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< title > Identification of the Stewart Platform using Simscape< / title >
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< meta name = "author" content = "Dehaeze Thomas" / >
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< h1 class = "title" > Identification of the Stewart Platform using Simscape< / 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 = "#orge8b6206" > 1. Modal Analysis of the Stewart Platform< / a >
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< ul >
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< li > < a href = "#org40f9c57" > 1.1. Initialize the Stewart Platform< / a > < / li >
< li > < a href = "#orgd9529ee" > 1.2. Identification< / a > < / li >
< li > < a href = "#orgbdba4a6" > 1.3. Coordinate transformation< / a > < / li >
< li > < a href = "#org11e3698" > 1.4. Analysis< / a > < / li >
< li > < a href = "#org1db5fc4" > 1.5. Visualizing the modes< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#orgfeed9a3" > 2. Transmissibility Analysis< / a >
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< ul >
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< li > < a href = "#org5ba3096" > 2.1. Initialize the Stewart platform< / a > < / li >
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< li > < a href = "#org279dcc8" > 2.2. Transmissibility< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#org3ad92e9" > 3. Compliance Analysis< / a >
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< ul >
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< li > < a href = "#orgc957431" > 3.1. Initialize the Stewart platform< / a > < / li >
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< li > < a href = "#org26cb46a" > 3.2. Compliance< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#org51e266f" > 4. Functions< / a >
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< ul >
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< li > < a href = "#org25ca725" > 4.1. Compute the Transmissibility< / a >
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< ul >
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< li > < a href = "#orgafb57d0" > Function description< / a > < / li >
< li > < a href = "#orga00af61" > Optional Parameters< / a > < / li >
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< li > < a href = "#org17a8811" > Identification of the Transmissibility Matrix< / a > < / li >
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< li > < a href = "#orgbc9a383" > Computation of the Frobenius norm< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#orgb6e05b3" > 4.2. Compute the Compliance< / a >
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< ul >
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< li > < a href = "#org210c0ca" > Function description< / a > < / li >
< li > < a href = "#org24feeb1" > Optional Parameters< / a > < / li >
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< li > < a href = "#org2c35042" > Identification of the Compliance Matrix< / a > < / li >
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< li > < a href = "#orgb002200" > Computation of the Frobenius norm< / a > < / li >
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< / ul >
< / li >
< / ul >
< / li >
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< p >
In this document, we discuss the various methods to identify the behavior of the Stewart platform.
< / p >
< ul class = "org-ul" >
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< li > < a href = "#orgd142bb4" > 1< / a > < / li >
< li > < a href = "#org5213401" > 2< / a > < / li >
< li > < a href = "#org39baa25" > 3< / a > < / li >
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< / ul >
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< div id = "outline-container-orge8b6206" class = "outline-2" >
< h2 id = "orge8b6206" > < span class = "section-number-2" > 1< / span > Modal Analysis of the Stewart Platform< / h2 >
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< div class = "outline-text-2" id = "text-1" >
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< p >
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< a id = "orgd142bb4" > < / a >
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< / p >
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< / div >
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< div id = "outline-container-org40f9c57" class = "outline-3" >
< h3 id = "org40f9c57" > < span class = "section-number-3" > 1.1< / span > Initialize the Stewart Platform< / h3 >
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< pre class = "src src-matlab" > stewart = initializeStewartPlatform();
stewart = initializeFramesPositions(stewart);
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);
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< / pre >
< / div >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > ground = initializeGround(< span class = "org-string" > 'type'< / span > , < span class = "org-string" > 'none'< / span > );
payload = initializePayload(< span class = "org-string" > 'type'< / span > , < span class = "org-string" > 'none'< / span > );
controller = initializeController(< span class = "org-string" > 'type'< / span > , < span class = "org-string" > 'open-loop'< / span > );
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< div id = "outline-container-orgd9529ee" class = "outline-3" >
< h3 id = "orgd9529ee" > < span class = "section-number-3" > 1.2< / span > Identification< / h3 >
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< div class = "outline-text-3" id = "text-1-2" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > < span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Options for Linearized< / span > < / span >
options = linearizeOptions;
options.SampleTime = 0;
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< 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 > ;
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< span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Input/Output definition< / span > < / span >
clear io; io_i = 1;
io(io_i) = linio([mdl, < span class = "org-string" > '/Controller'< / span > ], 1, < span class = "org-string" > 'openinput'< / span > ); io_i = io_i < span class = "org-type" > +< / span > 1; < span class = "org-comment" > % Actuator Force Inputs [N]< / span >
io(io_i) = linio([mdl, < span class = "org-string" > '/Relative Motion Sensor'< / span > ], 1, < span class = "org-string" > 'openoutput'< / span > ); io_i = io_i < span class = "org-type" > +< / span > 1; < span class = "org-comment" > % Position/Orientation of {B} w.r.t. {A}< / span >
io(io_i) = linio([mdl, < span class = "org-string" > '/Relative Motion Sensor'< / span > ], 2, < span class = "org-string" > 'openoutput'< / span > ); io_i = io_i < span class = "org-type" > +< / span > 1; < span class = "org-comment" > % Velocity of {B} w.r.t. {A}< / span >
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< span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Run the linearization< / span > < / span >
G = linearize(mdl, io);
< span class = "org-comment" > % G.InputName = {'tau1', 'tau2', 'tau3', 'tau4', 'tau5', 'tau6'};< / span >
< span class = "org-comment" > % G.OutputName = {'Xdx', 'Xdy', 'Xdz', 'Xrx', 'Xry', 'Xrz', 'Vdx', 'Vdy', 'Vdz', 'Vrx', 'Vry', 'Vrz'};< / span >
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< / pre >
< / div >
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< p >
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Let’ s check the size of < code > G< / code > :
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< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > size(G)
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< / pre >
< / div >
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< pre class = "example" >
size(G)
State-space model with 12 outputs, 6 inputs, and 18 states.
