Add study of active damping techniques
This commit is contained in:
178
matlab/active_damping_dvf.m
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178
matlab/active_damping_dvf.m
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@@ -0,0 +1,178 @@
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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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open('simulink/stewart_active_damping.slx')
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% Identification of the Dynamics with perfect Joints
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% We first initialize the Stewart platform without joint stiffness.
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stewart = initializeFramesPositions('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, 'disable', true);
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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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% And we identify the dynamics from force actuators to force sensors.
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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_active_damping';
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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, '/F'], 1, 'openinput'); io_i = io_i + 1; % Actuator Force Inputs [N]
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io(io_i) = linio([mdl, '/Dm'], 1, 'openoutput'); io_i = io_i + 1; % Relative Displacement Outputs [N]
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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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% The transfer function from actuator forces to relative motion sensors is shown in Figure [[fig:dvf_plant_coupling]].
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freqs = logspace(1, 3, 1000);
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figure;
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ax1 = subplot(2, 1, 1);
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hold on;
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for i = 2:6
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set(gca,'ColorOrderIndex',2);
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plot(freqs, abs(squeeze(freqresp(G(['Dm', num2str(i)], 'F1'), freqs, 'Hz'))));
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end
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set(gca,'ColorOrderIndex',1);
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plot(freqs, abs(squeeze(freqresp(G('Dm1', 'F1'), 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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ax2 = subplot(2, 1, 2);
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hold on;
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for i = 2:6
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set(gca,'ColorOrderIndex',2);
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p2 = plot(freqs, 180/pi*angle(squeeze(freqresp(G(['Dm', num2str(i)], 'F1'), freqs, 'Hz'))));
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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(G('Dm1', 'F1'), 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], {'$D_{m,i}/F_i$', '$D_{m,j}/F_i$'})
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linkaxes([ax1,ax2],'x');
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% Effect of the Flexible Joint stiffness on the Dynamics
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% We add some stiffness and damping in the flexible joints and we re-identify the dynamics.
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stewart = initializeJointDynamics(stewart);
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Gf = linearize(mdl, io, options);
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Gf.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
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Gf.OutputName = {'Dm1', 'Dm2', 'Dm3', 'Dm4', 'Dm5', 'Dm6'};
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% The new dynamics from force actuator to relative motion sensor is shown in Figure [[fig:dvf_plant_flexible_joint_decentralized]].
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freqs = logspace(1, 3, 1000);
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figure;
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ax1 = subplot(2, 1, 1);
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hold on;
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plot(freqs, abs(squeeze(freqresp(G( 'Dm1', 'F1'), freqs, 'Hz'))));
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plot(freqs, abs(squeeze(freqresp(Gf('Dm1', 'F1'), 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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ax2 = subplot(2, 1, 2);
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hold on;
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plot(freqs, 180/pi*angle(squeeze(freqresp(G( 'Dm1', 'F1'), freqs, 'Hz'))), 'DisplayName', 'Perfect Joints');
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plot(freqs, 180/pi*angle(squeeze(freqresp(Gf('Dm1', 'F1'), freqs, 'Hz'))), 'DisplayName', 'Flexible Joints');
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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('location', 'northeast');
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linkaxes([ax1,ax2],'x');
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% Obtained Damping
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% The control is a performed in a decentralized manner.
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% The $6 \times 6$ control is a diagonal matrix with pure derivative action on the diagonal:
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% \[ K(s) = g
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% \begin{bmatrix}
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% s & & \\
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% & \ddots & \\
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% & & s
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% \end{bmatrix} \]
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% The root locus is shown in figure [[fig:root_locus_dvf_rot_stiffness]] and the obtained pole damping function of the control gain is shown in figure [[fig:pole_damping_gain_dvf_rot_stiffness]].
