%% Clear Workspace and Close figures clear; close all; clc; %% Intialize Laplace variable s = zpk('s'); %% Path for functions, data and scripts addpath('./src/'); % Path for functions %% Colors for the figures colors = colororder; %% Initialize Frequency Vector freqs = logspace(0, 3, 1000); %% Compute Equation of motion l = 1; h=2; la = 0.5; % Horizontal position of actuators [m] ha = 0.2; % Vertical of actuators [m] m = 40; % Payload mass [kg] I = 5; % Payload rotational inertia [kg m^2] c = 2e2; % Actuator Damping [N/(m/s)] k = 1e6; % Actuator Stiffness [N/m] % Unit vectors of the actuators s1 = [1;0]; s2 = [0;1]; s3 = [0;1]; % Stiffnesss and Damping matrices of the struts Kr = diag([k,k,k]); Cr = diag([c,c,c]); % Location of the joints with respect to the center of mass Mb1 = [-l/2;-ha]; Mb2 = [-la; -h/2]; Mb3 = [ la; -h/2]; % Jacobian matrix (Center of Mass) J_CoM = [s1', Mb1(1)*s1(2)-Mb1(2)*s1(1); s2', Mb2(1)*s2(2)-Mb2(2)*s2(1); s3', Mb3(1)*s3(2)-Mb3(2)*s3(1)]; % Mass Matrix in frame {M} M = diag([m,m,I]); % Stiffness Matrix in frame {M} K = J_CoM'*Kr*J_CoM; % Damping Matrix in frame {M} C = J_CoM'*Cr*J_CoM; % Plant in the frame of the struts G_L = J_CoM*inv(M*s^2 + C*s + K)*J_CoM'; figure; tiledlayout(3, 3, 'TileSpacing', 'Compact', 'Padding', 'None'); for out_i = 1:3 for in_i = 1:3 nexttile; plot(freqs, abs(squeeze(freqresp(G_L(out_i,in_i), freqs, 'Hz'))), 'k-', ... 'DisplayName', sprintf('$\\mathcal{L}_%i/\\tau_%i$', out_i, in_i)); set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); xlim([freqs(1), freqs(end)]); ylim([2e-8, 4e-5]); xticks([1e0, 1e1, 1e2]) yticks([1e-7, 1e-6, 1e-5]) leg = legend('location', 'northeast', 'FontSize', 8); leg.ItemTokenSize(1) = 18; if in_i == 1 ylabel('Mag. [m/N]') else set(gca, 'YTickLabel',[]); end if out_i == 3 xlabel('Frequency [Hz]') else set(gca, 'XTickLabel',[]); end end end %% Jacobian Decoupling - Center of Mass G_CoM = pinv(J_CoM)*G_L*pinv(J_CoM'); G_CoM.InputName = {'Fx', 'Fy', 'Mz'}; G_CoM.OutputName = {'Dx', 'Dy', 'Rz'}; figure; hold on; plot(freqs, abs(squeeze(freqresp(G_CoM(1, 3), freqs, 'Hz'))), 'color', [0,0,0,0.2], ... 'DisplayName', '$D_{x,\{M\}}/M_{z,\{M\}}$'); plot(freqs, abs(squeeze(freqresp(G_CoM(3, 1), freqs, 'Hz'))), 'color', [0,0,0,0.2], ... 'DisplayName', '$R_{z,\{M\}}/F_{x,\{M\}}$'); plot(freqs, abs(squeeze(freqresp(G_CoM(1, 1), freqs, 'Hz'))), 'color', colors(1,:), 'DisplayName', '$D_{x,\{M\}}/F_{x,\{M\}}$'); plot(freqs, abs(squeeze(freqresp(G_CoM(2, 2), freqs, 'Hz'))), 'color', colors(2,:), 'DisplayName', '$D_{y,\{M\}}/F_{y,\{M\}}$'); plot(freqs, abs(squeeze(freqresp(G_CoM(3, 3), freqs, 'Hz'))), 'color', colors(3,:), 