Transmissibility and Compliance comp IFF/DVF/OL
This commit is contained in:
@@ -91,7 +91,7 @@ To run the script, open the Simulink Project, and type =run active_damping_inert
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#+end_src
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#+begin_src matlab
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ground = initializeGround('type', 'none');
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ground = initializeGround('type', 'rigid', 'rot_point', stewart.platform_F.FO_A);
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payload = initializePayload('type', 'none');
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#+end_src
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@@ -321,10 +321,14 @@ We first initialize the Stewart platform without joint stiffness.
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#+end_src
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#+begin_src matlab
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ground = initializeGround('type', 'none');
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ground = initializeGround('type', 'rigid', 'rot_point', stewart.platform_F.FO_A);
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payload = initializePayload('type', 'none');
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#+end_src
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#+begin_src matlab
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controller = initializeController('type', 'open-loop');
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#+end_src
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And we identify the dynamics from force actuators to force sensors.
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#+begin_src matlab
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%% Options for Linearized
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@@ -590,7 +594,7 @@ We first initialize the Stewart platform without joint stiffness.
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#+end_src
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#+begin_src matlab
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ground = initializeGround('type', 'none');
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ground = initializeGround('type', 'rigid', 'rot_point', stewart.platform_F.FO_A);
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payload = initializePayload('type', 'none');
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#+end_src
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@@ -773,3 +777,170 @@ The root locus is shown in figure [[fig:root_locus_dvf_rot_stiffness]].
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#+begin_important
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Joint stiffness does increase the resonance frequencies of the system but does not change the attainable damping when using relative motion sensors.
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#+end_important
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* Compliance and Transmissibility Comparison
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** Initialization
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We first initialize the Stewart platform without joint stiffness.
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#+begin_src matlab
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stewart = initializeStewartPlatform();
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stewart = initializeFramesPositions(stewart, 'H', 90e-3, 'MO_B', 45e-3);
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stewart = generateGeneralConfiguration(stewart);
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stewart = computeJointsPose(stewart);
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stewart = initializeStrutDynamics(stewart);
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stewart = initializeJointDynamics(stewart, 'type_F', 'universal_p', 'type_M', 'spherical_p');
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stewart = initializeCylindricalPlatforms(stewart);
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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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#+end_src
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The rotation point of the ground is located at the origin of frame $\{A\}$.
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#+begin_src matlab
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ground = initializeGround('type', 'rigid', 'rot_point', stewart.platform_F.FO_A);
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payload = initializePayload('type', 'none');
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#+end_src
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** Identification
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Let's first identify the transmissibility and compliance in the open-loop case.
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#+begin_src matlab
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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, freqs] = computeCompliance();
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#+end_src
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Now, let's identify the transmissibility and compliance for the Integral Force Feedback architecture.
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#+begin_src matlab
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controller = initializeController('type', 'iff');
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G_iff = (2e4/s)*eye(6);
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[T_iff, T_norm_iff, ~] = computeTransmissibility();
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[C_iff, C_norm_iff, ~] = computeCompliance();
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#+end_src
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And for the Direct Velocity Feedback.
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#+begin_src matlab
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controller = initializeController('type', 'dvf');
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G_dvf = 1e4*s/(1+s/2/pi/5000)*eye(6);
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[T_dvf, T_norm_dvf, ~] = computeTransmissibility();
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[C_dvf, C_norm_dvf, ~] = computeCompliance();
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#+end_src
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** Results
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#+begin_src matlab :exports none
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p_handle = zeros(6*6,1);
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fig = figure;
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for ix = 1:6
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for iy = 1:6
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p_handle((ix-1)*6 + iy) = subplot(6, 6, (ix-1)*6 + iy);
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hold on;
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set(gca,'ColorOrderIndex',1);
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plot(freqs, abs(squeeze(freqresp(T_ol(ix, iy), freqs, 'Hz'))));
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set(gca,'ColorOrderIndex',2);
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plot(freqs, abs(squeeze(freqresp(T_iff(ix, iy), freqs, 'Hz'))));
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set(gca,'ColorOrderIndex',3);
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plot(freqs, abs(squeeze(freqresp(T_dvf(ix, iy), freqs, 'Hz'))));
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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if ix < 6
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xticklabels({});
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end
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if iy > 1
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yticklabels({});
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end
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end
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end
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linkaxes(p_handle, 'xy')
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xlim([freqs(1), freqs(end)]);
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han = axes(fig, 'visible', 'off');
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han.XLabel.Visible = 'on';
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han.YLabel.Visible = 'on';
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ylabel(han, 'Frequency [Hz]');
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xlabel(han, 'Transmissibility');
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#+end_src
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#+header: :tangle no :exports results :results none :noweb yes
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#+begin_src matlab :var filepath="figs/transmissibility_iff_dvf.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
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<<plt-matlab>>
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#+end_src
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#+name: fig:transmissibility_iff_dvf
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#+caption: Obtained transmissibility for Open-Loop Control (Blue), Integral Force Feedback (Red) and Direct Velocity Feedback (Yellow) ([[./figs/transmissibility_iff_dvf.png][png]], [[./figs/transmissibility_iff_dvf.pdf][pdf]])
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[[file:figs/transmissibility_iff_dvf.png]]
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#+begin_src matlab :exports none
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p_handle = zeros(6*6,1);
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fig = figure;
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for ix = 1:6
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for iy = 1:6
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p_handle((ix-1)*6 + iy) = subplot(6, 6, (ix-1)*6 + iy);
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hold on;
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set(gca,'ColorOrderIndex',1);
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plot(freqs, abs(squeeze(freqresp(C_ol(ix, iy), freqs, 'Hz'))));
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set(gca,'ColorOrderIndex',2);
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plot(freqs, abs(squeeze(freqresp(C_iff(ix, iy), freqs, 'Hz'))));
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set(gca,'ColorOrderIndex',3);
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plot(freqs, abs(squeeze(freqresp(C_dvf(ix, iy), freqs, 'Hz'))));
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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if ix < 6
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xticklabels({});
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end
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if iy > 1
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yticklabels({});
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end
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end
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end
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linkaxes(p_handle, 'xy')
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xlim([freqs(1), freqs(end)]);
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han = axes(fig, 'visible', 'off');
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han.XLabel.Visible = 'on';
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han.YLabel.Visible = 'on';
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ylabel(han, 'Frequency [Hz]');
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xlabel(han, 'Compliance');
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#+end_src
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#+header: :tangle no :exports results :results none :noweb yes
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#+begin_src matlab :var filepath="figs/compliance_iff_dvf.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
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<<plt-matlab>>
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#+end_src
