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< a accesskey = "h" href = "./index.html" > UP < / a >
|
< a accesskey = "H" href = "./index.html" > HOME < / a >
< / div > < div id = "content" >
< h1 class = "title" > Stewart Platform - Tracking Control< / 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 = "#orgd7b25e5" > 1. Decentralized Control Architecture using Strut Length< / a >
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< ul >
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< li > < a href = "#orgf22ae1f" > 1.1. Control Schematic< / a > < / li >
< li > < a href = "#orgfbc962f" > 1.2. Initialize the Stewart platform< / a > < / li >
< li > < a href = "#org5b50e6c" > 1.3. Identification of the plant< / a > < / li >
< li > < a href = "#org127af6e" > 1.4. Plant Analysis< / a > < / li >
< li > < a href = "#org64fe247" > 1.5. Controller Design< / a > < / li >
< li > < a href = "#org30afd72" > 1.6. Simulation< / a > < / li >
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< li > < a href = "#org974b430" > 1.7. Results< / a > < / li >
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< li > < a href = "#orge3e2e02" > 1.8. Conclusion< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#orga519721" > 2. Centralized Control Architecture using Pose Measurement< / a >
< ul >
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< li > < a href = "#org3846e3e" > 2.1. Control Schematic< / a > < / li >
< li > < a href = "#orga001fab" > 2.2. Initialize the Stewart platform< / a > < / li >
< li > < a href = "#orgdff5afa" > 2.3. Identification of the plant< / a > < / li >
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< li > < a href = "#org2223469" > 2.4. Diagonal Control - Leg’ s Frame< / a >
< ul >
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< li > < a href = "#org224a0bb" > 2.4.1. Control Architecture< / a > < / li >
< li > < a href = "#org7cd17a1" > 2.4.2. Plant Analysis< / a > < / li >
< li > < a href = "#orgfe68d27" > 2.4.3. Controller Design< / a > < / li >
< li > < a href = "#org2481134" > 2.4.4. Simulation< / a > < / li >
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< / ul >
< / li >
< li > < a href = "#org26a8857" > 2.5. Diagonal Control - Cartesian Frame< / a >
< ul >
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< li > < a href = "#org0831ba6" > 2.5.1. Control Architecture< / a > < / li >
< li > < a href = "#orga750816" > 2.5.2. Plant Analysis< / a > < / li >
< li > < a href = "#org925664d" > 2.5.3. Controller Design< / a > < / li >
< li > < a href = "#org42d3b74" > 2.5.4. Simulation< / a > < / li >
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< / ul >
< / li >
< li > < a href = "#orgad7bc54" > 2.6. Diagonal Control - Steady State Decoupling< / a >
< ul >
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< li > < a href = "#org51231d3" > 2.6.1. Control Architecture< / a > < / li >
< li > < a href = "#org0328e3c" > 2.6.2. Plant Analysis< / a > < / li >
< li > < a href = "#orge8a14d5" > 2.6.3. Controller Design< / a > < / li >
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< / ul >
< / li >
< li > < a href = "#orga2eadeb" > 2.7. Comparison< / a >
< ul >
< li > < a href = "#org09ae901" > 2.7.1. Obtained MIMO Controllers< / a > < / li >
< li > < a href = "#org23ae479" > 2.7.2. Simulation Results< / a > < / li >
< / ul >
< / li >
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< li > < a href = "#org94c3e48" > 2.8. Conclusion< / a > < / li >
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< / ul >
< / li >
< li > < a href = "#org4b8c360" > 3. Hybrid Control Architecture - HAC-LAC with relative DVF< / a >
< ul >
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< li > < a href = "#org2183826" > 3.1. Control Schematic< / a > < / li >
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< / ul >
< / li >
< li > < a href = "#org445f7a9" > 4. Position Error computation< / a > < / li >
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< / ul >
< / div >
< / div >
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< p >
Let’ s suppose the control objective is to position \(\bm{\mathcal{X}}\) of the mobile platform of the Stewart platform such that it is following some reference position \(\bm{r}_\mathcal{X}\).
< / p >
< p >
Depending of the measured quantity, different control architectures can be used:
< / p >
< ul class = "org-ul" >
< li > If the struts length \(\bm{\mathcal{L}}\) is measured, a decentralized control architecture can be used (Section < a href = "#orgea7df6c" > 1< / a > )< / li >
< li > If the pose of the mobile platform \(\bm{\mathcal{X}}\) is directly measured, a centralized control architecture can be used (Section < a href = "#org48604d1" > 2< / a > )< / li >
< li > If both \(\bm{\mathcal{L}}\) and \(\bm{\mathcal{X}}\) are measured, we can use an hybrid control architecture (Section < a href = "#org14e3e5f" > 3< / a > )< / li >
< / ul >
< div id = "outline-container-orgd7b25e5" class = "outline-2" >
< h2 id = "orgd7b25e5" > < span class = "section-number-2" > 1< / span > Decentralized Control Architecture using Strut Length< / h2 >
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< div class = "outline-text-2" id = "text-1" >
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< p >
< a id = "orgea7df6c" > < / a >
< / p >
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< / div >
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< div id = "outline-container-orgf22ae1f" class = "outline-3" >
< h3 id = "orgf22ae1f" > < span class = "section-number-3" > 1.1< / span > Control Schematic< / h3 >
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< div class = "outline-text-3" id = "text-1-1" >
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< p >
The control architecture is shown in Figure < a href = "#org4f704a1" > 1< / a > .
< / p >
< p >
The required leg length \(\bm{r}_\mathcal{L}\) is computed from the reference path \(\bm{r}_\mathcal{X}\) using the inverse kinematics.
< / p >
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< p >
Then, a diagonal (decentralized) controller \(\bm{K}_\mathcal{L}\) is used such that each leg lengths stays close to its required length.
< / p >
< div id = "org4f704a1" class = "figure" >
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< p > < img src = "figs/control_measure_rotating_2dof.png" alt = "control_measure_rotating_2dof.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 1: < / span > Decentralized control for reference tracking< / p >
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< / div >
< / div >
< / div >
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< div id = "outline-container-orgfbc962f" class = "outline-3" >
< h3 id = "orgfbc962f" > < span class = "section-number-3" > 1.2< / span > Initialize the Stewart platform< / h3 >
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< div class = "outline-text-3" id = "text-1-2" >
< div class = "org-src-container" >
< 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);
< / 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" > 'rigid'< / span > );
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payload = initializePayload(< span class = "org-string" > 'type'< / span > , < span class = "org-string" > 'none'< / span > );
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controller = initializeController(< span class = "org-string" > 'type'< / span > , < span class = "org-string" > 'open-loop'< / span > );
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > disturbances = initializeDisturbances();
references = initializeReferences(stewart);
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< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-org5b50e6c" class = "outline-3" >
< h3 id = "org5b50e6c" > < span class = "section-number-3" > 1.3< / span > Identification of the plant< / h3 >
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< div class = "outline-text-3" id = "text-1-3" >
< p >
Let’ s identify the transfer function from \(\bm{\tau}\) to \(\bm{L}\).