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gains = logspace(0, 5, 1000);
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figure;
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hold on;
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plot(real(pole(G)), imag(pole(G)), 'x');
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plot(real(pole(Gf)), imag(pole(Gf)), 'x');
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set(gca,'ColorOrderIndex',1);
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plot(real(tzero(G)), imag(tzero(G)), 'o');
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plot(real(tzero(Gf)), imag(tzero(Gf)), 'o');
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for i = 1:length(gains)
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cl_poles = pole(feedback(G, (gains(i)*s)*eye(6)));
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set(gca,'ColorOrderIndex',1);
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plot(real(cl_poles), imag(cl_poles), '.');
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cl_poles = pole(feedback(Gf, (gains(i)*s)*eye(6)));
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set(gca,'ColorOrderIndex',2);
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plot(real(cl_poles), imag(cl_poles), '.');
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end
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ylim([0,inf]);
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xlim([-3000,0]);
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xlabel('Real Part')
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ylabel('Imaginary Part')
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axis square
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% #+name: fig:root_locus_dvf_rot_stiffness
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% #+caption: Root Locus plot with Direct Velocity Feedback when considering the Stiffness of flexible joints ([[./figs/root_locus_dvf_rot_stiffness.png][png]], [[./figs/root_locus_dvf_rot_stiffness.pdf][pdf]])
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% [[file:figs/root_locus_dvf_rot_stiffness.png]]
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gains = logspace(0, 5, 1000);
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figure;
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hold on;
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for i = 1:length(gains)
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set(gca,'ColorOrderIndex',1);
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cl_poles = pole(feedback(G, (gains(i)*s)*eye(6)));
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poles_damp = phase(cl_poles(imag(cl_poles)>0)) - pi/2;
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plot(gains(i)*ones(size(poles_damp)), poles_damp, '.');
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set(gca,'ColorOrderIndex',2);
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cl_poles = pole(feedback(Gf, (gains(i)*s)*eye(6)));
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poles_damp = phase(cl_poles(imag(cl_poles)>0)) - pi/2;
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plot(gains(i)*ones(size(poles_damp)), poles_damp, '.');
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end
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xlabel('Control Gain');
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ylabel('Damping of the Poles');
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set(gca, 'XScale', 'log');
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ylim([0,pi/2]);
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178
matlab/active_damping_iff.m
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178
matlab/active_damping_iff.m
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@@ -0,0 +1,178 @@
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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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open('simulink/stewart_active_damping.slx')
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% Identification of the Dynamics with perfect Joints
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% We first initialize the Stewart platform without joint stiffness.
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stewart = initializeFramesPositions('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, 'disable', true);
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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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% And we identify the dynamics from force actuators to force sensors.
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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_active_damping';
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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, '/F'], 1, 'openinput'); io_i = io_i + 1; % Actuator Force Inputs [N]
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io(io_i) = linio([mdl, '/Fm'], 1, 'openoutput'); io_i = io_i + 1; % Force Sensor Outputs [N]
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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 = {'Fm1', 'Fm2', 'Fm3', 'Fm4', 'Fm5', 'Fm6'};
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% The transfer function from actuator forces to force sensors is shown in Figure [[fig:iff_plant_coupling]].
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freqs = logspace(1, 3, 1000);
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figure;
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ax1 = subplot(2, 1, 1);
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hold on;
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for i = 2:6
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set(gca,'ColorOrderIndex',2);
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plot(freqs, abs(squeeze(freqresp(G(['Fm', num2str(i)], 'F1'), freqs, 'Hz'))));
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end
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set(gca,'ColorOrderIndex',1);
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plot(freqs, abs(squeeze(freqresp(G('Fm1', 'F1'), 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 [N/N]'); set(gca, 'XTickLabel',[]);
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ax2 = subplot(2, 1, 2);
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hold on;
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for i = 2:6
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set(gca,'ColorOrderIndex',2);
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p2 = plot(freqs, 180/pi*angle(squeeze(freqresp(G(['Fm', num2str(i)], 'F1'), freqs, 'Hz'))));
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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(G('Fm1', 'F1'), 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], {'$F_{m,i}/F_i$', '$F_{m,j}/F_i$'})
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linkaxes([ax1,ax2],'x');
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% Effect of the Flexible Joint stiffness on the Dynamics
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% We add some stiffness and damping in the flexible joints and we re-identify the dynamics.