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#+name: fig:compliance_iff_dvf
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#+caption: Obtained compliance for Open-Loop Control (Blue), Integral Force Feedback (Red) and Direct Velocity Feedback (Yellow) ([[./figs/compliance_iff_dvf.png][png]], [[./figs/compliance_iff_dvf.pdf][pdf]])
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[[file:figs/compliance_iff_dvf.png]]
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#+begin_src matlab :exports none
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figure;
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subplot(1,2,1);
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hold on;
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plot(freqs, T_norm_ol)
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plot(freqs, T_norm_iff)
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plot(freqs, T_norm_dvf)
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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xlabel('Frequency [Hz]');
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ylabel('Transmissibility - Frobenius Norm');
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subplot(1,2,2);
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hold on;
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plot(freqs, C_norm_ol, 'DisplayName', 'OL')
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plot(freqs, C_norm_iff, 'DisplayName', 'IFF')
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plot(freqs, C_norm_dvf, 'DisplayName', 'DVF')
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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xlabel('Frequency [Hz]');
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ylabel('Compliance - Frobenius Norm');
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legend();
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#+end_src
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#+header: :tangle no :exports results :results none :noweb yes
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#+begin_src matlab :var filepath="figs/frobenius_norm_T_C_iff_dvf.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
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<<plt-matlab>>
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#+end_src
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#+name: fig:frobenius_norm_T_C_iff_dvf
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#+caption: Frobenius norm of the Transmissibility and Compliance Matrices ([[./figs/frobenius_norm_T_C_iff_dvf.png][png]], [[./figs/frobenius_norm_T_C_iff_dvf.pdf][pdf]])
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[[file:figs/frobenius_norm_T_C_iff_dvf.png]]
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@@ -259,241 +259,27 @@ A lead is added around the crossover frequency which is set to be around 500Hz.
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linkaxes([ax1,ax2],'x');
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#+end_src
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* Transmissibility Analysis
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** Matlab Init :noexport:
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#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
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<<matlab-dir>>
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#+end_src
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#+begin_src matlab :exports none :results silent :noweb yes
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<<matlab-init>>
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#+end_src
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#+begin_src matlab
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simulinkproject('../');
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#+end_src
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#+begin_src matlab
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open('stewart_platform_model.slx')
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#+end_src
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** Initialize the Stewart platform
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#+begin_src matlab
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stewart = initializeStewartPlatform();
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stewart = initializeFramesPositions(stewart, 'H', 90e-3, 'MO_B', 45e-3);
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stewart = generateGeneralConfiguration(stewart);
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stewart = computeJointsPose(stewart);
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stewart = initializeStrutDynamics(stewart);
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stewart = initializeJointDynamics(stewart, 'type_F', 'universal_p', 'type_M', 'spherical_p');
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stewart = initializeCylindricalPlatforms(stewart);
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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', 'accelerometer', 'freq', 5e3);
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#+end_src
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We set the rotation point of the ground to be at the same point at frames $\{A\}$ and $\{B\}$.
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#+begin_src matlab
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ground = initializeGround('type', 'rigid', 'rot_point', stewart.platform_F.FO_A);
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payload = initializePayload('type', 'rigid');
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#+end_src
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** Transmissibility
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#+begin_src matlab
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%% Options for Linearized
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options = linearizeOptions;
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options.SampleTime = 0;
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%% Name of the Simulink File
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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, '/Disturbances/D_w'], 1, 'openinput'); io_i = io_i + 1; % Base Motion [m, rad]
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io(io_i) = linio([mdl, '/Absolute Motion Sensor'], 1, 'openoutput'); io_i = io_i + 1; % Absolute Motion [m, rad]
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%% Run the linearization
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T = linearize(mdl, io, options);
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T.InputName = {'Wdx', 'Wdy', 'Wdz', 'Wrx', 'Wry', 'Wrz'};
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T.OutputName = {'Edx', 'Edy', 'Edz', 'Erx', 'Ery', 'Erz'};
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#+end_src
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#+begin_src matlab
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freqs = logspace(1, 4, 1000);
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figure;
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for ix = 1:6
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for iy = 1:6
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subplot(6, 6, (ix-1)*6 + iy);
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hold on;
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plot(freqs, abs(squeeze(freqresp(T(ix, iy), freqs, 'Hz'))), 'k-');
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylim([1e-5, 10]);
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xlim([freqs(1), freqs(end)]);
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if ix < 6
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xticklabels({});
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end
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if iy > 1
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yticklabels({});
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end
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end
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end
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#+end_src
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From cite:preumont07_six_axis_singl_stage_activ, 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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\begin{align*}
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\| \bm{T}(\omega) \| &= \sqrt{\text{Trace}[\bm{T}(\omega) \bm{T}(\omega)^H]}\\
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&= \sqrt{\Sigma_{i=1}^6 \Sigma_{j=1}^6 |T_{ij}|^2}
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\end{align*}
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#+begin_src matlab
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freqs = logspace(1, 4, 1000);
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T_norm = zeros(length(freqs), 1);
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for i = 1:length(freqs)
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T_norm(i) = sqrt(trace(freqresp(T, freqs(i), 'Hz')*freqresp(T, freqs(i), 'Hz')'));
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end
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#+end_src
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And we normalize by a factor $\sqrt{6}$ to obtain a performance metric comparable to the transmissibility of a one-axis isolator:
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\[ \Gamma(\omega) = \|\bm{T}(\omega)\| / \sqrt{6} \]
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#+begin_src matlab
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Gamma = T_norm/sqrt(6);
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#+end_src
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#+begin_src matlab
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figure;
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plot(freqs, Gamma)
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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#+end_src
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* Compliance Analysis
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** Matlab Init :noexport:
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#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
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<<matlab-dir>>
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#+end_src
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#+begin_src matlab :exports none :results silent :noweb yes
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<<matlab-init>>
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#+end_src
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#+begin_src matlab
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simulinkproject('../');
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#+end_src
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#+begin_src matlab
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open('stewart_platform_model.slx')
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#+end_src
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** Initialize the Stewart platform
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#+begin_src matlab
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stewart = initializeStewartPlatform();
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stewart = initializeFramesPositions(stewart, 'H', 90e-3, 'MO_B', 45e-3);
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stewart = generateGeneralConfiguration(stewart);
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stewart = computeJointsPose(stewart);
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stewart = initializeStrutDynamics(stewart);
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stewart = initializeJointDynamics(stewart, 'type_F', 'universal_p', 'type_M', 'spherical_p');
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stewart = initializeCylindricalPlatforms(stewart);
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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', 'accelerometer', 'freq', 5e3);
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#+end_src
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We set the rotation point of the ground to be at the same point at frames $\{A\}$ and $\{B\}$.
