From 73de5c5adc71187b1b254631a690358c80266f0c Mon Sep 17 00:00:00 2001 From: Thomas Dehaeze Date: Fri, 28 Mar 2025 14:31:40 +0100 Subject: [PATCH] Christophe's review --- figs/nhexa_comp_multi_body_analytical.pdf | 914 +++++++++--------- figs/nhexa_comp_multi_body_analytical.png | Bin 152210 -> 164653 bytes .../nhexa_decentralized_hac_iff_loop_gain.pdf | Bin 46032 -> 48288 bytes .../nhexa_decentralized_hac_iff_loop_gain.png | Bin 65825 -> 70838 bytes ...nhexa_decentralized_hac_iff_root_locus.pdf | Bin 110386 -> 112395 bytes ...nhexa_decentralized_hac_iff_root_locus.png | Bin 31179 -> 36039 bytes ..._forward_kinematics_approximate_errors.pdf | Bin 25587 -> 24195 bytes ..._forward_kinematics_approximate_errors.png | Bin 80338 -> 92098 bytes matlab/mat/nano_hexapod.mat | Bin 2406 -> 2400 bytes matlab/mat/nano_hexapod_model_controller.mat | Bin 261 -> 235 bytes matlab/nhexa_1_stewart_platform.m | 64 +- matlab/nhexa_2_model.m | 24 +- matlab/nhexa_3_control.m | 40 +- matlab/src/generateGeneralConfiguration.m | 4 +- matlab/src/initializeSimplifiedNanoHexapod.m | 4 +- simscape-nano-hexapod.org | 170 +++- simscape-nano-hexapod.pdf | Bin 4612031 -> 4628852 bytes simscape-nano-hexapod.tex | 32 +- 18 files changed, 671 insertions(+), 581 deletions(-) diff --git a/figs/nhexa_comp_multi_body_analytical.pdf b/figs/nhexa_comp_multi_body_analytical.pdf index 89d228c..38e5c8a 100644 --- a/figs/nhexa_comp_multi_body_analytical.pdf +++ b/figs/nhexa_comp_multi_body_analytical.pdf @@ -3,7 +3,7 @@ 1 0 obj << /Producer (Apache FOP Version 2.4.0-SNAPSHOT: PDFDocumentGraphics2D) -/CreationDate (D:20250212102338+01'00') +/CreationDate (D:20250328112929+01'00') >> endobj 2 0 obj @@ -778,163 +778,156 @@ x ugzE>7{wn/|[d*_Gz'pQكsiJ(߉11ɛ=(S_Grfq_ ^_hsw'f`STˍ#/7s5%,"bgg{~c!-C7 W[zERXW5-%W9_n;v"|o7n/8|I_%~r&"Co7wB_~vvrcu*E闛4o7w3L_$~+vRƜǜCP[Gxbd䫍?evm<= YZ%tϢ6b9w~= `YB}PM=uzmQ,j:gew=Kx,{}PtY}ƟEvϲ6_nQ,j gMw=fYcޣ|M?u{mQ,n㏺gw]t,MVFӱ7Y?5a?5aMǂNgo}b%|VN pZwP|8h:m?|0՚.lp|0R{2YVƊb_[n#`=jwƿ ֿ7W#<<7z`l?cߍ #h1? 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vc%0*7U|_fbk&rte%m(RmK+?y8MIR@UJ^?7-zaTYF*C#)}04fduB8m+m#xNWi delta 115 zcmV-(0F3|Z0fhpPG#FQ9WFSUmVjwXxATcmHG&DLjHy|=FGBuG=BavVQk#rCYeE;{m&2qb+hSoEnP=@WqRT~d=%fi6tU$*f9EQSi(wNlgcG6x{NQ Vl2aAjQd3is5|gum>HtI}7x}@5D`Wrw diff --git a/matlab/nhexa_1_stewart_platform.m b/matlab/nhexa_1_stewart_platform.m index 5b21bc8..2b7b84a 100644 --- a/matlab/nhexa_1_stewart_platform.m +++ b/matlab/nhexa_1_stewart_platform.m @@ -18,55 +18,75 @@ mdl = 'nano_hexapod_model'; %% Colors for the figures colors = colororder; -%% Frequency Vector +%% Frequency Vector [Hz] 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 accuracy of the Jacobian-based forward kinematics solution was estimated by a simple analysis. +% For a series of platform positions, 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 eqref:eq:nhexa_forward_kinematics_approximate, from which the error between the estimated and true poses can be calculated, both in terms of position $\epsilon_D$ and orientation $\epsilon_R$. -% The estimation errors in the $x$, $y$, and $z$ directions are shown in Figure ref:fig:nhexa_forward_kinematics_approximate_errors. +% For motion strokes from $1\,\mu m$ to $10\,mm$, the errors are estimated for all direction of motion, and the worst case errors 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. +% Since the maximum required stroke of the nano-hexapod ($\approx 100\,\mu m$) is three orders of magnitude smaller than its overall 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); +stewart = initializeSimplifiedNanoHexapod('H', 100e-3); -Xrs = logspace(-6, -1, 100); % Wanted X translation of the mobile platform [m] +Xrs = logspace(-6, -1, 10); % Wanted X translation of the mobile platform [m] +phis = linspace(-pi, pi, 100); % Tested azimutal angles [rad] +thetas = linspace(0, pi, 100); % Tested polar angles [rad] % Compute the strut exact length for each X-position -Ls_exact = zeros(6, length(Xrs)); +Xrs_errors = zeros(1, length(Xrs)); % Maximum distance error [m] +Rrs_errors = zeros(1, length(Xrs)); % Maximum angular error [rad] 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); + Xrs_error_min = 0; + Rrs_error_min = 0; + for theta = thetas + for phi = phis + ix = [sin(theta)*cos(phi); sin(theta)*sin(phi);cos(theta)]; % Unit vector for the displacement direction + [~, L_exact] = inverseKinematics(stewart, 'AP', Xrs(i)*ix); % Compute exact strut length for the wanted position + Xrs_approx = inv(stewart.geometry.J)*L_exact; % Approximate the position using the Jacobian + Xrs_error = norm(Xrs(i)*ix - Xrs_approx(1:3), 2); % Compute the position estimation error + Rrs_error = norm(Xrs_approx(4:6), 2); % Compute the angular estimation error + if Xrs_error > Xrs_error_min + Xrs_error_min = Xrs_error; + end + if Rrs_error > Rrs_error_min + Rrs_error_min = Rrs_error; + end + end + end + Xrs_errors(i) = Xrs_error_min; + Rrs_errors(i) = Rrs_error_min; end %% Errors associated with the use of the Jacobian matrix to solve the forward kinematic problem figure; +yyaxis left 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'); +plot(1e6*Xrs, 1e9*Xrs_errors, 'DisplayName', '$\epsilon_D$'); +plot(1e6*Xrs, 1e6*Xrs, '--', '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'); +xlabel('Motion Stroke'); 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({'$1nm$', '$10nm$', '$100nm$', '$1\mu m$', '$10\mu m$'}); + +yyaxis right +plot(1e6*Xrs, 1e9*Rrs_errors, 'DisplayName', '$\epsilon_R$'); +set(gca, 'YScale', 'log'); +ylim([1, 1e4]); +yticks([1, 10, 100, 1000, 10000]); +yticklabels({'$1$nrad', '$10$nrad', '$100$nrad', '$1\mu$rad', '$10\mu$rad'}); diff --git a/matlab/nhexa_2_model.m b/matlab/nhexa_2_model.m index 2bbcdc6..d30ad6a 100644 --- a/matlab/nhexa_2_model.m +++ b/matlab/nhexa_2_model.m @@ -18,7 +18,7 @@ mdl = 'nano_hexapod_model'; %% Colors for the figures colors = colororder; -%% Frequency Vector +%% Frequency Vector [Hz] freqs = logspace(0, 3, 1000); % Validation of the multi-body model @@ -44,13 +44,13 @@ freqs = logspace(0, 3, 1000); % [[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. +% The validation of the multi-body model was 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. +% 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} @@ -60,9 +60,9 @@ freqs = logspace(0, 3, 1000); % \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. +% The transfer functions from the actuator forces to the 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. +% The developed 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. @@ -123,7 +123,7 @@ 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], ... +plot(freqs, abs(squeeze(freqresp(G_simscape(i,1), freqs, 'Hz'))), 'color', [colors(i,:), 0.5], 'linewidth', 2.5, ... 'DisplayName', sprintf('$l_%i/f_1$ - Multi-Body', i)) end for i = 1:6 @@ -140,7 +140,7 @@ 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]); + plot(freqs, 180/pi*angle(squeeze(freqresp(G_simscape(i,1), freqs, 'Hz'))), 'color', [colors(i,:),0.5], 'linewidth', 2.5); end for i = 1:6 plot(freqs, 180/pi*angle(squeeze(freqresp(G_analytical(i,1), freqs, 'Hz'))), '--', 'color', colors(i,:)); @@ -157,13 +157,13 @@ xlim([freqs(1), freqs(end)]); % Nano Hexapod 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. +% Following the validation of the multi-body model, a detailed analysis of the nano-hexapod dynamics was performed. +% The model parameters were set according to the specifications outlined in Section ref:ssec:nhexa_model_def, with a payload mass of $10\,kg$. +% The transfer functions from actuator forces $\bm{f}$ to both strut displacements $\bm{\mathcal{L}}$ and force measurements $\bm{f}_n$ were 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. +% While the system has six degrees of freedom, only four distinct resonance frequencies were 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. @@ -173,7 +173,7 @@ xlim([freqs(1), freqs(end)]); % 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. +% The inclusion of parallel stiffness introduces an additional complex conjugate zero at low frequency, which was previously observed in the three-degree-of-freedom rotating model. %% Multi-Body model of the Nano-Hexapod diff --git a/matlab/nhexa_3_control.m b/matlab/nhexa_3_control.m index 83a31fa..9a82ae8 100644 --- a/matlab/nhexa_3_control.m +++ b/matlab/nhexa_3_control.m @@ -18,22 +18,22 @@ mdl = 'nano_hexapod_model'; %% Colors for the figures colors = colororder; -%% Frequency Vector +%% Frequency Vector [Hz] 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. +% Alternatively, control can be implemented directly in Cartesian space, as illustrated 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 using the transpose of the inverse Jacobian matrix eqref:eq:nhexa_jacobian_forces. % 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. +% A key advantage of this approach is that the control performance can be tuned individually 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. +% For the conceptual validation of the nano-hexapod, control in the strut space was 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. @@ -166,12 +166,12 @@ xlim([freqs(1), freqs(end)]); % \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. +% In this section, the stiffness in parallel with the force sensor was omitted since the Stewart platform is not subjected to rotation. +% The effect of this parallel stiffness is examined in the next section when the platform is integrated into the complete NASS. -% 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 $\infty$. +% 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. +% This property is particularly valuable because 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. @@ -266,9 +266,9 @@ xlim([1, 1e3]); % [[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. +% Figure ref:fig:nhexa_decentralized_hac_iff_plant_undamped shows the original transfer functions from actuator forces $\bm{f}$ to strut errors $\bm{\epsilon}_{\mathcal{L}}$, which are characterized by pronounced resonant peaks. +% When the decentralized IFF is implemented, the transfer functions from modified inputs $\bm{f}^{\prime}$ to strut errors $\bm{\epsilon}_{\mathcal{L}}$ exhibit significantly attenuated resonances (Figure ref:fig:nhexa_decentralized_hac_iff_plant_damped). +% This damping of structural resonances serves two purposes: it reduces vibrations near resonances and simplifies the design of the high authority controller by providing simpler plant dynamics. %% Identify the IFF Plant @@ -395,8 +395,8 @@ xlim([freqs(1), freqs(end)]); % #+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. +% Based upon the damped plant dynamics shown in Figure ref:fig:nhexa_decentralized_hac_iff_plant_damped, a high authority controller was 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 against 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} @@ -408,9 +408,10 @@ xlim([freqs(1), freqs(end)]); % \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 $\infty$. +% Such characteristic loci represent the eigenvalues of the loop gain matrix $\bm{G}(j\omega)\bm{K}(j\omega)$ plotted in the complex plane as the 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. +% As shown in Figure ref:fig:nhexa_decentralized_hac_iff_root_locus, all loci remain to the right of the $-1$ point, validating the stability of the closed-loop system. +% Additionally, the distance of the loci from the $-1$ point provides information about stability margins of the coupled system. %% High Authority Controller - Mid Stiffness Nano-Hexapod @@ -442,6 +443,7 @@ Lmimo = squeeze(freqresp(G_hac_struts*Khac, freqs, 'Hz')); for i_f = 2:length(freqs) Ldet(:, i_f) = eig(squeeze(Lmimo(:,:,i_f))); end +mod_margin = min(min(abs(Ldet + ones(size(Ldet))))); figure; hold on; @@ -454,6 +456,8 @@ plot(real(squeeze(Ldet(i,:))), -imag(squeeze(Ldet(i,:))), ... 