< / span > , < span class = "org-string" > 'F6'< / span > };
G_iff.OutputName = {< span class = "org-string" > 'Dx'< / span > , < span class = "org-string" > 'Dy'< / span > , < span class = "org-string" > 'Dz'< / span > , < span class = "org-string" > 'Rx'< / span > , < span class = "org-string" > 'Ry'< / span > , < span class = "org-string" > 'Rz'< / span > };
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Gc_iff = minreal(G_iff)< span class = "org-type" > /< / span > stewart.kinematics.J< span class = "org-type" > '< / span > ;
Gc_iff.InputName = {< span class = "org-string" > 'Fx'< / span > , < span class = "org-string" > 'Fy'< / span > , < span class = "org-string" > 'Fz'< / span > , < span class = "org-string" > 'Mx'< / span > , < span class = "org-string" > 'My'< / span > , < span class = "org-string" > 'Mz'< / span > };
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< / pre >
< / div >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > wc = 2< span class = "org-type" > *< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > *< / span > 100; < span class = "org-comment" > % Wanted Bandwidth [rad/s]< / span >
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h = 1.2;
H_lead = 1< span class = "org-type" > /< / span > h< span class = "org-type" > *< / span > (1 < span class = "org-type" > +< / span > s< span class = "org-type" > /< / span > (wc< span class = "org-type" > /< / span > h))< span class = "org-type" > /< / span > (1 < span class = "org-type" > +< / span > s< span class = "org-type" > /< / span > (wc< span class = "org-type" > *< / span > h));
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Kd_iff = diag(1< span class = "org-type" > ./< / span > abs(diag(freqresp(1< span class = "org-type" > /< / span > s< span class = "org-type" > *< / span > Gc_iff, wc)))) < span class = "org-type" > .*< / span > H_lead < span class = "org-type" > .*< / span > 1< span class = "org-type" > /< / span > s;
K_hac_iff = inv(stewart.kinematics.J< span class = "org-type" > '< / span > )< span class = "org-type" > *< / span > Kd_iff;
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< / pre >
< / div >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > controller = initializeController(< span class = "org-string" > 'type'< / span > , < span class = "org-string" > 'hac-iff'< / span > );
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< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-org757f9e9" class = "outline-3" >
< h3 id = "org757f9e9" > < span class = "section-number-3" > 4.3< / span > HAC-DVF< / h3 >
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< div class = "outline-text-3" id = "text-4-3" >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > controller = initializeController(< span class = "org-string" > 'type'< / span > , < span class = "org-string" > 'dvf'< / span > );
K_dvf = < span class = "org-type" > -< / span > 1e4< span class = "org-type" > *< / span > s< span class = "org-type" > /< / span > (1< span class = "org-type" > +< / span > s< span class = "org-type" > /< / span > 2< span class = "org-type" > /< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > /< / span > 5000)< span class = "org-type" > *< / span > eye(6);
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< span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Name of the Simulink File< / span > < / span >
mdl = < span class = "org-string" > 'stewart_platform_model'< / span > ;
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< span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Input/Output definition< / span > < / span >
clear io; io_i = 1;
io(io_i) = linio([mdl, < span class = "org-string" > '/Controller'< / span > ], 1, < span class = "org-string" > 'input'< / span > ); io_i = io_i < span class = "org-type" > +< / span > 1; < span class = "org-comment" > % Actuator Force Inputs [N]< / span >
io(io_i) = linio([mdl, < span class = "org-string" > '/Absolute Motion Sensor'< / span > ], 1, < span class = "org-string" > 'openoutput'< / span > ); io_i = io_i < span class = "org-type" > +< / span > 1; < span class = "org-comment" > % Absolute Sensor [m, rad]< / span >
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< span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Run the linearization< / span > < / span >
G_dvf = linearize(mdl, io);
G_dvf.InputName = {< span class = "org-string" > 'F1'< / span > , < span class = "org-string" > 'F2'< / span > , < span class = "org-string" > 'F3'< / span > , < span class = "org-string" > 'F4'< / span > , < span class = "org-string" > 'F5'< / span > , < span class = "org-string" > 'F6'< / span > };