'org_babel_eoe'
ans =
'org_babel_eoe'
< / pre >
< p >
We expect to have only 12 states (corresponding to the 6dof of the mobile platform).
< / p >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > Gm = minreal(G);
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< / pre >
< / div >
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< pre class = "example" >
Gm = minreal(G);
6 states removed.
< / pre >
< p >
And indeed, we obtain 12 states.
< / p >
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< / div >
< / div >
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< div id = "outline-container-orgbdba4a6" class = "outline-3" >
< h3 id = "orgbdba4a6" > < span class = "section-number-3" > 1.3< / span > Coordinate transformation< / h3 >
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< div class = "outline-text-3" id = "text-1-3" >
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< p >
We can perform the following transformation using the < code > ss2ss< / code > command.
< / p >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > Gt = ss2ss(Gm, Gm.C);
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< / pre >
< / div >
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< p >
Then, the < code > C< / code > matrix of < code > Gt< / code > is the unity matrix which means that the states of the state space model are equal to the measurements \(\bm{Y}\).
< / p >
< p >
The measurements are the 6 displacement and 6 velocities of mobile platform with respect to \(\{B\}\).
< / p >
< p >
We could perform the transformation by hand:
< / p >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > At = Gm.C< span class = "org-type" > *< / span > Gm.A< span class = "org-type" > *< / span > pinv(Gm.C);
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Bt = Gm.C< span class = "org-type" > *< / span > Gm.B;
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Ct = eye(12);
Dt = zeros(12, 6);
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Gt = ss(At, Bt, Ct, Dt);
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< div id = "outline-container-org11e3698" class = "outline-3" >
< h3 id = "org11e3698" > < span class = "section-number-3" > 1.4< / span > Analysis< / h3 >
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< pre class = "src src-matlab" > [V,D] = eig(Gt.A);
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< / pre >
< / div >
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< table border = "2" cellspacing = "0" cellpadding = "6" rules = "groups" frame = "hsides" >
< colgroup >
< col class = "org-right" / >
< col class = "org-right" / >
< col class = "org-right" / >
< / colgroup >
< thead >
< tr >
< th scope = "col" class = "org-right" > Mode Number< / th >
< th scope = "col" class = "org-right" > Resonance Frequency [Hz]< / th >
< th scope = "col" class = "org-right" > Damping Ratio [%]< / th >
< / tr >
< / thead >
< tbody >
< tr >
< td class = "org-right" > 1.0< / td >
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< td class = "org-right" > 780.6< / td >
< td class = "org-right" > 0.4< / td >
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< / tr >
< tr >
< td class = "org-right" > 2.0< / td >
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< td class = "org-right" > 780.6< / td >
< td class = "org-right" > 0.3< / td >
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< / tr >
< tr >
< td class = "org-right" > 3.0< / td >
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< td class = "org-right" > 903.9< / td >
< td class = "org-right" > 0.3< / td >
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< / tr >
< tr >
< td class = "org-right" > 4.0< / td >
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< td class = "org-right" > 1061.4< / td >
< td class = "org-right" > 0.3< / td >
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< / tr >
< tr >
< td class = "org-right" > 5.0< / td >
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< td class = "org-right" > 1061.4< / td >
< td class = "org-right" > 0.2< / td >
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< / tr >
< tr >
< td class = "org-right" > 6.0< / td >
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< td class = "org-right" > 1269.6< / td >
< td class = "org-right" > 0.2< / td >
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< / tr >
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< div id = "outline-container-org1db5fc4" class = "outline-3" >
< h3 id = "org1db5fc4" > < span class = "section-number-3" > 1.5< / span > Visualizing the modes< / h3 >
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< div class = "outline-text-3" id = "text-1-5" >
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< p >
To visualize the i’ th mode, we may excite the system using the inputs \(U_i\) such that \(B U_i\) is co-linear to \(\xi_i\) (the mode we want to excite).
< / p >
< p >
\[ U(t) = e^{\alpha t} ( ) \]
< / p >
< p >
Let’ s first sort the modes and just take the modes corresponding to a eigenvalue with a positive imaginary part.
< / p >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > ws = imag(diag(D));
[ws,I] = sort(ws)
ws = ws(7< span class = "org-type" > :< / span > end); I = I(7< span class = "org-type" > :< / span > end);
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< / pre >
< / div >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > < span class = "org-keyword" > for< / span > < span class = "org-variable-name" > < span class = "org-constant" > i< / span > < / span > = < span class = "org-constant" > 1:length(ws)< / span >
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< / pre >
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< / div >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > i_mode = I(< span class = "org-constant" > i< / span > ); < span class = "org-comment" > % the argument is the i'th mode< / span >
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< / pre >
< / div >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > lambda_i = D(i_mode, i_mode);
xi_i = V(< span class = "org-type" > :< / span > ,i_mode);
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a_i = real(lambda_i);
b_i = imag(lambda_i);
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< / pre >
< / div >
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< p >
Let do 10 periods of the mode.
< / p >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > t = linspace(0, 10< span class = "org-type" > /< / span > (imag(lambda_i)< span class = "org-type" > /< / span > 2< span class = "org-type" > /< / span > < span class = "org-constant" > pi< / span > ), 1000);
U_i = pinv(Gt.B) < span class = "org-type" > *< / span > real(xi_i < span class = "org-type" > *< / span > lambda_i < span class = "org-type" > *< / span > (cos(b_i < span class = "org-type" > *< / span > t) < span class = "org-type" > +< / span > 1< span class = "org-constant" > i< / span > < span class = "org-type" > *< / span > sin(b_i < span class = "org-type" > *< / span > t)));
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< / pre >
< / div >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > U = timeseries(U_i, t);
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< / pre >
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< / div >
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< p >
Simulation:
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > load(< span class = "org-string" > 'mat/conf_simscape.mat'< / span > );
< span class = "org-matlab-simulink-keyword" > set_param< / span > (< span class = "org-variable-name" > conf_simscape< / span > , < span class = "org-string" > 'StopTime'< / span > , num2str(t(< span class = "org-variable-name" > end< / span > )));
< span class = "org-matlab-simulink-keyword" > sim< / span > (mdl);
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< / pre >
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< / div >
< p >
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Save the movie of the mode shape.
