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Thomas Dehaeze 2025-02-12 11:42:04 +01:00
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matlab/mat/nano_hexapod.mat Normal file
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matlab/mat/nano_hexapod_model_conf_log.mat Normal file
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matlab/mat/nano_hexapod_model_controller.mat Normal file
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matlab/nano_hexapod_model.slx
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matlab/nhexa_1_stewart_platform.m Normal file
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%% Clear Workspace and Close figures
clear; close all; clc;
%% Intialize Laplace variable
s = zpk('s');
%% Path for functions, data and scripts
addpath('./mat/'); % Path for Data
addpath('./src/'); % Path for functions
addpath('./subsystems/'); % Path for Subsystems Simulink files
%% Data directory
data_dir = './mat/';
% Simulink Model name
mdl = 'nano_hexapod_model';
%% Colors for the figures
colors = colororder;
%% Frequency Vector
freqs = logspace(0, 3, 1000);
% Range validity of the approximate inverse kinematics
% The accuracy of the Jacobian-based forward kinematics solution was estimated through a systematic error analysis.
% For a series of platform positions along the $x\text{-axis}$, the exact strut lengths are computed using the analytical inverse kinematics equation eqref:eq:nhexa_inverse_kinematics.
% These strut lengths are then used with the Jacobian to estimate the platform pose, from which the error between the estimated and true poses can be calculated.
% The estimation errors in the $x$, $y$, and $z$ directions are shown in Figure ref:fig:nhexa_forward_kinematics_approximate_errors.
% The results demonstrate that for displacements up to approximately $1\,\%$ of the hexapod's size (which corresponds to $100\,\mu m$ as the size of the Stewart platform is here $\approx 100\,mm$), the Jacobian approximation provides excellent accuracy.
% This finding has particular significance for the Nano-hexapod application.
% Since the maximum required stroke ($\approx 100\,\mu m$) is three orders of magnitude smaller than the stewart platform size ($\approx 100\,mm$), the Jacobian matrix can be considered constant throughout the workspace.
% It can be computed once at the rest position and used for both forward and inverse kinematics with high accuracy.
%% Estimate the errors associated with approximate forward kinematics using the Jacobian matrix
stewart = initializeSimplifiedNanoHexapod('H', 100e-3, 'MO_B', 0);
Xrs = logspace(-6, -1, 100); % Wanted X translation of the mobile platform [m]
% Compute the strut exact length for each X-position
Ls_exact = zeros(6, length(Xrs));
for i = 1:length(Xrs)
[~, L_exact(:, i)] = inverseKinematics(stewart, 'AP', [Xrs(i); 0; 0]);
end
% From the strut length, compute the stewart pose using the Jacobian matrix
Xrs_approx = zeros(6, length(Xrs));
for i = 1:length(Xrs)
Xrs_approx(:, i) = inv(stewart.geometry.J)*L_exact(:, i);
end
%% Errors associated with the use of the Jacobian matrix to solve the forward kinematic problem
figure;
hold on;
plot(1e6*Xrs, 1e9*abs(Xrs - Xrs_approx(1, :)), 'DisplayName', '$\epsilon_x$');
plot(1e6*Xrs, 1e9*abs(Xrs_approx(2, :)), 'DisplayName', '$\epsilon_y$');
plot(1e6*Xrs, 1e9*abs(Xrs_approx(3, :)), 'DisplayName', '$\epsilon_z$');
plot(1e6*Xrs, 1e6*Xrs, 'k--', 'DisplayName', '$0.1\%$ error');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
leg = legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 1);
leg.ItemTokenSize(1) = 15;
xlim([1, 1e4]); ylim([1, 1e4]);
xlabel('$D_x$ motion');
ylabel('Kinematic Errors');
xticks([1, 10, 100, 1000, 10000]);
yticks([1, 10, 100, 1000, 10000]);
xticklabels({'$1\mu m$', '$10\mu m$', '$100\mu m$', '$1mm$', '$10mm$'});
yticklabels({'$1nm$', '$10nm$', '$100nm$', '$1\mu m$', '$10\mu m$'});

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%% Clear Workspace and Close figures
clear; close all; clc;
%% Intialize Laplace variable
s = zpk('s');
%% Path for functions, data and scripts
addpath('./mat/'); % Path for Data
addpath('./src/'); % Path for functions
addpath('./subsystems/'); % Path for Subsystems Simulink files
%% Data directory
data_dir = './mat/';
% Simulink Model name
mdl = 'nano_hexapod_model';
%% Colors for the figures
colors = colororder;
%% Frequency Vector
freqs = logspace(0, 3, 1000);
% Validation of the multi-body model
% <<ssec:nhexa_model_validation>>
% The developed multi-body model of the Stewart platform is represented schematically in Figure ref:fig:nhexa_stewart_model_input_outputs, highlighting the key inputs and outputs: actuator forces $\bm{f}$, force sensor measurements $\bm{f}_n$, and relative displacement measurements $\bm{\mathcal{L}}$.
% The frames $\{F\}$ and $\{M\}$ serve as interfaces for integration with other elements in the multi-body system.
