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< title > Stewart Platform - Vibration Isolation< / title >
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< h1 class = "title" > Stewart Platform - Vibration Isolation< / h1 >
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< div id = "table-of-contents" >
< h2 > Table of Contents< / h2 >
< div id = "text-table-of-contents" >
< ul >
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< li > < a href = "#orgf86b757" > 1. HAC-LAC (Cascade) Control - Integral Control< / a >
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< ul >
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< li > < a href = "#org3a9f4d4" > 1.1. Introduction< / a > < / li >
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< li > < a href = "#org72e0133" > 1.2. Initialization< / a > < / li >
< li > < a href = "#orgb6e0288" > 1.3. Identification< / a >
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< ul >
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< li > < a href = "#orgd37eab6" > 1.3.1. HAC - Without LAC< / a > < / li >
< li > < a href = "#org451f819" > 1.3.2. HAC - IFF< / a > < / li >
< li > < a href = "#org36fc937" > 1.3.3. HAC - DVF< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#org4d7a6d8" > 1.4. Control Architecture< / a > < / li >
< li > < a href = "#org3e1b1b7" > 1.5. 6x6 Plant Comparison< / a > < / li >
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< li > < a href = "#orgd88e4bb" > 1.6. HAC - DVF< / a >
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< ul >
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< li > < a href = "#org948a0ef" > 1.6.1. Plant< / a > < / li >
< li > < a href = "#org2c7f7fe" > 1.6.2. Controller Design< / a > < / li >
< li > < a href = "#orgab6b6ba" > 1.6.3. Obtained Performance< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#org72402cd" > 1.7. HAC - IFF< / a >
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< ul >
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< li > < a href = "#org6228267" > 1.7.1. Plant< / a > < / li >
< li > < a href = "#org46e2ce2" > 1.7.2. Controller Design< / a > < / li >
< li > < a href = "#org3e940d0" > 1.7.3. Obtained Performance< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#org81c1767" > 1.8. Comparison< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#org6f94eba" > 2. MIMO Analysis< / a >
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< ul >
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< li > < a href = "#org389b32d" > 2.1. Initialization< / a > < / li >
< li > < a href = "#orgab59403" > 2.2. Identification< / a >
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< ul >
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< li > < a href = "#org6a97945" > 2.2.1. HAC - Without LAC< / a > < / li >
< li > < a href = "#org2c0fb88" > 2.2.2. HAC - DVF< / a > < / li >
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< li > < a href = "#orgf606814" > 2.2.3. Cartesian Frame< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#org8349fa6" > 2.3. Singular Value Decomposition< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#orgc8479b7" > 3. Diagonal Control based on the damped plant< / a >
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< ul >
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< li > < a href = "#org0bf14d6" > 3.1. Initialization< / a > < / li >
< li > < a href = "#org1540d5b" > 3.2. Identification< / a > < / li >
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< li > < a href = "#orgae85e0d" > 3.3. Steady State Decoupling< / a >
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< ul >
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< li > < a href = "#org1e2bbe7" > 3.3.1. Pre-Compensator Design< / a > < / li >
< li > < a href = "#org077e6f6" > 3.3.2. Diagonal Control Design< / a > < / li >
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< li > < a href = "#org2ef5df1" > 3.3.3. Results< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#orgad35bf9" > 3.4. Decoupling at Crossover< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#org846cef9" > 4. Time Domain Simulation< / a >
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< ul >
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< li > < a href = "#org323700d" > 4.1. Initialization< / a > < / li >
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< li > < a href = "#org8dbc004" > 4.2. HAC IFF< / a > < / li >
< li > < a href = "#org7dc4716" > 4.3. HAC-DVF< / a > < / li >
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< li > < a href = "#org1230d9e" > 4.4. Results< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#org69ebad1" > 5. Functions< / a >
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< ul >
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< li > < a href = "#orgc7bcc65" > 5.1. < code > initializeController< / code > : Initialize the Controller< / a >
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< ul >
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< li > < a href = "#orgf672f64" > Function description< / a > < / li >
< li > < a href = "#org941466e" > Optional Parameters< / a > < / li >
< li > < a href = "#org65d3a7d" > Structure initialization< / a > < / li >
< li > < a href = "#org32be93f" > Add Type< / a > < / li >
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< div id = "outline-container-orgf86b757" class = "outline-2" >
< h2 id = "orgf86b757" > < span class = "section-number-2" > 1< / span > HAC-LAC (Cascade) Control - Integral Control< / h2 >
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< div class = "outline-text-2" id = "text-1" >
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< / div >
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< div id = "outline-container-org3a9f4d4" class = "outline-3" >
< h3 id = "org3a9f4d4" > < span class = "section-number-3" > 1.1< / span > Introduction< / h3 >
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< div class = "outline-text-3" id = "text-1-1" >
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< p >
In this section, we wish to study the use of the High Authority Control - Low Authority Control (HAC-LAC) architecture on the Stewart platform.
< / p >
< p >
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The control architectures are shown in Figures < a href = "#org63523c7" > 1< / a > and < a href = "#org7ad8618" > 2< / a > .
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< / p >
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< p >
First, the LAC loop is closed (the LAC control is described < a href = "active-damping.html" > here< / a > ), and then the HAC controller is designed and the outer loop is closed.
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< / p >
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< div id = "org63523c7" class = "figure" >
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< p > < img src = "figs/control_arch_hac_iff.png" alt = "control_arch_hac_iff.png" / >
< / p >
< p > < span class = "figure-number" > Figure 1: < / span > HAC-LAC architecture with IFF< / p >
< / div >
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< div id = "org7ad8618" class = "figure" >
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< p > < img src = "figs/control_arch_hac_dvf.png" alt = "control_arch_hac_dvf.png" / >
< / p >
< p > < span class = "figure-number" > Figure 2: < / span > HAC-LAC architecture with DVF< / p >
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< div id = "outline-container-org72e0133" class = "outline-3" >
< h3 id = "org72e0133" > < span class = "section-number-3" > 1.2< / span > Initialization< / h3 >
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< div class = "outline-text-3" id = "text-1-2" >
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< p >
We first initialize the Stewart platform.