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#+begin_src matlab
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ground = initializeGround('type', 'none');
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payload = initializePayload('type', 'rigid');
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#+end_src
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** Compliance
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#+begin_src matlab
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%% Options for Linearized
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options = linearizeOptions;
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options.SampleTime = 0;
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%% Name of the Simulink File
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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, '/Disturbances/F_ext'], 1, 'openinput'); io_i = io_i + 1; % Base Motion [m, rad]
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io(io_i) = linio([mdl, '/Absolute Motion Sensor'], 1, 'openoutput'); io_i = io_i + 1; % Absolute Motion [m, rad]
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%% Run the linearization
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C = linearize(mdl, io, options);
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C.InputName = {'Fdx', 'Fdy', 'Fdz', 'Mdx', 'Mdy', 'Mdz'};
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C.OutputName = {'Edx', 'Edy', 'Edz', 'Erx', 'Ery', 'Erz'};
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#+end_src
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#+begin_src matlab
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freqs = logspace(1, 4, 1000);
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figure;
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for ix = 1:6
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for iy = 1:6
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subplot(6, 6, (ix-1)*6 + iy);
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hold on;
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plot(freqs, abs(squeeze(freqresp(C(ix, iy), freqs, 'Hz'))), 'k-');
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylim([1e-10, 1e-3]);
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xlim([freqs(1), freqs(end)]);
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if ix < 6
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xticklabels({});
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end
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if iy > 1
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yticklabels({});
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end
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end
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end
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#+end_src
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We can try to use the Frobenius norm to obtain a scalar value representing the 6-dof compliance of the Stewart platform.
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#+begin_src matlab
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freqs = logspace(1, 4, 1000);
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C_norm = zeros(length(freqs), 1);
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for i = 1:length(freqs)
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C_norm(i) = sqrt(trace(freqresp(C, freqs(i), 'Hz')*freqresp(C, freqs(i), 'Hz')'));
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end
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#+end_src
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#+begin_src matlab
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figure;
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plot(freqs, C_norm)
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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#+end_src
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* Functions
|
||||
** Compute the Transmissibility
|
||||
** =initializeController=: Initialize the Controller
|
||||
:PROPERTIES:
|
||||
:header-args:matlab+: :tangle ../src/computeTransmissibility.m
|
||||
:header-args:matlab+: :tangle ../src/initializeController.m
|
||||
:header-args:matlab+: :comments none :mkdirp yes :eval no
|
||||
:END:
|
||||
<<sec:computeTransmissibility>>
|
||||
<<sec:initializeController>>
|
||||
|
||||
*** Function description
|
||||
:PROPERTIES:
|
||||
:UNNUMBERED: t
|
||||
:END:
|
||||
#+begin_src matlab
|
||||
function [T, T_norm, freqs] = computeTransmissibility(args)
|
||||
% computeTransmissibility -
|
||||
function [controller] = initializeController(args)
|
||||
% initializeController - Initialize the Controller
|
||||
%
|
||||
% Syntax: [T, T_norm, freqs] = computeTransmissibility(args)
|
||||
% Syntax: [] = initializeController(args)
|
||||
%
|
||||
% Inputs:
|
||||
% - args - Structure with the following fields:
|
||||
% - plots [true/false] - Should plot the transmissilibty matrix and its Frobenius norm
|
||||
% - freqs [] - Frequency vector to estimate the Frobenius norm
|
||||
%
|
||||
% Outputs:
|
||||
% - T [6x6 ss] - Transmissibility matrix
|
||||
% - T_norm [length(freqs)x1] - Frobenius norm of the Transmissibility matrix
|
||||
% - freqs [length(freqs)x1] - Frequency vector in [Hz]
|
||||
% - args - Can have the following fields:
|
||||
#+end_src
|
||||
|
||||
*** Optional Parameters
|
||||
@@ -501,216 +287,30 @@ We can try to use the Frobenius norm to obtain a scalar value representing the 6
|
||||
:UNNUMBERED: t
|
||||
:END:
|
||||
#+begin_src matlab
|
||||
arguments
|
||||
args.plots logical {mustBeNumericOrLogical} = false
|
||||
args.freqs double {mustBeNumeric, mustBeNonnegative} = logspace(1,4,1000)
|
||||
end
|
||||
arguments
|
||||
args.type char {mustBeMember(args.type, {'open-loop', 'iff', 'dvf'})} = 'open-loop'
|
||||
end
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
freqs = args.freqs;
|
||||
#+end_src
|
||||
|
||||
*** Identification of the Transmissibility Matrix
|
||||
*** Structure initialization
|
||||
:PROPERTIES:
|
||||
:UNNUMBERED: t
|
||||
:END:
|
||||
#+begin_src matlab
|
||||
%% Options for Linearized
|
||||
options = linearizeOptions;
|
||||
options.SampleTime = 0;
|
||||
|
||||
%% Name of the Simulink File
|
||||
mdl = 'stewart_platform_model';
|
||||
|
||||
%% Input/Output definition
|
||||
clear io; io_i = 1;
|
||||
io(io_i) = linio([mdl, '/Disturbances/D_w'], 1, 'openinput'); io_i = io_i + 1; % Base Motion [m, rad]
|
||||
io(io_i) = linio([mdl, '/Absolute Motion Sensor'], 1, 'openoutput'); io_i = io_i + 1; % Absolute Motion [m, rad]
|
||||
|
||||
%% Run the linearization
|
||||
T = linearize(mdl, io, options);
|
||||
T.InputName = {'Wdx', 'Wdy', 'Wdz', 'Wrx', 'Wry', 'Wrz'};
|
||||
T.OutputName = {'Edx', 'Edy', 'Edz', 'Erx', 'Ery', 'Erz'};
|
||||
controller = struct();
|
||||
#+end_src
|
||||
|
||||
If wanted, the 6x6 transmissibility matrix is plotted.
