500 lines
14 KiB
Matlab
500 lines
14 KiB
Matlab
%% 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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addpath('STEP');
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% Jacobian
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% First, the position of the "joints" (points of force application) are estimated and the Jacobian computed.
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open('drone_platform_jacobian.slx');
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sim('drone_platform_jacobian');
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Aa = [a1.Data(1,:);
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a2.Data(1,:);
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a3.Data(1,:);
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a4.Data(1,:);
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a5.Data(1,:);
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a6.Data(1,:)]';
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Ab = [b1.Data(1,:);
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b2.Data(1,:);
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b3.Data(1,:);
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b4.Data(1,:);
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b5.Data(1,:);
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b6.Data(1,:)]';
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As = (Ab - Aa)./vecnorm(Ab - Aa);
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l = vecnorm(Ab - Aa)';
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J = [As' , cross(Ab, As)'];
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save('./jacobian.mat', 'Aa', 'Ab', 'As', 'l', 'J');
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% Simscape Model
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open('drone_platform.slx');
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% Definition of spring parameters
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kx = 0.5*1e3/3; % [N/m]
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ky = 0.5*1e3/3;
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kz = 1e3/3;
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cx = 0.025; % [Nm/rad]
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cy = 0.025;
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cz = 0.025;
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% We load the Jacobian.
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load('./jacobian.mat', 'Aa', 'Ab', 'As', 'l', 'J');
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% Identification of the plant
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% The dynamics is identified from forces applied by each legs to the measured acceleration of the top platform.
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%% Name of the Simulink File
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mdl = 'drone_platform';
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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, '/Dw'], 1, 'openinput'); io_i = io_i + 1;
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io(io_i) = linio([mdl, '/u'], 1, 'openinput'); io_i = io_i + 1;
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io(io_i) = linio([mdl, '/Inertial Sensor'], 1, 'openoutput'); io_i = io_i + 1;
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G = linearize(mdl, io);
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G.InputName = {'Dwx', 'Dwy', 'Dwz', 'Rwx', 'Rwy', 'Rwz', ...
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'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
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G.OutputName = {'Ax', 'Ay', 'Az', 'Arx', 'Ary', 'Arz'};
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% There are 24 states (6dof for the bottom platform + 6dof for the top platform).
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size(G)
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% #+RESULTS:
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% : State-space model with 6 outputs, 12 inputs, and 24 states.
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% G = G*blkdiag(inv(J), eye(6));
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% G.InputName = {'Dw1', 'Dw2', 'Dw3', 'Dw4', 'Dw5', 'Dw6', ...
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% 'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
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% Thanks to the Jacobian, we compute the transfer functions in the frame of the legs and in an inertial frame.
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Gx = G*blkdiag(eye(6), inv(J'));
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Gx.InputName = {'Dwx', 'Dwy', 'Dwz', 'Rwx', 'Rwy', 'Rwz', ...
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'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
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Gl = J*G;
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Gl.OutputName = {'A1', 'A2', 'A3', 'A4', 'A5', 'A6'};
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% Obtained Dynamics
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freqs = logspace(-1, 2, 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(Gx('Ax', 'Fx'), freqs, 'Hz'))), 'DisplayName', '$A_x/F_x$');
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plot(freqs, abs(squeeze(freqresp(Gx('Ay', 'Fy'), freqs, 'Hz'))), 'DisplayName', '$A_y/F_y$');
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plot(freqs, abs(squeeze(freqresp(Gx('Az', 'Fz'), freqs, 'Hz'))), 'DisplayName', '$A_z/F_z$');
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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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legend('location', 'southeast');
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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(Gx('Ax', 'Fx'), freqs, 'Hz'))));
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plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Ay', 'Fy'), freqs, 'Hz'))));
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plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Az', 'Fz'), 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([-360:90:360]);
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linkaxes([ax1,ax2],'x');
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% #+name: fig:stewart_platform_translations
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% #+caption: Stewart Platform Plant from forces applied by the legs to the acceleration of the platform
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% #+RESULTS:
