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< h1 class = "title" > Stewart Platform - Dynamics Study< / h1 >
< div id = "table-of-contents" >
< h2 > Table of Contents< / h2 >
< div id = "text-table-of-contents" >
< ul >
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< li > < a href = "#orgc59e712" > 1. Compare external forces and forces applied by the actuators< / a >
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< ul >
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< li > < a href = "#org4509b7d" > 1.1. Comparison with fixed support< / a > < / li >
< li > < a href = "#org8662186" > 1.2. Comparison with a flexible support< / a > < / li >
< li > < a href = "#orgb87f273" > 1.3. Conclusion< / a > < / li >
< / ul >
< / li >
< li > < a href = "#org81ab204" > 2. Comparison of the static transfer function and the Compliance matrix< / a >
< ul >
< li > < a href = "#orge7e7242" > 2.1. Analysis< / a > < / li >
< li > < a href = "#org230655f" > 2.2. Conclusion< / a > < / li >
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< / ul >
< / li >
< / ul >
< / div >
< / div >
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< div id = "outline-container-orgc59e712" class = "outline-2" >
< h2 id = "orgc59e712" > < span class = "section-number-2" > 1< / span > Compare external forces and forces applied by the actuators< / h2 >
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< div class = "outline-text-2" id = "text-1" >
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< / div >
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< div id = "outline-container-org4509b7d" class = "outline-3" >
< h3 id = "org4509b7d" > < span class = "section-number-3" > 1.1< / span > Comparison with fixed support< / h3 >
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< div class = "outline-text-3" id = "text-1-1" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > stewart = initializeStewartPlatform();
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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);
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stewart = generateGeneralConfiguration(stewart);
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stewart = computeJointsPose(stewart);
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stewart = initializeStrutDynamics(stewart);
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stewart = initializeJointDynamics(stewart, < span class = "org-string" > 'type_F'< / span > , < span class = "org-string" > 'universal_p'< / span > , < span class = "org-string" > 'type_M'< / span > , < span class = "org-string" > 'spherical_p'< / span > );
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stewart = initializeCylindricalPlatforms(stewart);
stewart = initializeCylindricalStruts(stewart);
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stewart = computeJacobian(stewart);
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stewart = initializeStewartPose(stewart);
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stewart = initializeInertialSensor(stewart, < 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" > ground = initializeGround(< span class = "org-string" > 'type'< / span > , < span class = "org-string" > 'none'< / span > );
payload = initializePayload(< span class = "org-string" > 'type'< / span > , < span class = "org-string" > 'none'< / span > );
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< / pre >
< / div >
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< p >
Estimation of the transfer function from \(\bm{\tau}\) to \(\mathcal{\bm{X}}\):
< / p >
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< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Options for Linearized< / span > < / span >
options = linearizeOptions;
options.SampleTime = 0;
< span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Name of the Simulink File< / span > < / span >
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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;
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io(io_i) = linio([mdl, < span class = "org-string" > '/Controller'< / span > ], 1, < span class = "org-string" > 'openinput'< / 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" > '/Relative Motion Sensor'< / span > ], 1, < span class = "org-string" > 'openoutput'< / span > ); io_i = io_i < span class = "org-type" > +< / span > 1; < span class = "org-comment" > % Position/Orientation of {B} w.r.t. {A}< / span >
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< span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Run the linearization< / span > < / span >