% A three-dimensional visualization of the model is presented in Figure ref:fig:nhexa_simscape_screenshot.
% #+attr_latex: :options [b]{0.6\linewidth}
% #+begin_minipage
% #+name: fig:nhexa_stewart_model_input_outputs
% #+caption: Nano-Hexapod plant with inputs and outputs. Frames $\{F\}$ and $\{M\}$ can be connected to other elements in the multi-body models.
% #+attr_latex: :scale 1 :float nil
% [[file:figs/nhexa_stewart_model_input_outputs.png]]
% #+end_minipage
% \hfill
% #+attr_latex: :options [b]{0.35\linewidth}
% #+begin_minipage
% #+name: fig:nhexa_simscape_screenshot
% #+caption: 3D representation of the multi-body model
% #+attr_latex: :width 0.90\linewidth :float nil
% [[file:figs/nhexa_simscape_screenshot.jpg]]
% #+end_minipage
% The validation of the multi-body model is performed using the simplest Stewart platform configuration, enabling direct comparison with the analytical transfer functions derived in Section ref:ssec:nhexa_stewart_platform_dynamics.
% This configuration consists of massless universal joints at the base, massless spherical joints at the top platform, and massless struts with stiffness $k_a = 1\,\text{N}/\mu\text{m}$ and damping $c_a = 10\,\text{N}/({\text{m}/\text{s}})$.
% The geometric parameters remain as specified in Table ref:tab:nhexa_actuator_parameters.
% While the moving platform itself is considered massless, a $10\,\text{kg}$ cylindrical payload is mounted on top with a radius of $r = 110\,mm$ and a height $h = 300\,mm$.
% For the analytical model, the stiffness, damping and mass matrices are defined in eqref:eq:nhexa_analytical_matrices.
% \begin{subequations}\label{eq:nhexa_analytical_matrices}
% \begin{align}
% \bm{\mathcal{K}} &= \text{diag}(k_a,\ k_a,\ k_a,\ k_a,\ k_a,\ k_a) \\
% \bm{\mathcal{C}} &= \text{diag}(c_a,\ c_a,\ c_a,\ c_a,\ c_a,\ c_a) \\
% \bm{M} &= \text{diag}\left(m,\ m,\ m,\ \frac{1}{12}m(3r^2 + h^2),\ \frac{1}{12}m(3r^2 + h^2),\ \frac{1}{2}mr^2\right)
% \end{align}
% \end{subequations}
% The transfer functions from actuator forces to strut displacements are computed using these matrices according to equation eqref:eq:nhexa_transfer_function_struts.
% These analytical transfer functions are then compared with those extracted from the multi-body model.
% The multi-body model yields a state-space representation with 12 states, corresponding to the six degrees of freedom of the moving platform.
% Figure ref:fig:nhexa_comp_multi_body_analytical presents a comparison between the analytical and multi-body transfer functions, specifically showing the response from the first actuator force to all six strut displacements.
% The close agreement between both approaches across the frequency spectrum validates the multi-body model's accuracy in capturing the system's dynamic behavior.
%% Plant using Analytical Equations
% Stewart platform definition
k = 1e6; % Actuator stiffness [N/m]
c = 1e1; % Actuator damping [N/(m/s)]
stewart = initializeSimplifiedNanoHexapod(...
'Mpm', 1e-3, ...
'actuator_type', '1dof', ...
'actuator_k', k, ...
'actuator_kp', 0, ...
'actuator_c', c ...
);
% Payload: Cylinder
h = 300e-3; % Height of the cylinder [m]
r = 110e-3; % Radius of the cylinder [m]
m = 10; % Mass of the payload [kg]
initializeSample('type', 'cylindrical', 'm', m, 'H', h, 'R', r);
% Mass Matrix
M = zeros(6,6);
M(1,1) = m;
M(2,2) = m;
M(3,3) = m;
M(4,4) = 1/12*m*(3*r^2 + h^2);
M(5,5) = 1/12*m*(3*r^2 + h^2);
M(6,6) = 1/2*m*r^2;
% Stiffness and Damping matrices
K = k*eye(6);
C = c*eye(6);
% Compute plant in the frame of the struts
G_analytical = inv(ss(inv(stewart.geometry.J')*M*inv(stewart.geometry.J)*s^2 + C*s + K));
% Compare with Simscape model
initializeLoggingConfiguration('log', 'none');
initializeController('type', 'open-loop');
% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Controller'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs [N]
io(io_i) = linio([mdl, '/plant'], 2, 'openoutput', [], 'dL'); io_i = io_i + 1; % Encoders [m]
G_simscape = linearize(mdl, io);
G_simscape.InputName = {'f1', 'f2', 'f3', 'f4', 'f5', 'f6'};
G_simscape.OutputName = {'dL1', 'dL2', 'dL3', 'dL4', 'dL5', 'dL6'};
%% Comparison of the analytical transfer functions and the multi-body model
figure;
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
for i = 1:6
plot(freqs, abs(squeeze(freqresp(G_simscape(i,1), freqs, 'Hz'))), 'color', [colors(i,:), 0.5], ...
