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224
A5-simscape-nano-hexapod/nhexa_2_model.m
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224
A5-simscape-nano-hexapod/nhexa_2_model.m
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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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%% Path for functions, data and scripts
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addpath('./mat/'); % Path for Data
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addpath('./src/'); % Path for functions
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addpath('./subsystems/'); % Path for Subsystems Simulink files
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%% Data directory
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data_dir = './mat/';
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% Simulink Model name
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mdl = 'nano_hexapod_model';
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%% Colors for the figures
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colors = colororder;
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%% Frequency Vector [Hz]
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freqs = logspace(0, 3, 1000);
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%% Plant using Analytical Equations
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% Stewart platform definition
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k = 1e6; % Actuator stiffness [N/m]
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c = 1e1; % Actuator damping [N/(m/s)]
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stewart = initializeSimplifiedNanoHexapod(...
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'Mpm', 1e-3, ...
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'actuator_type', '1dof', ...
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'actuator_k', k, ...
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'actuator_kp', 0, ...
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'actuator_c', c ...
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);
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% Payload: Cylinder
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h = 300e-3; % Height of the cylinder [m]
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r = 110e-3; % Radius of the cylinder [m]
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m = 10; % Mass of the payload [kg]
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initializeSample('type', 'cylindrical', 'm', m, 'H', h, 'R', r);
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% Mass Matrix
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M = zeros(6,6);
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M(1,1) = m;
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M(2,2) = m;
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M(3,3) = m;
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M(4,4) = 1/12*m*(3*r^2 + h^2);
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M(5,5) = 1/12*m*(3*r^2 + h^2);
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M(6,6) = 1/2*m*r^2;
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% Stiffness and Damping matrices
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K = k*eye(6);
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C = c*eye(6);
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% Compute plant in the frame of the struts
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G_analytical = inv(ss(inv(stewart.geometry.J')*M*inv(stewart.geometry.J)*s^2 + C*s + K));
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% Compare with Simscape model
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initializeLoggingConfiguration('log', 'none');
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initializeController('type', 'open-loop');
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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 Inputs [N]
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io(io_i) = linio([mdl, '/plant'], 2, 'openoutput', [], 'dL'); io_i = io_i + 1; % Encoders [m]
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G_simscape = linearize(mdl, io);
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G_simscape.InputName = {'f1', 'f2', 'f3', 'f4', 'f5', 'f6'};
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G_simscape.OutputName = {'dL1', 'dL2', 'dL3', 'dL4', 'dL5', 'dL6'};
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%% Comparison of the analytical transfer functions and the multi-body model
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figure;
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tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
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ax1 = nexttile([2,1]);
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hold on;
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for i = 1:6
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plot(freqs, abs(squeeze(freqresp(G_simscape(i,1), freqs, 'Hz'))), 'color', [colors(i,:), 0.5], 'linewidth', 2.5, ...
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'DisplayName', sprintf('$l_%i/f_1$ - Multi-Body', i))
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end
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for i = 1:6
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plot(freqs, abs(squeeze(freqresp(G_analytical(i,1), freqs, 'Hz'))), '--', 'color', [colors(i,:)], ...
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'DisplayName', sprintf('$l_%i/f_1$ - Analytical', i))
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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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ylim([1e-9, 1e-4]);
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leg = legend('location', 'northwest', 'FontSize', 6, 'NumColumns', 1);
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leg.ItemTokenSize(1) = 15;
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ax2 = nexttile;
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hold on;
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for i = 1:6
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plot(freqs, 180/pi*angle(squeeze(freqresp(G_simscape(i,1), freqs, 'Hz'))), 'color', [colors(i,:),0.5], 'linewidth', 2.5);
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end
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for i = 1:6
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plot(freqs, 180/pi*angle(squeeze(freqresp(G_analytical(i,1), freqs, 'Hz'))), '--', 'color', colors(i,:));
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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([-180, -90, 0, 90, 180]);
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linkaxes([ax1,ax2],'x');
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xlim([freqs(1), freqs(end)]);
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%% Multi-Body model of the Nano-Hexapod
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% Initialize 1DoF
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initializeSimplifiedNanoHexapod('flex_type_F', '2dof', 'flex_type_M', '3dof', 'actuator_type', '1dof');
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initializeSample('type', 'cylindrical', 'm', 10, 'H', 300e-3);
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initializeLoggingConfiguration('log', 'none');
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initializeController('type', 'open-loop');
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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 Inputs [N]
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io(io_i) = linio([mdl, '/plant'], 2, 'openoutput', [], 'dL'); io_i = io_i + 1; % Encoders [m]
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io(io_i) = linio([mdl, '/plant'], 2, 'openoutput', [], 'fn'); io_i = io_i + 1; % Force Sensors [N]
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% With no payload
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G = linearize(mdl, io);
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G.InputName = {'f1', 'f2', 'f3', 'f4', 'f5', 'f6'};
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G.OutputName = {'dL1', 'dL2', 'dL3', 'dL4', 'dL5', 'dL6', ...
