337 lines
12 KiB
Mathematica
337 lines
12 KiB
Mathematica
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%% Clear Workspace and Close figures
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clear; close all; clc;
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%% Intialize Laplace variable
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s = zpk('s');
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simulinkproject('../');
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% Cube's center at the Center of Mass of the mobile platform
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% Let's create a Cubic Stewart Platform where the *Center of Mass of the mobile platform is located at the center of the cube*.
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% We define the size of the Stewart platform and the position of frames $\{A\}$ and $\{B\}$.
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H = 200e-3; % height of the Stewart platform [m]
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MO_B = -10e-3; % Position {B} with respect to {M} [m]
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% Now, we set the cube's parameters such that the center of the cube is coincident with $\{A\}$ and $\{B\}$.
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Hc = 2.5*H; % Size of the useful part of the cube [m]
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FOc = H + MO_B; % Center of the cube with respect to {F}
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stewart = initializeStewartPlatform();
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stewart = initializeFramesPositions(stewart, 'H', H, 'MO_B', MO_B);
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stewart = generateCubicConfiguration(stewart, 'Hc', Hc, 'FOc', FOc, 'FHa', 25e-3, 'MHb', 25e-3);
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stewart = computeJointsPose(stewart);
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stewart = initializeStrutDynamics(stewart, 'K', 1e6*ones(6,1), 'C', 1e1*ones(6,1));
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stewart = initializeJointDynamics(stewart, 'type_F', 'universal', 'type_M', 'spherical');
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stewart = computeJacobian(stewart);
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stewart = initializeStewartPose(stewart);
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% Now we set the geometry and mass of the mobile platform such that its center of mass is coincident with $\{A\}$ and $\{B\}$.
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stewart = initializeCylindricalPlatforms(stewart, 'Fpr', 1.2*max(vecnorm(stewart.platform_F.Fa)), ...
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'Mpm', 10, ...
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'Mph', 20e-3, ...
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'Mpr', 1.2*max(vecnorm(stewart.platform_M.Mb)));
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% And we set small mass for the struts.
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stewart = initializeCylindricalStruts(stewart, 'Fsm', 1e-3, 'Msm', 1e-3);
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stewart = initializeInertialSensor(stewart);
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% No flexibility below the Stewart platform and no payload.
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ground = initializeGround('type', 'none');
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payload = initializePayload('type', 'none');
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% The obtain geometry is shown in figure [[fig:stewart_cubic_conf_decouple_dynamics]].
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displayArchitecture(stewart, 'labels', false, 'view', 'all');
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% #+name: fig:stewart_cubic_conf_decouple_dynamics
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% #+caption: Geometry used for the simulations - The cube's center, the frames $\{A\}$ and $\{B\}$ and the Center of mass of the mobile platform are coincident ([[./figs/stewart_cubic_conf_decouple_dynamics.png][png]], [[./figs/stewart_cubic_conf_decouple_dynamics.pdf][pdf]])
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% [[file:figs/stewart_cubic_conf_decouple_dynamics.png]]
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% We now identify the dynamics from forces applied in each strut $\bm{\tau}$ to the displacement of each strut $d \bm{\mathcal{L}}$.
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open('stewart_platform_model.slx')
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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, '/Controller'], 1, 'openinput'); io_i = io_i + 1; % Actuator Force Inputs [N]
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io(io_i) = linio([mdl, '/Stewart Platform'], 1, 'openoutput', [], 'dLm'); io_i = io_i + 1; % Relative Displacement Outputs [m]
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%% Run the linearization
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G = linearize(mdl, io, options);
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G.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
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G.OutputName = {'Dm1', 'Dm2', 'Dm3', 'Dm4', 'Dm5', 'Dm6'};
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% Now, thanks to the Jacobian (Figure [[fig:local_to_cartesian_coordinates]]), we compute the transfer function from $\bm{\mathcal{F}}$ to $\bm{\mathcal{X}}$.
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Gc = inv(stewart.kinematics.J)*G*inv(stewart.kinematics.J');
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Gc.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
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Gc.OutputName = {'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'};
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% The obtain dynamics $\bm{G}_{c}(s) = \bm{J}^{-T} \bm{G}(s) \bm{J}^{-1}$ is shown in Figure [[fig:stewart_cubic_decoupled_dynamics_cartesian]].