< / p >
< div class = "org-src-container" >
< 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" > '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);
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" > 'L1'< / span > , < span class = "org-string" > 'L2'< / span > , < span class = "org-string" > 'L3'< / span > , < span class = "org-string" > 'L4'< / span > , < span class = "org-string" > 'L5'< / span > , < span class = "org-string" > 'L6'< / span > };
< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-org127af6e" class = "outline-3" >
< h3 id = "org127af6e" > < span class = "section-number-3" > 1.4< / span > Plant Analysis< / h3 >
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< div class = "outline-text-3" id = "text-1-4" >
< p >
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The diagonal terms of the plant is shown in Figure < a href = "#org8c82316" > 2< / a > .
< / p >
< p >
All the diagonal terms are equal.
< / p >
< div id = "org8c82316" class = "figure" >
< p > < img src = "figs/plant_decentralized_diagonal.png" alt = "plant_decentralized_diagonal.png" / >
< / p >
< p > < span class = "figure-number" > Figure 2: < / span > Diagonal Elements of the Plant (< a href = "./figs/plant_decentralized_diagonal.png" > png< / a > , < a href = "./figs/plant_decentralized_diagonal.pdf" > pdf< / a > )< / p >
< / div >
< p >
The off-diagonal terms are shown in Figure < a href = "#org2ba5482" > 3< / a > .
< / p >
< p >
We see that the plant is decoupled at low frequency which indicate that decentralized control may be a good idea.
< / p >
< div id = "org2ba5482" class = "figure" >
< p > < img src = "figs/plant_decentralized_off_diagonal.png" alt = "plant_decentralized_off_diagonal.png" / >
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< / p >
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< p > < span class = "figure-number" > Figure 3: < / span > Diagonal Elements of the Plant (< a href = "./figs/plant_decentralized_off_diagonal.png" > png< / a > , < a href = "./figs/plant_decentralized_off_diagonal.pdf" > pdf< / a > )< / p >
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< / div >
< / div >
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< / div >
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< div id = "outline-container-org64fe247" class = "outline-3" >
< h3 id = "org64fe247" > < span class = "section-number-3" > 1.5< / span > Controller Design< / h3 >
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< div class = "outline-text-3" id = "text-1-5" >
< p >
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The controller consists of:
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< / p >
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< ul class = "org-ul" >
< li > A pure integrator< / li >
< li > A lead around the crossover frequency to increase the phase margin< / li >
< li > A low pass filter with a cut-off frequency 3 times the crossover to increase the gain margin< / li >
< / ul >
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< p >
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The obtained loop gains corresponding to the diagonal elements are shown in Figure < a href = "#org08e591a" > 4< / a > .
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > wc = 2< span class = "org-type" > *< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > *< / span > 30;
Kl = diag(1< span class = "org-type" > ./< / span > diag(abs(freqresp(G, wc)))) < span class = "org-type" > *< / span > wc< span class = "org-type" > /< / span > s < span class = "org-type" > *< / span > 1< span class = "org-type" > /< / span > (1 < span class = "org-type" > +< / span > s< span class = "org-type" > /< / span > 3< span class = "org-type" > /< / span > wc);
< / pre >
< / div >
< div id = "org08e591a" class = "figure" >
< p > < img src = "figs/loop_gain_decentralized_L.png" alt = "loop_gain_decentralized_L.png" / >
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< / p >
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< p > < span class = "figure-number" > Figure 4: < / span > Loop Gain of the diagonal elements (< a href = "./figs/loop_gain_decentralized_L.png" > png< / a > , < a href = "./figs/loop_gain_decentralized_L.pdf" > pdf< / a > )< / p >
< / div >
< / div >
< / div >
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< div id = "outline-container-org30afd72" class = "outline-3" >
< h3 id = "org30afd72" > < span class = "section-number-3" > 1.6< / span > Simulation< / h3 >
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< div class = "outline-text-3" id = "text-1-6" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > t = linspace(0, 10, 1000);
r = zeros(6, length(t));
r(1, < span class = "org-type" > :< / span > ) = 5e< span class = "org-type" > -< / span > 3< span class = "org-type" > *< / span > sin(2< span class = "org-type" > *< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > *< / span > t);
references = initializeReferences(stewart, < span class = "org-string" > 't'< / span > , t, < span class = "org-string" > 'r'< / span > , r);
< / pre >
< / div >
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< div class = "org-src-container" >
< pre class = "src src-matlab" > controller = initializeController(< span class = "org-string" > 'type'< / span > , < span class = "org-string" > 'ref-track-L'< / span > );
< / pre >
< / div >
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< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-matlab-simulink-keyword" > sim< / span > (< span class = "org-string" > 'stewart_platform_model'< / span > )
simout_D = simout;
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< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-org974b430" class = "outline-3" >
< h3 id = "org974b430" > < span class = "section-number-3" > 1.7< / span > Results< / h3 >
< div class = "outline-text-3" id = "text-1-7" >
< div id = "org1ac9124" class = "figure" >
< p > < img src = "figs/decentralized_control_Ex.png" alt = "decentralized_control_Ex.png" / >
< / p >
< p > < span class = "figure-number" > Figure 5: < / span > Position error \(\bm{\epsilon}_\mathcal{X}\) (< a href = "./figs/decentralized_control_Ex.png" > png< / a > , < a href = "./figs/decentralized_control_Ex.pdf" > pdf< / a > )< / p >
< / div >
< div id = "org10eb8ae" class = "figure" >
< p > < img src = "figs/decentralized_control_El.png" alt = "decentralized_control_El.png" / >
< / p >
< p > < span class = "figure-number" > Figure 6: < / span > Position error \(\bm{\epsilon}_\mathcal{L}\) (< a href = "./figs/decentralized_control_El.png" > png< / a > , < a href = "./figs/decentralized_control_El.pdf" > pdf< / a > )< / p >
< / div >
< / div >
< / div >
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< div id = "outline-container-orge3e2e02" class = "outline-3" >
< h3 id = "orge3e2e02" > < span class = "section-number-3" > 1.8< / span > Conclusion< / h3 >
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< div class = "outline-text-3" id = "text-1-8" >
< p >
Such control architecture is easy to implement and give good results.