|
||||
#+begin_src matlab
|
||||
p_handle = zeros(6*6,1);
|
||||
|
||||
if args.plots
|
||||
fig = figure;
|
||||
for ix = 1:6
|
||||
for iy = 1:6
|
||||
p_handle((ix-1)*6 + iy) = subplot(6, 6, (ix-1)*6 + iy);
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(T(ix, iy), freqs, 'Hz'))), 'k-');
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
if ix < 6
|
||||
xticklabels({});
|
||||
end
|
||||
if iy > 1
|
||||
yticklabels({});
|
||||
end
|
||||
end
|
||||
end
|
||||
|
||||
linkaxes(p_handle, 'xy')
|
||||
xlim([freqs(1), freqs(end)]);
|
||||
ylim([1e-5, 1e2]);
|
||||
|
||||
han = axes(fig, 'visible', 'off');
|
||||
han.XLabel.Visible = 'on';
|
||||
han.YLabel.Visible = 'on';
|
||||
ylabel(han, 'Frequency [Hz]');
|
||||
xlabel(han, 'Transmissibility [m/m]');
|
||||
end
|
||||
#+end_src
|
||||
|
||||
*** Computation of the Frobenius norm
|
||||
*** Add Type
|
||||
:PROPERTIES:
|
||||
:UNNUMBERED: t
|
||||
:END:
|
||||
#+begin_src matlab
|
||||
T_norm = zeros(length(freqs), 1);
|
||||
|
||||
for i = 1:length(freqs)
|
||||
T_norm(i) = sqrt(trace(freqresp(T, freqs(i), 'Hz')*freqresp(T, freqs(i), 'Hz')'));
|
||||
end
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
T_norm = T_norm/sqrt(6);
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
if args.plots
|
||||
figure;
|
||||
plot(freqs, T_norm)
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
xlabel('Frequency [Hz]');
|
||||
ylabel('Transmissibility - Frobenius Norm');
|
||||
end
|
||||
#+end_src
|
||||
|
||||
** Compute the Compliance
|
||||
:PROPERTIES:
|
||||
:header-args:matlab+: :tangle ../src/computeCompliance.m
|
||||
:header-args:matlab+: :comments none :mkdirp yes :eval no
|
||||
:END:
|
||||
<<sec:computeCompliance>>
|
||||
|
||||
*** Function description
|
||||
:PROPERTIES:
|
||||
:UNNUMBERED: t
|
||||
:END:
|
||||
#+begin_src matlab
|
||||
function [C, C_norm, freqs] = computeCompliance(args)
|
||||
% computeCompliance -
|
||||
%
|
||||
% Syntax: [C, C_norm, freqs] = computeCompliance(args)
|
||||
%
|
||||
% Inputs:
|
||||
% - args - Structure with the following fields:
|
||||
% - plots [true/false] - Should plot the transmissilibty matrix and its Frobenius norm
|
||||
% - freqs [] - Frequency vector to estimate the Frobenius norm
|
||||
%
|
||||
% Outputs:
|
||||
% - C [6x6 ss] - Compliance matrix
|
||||
% - C_norm [length(freqs)x1] - Frobenius norm of the Compliance matrix
|
||||
% - freqs [length(freqs)x1] - Frequency vector in [Hz]
|
||||
#+end_src
|
||||
|
||||
*** Optional Parameters
|
||||
:PROPERTIES:
|
||||
:UNNUMBERED: t
|
||||
:END:
|
||||
#+begin_src matlab
|
||||
arguments
|
||||
args.plots logical {mustBeNumericOrLogical} = false
|
||||
args.freqs double {mustBeNumeric, mustBeNonnegative} = logspace(1,4,1000)
|
||||
end
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
freqs = args.freqs;
|
||||
#+end_src
|
||||
|
||||
*** Identification of the Compliance Matrix
|
||||
:PROPERTIES:
|
||||
:UNNUMBERED: t
|
||||
:END:
|
||||
#+begin_src matlab
|
||||
%% Options for Linearized
|
||||
options = linearizeOptions;
|
||||
options.SampleTime = 0;
|
||||
|
||||
%% Name of the Simulink File
|
||||
mdl = 'stewart_platform_model';
|
||||
|
||||
%% Input/Output definition
|
||||
clear io; io_i = 1;
|
||||
io(io_i) = linio([mdl, '/Disturbances/F_ext'], 1, 'openinput'); io_i = io_i + 1; % External forces [N, N*m]
|
||||
io(io_i) = linio([mdl, '/Absolute Motion Sensor'], 1, 'openoutput'); io_i = io_i + 1; % Absolute Motion [m, rad]
|
||||
|
||||
%% Run the linearization
|
||||
C = linearize(mdl, io, options);
|
||||
C.InputName = {'Fdx', 'Fdy', 'Fdz', 'Mdx', 'Mdy', 'Mdz'};
|
||||
C.OutputName = {'Edx', 'Edy', 'Edz', 'Erx', 'Ery', 'Erz'};
|
||||
#+end_src
|
||||
|
||||
If wanted, the 6x6 transmissibility matrix is plotted.
