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< title > Stewart Platform - Decentralized Active Damping< / title >
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< h1 class = "title" > Stewart Platform - Decentralized Active Damping< / 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 = "#orgddaf52f" > 1. Inertial Control< / a >
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< ul >
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< li > < a href = "#org933440d" > 1.1. Identification of the Dynamics< / a > < / li >
< li > < a href = "#orged6d23c" > 1.2. Effect of the Flexible Joint stiffness and Actuator amplification on the Dynamics< / a > < / li >
< li > < a href = "#org533c409" > 1.3. Obtained Damping< / a > < / li >
< li > < a href = "#orgc76021e" > 1.4. Conclusion< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#orgf8ed544" > 2. Integral Force Feedback< / a >
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< ul >
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< li > < a href = "#org7b81fe5" > 2.1. Identification of the Dynamics with perfect Joints< / a > < / li >
< li > < a href = "#org3dca396" > 2.2. Effect of the Flexible Joint stiffness and Actuator amplification on the Dynamics< / a > < / li >
< li > < a href = "#org7044ed4" > 2.3. Obtained Damping< / a > < / li >
< li > < a href = "#org9c769b9" > 2.4. Conclusion< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#orgabec4e1" > 3. Direct Velocity Feedback< / a >
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< ul >
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< li > < a href = "#orga2d019b" > 3.1. Identification of the Dynamics with perfect Joints< / a > < / li >
< li > < a href = "#org2875dd1" > 3.2. Effect of the Flexible Joint stiffness and Actuator amplification on the Dynamics< / a > < / li >
< li > < a href = "#org0cea759" > 3.3. Obtained Damping< / a > < / li >
< li > < a href = "#orga866100" > 3.4. Conclusion< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#orgc7e2089" > 4. Compliance and Transmissibility Comparison< / a >
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< ul >
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< li > < a href = "#org6ec3b9e" > 4.1. Initialization< / a > < / li >
< li > < a href = "#orge87554a" > 4.2. Identification< / a > < / li >
< li > < a href = "#org1d70ccd" > 4.3. Results< / a > < / li >
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< / ul >
< / li >
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< / div >
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< p >
The following decentralized active damping techniques are briefly studied:
< / p >
< ul class = "org-ul" >
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< li > Inertial Control (proportional feedback of the absolute velocity): Section < a href = "#org709d56c" > 1< / a > < / li >
< li > Integral Force Feedback: Section < a href = "#org1f0d316" > 2< / a > < / li >
< li > Direct feedback of the relative velocity of each strut: Section < a href = "#org63027d0" > 3< / a > < / li >
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< / ul >
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< div id = "outline-container-orgddaf52f" class = "outline-2" >
< h2 id = "orgddaf52f" > < span class = "section-number-2" > 1< / span > Inertial Control< / h2 >
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< div class = "outline-text-2" id = "text-1" >
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< p >
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< a id = "org709d56c" > < / a >
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< / p >
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< div class = "note" id = "org8a14d8c" >
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< p >
The Matlab script corresponding to this section is accessible < a href = "../matlab/active_damping_inertial.m" > here< / a > .
< / p >
< p >
To run the script, open the Simulink Project, and type < code > run active_damping_inertial.m< / code > .
< / p >
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< div id = "outline-container-org933440d" class = "outline-3" >
< h3 id = "org933440d" > < span class = "section-number-3" > 1.1< / span > Identification of the Dynamics< / 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();
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_p'< / span > , < span class = "org-string" > 'type_M'< / span > , < span class = "org-string" > 'spherical_p'< / 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" > 'accelerometer'< / span > , < span class = "org-string" > 'freq'< / span > , 5e3);
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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 > );
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" > %% 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 >
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" > '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" > '/Stewart Platform'< / span > ], 1, < span class = "org-string" > 'openoutput'< / span > , [], < span class = "org-string" > 'Vm'< / span > ); io_i = io_i < span class = "org-type" > +< / span > 1; < span class = "org-comment" > % Absolute velocity of each leg [m/s]< / span >
< 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" > 'Vm1'< / span > , < span class = "org-string" > 'Vm2'< / span > , < span class = "org-string" > 'Vm3'< / span > , < span class = "org-string" > 'Vm4'< / span > , < span class = "org-string" > 'Vm5'< / span > , < span class = "org-string" > 'Vm6'< / span > };
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< / pre >
< / div >
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< p >
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The transfer function from actuator forces to force sensors is shown in Figure < a href = "#org5cd47c9" > 1< / a > .