G = linearize(mdl, io, options);
G.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.OutputName = {< span class = "org-string" > 'Edx'< / span > , < span class = "org-string" > 'Edy'< / span > , < span class = "org-string" > 'Edz'< / span > , < span class = "org-string" > 'Erx'< / span > , < span class = "org-string" > 'Ery'< / span > , < span class = "org-string" > 'Erz'< / span > };
< / pre >
< / div >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > Gc = minreal(G< span class = "org-type" > *< / span > inv(stewart.kinematics.J< span class = "org-type" > '< / span > ));
Gc.InputName = {< span class = "org-string" > 'Fnx'< / span > , < span class = "org-string" > 'Fny'< / span > , < span class = "org-string" > 'Fnz'< / span > , < span class = "org-string" > 'Mnx'< / span > , < span class = "org-string" > 'Mny'< / span > , < span class = "org-string" > 'Mnz'< / span > };
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< / pre >
< / div >
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< p >
Estimation of the transfer function from \(\bm{\mathcal{F}}_{\text{ext}}\) to \(\mathcal{\bm{X}}\):
< / p >
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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" > %% Input/Output definition< / span > < / span >
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clear io; io_i = 1;
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io(io_i) = linio([mdl, < span class = "org-string" > '/Disturbances'< / span > ], 1, < span class = "org-string" > 'openinput'< / span > , [], < span class = "org-string" > 'F_ext'< / span > ); io_i = io_i < span class = "org-type" > +< / span > 1; < span class = "org-comment" > % External forces/torques applied on {B}< / span >
io(io_i) = linio([mdl, < span class = "org-string" > '/Relative Motion Sensor'< / span > ], 1, < span class = "org-string" > 'openoutput'< / span > ); io_i = io_i < span class = "org-type" > +< / span > 1; < span class = "org-comment" > % Position/Orientation of {B} w.r.t. {A}< / span >
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< span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Run the linearization< / span > < / span >
Gd = linearize(mdl, io, options);
Gd.InputName = {< span class = "org-string" > 'Fex'< / span > , < span class = "org-string" > 'Fey'< / span > , < span class = "org-string" > 'Fez'< / span > , < span class = "org-string" > 'Mex'< / span > , < span class = "org-string" > 'Mey'< / span > , < span class = "org-string" > 'Mez'< / span > };
Gd.OutputName = {< span class = "org-string" > 'Edx'< / span > , < span class = "org-string" > 'Edy'< / span > , < span class = "org-string" > 'Edz'< / span > , < span class = "org-string" > 'Erx'< / span > , < span class = "org-string" > 'Ery'< / span > , < span class = "org-string" > 'Erz'< / span > };
< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-org8662186" class = "outline-3" >
< h3 id = "org8662186" > < span class = "section-number-3" > 1.2< / span > Comparison with a flexible support< / h3 >
< div class = "outline-text-3" id = "text-1-2" >
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< p >
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We redo the identification for when the Stewart platform is on a flexible support.
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< / 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" > 'flexible'< / span > );
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< / pre >
< / div >
< p >
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Estimation of the transfer function from \(\bm{\tau}\) to \(\mathcal{\bm{X}}\):
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< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Options for Linearized< / span > < / span >
options = linearizeOptions;
options.SampleTime = 0;
< span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Name of the Simulink File< / span > < / span >
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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;
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io(io_i) = linio([mdl, < span class = "org-string" > '/Controller'< / span > ], 1, < span class = "org-string" > 'openinput'< / 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" > '/Relative Motion Sensor'< / span > ], 1, < span class = "org-string" > 'openoutput'< / span > ); io_i = io_i < span class = "org-type" > +< / span > 1; < span class = "org-comment" > % Position/Orientation of {B} w.r.t. {A}< / span >
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< span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Run the linearization< / span > < / span >