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freqs = logspace(1, 3, 500);
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figure;
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ax1 = subplot(2, 2, 1);
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hold on;
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for i = 1:6
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for j = i+1:6
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plot(freqs, abs(squeeze(freqresp(Gc(i, j), freqs, 'Hz'))), 'k-');
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end
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end
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set(gca,'ColorOrderIndex',1);
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plot(freqs, abs(squeeze(freqresp(Gc(1, 1), freqs, 'Hz'))));
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plot(freqs, abs(squeeze(freqresp(Gc(2, 2), freqs, 'Hz'))));
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plot(freqs, abs(squeeze(freqresp(Gc(3, 3), freqs, 'Hz'))));
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
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ax3 = subplot(2, 2, 3);
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hold on;
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for i = 1:6
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for j = i+1:6
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p4 = plot(freqs, 180/pi*angle(squeeze(freqresp(Gc(i, j), freqs, 'Hz'))), 'k-');
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end
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end
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set(gca,'ColorOrderIndex',1);
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p1 = plot(freqs, 180/pi*angle(squeeze(freqresp(Gc(1, 1), freqs, 'Hz'))));
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p2 = plot(freqs, 180/pi*angle(squeeze(freqresp(Gc(2, 2), freqs, 'Hz'))));
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p3 = plot(freqs, 180/pi*angle(squeeze(freqresp(Gc(3, 3), freqs, 'Hz'))));
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
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ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
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ylim([-180, 180]);
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yticks([-180, -90, 0, 90, 180]);
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legend([p1, p2, p3, p4], {'$D_x/F_x$','$D_y/F_y$', '$D_z/F_z$', '$D_i/F_j$'})
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ax2 = subplot(2, 2, 2);
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hold on;
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for i = 1:6
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for j = i+1:6
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plot(freqs, abs(squeeze(freqresp(Gc(i, j), freqs, 'Hz'))), 'k-');
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end
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end
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set(gca,'ColorOrderIndex',1);
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plot(freqs, abs(squeeze(freqresp(Gc(4, 4), freqs, 'Hz'))));
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plot(freqs, abs(squeeze(freqresp(Gc(5, 5), freqs, 'Hz'))));
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plot(freqs, abs(squeeze(freqresp(Gc(6, 6), freqs, 'Hz'))));
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
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ax4 = subplot(2, 2, 4);
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hold on;
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for i = 1:6
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for j = i+1:6
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p4 = plot(freqs, 180/pi*angle(squeeze(freqresp(Gc(i, j), freqs, 'Hz'))), 'k-');
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end
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end
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set(gca,'ColorOrderIndex',1);
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p1 = plot(freqs, 180/pi*angle(squeeze(freqresp(Gc(4, 4), freqs, 'Hz'))));
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p2 = plot(freqs, 180/pi*angle(squeeze(freqresp(Gc(5, 5), freqs, 'Hz'))));
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p3 = plot(freqs, 180/pi*angle(squeeze(freqresp(Gc(6, 6), freqs, 'Hz'))));
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
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ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
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ylim([-180, 180]);
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yticks([-180, -90, 0, 90, 180]);
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legend([p1, p2, p3, p4], {'$R_x/M_x$','$R_y/M_y$', '$R_z/M_z$', '$R_i/M_j$'})
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linkaxes([ax1,ax2,ax3,ax4],'x');
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% Cube's center not coincident with the Mass of the Mobile platform
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% Let's create a Stewart platform with a cubic architecture where the cube's center is at the center of the Stewart platform.
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H = 200e-3; % height of the Stewart platform [m]
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MO_B = -100e-3; % Position {B} with respect to {M} [m]
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% Now, we set the cube's parameters such that the center of the cube is coincident with $\{A\}$ and $\{B\}$.
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Hc = 2.5*H; % Size of the useful part of the cube [m]
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FOc = H + MO_B; % Center of the cube with respect to {F}
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stewart = initializeStewartPlatform();
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stewart = initializeFramesPositions(stewart, 'H', H, 'MO_B', MO_B);
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stewart = generateCubicConfiguration(stewart, 'Hc', Hc, 'FOc', FOc, 'FHa', 25e-3, 'MHb', 25e-3);
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stewart = computeJointsPose(stewart);
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stewart = initializeStrutDynamics(stewart, 'K', 1e6*ones(6,1), 'C', 1e1*ones(6,1));
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stewart = initializeJointDynamics(stewart, 'type_F', 'universal', 'type_M', 'spherical');
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stewart = computeJacobian(stewart);
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stewart = initializeStewartPose(stewart);
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% However, the Center of Mass of the mobile platform is *not* located at the cube's center.
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stewart = initializeCylindricalPlatforms(stewart, 'Fpr', 1.2*max(vecnorm(stewart.platform_F.Fa)), ...
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'Mpm', 10, ...
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'Mph', 20e-3, ...
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'Mpr', 1.2*max(vecnorm(stewart.platform_M.Mb)));
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% And we set small mass for the struts.
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stewart = initializeCylindricalStruts(stewart, 'Fsm', 1e-3, 'Msm', 1e-3);
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stewart = initializeInertialSensor(stewart);
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% No flexibility below the Stewart platform and no payload.
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ground = initializeGround('type', 'none');
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payload = initializePayload('type', 'none');
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% The obtain geometry is shown in figure [[fig:stewart_cubic_conf_mass_above]].
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displayArchitecture(stewart, 'labels', false, 'view', 'all');
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% #+name: fig:stewart_cubic_conf_mass_above
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% #+caption: Geometry used for the simulations - The cube's center is coincident with the frames $\{A\}$ and $\{B\}$ but not with the Center of mass of the mobile platform ([[./figs/stewart_cubic_conf_mass_above.png][png]], [[./figs/stewart_cubic_conf_mass_above.pdf][pdf]])
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% [[file:figs/stewart_cubic_conf_mass_above.png]]
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% We now identify the dynamics from forces applied in each strut $\bm{\tau}$ to the displacement of each strut $d \bm{\mathcal{L}}$.