'DisplayName', sprintf('$l_%i/f_1$ - Multi-Body', i))
end
for i = 1:6
plot(freqs, abs(squeeze(freqresp(G_analytical(i,1), freqs, 'Hz'))), '--', 'color', [colors(i,:)], ...
'DisplayName', sprintf('$l_%i/f_1$ - Analytical', i))
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
ylim([1e-9, 1e-4]);
leg = legend('location', 'northwest', 'FontSize', 6, 'NumColumns', 1);
leg.ItemTokenSize(1) = 15;
ax2 = nexttile;
hold on;
for i = 1:6
plot(freqs, 180/pi*angle(squeeze(freqresp(G_simscape(i,1), freqs, 'Hz'))), 'color', [colors(i,:),0.5]);
end
for i = 1:6
plot(freqs, 180/pi*angle(squeeze(freqresp(G_analytical(i,1), freqs, 'Hz'))), '--', 'color', colors(i,:));
end
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]);
linkaxes([ax1,ax2],'x');
xlim([freqs(1), freqs(end)]);
% Nano Hexapod Dynamics
% <<ssec:nhexa_model_dynamics>>
% Following the validation of the multi-body model, a detailed analysis of the nano-hexapod dynamics has been performed.
% The model parameters are set according to the specifications outlined in Section ref:ssec:nhexa_model_def, with a payload mass of $10\,kg$.
% Transfer functions from actuator forces $\bm{f}$ to both strut displacements $\bm{\mathcal{L}}$ and force measurements $\bm{f}_n$ are derived from the multi-body model.
% The transfer functions relating actuator forces to strut displacements are presented in Figure ref:fig:nhexa_multi_body_plant_dL.
% Due to the system's symmetrical design and identical strut configurations, all diagonal terms (transfer functions from force $f_i$ to displacement $l_i$ of the same strut) exhibit identical behavior.
% While the system possesses six degrees of freedom, only four distinct resonance frequencies are observed in the frequency response.
% This reduction from six to four observable modes is attributed to the system's symmetry, where two pairs of resonances occur at identical frequencies.
% The system's behavior can be characterized in three frequency regions.
% At low frequencies, well below the first resonance, the plant demonstrates good decoupling between actuators, with the response dominated by the strut stiffness: $\bm{G}(j\omega) \xrightarrow[\omega \to 0]{} \bm{\mathcal{K}}^{-1}$.
% In the mid-frequency range, the system exhibits coupled dynamics through its resonant modes, reflecting the complex interactions between the platform's degrees of freedom.
% At high frequencies, above the highest resonance, the response is governed by the payload's inertia mapped to the strut coordinates: $\bm{G}(j\omega) \xrightarrow[\omega \to \infty]{} \bm{J} \bm{M}^{-T} \bm{J}^T \frac{-1}{\omega^2}$
% The force sensor transfer functions, shown in Figure ref:fig:nhexa_multi_body_plant_fm, display characteristics typical of collocated actuator-sensor pairs.
% Each actuator's transfer function to its associated force sensor exhibits alternating complex conjugate poles and zeros.
% The inclusion of parallel stiffness introduces an additional complex conjugate zero at low frequency, a feature previously observed in the three-degree-of-freedom rotating model.
%% Multi-Body model of the Nano-Hexapod
% Initialize 1DoF
initializeSimplifiedNanoHexapod('flex_type_F', '2dof', 'flex_type_M', '3dof', 'actuator_type', '1dof');
initializeSample('type', 'cylindrical', 'm', 10, 'H', 300e-3);
initializeLoggingConfiguration('log', 'none');
initializeController('type', 'open-loop');
% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Controller'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs [N]
io(io_i) = linio([mdl, '/plant'], 2, 'openoutput', [], 'dL'); io_i = io_i + 1; % Encoders [m]
io(io_i) = linio([mdl, '/plant'], 2, 'openoutput', [], 'fn'); io_i = io_i + 1; % Force Sensors [N]
% With no payload
G = linearize(mdl, io);
G.InputName = {'f1', 'f2', 'f3', 'f4', 'f5', 'f6'};
G.OutputName = {'dL1', 'dL2', 'dL3', 'dL4', 'dL5', 'dL6', ...
'fn1', 'fn2', 'fn3', 'fn4', 'fn5', 'fn6'};
%% Multi-Body model of the Nano-Hexapod without parallel stiffness
% Initialize 1DoF
initializeSimplifiedNanoHexapod('flex_type_F', '2dof', 'flex_type_M', '3dof', 'actuator_type', '1dof', 'actuator_kp', 0);
% With no payload
G_no_kp = linearize(mdl, io);
G_no_kp.InputName = {'f1', 'f2', 'f3', 'f4', 'f5', 'f6'};
G_no_kp.OutputName = {'dL1', 'dL2', 'dL3', 'dL4', 'dL5', 'dL6', ...
'fn1', 'fn2', 'fn3', 'fn4', 'fn5', 'fn6'};
%% Transfer function from actuator force inputs to displacement of each strut
figure;
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
for i = 1:5
for j = i+1:6
plot(freqs, abs(squeeze(freqresp(G(i,j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ...