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< / p >
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< div id = "org5cd47c9" class = "figure" >
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< p > < img src = "figs/inertial_plant_coupling.png" alt = "inertial_plant_coupling.png" / >
< / p >
< p > < span class = "figure-number" > Figure 1: < / span > Transfer function from the Actuator force \(F_{i}\) to the absolute velocity of the same leg \(v_{m,i}\) and to the absolute velocity of the other legs \(v_{m,j}\) with \(i \neq j\) in grey (< a href = "./figs/inertial_plant_coupling.png" > png< / a > , < a href = "./figs/inertial_plant_coupling.pdf" > pdf< / a > )< / p >
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< div id = "outline-container-orged6d23c" class = "outline-3" >
< h3 id = "orged6d23c" > < span class = "section-number-3" > 1.2< / span > Effect of the Flexible Joint stiffness and Actuator amplification on the Dynamics< / h3 >
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< div class = "outline-text-3" id = "text-1-2" >
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< p >
We add some stiffness and damping in the flexible joints and we re-identify the dynamics.
< / p >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > 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 > );
Gf = linearize(mdl, io, options);
Gf.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 > };
Gf.OutputName = {< span class = "org-string" > 'Vm1'< / span > , < span class = "org-string" > 'Vm2'< / span > , < span class = "org-string" > 'Vm3'< / span > , < span class = "org-string" > 'Vm4'< / span > , < span class = "org-string" > 'Vm5'< / span > , < span class = "org-string" > 'Vm6'< / span > };
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< / pre >
< / div >
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< p >
We now use the amplified actuators and re-identify the dynamics
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > stewart = initializeAmplifiedStrutDynamics(stewart);
Ga = linearize(mdl, io, options);
Ga.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 > };
Ga.OutputName = {< span class = "org-string" > 'Vm1'< / span > , < span class = "org-string" > 'Vm2'< / span > , < span class = "org-string" > 'Vm3'< / span > , < span class = "org-string" > 'Vm4'< / span > , < span class = "org-string" > 'Vm5'< / span > , < span class = "org-string" > 'Vm6'< / span > };
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< / pre >
< / div >
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< p >
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The new dynamics from force actuator to force sensor is shown in Figure < a href = "#org8fcc87b" > 2< / a > .
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< / p >
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< div id = "org8fcc87b" class = "figure" >
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< p > < img src = "figs/inertial_plant_flexible_joint_decentralized.png" alt = "inertial_plant_flexible_joint_decentralized.png" / >
< / p >
< p > < span class = "figure-number" > Figure 2: < / span > Transfer function from the Actuator force \(F_{i}\) to the absolute velocity sensor \(v_{m,i}\) (< a href = "./figs/inertial_plant_flexible_joint_decentralized.png" > png< / a > , < a href = "./figs/inertial_plant_flexible_joint_decentralized.pdf" > pdf< / a > )< / p >
< / div >
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< / div >
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< div id = "outline-container-org533c409" class = "outline-3" >
< h3 id = "org533c409" > < span class = "section-number-3" > 1.3< / span > Obtained Damping< / h3 >
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< div class = "outline-text-3" id = "text-1-3" >
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< p >
The control is a performed in a decentralized manner.
The \(6 \times 6\) control is a diagonal matrix with pure proportional action on the diagonal:
\[ K(s) = g
\begin{bmatrix}
1 & & 0 \\
& \ddots & \\
0 & & 1
\end{bmatrix} \]
< / p >
< p >
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The root locus is shown in figure < a href = "#orgaea8656" > 3< / a > .
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< / p >
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< div id = "orgaea8656" class = "figure" >
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< p > < img src = "figs/root_locus_inertial_rot_stiffness.png" alt = "root_locus_inertial_rot_stiffness.png" / >
< / p >
< p > < span class = "figure-number" > Figure 3: < / span > Root Locus plot with Decentralized Inertial Control when considering the stiffness of flexible joints (< a href = "./figs/root_locus_inertial_rot_stiffness.png" > png< / a > , < a href = "./figs/root_locus_inertial_rot_stiffness.pdf" > pdf< / a > )< / p >
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< h3 id = "orgc76021e" > < span class = "section-number-3" > 1.4< / span > Conclusion< / h3 >
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< div class = "outline-text-3" id = "text-1-4" >
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< div class = "important" id = "org37b8ef0" >
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< p >
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We do not have guaranteed stability with Inertial control. This is because of the flexibility inside the internal sensor.
