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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 >
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< li > < a href = "#org55e0dad" > 1.3. Conclusion< / a > < / li >
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< / 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 >
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< li > < a href = "#org9ee3939" > 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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< p >
In this section, we wish to compare the effect of forces/torques applied by the actuators with the effect of external forces/torques on the displacement of the mobile platform.
< / p >
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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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< p >
Let’ s generate a Stewart platform.
< / p >
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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, 'H', 90e-3, 'MO_B', 45e-3);
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stewart = generateGeneralConfiguration(stewart);
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stewart = computeJointsPose(stewart);
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stewart = initializeStrutDynamics(stewart);
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stewart = initializeJointDynamics(stewart, 'type_F', 'universal_p', 'type_M', 'spherical_p');
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stewart = initializeCylindricalPlatforms(stewart);
stewart = initializeCylindricalStruts(stewart);
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stewart = computeJacobian(stewart);
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stewart = initializeStewartPose(stewart);
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stewart = initializeInertialSensor(stewart, 'type', 'none');
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< / pre >
< / div >
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< p >
We don’ t put any flexibility below the Stewart platform such that < b > its base is fixed to an inertial frame< / b > .
We also don’ t put any payload on top of the Stewart platform.
< / p >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > ground = initializeGround('type', 'none');
payload = initializePayload('type', 'none');
controller = initializeController('type', 'open-loop');
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< / pre >
< / div >
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< p >
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The transfer function from actuator forces \(\bm{\tau}\) to the relative displacement of the mobile platform \(\mathcal{\bm{X}}\) is extracted.
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< / p >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > %% Options for Linearized
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options = linearizeOptions;
options.SampleTime = 0;
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%% Name of the Simulink File
mdl = 'stewart_platform_model';
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%% Input/Output definition
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clear io; io_i = 1;
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io(io_i) = linio([mdl, '/Controller'], 1, 'openinput'); io_i = io_i + 1; % Actuator Force Inputs [N]
io(io_i) = linio([mdl, '/Relative Motion Sensor'], 1, 'openoutput'); io_i = io_i + 1; % Position/Orientation of {B} w.r.t. {A}
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%% Run the linearization
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G = linearize(mdl, io, options);
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G.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
G.OutputName = {'Edx', 'Edy', 'Edz', 'Erx', 'Ery', 'Erz'};
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< / pre >
< / div >
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< p >
Using the Jacobian matrix, we compute the transfer function from force/torques applied by the actuators on the frame \(\{B\}\) fixed to the mobile platform:
< / p >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > Gc = minreal(G*inv(stewart.kinematics.J'));
Gc.InputName = {'Fnx', 'Fny', 'Fnz', 'Mnx', 'Mny', 'Mnz'};
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< / pre >
< / div >
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< p >
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We also extract the transfer function from external forces \(\bm{\mathcal{F}}_{\text{ext}}\) on the frame \(\{B\}\) fixed to the mobile platform to the relative displacement \(\mathcal{\bm{X}}\) of \(\{B\}\) with respect to frame \(\{A\}\):
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< / p >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > %% Input/Output definition
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clear io; io_i = 1;
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io(io_i) = linio([mdl, '/Disturbances'], 1, 'openinput', [], 'F_ext'); io_i = io_i + 1; % External forces/torques applied on {B}
io(io_i) = linio([mdl, '/Relative Motion Sensor'], 1, 'openoutput'); io_i = io_i + 1; % Position/Orientation of {B} w.r.t. {A}
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%% Run the linearization
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Gd = linearize(mdl, io, options);
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Gd.InputName = {'Fex', 'Fey', 'Fez', 'Mex', 'Mey', 'Mez'};
Gd.OutputName = {'Edx', 'Edy', 'Edz', 'Erx', 'Ery', 'Erz'};
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< / pre >
< / div >
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< p >
The comparison of the two transfer functions is shown in Figure < a href = "#orgbf9a54a" > 1< / a > .