< / p >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > stewart = initializeStewartPlatform();
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stewart = initializeFramesPositions(stewart, 'H', 90e-3, 'MO_B', 45e-3);
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stewart = generateGeneralConfiguration(stewart);
stewart = computeJointsPose(stewart);
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stewart = initializeStrutDynamics(stewart);
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stewart = initializeJointDynamics(stewart, 'type_F', 'universal', 'type_M', 'spherical');
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stewart = initializeCylindricalPlatforms(stewart);
stewart = initializeCylindricalStruts(stewart);
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stewart = computeJacobian(stewart);
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stewart = initializeStewartPose(stewart);
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stewart = initializeInertialSensor(stewart, 'type', 'none');
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< / pre >
< / div >
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< p >
The rotation point of the ground is located at the origin of frame \(\{A\}\).
< / p >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > ground = initializeGround('type', 'rigid', 'rot_point', stewart.platform_F.FO_A);
payload = initializePayload('type', 'none');
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< div id = "outline-container-orgb6e0288" class = "outline-3" >
< h3 id = "orgb6e0288" > < span class = "section-number-3" > 1.3< / span > Identification< / h3 >
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< div class = "outline-text-3" id = "text-1-3" >
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< p >
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We identify the transfer function from the actuator forces \(\bm{\tau}\) to the absolute displacement of the mobile platform \(\bm{\mathcal{X}}\) in three different cases:
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< / p >
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< ul class = "org-ul" >
< li > Open Loop plant< / li >
< li > Already damped plant using Integral Force Feedback< / li >
< li > Already damped plant using Direct velocity feedback< / li >
< / ul >
< / div >
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< div id = "outline-container-orgd37eab6" class = "outline-4" >
< h4 id = "orgd37eab6" > < span class = "section-number-4" > 1.3.1< / span > HAC - Without LAC< / h4 >
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< div class = "outline-text-4" id = "text-1-3-1" >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > controller = initializeController('type', 'open-loop');
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< / pre >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > %% Name of the Simulink File
mdl = 'stewart_platform_model';
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%% Input/Output definition
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clear io; io_i = 1;
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io(io_i) = linio([mdl, '/Controller'], 1, 'input'); io_i = io_i + 1; % Actuator Force Inputs [N]
io(io_i) = linio([mdl, '/Absolute Motion Sensor'], 1, 'openoutput'); io_i = io_i + 1; % Absolute Sensor [m, rad]
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%% Run the linearization
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G_ol = linearize(mdl, io);
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G_ol.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
G_ol.OutputName = {'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'};
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< / pre >
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< div id = "outline-container-org451f819" class = "outline-4" >
< h4 id = "org451f819" > < span class = "section-number-4" > 1.3.2< / span > HAC - IFF< / h4 >
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< div class = "outline-text-4" id = "text-1-3-2" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > controller = initializeController('type', 'iff');
K_iff = -(1e4/s)*eye(6);
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< / pre >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > %% Name of the Simulink File
mdl = 'stewart_platform_model';
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%% Input/Output definition
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clear io; io_i = 1;
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io(io_i) = linio([mdl, '/Controller'], 1, 'input'); io_i = io_i + 1; % Actuator Force Inputs [N]
io(io_i) = linio([mdl, '/Absolute Motion Sensor'], 1, 'openoutput'); io_i = io_i + 1; % Absolute Sensor [m, rad]
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%% Run the linearization
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G_iff = linearize(mdl, io);
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G_iff.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
G_iff.OutputName = {'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'};
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< / pre >
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< div id = "outline-container-org36fc937" class = "outline-4" >
< h4 id = "org36fc937" > < span class = "section-number-4" > 1.3.3< / span > HAC - DVF< / h4 >
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< div class = "outline-text-4" id = "text-1-3-3" >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > controller = initializeController('type', 'dvf');
K_dvf = -1e4*s/(1+s/2/pi/5000)*eye(6);
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< / pre >
< / div >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > %% Name of the Simulink File
mdl = 'stewart_platform_model';
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%% Input/Output definition
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clear io; io_i = 1;
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io(io_i) = linio([mdl, '/Controller'], 1, 'input'); io_i = io_i + 1; % Actuator Force Inputs [N]
io(io_i) = linio([mdl, '/Absolute Motion Sensor'], 1, 'openoutput'); io_i = io_i + 1; % Absolute Sensor [m, rad]
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%% Run the linearization
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G_dvf = linearize(mdl, io);
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G_dvf.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
G_dvf.OutputName = {'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'};
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< / pre >
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< div id = "outline-container-org4d7a6d8" class = "outline-3" >
< h3 id = "org4d7a6d8" > < span class = "section-number-3" > 1.4< / span > Control Architecture< / h3 >
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< div class = "outline-text-3" id = "text-1-4" >
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< p >
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We use the Jacobian to express the actuator forces in the cartesian frame, and thus we obtain the transfer functions from \(\bm{\mathcal{F}}\) to \(\bm{\mathcal{X}}\).
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > Gc_ol = minreal(G_ol)/stewart.kinematics.J';
Gc_ol.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
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Gc_iff = minreal(G_iff)/stewart.kinematics.J';
Gc_iff.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
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Gc_dvf = minreal(G_dvf)/stewart.kinematics.J';
Gc_dvf.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
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< / pre >
< / div >
< p >
We then design a controller based on the transfer functions from \(\bm{\mathcal{F}}\) to \(\bm{\mathcal{X}}\), finally, we will pre-multiply the controller by \(\bm{J}^{-T}\).