The inverse kinematics is easy to compute.
< / p >
< p >
However, as \(\mathcal{X}\) is not directly measured, it is possible that important positioning errors are due to finite stiffness of the joints and other imperfections.
< / p >
< / div >
< / div >
< / div >
< div id = "outline-container-orga519721" class = "outline-2" >
< h2 id = "orga519721" > < span class = "section-number-2" > 2< / span > Centralized Control Architecture using Pose Measurement< / h2 >
< div class = "outline-text-2" id = "text-2" >
< p >
< a id = "org48604d1" > < / a >
< / p >
< / div >
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< div id = "outline-container-org3846e3e" class = "outline-3" >
< h3 id = "org3846e3e" > < span class = "section-number-3" > 2.1< / span > Control Schematic< / h3 >
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< div class = "outline-text-3" id = "text-2-1" >
< p >
The centralized controller takes the position error \(\bm{\epsilon}_\mathcal{X}\) as an inputs and generate actuator forces \(\bm{\tau}\) (see Figure < a href = "#org97ec686" > 7< / a > ).
The signals are:
< / p >
< ul class = "org-ul" >
< li > reference path \(\bm{r}_\mathcal{X} = \begin{bmatrix} \epsilon_x & \epsilon_y & \epsilon_z & \epsilon_{R_x} & \epsilon_{R_y} & \epsilon_{R_z} \end{bmatrix}\)< / li >
< li > tracking error \(\bm{\epsilon}_\mathcal{X} = \begin{bmatrix} \epsilon_x & \epsilon_y & \epsilon_z & \epsilon_{R_x} & \epsilon_{R_y} & \epsilon_{R_z} \end{bmatrix}\)< / li >
< li > actuator forces \(\bm{\tau} = \begin{bmatrix} \tau_1 & \tau_2 & \tau_3 & \tau_4 & \tau_5 & \tau_6 \end{bmatrix}\)< / li >
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< li > payload pose \(\bm{\mathcal{X}} = \begin{bmatrix} x & y & z & R_x & R_y & R_z \end{bmatrix}\)< / li >
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< / ul >
< div id = "org97ec686" class = "figure" >
< p > < img src = "figs/centralized_reference_tracking.png" alt = "centralized_reference_tracking.png" / >
< / p >
< p > < span class = "figure-number" > Figure 7: < / span > Centralized Controller< / p >
< / div >
< p >
Instead of designing a full MIMO controller \(K\), we first try to make the plant more diagonal by pre- or post-multiplying some constant matrix, then we design a diagonal controller.
< / p >
< p >
We can think of two ways to make the plant more diagonal that are described in sections < a href = "#org31fd942" > 2.4< / a > and < a href = "#orgfd201c3" > 2.5< / a > .
< / p >
< div class = "important" >
< p >
Note here that the subtraction shown in Figure < a href = "#org97ec686" > 7< / a > is not a real subtraction.
It is indeed a more complex computation explained in section < a href = "#org5f61540" > 4< / a > .
< / p >
< / div >
< / div >
< / div >
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< div id = "outline-container-orga001fab" class = "outline-3" >
< h3 id = "orga001fab" > < span class = "section-number-3" > 2.2< / span > Initialize the Stewart platform< / h3 >
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< div class = "outline-text-3" id = "text-2-2" >
< div class = "org-src-container" >
< 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);
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > ground = initializeGround(< span class = "org-string" > 'type'< / span > , < span class = "org-string" > 'rigid'< / span > );
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 > );
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > disturbances = initializeDisturbances();
references = initializeReferences(stewart);
< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-orgdff5afa" class = "outline-3" >
< h3 id = "orgdff5afa" > < span class = "section-number-3" > 2.3< / span > Identification of the plant< / h3 >
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< div class = "outline-text-3" id = "text-2-3" >
< p >
Let’ s identify the transfer function from \(\bm{\tau}\) to \(\bm{L}\).
< / p >
< div class = "org-src-container" >
< 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" > '/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" > % Relative Displacement Outputs [m]< / 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" > 'Dx'< / span > , < span class = "org-string" > 'Dy'< / span > , < span class = "org-string" > 'Dz'< / span > , < span class = "org-string" > 'Rx'< / span > , < span class = "org-string" > 'Ry'< / span > , < span class = "org-string" > 'Rz'< / span > };
< / pre >
< / div >
< / div >
< / div >
< div id = "outline-container-org2223469" class = "outline-3" >
< h3 id = "org2223469" > < span class = "section-number-3" > 2.4< / span > Diagonal Control - Leg’ s Frame< / h3 >
< div class = "outline-text-3" id = "text-2-4" >
< p >
< a id = "org31fd942" > < / a >
< / p >
< / div >
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< div id = "outline-container-org224a0bb" class = "outline-4" >
< h4 id = "org224a0bb" > < span class = "section-number-4" > 2.4.1< / span > Control Architecture< / h4 >
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< div class = "outline-text-4" id = "text-2-4-1" >
< p >
The pose error \(\bm{\epsilon}_\mathcal{X}\) is first converted in the frame of the leg by using the Jacobian matrix.
Then a diagonal controller \(\bm{K}_\mathcal{L}\) is designed.
The final implemented controller is \(\bm{K} = \bm{K}_\mathcal{L} \cdot \bm{J}\) as shown in Figure < a href = "#orgb1f5ad2" > 8< / a >
< / p >
< p >
Note here that the transformation from the pose error \(\bm{\epsilon}_\mathcal{X}\) to the leg’ s length errors by using the Jacobian matrix is only valid for small errors.
< / p >
< div id = "orgb1f5ad2" class = "figure" >
< p > < img src = "figs/centralized_reference_tracking_L.png" alt = "centralized_reference_tracking_L.png" / >
< / p >
< p > < span class = "figure-number" > Figure 8: < / span > Controller in the frame of the legs< / p >
< / div >
< / div >
< / div >
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< div id = "outline-container-org7cd17a1" class = "outline-4" >
< h4 id = "org7cd17a1" > < span class = "section-number-4" > 2.4.2< / span > Plant Analysis< / h4 >
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< div class = "outline-text-4" id = "text-2-4-2" >
< p >
We now multiply the plant by the Jacobian matrix as shown in Figure < a href = "#orgb1f5ad2" > 8< / a > to obtain a more diagonal plant.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > Gl = stewart.kinematics.J< span class = "org-type" > *< / span > G;
Gl.OutputName = {< span class = "org-string" > 'D1'< / span > , < span class = "org-string" > 'D2'< / span > , < span class = "org-string" > 'D3'< / span > , < span class = "org-string" > 'D4'< / span > , < span class = "org-string" > 'D5'< / span > , < span class = "org-string" > 'D6'< / span > };
< / pre >
< / div >
< div id = "org6658ce5" class = "figure" >
< p > < img src = "figs/plant_centralized_diagonal_L.png" alt = "plant_centralized_diagonal_L.png" / >
< / p >
< p > < span class = "figure-number" > Figure 9: < / span > Diagonal Elements of the plant \(\bm{J} \bm{G}\) (< a href = "./figs/plant_centralized_diagonal_L.png" > png< / a > , < a href = "./figs/plant_centralized_diagonal_L.pdf" > pdf< / a > )< / p >
< / div >
< p >
All the diagonal elements are identical.