< / p >
< div id = "orgbf9a54a" class = "figure" >
< p > < img src = "figs/comparison_Fext_F_fixed_base.png" alt = "comparison_Fext_F_fixed_base.png" / >
< / p >
< p > < span class = "figure-number" > Figure 1: < / span > Comparison of the transfer functions from \(\bm{\mathcal{F}}\) to \(\mathcal{\bm{X}}\) and from \(\bm{\mathcal{F}}_{\text{ext}}\) to \(\mathcal{\bm{X}}\) (< a href = "./figs/comparison_Fext_F_fixed_base.png" > png< / a > , < a href = "./figs/comparison_Fext_F_fixed_base.pdf" > pdf< / a > )< / p >
< / div >
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< p >
This can be understood from figure < a href = "#org8bd3e63" > 2< / a > where \(\mathcal{F}_{x}\) and \(\mathcal{F}_{x,\text{ext}}\) have clearly the same effect on \(\mathcal{X}_{x}\).
< / p >
< div id = "org8bd3e63" class = "figure" >
< p > < img src = "figs/1dof_actuator_external_forces.png" alt = "1dof_actuator_external_forces.png" / >
< / p >
< p > < span class = "figure-number" > Figure 2: < / span > Schematic representation of the stewart platform on a rigid support< / p >
< / div >
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< / 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 now add a flexible support under the Stewart platform.
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< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > ground = initializeGround('type', 'flexible');
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< / pre >
< / div >
< p >
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And we perform again the identification.
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< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > %% Input/Output definition
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clear io; io_i = 1;
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io(io_i) = linio([mdl, '/Controller'], 1, 'openinput'); io_i = io_i + 1; % Actuator Force Inputs [N]
io(io_i) = linio([mdl, '/Relative Motion Sensor'], 1, 'openoutput'); io_i = io_i + 1; % Position/Orientation of {B} w.r.t. {A}
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%% Run the linearization
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G = linearize(mdl, io, options);
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G.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
G.OutputName = {'Edx', 'Edy', 'Edz', 'Erx', 'Ery', 'Erz'};
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Gc = minreal(G*inv(stewart.kinematics.J'));
Gc.InputName = {'Fnx', 'Fny', 'Fnz', 'Mnx', 'Mny', 'Mnz'};
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%% Input/Output definition
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clear io; io_i = 1;
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io(io_i) = linio([mdl, '/Disturbances'], 1, 'openinput', [], 'F_ext'); io_i = io_i + 1; % External forces/torques applied on {B}
io(io_i) = linio([mdl, '/Relative Motion Sensor'], 1, 'openoutput'); io_i = io_i + 1; % Position/Orientation of {B} w.r.t. {A}
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%% Run the linearization
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Gd = linearize(mdl, io, options);
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Gd.InputName = {'Fex', 'Fey', 'Fez', 'Mex', 'Mey', 'Mez'};
Gd.OutputName = {'Edx', 'Edy', 'Edz', 'Erx', 'Ery', 'Erz'};
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< / pre >
< / div >
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< p >
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The comparison between the obtained transfer functions is shown in Figure < a href = "#orga2f2bd5" > 3< / a > .
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< / p >
< div id = "orga2f2bd5" class = "figure" >
< p > < img src = "figs/comparison_Fext_F_flexible_base.png" alt = "comparison_Fext_F_flexible_base.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 3: < / span > Comparison of the transfer functions from \(\bm{\mathcal{F}}\) to \(\mathcal{\bm{X}}\) and from \(\bm{\mathcal{F}}_{\text{ext}}\) to \(\mathcal{\bm{X}}\) (< a href = "./figs/comparison_Fext_F_flexible_base.png" > png< / a > , < a href = "./figs/comparison_Fext_F_flexible_base.pdf" > pdf< / a > )< / p >
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< / div >
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< p >
The addition of a flexible support can be schematically represented in Figure < a href = "#orgee3ecbe" > 4< / a > .
We see that \(\mathcal{F}_{x}\) applies a force both on \(m\) and \(m^{\prime}\) whereas \(\mathcal{F}_{x,\text{ext}}\) only applies a force on \(m\).