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% [[file:figs/stewart_platform_translations.png]]
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freqs = logspace(-1, 2, 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(Gx('Arx', 'Mx'), freqs, 'Hz'))), 'DisplayName', '$A_{R_x}/M_x$');
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plot(freqs, abs(squeeze(freqresp(Gx('Ary', 'My'), freqs, 'Hz'))), 'DisplayName', '$A_{R_y}/M_y$');
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plot(freqs, abs(squeeze(freqresp(Gx('Arz', 'Mz'), freqs, 'Hz'))), 'DisplayName', '$A_{R_z}/M_z$');
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Amplitude [rad/(Nm)]'); set(gca, 'XTickLabel',[]);
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legend('location', 'southeast');
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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(Gx('Arx', 'Mx'), freqs, 'Hz'))));
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plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Ary', 'My'), freqs, 'Hz'))));
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plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Arz', 'Mz'), 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([-360:90:360]);
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linkaxes([ax1,ax2],'x');
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% #+name: fig:stewart_platform_rotations
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% #+caption: Stewart Platform Plant from torques applied by the legs to the angular acceleration of the platform
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% #+RESULTS:
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% [[file:figs/stewart_platform_rotations.png]]
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freqs = logspace(-1, 2, 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 out_i = 1:5
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for in_i = i+1:6
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plot(freqs, abs(squeeze(freqresp(Gl(sprintf('A%i', out_i), sprintf('F%i', in_i)), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2]);
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end
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end
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for ch_i = 1:6
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plot(freqs, abs(squeeze(freqresp(Gl(sprintf('A%i', ch_i), sprintf('F%i', ch_i)), freqs, 'Hz'))));
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end
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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 ch_i = 1:6
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plot(freqs, 180/pi*angle(squeeze(freqresp(Gl(sprintf('A%i', ch_i), sprintf('F%i', ch_i)), freqs, 'Hz'))));
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end
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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([-360:90:360]);
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linkaxes([ax1,ax2],'x');
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% #+name: fig:stewart_platform_legs
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% #+caption: Stewart Platform Plant from forces applied by the legs to displacement of the legs
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% #+RESULTS:
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% [[file:figs/stewart_platform_legs.png]]
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freqs = logspace(-1, 2, 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(Gx('Ax', 'Dwx')/s^2, freqs, 'Hz'))), 'DisplayName', '$D_x/D_{w,x}$');
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% plot(freqs, abs(squeeze(freqresp(Gx('Ay', 'Dwy')/s^2, freqs, 'Hz'))), 'DisplayName', '$D_y/D_{w,y}$');
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% plot(freqs, abs(squeeze(freqresp(Gx('Az', 'Dwz')/s^2, freqs, 'Hz'))), 'DisplayName', '$D_z/D_{w,z}$');
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set(gca,'ColorOrderIndex',1)
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plot(freqs, abs(squeeze(freqresp(TR(1,1), freqs, 'Hz'))), '--', 'DisplayName', '$D_x/D_{w,x}$');
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plot(freqs, abs(squeeze(freqresp(TR(2,2), freqs, 'Hz'))), '--', 'DisplayName', '$D_x/D_{w,x}$');
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plot(freqs, abs(squeeze(freqresp(TR(3,3), freqs, 'Hz'))), '--', 'DisplayName', '$D_x/D_{w,x}$');
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Transmissibility - Translations'); xlabel('Frequency [Hz]');
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legend('location', 'northeast');
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ax2 = subplot(2, 1, 2);
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hold on;
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% plot(freqs, abs(squeeze(freqresp(Gx('Arx', 'Rwx')/s^2, freqs, 'Hz'))), 'DisplayName', '$R_x/R_{w,x}$');
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% plot(freqs, abs(squeeze(freqresp(Gx('Ary', 'Rwy')/s^2, freqs, 'Hz'))), 'DisplayName', '$R_y/R_{w,y}$');
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% plot(freqs, abs(squeeze(freqresp(Gx('Arz', 'Rwz')/s^2, freqs, 'Hz'))), 'DisplayName', '$R_z/R_{w,z}$');
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set(gca,'ColorOrderIndex',1)
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plot(freqs, abs(squeeze(freqresp(TR(4,4), freqs, 'Hz'))), '--', 'DisplayName', '$D_x/D_{w,x}$');
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plot(freqs, abs(squeeze(freqresp(TR(5,5), freqs, 'Hz'))), '--', 'DisplayName', '$D_x/D_{w,x}$');
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plot(freqs, abs(squeeze(freqresp(TR(6,6), freqs, 'Hz'))), '--', 'DisplayName', '$D_x/D_{w,x}$');
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Transmissibility - Rotations'); xlabel('Frequency [Hz]');
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legend('location', 'northeast');
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linkaxes([ax1,ax2],'x');
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% Real Approximation of $G$ at the decoupling frequency
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% Let's compute a real approximation of the complex matrix $H_1$ which corresponds to the the transfer function $G_c(j\omega_c)$ from forces applied by the actuators to the measured acceleration of the top platform evaluated at the frequency $\omega_c$.
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wc = 2*pi*20; % Decoupling frequency [rad/s]
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Gc = G({'Ax', 'Ay', 'Az', 'Arx', 'Ary', 'Arz'}, ...