'DisplayName', '$R_{z,\{M\}}/M_{z,\{M\}}$'); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); xlabel('Frequency [Hz]'); ylabel('Magnitude'); ylim([1e-10, 1e-3]); leg = legend('location', 'southwest', 'FontSize', 8); leg.ItemTokenSize(1) = 18; %% Jacobian Decoupling - Center of Mass % Location of the joints with respect to the center of stiffness Mb1 = [-l/2; 0]; Mb2 = [-la; -h/2+ha]; Mb3 = [ la; -h/2+ha]; % Jacobian matrix (Center of Stiffness) J_CoK = [s1', Mb1(1)*s1(2)-Mb1(2)*s1(1); s2', Mb2(1)*s2(2)-Mb2(2)*s2(1); s3', Mb3(1)*s3(2)-Mb3(2)*s3(1)]; G_CoK = pinv(J_CoK)*G_L*pinv(J_CoK'); G_CoK.InputName = {'Fx', 'Fy', 'Mz'}; G_CoK.OutputName = {'Dx', 'Dy', 'Rz'}; figure; hold on; plot(freqs, abs(squeeze(freqresp(G_CoK(1, 1), freqs, 'Hz'))), 'color', colors(1,:), 'DisplayName', '$D_{x,\{K\}}/F_{x,\{K\}}$'); plot(freqs, abs(squeeze(freqresp(G_CoK(2, 2), freqs, 'Hz'))), 'color', colors(2,:), 'DisplayName', '$D_{y,\{K\}}/F_{y,\{K\}}$'); plot(freqs, abs(squeeze(freqresp(G_CoK(3, 3), freqs, 'Hz'))), 'color', colors(3,:), 'DisplayName', '$R_{z,\{K\}}/M_{z,\{K\}}$'); plot(freqs, abs(squeeze(freqresp(G_CoK(1, 3), freqs, 'Hz'))), 'color', [0,0,0,0.2], ... 'DisplayName', '$D_{x,\{K\}}/M_{z,\{K\}}$'); plot(freqs, abs(squeeze(freqresp(G_CoK(3, 1), freqs, 'Hz'))), 'color', [0,0,0,0.2], ... 'DisplayName', '$R_{z,\{K\}}/F_{x,\{K\}}$'); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); xlabel('Frequency [Hz]'); ylabel('Kagnitude'); ylim([1e-10, 1e-3]); leg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2); leg.ItemTokenSize(1) = 18; %% Modal decoupling % Compute the eigen vectors [phi, wi] = eig(M\K); % Sort the eigen vectors by increasing associated frequency [~, i] = sort(diag(wi)); phi = phi(:, i); % Plant in the modal space Gm = inv(phi)*inv(J_CoM)*G_L*inv(J_CoM')*inv(phi'); %% Modal decoupled plant figure; hold on; plot(freqs, abs(squeeze(freqresp(Gm(1,1), freqs, 'Hz'))), 'color', colors(1,:), 'DisplayName', '$\mathcal{X}_{m,1}/\tau_{m,1}$'); plot(freqs, abs(squeeze(freqresp(Gm(2,2), freqs, 'Hz'))), 'color', colors(2,:), 'DisplayName', '$\mathcal{X}_{m,2}/\tau_{m,2}$'); plot(freqs, abs(squeeze(freqresp(Gm(3,3), freqs, 'Hz'))), 'color', colors(3,:), 'DisplayName', '$\mathcal{X}_{m,3}/\tau_{m,3}$'); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); xlabel('Frequency [Hz]'); ylabel('Magnitude'); ylim([1e-8, 1e-4]); leg = legend('location', 'northeast', 'FontSize', 8); leg.ItemTokenSize(1) = 18; %% SVD Decoupling wc = 2*pi*100; % Decoupling frequency [rad/s] % System's response at the decoupling frequency H1 = evalfr(G_L, j*wc); % Real approximation of G(j.wc) D = pinv(real(H1'*H1)); H1 = pinv(D*real(H1'*diag(exp(j*angle(diag(H1*D*H1.'))