This will simplify the design of the controller as all the elements of the diagonal controller can be made identical.
< / p >
< p >
The off-diagonal terms of the controller are shown in Figure < a href = "#orgba050e4" > 10< / a > .
< / p >
< div id = "orgba050e4" class = "figure" >
< p > < img src = "figs/plant_centralized_off_diagonal_L.png" alt = "plant_centralized_off_diagonal_L.png" / >
< / p >
< p > < span class = "figure-number" > Figure 10: < / span > Off Diagonal Elements of the plant \(\bm{J} \bm{G}\) (< a href = "./figs/plant_centralized_off_diagonal_L.png" > png< / a > , < a href = "./figs/plant_centralized_off_diagonal_L.pdf" > pdf< / a > )< / p >
< / div >
< p >
We can see that this < b > totally decouples the system at low frequency< / b > .
< / p >
< p >
This was expected since:
\[ \bm{G}(\omega = 0) = \frac{\delta\bm{\mathcal{X}}}{\delta\bm{\tau}}(\omega = 0) = \bm{J}^{-1} \frac{\delta\bm{\mathcal{L}}}{\delta\bm{\tau}}(\omega = 0) = \bm{J}^{-1} \text{diag}(\mathcal{K}_1^{-1} \ \dots \ \mathcal{K}_6^{-1}) \]
< / p >
< p >
Thus \(J \cdot G(\omega = 0) = J \cdot \frac{\delta\bm{\mathcal{X}}}{\delta\bm{\tau}}(\omega = 0)\) is a diagonal matrix containing the inverse of the joint’ s stiffness.
< / p >
< / div >
< / div >
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< div id = "outline-container-orgfe68d27" class = "outline-4" >
< h4 id = "orgfe68d27" > < span class = "section-number-4" > 2.4.3< / span > Controller Design< / h4 >
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< div class = "outline-text-4" id = "text-2-4-3" >
< p >
The controller consists of:
< / p >
< ul class = "org-ul" >
< li > A pure integrator< / li >
< li > A lead around the crossover frequency to increase the phase margin< / li >
< li > A low pass filter with a cut-off frequency 3 times the crossover to increase the gain margin< / li >
< / ul >
< p >
The obtained loop gains corresponding to the diagonal elements are shown in Figure < a href = "#orga803083" > 11< / a > .
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > wc = 2< span class = "org-type" > *< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > *< / span > 30;
Kl = diag(1< span class = "org-type" > ./< / span > diag(abs(freqresp(Gl, wc)))) < span class = "org-type" > *< / span > wc< span class = "org-type" > /< / span > s < span class = "org-type" > *< / span > 1< span class = "org-type" > /< / span > (1 < span class = "org-type" > +< / span > s< span class = "org-type" > /< / span > 3< span class = "org-type" > /< / span > wc);
< / pre >
< / div >
< div id = "orga803083" class = "figure" >
< p > < img src = "figs/loop_gain_centralized_L.png" alt = "loop_gain_centralized_L.png" / >
< / p >
< p > < span class = "figure-number" > Figure 11: < / span > Loop Gain of the diagonal elements (< a href = "./figs/loop_gain_centralized_L.png" > png< / a > , < a href = "./figs/loop_gain_centralized_L.pdf" > pdf< / a > )< / p >
< / div >
< p >
The controller \(\bm{K} = \bm{K}_\mathcal{L} \bm{J}\) is computed.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > K = Kl< span class = "org-type" > *< / span > stewart.kinematics.J;
< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-org2481134" class = "outline-4" >
< h4 id = "org2481134" > < span class = "section-number-4" > 2.4.4< / span > Simulation< / h4 >
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< div class = "outline-text-4" id = "text-2-4-4" >
< p >
We specify the reference path to follow.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > t = linspace(0, 10, 1000);
r = zeros(6, length(t));
r(1, < span class = "org-type" > :< / span > ) = 5e< span class = "org-type" > -< / span > 3< span class = "org-type" > *< / span > sin(2< span class = "org-type" > *< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > *< / span > t);
references = initializeReferences(stewart, < span class = "org-string" > 't'< / span > , t, < span class = "org-string" > 'r'< / span > , r);
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > controller = initializeController(< span class = "org-string" > 'type'< / span > , < span class = "org-string" > 'ref-track-X'< / span > );
< / pre >
< / div >
< p >
We run the simulation and we save the results.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-matlab-simulink-keyword" > sim< / span > (< span class = "org-string" > 'stewart_platform_model'< / span > )
simout_L = simout;
< / pre >
< / div >
< / div >
< / div >
< / div >
< div id = "outline-container-org26a8857" class = "outline-3" >
< h3 id = "org26a8857" > < span class = "section-number-3" > 2.5< / span > Diagonal Control - Cartesian Frame< / h3 >
< div class = "outline-text-3" id = "text-2-5" >
< p >
< a id = "orgfd201c3" > < / a >
< / p >
< / div >
2020-03-11 19:01:50 +01:00
< div id = "outline-container-org0831ba6" class = "outline-4" >
< h4 id = "org0831ba6" > < span class = "section-number-4" > 2.5.1< / span > Control Architecture< / h4 >
2020-03-11 19:00:22 +01:00
< div class = "outline-text-4" id = "text-2-5-1" >
< p >
A diagonal controller \(\bm{K}_\mathcal{X}\) take the pose error \(\bm{\epsilon}_\mathcal{X}\) and generate cartesian forces \(\bm{\mathcal{F}}\) that are then converted to actuators forces using the Jacobian as shown in Figure e < a href = "#org6b158db" > 12< / a > .
< / p >
< p >
The final implemented controller is \(\bm{K} = \bm{J}^{-T} \cdot \bm{K}_\mathcal{X}\).