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{'F1', 'F2', 'F3', 'F4', 'F5', 'F6'}); % Transfer function to find a real approximation
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H1 = evalfr(Gc, j*wc);
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% The real approximation is computed as follows:
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D = pinv(real(H1'*H1));
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H1 = inv(D*real(H1'*diag(exp(j*angle(diag(H1*D*H1.'))/2))));
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% Verification of the decoupling using the "Gershgorin Radii"
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% First, the Singular Value Decomposition of $H_1$ is performed:
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% \[ H_1 = U \Sigma V^H \]
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[U,S,V] = svd(H1);
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% Then, the "Gershgorin Radii" is computed for the plant $G_c(s)$ and the "SVD Decoupled Plant" $G_d(s)$:
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% \[ G_d(s) = U^T G_c(s) V \]
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% This is computed over the following frequencies.
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freqs = logspace(-2, 2, 1000); % [Hz]
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% Gershgorin Radii for the coupled plant:
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Gr_coupled = zeros(length(freqs), size(Gc,2));
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H = abs(squeeze(freqresp(Gc, freqs, 'Hz')));
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for out_i = 1:size(Gc,2)
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Gr_coupled(:, out_i) = squeeze((sum(H(out_i,:,:)) - H(out_i,out_i,:))./H(out_i, out_i, :));
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end
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% Gershgorin Radii for the decoupled plant using SVD:
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Gd = U'*Gc*V;
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Gr_decoupled = zeros(length(freqs), size(Gd,2));
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H = abs(squeeze(freqresp(Gd, freqs, 'Hz')));
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for out_i = 1:size(Gd,2)
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Gr_decoupled(:, out_i) = squeeze((sum(H(out_i,:,:)) - H(out_i,out_i,:))./H(out_i, out_i, :));
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end
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% Gershgorin Radii for the decoupled plant using the Jacobian:
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Gj = Gc*inv(J');
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Gr_jacobian = zeros(length(freqs), size(Gj,2));
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H = abs(squeeze(freqresp(Gj, freqs, 'Hz')));
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for out_i = 1:size(Gj,2)
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Gr_jacobian(:, out_i) = squeeze((sum(H(out_i,:,:)) - H(out_i,out_i,:))./H(out_i, out_i, :));
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end
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figure;
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hold on;
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plot(freqs, Gr_coupled(:,1), 'DisplayName', 'Coupled');
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plot(freqs, Gr_decoupled(:,1), 'DisplayName', 'SVD');
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plot(freqs, Gr_jacobian(:,1), 'DisplayName', 'Jacobian');
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for in_i = 2:6
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set(gca,'ColorOrderIndex',1)
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plot(freqs, Gr_coupled(:,in_i), 'HandleVisibility', 'off');
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set(gca,'ColorOrderIndex',2)
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plot(freqs, Gr_decoupled(:,in_i), 'HandleVisibility', 'off');
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set(gca,'ColorOrderIndex',3)
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plot(freqs, Gr_jacobian(:,in_i), 'HandleVisibility', 'off');
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end
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plot(freqs, 0.5*ones(size(freqs)), 'k--', 'DisplayName', 'Limit')
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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hold off;
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xlabel('Frequency (Hz)'); ylabel('Gershgorin Radii')
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legend('location', 'northeast');
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% Decoupled Plant
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% Let's see the bode plot of the decoupled plant $G_d(s)$.
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% \[ G_d(s) = U^T G_c(s) V \]
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freqs = logspace(-1, 2, 1000);
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figure;
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hold on;
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for ch_i = 1:6
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plot(freqs, abs(squeeze(freqresp(Gd(ch_i, ch_i), freqs, 'Hz'))), ...
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'DisplayName', sprintf('$G(%i, %i)$', ch_i, ch_i));
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end
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for in_i = 1:5
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for out_i = in_i+1:6
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plot(freqs, abs(squeeze(freqresp(Gd(out_i, in_i), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ...
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'HandleVisibility', 'off');
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end
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end
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Amplitude'); xlabel('Frequency [Hz]');
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legend('location', 'southeast');
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% #+name: fig:simscape_model_decoupled_plant_svd
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% #+caption: Decoupled Plant using SVD
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% #+RESULTS:
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% [[file:figs/simscape_model_decoupled_plant_svd.png]]
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freqs = logspace(-1, 2, 1000);
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figure;
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hold on;
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for ch_i = 1:6
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plot(freqs, abs(squeeze(freqresp(Gj(ch_i, ch_i), freqs, 'Hz'))), ...
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'DisplayName', sprintf('$G(%i, %i)$', ch_i, ch_i));
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end
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for in_i = 1:5
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for out_i = in_i+1:6
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plot(freqs, abs(squeeze(freqresp(Gj(out_i, in_i), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ...
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'HandleVisibility', 'off');
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end
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end
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Amplitude'); xlabel('Frequency [Hz]');
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legend('location', 'southeast');
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% Diagonal Controller
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% The controller $K$ is a diagonal controller consisting a low pass filters with a crossover frequency $\omega_c$ and a DC gain $C_g$.