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< h2 id = "orgf8ed544" > < span class = "section-number-2" > 2< / span > Integral Force Feedback< / h2 >
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< div class = "outline-text-2" id = "text-2" >
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< p >
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< a id = "org1f0d316" > < / a >
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< / p >
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< div class = "note" id = "org54cec4b" >
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< p >
The Matlab script corresponding to this section is accessible < a href = "../matlab/active_damping_iff.m" > here< / a > .
< / p >
< p >
To run the script, open the Simulink Project, and type < code > run active_damping_iff.m< / code > .
< / p >
< / div >
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< div id = "outline-container-org7b81fe5" class = "outline-3" >
< h3 id = "org7b81fe5" > < span class = "section-number-3" > 2.1< / span > Identification of the Dynamics with perfect Joints< / h3 >
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< div class = "outline-text-3" id = "text-2-1" >
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< p >
We first initialize the Stewart platform without joint stiffness.
< / 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_p'< / span > , < span class = "org-string" > 'type_M'< / span > , < span class = "org-string" > 'spherical_p'< / 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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< 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 > );
controller = initializeController(< span class = "org-string" > 'type'< / span > , < span class = "org-string" > 'open-loop'< / span > );
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< / pre >
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< p >
And we identify the dynamics from force actuators to force sensors.
< / 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" > %% 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" > '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" > '/Stewart Platform'< / span > ], 1, < span class = "org-string" > 'openoutput'< / span > , [], < span class = "org-string" > 'Taum'< / span > ); io_i = io_i < span class = "org-type" > +< / span > 1; < span class = "org-comment" > % Force Sensor Outputs [N]< / span >
< span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Run the linearization< / span > < / span >
G = linearize(mdl, io);
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" > 'Fm1'< / span > , < span class = "org-string" > 'Fm2'< / span > , < span class = "org-string" > 'Fm3'< / span > , < span class = "org-string" > 'Fm4'< / span > , < span class = "org-string" > 'Fm5'< / span > , < span class = "org-string" > 'Fm6'< / span > };
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< / pre >
< / div >
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< p >
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The transfer function from actuator forces to force sensors is shown in Figure < a href = "#org3b3de64" > 4< / a > .
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< / p >
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< div id = "org3b3de64" class = "figure" >
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< p > < img src = "figs/iff_plant_coupling.png" alt = "iff_plant_coupling.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 4: < / span > Transfer function from the Actuator force \(F_{i}\) to the Force sensor of the same leg \(F_{m,i}\) and to the force sensor of the other legs \(F_{m,j}\) with \(i \neq j\) in grey (< a href = "./figs/iff_plant_coupling.png" > png< / a > , < a href = "./figs/iff_plant_coupling.pdf" > pdf< / a > )< / p >
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< / div >
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< / div >
< / div >
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< div id = "outline-container-org3dca396" class = "outline-3" >
< h3 id = "org3dca396" > < span class = "section-number-3" > 2.2< / span > Effect of the Flexible Joint stiffness and Actuator amplification on the Dynamics< / h3 >
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< div class = "outline-text-3" id = "text-2-2" >
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< p >
We add some stiffness and damping in the flexible joints and we re-identify the dynamics.
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > 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 > );
Gf = linearize(mdl, io);
Gf.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 > };
Gf.OutputName = {< span class = "org-string" > 'Fm1'< / span > , < span class = "org-string" > 'Fm2'< / span > , < span class = "org-string" > 'Fm3'< / span > , < span class = "org-string" > 'Fm4'< / span > , < span class = "org-string" > 'Fm5'< / span > , < span class = "org-string" > 'Fm6'< / span > };
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< / pre >
< / div >
< p >
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We now use the amplified actuators and re-identify the dynamics
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > stewart = initializeAmplifiedStrutDynamics(stewart);
Ga = linearize(mdl, io);
Ga.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 > };
Ga.OutputName = {< span class = "org-string" > 'Fm1'< / span > , < span class = "org-string" > 'Fm2'< / span > , < span class = "org-string" > 'Fm3'< / span > , < span class = "org-string" > 'Fm4'< / span > , < span class = "org-string" > 'Fm5'< / span > , < span class = "org-string" > 'Fm6'< / span > };
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< / pre >
< / div >
< p >
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The new dynamics from force actuator to force sensor is shown in Figure < a href = "#org1902b8f" > 5< / a > .