G = linearize(mdl, io, options);
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G.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 > };
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G.OutputName = {< span class = "org-string" > 'Edx'< / span > , < span class = "org-string" > 'Edy'< / span > , < span class = "org-string" > 'Edz'< / span > , < span class = "org-string" > 'Erx'< / span > , < span class = "org-string" > 'Ery'< / span > , < span class = "org-string" > 'Erz'< / span > };
< / pre >
< / div >
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< div class = "org-src-container" >
< pre class = "src src-matlab" > Gc = minreal(G< span class = "org-type" > *< / span > inv(stewart.kinematics.J< span class = "org-type" > '< / span > ));
Gc.InputName = {< span class = "org-string" > 'Fnx'< / span > , < span class = "org-string" > 'Fny'< / span > , < span class = "org-string" > 'Fnz'< / span > , < span class = "org-string" > 'Mnx'< / span > , < span class = "org-string" > 'Mny'< / span > , < span class = "org-string" > 'Mnz'< / span > };
< / pre >
< / div >
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< p >
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Estimation of the transfer function from \(\bm{\mathcal{F}}_{\text{ext}}\) to \(\mathcal{\bm{X}}\):
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< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > < span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Input/Output definition< / span > < / span >
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clear io; io_i = 1;
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io(io_i) = linio([mdl, < span class = "org-string" > '/Disturbances'< / span > ], 1, < span class = "org-string" > 'openinput'< / span > , [], < span class = "org-string" > 'F_ext'< / span > ); io_i = io_i < span class = "org-type" > +< / span > 1; < span class = "org-comment" > % External forces/torques applied on {B}< / span >
io(io_i) = linio([mdl, < span class = "org-string" > '/Relative Motion Sensor'< / span > ], 1, < span class = "org-string" > 'openoutput'< / span > ); io_i = io_i < span class = "org-type" > +< / span > 1; < span class = "org-comment" > % Position/Orientation of {B} w.r.t. {A}< / span >
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< span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Run the linearization< / span > < / span >
Gd = linearize(mdl, io, options);
Gd.InputName = {< span class = "org-string" > 'Fex'< / span > , < span class = "org-string" > 'Fey'< / span > , < span class = "org-string" > 'Fez'< / span > , < span class = "org-string" > 'Mex'< / span > , < span class = "org-string" > 'Mey'< / span > , < span class = "org-string" > 'Mez'< / span > };
Gd.OutputName = {< span class = "org-string" > 'Edx'< / span > , < span class = "org-string" > 'Edy'< / span > , < span class = "org-string" > 'Edz'< / span > , < span class = "org-string" > 'Erx'< / span > , < span class = "org-string" > 'Ery'< / span > , < span class = "org-string" > 'Erz'< / span > };
< / pre >
< / div >
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< / div >
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< div id = "outline-container-orgb87f273" class = "outline-3" >
< h3 id = "orgb87f273" > < span class = "section-number-3" > 1.3< / span > Conclusion< / h3 >
< div class = "outline-text-3" id = "text-1-3" >
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< div class = "important" >
< p >
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The transfer function from forces/torques applied by the actuators on the payload \(\bm{\mathcal{F}} = \bm{J}^T \bm{\tau}\) to the pose of the mobile platform \(\bm{\mathcal{X}}\) is the same as the transfer function from external forces/torques to \(\bm{\mathcal{X}}\) as long as the Stewart platform’ s base is fixed.
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< / p >
< / div >
< / div >
< / div >
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< / div >
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< div id = "outline-container-org81ab204" class = "outline-2" >
< h2 id = "org81ab204" > < span class = "section-number-2" > 2< / span > Comparison of the static transfer function and the Compliance matrix< / h2 >
< div class = "outline-text-2" id = "text-2" >
< / div >
< div id = "outline-container-orge7e7242" class = "outline-3" >
< h3 id = "orge7e7242" > < span class = "section-number-3" > 2.1< / span > Analysis< / h3 >
< div class = "outline-text-3" id = "text-2-1" >
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< p >
Initialization of the Stewart platform.