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< div id = "outline-container-org3e1b1b7" class = "outline-3" >
< h3 id = "org3e1b1b7" > < span class = "section-number-3" > 1.5< / span > 6x6 Plant Comparison< / h3 >
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< div class = "outline-text-3" id = "text-1-5" >
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< div id = "orgbee4071" class = "figure" >
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< p > < img src = "figs/hac_lac_coupling_jacobian.png" alt = "hac_lac_coupling_jacobian.png" / >
< / p >
< p > < span class = "figure-number" > Figure 3: < / span > Norm of the transfer functions from \(\bm{\mathcal{F}}\) to \(\bm{\mathcal{X}}\) (< a href = "./figs/hac_lac_coupling_jacobian.png" > png< / a > , < a href = "./figs/hac_lac_coupling_jacobian.pdf" > pdf< / a > )< / p >
< / div >
< / div >
< / div >
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< div id = "outline-container-orgd88e4bb" class = "outline-3" >
< h3 id = "orgd88e4bb" > < span class = "section-number-3" > 1.6< / span > HAC - DVF< / h3 >
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< div class = "outline-text-3" id = "text-1-6" >
< / div >
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< div id = "outline-container-org948a0ef" class = "outline-4" >
< h4 id = "org948a0ef" > < span class = "section-number-4" > 1.6.1< / span > Plant< / h4 >
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< div class = "outline-text-4" id = "text-1-6-1" >
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< div id = "org487a558" class = "figure" >
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< p > < img src = "figs/hac_lac_plant_dvf.png" alt = "hac_lac_plant_dvf.png" / >
< / p >
< p > < span class = "figure-number" > Figure 4: < / span > Diagonal elements of the plant for HAC control when DVF is previously applied (< a href = "./figs/hac_lac_plant_dvf.png" > png< / a > , < a href = "./figs/hac_lac_plant_dvf.pdf" > pdf< / a > )< / p >
< / div >
< / div >
< / div >
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< div id = "outline-container-org2c7f7fe" class = "outline-4" >
< h4 id = "org2c7f7fe" > < span class = "section-number-4" > 1.6.2< / span > Controller Design< / h4 >
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< div class = "outline-text-4" id = "text-1-6-2" >
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< p >
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We design a diagonal controller with equal bandwidth for the 6 terms.
The controller is a pure integrator with a small lead near the crossover.
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< / p >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > wc = 2*pi*300; % Wanted Bandwidth [rad/s]
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h = 1.2;
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H_lead = 1/h*(1 + s/(wc/h))/(1 + s/(wc*h));
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Kd_dvf = diag(1./abs(diag(freqresp(1/s*Gc_dvf, wc)))) .* H_lead .* 1/s;
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< / pre >
< / div >
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< div id = "org5d0e2e3" class = "figure" >
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< p > < img src = "figs/hac_lac_loop_gain_dvf.png" alt = "hac_lac_loop_gain_dvf.png" / >
< / p >
< p > < span class = "figure-number" > Figure 5: < / span > Diagonal elements of the Loop Gain for the HAC control (< a href = "./figs/hac_lac_loop_gain_dvf.png" > png< / a > , < a href = "./figs/hac_lac_loop_gain_dvf.pdf" > pdf< / a > )< / p >
< / div >
< p >
Finally, we pre-multiply the diagonal controller by \(\bm{J}^{-T}\) prior implementation.
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > K_hac_dvf = inv(stewart.kinematics.J')*Kd_dvf;
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< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-orgab6b6ba" class = "outline-4" >
< h4 id = "orgab6b6ba" > < span class = "section-number-4" > 1.6.3< / span > Obtained Performance< / h4 >
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< div class = "outline-text-4" id = "text-1-6-3" >
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< p >
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We identify the transmissibility and compliance of the system.
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< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > controller = initializeController('type', 'open-loop');
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[T_ol, T_norm_ol, freqs] = computeTransmissibility();
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[C_ol, C_norm_ol, ~] = computeCompliance();
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< / pre >
< / div >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > controller = initializeController('type', 'dvf');
[T_dvf, T_norm_dvf, ~] = computeTransmissibility();
[C_dvf, C_norm_dvf, ~] = computeCompliance();
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< / pre >
< / div >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > controller = initializeController('type', 'hac-dvf');
[T_hac_dvf, T_norm_hac_dvf, ~] = computeTransmissibility();
[C_hac_dvf, C_norm_hac_dvf, ~] = computeCompliance();
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< / pre >
< / div >
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< div id = "orga06a318" class = "figure" >
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< p > < img src = "figs/hac_lac_C_T_dvf.png" alt = "hac_lac_C_T_dvf.png" / >
< / p >
< p > < span class = "figure-number" > Figure 6: < / span > Obtained Compliance and Transmissibility (< a href = "./figs/hac_lac_C_T_dvf.png" > png< / a > , < a href = "./figs/hac_lac_C_T_dvf.pdf" > pdf< / a > )< / p >
< / div >
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< / div >
< / div >
< / div >
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< div id = "outline-container-org72402cd" class = "outline-3" >
< h3 id = "org72402cd" > < span class = "section-number-3" > 1.7< / span > HAC - IFF< / h3 >
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< div class = "outline-text-3" id = "text-1-7" >
< / div >
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< div id = "outline-container-org6228267" class = "outline-4" >
< h4 id = "org6228267" > < span class = "section-number-4" > 1.7.1< / span > Plant< / h4 >
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< div class = "outline-text-4" id = "text-1-7-1" >
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< div id = "org0fc8dea" class = "figure" >
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< p > < img src = "figs/hac_lac_plant_iff.png" alt = "hac_lac_plant_iff.png" / >
< / p >
< p > < span class = "figure-number" > Figure 7: < / span > Diagonal elements of the plant for HAC control when IFF is previously applied (< a href = "./figs/hac_lac_plant_iff.png" > png< / a > , < a href = "./figs/hac_lac_plant_iff.pdf" > pdf< / a > )< / p >
< / div >
< / div >
< / div >
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< div id = "outline-container-org46e2ce2" class = "outline-4" >
< h4 id = "org46e2ce2" > < span class = "section-number-4" > 1.7.2< / span > Controller Design< / h4 >
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< div class = "outline-text-4" id = "text-1-7-2" >
< p >
We design a diagonal controller with equal bandwidth for the 6 terms.
The controller is a pure integrator with a small lead near the crossover.
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > wc = 2*pi*300; % Wanted Bandwidth [rad/s]
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h = 1.2;
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H_lead = 1/h*(1 + s/(wc/h))/(1 + s/(wc*h));
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Kd_iff = diag(1./abs(diag(freqresp(1/s*Gc_iff, wc)))) .* H_lead .* 1/s;
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< / pre >
< / div >
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< div id = "org211b595" class = "figure" >
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< p > < img src = "figs/hac_lac_loop_gain_iff.png" alt = "hac_lac_loop_gain_iff.png" / >
< / p >
< p > < span class = "figure-number" > Figure 8: < / span > Diagonal elements of the Loop Gain for the HAC control (< a href = "./figs/hac_lac_loop_gain_iff.png" > png< / a > , < a href = "./figs/hac_lac_loop_gain_iff.pdf" > pdf< / a > )< / p >
< / div >
< p >
Finally, we pre-multiply the diagonal controller by \(\bm{J}^{-T}\) prior implementation.