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'fn1', 'fn2', 'fn3', 'fn4', 'fn5', 'fn6'};
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%% Multi-Body model of the Nano-Hexapod without parallel stiffness
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% Initialize 1DoF
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initializeSimplifiedNanoHexapod('flex_type_F', '2dof', 'flex_type_M', '3dof', 'actuator_type', '1dof', 'actuator_kp', 0);
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% With no payload
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G_no_kp = linearize(mdl, io);
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G_no_kp.InputName = {'f1', 'f2', 'f3', 'f4', 'f5', 'f6'};
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G_no_kp.OutputName = {'dL1', 'dL2', 'dL3', 'dL4', 'dL5', 'dL6', ...
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'fn1', 'fn2', 'fn3', 'fn4', 'fn5', 'fn6'};
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%% Transfer function from actuator force inputs to displacement of each strut
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figure;
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tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
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ax1 = nexttile([2,1]);
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hold on;
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for i = 1:5
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for j = i+1:6
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plot(freqs, abs(squeeze(freqresp(G(i,j), 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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plot(freqs, abs(squeeze(freqresp(G(1,1), freqs, 'Hz'))), 'color', colors(1,:), ...
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'DisplayName', '$l_i/f_i$')
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for i = 2:6
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plot(freqs, abs(squeeze(freqresp(G(i,i), freqs, 'Hz'))), 'color', colors(1,:), ...
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'HandleVisibility', 'off');
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end
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plot(freqs, abs(squeeze(freqresp(G(1,2), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ...
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'DisplayName', '$l_i/f_j$')
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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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ylim([1e-9, 1e-4]);
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leg = legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 1);
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leg.ItemTokenSize(1) = 15;
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ax2 = nexttile;
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hold on;
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for i = 1:6
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plot(freqs, 180/pi*angle(squeeze(freqresp(G(i,i), freqs, 'Hz'))), 'color', [colors(1,:),0.5]);
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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([-180, -90, 0, 90, 180]);
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linkaxes([ax1,ax2],'x');
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xlim([freqs(1), freqs(end)]);
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%% Transfer function from actuator force inputs to force sensor in each strut
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figure;
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tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
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ax1 = nexttile([2,1]);
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hold on;
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for i = 1:5
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for j = i+1:6
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plot(freqs, abs(squeeze(freqresp(G(6+i,j), 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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plot(freqs, abs(squeeze(freqresp(G(7,1), freqs, 'Hz'))), 'color', colors(1,:), ...
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'DisplayName', '$f_{ni}/f_i$')
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plot(freqs, abs(squeeze(freqresp(G_no_kp(7,1), freqs, 'Hz'))), 'color', colors(2,:), ...
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'DisplayName', '$f_{ni}/f_i$ (no $k_p$)')
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for i = 2:6
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plot(freqs, abs(squeeze(freqresp(G(6+i,i), freqs, 'Hz'))), 'color', colors(1,:), ...
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'HandleVisibility', 'off');
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plot(freqs, abs(squeeze(freqresp(G_no_kp(6+i,i), freqs, 'Hz'))), 'color', colors(2,:), ...
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'HandleVisibility', 'off');
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end
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plot(freqs, abs(squeeze(freqresp(G(7,2), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ...
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'DisplayName', '$f_{ni}/f_j$')
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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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ylim([1e-4, 1e2]);
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leg = legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 1);
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leg.ItemTokenSize(1) = 15;
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ax2 = nexttile;
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hold on;
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for i = 1:6
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plot(freqs, 180/pi*angle(squeeze(freqresp(G(6+i,i), freqs, 'Hz'))), 'color', colors(1,:));
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plot(freqs, 180/pi*angle(squeeze(freqresp(G_no_kp(6+i,i), freqs, 'Hz'))), 'color', colors(2,:));
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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([-180, -90, 0, 90, 180]);
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linkaxes([ax1,ax2],'x');
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xlim([freqs(1), freqs(end)]);
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