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > stewart = initializeStewartPlatform();
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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);
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stewart = generateGeneralConfiguration(stewart);
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stewart = computeJointsPose(stewart);
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stewart = initializeStrutDynamics(stewart);
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stewart = initializeJointDynamics(stewart, < span class = "org-string" > 'type_F'< / span > , < span class = "org-string" > 'universal_p'< / span > , < span class = "org-string" > 'type_M'< / span > , < span class = "org-string" > 'spherical_p'< / span > );
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stewart = initializeCylindricalPlatforms(stewart);
stewart = initializeCylindricalStruts(stewart);
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stewart = computeJacobian(stewart);
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stewart = initializeStewartPose(stewart);
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stewart = initializeInertialSensor(stewart, < span class = "org-string" > 'type'< / span > , < span class = "org-string" > 'none'< / span > );
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > ground = initializeGround(< span class = "org-string" > 'type'< / span > , < span class = "org-string" > 'none'< / span > );
payload = initializePayload(< span class = "org-string" > 'type'< / span > , < span class = "org-string" > 'none'< / span > );
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< / pre >
< / div >
< p >
Estimation of the transfer function from \(\mathcal{\bm{F}}\) to \(\mathcal{\bm{X}}\):
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Options for Linearized< / span > < / span >
options = linearizeOptions;
options.SampleTime = 0;
< span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Name of the Simulink File< / span > < / span >
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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" > '/F'< / span > ], 1, < span class = "org-string" > 'openinput'< / span > ); io_i = io_i < span class = "org-type" > +< / span > 1;
io(io_i) = linio([mdl, < span class = "org-string" > '/X'< / span > ], 1, < span class = "org-string" > 'openoutput'< / span > ); io_i = io_i < span class = "org-type" > +< / span > 1;
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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" > 'openinput'< / 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" > '/Relative Motion Sensor'< / span > ], 1, < span class = "org-string" > 'openoutput'< / span > ); io_i = io_i < span class = "org-type" > +< / span > 1; < span class = "org-comment" > % Position/Orientation of {B} w.r.t. {A}< / span >
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< span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Run the linearization< / span > < / span >
G = linearize(mdl, io, options);
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G.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 > };
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G.OutputName = {< span class = "org-string" > 'Edx'< / span > , < span class = "org-string" > 'Edy'< / span > , < span class = "org-string" > 'Edz'< / span > , < span class = "org-string" > 'Erx'< / span > , < span class = "org-string" > 'Ery'< / span > , < span class = "org-string" > 'Erz'< / span > };
< / pre >
< / div >
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< div class = "org-src-container" >
< pre class = "src src-matlab" > Gc = minreal(G< span class = "org-type" > *< / span > inv(stewart.kinematics.J< span class = "org-type" > '< / span > ));
Gc.InputName = {< span class = "org-string" > 'Fnx'< / span > , < span class = "org-string" > 'Fny'< / span > , < span class = "org-string" > 'Fnz'< / span > , < span class = "org-string" > 'Mnx'< / span > , < span class = "org-string" > 'Mny'< / span > , < span class = "org-string" > 'Mnz'< / span > };
< / pre >
< / div >
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< p >
Let’ s first look at the low frequency transfer function matrix from \(\mathcal{\bm{F}}\) to \(\mathcal{\bm{X}}\).
< / p >
< table border = "2" cellspacing = "0" cellpadding = "6" rules = "groups" frame = "hsides" >
< colgroup >
< col class = "org-right" / >
< col class = "org-right" / >
< col class = "org-right" / >
< col class = "org-right" / >
< col class = "org-right" / >
< col class = "org-right" / >
< / colgroup >
< tbody >
< tr >
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< td class = "org-right" > 4.7e-08< / td >
< td class = "org-right" > -7.2e-19< / td >
< td class = "org-right" > 5.0e-18< / td >
< td class = "org-right" > -8.9e-18< / td >
< td class = "org-right" > 3.2e-07< / td >
< td class = "org-right" > 9.9e-18< / td >
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< / tr >
< tr >
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< td class = "org-right" > 4.7e-18< / td >
< td class = "org-right" > 4.7e-08< / td >
< td class = "org-right" > -5.7e-18< / td >
< td class = "org-right" > -3.2e-07< / td >
< td class = "org-right" > -1.6e-17< / td >
< td class = "org-right" > -1.7e-17< / td >
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< / tr >
< tr >
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< td class = "org-right" > 3.3e-18< / td >
< td class = "org-right" > -6.3e-18< / td >
< td class = "org-right" > 2.1e-08< / td >
< td class = "org-right" > 4.4e-17< / td >
< td class = "org-right" > 6.6e-18< / td >
< td class = "org-right" > 7.4e-18< / td >
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< / tr >
< tr >
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< td class = "org-right" > -3.2e-17< / td >
< td class = "org-right" > -3.2e-07< / td >
< td class = "org-right" > 6.2e-18< / td >
< td class = "org-right" > 5.2e-06< / td >
< td class = "org-right" > -3.5e-16< / td >
< td class = "org-right" > 6.3e-17< / td >
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< / tr >
< tr >
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< td class = "org-right" > 3.2e-07< / td >
< td class = "org-right" > 2.7e-17< / td >
< td class = "org-right" > 4.8e-17< / td >
< td class = "org-right" > -4.5e-16< / td >
< td class = "org-right" > 5.2e-06< / td >
< td class = "org-right" > -1.2e-19< / td >
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< / tr >
< tr >
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< td class = "org-right" > 4.0e-17< / td >
< td class = "org-right" > -9.5e-17< / td >
< td class = "org-right" > 8.4e-18< / td >
< td class = "org-right" > 4.3e-16< / td >
< td class = "org-right" > 5.8e-16< / td >
< td class = "org-right" > 1.7e-06< / td >
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< / tr >
< / tbody >
< / table >
< p >
And now at the Compliance matrix.