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< / p >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > smwritevideo(mdl, sprintf(< span class = "org-string" > 'figs/mode%i'< / span > , < span class = "org-constant" > i< / span > ), ...
< span class = "org-string" > 'PlaybackSpeedRatio'< / span > , 1< span class = "org-type" > /< / span > (b_i< span class = "org-type" > /< / span > 2< span class = "org-type" > /< / span > < span class = "org-constant" > pi< / span > ), ...
< span class = "org-string" > 'FrameRate'< / span > , 30, ...
< span class = "org-string" > 'FrameSize'< / span > , [800, 400]);
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< / pre >
< / div >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > < span class = "org-keyword" > end< / span >
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< / pre >
< / div >
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< div id = "orgd5bd1cd" class = "figure" >
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< p > < img src = "figs/mode1.gif" alt = "mode1.gif" / >
< / p >
< p > < span class = "figure-number" > Figure 1: < / span > Identified mode - 1< / p >
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< / div >
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< div id = "org5c59f9a" class = "figure" >
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< p > < img src = "figs/mode3.gif" alt = "mode3.gif" / >
< / p >
< p > < span class = "figure-number" > Figure 2: < / span > Identified mode - 3< / p >
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< / div >
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< div id = "org0f2e8c4" class = "figure" >
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< p > < img src = "figs/mode5.gif" alt = "mode5.gif" / >
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< / p >
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< p > < span class = "figure-number" > Figure 3: < / span > Identified mode - 5< / p >
< / div >
< / div >
< / div >
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< / div >
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< div id = "outline-container-orgfeed9a3" class = "outline-2" >
< h2 id = "orgfeed9a3" > < span class = "section-number-2" > 2< / span > Transmissibility Analysis< / h2 >
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< div class = "outline-text-2" id = "text-2" >
< p >
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< a id = "org5213401" > < / a >
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< / p >
< / div >
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< div id = "outline-container-org5ba3096" class = "outline-3" >
< h3 id = "org5ba3096" > < span class = "section-number-3" > 2.1< / span > Initialize the Stewart platform< / h3 >
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< div class = "outline-text-3" id = "text-2-1" >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > stewart = initializeStewartPlatform();
stewart = initializeFramesPositions(stewart, < span class = "org-string" > 'H'< / span > , 90e< span class = "org-type" > -< / span > 3, < span class = "org-string" > 'MO_B'< / span > , 45e< span class = "org-type" > -< / span > 3);
stewart = generateGeneralConfiguration(stewart);
stewart = computeJointsPose(stewart);
stewart = initializeStrutDynamics(stewart);
stewart = initializeJointDynamics(stewart, < span class = "org-string" > 'type_F'< / span > , < span class = "org-string" > 'universal_p'< / span > , < span class = "org-string" > 'type_M'< / span > , < span class = "org-string" > 'spherical_p'< / span > );
stewart = initializeCylindricalPlatforms(stewart);
stewart = initializeCylindricalStruts(stewart);
stewart = computeJacobian(stewart);
stewart = initializeStewartPose(stewart);
stewart = initializeInertialSensor(stewart, < span class = "org-string" > 'type'< / span > , < span class = "org-string" > 'accelerometer'< / span > , < span class = "org-string" > 'freq'< / span > , 5e3);
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< / pre >
< / div >
< p >
We set the rotation point of the ground to be at the same point at frames \(\{A\}\) and \(\{B\}\).
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > ground = initializeGround(< span class = "org-string" > 'type'< / span > , < span class = "org-string" > 'rigid'< / span > , < span class = "org-string" > 'rot_point'< / span > , stewart.platform_F.FO_A);
payload = initializePayload(< span class = "org-string" > 'type'< / span > , < span class = "org-string" > 'rigid'< / span > );
controller = initializeController(< span class = "org-string" > 'type'< / span > , < span class = "org-string" > 'open-loop'< / span > );
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< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-org279dcc8" class = "outline-3" >
< h3 id = "org279dcc8" > < span class = "section-number-3" > 2.2< / span > Transmissibility< / h3 >
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< div class = "outline-text-3" id = "text-2-2" >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > < span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Options for Linearized< / span > < / span >
options = linearizeOptions;
options.SampleTime = 0;
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< 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 > ;
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< span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Input/Output definition< / span > < / span >
clear io; io_i = 1;
io(io_i) = linio([mdl, < span class = "org-string" > '/Disturbances/D_w'< / span > ], 1, < span class = "org-string" > 'openinput'< / span > ); io_i = io_i < span class = "org-type" > +< / span > 1; < span class = "org-comment" > % Base Motion [m, rad]< / span >
io(io_i) = linio([mdl, < span class = "org-string" > '/Absolute Motion Sensor'< / span > ], 1, < span class = "org-string" > 'openoutput'< / span > ); io_i = io_i < span class = "org-type" > +< / span > 1; < span class = "org-comment" > % Absolute Motion [m, rad]< / span >
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< span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Run the linearization< / span > < / span >
T = linearize(mdl, io, options);
T.InputName = {< span class = "org-string" > 'Wdx'< / span > , < span class = "org-string" > 'Wdy'< / span > , < span class = "org-string" > 'Wdz'< / span > , < span class = "org-string" > 'Wrx'< / span > , < span class = "org-string" > 'Wry'< / span > , < span class = "org-string" > 'Wrz'< / span > };
T.OutputName = {< span class = "org-string" > 'Edx'< / span > , < span class = "org-string" > 'Edy'< / span > , < span class = "org-string" > 'Edz'< / span > , < span class = "org-string" > 'Erx'< / span > , < span class = "org-string" > 'Ery'< / span > , < span class = "org-string" > 'Erz'< / span > };