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< / p >
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< div id = "org1902b8f" class = "figure" >
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< p > < img src = "figs/iff_plant_flexible_joint_decentralized.png" alt = "iff_plant_flexible_joint_decentralized.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 5: < / span > Transfer function from the Actuator force \(F_{i}\) to the force sensor \(F_{m,i}\) (< a href = "./figs/iff_plant_flexible_joint_decentralized.png" > png< / a > , < a href = "./figs/iff_plant_flexible_joint_decentralized.pdf" > pdf< / a > )< / p >
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< / div >
< / div >
< / div >
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< div id = "outline-container-org7044ed4" class = "outline-3" >
< h3 id = "org7044ed4" > < span class = "section-number-3" > 2.3< / span > Obtained Damping< / h3 >
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< div class = "outline-text-3" id = "text-2-3" >
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< p >
The control is a performed in a decentralized manner.
The \(6 \times 6\) control is a diagonal matrix with pure integration action on the diagonal:
\[ K(s) = g
\begin{bmatrix}
\frac{1}{s} & & 0 \\
& \ddots & \\
0 & & \frac{1}{s}
\end{bmatrix} \]
< / p >
< p >
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The root locus is shown in figure < a href = "#orgce5d8d8" > 6< / a > and the obtained pole damping function of the control gain is shown in figure < a href = "#org67419cd" > 7< / a > .
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< / p >
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< div id = "orgce5d8d8" class = "figure" >
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< p > < img src = "figs/root_locus_iff_rot_stiffness.png" alt = "root_locus_iff_rot_stiffness.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 6: < / span > Root Locus plot with Decentralized Integral Force Feedback when considering the stiffness of flexible joints (< a href = "./figs/root_locus_iff_rot_stiffness.png" > png< / a > , < a href = "./figs/root_locus_iff_rot_stiffness.pdf" > pdf< / a > )< / p >
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< / div >
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< div id = "org67419cd" class = "figure" >
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< p > < img src = "figs/pole_damping_gain_iff_rot_stiffness.png" alt = "pole_damping_gain_iff_rot_stiffness.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 7: < / span > Damping of the poles with respect to the gain of the Decentralized Integral Force Feedback when considering the stiffness of flexible joints (< a href = "./figs/pole_damping_gain_iff_rot_stiffness.png" > png< / a > , < a href = "./figs/pole_damping_gain_iff_rot_stiffness.pdf" > pdf< / a > )< / p >
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< / div >
< / div >
< / div >
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< div id = "outline-container-org9c769b9" class = "outline-3" >
< h3 id = "org9c769b9" > < span class = "section-number-3" > 2.4< / span > Conclusion< / h3 >
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< div class = "outline-text-3" id = "text-2-4" >
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< div class = "important" id = "orged36719" >
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< p >
The joint stiffness has a huge impact on the attainable active damping performance when using force sensors.
Thus, if Integral Force Feedback is to be used in a Stewart platform with flexible joints, the rotational stiffness of the joints should be minimized.
< / p >
< / div >
< / div >
< / div >
< / div >
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< div id = "outline-container-orgabec4e1" class = "outline-2" >
< h2 id = "orgabec4e1" > < span class = "section-number-2" > 3< / span > Direct Velocity Feedback< / h2 >
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< div class = "outline-text-2" id = "text-3" >
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< p >
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< a id = "org63027d0" > < / a >
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< / p >
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< div class = "note" id = "orgfb739d8" >
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< p >
The Matlab script corresponding to this section is accessible < a href = "../matlab/active_damping_dvf.m" > here< / a > .
< / p >
< p >
To run the script, open the Simulink Project, and type < code > run active_damping_dvf.m< / code > .
< / p >
< / div >
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< / div >
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< div id = "outline-container-orga2d019b" class = "outline-3" >
< h3 id = "orga2d019b" > < span class = "section-number-3" > 3.1< / span > Identification of the Dynamics with perfect Joints< / h3 >
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< div class = "outline-text-3" id = "text-3-1" >
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< p >
We first initialize the Stewart platform without joint stiffness.