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > K_hac_iff = inv(stewart.kinematics.J')*Kd_iff;
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< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-org3e940d0" class = "outline-4" >
< h4 id = "org3e940d0" > < span class = "section-number-4" > 1.7.3< / span > Obtained Performance< / h4 >
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< div class = "outline-text-4" id = "text-1-7-3" >
< p >
We identify the transmissibility and compliance of the system.
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > controller = initializeController('type', 'open-loop');
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[T_ol, T_norm_ol, freqs] = computeTransmissibility();
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[C_ol, C_norm_ol, ~] = computeCompliance();
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< / pre >
< / div >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > controller = initializeController('type', 'iff');
[T_iff, T_norm_iff, ~] = computeTransmissibility();
[C_iff, C_norm_iff, ~] = computeCompliance();
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< / pre >
< / div >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > controller = initializeController('type', 'hac-iff');
[T_hac_iff, T_norm_hac_iff, ~] = computeTransmissibility();
[C_hac_iff, C_norm_hac_iff, ~] = computeCompliance();
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< / pre >
< / div >
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< div id = "orgc47bfe0" class = "figure" >
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< p > < img src = "figs/hac_lac_C_T_iff.png" alt = "hac_lac_C_T_iff.png" / >
< / p >
< p > < span class = "figure-number" > Figure 9: < / span > Obtained Compliance and Transmissibility (< a href = "./figs/hac_lac_C_T_iff.png" > png< / a > , < a href = "./figs/hac_lac_C_T_iff.pdf" > pdf< / a > )< / p >
< / div >
< / div >
< / div >
< / div >
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< div id = "outline-container-org81c1767" class = "outline-3" >
< h3 id = "org81c1767" > < span class = "section-number-3" > 1.8< / span > Comparison< / h3 >
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< div class = "outline-text-3" id = "text-1-8" >
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< div id = "orgf042a3f" class = "figure" >
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< p > < img src = "figs/hac_lac_C_full_comparison.png" alt = "hac_lac_C_full_comparison.png" / >
< / p >
< p > < span class = "figure-number" > Figure 10: < / span > Comparison of the norm of the Compliance matrices for the HAC-LAC architecture (< a href = "./figs/hac_lac_C_full_comparison.png" > png< / a > , < a href = "./figs/hac_lac_C_full_comparison.pdf" > pdf< / a > )< / p >
< / div >
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< div id = "org5a9df7e" class = "figure" >
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< p > < img src = "figs/hac_lac_T_full_comparison.png" alt = "hac_lac_T_full_comparison.png" / >
< / p >
< p > < span class = "figure-number" > Figure 11: < / span > Comparison of the norm of the Transmissibility matrices for the HAC-LAC architecture (< a href = "./figs/hac_lac_T_full_comparison.png" > png< / a > , < a href = "./figs/hac_lac_T_full_comparison.pdf" > pdf< / a > )< / p >
< / div >
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< div id = "orgdc12a99" class = "figure" >
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< p > < img src = "figs/hac_lac_C_T_comparison.png" alt = "hac_lac_C_T_comparison.png" / >
< / p >
< p > < span class = "figure-number" > Figure 12: < / span > Comparison of the Frobenius norm of the Compliance and Transmissibility for the HAC-LAC architecture with both IFF and DVF (< a href = "./figs/hac_lac_C_T_comparison.png" > png< / a > , < a href = "./figs/hac_lac_C_T_comparison.pdf" > pdf< / a > )< / p >
< / div >
< / div >
< / div >
< / div >
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< div id = "outline-container-org6f94eba" class = "outline-2" >
< h2 id = "org6f94eba" > < span class = "section-number-2" > 2< / span > MIMO Analysis< / h2 >
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< div class = "outline-text-2" id = "text-2" >
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< p >
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Let’ s define the system as shown in figure < a href = "#orgac8f77c" > 13< / a > .
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< / p >
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< div id = "orgac8f77c" class = "figure" >
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< p > < img src = "figs/general_control_names.png" alt = "general_control_names.png" / >
< / p >
< p > < span class = "figure-number" > Figure 13: < / span > General Control Architecture< / p >
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< / div >
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< table id = "orgc2ee4b0" border = "2" cellspacing = "0" cellpadding = "6" rules = "groups" frame = "hsides" >
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< caption class = "t-above" > < span class = "table-number" > Table 1:< / span > Signals definition for the generalized plant< / caption >
< colgroup >
< col class = "org-left" / >
< col class = "org-left" / >
< col class = "org-left" / >
< / colgroup >
< thead >
< tr >
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< th scope = "col" class = "org-left" >   < / th >
< th scope = "col" class = "org-left" > < b > Symbol< / b > < / th >
< th scope = "col" class = "org-left" > < b > Meaning< / b > < / th >
< / tr >
< / thead >
< tbody >
< tr >
< td class = "org-left" > < b > Exogenous Inputs< / b > < / td >
< td class = "org-left" > \(\bm{\mathcal{X}}_w\)< / td >
< td class = "org-left" > Ground motion< / td >
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< / tr >
< tr >
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< td class = "org-left" >   < / td >
< td class = "org-left" > \(\bm{\mathcal{F}}_d\)< / td >
< td class = "org-left" > External Forces applied to the Payload< / td >
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< / tr >
< tr >
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< td class = "org-left" >   < / td >
< td class = "org-left" > \(\bm{r}\)< / td >
< td class = "org-left" > Reference signal for tracking< / td >
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< / tr >
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< / tbody >
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< tbody >
< tr >
< td class = "org-left" > < b > Exogenous Outputs< / b > < / td >
< td class = "org-left" > \(\bm{\mathcal{X}}\)< / td >
< td class = "org-left" > Absolute Motion of the Payload< / td >
< / tr >
< tr >
< td class = "org-left" >   < / td >
< td class = "org-left" > \(\bm{\tau}\)< / td >
< td class = "org-left" > Actuator Rate< / td >