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stewart = initializeJointDynamics(stewart);
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Gf = linearize(mdl, io, options);
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Gf.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
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Gf.OutputName = {'Fm1', 'Fm2', 'Fm3', 'Fm4', 'Fm5', 'Fm6'};
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% The new dynamics from force actuator to force sensor is shown in Figure [[fig:iff_plant_flexible_joint_decentralized]].
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freqs = logspace(1, 3, 1000);
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figure;
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ax1 = subplot(2, 1, 1);
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hold on;
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plot(freqs, abs(squeeze(freqresp(G( 'Fm1', 'F1'), freqs, 'Hz'))));
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plot(freqs, abs(squeeze(freqresp(Gf('Fm1', 'F1'), 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 [N/N]'); set(gca, 'XTickLabel',[]);
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ax2 = subplot(2, 1, 2);
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hold on;
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plot(freqs, 180/pi*angle(squeeze(freqresp(G( 'Fm1', 'F1'), freqs, 'Hz'))), 'DisplayName', 'Perfect Joints');
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plot(freqs, 180/pi*angle(squeeze(freqresp(Gf('Fm1', 'F1'), freqs, 'Hz'))), 'DisplayName', 'Flexible Joints');
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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('location', 'southwest')
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linkaxes([ax1,ax2],'x');
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% Obtained Damping
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% The control is a performed in a decentralized manner.
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% The $6 \times 6$ control is a diagonal matrix with pure integration action on the diagonal:
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% \[ K(s) = g
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% \begin{bmatrix}
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% \frac{1}{s} & & 0 \\
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% & \ddots & \\
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% 0 & & \frac{1}{s}
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% \end{bmatrix} \]
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% The root locus is shown in figure [[fig:root_locus_iff_rot_stiffness]] and the obtained pole damping function of the control gain is shown in figure [[fig:pole_damping_gain_iff_rot_stiffness]].
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gains = logspace(0, 5, 1000);
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figure;
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hold on;
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plot(real(pole(G)), imag(pole(G)), 'x');
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plot(real(pole(Gf)), imag(pole(Gf)), 'x');
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set(gca,'ColorOrderIndex',1);
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plot(real(tzero(G)), imag(tzero(G)), 'o');
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plot(real(tzero(Gf)), imag(tzero(Gf)), 'o');
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for i = 1:length(gains)
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cl_poles = pole(feedback(G, (gains(i)/s)*eye(6)));
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set(gca,'ColorOrderIndex',1);
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plot(real(cl_poles), imag(cl_poles), '.');
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cl_poles = pole(feedback(Gf, (gains(i)/s)*eye(6)));
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set(gca,'ColorOrderIndex',2);
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plot(real(cl_poles), imag(cl_poles), '.');
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end
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ylim([0,inf]);
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xlim([-3000,0]);
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xlabel('Real Part')
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ylabel('Imaginary Part')
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axis square
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% #+name: fig:root_locus_iff_rot_stiffness