'HandleVisibility', 'off');
end
end
plot(freqs, abs(squeeze(freqresp(G(1,1), freqs, 'Hz'))), 'color', colors(1,:), ...
'DisplayName', '$l_i/f_i$')
for i = 2:6
plot(freqs, abs(squeeze(freqresp(G(i,i), freqs, 'Hz'))), 'color', colors(1,:), ...
'HandleVisibility', 'off');
end
plot(freqs, abs(squeeze(freqresp(G(1,2), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ...
'DisplayName', '$l_i/f_j$')
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
ylim([1e-9, 1e-4]);
leg = legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 1);
leg.ItemTokenSize(1) = 15;
ax2 = nexttile;
hold on;
for i = 1:6
plot(freqs, 180/pi*angle(squeeze(freqresp(G(i,i), freqs, 'Hz'))), 'color', [colors(1,:),0.5]);
end
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]);
linkaxes([ax1,ax2],'x');
xlim([freqs(1), freqs(end)]);
%% Transfer function from actuator force inputs to force sensor in each strut
figure;
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
for i = 1:5
for j = i+1:6
plot(freqs, abs(squeeze(freqresp(G(6+i,j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ...
'HandleVisibility', 'off');
end
end
plot(freqs, abs(squeeze(freqresp(G(7,1), freqs, 'Hz'))), 'color', colors(1,:), ...
'DisplayName', '$f_{ni}/f_i$')
plot(freqs, abs(squeeze(freqresp(G_no_kp(7,1), freqs, 'Hz'))), 'color', colors(2,:), ...
'DisplayName', '$f_{ni}/f_i$ (no $k_p$)')
for i = 2:6
plot(freqs, abs(squeeze(freqresp(G(6+i,i), freqs, 'Hz'))), 'color', colors(1,:), ...
'HandleVisibility', 'off');
plot(freqs, abs(squeeze(freqresp(G_no_kp(6+i,i), freqs, 'Hz'))), 'color', colors(2,:), ...
'HandleVisibility', 'off');
end
plot(freqs, abs(squeeze(freqresp(G(7,2), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ...
'DisplayName', '$f_{ni}/f_j$')
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [N/N]'); set(gca, 'XTickLabel',[]);
ylim([1e-4, 1e2]);
leg = legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 1);
leg.ItemTokenSize(1) = 15;
ax2 = nexttile;
hold on;
for i = 1:6
plot(freqs, 180/pi*angle(squeeze(freqresp(G(6+i,i), freqs, 'Hz'))), 'color', colors(1,:));
plot(freqs, 180/pi*angle(squeeze(freqresp(G_no_kp(6+i,i), freqs, 'Hz'))), 'color', colors(2,:));
end
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]);
linkaxes([ax1,ax2],'x');
xlim([freqs(1), freqs(end)]);

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matlab/nhexa_3_control.m Normal file
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%% Clear Workspace and Close figures
clear; close all; clc;
%% Intialize Laplace variable
s = zpk('s');
%% Path for functions, data and scripts
addpath('./mat/'); % Path for Data
addpath('./src/'); % Path for functions
addpath('./subsystems/'); % Path for Subsystems Simulink files
%% Data directory
data_dir = './mat/';
% Simulink Model name
mdl = 'nano_hexapod_model';
%% Colors for the figures
colors = colororder;
%% Frequency Vector
freqs = logspace(0, 3, 1000);
% Control in Cartesian Space
% Alternatively, control can be implemented directly in Cartesian space, as shown in Figure ref:fig:nhexa_control_cartesian.
% Here, the controller processes Cartesian errors $\bm{\epsilon}_{\mathcal{X}}$ to generate forces and torques $\bm{\mathcal{F}}$, which are then mapped to actuator forces through the transpose of the inverse Jacobian matrix.
% The plant behavior in Cartesian space, illustrated in Figure ref:fig:nhexa_plant_frame_cartesian, reveals interesting characteristics.
% Some degrees of freedom, particularly the vertical translation and rotation about the vertical axis, exhibit simpler second-order dynamics.
% A key advantage of this approach is that control performance can be individually tuned for each direction.
% This is particularly valuable when performance requirements differ between degrees of freedom - for instance, when higher positioning accuracy is required vertically than horizontally, or when certain rotational degrees of freedom can tolerate larger errors than others.
% However, significant coupling exists between certain degrees of freedom, particularly between rotations and translations (e.g., $\epsilon_{R_x}/\mathcal{F}_y$ or $\epsilon_{D_y}/\bm\mathcal{M}_x$).
% For the conceptual validation of the nano-hexapod, control in the strut space has been selected due to its simpler implementation and the beneficial decoupling properties observed at low frequencies.
% More sophisticated control strategies will be explored during the detailed design phase.