< / p >
< table border = "2" cellspacing = "0" cellpadding = "6" rules = "groups" frame = "hsides" >
< colgroup >
< col class = "org-right" / >
< col class = "org-right" / >
< col class = "org-right" / >
< col class = "org-right" / >
< col class = "org-right" / >
< col class = "org-right" / >
< / colgroup >
< tbody >
< tr >
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< td class = "org-right" > 4.7e-08< / td >
< td class = "org-right" > -2.0e-24< / td >
< td class = "org-right" > 7.4e-25< / td >
< td class = "org-right" > 5.9e-23< / td >
< td class = "org-right" > 3.2e-07< / td >
< td class = "org-right" > 5.9e-24< / td >
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< / tr >
< tr >
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< td class = "org-right" > -7.1e-25< / td >
< td class = "org-right" > 4.7e-08< / td >
< td class = "org-right" > 2.9e-25< / td >
< td class = "org-right" > -3.2e-07< / td >
< td class = "org-right" > -5.4e-24< / td >
< td class = "org-right" > -3.3e-23< / td >
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< / tr >
< tr >
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< td class = "org-right" > 7.9e-26< / td >
< td class = "org-right" > -6.4e-25< / td >
< td class = "org-right" > 2.1e-08< / td >
< td class = "org-right" > 1.9e-23< / td >
< td class = "org-right" > 5.3e-25< / td >
< td class = "org-right" > -6.5e-40< / td >
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< / tr >
< tr >
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< td class = "org-right" > 1.4e-23< / td >
< td class = "org-right" > -3.2e-07< / td >
< td class = "org-right" > 1.3e-23< / td >
< td class = "org-right" > 5.2e-06< / td >
< td class = "org-right" > 4.9e-22< / td >
< td class = "org-right" > -3.8e-24< / td >
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< / tr >
< tr >
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< td class = "org-right" > 3.2e-07< / td >
< td class = "org-right" > 7.6e-24< / td >
< td class = "org-right" > 1.2e-23< / td >
< td class = "org-right" > 6.9e-22< / td >
< td class = "org-right" > 5.2e-06< / td >
< td class = "org-right" > -2.6e-22< / td >
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< / tr >
< tr >
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< td class = "org-right" > 7.3e-24< / td >
< td class = "org-right" > -3.2e-23< / td >
< td class = "org-right" > -1.6e-39< / td >
< td class = "org-right" > 9.9e-23< / td >
< td class = "org-right" > -3.3e-22< / td >
< td class = "org-right" > 1.7e-06< / td >
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< / tr >
< / tbody >
< / table >
< / div >
< / div >
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< div id = "outline-container-org230655f" class = "outline-3" >
< h3 id = "org230655f" > < span class = "section-number-3" > 2.2< / span > Conclusion< / h3 >
< div class = "outline-text-3" id = "text-2-2" >
< div class = "important" >
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< p >
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The low frequency transfer function matrix from \(\mathcal{\bm{F}}\) to \(\mathcal{\bm{X}}\) corresponds to the compliance matrix of the Stewart platform.
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< / p >
< / div >
< / div >
< / div >
< / div >
< / div >
< div id = "postamble" class = "status" >
< p class = "author" > Author: Dehaeze Thomas< / p >
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< p class = "date" > Created: 2020-02-13 jeu. 15:19< / p >
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< / div >
< / body >
< / html >