< / 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_p'< / span > , < span class = "org-string" > 'type_M'< / span > , < span class = "org-string" > 'spherical_p'< / 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 >
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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 > );
controller = initializeController(< span class = "org-string" > 'type'< / span > , < span class = "org-string" > 'open-loop'< / span > );
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< / pre >
< / div >
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< p >
And we identify the dynamics from force actuators to force sensors.
< / 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" > %% 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 >
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" > '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" > '/Stewart Platform'< / span > ], 1, < span class = "org-string" > 'openoutput'< / span > , [], < span class = "org-string" > 'dLm'< / span > ); io_i = io_i < span class = "org-type" > +< / span > 1; < span class = "org-comment" > % Relative Displacement Outputs [m]< / span >
< 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" > 'Dm1'< / span > , < span class = "org-string" > 'Dm2'< / span > , < span class = "org-string" > 'Dm3'< / span > , < span class = "org-string" > 'Dm4'< / span > , < span class = "org-string" > 'Dm5'< / span > , < span class = "org-string" > 'Dm6'< / span > };
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< / pre >
< / div >
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< p >
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The transfer function from actuator forces to relative motion sensors is shown in Figure < a href = "#org34c24ba" > 8< / a > .
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< / p >
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< div id = "org34c24ba" class = "figure" >
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< p > < img src = "figs/dvf_plant_coupling.png" alt = "dvf_plant_coupling.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 8: < / span > Transfer function from the Actuator force \(F_{i}\) to the Relative Motion Sensor \(D_{m,j}\) with \(i \neq j\) (< a href = "./figs/dvf_plant_coupling.png" > png< / a > , < a href = "./figs/dvf_plant_coupling.pdf" > pdf< / a > )< / p >
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< / div >
< / div >
< / div >
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< div id = "outline-container-org2875dd1" class = "outline-3" >
< h3 id = "org2875dd1" > < span class = "section-number-3" > 3.2< / span > Effect of the Flexible Joint stiffness and Actuator amplification on the Dynamics< / h3 >
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< div class = "outline-text-3" id = "text-3-2" >
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< p >
We add some stiffness and damping in the flexible joints and we re-identify the dynamics.
< / p >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > 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 > );
Gf = linearize(mdl, io, options);
Gf.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 > };
Gf.OutputName = {< span class = "org-string" > 'Dm1'< / span > , < span class = "org-string" > 'Dm2'< / span > , < span class = "org-string" > 'Dm3'< / span > , < span class = "org-string" > 'Dm4'< / span > , < span class = "org-string" > 'Dm5'< / span > , < span class = "org-string" > 'Dm6'< / span > };
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< / pre >
< / div >
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< p >
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We now use the amplified actuators and re-identify the dynamics
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > stewart = initializeAmplifiedStrutDynamics(stewart);
Ga = linearize(mdl, io, options);
Ga.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 > };
Ga.OutputName = {< span class = "org-string" > 'Dm1'< / span > , < span class = "org-string" > 'Dm2'< / span > , < span class = "org-string" > 'Dm3'< / span > , < span class = "org-string" > 'Dm4'< / span > , < span class = "org-string" > 'Dm5'< / span > , < span class = "org-string" > 'Dm6'< / span > };
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< / pre >
< / div >
< p >
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The new dynamics from force actuator to relative motion sensor is shown in Figure < a href = "#orgba524e3" > 9< / a > .
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< / p >
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< div id = "orgba524e3" class = "figure" >
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< p > < img src = "figs/dvf_plant_flexible_joint_decentralized.png" alt = "dvf_plant_flexible_joint_decentralized.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 9: < / span > Transfer function from the Actuator force \(F_{i}\) to the relative displacement sensor \(D_{m,i}\) (< a href = "./figs/dvf_plant_flexible_joint_decentralized.png" > png< / a > , < a href = "./figs/dvf_plant_flexible_joint_decentralized.pdf" > pdf< / a > )< / p >
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< / div >
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< / div >
< / div >
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< div id = "outline-container-org0cea759" class = "outline-3" >
< h3 id = "org0cea759" > < span class = "section-number-3" > 3.3< / span > Obtained Damping< / h3 >
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< div class = "outline-text-3" id = "text-3-3" >
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< p >
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The control is a performed in a decentralized manner.