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< / pre >
< / div >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > freqs = logspace(1, 4, 1000);
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< span class = "org-type" > figure< / span > ;
< span class = "org-keyword" > for< / span > < span class = "org-variable-name" > ix< / span > = < span class = "org-constant" > 1:6< / span >
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< span class = "org-keyword" > for< / span > < span class = "org-variable-name" > iy< / span > = < span class = "org-constant" > 1:6< / span >
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subplot(6, 6, (ix< span class = "org-type" > -< / span > 1)< span class = "org-type" > *< / span > 6 < span class = "org-type" > +< / span > iy);
hold on;
plot(freqs, abs(squeeze(freqresp(T(ix, iy), freqs, < span class = "org-string" > 'Hz'< / span > ))), < span class = "org-string" > 'k-'< / span > );
< span class = "org-type" > set< / span > (< span class = "org-variable-name" > gca< / span > , < span class = "org-string" > 'XScale'< / span > , < span class = "org-string" > 'log'< / span > ); < span class = "org-type" > set< / span > (< span class = "org-variable-name" > gca< / span > , < span class = "org-string" > 'YScale'< / span > , < span class = "org-string" > 'log'< / span > );
ylim([1e< span class = "org-type" > -< / span > 5, 10]);
xlim([freqs(1), freqs(end)]);
< span class = "org-keyword" > if< / span > ix < span class = "org-type" > < < / span > 6
xticklabels({});
< span class = "org-keyword" > end< / span >
< span class = "org-keyword" > if< / span > iy < span class = "org-type" > > < / span > 1
yticklabels({});
< span class = "org-keyword" > end< / span >
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< span class = "org-keyword" > end< / span >
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< span class = "org-keyword" > end< / span >
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< / pre >
< / div >
< p >
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From (< a href = "#citeproc_bib_item_1" > Preumont et al. 2007< / a > ), one can use the Frobenius norm of the transmissibility matrix to obtain a scalar indicator of the transmissibility performance of the system:
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< / p >
\begin{align*}
\| \bm{T}(\omega) \| & = \sqrt{\text{Trace}[\bm{T}(\omega) \bm{T}(\omega)^H]}\\
& = \sqrt{\Sigma_{i=1}^6 \Sigma_{j=1}^6 |T_{ij}|^2}
\end{align*}
< div class = "org-src-container" >
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< pre class = "src src-matlab" > freqs = logspace(1, 4, 1000);
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T_norm = zeros(length(freqs), 1);
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< span class = "org-keyword" > for< / span > < span class = "org-variable-name" > < span class = "org-constant" > i< / span > < / span > = < span class = "org-constant" > 1:length(freqs)< / span >
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T_norm(< span class = "org-constant" > i< / span > ) = sqrt(trace(freqresp(T, freqs(< span class = "org-constant" > i< / span > ), < span class = "org-string" > 'Hz'< / span > )< span class = "org-type" > *< / span > freqresp(T, freqs(< span class = "org-constant" > i< / span > ), < span class = "org-string" > 'Hz'< / span > )< span class = "org-type" > '< / span > ));
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< span class = "org-keyword" > end< / span >
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< / pre >
< / div >
< p >
And we normalize by a factor \(\sqrt{6}\) to obtain a performance metric comparable to the transmissibility of a one-axis isolator:
\[ \Gamma(\omega) = \|\bm{T}(\omega)\| / \sqrt{6} \]
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > Gamma = T_norm< span class = "org-type" > /< / span > sqrt(6);
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< / pre >
< / div >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > < span class = "org-type" > figure< / span > ;
plot(freqs, Gamma)
< span class = "org-type" > set< / span > (< span class = "org-variable-name" > gca< / span > , < span class = "org-string" > 'XScale'< / span > , < span class = "org-string" > 'log'< / span > ); < span class = "org-type" > set< / span > (< span class = "org-variable-name" > gca< / span > , < span class = "org-string" > 'YScale'< / span > , < span class = "org-string" > 'log'< / span > );
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< / pre >
< / div >
< / div >
< / div >
< / div >
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< div id = "outline-container-org3ad92e9" class = "outline-2" >
< h2 id = "org3ad92e9" > < span class = "section-number-2" > 3< / span > Compliance Analysis< / h2 >
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< div class = "outline-text-2" id = "text-3" >
< p >
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< a id = "org39baa25" > < / a >
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< / p >
< / div >
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< div id = "outline-container-orgc957431" class = "outline-3" >
< h3 id = "orgc957431" > < span class = "section-number-3" > 3.1< / span > Initialize the Stewart platform< / h3 >
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< div class = "outline-text-3" id = "text-3-1" >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > stewart = initializeStewartPlatform();
stewart = initializeFramesPositions(stewart, < span class = "org-string" > 'H'< / span > , 90e< span class = "org-type" > -< / span > 3, < span class = "org-string" > 'MO_B'< / span > , 45e< span class = "org-type" > -< / span > 3);
stewart = generateGeneralConfiguration(stewart);
stewart = computeJointsPose(stewart);
stewart = initializeStrutDynamics(stewart);
stewart = initializeJointDynamics(stewart, < span class = "org-string" > 'type_F'< / span > , < span class = "org-string" > 'universal_p'< / span > , < span class = "org-string" > 'type_M'< / span > , < span class = "org-string" > 'spherical_p'< / span > );
stewart = initializeCylindricalPlatforms(stewart);
stewart = initializeCylindricalStruts(stewart);
stewart = computeJacobian(stewart);
stewart = initializeStewartPose(stewart);
stewart = initializeInertialSensor(stewart, < span class = "org-string" > 'type'< / span > , < span class = "org-string" > 'accelerometer'< / span > , < span class = "org-string" > 'freq'< / span > , 5e3);
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< / pre >
< / div >
< p >
We set the rotation point of the ground to be at the same point at frames \(\{A\}\) and \(\{B\}\).