/2)))); [U,S,V] = svd(H1); Gsvd = inv(U)*G_L*inv(V'); figure; hold on; for i_in = 1:3 for i_out = [i_in+1:3] plot(freqs, abs(squeeze(freqresp(Gsvd(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ... 'HandleVisibility', 'off'); end end plot(freqs, abs(squeeze(freqresp(Gsvd(1, 2), freqs, 'Hz'))), 'color', [0,0,0,0.2], ... 'DisplayName', '$G_{SVD}(i,j)\ i \neq j$'); set(gca,'ColorOrderIndex',1) for i_in_out = 1:3 plot(freqs, abs(squeeze(freqresp(Gsvd(i_in_out, i_in_out), freqs, 'Hz'))), 'DisplayName', sprintf('$G_{SVD}(%d,%d)$', i_in_out, i_in_out)); end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); xlabel('Frequency [Hz]'); ylabel('Magnitude'); ylim([1e-10, 2e-4]); leg = legend('location', 'northeast', 'FontSize', 8); leg.ItemTokenSize(1) = 18; %% Simscape model with relative motion sensor at alternative positions mdl = 'detail_control_decoupling_test_model'; open(mdl) deq = 0.2; % Length of the actuators [m] % Input/Output definition clear io; io_i = 1; io(io_i) = linio([mdl, '/F1'], 1, 'openinput'); io_i = io_i + 1; io(io_i) = linio([mdl, '/F2'], 1, 'openinput'); io_i = io_i + 1; io(io_i) = linio([mdl, '/F3'], 1, 'openinput'); io_i = io_i + 1; io(io_i) = linio([mdl, '/Payload'], 1, 'openoutput'); io_i = io_i + 1; io(io_i) = linio([mdl, '/Payload'], 2, 'openoutput'); io_i = io_i + 1; io(io_i) = linio([mdl, '/Payload'], 3, 'openoutput'); io_i = io_i + 1; G_L_alt = linearize(mdl, io); G_L_alt.InputName = {'F1', 'F2', 'F3'}; G_L_alt.OutputName = {'d1', 'd2', 'd32'}; % SVD Decoupling with the new plant wc = 2*pi*100; % Decoupling frequency [rad/s] % System's response at the decoupling frequency H1 = evalfr(G_L_alt, j*wc); % Real approximation of G(j.wc) D = pinv(real(H1'*H1)); H1 = pinv(D*real(H1'*diag(exp(j*angle(diag(H1*D*H1.'))/2)))); [U,S,V] = svd(H1); Gsvd_alt = inv(U)*G_L_alt*inv(V'); %% Obtained plant after SVD decoupling - Relative motion sensors are not collocated with the actuators figure; hold on; for i_in = 1:3 for i_out = [i_in+1:3] plot(freqs, abs(squeeze(freqresp(Gsvd_alt(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ... 'HandleVisibility', 'off'); end end plot(freqs, abs(squeeze(freqresp(Gsvd_alt(1, 2), freqs, 'Hz'))), 'color', [0,0,0,0.2], ... 'DisplayName', '$G_{SVD}(i,j)\ i \neq j$'); set(gca,'ColorOrderIndex',1) for i_in_out = 1:3 plot(freqs, abs(squeeze(freqresp(Gsvd_alt(i_in_out, i_in_out), freqs, 'Hz'))), 'DisplayName', sprintf('$G_{SVD}(%d,%d)$', i_in_out, i_in_out)); end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); xlabel('Frequency [Hz]'); ylabel('Magnitude'); ylim([5e-11, 7e-5]); leg = legend('location', 'southwest', 'FontSize', 8); leg.ItemTokenSize(1) = 18;