G_dvf.OutputName = {< span class = "org-string" > 'Dx'< / span > , < span class = "org-string" > 'Dy'< / span > , < span class = "org-string" > 'Dz'< / span > , < span class = "org-string" > 'Rx'< / span > , < span class = "org-string" > 'Ry'< / span > , < span class = "org-string" > 'Rz'< / span > };
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Gc_dvf = minreal(G_dvf)< span class = "org-type" > /< / span > stewart.kinematics.J< span class = "org-type" > '< / span > ;
Gc_dvf.InputName = {< span class = "org-string" > 'Fx'< / span > , < span class = "org-string" > 'Fy'< / span > , < span class = "org-string" > 'Fz'< / span > , < span class = "org-string" > 'Mx'< / span > , < span class = "org-string" > 'My'< / span > , < span class = "org-string" > 'Mz'< / span > };
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< / pre >
< / div >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > wc = 2< span class = "org-type" > *< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > *< / span > 100; < span class = "org-comment" > % Wanted Bandwidth [rad/s]< / span >
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h = 1.2;
H_lead = 1< span class = "org-type" > /< / span > h< span class = "org-type" > *< / span > (1 < span class = "org-type" > +< / span > s< span class = "org-type" > /< / span > (wc< span class = "org-type" > /< / span > h))< span class = "org-type" > /< / span > (1 < span class = "org-type" > +< / span > s< span class = "org-type" > /< / span > (wc< span class = "org-type" > *< / span > h));
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Kd_dvf = diag(1< span class = "org-type" > ./< / span > abs(diag(freqresp(1< span class = "org-type" > /< / span > s< span class = "org-type" > *< / span > Gc_dvf, wc)))) < span class = "org-type" > .*< / span > H_lead < span class = "org-type" > .*< / span > 1< span class = "org-type" > /< / span > s;
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K_hac_dvf = inv(stewart.kinematics.J< span class = "org-type" > '< / span > )< span class = "org-type" > *< / span > Kd_dvf;
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< / pre >
< / div >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > controller = initializeController(< span class = "org-string" > 'type'< / span > , < span class = "org-string" > 'hac-dvf'< / span > );
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< div id = "outline-container-org620278d" class = "outline-3" >
< h3 id = "org620278d" > < span class = "section-number-3" > 4.4< / span > Results< / h3 >
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< div class = "outline-text-3" id = "text-4-4" >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > < span class = "org-type" > figure< / span > ;
subplot(1, 2, 1);
hold on;
plot(Eg.Time, Eg.Data(< span class = "org-type" > :< / span > , 1), < span class = "org-string" > 'DisplayName'< / span > , < span class = "org-string" > 'X'< / span > );
plot(Eg.Time, Eg.Data(< span class = "org-type" > :< / span > , 2), < span class = "org-string" > 'DisplayName'< / span > , < span class = "org-string" > 'Y'< / span > );
plot(Eg.Time, Eg.Data(< span class = "org-type" > :< / span > , 3), < span class = "org-string" > 'DisplayName'< / span > , < span class = "org-string" > 'Z'< / span > );
hold off;
xlabel(< span class = "org-string" > 'Time [s]'< / span > );
ylabel(< span class = "org-string" > 'Position error [m]'< / span > );
legend();
subplot(1, 2, 2);
hold on;
plot(simout.Xa.Time, simout.Xa.Data(< span class = "org-type" > :< / span > , 1));
plot(simout.Xa.Time, simout.Xa.Data(< span class = "org-type" > :< / span > , 2));
plot(simout.Xa.Time, simout.Xa.Data(< span class = "org-type" > :< / span > , 3));
hold off;
xlabel(< span class = "org-string" > 'Time [s]'< / span > );
ylabel(< span class = "org-string" > 'Orientation error [rad]'< / span > );
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< / pre >
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< div id = "outline-container-org6a9c87c" class = "outline-2" >
< h2 id = "org6a9c87c" > < span class = "section-number-2" > 5< / span > Functions< / h2 >
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< div class = "outline-text-2" id = "text-5" >
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< div id = "outline-container-orgd1492e7" class = "outline-3" >
< h3 id = "orgd1492e7" > < span class = "section-number-3" > 5.1< / span > < code > initializeController< / code > : Initialize the Controller< / h3 >
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< div class = "outline-text-3" id = "text-5-1" >
< p >
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< a id = "orgc575d0d" > < / a >