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > ground = initializeGround(< span class = "org-string" > 'type'< / span > , < span class = "org-string" > 'none'< / span > );
payload = initializePayload(< span class = "org-string" > 'type'< / span > , < span class = "org-string" > 'rigid'< / span > );
controller = initializeController(< span class = "org-string" > 'type'< / span > , < span class = "org-string" > 'open-loop'< / span > );
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< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-org26cb46a" class = "outline-3" >
< h3 id = "org26cb46a" > < span class = "section-number-3" > 3.2< / span > Compliance< / 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" > < span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Options for Linearized< / span > < / span >
options = linearizeOptions;
options.SampleTime = 0;
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< 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 > ;
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< span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Input/Output definition< / span > < / span >
clear io; io_i = 1;
io(io_i) = linio([mdl, < span class = "org-string" > '/Disturbances/F_ext'< / span > ], 1, < span class = "org-string" > 'openinput'< / span > ); io_i = io_i < span class = "org-type" > +< / span > 1; < span class = "org-comment" > % Base Motion [m, rad]< / span >
io(io_i) = linio([mdl, < span class = "org-string" > '/Absolute Motion Sensor'< / span > ], 1, < span class = "org-string" > 'openoutput'< / span > ); io_i = io_i < span class = "org-type" > +< / span > 1; < span class = "org-comment" > % Absolute Motion [m, rad]< / span >
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< span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Run the linearization< / span > < / span >
C = linearize(mdl, io, options);
C.InputName = {< span class = "org-string" > 'Fdx'< / span > , < span class = "org-string" > 'Fdy'< / span > , < span class = "org-string" > 'Fdz'< / span > , < span class = "org-string" > 'Mdx'< / span > , < span class = "org-string" > 'Mdy'< / span > , < span class = "org-string" > 'Mdz'< / span > };
C.OutputName = {< span class = "org-string" > 'Edx'< / span > , < span class = "org-string" > 'Edy'< / span > , < span class = "org-string" > 'Edz'< / span > , < span class = "org-string" > 'Erx'< / span > , < span class = "org-string" > 'Ery'< / span > , < span class = "org-string" > 'Erz'< / span > };
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< / pre >
< / div >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > freqs = logspace(1, 4, 1000);
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< span class = "org-type" > figure< / span > ;
< span class = "org-keyword" > for< / span > < span class = "org-variable-name" > ix< / span > = < span class = "org-constant" > 1:6< / span >
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< span class = "org-keyword" > for< / span > < span class = "org-variable-name" > iy< / span > = < span class = "org-constant" > 1:6< / span >
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subplot(6, 6, (ix< span class = "org-type" > -< / span > 1)< span class = "org-type" > *< / span > 6 < span class = "org-type" > +< / span > iy);
hold on;
plot(freqs, abs(squeeze(freqresp(C(ix, iy), freqs, < span class = "org-string" > 'Hz'< / span > ))), < span class = "org-string" > 'k-'< / span > );
< span class = "org-type" > set< / span > (< span class = "org-variable-name" > gca< / span > , < span class = "org-string" > 'XScale'< / span > , < span class = "org-string" > 'log'< / span > ); < span class = "org-type" > set< / span > (< span class = "org-variable-name" > gca< / span > , < span class = "org-string" > 'YScale'< / span > , < span class = "org-string" > 'log'< / span > );
ylim([1e< span class = "org-type" > -< / span > 10, 1e< span class = "org-type" > -< / span > 3]);
xlim([freqs(1), freqs(end)]);
< span class = "org-keyword" > if< / span > ix < span class = "org-type" > < < / span > 6
xticklabels({});
< span class = "org-keyword" > end< / span >
< span class = "org-keyword" > if< / span > iy < span class = "org-type" > > < / span > 1
yticklabels({});
< span class = "org-keyword" > end< / span >
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< span class = "org-keyword" > end< / span >
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< span class = "org-keyword" > end< / span >
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< / pre >
< / div >
< p >
We can try to use the Frobenius norm to obtain a scalar value representing the 6-dof compliance of the Stewart platform.
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > freqs = logspace(1, 4, 1000);
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C_norm = zeros(length(freqs), 1);
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< span class = "org-keyword" > for< / span > < span class = "org-variable-name" > < span class = "org-constant" > i< / span > < / span > = < span class = "org-constant" > 1:length(freqs)< / span >
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C_norm(< span class = "org-constant" > i< / span > ) = sqrt(trace(freqresp(C, freqs(< span class = "org-constant" > i< / span > ), < span class = "org-string" > 'Hz'< / span > )< span class = "org-type" > *< / span > freqresp(C, freqs(< span class = "org-constant" > i< / span > ), < span class = "org-string" > 'Hz'< / span > )< span class = "org-type" > '< / span > ));
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< span class = "org-keyword" > end< / span >
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< / pre >
< / div >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > < span class = "org-type" > figure< / span > ;
plot(freqs, C_norm)
< span class = "org-type" > set< / span > (< span class = "org-variable-name" > gca< / span > , < span class = "org-string" > 'XScale'< / span > , < span class = "org-string" > 'log'< / span > ); < span class = "org-type" > set< / span > (< span class = "org-variable-name" > gca< / span > , < span class = "org-string" > 'YScale'< / span > , < span class = "org-string" > 'log'< / span > );
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< / pre >
< / div >
< / div >
< / div >
< / div >
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< div id = "outline-container-org51e266f" class = "outline-2" >
< h2 id = "org51e266f" > < span class = "section-number-2" > 4< / span > Functions< / h2 >
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< div class = "outline-text-2" id = "text-4" >
< / div >
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< div id = "outline-container-org25ca725" class = "outline-3" >
< h3 id = "org25ca725" > < span class = "section-number-3" > 4.1< / span > Compute the Transmissibility< / h3 >
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< div class = "outline-text-3" id = "text-4-1" >
< p >
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< a id = "org78f2be2" > < / a >
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< / p >
< / div >
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< div id = "outline-container-orgafb57d0" class = "outline-4" >
< h4 id = "orgafb57d0" > Function description< / h4 >