|
||||
#+begin_src matlab
|
||||
p_handle = zeros(6*6,1);
|
||||
|
||||
if args.plots
|
||||
fig = figure;
|
||||
for ix = 1:6
|
||||
for iy = 1:6
|
||||
p_handle((ix-1)*6 + iy) = subplot(6, 6, (ix-1)*6 + iy);
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(C(ix, iy), freqs, 'Hz'))), 'k-');
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
if ix < 6
|
||||
xticklabels({});
|
||||
end
|
||||
if iy > 1
|
||||
yticklabels({});
|
||||
end
|
||||
end
|
||||
end
|
||||
|
||||
linkaxes(p_handle, 'xy')
|
||||
xlim([freqs(1), freqs(end)]);
|
||||
|
||||
han = axes(fig, 'visible', 'off');
|
||||
han.XLabel.Visible = 'on';
|
||||
han.YLabel.Visible = 'on';
|
||||
xlabel(han, 'Frequency [Hz]');
|
||||
ylabel(han, 'Compliance [m/N, rad/(N*m)]');
|
||||
end
|
||||
#+end_src
|
||||
|
||||
*** Computation of the Frobenius norm
|
||||
:PROPERTIES:
|
||||
:UNNUMBERED: t
|
||||
:END:
|
||||
#+begin_src matlab
|
||||
freqs = args.freqs;
|
||||
|
||||
C_norm = zeros(length(freqs), 1);
|
||||
|
||||
for i = 1:length(freqs)
|
||||
C_norm(i) = sqrt(trace(freqresp(C, freqs(i), 'Hz')*freqresp(C, freqs(i), 'Hz')'));
|
||||
end
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
if args.plots
|
||||
figure;
|
||||
plot(freqs, C_norm)
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
xlabel('Frequency [Hz]');
|
||||
ylabel('Compliance - Frobenius Norm');
|
||||
switch args.type
|
||||
case 'open-loop'
|
||||
controller.type = 0;
|
||||
case 'iff'
|
||||
controller.type = 1;
|
||||
case 'dvf'
|
||||
controller.type = 2;
|
||||
end
|
||||
#+end_src
|
||||
|
||||
@@ -338,11 +338,6 @@ Estimation of the transfer function from $\mathcal{\bm{F}}$ to $\mathcal{\bm{X}}
|
||||
%% Name of the Simulink File
|
||||
mdl = 'stewart_platform_model';
|
||||
|
||||
%% Input/Output definition
|
||||
clear io; io_i = 1;
|
||||
io(io_i) = linio([mdl, '/F'], 1, 'openinput'); io_i = io_i + 1;
|
||||
io(io_i) = linio([mdl, '/X'], 1, 'openoutput'); io_i = io_i + 1;
|
||||
|
||||
%% Input/Output definition
|
||||
clear io; io_i = 1;
|
||||
io(io_i) = linio([mdl, '/Controller'], 1, 'openinput'); io_i = io_i + 1; % Actuator Force Inputs [N]
|
||||
|
||||
@@ -39,8 +39,14 @@
|
||||
:END:
|
||||
|
||||
* Introduction :ignore:
|
||||
In this document, we discuss the various methods to identify the behavior of the Stewart platform.
|
||||
|
||||
- [[sec:modal_analysis]]
|
||||
- [[sec:transmissibility]]
|
||||
- [[sec:compliance]]
|
||||
|
||||
* Modal Analysis of the Stewart Platform
|
||||
<<sec:modal_analysis>>
|
||||
** Introduction :ignore:
|
||||
** Matlab Init :noexport:ignore:
|
||||
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
|
||||
@@ -239,3 +245,463 @@ Save the movie of the mode shape.
|
||||
#+caption: Identified mode - 5
|
||||
[[file:figs/mode5.gif]]
|
||||
|
||||
* Transmissibility Analysis
|
||||
<<sec:transmissibility>>
|
||||
** Introduction :ignore:
|
||||
** Matlab Init :noexport:
|
||||
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
|
||||
<<matlab-dir>>
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :exports none :results silent :noweb yes
|
||||
<<matlab-init>>
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
simulinkproject('../');
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
open('stewart_platform_model.slx')
|
||||
#+end_src
|
||||
|
||||
** Initialize the Stewart platform
|
||||
#+begin_src matlab
|
||||
stewart = initializeStewartPlatform();
|
||||
stewart = initializeFramesPositions(stewart, 'H', 90e-3, 'MO_B', 45e-3);
|
||||
stewart = generateGeneralConfiguration(stewart);
|
||||
stewart = computeJointsPose(stewart);
|
||||
stewart = initializeStrutDynamics(stewart);
|
||||
stewart = initializeJointDynamics(stewart, 'type_F', 'universal_p', 'type_M', 'spherical_p');
|
||||
stewart = initializeCylindricalPlatforms(stewart);
|
||||
stewart = initializeCylindricalStruts(stewart);
|
||||
stewart = computeJacobian(stewart);
|
||||
stewart = initializeStewartPose(stewart);
|
||||
stewart = initializeInertialSensor(stewart, 'type', 'accelerometer', 'freq', 5e3);
|
||||
#+end_src
|
||||
|
||||
We set the rotation point of the ground to be at the same point at frames $\{A\}$ and $\{B\}$.