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wc = 2*pi*0.1; % Crossover Frequency [rad/s]
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C_g = 50; % DC Gain
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K = eye(6)*C_g/(s+wc);
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% #+RESULTS:
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% [[file:figs/centralized_control.png]]
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G_cen = feedback(G, inv(J')*K, [7:12], [1:6]);
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% #+RESULTS:
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% [[file:figs/svd_control.png]]
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% SVD Control
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G_svd = feedback(G, pinv(V')*K*pinv(U), [7:12], [1:6]);
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% Results
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% Let's first verify the stability of the closed-loop systems:
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isstable(G_cen)
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% #+RESULTS:
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% : ans =
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% : logical
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% : 1
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isstable(G_svd)
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% #+RESULTS:
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% : ans =
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% : logical
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% : 0
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% The obtained transmissibility in Open-loop, for the centralized control as well as for the SVD control are shown in Figure [[fig:stewart_platform_simscape_cl_transmissibility]].
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freqs = logspace(-3, 3, 1000);
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figure
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ax1 = subplot(2, 3, 1);
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hold on;
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plot(freqs, abs(squeeze(freqresp(G( 'Ax', 'Dwx')/s^2, freqs, 'Hz'))), 'DisplayName', 'Open-Loop');
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plot(freqs, abs(squeeze(freqresp(G_cen('Ax', 'Dwx')/s^2, freqs, 'Hz'))), 'DisplayName', 'Centralized');
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plot(freqs, abs(squeeze(freqresp(G_svd('Ax', 'Dwx')/s^2, freqs, 'Hz'))), 'DisplayName', 'SVD');
|
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hold off;
|
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
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ylabel('Transmissibility - $D_x/D_{w,x}$'); xlabel('Frequency [Hz]');
|
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legend('location', 'southwest');
|
|
|
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ax2 = subplot(2, 3, 2);
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hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G( 'Ay', 'Dwy')/s^2, freqs, 'Hz'))));
|
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plot(freqs, abs(squeeze(freqresp(G_cen('Ay', 'Dwy')/s^2, freqs, 'Hz'))));
|
|
plot(freqs, abs(squeeze(freqresp(G_svd('Ay', 'Dwy')/s^2, freqs, 'Hz'))));
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Transmissibility - $D_y/D_{w,y}$'); xlabel('Frequency [Hz]');
|
|
|
|
ax3 = subplot(2, 3, 3);
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G( 'Az', 'Dwz')/s^2, freqs, 'Hz'))));
|
|
plot(freqs, abs(squeeze(freqresp(G_cen('Az', 'Dwz')/s^2, freqs, 'Hz'))));
|
|
plot(freqs, abs(squeeze(freqresp(G_svd('Az', 'Dwz')/s^2, freqs, 'Hz'))));
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Transmissibility - $D_z/D_{w,z}$'); xlabel('Frequency [Hz]');
|
|
|
|
ax4 = subplot(2, 3, 4);
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G( 'Arx', 'Rwx')/s^2, freqs, 'Hz'))));
|
|
plot(freqs, abs(squeeze(freqresp(G_cen('Arx', 'Rwx')/s^2, freqs, 'Hz'))));
|
|
plot(freqs, abs(squeeze(freqresp(G_svd('Arx', 'Rwx')/s^2, freqs, 'Hz'))));
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Transmissibility - $R_x/R_{w,x}$'); xlabel('Frequency [Hz]');
|
|
|
|
ax5 = subplot(2, 3, 5);
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G( 'Ary', 'Rwy')/s^2, freqs, 'Hz'))));
|
|
plot(freqs, abs(squeeze(freqresp(G_cen('Ary', 'Rwy')/s^2, freqs, 'Hz'))));
|
|
plot(freqs, abs(squeeze(freqresp(G_svd('Ary', 'Rwy')/s^2, freqs, 'Hz'))));
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Transmissibility - $R_y/R_{w,y}$'); xlabel('Frequency [Hz]');
|
|
|
|
ax6 = subplot(2, 3, 6);
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G( 'Arz', 'Rwz')/s^2, freqs, 'Hz'))));
|
|
plot(freqs, abs(squeeze(freqresp(G_cen('Arz', 'Rwz')/s^2, freqs, 'Hz'))));
|
|
plot(freqs, abs(squeeze(freqresp(G_svd('Arz', 'Rwz')/s^2, freqs, 'Hz'))));
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Transmissibility - $R_z/R_{w,z}$'); xlabel('Frequency [Hz]');
|
|
|
|
linkaxes([ax1,ax2,ax3,ax4,ax5,ax6],'x');
|
|
xlim([freqs(1), freqs(end)]);
|