< div class = "outline-text-4" id = "text-orgafb57d0" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > < span class = "org-keyword" > function< / span > < span class = "org-variable-name" > [T, T_norm, freqs]< / span > = < span class = "org-function-name" > computeTransmissibility< / span > (< span class = "org-variable-name" > args< / span > )
< span class = "org-comment" > % computeTransmissibility -< / span >
< span class = "org-comment" > %< / span >
< span class = "org-comment" > % Syntax: [T, T_norm, freqs] = computeTransmissibility(args)< / span >
< span class = "org-comment" > %< / span >
< span class = "org-comment" > % Inputs:< / span >
< span class = "org-comment" > % - args - Structure with the following fields:< / span >
< span class = "org-comment" > % - plots [true/false] - Should plot the transmissilibty matrix and its Frobenius norm< / span >
< span class = "org-comment" > % - freqs [] - Frequency vector to estimate the Frobenius norm< / span >
< span class = "org-comment" > %< / span >
< span class = "org-comment" > % Outputs:< / span >
< span class = "org-comment" > % - T [6x6 ss] - Transmissibility matrix< / span >
< span class = "org-comment" > % - T_norm [length(freqs)x1] - Frobenius norm of the Transmissibility matrix< / span >
< span class = "org-comment" > % - freqs [length(freqs)x1] - Frequency vector in [Hz]< / span >
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< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-orga00af61" class = "outline-4" >
< h4 id = "orga00af61" > Optional Parameters< / h4 >
< div class = "outline-text-4" id = "text-orga00af61" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > < span class = "org-keyword" > arguments< / span >
< span class = "org-variable-name" > args< / span > .plots logical {mustBeNumericOrLogical} = < span class = "org-constant" > false< / span >
< span class = "org-variable-name" > args< / span > .freqs double {mustBeNumeric, mustBeNonnegative} = logspace(1,4,1000)
< span class = "org-keyword" > end< / span >
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< / pre >
< / div >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > freqs = args.freqs;
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< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-org17a8811" class = "outline-4" >
< h4 id = "org17a8811" > Identification of the Transmissibility Matrix< / h4 >
< div class = "outline-text-4" id = "text-org17a8811" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > < span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Options for Linearized< / span > < / span >
options = linearizeOptions;
options.SampleTime = 0;
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< 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 > ;
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< span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Input/Output definition< / span > < / span >
clear io; io_i = 1;
io(io_i) = linio([mdl, < span class = "org-string" > '/Disturbances/D_w'< / span > ], 1, < span class = "org-string" > 'openinput'< / span > ); io_i = io_i < span class = "org-type" > +< / span > 1; < span class = "org-comment" > % Base Motion [m, rad]< / span >
io(io_i) = linio([mdl, < span class = "org-string" > '/Absolute Motion Sensor'< / span > ], 1, < span class = "org-string" > 'output'< / span > ); io_i = io_i < span class = "org-type" > +< / span > 1; < span class = "org-comment" > % Absolute Motion [m, rad]< / span >
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< span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Run the linearization< / span > < / span >
T = linearize(mdl, io, options);
T.InputName = {< span class = "org-string" > 'Wdx'< / span > , < span class = "org-string" > 'Wdy'< / span > , < span class = "org-string" > 'Wdz'< / span > , < span class = "org-string" > 'Wrx'< / span > , < span class = "org-string" > 'Wry'< / span > , < span class = "org-string" > 'Wrz'< / span > };
T.OutputName = {< span class = "org-string" > 'Edx'< / span > , < span class = "org-string" > 'Edy'< / span > , < span class = "org-string" > 'Edz'< / span > , < span class = "org-string" > 'Erx'< / span > , < span class = "org-string" > 'Ery'< / span > , < span class = "org-string" > 'Erz'< / span > };
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< / pre >
< / div >
< p >
If wanted, the 6x6 transmissibility matrix is plotted.
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > p_handle = zeros(6< span class = "org-type" > *< / span > 6,1);
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< span class = "org-keyword" > if< / span > args.plots
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fig = < span class = "org-type" > figure< / span > ;
< span class = "org-keyword" > for< / span > < span class = "org-variable-name" > ix< / span > = < span class = "org-constant" > 1:6< / span >
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< span class = "org-keyword" > for< / span > < span class = "org-variable-name" > iy< / span > = < span class = "org-constant" > 1:6< / span >
p_handle((ix< span class = "org-type" > -< / span > 1)< span class = "org-type" > *< / span > 6 < span class = "org-type" > +< / span > iy) = subplot(6, 6, (ix< span class = "org-type" > -< / span > 1)< span class = "org-type" > *< / span > 6 < span class = "org-type" > +< / span > iy);
hold on;
plot(freqs, abs(squeeze(freqresp(T(ix, iy), freqs, < span class = "org-string" > 'Hz'< / span > ))), < span class = "org-string" > 'k-'< / span > );
< span class = "org-type" > set< / span > (< span class = "org-variable-name" > gca< / span > , < span class = "org-string" > 'XScale'< / span > , < span class = "org-string" > 'log'< / span > ); < span class = "org-type" > set< / span > (< span class = "org-variable-name" > gca< / span > , < span class = "org-string" > 'YScale'< / span > , < span class = "org-string" > 'log'< / span > );
< span class = "org-keyword" > if< / span > ix < span class = "org-type" > < < / span > 6
xticklabels({});
< span class = "org-keyword" > end< / span >
< span class = "org-keyword" > if< / span > iy < span class = "org-type" > > < / span > 1
yticklabels({});
< span class = "org-keyword" > end< / span >
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< span class = "org-keyword" > end< / span >
< span class = "org-keyword" > end< / span >
linkaxes(p_handle, < span class = "org-string" > 'xy'< / span > )
xlim([freqs(1), freqs(end)]);
ylim([1e< span class = "org-type" > -< / span > 5, 1e2]);
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han = < span class = "org-type" > axes< / span > (fig, < span class = "org-string" > 'visible'< / span > , < span class = "org-string" > 'off'< / span > );
han.XLabel.Visible = < span class = "org-string" > 'on'< / span > ;
han.YLabel.Visible = < span class = "org-string" > 'on'< / span > ;
xlabel(han, < span class = "org-string" > 'Frequency [Hz]'< / span > );
ylabel(han, < span class = "org-string" > 'Transmissibility [m/m]'< / span > );
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< span class = "org-keyword" > end< / span >
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< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-orgbc9a383" class = "outline-4" >