|
||||
#+begin_src matlab
|
||||
ground = initializeGround('type', 'rigid', 'rot_point', stewart.platform_F.FO_A);
|
||||
payload = initializePayload('type', 'rigid');
|
||||
#+end_src
|
||||
|
||||
** Transmissibility
|
||||
#+begin_src matlab
|
||||
%% Options for Linearized
|
||||
options = linearizeOptions;
|
||||
options.SampleTime = 0;
|
||||
|
||||
%% Name of the Simulink File
|
||||
mdl = 'stewart_platform_model';
|
||||
|
||||
%% Input/Output definition
|
||||
clear io; io_i = 1;
|
||||
io(io_i) = linio([mdl, '/Disturbances/D_w'], 1, 'openinput'); io_i = io_i + 1; % Base Motion [m, rad]
|
||||
io(io_i) = linio([mdl, '/Absolute Motion Sensor'], 1, 'openoutput'); io_i = io_i + 1; % Absolute Motion [m, rad]
|
||||
|
||||
%% Run the linearization
|
||||
T = linearize(mdl, io, options);
|
||||
T.InputName = {'Wdx', 'Wdy', 'Wdz', 'Wrx', 'Wry', 'Wrz'};
|
||||
T.OutputName = {'Edx', 'Edy', 'Edz', 'Erx', 'Ery', 'Erz'};
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
freqs = logspace(1, 4, 1000);
|
||||
|
||||
figure;
|
||||
for ix = 1:6
|
||||
for iy = 1:6
|
||||
subplot(6, 6, (ix-1)*6 + iy);
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(T(ix, iy), freqs, 'Hz'))), 'k-');
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylim([1e-5, 10]);
|
||||
xlim([freqs(1), freqs(end)]);
|
||||
if ix < 6
|
||||
xticklabels({});
|
||||
end
|
||||
if iy > 1
|
||||
yticklabels({});
|
||||
end
|
||||
end
|
||||
end
|
||||
#+end_src
|
||||
|
||||
From cite:preumont07_six_axis_singl_stage_activ, one can use the Frobenius norm of the transmissibility matrix to obtain a scalar indicator of the transmissibility performance of the system:
|
||||
\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*}
|
||||
|
||||
#+begin_src matlab
|
||||
freqs = logspace(1, 4, 1000);
|
||||
|
||||
T_norm = zeros(length(freqs), 1);
|
||||
|
||||
for i = 1:length(freqs)
|
||||
T_norm(i) = sqrt(trace(freqresp(T, freqs(i), 'Hz')*freqresp(T, freqs(i), 'Hz')'));
|
||||
end
|
||||
#+end_src
|
||||
|
||||
And we normalize by a factor $\sqrt{6}$ to obtain a performance metric comparable to the transmissibility of a one-axis isolator:
|
||||
\[ \Gamma(\omega) = \|\bm{T}(\omega)\| / \sqrt{6} \]
|
||||
|
||||
#+begin_src matlab
|
||||
Gamma = T_norm/sqrt(6);
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
figure;
|
||||
plot(freqs, Gamma)
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
#+end_src
|
||||
|
||||
* Compliance Analysis
|
||||
<<sec:compliance>>
|
||||
** Introduction :ignore:
|
||||
** Matlab Init :noexport:
|
||||
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
|
||||
<<matlab-dir>>
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :exports none :results silent :noweb yes
|
||||
<<matlab-init>>
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
simulinkproject('../');
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
open('stewart_platform_model.slx')
|
||||
#+end_src
|
||||
|
||||
** Initialize the Stewart platform
|
||||
#+begin_src matlab
|
||||
stewart = initializeStewartPlatform();
|
||||
stewart = initializeFramesPositions(stewart, 'H', 90e-3, 'MO_B', 45e-3);
|
||||
stewart = generateGeneralConfiguration(stewart);
|
||||
stewart = computeJointsPose(stewart);
|
||||
stewart = initializeStrutDynamics(stewart);
|
||||
stewart = initializeJointDynamics(stewart, 'type_F', 'universal_p', 'type_M', 'spherical_p');
|
||||
stewart = initializeCylindricalPlatforms(stewart);
|
||||
stewart = initializeCylindricalStruts(stewart);
|
||||
stewart = computeJacobian(stewart);
|
||||
stewart = initializeStewartPose(stewart);
|
||||
stewart = initializeInertialSensor(stewart, 'type', 'accelerometer', 'freq', 5e3);
|
||||
#+end_src
|
||||
|
||||
We set the rotation point of the ground to be at the same point at frames $\{A\}$ and $\{B\}$.