And thus \(\mathcal{F}_{x}\) and \(\mathcal{F}_{x,\text{ext}}\) have clearly < b > not< / b > the same effect on \(\mathcal{X}_{x}\).
< / p >
< div id = "orgee3ecbe" class = "figure" >
< p > < img src = "figs/2dof_actuator_external_forces.png" alt = "2dof_actuator_external_forces.png" / >
< / p >
< p > < span class = "figure-number" > Figure 4: < / span > Schematic representation of the stewart platform on top of a flexible support< / p >
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< / div >
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< / div >
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< / div >
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< div id = "outline-container-org55e0dad" class = "outline-3" >
< h3 id = "org55e0dad" > < span class = "section-number-3" > 1.3< / span > Conclusion< / h3 >
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< 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" >
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< p >
In this section, we see how the Compliance matrix of the Stewart platform is linked to the static relation between \(\mathcal{\bm{F}}\) to \(\mathcal{\bm{X}}\).
< / p >
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< / div >
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< 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, 'H', 90e-3, 'MO_B', 45e-3);
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stewart = generateGeneralConfiguration(stewart);
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stewart = computeJointsPose(stewart);
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stewart = initializeStrutDynamics(stewart);
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stewart = initializeJointDynamics(stewart, 'type_F', 'universal_p', 'type_M', 'spherical_p');
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stewart = initializeCylindricalPlatforms(stewart);
stewart = initializeCylindricalStruts(stewart);
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stewart = computeJacobian(stewart);
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stewart = initializeStewartPose(stewart);
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stewart = initializeInertialSensor(stewart, 'type', 'none');
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< / pre >
< / div >
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< p >
No flexibility below the Stewart platform and no payload.
< / p >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > ground = initializeGround('type', 'none');
payload = initializePayload('type', 'none');
controller = initializeController('type', 'open-loop');
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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" >
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< pre class = "src src-matlab" > %% Options for Linearized
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options = linearizeOptions;
options.SampleTime = 0;
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%% Name of the Simulink File
mdl = 'stewart_platform_model';
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%% Input/Output definition
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clear io; io_i = 1;
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io(io_i) = linio([mdl, '/Controller'], 1, 'openinput'); io_i = io_i + 1; % Actuator Force Inputs [N]
io(io_i) = linio([mdl, '/Relative Motion Sensor'], 1, 'openoutput'); io_i = io_i + 1; % Position/Orientation of {B} w.r.t. {A}
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%% Run the linearization
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G = linearize(mdl, io, options);
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G.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
G.OutputName = {'Edx', 'Edy', 'Edz', 'Erx', 'Ery', 'Erz'};
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< / pre >
< / div >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > Gc = minreal(G*inv(stewart.kinematics.J'));
Gc.InputName = {'Fnx', 'Fny', 'Fnz', 'Mnx', 'Mny', 'Mnz'};
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< / 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 >
2020-02-13 15:19:30 +01:00
< 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 >
2020-02-13 15:19:30 +01:00
< 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 >
2020-02-13 15:19:30 +01:00
< 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 >
2020-02-13 15:19:30 +01:00
< 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 >
2020-01-22 16:31:44 +01:00
< / tr >
< tr >
2020-02-13 15:19:30 +01:00
< 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 >
2020-02-13 15:19:30 +01:00
< 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 >
2020-08-05 13:28:14 +02:00
< div id = "outline-container-org9ee3939" class = "outline-3" >
< h3 id = "org9ee3939" > < span class = "section-number-3" > 2.2< / span > Conclusion< / h3 >
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< 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.
2020-01-22 16:31:44 +01:00
< / p >
< / div >
< / div >
< / div >
< / div >
< / div >
< div id = "postamble" class = "status" >
< p class = "author" > Author: Dehaeze Thomas< / p >
2020-08-05 13:28:14 +02:00
< p class = "date" > Created: 2020-08-05 mer. 13:27< / p >
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< / div >
< / body >
< / html >