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% #+caption: Root Locus plot with Decentralized Integral Force Feedback when considering the stiffness of flexible joints ([[./figs/root_locus_iff_rot_stiffness.png][png]], [[./figs/root_locus_iff_rot_stiffness.pdf][pdf]])
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% [[file:figs/root_locus_iff_rot_stiffness.png]]
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gains = logspace(0, 5, 1000);
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figure;
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hold on;
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for i = 1:length(gains)
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set(gca,'ColorOrderIndex',1);
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cl_poles = pole(feedback(G, (gains(i)/s)*eye(6)));
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poles_damp = phase(cl_poles(imag(cl_poles)>0)) - pi/2;
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plot(gains(i)*ones(size(poles_damp)), poles_damp, '.');
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set(gca,'ColorOrderIndex',2);
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cl_poles = pole(feedback(Gf, (gains(i)/s)*eye(6)));
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poles_damp = phase(cl_poles(imag(cl_poles)>0)) - pi/2;
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plot(gains(i)*ones(size(poles_damp)), poles_damp, '.');
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end
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xlabel('Control Gain');
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ylabel('Damping of the Poles');
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set(gca, 'XScale', 'log');
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ylim([0,pi/2]);
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173
matlab/active_damping_inertial.m
Normal file
173
matlab/active_damping_inertial.m
Normal file
@@ -0,0 +1,173 @@
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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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open('simulink/stewart_active_damping.slx')
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% Identification of the Dynamics
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stewart = initializeFramesPositions('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, 'disable', true);
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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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%% 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_active_damping';
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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, '/F'], 1, 'openinput'); io_i = io_i + 1; % Actuator Force Inputs [N]
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io(io_i) = linio([mdl, '/Vm'], 1, 'openoutput'); io_i = io_i + 1; % Absolute velocity of each leg [m/s]
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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'};
|
||||
G.OutputName = {'Vm1', 'Vm2', 'Vm3', 'Vm4', 'Vm5', 'Vm6'};
|
||||
|
||||
|
||||
|
||||
% The transfer function from actuator forces to force sensors is shown in Figure [[fig:inertial_plant_coupling]].
|
||||
|
||||
freqs = logspace(1, 3, 1000);
|
||||
|
||||
figure;
|
||||
|
||||
ax1 = subplot(2, 1, 1);
|
||||
hold on;
|
||||
for i = 2:6
|
||||
set(gca,'ColorOrderIndex',2);
|
||||
plot(freqs, abs(squeeze(freqresp(G(['Vm', num2str(i)], 'F1'), freqs, 'Hz'))));
|
||||
end
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
plot(freqs, abs(squeeze(freqresp(G('Vm1', 'F1'), freqs, 'Hz'))));
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Amplitude [$\frac{m/s}{N}$]'); set(gca, 'XTickLabel',[]);
|
||||
|
||||
ax2 = subplot(2, 1, 2);
|
||||
hold on;
|
||||
for i = 2:6
|
||||
set(gca,'ColorOrderIndex',2);
|
||||
p2 = plot(freqs, 180/pi*angle(squeeze(freqresp(G(['Vm', num2str(i)], 'F1'), freqs, 'Hz'))));
|
||||
end
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
p1 = plot(freqs, 180/pi*angle(squeeze(freqresp(G('Vm1', 'F1'), 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], {'$F_{m,i}/F_i$', '$F_{m,j}/F_i$'})
|
||||
|
||||
linkaxes([ax1,ax2],'x');
|
||||
|
||||
% Effect of the Flexible Joint stiffness on the Dynamics
|
||||
% We add some stiffness and damping in the flexible joints and we re-identify the dynamics.
|
||||
|
||||
stewart = initializeJointDynamics(stewart);
|
||||
Gf = linearize(mdl, io, options);
|
||||
Gf.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
|
||||
Gf.OutputName = {'Vm1', 'Vm2', 'Vm3', 'Vm4', 'Vm5', 'Vm6'};
|
||||
|
||||
|
||||
|
||||
% The new dynamics from force actuator to force sensor is shown in Figure [[fig:inertial_plant_flexible_joint_decentralized]].