|
||||
#+begin_src matlab
|
||||
ground = initializeGround('type', 'none');
|
||||
payload = initializePayload('type', 'rigid');
|
||||
#+end_src
|
||||
|
||||
** Compliance
|
||||
#+begin_src matlab
|
||||
%% Options for Linearized
|
||||
options = linearizeOptions;
|
||||
options.SampleTime = 0;
|
||||
|
||||
%% Name of the Simulink File
|
||||
mdl = 'stewart_platform_model';
|
||||
|
||||
%% Input/Output definition
|
||||
clear io; io_i = 1;
|
||||
io(io_i) = linio([mdl, '/Disturbances/F_ext'], 1, 'openinput'); io_i = io_i + 1; % Base Motion [m, rad]
|
||||
io(io_i) = linio([mdl, '/Absolute Motion Sensor'], 1, 'openoutput'); io_i = io_i + 1; % Absolute Motion [m, rad]
|
||||
|
||||
%% Run the linearization
|
||||
C = linearize(mdl, io, options);
|
||||
C.InputName = {'Fdx', 'Fdy', 'Fdz', 'Mdx', 'Mdy', 'Mdz'};
|
||||
C.OutputName = {'Edx', 'Edy', 'Edz', 'Erx', 'Ery', 'Erz'};
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
freqs = logspace(1, 4, 1000);
|
||||
|
||||
figure;
|
||||
for ix = 1:6
|
||||
for iy = 1:6
|
||||
subplot(6, 6, (ix-1)*6 + iy);
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(C(ix, iy), freqs, 'Hz'))), 'k-');
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylim([1e-10, 1e-3]);
|
||||
xlim([freqs(1), freqs(end)]);
|
||||
if ix < 6
|
||||
xticklabels({});
|
||||
end
|
||||
if iy > 1
|
||||
yticklabels({});
|
||||
end
|
||||
end
|
||||
end
|
||||
#+end_src
|
||||
|
||||
We can try to use the Frobenius norm to obtain a scalar value representing the 6-dof compliance of the Stewart platform.
|
||||
|
||||
#+begin_src matlab
|
||||
freqs = logspace(1, 4, 1000);
|
||||
|
||||
C_norm = zeros(length(freqs), 1);
|
||||
|
||||
for i = 1:length(freqs)
|
||||
C_norm(i) = sqrt(trace(freqresp(C, freqs(i), 'Hz')*freqresp(C, freqs(i), 'Hz')'));
|
||||
end
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
figure;
|
||||
plot(freqs, C_norm)
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
#+end_src
|
||||
|
||||
* Functions
|
||||
** Compute the Transmissibility
|
||||
:PROPERTIES:
|
||||
:header-args:matlab+: :tangle ../src/computeTransmissibility.m
|
||||
:header-args:matlab+: :comments none :mkdirp yes :eval no
|
||||
:END:
|
||||
<<sec:computeTransmissibility>>
|
||||
|
||||
*** Function description
|
||||
:PROPERTIES:
|
||||
:UNNUMBERED: t
|
||||
:END:
|
||||
#+begin_src matlab
|
||||
function [T, T_norm, freqs] = computeTransmissibility(args)
|
||||
% computeTransmissibility -
|
||||
%
|
||||
% Syntax: [T, T_norm, freqs] = computeTransmissibility(args)
|
||||
%
|
||||
% Inputs:
|
||||
% - args - Structure with the following fields:
|
||||
% - plots [true/false] - Should plot the transmissilibty matrix and its Frobenius norm
|
||||
% - freqs [] - Frequency vector to estimate the Frobenius norm
|
||||
%
|
||||
% Outputs:
|
||||
% - T [6x6 ss] - Transmissibility matrix
|
||||
% - T_norm [length(freqs)x1] - Frobenius norm of the Transmissibility matrix
|
||||
% - freqs [length(freqs)x1] - Frequency vector in [Hz]
|
||||
#+end_src
|
||||
|
||||
*** Optional Parameters
|
||||
:PROPERTIES:
|
||||
:UNNUMBERED: t
|
||||
:END:
|
||||
#+begin_src matlab
|
||||
arguments
|
||||
args.plots logical {mustBeNumericOrLogical} = false
|
||||
args.freqs double {mustBeNumeric, mustBeNonnegative} = logspace(1,4,1000)
|
||||
end
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
freqs = args.freqs;
|
||||
#+end_src
|
||||
|
||||
*** Identification of the Transmissibility Matrix
|
||||
:PROPERTIES:
|
||||
:UNNUMBERED: t
|
||||
:END:
|
||||
#+begin_src matlab
|
||||
%% Options for Linearized
|
||||
options = linearizeOptions;
|
||||
options.SampleTime = 0;
|
||||
|
||||
%% Name of the Simulink File
|
||||
mdl = 'stewart_platform_model';
|
||||
|
||||
%% Input/Output definition
|
||||
clear io; io_i = 1;
|
||||
io(io_i) = linio([mdl, '/Disturbances/D_w'], 1, 'openinput'); io_i = io_i + 1; % Base Motion [m, rad]
|
||||
io(io_i) = linio([mdl, '/Absolute Motion Sensor'], 1, 'openoutput'); io_i = io_i + 1; % Absolute Motion [m, rad]
|
||||
|
||||
%% Run the linearization
|
||||
T = linearize(mdl, io, options);
|
||||
T.InputName = {'Wdx', 'Wdy', 'Wdz', 'Wrx', 'Wry', 'Wrz'};
|
||||
T.OutputName = {'Edx', 'Edy', 'Edz', 'Erx', 'Ery', 'Erz'};
|
||||
#+end_src
|
||||
|
||||
If wanted, the 6x6 transmissibility matrix is plotted.