< / tr >
< / tbody >
< tbody >
< tr >
< td class = "org-left" > < b > Sensed Outputs< / b > < / td >
< td class = "org-left" > \(\bm{\tau}_m\)< / td >
< td class = "org-left" > Force Sensors in each leg< / td >
< / tr >
< tr >
< td class = "org-left" >   < / td >
< td class = "org-left" > \(\delta \bm{\mathcal{L}}_m\)< / td >
< td class = "org-left" > Measured displacement of each leg< / td >
< / tr >
< tr >
< td class = "org-left" >   < / td >
< td class = "org-left" > \(\bm{\mathcal{X}}\)< / td >
< td class = "org-left" > Absolute Motion of the Payload< / td >
< / tr >
< / tbody >
< tbody >
< tr >
< td class = "org-left" > < b > Control Signals< / b > < / td >
< td class = "org-left" > \(\bm{\tau}\)< / td >
< td class = "org-left" > Actuator Inputs< / td >
< / tr >
< / tbody >
< / table >
< / div >
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< div id = "outline-container-org389b32d" class = "outline-3" >
< h3 id = "org389b32d" > < span class = "section-number-3" > 2.1< / span > Initialization< / h3 >
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< div class = "outline-text-3" id = "text-2-1" >
< p >
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We first initialize the Stewart platform.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > stewart = initializeStewartPlatform();
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stewart = initializeFramesPositions(stewart, 'H', 90e-3, 'MO_B', 45e-3);
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stewart = generateGeneralConfiguration(stewart);
stewart = computeJointsPose(stewart);
stewart = initializeStrutDynamics(stewart);
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stewart = initializeJointDynamics(stewart, 'type_F', 'universal', 'type_M', 'spherical');
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stewart = initializeCylindricalPlatforms(stewart);
stewart = initializeCylindricalStruts(stewart);
stewart = computeJacobian(stewart);
stewart = initializeStewartPose(stewart);
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stewart = initializeInertialSensor(stewart, 'type', 'none');
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< / pre >
< / div >
< p >
The rotation point of the ground is located at the origin of frame \(\{A\}\).
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > ground = initializeGround('type', 'rigid', 'rot_point', stewart.platform_F.FO_A);
payload = initializePayload('type', 'none');
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< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-orgab59403" class = "outline-3" >
< h3 id = "orgab59403" > < span class = "section-number-3" > 2.2< / span > Identification< / h3 >
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< div class = "outline-text-3" id = "text-2-2" >
< / div >
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< div id = "outline-container-org6a97945" class = "outline-4" >
< h4 id = "org6a97945" > < span class = "section-number-4" > 2.2.1< / span > HAC - Without LAC< / h4 >
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< div class = "outline-text-4" id = "text-2-2-1" >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > controller = initializeController('type', 'open-loop');
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< / pre >
< / div >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > %% Name of the Simulink File
mdl = 'stewart_platform_model';
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%% Input/Output definition
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clear io; io_i = 1;
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io(io_i) = linio([mdl, '/Controller'], 1, 'input'); io_i = io_i + 1; % Actuator Force Inputs [N]
io(io_i) = linio([mdl, '/Absolute Motion Sensor'], 1, 'openoutput'); io_i = io_i + 1; % Absolute Sensor [m, rad]
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%% Run the linearization
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G_ol = linearize(mdl, io);
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G_ol.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
G_ol.OutputName = {'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'};
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< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-org2c0fb88" class = "outline-4" >
< h4 id = "org2c0fb88" > < span class = "section-number-4" > 2.2.2< / span > HAC - DVF< / h4 >
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< div class = "outline-text-4" id = "text-2-2-2" >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > controller = initializeController('type', 'dvf');
K_dvf = -1e4*s/(1+s/2/pi/5000)*eye(6);
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< / pre >
< / div >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > %% Name of the Simulink File
mdl = 'stewart_platform_model';
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%% Input/Output definition
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clear io; io_i = 1;
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io(io_i) = linio([mdl, '/Controller'], 1, 'input'); io_i = io_i + 1; % Actuator Force Inputs [N]
io(io_i) = linio([mdl, '/Absolute Motion Sensor'], 1, 'openoutput'); io_i = io_i + 1; % Absolute Sensor [m, rad]
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%% Run the linearization
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G_dvf = linearize(mdl, io);
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G_dvf.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
G_dvf.OutputName = {'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'};
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< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-orgf606814" class = "outline-4" >
< h4 id = "orgf606814" > < span class = "section-number-4" > 2.2.3< / span > Cartesian Frame< / h4 >
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< div class = "outline-text-4" id = "text-2-2-3" >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > Gc_ol = minreal(G_ol)/stewart.kinematics.J';
Gc_ol.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
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Gc_dvf = minreal(G_dvf)/stewart.kinematics.J';
Gc_dvf.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
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< / pre >
< / div >
< / div >
< / div >
< / div >
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< div id = "outline-container-org8349fa6" class = "outline-3" >
< h3 id = "org8349fa6" > < span class = "section-number-3" > 2.3< / span > Singular Value Decomposition< / h3 >
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< div class = "outline-text-3" id = "text-2-3" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > freqs = logspace(1, 4, 1000);
U_ol = zeros(6,6,length(freqs));
S_ol = zeros(6,length(freqs));
V_ol = zeros(6,6,length(freqs));
U_dvf = zeros(6,6,length(freqs));
S_dvf = zeros(6,length(freqs));