< / p >
< div id = "org6b158db" class = "figure" >
< p > < img src = "figs/centralized_reference_tracking_X.png" alt = "centralized_reference_tracking_X.png" / >
< / p >
< p > < span class = "figure-number" > Figure 12: < / span > Controller in the cartesian frame< / p >
< / div >
< / div >
< / div >
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< div id = "outline-container-orga750816" class = "outline-4" >
< h4 id = "orga750816" > < span class = "section-number-4" > 2.5.2< / span > Plant Analysis< / h4 >
2020-03-11 19:00:22 +01:00
< div class = "outline-text-4" id = "text-2-5-2" >
< p >
We now multiply the plant by the Jacobian matrix as shown in Figure < a href = "#org6b158db" > 12< / a > to obtain a more diagonal plant.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > Gx = G< span class = "org-type" > *< / span > inv(stewart.kinematics.J< span class = "org-type" > '< / span > );
Gx.InputName = {< span class = "org-string" > 'Fx'< / span > , < span class = "org-string" > 'Fy'< / span > , < span class = "org-string" > 'Fz'< / span > , < span class = "org-string" > 'Mx'< / span > , < span class = "org-string" > 'My'< / span > , < span class = "org-string" > 'Mz'< / span > };
< / pre >
< / div >
< div id = "org2b61181" class = "figure" >
< p > < img src = "figs/plant_centralized_diagonal_X.png" alt = "plant_centralized_diagonal_X.png" / >
< / p >
< p > < span class = "figure-number" > Figure 13: < / span > Diagonal Elements of the plant \(\bm{G} \bm{J}^{-T}\) (< a href = "./figs/plant_centralized_diagonal_X.png" > png< / a > , < a href = "./figs/plant_centralized_diagonal_X.pdf" > pdf< / a > )< / p >
< / div >
< p >
The diagonal terms are not the same.
The resonances of the system are “ decoupled” .
For instance, the vertical resonance of the system is only present on the diagonal term corresponding to \(D_z/\mathcal{F}_z\).
< / p >
< div id = "org1ff5b9c" class = "figure" >
< p > < img src = "figs/plant_centralized_off_diagonal_X.png" alt = "plant_centralized_off_diagonal_X.png" / >
< / p >
< p > < span class = "figure-number" > Figure 14: < / span > Off Diagonal Elements of the plant \(\bm{G} \bm{J}^{-T}\) (< a href = "./figs/plant_centralized_off_diagonal_X.png" > png< / a > , < a href = "./figs/plant_centralized_off_diagonal_X.pdf" > pdf< / a > )< / p >
< / div >
< p >
Here the system is almost decoupled at all frequencies except for the transfer functions \(\frac{R_y}{\mathcal{F}_x}\) and \(\frac{R_x}{\mathcal{F}_y}\).
< / p >
< p >
This is due to the fact that the compliance matrix of the Stewart platform is not diagonal.
< / 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 >
< td class = "org-right" > 4.75e-08< / td >
< td class = "org-right" > -1.9751e-24< / td >
< td class = "org-right" > 7.3536e-25< / td >
< td class = "org-right" > 5.915e-23< / td >
< td class = "org-right" > 3.2093e-07< / td >
< td class = "org-right" > 5.8696e-24< / td >
< / tr >
< tr >
< td class = "org-right" > -7.1302e-25< / td >
< td class = "org-right" > 4.75e-08< / td >
< td class = "org-right" > 2.8866e-25< / td >
< td class = "org-right" > -3.2093e-07< / td >
< td class = "org-right" > -5.38e-24< / td >
< td class = "org-right" > -3.2725e-23< / td >
< / tr >
< tr >
< td class = "org-right" > 7.9012e-26< / td >
< td class = "org-right" > -6.3991e-25< / td >
< td class = "org-right" > 2.099e-08< / td >
< td class = "org-right" > 1.9073e-23< / td >
< td class = "org-right" > 5.3384e-25< / td >
< td class = "org-right" > -6.4867e-40< / td >
< / tr >
< tr >
< td class = "org-right" > 1.3724e-23< / td >
< td class = "org-right" > -3.2093e-07< / td >
< td class = "org-right" > 1.2799e-23< / td >
< td class = "org-right" > 5.1863e-06< / td >
< td class = "org-right" > 4.9412e-22< / td >
< td class = "org-right" > -3.8269e-24< / td >
< / tr >
< tr >
< td class = "org-right" > 3.2093e-07< / td >
< td class = "org-right" > 7.6013e-24< / td >
< td class = "org-right" > 1.2239e-23< / td >
< td class = "org-right" > 6.8886e-22< / td >
< td class = "org-right" > 5.1863e-06< / td >
< td class = "org-right" > -2.6025e-22< / td >
< / tr >
< tr >
< td class = "org-right" > 7.337e-24< / td >
< td class = "org-right" > -3.2395e-23< / td >
< td class = "org-right" > -1.571e-39< / td >
< td class = "org-right" > 9.927e-23< / td >
< td class = "org-right" > -3.2531e-22< / td >
< td class = "org-right" > 1.7073e-06< / td >
< / tr >
< / tbody >
< / table >
< p >
One way to have this compliance matrix diagonal (and thus having a decoupled plant at DC) is to use a < b > cubic architecture< / b > with the center of the cube’ s coincident with frame \(\{A\}\).
< / p >
< p >
This control architecture can also give a dynamically decoupled plant if the Center of mass of the payload is also coincident with frame \(\{A\}\).
< / p >
< / div >
< / div >
2020-03-11 19:01:50 +01:00
< div id = "outline-container-org925664d" class = "outline-4" >
< h4 id = "org925664d" > < span class = "section-number-4" > 2.5.3< / span > Controller Design< / h4 >
2020-03-11 19:00:22 +01:00
< div class = "outline-text-4" id = "text-2-5-3" >
< p >
The controller consists of:
< / p >
< ul class = "org-ul" >
< li > A pure integrator< / li >
< li > A lead around the crossover frequency to increase the phase margin< / li >
< li > A low pass filter with a cut-off frequency 3 times the crossover to increase the gain margin< / li >
< / ul >
< p >
The obtained loop gains corresponding to the diagonal elements are shown in Figure < a href = "#org9051c86" > 15< / a > .