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open('stewart_platform_model.slx')
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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, '/Controller'], 1, 'openinput'); io_i = io_i + 1; % Actuator Force Inputs [N]
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io(io_i) = linio([mdl, '/Stewart Platform'], 1, 'openoutput', [], 'dLm'); io_i = io_i + 1; % Relative Displacement Outputs [m]
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%% Run the linearization
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G = linearize(mdl, io, options);
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G.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
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G.OutputName = {'Dm1', 'Dm2', 'Dm3', 'Dm4', 'Dm5', 'Dm6'};
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% And we use the Jacobian to compute the transfer function from $\bm{\mathcal{F}}$ to $\bm{\mathcal{X}}$.
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Gc = inv(stewart.kinematics.J)*G*inv(stewart.kinematics.J');
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Gc.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
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Gc.OutputName = {'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'};
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% The obtain dynamics $\bm{G}_{c}(s) = \bm{J}^{-T} \bm{G}(s) \bm{J}^{-1}$ is shown in Figure [[fig:stewart_conf_coupling_mass_matrix]].
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freqs = logspace(1, 3, 500);
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figure;
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ax1 = subplot(2, 2, 1);
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hold on;
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for i = 1:6
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for j = i+1:6
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plot(freqs, abs(squeeze(freqresp(Gc(i, j), freqs, 'Hz'))), 'k-');
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end
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end
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set(gca,'ColorOrderIndex',1);
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plot(freqs, abs(squeeze(freqresp(Gc(1, 1), freqs, 'Hz'))));
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plot(freqs, abs(squeeze(freqresp(Gc(2, 2), freqs, 'Hz'))));
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plot(freqs, abs(squeeze(freqresp(Gc(3, 3), freqs, 'Hz'))));
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
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ax3 = subplot(2, 2, 3);
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hold on;
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for i = 1:6
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for j = i+1:6
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p4 = plot(freqs, 180/pi*angle(squeeze(freqresp(Gc(i, j), freqs, 'Hz'))), 'k-');
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end
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end
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set(gca,'ColorOrderIndex',1);
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p1 = plot(freqs, 180/pi*angle(squeeze(freqresp(Gc(1, 1), freqs, 'Hz'))));
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p2 = plot(freqs, 180/pi*angle(squeeze(freqresp(Gc(2, 2), freqs, 'Hz'))));
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p3 = plot(freqs, 180/pi*angle(squeeze(freqresp(Gc(3, 3), freqs, 'Hz'))));
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
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ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
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ylim([-180, 180]);
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yticks([-180, -90, 0, 90, 180]);
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legend([p1, p2, p3, p4], {'$D_x/F_x$','$D_y/F_y$', '$D_z/F_z$', '$D_i/F_j$'})
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ax2 = subplot(2, 2, 2);
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hold on;
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for i = 1:6
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for j = i+1:6
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plot(freqs, abs(squeeze(freqresp(Gc(i, j), freqs, 'Hz'))), 'k-');
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||
|
|
end
|
||
|
|
end
|
||
|
|
set(gca,'ColorOrderIndex',1);
|
||
|
|
plot(freqs, abs(squeeze(freqresp(Gc(4, 4), freqs, 'Hz'))));
|
||
|
|
plot(freqs, abs(squeeze(freqresp(Gc(5, 5), freqs, 'Hz'))));
|
||
|
|
plot(freqs, abs(squeeze(freqresp(Gc(6, 6), freqs, 'Hz'))));
|
||
|
|
hold off;
|
||
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||
|
|
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
|
||
|
|
|
||
|
|
ax4 = subplot(2, 2, 4);
|
||
|
|
hold on;
|
||
|
|
for i = 1:6
|
||
|
|
for j = i+1:6
|
||
|
|
p4 = plot(freqs, 180/pi*angle(squeeze(freqresp(Gc(i, j), freqs, 'Hz'))), 'k-');
|
||
|
|
end
|
||
|
|
end
|
||
|
|
set(gca,'ColorOrderIndex',1);
|
||
|
|
p1 = plot(freqs, 180/pi*angle(squeeze(freqresp(Gc(4, 4), freqs, 'Hz'))));
|
||
|
|
p2 = plot(freqs, 180/pi*angle(squeeze(freqresp(Gc(5, 5), freqs, 'Hz'))));
|
||
|
|
p3 = plot(freqs, 180/pi*angle(squeeze(freqresp(Gc(6, 6), freqs, 'Hz'))));
|
||
|
|
hold off;
|
||
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||
|
|
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
||
|
|
ylim([-180, 180]);
|
||
|
|
yticks([-180, -90, 0, 90, 180]);
|
||
|
|
legend([p1, p2, p3, p4], {'$R_x/M_x$','$R_y/M_y$', '$R_z/M_z$', '$R_i/M_j$'})
|
||
|
|
|
||
|
|
linkaxes([ax1,ax2,ax3,ax4],'x');
|