%% Identify plant from actuator forces to external metrology
stewart = initializeSimplifiedNanoHexapod();
initializeSample('type', 'cylindrical', 'm', 10, 'H', 300e-3);
initializeLoggingConfiguration('log', 'none');
initializeController('type', 'open-loop');
% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Controller'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs [N]
io(io_i) = linio([mdl, '/plant'], 1, 'openoutput'); io_i = io_i + 1; % External Metrology [m, rad]
% With no payload
G = linearize(mdl, io);
G.InputName = {'f1', 'f2', 'f3', 'f4', 'f5', 'f6'};
G.OutputName = {'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'};
%% Plant in the Cartesian Frame
G_cart = G*inv(stewart.geometry.J');
G_cart.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
%% Plant in the frame of the struts
G_struts = stewart.geometry.J*G;
G_struts.OutputName = {'D1', 'D2', 'D3', 'D4', 'D5', 'D6'};
%% Bode plot of the plant projected in the frame of the struts
figure;
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
for i = 1:5
for j = i+1:6
plot(freqs, abs(squeeze(freqresp(G_struts(i,j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ...
'HandleVisibility', 'off');
end
end
plot(freqs, abs(squeeze(freqresp(G_struts(1,1), freqs, 'Hz'))), 'color', colors(1,:), ...
'DisplayName', '$-\epsilon_{\mathcal{L}i}/f_i$')
for i = 2:6
plot(freqs, abs(squeeze(freqresp(G_struts(i,i), freqs, 'Hz'))), 'color', colors(1,:), ...
'HandleVisibility', 'off');
end
plot(freqs, abs(squeeze(freqresp(G_struts(1,2), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ...
'DisplayName', '$-\epsilon_{\mathcal{L}i}/f_j$')
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
ylim([1e-9, 1e-4]);
leg = legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 1);
leg.ItemTokenSize(1) = 15;
ax2 = nexttile;
hold on;
for i = 1:6
plot(freqs, 180/pi*angle(squeeze(freqresp(G_struts(i,i), freqs, 'Hz'))), 'color', [colors(1,:),0.5]);
end
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]);
linkaxes([ax1,ax2],'x');
xlim([freqs(1), freqs(end)]);
%% Bode plot of the plant projected in the Cartesian frame
figure;
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
for i = 1:5
for j = i+1:6
plot(freqs, abs(squeeze(freqresp(G_cart(i,j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ...
'HandleVisibility', 'off');
end
end
plot(freqs, abs(squeeze(freqresp(G_cart(1,1), freqs, 'Hz'))), 'color', colors(1,:), ...
'DisplayName', '$\epsilon_{D_x}/\mathcal{F}_x$ [m/N]')
plot(freqs, abs(squeeze(freqresp(G_cart(2,2), freqs, 'Hz'))), 'color', colors(2,:), ...
'DisplayName', '$\epsilon_{D_y}/\mathcal{F}_y$ [m/N]')
plot(freqs, abs(squeeze(freqresp(G_cart(3,3), freqs, 'Hz'))), 'color', colors(3,:), ...
'DisplayName', '$\epsilon_{D_z}/\mathcal{F}_z$ [m/N]')
plot(freqs, abs(squeeze(freqresp(G_cart(4,4), freqs, 'Hz'))), 'color', colors(4,:), ...
'DisplayName', '$\epsilon_{R_x}/\mathcal{M}_x$ [rad/Nm]')
plot(freqs, abs(squeeze(freqresp(G_cart(5,5), freqs, 'Hz'))), 'color', colors(5,:), ...
'DisplayName', '$\epsilon_{R_y}/\mathcal{M}_y$ [rad/Nm]')
plot(freqs, abs(squeeze(freqresp(G_cart(6,6), freqs, 'Hz'))), 'color', colors(6,:), ...
'DisplayName', '$\epsilon_{R_z}/\mathcal{M}_z$ [rad/Nm]')
plot(freqs, abs(squeeze(freqresp(G_cart(1,5), freqs, 'Hz'))), 'color', [0, 0, 0, 0.5], ...
'DisplayName', 'Coupling')
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude'); set(gca, 'XTickLabel',[]);
ylim([1e-9, 4e-3]);
leg = legend('location', 'southwest', 'FontSize', 7, 'NumColumns', 3);
leg.ItemTokenSize(1) = 15;
ax2 = nexttile;
hold on;
for i = 1:6
plot(freqs, 180/pi*angle(squeeze(freqresp(G_cart(i,i), freqs, 'Hz'))), 'color', colors(i,:));
end
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]);
linkaxes([ax1,ax2],'x');
xlim([freqs(1), freqs(end)]);
% #+name: fig:nhexa_decentralized_iff_schematic
% #+caption: Schematic of the implemented decentralized IFF controller. The damped plant has a new inputs $\bm{f}^{\prime}$
% #+RESULTS:
% [[file:figs/nhexa_decentralized_iff_schematic.png]]
% \begin{equation}\label{eq:nhexa_kiff}
% \bm{K}_{\text{IFF}}(s) = g \cdot \begin{bmatrix}
% K_{\text{IFF}}(s) & & 0 \\
% & \ddots & \\
% 0 & & K_{\text{IFF}}(s)
% \end{bmatrix}, \quad K_{\text{IFF}}(s) = \frac{1}{s}
% \end{equation}
% In this section, the stiffness in parallel with the force sensor has been omitted since the Stewart platform is not subjected to rotation.