V_dvf = zeros(6,6,length(freqs));
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for i = 1:length(freqs)
[U,S,V] = svd(freqresp(Gc_ol, freqs(i), 'Hz'));
U_ol(:,:,i) = U;
S_ol(:,i) = diag(S);
V_ol(:,:,i) = V;
[U,S,V] = svd(freqresp(Gc_dvf, freqs(i), 'Hz'));
U_dvf(:,:,i) = U;
S_dvf(:,i) = diag(S);
V_dvf(:,:,i) = V;
end
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< / pre >
< / div >
< / div >
< / div >
< / div >
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< div id = "outline-container-orgc8479b7" class = "outline-2" >
< h2 id = "orgc8479b7" > < span class = "section-number-2" > 3< / span > Diagonal Control based on the damped plant< / h2 >
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< div class = "outline-text-2" id = "text-3" >
< p >
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From (< a href = "#citeproc_bib_item_1" > Skogestad and Postlethwaite 2007< / a > ), a simple approach to multivariable control is the following two-step procedure:
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< / p >
< ol class = "org-ol" >
< li > < b > Design a pre-compensator< / b > \(W_1\), which counteracts the interactions in the plant and results in a new < b > shaped plant< / b > \(G_S(s) = G(s) W_1(s)\) which is < b > more diagonal and easier to control< / b > than the original plant \(G(s)\).< / li >
< li > < b > Design a diagonal controller< / b > \(K_S(s)\) for the shaped plant using methods similar to those for SISO systems.< / li >
< / ol >
< p >
The overall controller is then:
\[ K(s) = W_1(s)K_s(s) \]
< / p >
< p >
There are mainly three different cases:
< / p >
< ol class = "org-ol" >
< li > < b > Dynamic decoupling< / b > : \(G_S(s)\) is diagonal at all frequencies. For that we can choose \(W_1(s) = G^{-1}(s)\) and this is an inverse-based controller.< / li >
< li > < b > Steady-state decoupling< / b > : \(G_S(0)\) is diagonal. This can be obtained by selecting \(W_1(s) = G^{-1}(0)\).< / li >
< li > < b > Approximate decoupling at frequency \(\w_0\)< / b > : \(G_S(j\w_0)\) is as diagonal as possible. Decoupling the system at \(\w_0\) is a good choice because the effect on performance of reducing interaction is normally greatest at this frequency.< / li >
< / ol >
< / div >
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< div id = "outline-container-org0bf14d6" class = "outline-3" >
< h3 id = "org0bf14d6" > < span class = "section-number-3" > 3.1< / span > Initialization< / h3 >
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< div class = "outline-text-3" id = "text-3-1" >
< p >
We first initialize the Stewart platform.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > stewart = initializeStewartPlatform();
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stewart = initializeFramesPositions(stewart, 'H', 90e-3, 'MO_B', 45e-3);
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stewart = generateGeneralConfiguration(stewart);
stewart = computeJointsPose(stewart);
stewart = initializeStrutDynamics(stewart);
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stewart = initializeJointDynamics(stewart, 'type_F', 'universal', 'type_M', 'spherical');
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stewart = initializeCylindricalPlatforms(stewart);
stewart = initializeCylindricalStruts(stewart);
stewart = computeJacobian(stewart);
stewart = initializeStewartPose(stewart);
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stewart = initializeInertialSensor(stewart, 'type', 'none');
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< / pre >
< / div >
< p >
The rotation point of the ground is located at the origin of frame \(\{A\}\).
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > ground = initializeGround('type', 'rigid', 'rot_point', stewart.platform_F.FO_A);
payload = initializePayload('type', 'none');
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< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-org1540d5b" class = "outline-3" >
< h3 id = "org1540d5b" > < span class = "section-number-3" > 3.2< / span > Identification< / h3 >
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< div class = "outline-text-3" id = "text-3-2" >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > controller = initializeController('type', 'dvf');
K_dvf = -1e4*s/(1+s/2/pi/5000)*eye(6);
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< / pre >
< / div >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > %% Name of the Simulink File
mdl = 'stewart_platform_model';
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%% Input/Output definition
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clear io; io_i = 1;
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io(io_i) = linio([mdl, '/Controller'], 1, 'input'); io_i = io_i + 1; % Actuator Force Inputs [N]
io(io_i) = linio([mdl, '/Absolute Motion Sensor'], 1, 'openoutput'); io_i = io_i + 1; % Absolute Sensor [m, rad]
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%% Run the linearization
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G_dvf = linearize(mdl, io);
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G_dvf.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
G_dvf.OutputName = {'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'};
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< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-orgae85e0d" class = "outline-3" >
< h3 id = "orgae85e0d" > < span class = "section-number-3" > 3.3< / span > Steady State Decoupling< / h3 >
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< div class = "outline-text-3" id = "text-3-3" >
< / div >
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< div id = "outline-container-org1e2bbe7" class = "outline-4" >
< h4 id = "org1e2bbe7" > < span class = "section-number-4" > 3.3.1< / span > Pre-Compensator Design< / h4 >
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< div class = "outline-text-4" id = "text-3-3-1" >
< p >
We choose \(W_1 = G^{-1}(0)\).
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > W1 = inv(freqresp(G_dvf, 0));
< / pre >
< / div >
< p >
The (static) decoupled plant is \(G_s(s) = G(s) W_1\).
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > Gs = G_dvf*W1;
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< / pre >
< / div >
< p >
In the case of the Stewart platform, the pre-compensator for static decoupling is equal to \(\mathcal{K} \bm{J}\):
< / p >
\begin{align*}
W_1 & = \left( \frac{\bm{\mathcal{X}}}{\bm{\tau}}(s=0) \right)^{-1}\\
& = \left( \frac{\bm{\mathcal{X}}}{\bm{\tau}}(s=0) \bm{J}^T \right)^{-1}\\
& = \left( \bm{C} \bm{J}^T \right)^{-1}\\
& = \left( \bm{J}^{-1} \mathcal{K}^{-1} \right)^{-1}\\
& = \mathcal{K} \bm{J}
\end{align*}
< p >
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The static decoupled plant is schematic shown in Figure < a href = "#org76617c6" > 14< / a > and the bode plots of its diagonal elements are shown in Figure < a href = "#org96093e0" > 15< / a > .