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > wc = 2< span class = "org-type" > *< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > *< / span > 30;
Kx = diag(1< span class = "org-type" > ./< / span > diag(abs(freqresp(Gx, wc)))) < span class = "org-type" > *< / span > wc< span class = "org-type" > /< / span > s < span class = "org-type" > *< / span > 1< span class = "org-type" > /< / span > (1 < span class = "org-type" > +< / span > s< span class = "org-type" > /< / span > 3< span class = "org-type" > /< / span > wc);
< / pre >
< / div >
< div id = "org9051c86" class = "figure" >
< p > < img src = "figs/loop_gain_centralized_X.png" alt = "loop_gain_centralized_X.png" / >
< / p >
< p > < span class = "figure-number" > Figure 15: < / span > Loop Gain of the diagonal elements (< a href = "./figs/loop_gain_centralized_X.png" > png< / a > , < a href = "./figs/loop_gain_centralized_X.pdf" > pdf< / a > )< / p >
< / div >
< p >
The controller \(\bm{K} = \bm{J}^{-T} \bm{K}_\mathcal{X}\) is computed.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > K = inv(stewart.kinematics.J< span class = "org-type" > '< / span > )< span class = "org-type" > *< / span > Kx;
< / pre >
< / div >
< / div >
< / div >
2020-03-11 19:01:50 +01:00
< div id = "outline-container-org42d3b74" class = "outline-4" >
< h4 id = "org42d3b74" > < span class = "section-number-4" > 2.5.4< / span > Simulation< / h4 >
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< div class = "outline-text-4" id = "text-2-5-4" >
< p >
We specify the reference path to follow.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > t = linspace(0, 10, 1000);
r = zeros(6, length(t));
r(1, < span class = "org-type" > :< / span > ) = 5e< span class = "org-type" > -< / span > 3< span class = "org-type" > *< / span > sin(2< span class = "org-type" > *< / span > < span class = "org-constant" > pi< / span > < span class = "org-type" > *< / span > t);
references = initializeReferences(stewart, < span class = "org-string" > 't'< / span > , t, < span class = "org-string" > 'r'< / span > , r);
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > controller = initializeController(< span class = "org-string" > 'type'< / span > , < span class = "org-string" > 'ref-track-X'< / span > );
< / pre >
< / div >
< p >
We run the simulation and we save the results.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-matlab-simulink-keyword" > sim< / span > (< span class = "org-string" > 'stewart_platform_model'< / span > )
simout_X = simout;
< / pre >
< / div >
< / div >
< / div >
< / div >
< div id = "outline-container-orgad7bc54" class = "outline-3" >
< h3 id = "orgad7bc54" > < span class = "section-number-3" > 2.6< / span > Diagonal Control - Steady State Decoupling< / h3 >
< div class = "outline-text-3" id = "text-2-6" >
< p >
< a id = "org789ba4a" > < / a >
< / p >
< / div >
2020-03-11 19:01:50 +01:00
< div id = "outline-container-org51231d3" class = "outline-4" >
< h4 id = "org51231d3" > < span class = "section-number-4" > 2.6.1< / span > Control Architecture< / h4 >
2020-03-11 19:00:22 +01:00
< div class = "outline-text-4" id = "text-2-6-1" >
< p >
The plant \(\bm{G}\) is pre-multiply by \(\bm{G}^{-1}(\omega = 0)\) such that the “ shaped plant” \(\bm{G}_0 = \bm{G} \bm{G}^{-1}(\omega = 0)\) is diagonal at low frequency.
< / p >
< p >
Then a diagonal controller \(\bm{K}_0\) is designed.
< / p >
< p >
The control architecture is shown in Figure < a href = "#orgb226e44" > 16< / a > .
< / p >
< div id = "orgb226e44" class = "figure" >
< p > < img src = "figs/centralized_reference_tracking_S.png" alt = "centralized_reference_tracking_S.png" / >
< / p >
< p > < span class = "figure-number" > Figure 16: < / span > Static Decoupling of the Plant< / p >
< / div >
< / div >
< / div >
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< div id = "outline-container-org0328e3c" class = "outline-4" >
< h4 id = "org0328e3c" > < span class = "section-number-4" > 2.6.2< / span > Plant Analysis< / h4 >
2020-03-11 19:00:22 +01:00
< div class = "outline-text-4" id = "text-2-6-2" >
< p >
The plant is pre-multiplied by \(\bm{G}^{-1}(\omega = 0)\).
The diagonal elements of the shaped plant are shown in Figure < a href = "#orgc15aa83" > 17< / a > .
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > G0 = G< span class = "org-type" > *< / span > inv(freqresp(G, 0));
< / pre >
< / div >
< div id = "orgc15aa83" class = "figure" >
< p > < img src = "figs/plant_centralized_diagonal_SD.png" alt = "plant_centralized_diagonal_SD.png" / >
< / p >
< p > < span class = "figure-number" > Figure 17: < / span > Diagonal Elements of the plant \(\bm{G} \bm{G}^{-1}(\omega = 0)\) (< a href = "./figs/plant_centralized_diagonal_SD.png" > png< / a > , < a href = "./figs/plant_centralized_diagonal_SD.pdf" > pdf< / a > )< / p >
< / div >
< div id = "orga6b8b41" class = "figure" >
< p > < img src = "figs/plant_centralized_off_diagonal_SD.png" alt = "plant_centralized_off_diagonal_SD.png" / >
< / p >
< p > < span class = "figure-number" > Figure 18: < / span > Off Diagonal Elements of the plant \(\bm{G} \bm{J}^{-T}\) (< a href = "./figs/plant_centralized_off_diagonal_SD.png" > png< / a > , < a href = "./figs/plant_centralized_off_diagonal_SD.pdf" > pdf< / a > )< / p >
< / div >
< / div >
< / div >
2020-03-11 19:01:50 +01:00
< div id = "outline-container-orge8a14d5" class = "outline-4" >
< h4 id = "orge8a14d5" > < span class = "section-number-4" > 2.6.3< / span > Controller Design< / h4 >
2020-03-11 19:00:22 +01:00
< div class = "outline-text-4" id = "text-2-6-3" >
< p >
We have that:
\[ \bm{G}^{-1}(\omega = 0) = \left(\frac{\delta\bm{\mathcal{X}}}{\delta\bm{\tau}}(\omega = 0)\right)^{-1} = \left( \bm{J}^{-1} \frac{\delta\bm{\mathcal{L}}}{\delta\bm{\tau}}(\omega = 0) \right)^{-1} = \text{diag}(\mathcal{K}_1^{-1} \ \dots \ \mathcal{K}_6^{-1}) \bm{J} \]
< / p >
< p >
And because:
< / p >
< ul class = "org-ul" >
< li > all the leg stiffness are equal< / li >
< li > the controller equal to a \(\bm{K}_0(s) = k(s) \bm{I}_6\)< / li >
< / ul >
< p >
We have that \(\bm{K}_0(s)\) commutes with \(\bm{G}^{-1}(\omega = 0)\) and thus the overall controller \(\bm{K}\) is the same as the one obtain in section < a href = "#org31fd942" > 2.4< / a > .