< h4 id = "orgbc9a383" > Computation of the Frobenius norm< / h4 >
< div class = "outline-text-4" id = "text-orgbc9a383" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > T_norm = zeros(length(freqs), 1);
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< span class = "org-keyword" > for< / span > < span class = "org-variable-name" > < span class = "org-constant" > i< / span > < / span > = < span class = "org-constant" > 1:length(freqs)< / span >
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T_norm(< span class = "org-constant" > i< / span > ) = sqrt(trace(freqresp(T, freqs(< span class = "org-constant" > i< / span > ), < span class = "org-string" > 'Hz'< / span > )< span class = "org-type" > *< / span > freqresp(T, freqs(< span class = "org-constant" > i< / span > ), < span class = "org-string" > 'Hz'< / span > )< span class = "org-type" > '< / span > ));
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< span class = "org-keyword" > end< / span >
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< / pre >
< / div >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > T_norm = T_norm< span class = "org-type" > /< / span > sqrt(6);
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< / pre >
< / div >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > < span class = "org-keyword" > if< / span > args.plots
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< span class = "org-type" > figure< / span > ;
plot(freqs, T_norm)
< span class = "org-type" > set< / span > (< span class = "org-variable-name" > gca< / span > , < span class = "org-string" > 'XScale'< / span > , < span class = "org-string" > 'log'< / span > ); < span class = "org-type" > set< / span > (< span class = "org-variable-name" > gca< / span > , < span class = "org-string" > 'YScale'< / span > , < span class = "org-string" > 'log'< / span > );
xlabel(< span class = "org-string" > 'Frequency [Hz]'< / span > );
ylabel(< span class = "org-string" > 'Transmissibility - Frobenius Norm'< / span > );
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< span class = "org-keyword" > end< / span >
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< / pre >
< / div >
< / div >
< / div >
< / div >
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< div id = "outline-container-orgb6e05b3" class = "outline-3" >
< h3 id = "orgb6e05b3" > < span class = "section-number-3" > 4.2< / span > Compute the Compliance< / h3 >
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< div class = "outline-text-3" id = "text-4-2" >
< p >
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< a id = "org13d7e8a" > < / a >
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< / p >
< / div >
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< div id = "outline-container-org210c0ca" class = "outline-4" >
< h4 id = "org210c0ca" > Function description< / h4 >
< div class = "outline-text-4" id = "text-org210c0ca" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > < span class = "org-keyword" > function< / span > < span class = "org-variable-name" > [C, C_norm, freqs]< / span > = < span class = "org-function-name" > computeCompliance< / span > (< span class = "org-variable-name" > args< / span > )
< span class = "org-comment" > % computeCompliance -< / span >
< span class = "org-comment" > %< / span >
< span class = "org-comment" > % Syntax: [C, C_norm, freqs] = computeCompliance(args)< / span >
< span class = "org-comment" > %< / span >
< span class = "org-comment" > % Inputs:< / span >
< span class = "org-comment" > % - args - Structure with the following fields:< / span >
< span class = "org-comment" > % - plots [true/false] - Should plot the transmissilibty matrix and its Frobenius norm< / span >
< span class = "org-comment" > % - freqs [] - Frequency vector to estimate the Frobenius norm< / span >
< span class = "org-comment" > %< / span >
< span class = "org-comment" > % Outputs:< / span >
< span class = "org-comment" > % - C [6x6 ss] - Compliance matrix< / span >
< span class = "org-comment" > % - C_norm [length(freqs)x1] - Frobenius norm of the Compliance matrix< / span >
< span class = "org-comment" > % - freqs [length(freqs)x1] - Frequency vector in [Hz]< / span >
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< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-org24feeb1" class = "outline-4" >
< h4 id = "org24feeb1" > Optional Parameters< / h4 >
< div class = "outline-text-4" id = "text-org24feeb1" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > < span class = "org-keyword" > arguments< / span >
< span class = "org-variable-name" > args< / span > .plots logical {mustBeNumericOrLogical} = < span class = "org-constant" > false< / span >
< span class = "org-variable-name" > args< / span > .freqs double {mustBeNumeric, mustBeNonnegative} = logspace(1,4,1000)
< span class = "org-keyword" > end< / span >
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< / pre >
< / div >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > freqs = args.freqs;
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< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-org2c35042" class = "outline-4" >
< h4 id = "org2c35042" > Identification of the Compliance Matrix< / h4 >
< div class = "outline-text-4" id = "text-org2c35042" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > < span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Options for Linearized< / span > < / span >
options = linearizeOptions;
options.SampleTime = 0;
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< 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 > ;
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< span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Input/Output definition< / span > < / span >
clear io; io_i = 1;
io(io_i) = linio([mdl, < span class = "org-string" > '/Disturbances/F_ext'< / span > ], 1, < span class = "org-string" > 'openinput'< / span > ); io_i = io_i < span class = "org-type" > +< / span > 1; < span class = "org-comment" > % External forces [N, N*m]< / span >
io(io_i) = linio([mdl, < span class = "org-string" > '/Absolute Motion Sensor'< / span > ], 1, < span class = "org-string" > 'output'< / span > ); io_i = io_i < span class = "org-type" > +< / span > 1; < span class = "org-comment" > % Absolute Motion [m, rad]< / span >
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< span class = "org-matlab-cellbreak" > < span class = "org-comment" > %% Run the linearization< / span > < / span >
C = linearize(mdl, io, options);
C.InputName = {< span class = "org-string" > 'Fdx'< / span > , < span class = "org-string" > 'Fdy'< / span > , < span class = "org-string" > 'Fdz'< / span > , < span class = "org-string" > 'Mdx'< / span > , < span class = "org-string" > 'Mdy'< / span > , < span class = "org-string" > 'Mdz'< / span > };
C.OutputName = {< span class = "org-string" > 'Edx'< / span > , < span class = "org-string" > 'Edy'< / span > , < span class = "org-string" > 'Edz'< / span > , < span class = "org-string" > 'Erx'< / span > , < span class = "org-string" > 'Ery'< / span > , < span class = "org-string" > 'Erz'< / span > };
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< / pre >
< / div >
< p >
If wanted, the 6x6 transmissibility matrix is plotted.