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< / p >
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< div id = "org76617c6" class = "figure" >
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< p > < img src = "figs/control_arch_static_decoupling.png" alt = "control_arch_static_decoupling.png" / >
< / p >
< p > < span class = "figure-number" > Figure 14: < / span > Static Decoupling of the Stewart platform< / p >
< / div >
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< div id = "org96093e0" class = "figure" >
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< p > < img src = "figs/static_decoupling_diagonal_plant.png" alt = "static_decoupling_diagonal_plant.png" / >
< / p >
< p > < span class = "figure-number" > Figure 15: < / span > Bode plot of the diagonal elements of \(G_s(s)\) (< a href = "./figs/static_decoupling_diagonal_plant.png" > png< / a > , < a href = "./figs/static_decoupling_diagonal_plant.pdf" > pdf< / a > )< / p >
< / div >
< / div >
< / div >
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< div id = "outline-container-org077e6f6" class = "outline-4" >
< h4 id = "org077e6f6" > < span class = "section-number-4" > 3.3.2< / span > Diagonal Control Design< / h4 >
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< div class = "outline-text-4" id = "text-3-3-2" >
< p >
We design a diagonal controller \(K_s(s)\) that consist of a pure integrator and a lead around the crossover.
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > wc = 2*pi*300; % Wanted Bandwidth [rad/s]
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h = 1.5;
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H_lead = 1/h*(1 + s/(wc/h))/(1 + s/(wc*h));
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Ks_dvf = diag(1./abs(diag(freqresp(1/s*Gs, wc)))) .* H_lead .* 1/s;
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< / pre >
< / div >
< p >
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The overall controller is then \(K(s) = W_1 K_s(s)\) as shown in Figure < a href = "#org4d1ce48" > 16< / a > .
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< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > K_hac_dvf = W1 * Ks_dvf;
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< / pre >
< / div >
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< div id = "org4d1ce48" class = "figure" >
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< p > < img src = "figs/control_arch_static_decoupling_K.png" alt = "control_arch_static_decoupling_K.png" / >
< / p >
< p > < span class = "figure-number" > Figure 16: < / span > Controller including the static decoupling matrix< / p >
< / div >
< / div >
< / div >
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< div id = "outline-container-org2ef5df1" class = "outline-4" >
< h4 id = "org2ef5df1" > < span class = "section-number-4" > 3.3.3< / span > Results< / h4 >
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< div class = "outline-text-4" id = "text-3-3-3" >
< p >
We identify the transmissibility and compliance of the Stewart platform under open-loop and closed-loop control.
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > controller = initializeController('type', 'open-loop');
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[T_ol, T_norm_ol, freqs] = computeTransmissibility();
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[C_ol, C_norm_ol, ~] = computeCompliance();
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< / pre >
< / div >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > controller = initializeController('type', 'hac-dvf');
[T_hac_dvf, T_norm_hac_dvf, ~] = computeTransmissibility();
[C_hac_dvf, C_norm_hac_dvf, ~] = computeCompliance();
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< / pre >
< / div >
< p >
The results are shown in figure
< / p >
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< div id = "orge29798b" class = "figure" >
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< p > < img src = "figs/static_decoupling_C_T_frobenius_norm.png" alt = "static_decoupling_C_T_frobenius_norm.png" / >
< / p >
< p > < span class = "figure-number" > Figure 17: < / span > Frobenius norm of the Compliance and transmissibility matrices (< a href = "./figs/static_decoupling_C_T_frobenius_norm.png" > png< / a > , < a href = "./figs/static_decoupling_C_T_frobenius_norm.pdf" > pdf< / a > )< / p >
< / div >
< / div >
< / div >
< / div >
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< div id = "outline-container-orgad35bf9" class = "outline-3" >
< h3 id = "orgad35bf9" > < span class = "section-number-3" > 3.4< / span > Decoupling at Crossover< / h3 >
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< div class = "outline-text-3" id = "text-3-4" >
< ul class = "org-ul" >
< li class = "off" > < code > [  ]< / code > Find a method for real approximation of a complex matrix< / li >
< / ul >
< / div >
< / div >
< / div >
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< div id = "outline-container-org846cef9" class = "outline-2" >
< h2 id = "org846cef9" > < span class = "section-number-2" > 4< / span > Time Domain Simulation< / h2 >
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< div class = "outline-text-2" id = "text-4" >
< / div >
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< div id = "outline-container-org323700d" class = "outline-3" >
< h3 id = "org323700d" > < span class = "section-number-3" > 4.1< / span > Initialization< / h3 >
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< div class = "outline-text-3" id = "text-4-1" >
< p >
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We first initialize the Stewart platform.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > stewart = initializeStewartPlatform();
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stewart = initializeFramesPositions(stewart, 'H', 90e-3, 'MO_B', 45e-3);
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stewart = generateGeneralConfiguration(stewart);
stewart = computeJointsPose(stewart);
stewart = initializeStrutDynamics(stewart);
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stewart = initializeJointDynamics(stewart, 'type_F', 'universal', 'type_M', 'spherical');
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stewart = initializeCylindricalPlatforms(stewart);
stewart = initializeCylindricalStruts(stewart);
stewart = computeJacobian(stewart);
stewart = initializeStewartPose(stewart);
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stewart = initializeInertialSensor(stewart, 'type', 'none');
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< / pre >
< / div >
< p >
The rotation point of the ground is located at the origin of frame \(\{A\}\).