'HandleVisibility', 'off'); end plot(-1, 0, 'kx', 'HandleVisibility', 'off'); +patch(-1 + mod_margin*cos([0:0.1:2*pi+0.1]), mod_margin*sin([0:0.1:2*pi+0.1]), colors(5,:), 'linestyle', '--', 'EdgeColor','black', 'FaceAlpha', 0.5, 'HandleVisibility', 'off'); +text(-1,0.1, 'Robustness', 'FontSize', 8, 'horizontalalignment', 'center') hold off; set(gca, 'XScale', 'lin'); set(gca, 'YScale', 'lin'); xlabel('Real Part'); ylabel('Imaginary Part'); @@ -461,12 +465,16 @@ axis square xlim([-1.8, 0.2]); ylim([-1, 1]); %% Loop gain for the Decentralized HAC_IFF +i_fb = find(abs(squeeze(freqresp(G_hac_struts(1,1)*Khac(1,1), freqs, 'Hz')))<1, 1); + figure; tiledlayout(3, 1, 'TileSpacing', 'compact', 'Padding', 'None'); ax1 = nexttile([2,1]); hold on; +patch([freqs(1:i_fb), freqs(i_fb), freqs(1)], [abs(squeeze(freqresp(G_hac_struts(1,1)*Khac(1,1), [freqs(1:i_fb)], 'Hz'))); 1; 1], colors(5,:), 'EdgeColor','none', 'FaceAlpha', 0.5) plot(freqs, abs(squeeze(freqresp(G_hac_struts(1,1)*Khac(1,1), freqs, 'Hz')))); +text(1.2,2.5,{'Disturbance', 'rejection'}, 'FontSize', 8) hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); ylabel('Loop Gain'); set(gca, 'XTickLabel',[]); diff --git a/matlab/src/generateGeneralConfiguration.m b/matlab/src/generateGeneralConfiguration.m index 69b9786..5be7ee4 100644 --- a/matlab/src/generateGeneralConfiguration.m +++ b/matlab/src/generateGeneralConfiguration.m @@ -19,10 +19,10 @@ function [stewart] = generateGeneralConfiguration(stewart, args) arguments stewart - args.FH (1,1) double {mustBeNumeric, mustBePositive} = 15e-3 + args.FH (1,1) double {mustBeNumeric, mustBeNonnegative} = 15e-3 args.FR (1,1) double {mustBeNumeric, mustBePositive} = 115e-3; args.FTh (6,1) double {mustBeNumeric} = [-10, 10, 120-10, 120+10, 240-10, 240+10]*(pi/180); - args.MH (1,1) double {mustBeNumeric, mustBePositive} = 15e-3 + args.MH (1,1) double {mustBeNumeric, mustBeNonnegative} = 15e-3 args.MR (1,1) double {mustBeNumeric, mustBePositive} = 90e-3; args.MTh (6,1) double {mustBeNumeric} = [-60+10, 60-10, 60+10, 180-10, 180+10, -60-10]*(pi/180); end diff --git a/matlab/src/initializeSimplifiedNanoHexapod.m b/matlab/src/initializeSimplifiedNanoHexapod.m index a7202b1..524f1be 100644 --- a/matlab/src/initializeSimplifiedNanoHexapod.m +++ b/matlab/src/initializeSimplifiedNanoHexapod.m @@ -5,10 +5,10 @@ function [nano_hexapod] = initializeSimplifiedNanoHexapod(args) args.H (1,1) double {mustBeNumeric, mustBePositive} = 95e-3 % Height of the nano-hexapod [m] args.MO_B (1,1) double {mustBeNumeric} = 150e-3 % Height of {B} w.r.t. {M} [m] %% generateGeneralConfiguration - args.FH (1,1) double {mustBeNumeric, mustBePositive} = 15e-3 % Height of fixed joints [m] + args.FH (1,1) double {mustBeNumeric, mustBeNonnegative} = 15e-3 % Height of fixed joints [m] args.FR (1,1) double {mustBeNumeric, mustBePositive} = 120e-3 % Radius of fixed joints [m] args.FTh (6,1) double {mustBeNumeric} = [220, 320, 340, 80, 100, 200]*(pi/180) % Angles of fixed joints [rad] - args.MH (1,1) double {mustBeNumeric, mustBePositive} = 15e-3 % Height of mobile joints [m] + args.MH (1,1) double {mustBeNumeric, mustBeNonnegative} = 15e-3 % Height of mobile joints [m] args.MR (1,1) double {mustBeNumeric, mustBePositive} = 110e-3 % Radius of mobile joints [m] args.MTh (6,1) double {mustBeNumeric} = [255, 285, 15, 45, 135, 165]*(pi/180) % Angles of fixed joints [rad] %% Actuators diff --git a/simscape-nano-hexapod.org b/simscape-nano-hexapod.org