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< div id = "outline-container-orgae5eed0" class = "outline-4" >
< h4 id = "orgae5eed0" > Function description< / h4 >
< div class = "outline-text-4" id = "text-orgae5eed0" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > < span class = "org-keyword" > function< / span > < span class = "org-variable-name" > [controller]< / span > = < span class = "org-function-name" > initializeController< / span > (< span class = "org-variable-name" > args< / span > )
< span class = "org-comment" > % initializeController - Initialize the Controller< / span >
< span class = "org-comment" > %< / span >
< span class = "org-comment" > % Syntax: [] = initializeController(args)< / span >
< span class = "org-comment" > %< / span >
< span class = "org-comment" > % Inputs:< / span >
< span class = "org-comment" > % - args - Can have the following fields:< / span >
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< / pre >
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< div id = "outline-container-orgda07b57" class = "outline-4" >
< h4 id = "orgda07b57" > Optional Parameters< / h4 >
< div class = "outline-text-4" id = "text-orgda07b57" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > < span class = "org-keyword" > arguments< / span >
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< span class = "org-variable-name" > args< / span > .type char {mustBeMember(args.type, {< span class = "org-string" > 'open-loop'< / span > , < span class = "org-string" > 'iff'< / span > , < span class = "org-string" > 'dvf'< / span > , < span class = "org-string" > 'hac-iff'< / span > , < span class = "org-string" > 'hac-dvf'< / span > , < span class = "org-string" > 'ref-track-L'< / span > , < span class = "org-string" > 'ref-track-X'< / span > , < span class = "org-string" > 'ref-track-hac-dvf'< / span > })} = < span class = "org-string" > 'open-loop'< / span >
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< span class = "org-keyword" > end< / span >
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< / pre >
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< / div >
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< div id = "outline-container-orgdb009ab" class = "outline-4" >
< h4 id = "orgdb009ab" > Structure initialization< / h4 >
< div class = "outline-text-4" id = "text-orgdb009ab" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > controller = struct();
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< div id = "outline-container-org056c578" class = "outline-4" >
< h4 id = "org056c578" > Add Type< / h4 >
< div class = "outline-text-4" id = "text-org056c578" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > < span class = "org-keyword" > switch< / span > < span class = "org-constant" > args.type< / span >
< span class = "org-keyword" > case< / span > < span class = "org-string" > 'open-loop'< / span >
controller.type = 0;
< span class = "org-keyword" > case< / span > < span class = "org-string" > 'iff'< / span >
controller.type = 1;
< span class = "org-keyword" > case< / span > < span class = "org-string" > 'dvf'< / span >
controller.type = 2;
< span class = "org-keyword" > case< / span > < span class = "org-string" > 'hac-iff'< / span >
controller.type = 3;
< span class = "org-keyword" > case< / span > < span class = "org-string" > 'hac-dvf'< / span >
controller.type = 4;
< span class = "org-keyword" > case< / span > < span class = "org-string" > 'ref-track-L'< / span >
controller.type = 5;
< span class = "org-keyword" > case< / span > < span class = "org-string" > 'ref-track-X'< / span >
controller.type = 6;
< span class = "org-keyword" > case< / span > < span class = "org-string" > 'ref-track-hac-dvf'< / span >
controller.type = 7;
< span class = "org-keyword" > end< / span >
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< / pre >
< / div >
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< style > . csl-entry { text-indent : -1.5 em ; margin-left : 1.5 em ; } < / style > < h2 class = 'citeproc-org-bib-h2' > Bibliography< / h2 >
< div class = "csl-bib-body" >
< div class = "csl-entry" > < a name = "citeproc_bib_item_1" > < / a > Skogestad, Sigurd, and Ian Postlethwaite. 2007. < i > Multivariable Feedback Control: Analysis and Design< / i > . John Wiley.< / div >
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< / div >
< div id = "postamble" class = "status" >
< p class = "author" > Author: Dehaeze Thomas< / p >
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< p class = "date" > Created: 2021-01-08 ven. 15:52< / p >
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< / div >
< / body >
< / html >