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > ground = initializeGround('type', 'rigid', 'rot_point', stewart.platform_F.FO_A);
payload = initializePayload('type', 'none');
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< / pre >
< / div >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > load('./mat/motion_error_ol.mat', 'Eg')
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< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-org8dbc004" class = "outline-3" >
< h3 id = "org8dbc004" > < span class = "section-number-3" > 4.2< / span > HAC IFF< / h3 >
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< div class = "outline-text-3" id = "text-4-2" >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > controller = initializeController('type', 'iff');
K_iff = -(1e4/s)*eye(6);
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%% Name of the Simulink File
mdl = 'stewart_platform_model';
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%% Input/Output definition
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clear io; io_i = 1;
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io(io_i) = linio([mdl, '/Controller'], 1, 'input'); io_i = io_i + 1; % Actuator Force Inputs [N]
io(io_i) = linio([mdl, '/Absolute Motion Sensor'], 1, 'openoutput'); io_i = io_i + 1; % Absolute Sensor [m, rad]
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%% Run the linearization
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G_iff = linearize(mdl, io);
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G_iff.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
G_iff.OutputName = {'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'};
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Gc_iff = minreal(G_iff)/stewart.kinematics.J';
Gc_iff.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
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< / pre >
< / div >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > wc = 2*pi*100; % Wanted Bandwidth [rad/s]
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h = 1.2;
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H_lead = 1/h*(1 + s/(wc/h))/(1 + s/(wc*h));
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Kd_iff = diag(1./abs(diag(freqresp(1/s*Gc_iff, wc)))) .* H_lead .* 1/s;
K_hac_iff = inv(stewart.kinematics.J')*Kd_iff;
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< / pre >
< / div >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > controller = initializeController('type', 'hac-iff');
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< div id = "outline-container-org7dc4716" class = "outline-3" >
< h3 id = "org7dc4716" > < span class = "section-number-3" > 4.3< / span > HAC-DVF< / h3 >
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< div class = "outline-text-3" id = "text-4-3" >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > controller = initializeController('type', 'dvf');
K_dvf = -1e4*s/(1+s/2/pi/5000)*eye(6);
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%% Name of the Simulink File
mdl = 'stewart_platform_model';
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%% Input/Output definition
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clear io; io_i = 1;
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io(io_i) = linio([mdl, '/Controller'], 1, 'input'); io_i = io_i + 1; % Actuator Force Inputs [N]
io(io_i) = linio([mdl, '/Absolute Motion Sensor'], 1, 'openoutput'); io_i = io_i + 1; % Absolute Sensor [m, rad]
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%% Run the linearization
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G_dvf = linearize(mdl, io);
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G_dvf.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
G_dvf.OutputName = {'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'};
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Gc_dvf = minreal(G_dvf)/stewart.kinematics.J';
Gc_dvf.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
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< / pre >
< / div >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > wc = 2*pi*100; % Wanted Bandwidth [rad/s]
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h = 1.2;
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H_lead = 1/h*(1 + s/(wc/h))/(1 + s/(wc*h));
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Kd_dvf = diag(1./abs(diag(freqresp(1/s*Gc_dvf, wc)))) .* H_lead .* 1/s;
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K_hac_dvf = inv(stewart.kinematics.J')*Kd_dvf;
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< / pre >
< / div >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > controller = initializeController('type', 'hac-dvf');
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< div id = "outline-container-org1230d9e" class = "outline-3" >
< h3 id = "org1230d9e" > < span class = "section-number-3" > 4.4< / span > Results< / h3 >
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< div class = "outline-text-3" id = "text-4-4" >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > figure;
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subplot(1, 2, 1);
hold on;
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plot(Eg.Time, Eg.Data(:, 1), 'DisplayName', 'X');
plot(Eg.Time, Eg.Data(:, 2), 'DisplayName', 'Y');
plot(Eg.Time, Eg.Data(:, 3), 'DisplayName', 'Z');
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hold off;
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xlabel('Time [s]');
ylabel('Position error [m]');
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legend();
subplot(1, 2, 2);
hold on;
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plot(simout.Xa.Time, simout.Xa.Data(:, 1));
plot(simout.Xa.Time, simout.Xa.Data(:, 2));
plot(simout.Xa.Time, simout.Xa.Data(:, 3));
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hold off;
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xlabel('Time [s]');
ylabel('Orientation error [rad]');
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< div id = "outline-container-org69ebad1" class = "outline-2" >
< h2 id = "org69ebad1" > < span class = "section-number-2" > 5< / span > Functions< / h2 >
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< div class = "outline-text-2" id = "text-5" >
< / div >
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< div id = "outline-container-orgc7bcc65" class = "outline-3" >
< h3 id = "orgc7bcc65" > < span class = "section-number-3" > 5.1< / span > < code > initializeController< / code > : Initialize the Controller< / h3 >
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< div class = "outline-text-3" id = "text-5-1" >
< p >
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< a id = "org33a5401" > < / a >
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< / p >
< / div >
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< div id = "outline-container-orgf672f64" class = "outline-4" >
< h4 id = "orgf672f64" > Function description< / h4 >
< div class = "outline-text-4" id = "text-orgf672f64" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > function [controller] = initializeController(args)
% initializeController - Initialize the Controller
%
% Syntax: [] = initializeController(args)
%
% Inputs:
% - args - Can have the following fields:
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< div id = "outline-container-org941466e" class = "outline-4" >
< h4 id = "org941466e" > Optional Parameters< / h4 >
< div class = "outline-text-4" id = "text-org941466e" >
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< div class = "org-src-container" >
< pre class = "src src-matlab" > arguments
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args.type char {mustBeMember(args.type, {'open-loop', 'iff', 'dvf', 'hac-iff', 'hac-dvf', 'ref-track-L', 'ref-track-X', 'ref-track-hac-dvf'})} = 'open-loop'
end
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< div id = "outline-container-org65d3a7d" class = "outline-4" >
< h4 id = "org65d3a7d" > Structure initialization< / h4 >
< div class = "outline-text-4" id = "text-org65d3a7d" >
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< div class = "org-src-container" >
< pre class = "src src-matlab" > controller = struct();
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< div id = "outline-container-org32be93f" class = "outline-4" >
< h4 id = "org32be93f" > Add Type< / h4 >
< div class = "outline-text-4" id = "text-org32be93f" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > switch args.type
case 'open-loop'
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controller.type = 0;
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case 'iff'
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controller.type = 1;
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case 'dvf'
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controller.type = 2;
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case 'hac-iff'
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controller.type = 3;
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case 'hac-dvf'
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controller.type = 4;
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case 'ref-track-L'
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controller.type = 5;
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case 'ref-track-X'
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controller.type = 6;
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case 'ref-track-hac-dvf'
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controller.type = 7;
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end
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< style > . csl-entry { text-indent : -1.5 em ; margin-left : 1.5 em ; } < / style > < h2 class = 'citeproc-org-bib-h2' > Bibliography< / h2 >
< div class = "csl-bib-body" >
< div class = "csl-entry" > < a name = "citeproc_bib_item_1" > < / a > Skogestad, Sigurd, and Ian Postlethwaite. 2007. < i > Multivariable Feedback Control: Analysis and Design< / i > . John Wiley.< / div >
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< div id = "postamble" class = "status" >
< p class = "author" > Author: Dehaeze Thomas< / p >
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< p class = "date" > Created: 2020-08-05 mer. 13:27< / p >
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