< / p >
< / div >
< / div >
< / div >
< div id = "outline-container-orga2eadeb" class = "outline-3" >
< h3 id = "orga2eadeb" > < span class = "section-number-3" > 2.7< / span > Comparison< / h3 >
< div class = "outline-text-3" id = "text-2-7" >
< / div >
< div id = "outline-container-org09ae901" class = "outline-4" >
< h4 id = "org09ae901" > < span class = "section-number-4" > 2.7.1< / span > Obtained MIMO Controllers< / h4 >
< div class = "outline-text-4" id = "text-2-7-1" >
< div id = "orgf4c7f15" class = "figure" >
< p > < img src = "figs/centralized_control_comp_K.png" alt = "centralized_control_comp_K.png" / >
< / p >
< p > < span class = "figure-number" > Figure 19: < / span > Comparison of the MIMO controller \(\bm{K}\) for both centralized architectures (< a href = "./figs/centralized_control_comp_K.png" > png< / a > , < a href = "./figs/centralized_control_comp_K.pdf" > pdf< / a > )< / p >
< / div >
< / div >
< / div >
< div id = "outline-container-org23ae479" class = "outline-4" >
< h4 id = "org23ae479" > < span class = "section-number-4" > 2.7.2< / span > Simulation Results< / h4 >
< div class = "outline-text-4" id = "text-2-7-2" >
< p >
The position error \(\bm{\epsilon}_\mathcal{X}\) for both centralized architecture are shown in Figure < a href = "#org9fa8c8a" > 20< / a > .
The corresponding leg’ s length errors \(\bm{\epsilon}_\mathcal{L}\) are shown in Figure < a href = "#orgb139e02" > 21< / a > .
< / p >
< p >
Based on Figure < a href = "#org9fa8c8a" > 20< / a > , we can see that:
< / p >
< ul class = "org-ul" >
< li > There is some tracking error \(\epsilon_x\)< / li >
< li > The errors \(\epsilon_y\), \(\epsilon_{R_x}\) and \(\epsilon_{R_z}\) are quite negligible< / li >
< li > There is some error in the vertical position \(\epsilon_z\).
The frequency of the error \(\epsilon_z\) is twice the frequency of the reference path \(r_x\).< / li >
< li > There is some error \(\epsilon_{R_y}\).
This error is much lower when using the diagonal control in the frame of the leg instead of the cartesian frame.< / li >
< / ul >
< div id = "org9fa8c8a" class = "figure" >
< p > < img src = "figs/centralized_control_comp_Ex.png" alt = "centralized_control_comp_Ex.png" / >
< / p >
< p > < span class = "figure-number" > Figure 20: < / span > Comparison of the position error \(\bm{\epsilon}_\mathcal{X}\) for both centralized controllers (< a href = "./figs/centralized_control_comp_Ex.png" > png< / a > , < a href = "./figs/centralized_control_comp_Ex.pdf" > pdf< / a > )< / p >
< / div >
< div id = "orgb139e02" class = "figure" >
< p > < img src = "figs/centralized_control_comp_El.png" alt = "centralized_control_comp_El.png" / >
< / p >
< p > < span class = "figure-number" > Figure 21: < / span > Comparison of the leg’ s length error \(\bm{\epsilon}_\mathcal{L}\) for both centralized controllers (< a href = "./figs/centralized_control_comp_El.png" > png< / a > , < a href = "./figs/centralized_control_comp_El.pdf" > pdf< / a > )< / p >
< / div >
< / div >
< / div >
< / div >
2020-03-11 19:01:50 +01:00
< div id = "outline-container-org94c3e48" class = "outline-3" >
< h3 id = "org94c3e48" > < span class = "section-number-3" > 2.8< / span > Conclusion< / h3 >
2020-03-11 19:00:22 +01:00
< div class = "outline-text-3" id = "text-2-8" >
< p >
Both control architecture gives similar results even tough the control in the Leg’ s frame gives slightly better results.
< / p >
< p >
The main differences between the control architectures used in sections < a href = "#org31fd942" > 2.4< / a > and < a href = "#orgfd201c3" > 2.5< / a > are summarized in Table < a href = "#orgb1c0d5b" > 1< / a > .
< / p >
< table id = "orgb1c0d5b" border = "2" cellspacing = "0" cellpadding = "6" rules = "groups" frame = "hsides" >
< caption class = "t-above" > < span class = "table-number" > Table 1:< / span > Comparison of the two centralized control architectures< / caption >
< colgroup >
< col class = "org-left" / >
< col class = "org-left" / >
< col class = "org-left" / >
< col class = "org-left" / >
< / colgroup >
< thead >
< tr >
< th scope = "col" class = "org-left" >   < / th >
< th scope = "col" class = "org-left" > < b > Leg’ s Frame< / b > < / th >
< th scope = "col" class = "org-left" > < b > Cartesian Frame< / b > < / th >
< th scope = "col" class = "org-left" > < b > Static Decoupling< / b > < / th >
< / tr >
< / thead >
< tbody >
< tr >
< td class = "org-left" > < b > Plant Meaning< / b > < / td >
< td class = "org-left" > \(\delta\mathcal{L}_i/\tau_i\)< / td >
< td class = "org-left" > \(\delta\mathcal{X}_i/\mathcal{F}_i\)< / td >
< td class = "org-left" > No physical meaning< / td >
< / tr >
< tr >
< td class = "org-left" > < b > Obtained Decoupling< / b > < / td >
< td class = "org-left" > Decoupled at DC< / td >
< td class = "org-left" > Dynamical decoupling except few terms< / td >
< td class = "org-left" > Decoupled at DC< / td >
< / tr >
< tr >
< td class = "org-left" > < b > Diagonal Elements< / b > < / td >
< td class = "org-left" > Identical with all the Resonances< / td >
< td class = "org-left" > Different, resonances are cancel out< / td >
< td class = "org-left" > No Alternating poles and zeros< / td >
< / tr >
< tr >
< td class = "org-left" > < b > Mechanical Architecture< / b > < / td >
< td class = "org-left" > Architecture Independent< / td >
< td class = "org-left" > Better with Cubic Architecture< / td >
< td class = "org-left" >   < / td >
< / tr >
< tr >
< td class = "org-left" > < b > Robustness to Uncertainty< / b > < / td >
< td class = "org-left" > Good (only depends on \(J\))< / td >
< td class = "org-left" > Good (only depends on \(J\))< / td >
< td class = "org-left" > Bad (depends on the mass)< / td >
< / tr >
< / tbody >
< / table >
< p >
These decoupling methods only uses the Jacobian matrix which only depends on the Stewart platform geometry.