% The effect of this parallel stiffness will be examined in the next section when the platform is integrated into the complete NASS system.
% The Root Locus analysis, shown in Figure ref:fig:nhexa_decentralized_iff_root_locus, reveals the evolution of the closed-loop poles as the controller gain $g$ varies from $0$ to $\infty$.
% A key characteristic of force feedback control with collocated sensor-actuator pairs is observed: all closed-loop poles are bounded to the left-half plane, indicating guaranteed stability [[cite:&preumont08_trans_zeros_struc_contr_with]].
% This property is particularly valuable as the coupling is very large around resonance frequencies, enabling control of modes that would be difficult to include within the bandwidth using position feedback alone.
% The bode plot of an individual loop gain (i.e. the loop gain of $K_{\text{IFF}}(s) \cdot \frac{f_{ni}}{f_i}(s)$), presented in Figure ref:fig:nhexa_decentralized_iff_loop_gain, exhibits the typical characteristics of integral force feedback of having a phase bounded between $-90^o$ and $+90^o$.
% The loop-gain is high around the resonance frequencies, indicating that the decentralized IFF provides significant control authority over these modes.
% This high gain, combined with the bounded phase, enables effective damping of the resonant modes while maintaining stability.
%% Identify the IFF Plant
stewart = initializeSimplifiedNanoHexapod('actuator_kp', 0); % Ignoring parallel stiffness for now
initializeSample('type', 'cylindrical', 'm', 10, 'H', 300e-3);
initializeLoggingConfiguration('log', 'none');
initializeController('type', 'open-loop');
% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Controller'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs [N]
io(io_i) = linio([mdl, '/plant'], 2, 'openoutput', [], 'fn'); io_i = io_i + 1; % Force Sensors [N]
% With no payload
G_iff = linearize(mdl, io);
G_iff.InputName = {'f1', 'f2', 'f3', 'f4', 'f5', 'f6'};
G_iff.OutputName = {'fm1', 'fm2', 'fm3', 'fm4', 'fm5', 'fm6'};
%% IFF Controller Design
Kiff = -500/s * ... % Gain
eye(6); % Diagonal 6x6 controller (i.e. decentralized)
Kiff.InputName = {'fm1', 'fm2', 'fm3', 'fm4', 'fm5', 'fm6'};
Kiff.OutputName = {'f1', 'f2', 'f3', 'f4', 'f5', 'f6'};
%% Root Locus plot of the Decentralized IFF Control
gains = logspace(-2, 1, 200);
figure;
tiledlayout(1, 1, 'TileSpacing', 'compact', 'Padding', 'None');
nexttile();
hold on;
plot(real(pole(G_iff)), imag(pole(G_iff)), 'x', 'color', colors(1,:), ...
'DisplayName', '$g = 0$');
plot(real(tzero(G_iff)), imag(tzero(G_iff)), 'o', 'color', colors(1,:), ...
'HandleVisibility', 'off');
for g = gains
clpoles = pole(feedback(G_iff, g*Kiff, +1));
plot(real(clpoles), imag(clpoles), '.', 'color', colors(1,:), ...
'HandleVisibility', 'off');
end
% Optimal gain
clpoles = pole(feedback(G_iff, Kiff, +1));
plot(real(clpoles), imag(clpoles), 'kx', ...
'DisplayName', '$g_{opt}$');
hold off;
axis equal;
xlim([-600, 50]); ylim([-50, 600]);
xticks([-600:100:0]);
yticks([0:100:600]);
set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]);
xlabel('Real part'); ylabel('Imaginary part');
%% Loop gain for the Decentralized IFF
figure;
tiledlayout(3, 1, 'TileSpacing', 'compact', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
plot(freqs, abs(squeeze(freqresp(-G_iff(1,1)*Kiff(1,1), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Loop Gain'); set(gca, 'XTickLabel',[]);
ylim([1e-2, 1e2]);
% leg = legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 1);
% leg.ItemTokenSize(1) = 15;
ax2 = nexttile;
hold on;
plot(freqs, 180/pi*angle(squeeze(freqresp(-G_iff(1,1)*Kiff(1,1), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
yticks(-360:90:360);
ylim([-180, 180])
linkaxes([ax1,ax2],'x');
xlim([1, 1e3]);
% #+name: fig:nhexa_hac_iff_schematic
% #+caption: HAC-IFF control architecture with the High Authority Controller being implemented in the frame of the struts
% #+RESULTS:
% [[file:figs/nhexa_hac_iff_schematic.png]]
% The effect of decentralized IFF on the plant dynamics can be observed by comparing two sets of transfer functions.
% Figure ref:fig:nhexa_decentralized_hac_iff_plant_undamped shows the original transfer functions from actuator forces $\bm{f}$ to strut errors $\bm{\epsilon}_{\mathcal{L}}$, characterized by pronounced resonant peaks.
% When decentralized IFF is implemented, the transfer functions from modified inputs $\bm{f}^{\prime}$ to strut errors $\bm{\epsilon}_{\mathcal{L}}$, shown in Figure ref:fig:nhexa_decentralized_hac_iff_plant_damped, exhibit significantly attenuated resonances.