The \(6 \times 6\) control is a diagonal matrix with pure derivative action on the diagonal:
\[ K(s) = g
\begin{bmatrix}
s & & \\
& \ddots & \\
& & s
\end{bmatrix} \]
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< / p >
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< p >
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The root locus is shown in figure < a href = "#org0436b4d" > 10< / a > .
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< / p >
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< div id = "org0436b4d" class = "figure" >
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< p > < img src = "figs/root_locus_dvf_rot_stiffness.png" alt = "root_locus_dvf_rot_stiffness.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 10: < / span > Root Locus plot with Direct Velocity Feedback when considering the Stiffness of flexible joints (< a href = "./figs/root_locus_dvf_rot_stiffness.png" > png< / a > , < a href = "./figs/root_locus_dvf_rot_stiffness.pdf" > pdf< / a > )< / p >
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< / div >
< / div >
< / div >
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< div id = "outline-container-orga866100" class = "outline-3" >
< h3 id = "orga866100" > < span class = "section-number-3" > 3.4< / span > Conclusion< / h3 >
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< div class = "outline-text-3" id = "text-3-4" >
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< div class = "important" id = "org2640d3c" >
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< p >
Joint stiffness does increase the resonance frequencies of the system but does not change the attainable damping when using relative motion sensors.
< / p >
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< / div >
< / div >
< / div >
< / div >
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< h2 id = "orgc7e2089" > < span class = "section-number-2" > 4< / span > Compliance and Transmissibility Comparison< / h2 >
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< h3 id = "org6ec3b9e" > < span class = "section-number-3" > 4.1< / span > Initialization< / h3 >
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We first initialize the Stewart platform without joint stiffness.
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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_p'< / span > , < span class = "org-string" > 'type_M'< / span > , < span class = "org-string" > 'spherical_p'< / 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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The rotation point of the ground is located at the origin of frame \(\{A\}\).
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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 > );
controller = initializeController(< span class = "org-string" > 'type'< / span > , < span class = "org-string" > 'open-loop'< / span > );
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< h3 id = "orge87554a" > < span class = "section-number-3" > 4.2< / span > Identification< / h3 >
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Let’ s first identify the transmissibility and compliance in the open-loop case.
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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, freqs] = computeCompliance();
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Now, let’ s identify the transmissibility and compliance for the Integral Force Feedback architecture.
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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 = (1e4< span class = "org-type" > /< / span > s)< span class = "org-type" > *< / span > eye(6);
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[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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And for the Direct Velocity Feedback.
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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 = 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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[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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< h3 id = "org1d70ccd" > < span class = "section-number-3" > 4.3< / span > Results< / h3 >
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< p > < img src = "figs/transmissibility_iff_dvf.png" alt = "transmissibility_iff_dvf.png" / >
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< p > < span class = "figure-number" > Figure 11: < / span > Obtained transmissibility for Open-Loop Control (Blue), Integral Force Feedback (Red) and Direct Velocity Feedback (Yellow) (< a href = "./figs/transmissibility_iff_dvf.png" > png< / a > , < a href = "./figs/transmissibility_iff_dvf.pdf" > pdf< / a > )< / p >
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< p > < img src = "figs/compliance_iff_dvf.png" alt = "compliance_iff_dvf.png" / >
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< p > < span class = "figure-number" > Figure 12: < / span > Obtained compliance for Open-Loop Control (Blue), Integral Force Feedback (Red) and Direct Velocity Feedback (Yellow) (< a href = "./figs/compliance_iff_dvf.png" > png< / a > , < a href = "./figs/compliance_iff_dvf.pdf" > pdf< / a > )< / p >
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< p > < img src = "figs/frobenius_norm_T_C_iff_dvf.png" alt = "frobenius_norm_T_C_iff_dvf.png" / >
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< p > < span class = "figure-number" > Figure 13: < / span > Frobenius norm of the Transmissibility and Compliance Matrices (< a href = "./figs/frobenius_norm_T_C_iff_dvf.png" > png< / a > , < a href = "./figs/frobenius_norm_T_C_iff_dvf.pdf" > pdf< / a > )< / p >
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< p class = "author" > Author: Dehaeze Thomas< / p >
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< p class = "date" > Created: 2021-01-08 ven. 15:30< / p >
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