Thus, this method should be quite robust against parameter variation (e.g. the payload mass).
< / p >
< / div >
< / div >
< / div >
< div id = "outline-container-org4b8c360" class = "outline-2" >
< h2 id = "org4b8c360" > < span class = "section-number-2" > 3< / span > Hybrid Control Architecture - HAC-LAC with relative DVF< / h2 >
< div class = "outline-text-2" id = "text-3" >
< p >
< a id = "org14e3e5f" > < / a >
< / p >
< / div >
2020-03-11 19:01:50 +01:00
< div id = "outline-container-org2183826" class = "outline-3" >
< h3 id = "org2183826" > < span class = "section-number-3" > 3.1< / span > Control Schematic< / h3 >
2020-03-11 19:00:22 +01:00
< div class = "outline-text-3" id = "text-3-1" >
< div id = "org3a1b1db" class = "figure" >
< p > < img src = "figs/hybrid_reference_tracking.png" alt = "hybrid_reference_tracking.png" / >
< / p >
< p > < span class = "figure-number" > Figure 22: < / span > Centralized Controller< / p >
< / div >
< / div >
< / div >
< / div >
< div id = "outline-container-org445f7a9" class = "outline-2" >
< h2 id = "org445f7a9" > < span class = "section-number-2" > 4< / span > Position Error computation< / h2 >
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< a id = "org5f61540" > < / a >
Let’ s denote:
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< ul class = "org-ul" >
< li > \(\{W\}\) the initial fixed frame (base in which the measurement is done)< / li >
< li > \(\{R\}\) the reference frame corresponding to the wanted pose of the sample< / li >
< li > \(\{M\}\) the frame corresponding to the measured pose of the sample< / li >
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< p >
We have then computed:
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< ul class = "org-ul" >
< li > \({}^W\bm{T}_R\) which corresponds to the wanted pose of the sample with respect to the granite< / li >
< li > \({}^W\bm{T}_M\) which corresponds to the measured pose of the sample with respect to the granite< / li >
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< p >
Reference Position with respect to fixed frame {W}: \({}^WT_R\)
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< div class = "org-src-container" >
< pre class = "src src-matlab" > Dx = 0;
Dy = 0;
Dz = 0.1;
Rx = < span class = "org-constant" > pi< / span > ;
Ry = 0;
Rz = 0;
WTr = zeros(4,4);
R = [cos(Rz) < span class = "org-type" > -< / span > sin(Rz) 0;
sin(Rz) cos(Rz) 0;
0 0 1] < span class = "org-type" > *< / span > ...
[cos(Ry) 0 sin(Ry);
0 1 0;
< span class = "org-type" > -< / span > sin(Ry) 0 cos(Ry)] < span class = "org-type" > *< / span > ...
[1 0 0;
0 cos(Rx) < span class = "org-type" > -< / span > sin(Rx);
0 sin(Rx) cos(Rx)];
WTr(1< span class = "org-type" > :< / span > 3, 1< span class = "org-type" > :< / span > 3) = R;
WTr(1< span class = "org-type" > :< / span > 4, 4) = [Dx ; Dy ; Dz; 1];
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< p >
Measured Position with respect to fixed frame {W}: \({}^WT_M\)
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< div class = "org-src-container" >
< pre class = "src src-matlab" > Dx = 0;
Dy = 0;
Dz = 0;
Rx = < span class = "org-constant" > pi< / span > ;
Ry = 0;
Rz = 0;
WTm = zeros(4,4);
R = [cos(Rz) < span class = "org-type" > -< / span > sin(Rz) 0;
sin(Rz) cos(Rz) 0;
0 0 1] < span class = "org-type" > *< / span > ...
[cos(Ry) 0 sin(Ry);
0 1 0;
< span class = "org-type" > -< / span > sin(Ry) 0 cos(Ry)] < span class = "org-type" > *< / span > ...
[1 0 0;
0 cos(Rx) < span class = "org-type" > -< / span > sin(Rx);
0 sin(Rx) cos(Rx)];
WTm(1< span class = "org-type" > :< / span > 3, 1< span class = "org-type" > :< / span > 3) = R;
WTm(1< span class = "org-type" > :< / span > 4, 4) = [Dx ; Dy ; Dz; 1];
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< p >
We would like to compute \({}^M\bm{T}_R\) which corresponds to the wanted pose of the sample expressed in a frame attached to the top platform of the nano-hexapod (frame \(\{M\}\)).
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< p >
We have:
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\begin{align}
{}^M\bm{T}_R & = {}^M\bm{T}_W \cdot {}^W\bm{T}_R \\
& = {}^W{\bm{T}_M}^{-1} \cdot {}^W\bm{T}_R
\end{align}
< div class = "org-src-container" >
< pre class = "src src-matlab" > < span class = "org-comment" > % Error with respect to the top platform< / span >
MTr = [WTm(1< span class = "org-type" > :< / span > 3,1< span class = "org-type" > :< / span > 3)< span class = "org-type" > '< / span > , < span class = "org-type" > -< / span > WTm(1< span class = "org-type" > :< / span > 3,1< span class = "org-type" > :< / span > 3)< span class = "org-type" > '*< / span > WTm(1< span class = "org-type" > :< / span > 3,4) ; 0 0 0 1]< span class = "org-type" > *< / span > WTr;
T = MTr;
Edx = T(1, 4);
Edy = T(2, 4);
Edz = T(3, 4);
< span class = "org-comment" > % The angles obtained are u-v-w Euler angles (rotations in the moving frame)< / span >
Ery = atan2( T(1, 3), sqrt(T(1, 1)< span class = "org-type" > ^< / span > 2 < span class = "org-type" > +< / span > T(1, 2)< span class = "org-type" > ^< / span > 2));
Erx = atan2(< span class = "org-type" > -< / span > T(2, 3)< span class = "org-type" > /< / span > cos(Ery), T(3, 3)< span class = "org-type" > /< / span > cos(Ery));
Erz = atan2(< span class = "org-type" > -< / span > T(1, 2)< span class = "org-type" > /< / span > cos(Ery), T(1, 1)< span class = "org-type" > /< / span > cos(Ery));
[Edx, Edy, Edz, Erx, Ery, Erz]
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2020-02-28 17:35:44 +01:00
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< div id = "postamble" class = "status" >
< p class = "author" > Author: Dehaeze Thomas< / p >
2020-03-11 19:01:50 +01:00
< p class = "date" > Created: 2020-03-11 mer. 19:01< / p >
2020-02-28 17:35:44 +01:00
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