% This damping of structural resonances serves two purposes: it reduces vibrations in the vicinity of resonances and simplifies the design of the high authority controller by providing a simpler plant dynamics.
%% Identify the IFF Plant
initializeController('type', 'iff');
% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Controller'], 1, 'input'); io_i = io_i + 1; % Actuator Inputs [N]
io(io_i) = linio([mdl, '/plant'], 1, 'openoutput'); io_i = io_i + 1; % External Metrology [m,rad]
% With no payload
G_hac = linearize(mdl, io);
G_hac.InputName = {'f1', 'f2', 'f3', 'f4', 'f5', 'f6'};
G_hac.OutputName = {'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'};
%% Plant in the frame of the struts
G_hac_struts = stewart.geometry.J*G_hac;
G_hac_struts.OutputName = {'D1', 'D2', 'D3', 'D4', 'D5', 'D6'};
%% Bode plot of the plant projected in the frame of the struts
figure;
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
for i = 1:5
for j = i+1:6
plot(freqs, abs(squeeze(freqresp(G_struts(i,j), freqs, 'Hz'))), 'color', [0,0,0,0.1], ...
'HandleVisibility', 'off');
end
end
plot(freqs, abs(squeeze(freqresp(G_struts(1,1), freqs, 'Hz'))), 'color', colors(1,:), ...
'DisplayName', '$-\epsilon_{\mathcal{L}i}/f_i$')
for i = 2:6
plot(freqs, abs(squeeze(freqresp(G_struts(i,i), freqs, 'Hz'))), 'color', colors(1,:), ...
'HandleVisibility', 'off');
end
plot(freqs, abs(squeeze(freqresp(G_struts(1,2), freqs, 'Hz'))), 'color', [0,0,0,0.1], ...
'DisplayName', '$-\epsilon_{\mathcal{L}i}/f_j$')
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
ylim([1e-9, 1e-4]);
leg = legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 1);
leg.ItemTokenSize(1) = 15;
ax2 = nexttile;
hold on;
for i = 1:6
plot(freqs, 180/pi*angle(squeeze(freqresp(G_struts(i,i), freqs, 'Hz'))), 'color', colors(1,:));
end
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]);
linkaxes([ax1,ax2],'x');
xlim([freqs(1), freqs(end)]);
%% Bode plot of the plant projected in the frame of the struts
figure;
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
for i = 1:5
for j = i+1:6
plot(freqs, abs(squeeze(freqresp(G_hac_struts(i,j), freqs, 'Hz'))), 'color', [0,0,0,0.1], ...
'HandleVisibility', 'off');
end
end
plot(freqs, abs(squeeze(freqresp(G_struts(1,1), freqs, 'Hz'))), 'color', [colors(1,:), 0.2], ...
'DisplayName', '$-\epsilon_{\mathcal{L}i}/f_i$')
plot(freqs, abs(squeeze(freqresp(G_hac_struts(1,1), freqs, 'Hz'))), 'color', colors(2,:), ...
'DisplayName', '$-\epsilon_{\mathcal{L}i}/f_i^\prime$')
for i = 2:6
plot(freqs, abs(squeeze(freqresp(G_struts(i,i), freqs, 'Hz'))), 'color', [colors(1,:), 0.2], ...
'HandleVisibility', 'off');
plot(freqs, abs(squeeze(freqresp(G_hac_struts(i,i), freqs, 'Hz'))), 'color', colors(2,:), ...
'HandleVisibility', 'off');
end
plot(freqs, abs(squeeze(freqresp(G_hac_struts(1,2), freqs, 'Hz'))), 'color', [0,0,0,0.1], ...
'DisplayName', '$-\epsilon_{\mathcal{L}i}/f_j^\prime$')
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
ylim([1e-9, 1e-4]);
leg = legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 1);
leg.ItemTokenSize(1) = 15;
ax2 = nexttile;
hold on;
for i = 1:6
plot(freqs, 180/pi*angle(squeeze(freqresp(G_struts(i,i), freqs, 'Hz'))), 'color', [colors(1,:), 0.2]);
plot(freqs, 180/pi*angle(squeeze(freqresp(G_hac_struts(i,i), freqs, 'Hz'))), 'color', colors(2,:));
end
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]);
linkaxes([ax1,ax2],'x');
xlim([freqs(1), freqs(end)]);
% #+name: fig:nhexa_decentralized_hac_iff_plant
% #+caption: Plant in the frame of the strut for the High Authority Controller.
% #+attr_latex: :options [htbp]
% #+begin_figure
% #+attr_latex: :caption \subcaption{\label{fig:nhexa_decentralized_hac_iff_plant_undamped}Undamped plant in the frame of the struts}
% #+attr_latex: :options {0.48\textwidth}
% #+begin_subfigure
% #+attr_latex: :width 0.95\linewidth
% [[file:figs/nhexa_decentralized_hac_iff_plant_undamped.png]]
% #+end_subfigure
% #+attr_latex: :caption \subcaption{\label{fig:nhexa_decentralized_hac_iff_plant_damped}Damped plant with Decentralized IFF}
% #+attr_latex: :options {0.48\textwidth}
% #+begin_subfigure
% #+attr_latex: :width 0.95\linewidth
% [[file:figs/nhexa_decentralized_hac_iff_plant_damped.png]]
% #+end_subfigure
% #+end_figure
% Building upon the damped plant dynamics shown in Figure ref:fig:nhexa_decentralized_hac_iff_plant_damped, a high authority controller is designed with the structure given in eqref:eq:nhexa_khac.