|
||||
|
||||
freqs = logspace(1, 3, 1000);
|
||||
|
||||
figure;
|
||||
|
||||
ax1 = subplot(2, 1, 1);
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(G( 'Vm1', 'F1'), freqs, 'Hz'))));
|
||||
plot(freqs, abs(squeeze(freqresp(Gf('Vm1', 'F1'), freqs, 'Hz'))));
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Amplitude [$\frac{m/s}{N}$]'); set(gca, 'XTickLabel',[]);
|
||||
|
||||
ax2 = subplot(2, 1, 2);
|
||||
hold on;
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(G( 'Vm1', 'F1'), freqs, 'Hz'))), 'DisplayName', 'Perfect Joints');
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(Gf('Vm1', 'F1'), freqs, 'Hz'))), 'DisplayName', 'Flexible Joints');
|
||||
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('location', 'southwest')
|
||||
|
||||
linkaxes([ax1,ax2],'x');
|
||||
|
||||
% Obtained Damping
|
||||
% The control is a performed in a decentralized manner.
|
||||
% The $6 \times 6$ control is a diagonal matrix with pure proportional action on the diagonal:
|
||||
% \[ K(s) = g
|
||||
% \begin{bmatrix}
|
||||
% 1 & & 0 \\
|
||||
% & \ddots & \\
|
||||
% 0 & & 1
|
||||
% \end{bmatrix} \]
|
||||
|
||||
% The root locus is shown in figure [[fig:root_locus_inertial_rot_stiffness]] and the obtained pole damping function of the control gain is shown in figure [[fig:pole_damping_gain_inertial_rot_stiffness]].
|
||||
|
||||
gains = logspace(0, 5, 1000);
|
||||
|
||||
figure;
|
||||
hold on;
|
||||
plot(real(pole(G)), imag(pole(G)), 'x');
|
||||
plot(real(pole(Gf)), imag(pole(Gf)), 'x');
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
plot(real(tzero(G)), imag(tzero(G)), 'o');
|
||||
plot(real(tzero(Gf)), imag(tzero(Gf)), 'o');
|
||||
for i = 1:length(gains)
|
||||
cl_poles = pole(feedback(G, gains(i)*eye(6)));
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
plot(real(cl_poles), imag(cl_poles), '.');
|
||||
cl_poles = pole(feedback(Gf, gains(i)*eye(6)));
|
||||
set(gca,'ColorOrderIndex',2);
|
||||
plot(real(cl_poles), imag(cl_poles), '.');
|
||||
end
|
||||
ylim([0,2000]);
|
||||
xlim([-2000,0]);
|
||||
xlabel('Real Part')
|
||||
ylabel('Imaginary Part')
|
||||
axis square
|
||||
|
||||
|
||||
|
||||
% #+name: fig:root_locus_inertial_rot_stiffness
|
||||
% #+caption: Root Locus plot with Decentralized Inertial Control when considering the stiffness of flexible joints ([[./figs/root_locus_inertial_rot_stiffness.png][png]], [[./figs/root_locus_inertial_rot_stiffness.pdf][pdf]])
|
||||
% [[file:figs/root_locus_inertial_rot_stiffness.png]]
|
||||
|
||||
|
||||
gains = logspace(0, 5, 1000);
|
||||
|
||||
figure;
|
||||
hold on;
|
||||
for i = 1:length(gains)
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
cl_poles = pole(feedback(G, gains(i)*eye(6)));
|
||||
poles_damp = phase(cl_poles(imag(cl_poles)>0)) - pi/2;
|
||||
plot(gains(i)*ones(size(poles_damp)), poles_damp, '.');
|
||||
set(gca,'ColorOrderIndex',2);
|
||||
cl_poles = pole(feedback(Gf, gains(i)*eye(6)));
|
||||
poles_damp = phase(cl_poles(imag(cl_poles)>0)) - pi/2;
|
||||
plot(gains(i)*ones(size(poles_damp)), poles_damp, '.');
|
||||
end
|
||||
xlabel('Control Gain');
|
||||
ylabel('Damping of the Poles');
|
||||
set(gca, 'XScale', 'log');
|
||||
ylim([0,pi/2]);
|
||||
Reference in New Issue
Block a user