|
||||
#+begin_src matlab
|
||||
p_handle = zeros(6*6,1);
|
||||
|
||||
if args.plots
|
||||
fig = figure;
|
||||
for ix = 1:6
|
||||
for iy = 1:6
|
||||
p_handle((ix-1)*6 + iy) = subplot(6, 6, (ix-1)*6 + iy);
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(T(ix, iy), freqs, 'Hz'))), 'k-');
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
if ix < 6
|
||||
xticklabels({});
|
||||
end
|
||||
if iy > 1
|
||||
yticklabels({});
|
||||
end
|
||||
end
|
||||
end
|
||||
|
||||
linkaxes(p_handle, 'xy')
|
||||
xlim([freqs(1), freqs(end)]);
|
||||
ylim([1e-5, 1e2]);
|
||||
|
||||
han = axes(fig, 'visible', 'off');
|
||||
han.XLabel.Visible = 'on';
|
||||
han.YLabel.Visible = 'on';
|
||||
ylabel(han, 'Frequency [Hz]');
|
||||
xlabel(han, 'Transmissibility [m/m]');
|
||||
end
|
||||
#+end_src
|
||||
|
||||
*** Computation of the Frobenius norm
|
||||
:PROPERTIES:
|
||||
:UNNUMBERED: t
|
||||
:END:
|
||||
#+begin_src matlab
|
||||
T_norm = zeros(length(freqs), 1);
|
||||
|
||||
for i = 1:length(freqs)
|
||||
T_norm(i) = sqrt(trace(freqresp(T, freqs(i), 'Hz')*freqresp(T, freqs(i), 'Hz')'));
|
||||
end
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
T_norm = T_norm/sqrt(6);
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
if args.plots
|
||||
figure;
|
||||
plot(freqs, T_norm)
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
xlabel('Frequency [Hz]');
|
||||
ylabel('Transmissibility - Frobenius Norm');
|
||||
end
|
||||
#+end_src
|
||||
|
||||
** Compute the Compliance
|
||||
:PROPERTIES:
|
||||
:header-args:matlab+: :tangle ../src/computeCompliance.m
|
||||
:header-args:matlab+: :comments none :mkdirp yes :eval no
|
||||
:END:
|
||||
<<sec:computeCompliance>>
|
||||
|
||||
*** Function description
|
||||
:PROPERTIES:
|
||||
:UNNUMBERED: t
|
||||
:END:
|
||||
#+begin_src matlab
|
||||
function [C, C_norm, freqs] = computeCompliance(args)
|
||||
% computeCompliance -
|
||||
%
|
||||
% Syntax: [C, C_norm, freqs] = computeCompliance(args)
|
||||
%
|
||||
% Inputs:
|
||||
% - args - Structure with the following fields:
|
||||
% - plots [true/false] - Should plot the transmissilibty matrix and its Frobenius norm
|
||||
% - freqs [] - Frequency vector to estimate the Frobenius norm
|
||||
%
|
||||
% Outputs:
|
||||
% - C [6x6 ss] - Compliance matrix
|
||||
% - C_norm [length(freqs)x1] - Frobenius norm of the Compliance matrix
|
||||
% - freqs [length(freqs)x1] - Frequency vector in [Hz]
|
||||
#+end_src
|
||||
|
||||
*** Optional Parameters
|
||||
:PROPERTIES:
|
||||
:UNNUMBERED: t
|
||||
:END:
|
||||
#+begin_src matlab
|
||||
arguments
|
||||
args.plots logical {mustBeNumericOrLogical} = false
|
||||
args.freqs double {mustBeNumeric, mustBeNonnegative} = logspace(1,4,1000)
|
||||
end
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
freqs = args.freqs;
|
||||
#+end_src
|
||||
|
||||
*** Identification of the Compliance Matrix
|
||||
:PROPERTIES:
|
||||
:UNNUMBERED: t
|
||||
:END:
|
||||
#+begin_src matlab
|
||||
%% Options for Linearized
|
||||
options = linearizeOptions;
|
||||
options.SampleTime = 0;
|
||||
|
||||
%% Name of the Simulink File
|
||||
mdl = 'stewart_platform_model';
|
||||
|
||||
%% Input/Output definition
|
||||
clear io; io_i = 1;
|
||||
io(io_i) = linio([mdl, '/Disturbances/F_ext'], 1, 'openinput'); io_i = io_i + 1; % External forces [N, N*m]
|
||||
io(io_i) = linio([mdl, '/Absolute Motion Sensor'], 1, 'openoutput'); io_i = io_i + 1; % Absolute Motion [m, rad]
|
||||
|
||||
%% Run the linearization
|
||||
C = linearize(mdl, io, options);
|
||||
C.InputName = {'Fdx', 'Fdy', 'Fdz', 'Mdx', 'Mdy', 'Mdz'};
|
||||
C.OutputName = {'Edx', 'Edy', 'Edz', 'Erx', 'Ery', 'Erz'};
|
||||
#+end_src
|
||||
|
||||
If wanted, the 6x6 transmissibility matrix is plotted.
|
||||
#+begin_src matlab
|
||||
p_handle = zeros(6*6,1);
|
||||
|
||||
if args.plots
|
||||
fig = figure;
|
||||
for ix = 1:6
|
||||
for iy = 1:6
|
||||
p_handle((ix-1)*6 + iy) = subplot(6, 6, (ix-1)*6 + iy);
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(C(ix, iy), freqs, 'Hz'))), 'k-');
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
if ix < 6
|
||||
xticklabels({});
|
||||
end
|
||||
if iy > 1
|
||||
yticklabels({});
|
||||
end
|
||||
end
|
||||
end
|
||||
|
||||
linkaxes(p_handle, 'xy')
|
||||
xlim([freqs(1), freqs(end)]);
|
||||
|
||||
han = axes(fig, 'visible', 'off');
|
||||
han.XLabel.Visible = 'on';
|
||||
han.YLabel.Visible = 'on';
|
||||
xlabel(han, 'Frequency [Hz]');
|
||||
ylabel(han, 'Compliance [m/N, rad/(N*m)]');
|
||||
end
|
||||
#+end_src
|
||||
|
||||
*** Computation of the Frobenius norm
|
||||
:PROPERTIES:
|
||||
:UNNUMBERED: t
|
||||
:END:
|
||||
#+begin_src matlab
|
||||
freqs = args.freqs;
|
||||
|
||||
C_norm = zeros(length(freqs), 1);
|
||||
|
||||
for i = 1:length(freqs)
|
||||
C_norm(i) = sqrt(trace(freqresp(C, freqs(i), 'Hz')*freqresp(C, freqs(i), 'Hz')'));
|
||||
end
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
if args.plots
|
||||
figure;
|
||||
plot(freqs, C_norm)
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
xlabel('Frequency [Hz]');
|
||||
ylabel('Compliance - Frobenius Norm');
|
||||
end
|
||||
#+end_src
|
||||
|
||||
Reference in New Issue
Block a user