% The controller combines three elements: an integrator providing high gain at low frequencies, a lead compensator improving stability margins, and a low-pass filter for robustness to unmodeled high-frequency dynamics.
% The loop gain of an individual control channel is shown in Figure ref:fig:nhexa_decentralized_hac_iff_loop_gain.
% \begin{equation}\label{eq:nhexa_khac}
% \bm{K}_{\text{HAC}}(s) = \begin{bmatrix}
% K_{\text{HAC}}(s) & & 0 \\
% & \ddots & \\
% 0 & & K_{\text{HAC}}(s)
% \end{bmatrix}, \quad K_{\text{HAC}}(s) = g_0 \cdot \underbrace{\frac{\omega_c}{s}}_{\text{int}} \cdot \underbrace{\frac{1}{\sqrt{\alpha}}\frac{1 + \frac{s}{\omega_c/\sqrt{\alpha}}}{1 + \frac{s}{\omega_c\sqrt{\alpha}}}}_{\text{lead}} \cdot \underbrace{\frac{1}{1 + \frac{s}{\omega_0}}}_{\text{LPF}}
% \end{equation}
% The stability of the MIMO feedback loop is analyzed through the /characteristic loci/ method.
% Such characteristic loci, shown in Figure ref:fig:nhexa_decentralized_hac_iff_root_locus, represent the eigenvalues of the loop gain matrix $\bm{G}(j\omega)\bm{K}(j\omega)$ plotted in the complex plane as frequency varies from $0$ to $\infty$.
% For MIMO systems, this method generalizes the classical Nyquist stability criterion: with the open-loop system being stable, the closed-loop system is stable if none of the characteristic loci encircle the -1 point [[cite:&skogestad07_multiv_feedb_contr]].
% As seen in Figure ref:fig:nhexa_decentralized_hac_iff_root_locus, all loci remain to the right of the -1 point, confirming the stability of the closed-loop system. Additionally, the distance of the loci from the -1 point provides information about stability margins for the coupled system.
%% High Authority Controller - Mid Stiffness Nano-Hexapod
% Wanted crossover
wc = 2*pi*20; % [rad/s]
% Integrator
H_int = wc/s;
% Lead to increase phase margin
a = 2; % Amount of phase lead / width of the phase lead / high frequency gain
H_lead = 1/sqrt(a)*(1 + s/(wc/sqrt(a)))/(1 + s/(wc*sqrt(a)));
% Low Pass filter to increase robustness
H_lpf = 1/(1 + s/2/pi/200);
% Gain to have unitary crossover at 5Hz
H_gain = 1./abs(evalfr(G_hac_struts(1, 1), 1j*wc));
% Decentralized HAC
Khac = H_gain * ... % Gain
H_int * ... % Integrator
H_lpf * ... % Low Pass filter
eye(6); % 6x6 Diagonal
%% Plot of the eigenvalues of L in the complex plane
Ldet = zeros(6, length(freqs));
Lmimo = squeeze(freqresp(G_hac_struts*Khac, freqs, 'Hz'));
for i_f = 2:length(freqs)
Ldet(:, i_f) = eig(squeeze(Lmimo(:,:,i_f)));
end
figure;
hold on;
for i = 1:6
plot(real(squeeze(Ldet(i,:))), imag(squeeze(Ldet(i,:))), ...
'.', 'color', colors(1, :), ...
'HandleVisibility', 'off');
plot(real(squeeze(Ldet(i,:))), -imag(squeeze(Ldet(i,:))), ...
'.', 'color', colors(1, :), ...
'HandleVisibility', 'off');
end
plot(-1, 0, 'kx', 'HandleVisibility', 'off');
hold off;
set(gca, 'XScale', 'lin'); set(gca, 'YScale', 'lin');
xlabel('Real Part'); ylabel('Imaginary Part');
axis square
xlim([-1.8, 0.2]); ylim([-1, 1]);
%% Loop gain for the Decentralized HAC_IFF
figure;
tiledlayout(3, 1, 'TileSpacing', 'compact', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
plot(freqs, abs(squeeze(freqresp(G_hac_struts(1,1)*Khac(1,1), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Loop Gain'); set(gca, 'XTickLabel',[]);
ylim([1e-2, 1e2]);
ax2 = nexttile;
hold on;
plot(freqs, 180/pi*angle(squeeze(freqresp(G_hac_struts(1,1)*Khac(1,1), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
yticks(-360:90:360);
ylim([-180, 180])
linkaxes([ax1,ax2],'x');
xlim([1, 1e3]);

0
matlab/subsystems/nano_hexapod.slx → matlab/subsystems/nano_hexapod_simscape.slx
View File