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > p_handle = zeros(6< span class = "org-type" > *< / span > 6,1);
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< span class = "org-keyword" > if< / span > args.plots
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fig = < span class = "org-type" > figure< / span > ;
< span class = "org-keyword" > for< / span > < span class = "org-variable-name" > ix< / span > = < span class = "org-constant" > 1:6< / span >
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< span class = "org-keyword" > for< / span > < span class = "org-variable-name" > iy< / span > = < span class = "org-constant" > 1:6< / span >
p_handle((ix< span class = "org-type" > -< / span > 1)< span class = "org-type" > *< / span > 6 < span class = "org-type" > +< / span > iy) = subplot(6, 6, (ix< span class = "org-type" > -< / span > 1)< span class = "org-type" > *< / span > 6 < span class = "org-type" > +< / span > iy);
hold on;
plot(freqs, abs(squeeze(freqresp(C(ix, iy), freqs, < span class = "org-string" > 'Hz'< / span > ))), < span class = "org-string" > 'k-'< / span > );
< span class = "org-type" > set< / span > (< span class = "org-variable-name" > gca< / span > , < span class = "org-string" > 'XScale'< / span > , < span class = "org-string" > 'log'< / span > ); < span class = "org-type" > set< / span > (< span class = "org-variable-name" > gca< / span > , < span class = "org-string" > 'YScale'< / span > , < span class = "org-string" > 'log'< / span > );
< span class = "org-keyword" > if< / span > ix < span class = "org-type" > < < / span > 6
xticklabels({});
< span class = "org-keyword" > end< / span >
< span class = "org-keyword" > if< / span > iy < span class = "org-type" > > < / span > 1
yticklabels({});
< span class = "org-keyword" > end< / span >
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< span class = "org-keyword" > end< / span >
< span class = "org-keyword" > end< / span >
linkaxes(p_handle, < span class = "org-string" > 'xy'< / span > )
xlim([freqs(1), freqs(end)]);
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han = < span class = "org-type" > axes< / span > (fig, < span class = "org-string" > 'visible'< / span > , < span class = "org-string" > 'off'< / span > );
han.XLabel.Visible = < span class = "org-string" > 'on'< / span > ;
han.YLabel.Visible = < span class = "org-string" > 'on'< / span > ;
xlabel(han, < span class = "org-string" > 'Frequency [Hz]'< / span > );
ylabel(han, < span class = "org-string" > 'Compliance [m/N, rad/(N*m)]'< / span > );
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< span class = "org-keyword" > end< / span >
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< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-orgb002200" class = "outline-4" >
< h4 id = "orgb002200" > Computation of the Frobenius norm< / h4 >
< div class = "outline-text-4" id = "text-orgb002200" >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > freqs = args.freqs;
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C_norm = zeros(length(freqs), 1);
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< span class = "org-keyword" > for< / span > < span class = "org-variable-name" > < span class = "org-constant" > i< / span > < / span > = < span class = "org-constant" > 1:length(freqs)< / span >
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C_norm(< span class = "org-constant" > i< / span > ) = sqrt(trace(freqresp(C, freqs(< span class = "org-constant" > i< / span > ), < span class = "org-string" > 'Hz'< / span > )< span class = "org-type" > *< / span > freqresp(C, freqs(< span class = "org-constant" > i< / span > ), < span class = "org-string" > 'Hz'< / span > )< span class = "org-type" > '< / span > ));
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< span class = "org-keyword" > end< / span >
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< / pre >
< / div >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > < span class = "org-keyword" > if< / span > args.plots
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< span class = "org-type" > figure< / span > ;
plot(freqs, C_norm)
< span class = "org-type" > set< / span > (< span class = "org-variable-name" > gca< / span > , < span class = "org-string" > 'XScale'< / span > , < span class = "org-string" > 'log'< / span > ); < span class = "org-type" > set< / span > (< span class = "org-variable-name" > gca< / span > , < span class = "org-string" > 'YScale'< / span > , < span class = "org-string" > 'log'< / span > );
xlabel(< span class = "org-string" > 'Frequency [Hz]'< / span > );
ylabel(< span class = "org-string" > 'Compliance - Frobenius Norm'< / span > );
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< span class = "org-keyword" > end< / span >
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< / pre >
< / div >
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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 > Preumont, A., M. Horodinca, I. Romanescu, B. de Marneffe, M. Avraam, A. Deraemaeker, F. Bossens, and A. Abu Hanieh. 2007. “A Six-Axis Single-Stage Active Vibration Isolator Based on Stewart Platform.” < i > Journal of Sound and Vibration< / i > 300 (3-5):644– < a href = "https://doi.org/10.1016/j.jsv.2006.07.050" > https://doi.org/10.1016/j.jsv.2006.07.050< / a > .< / div >
< / div >
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< / div >
< / div >
< / div >
< / div >
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< / div >
< div id = "postamble" class = "status" >
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< p class = "author" > Author: Dehaeze Thomas< / p >
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< p class = "date" > Created: 2021-01-08 ven. 15:52< / p >
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< / div >
< / body >
< / html >