index 099d53a..dfc2c2c 100644 --- a/simscape-nano-hexapod.org +++ b/simscape-nano-hexapod.org @@ -404,7 +404,7 @@ However, the development of the Nano Active Stabilization System (NASS) now requ To develop this model, the architecture of the active platform must first be determined. The selection of an appropriate architecture begins with a review of existing positioning stages that incorporate active platforms similar to NASS (Section ref:ssec:nhexa_sample_stages). -This review reveals two distinctive features of the NASS that set it apart from existing systems: the fact that the active platform is continuously rotating and its requirement to accommodate high payload masses. +This review reveals two distinctive features of the NASS that set it apart from existing systems: the fact that the active platform is continuously rotating and its requirement to accommodate variable payload masses. In existing systems, the sample mass is typically negligible compared to the stage mass, whereas in NASS, the sample mass significantly influences the system's dynamic behavior. These distinctive requirements drive the selection of the active platform architecture. @@ -413,7 +413,7 @@ In Section ref:ssec:nhexa_active_platforms, different active platform configurat ** Sample Stages with Active Control <> -The positioning of samples with respect to to X-ray beam, that can be focused to sizes below 100 nanometers, presents significant challenges, because mechanical positioning systems are typically limited to micron-scale accuracy. +The positioning of samples with respect to X-ray beam, that can be focused to sizes below 100 nanometers, presents significant challenges, because mechanical positioning systems are typically limited to micron-scale accuracy. To overcome this limitation, external metrology systems have been implemented to measure sample positions with nanometer accuracy, enabling real-time feedback control for sample stabilization. A review of existing sample stages with active vibration control reveals various approaches to implementing such feedback systems. @@ -538,7 +538,7 @@ It must accommodate payloads up to 50 kg while maintaining high dynamical perfor For light samples, the typical design strategy of maximizing actuator stiffness works well because resonance frequencies in the kilohertz range can be achieved, enabling control bandwidths up to 100 Hz. However, achieving such resonance frequencies with a 50 kg payload would require unrealistic stiffness values of approximately $2000\,N/\mu m$. This limitation necessitates alternative control approaches, and the High Authority Control/Low Authority Control (HAC-LAC) strategy is proposed to address this challenge. -Consequently, the design must incorporate force sensors for active damping. +To this purpose, the design includes force sensors for active damping. Compliant mechanisms must also be used to eliminate friction and backlash, which would otherwise compromise the nano-positioning capabilities. Two primary categories of positioning platform architectures are considered: serial and parallel mechanisms. @@ -651,8 +651,8 @@ These theoretical foundations form the basis for subsequent design decisions and <> The Stewart platform consists of two rigid platforms connected by six parallel struts (Figure ref:fig:nhexa_stewart_architecture). -Each strut incorporates an active prismatic joint that allows for controlled length variation, with its ends attached to the fixed and mobile platforms through joints. -The typical configuration consists of a universal joint at one end and a spherical joint at the other, providing the necessary degrees of freedom[fn:1]. +Each strut is modelled with an active prismatic joint that allows for controlled length variation, with its ends attached to the fixed and mobile platforms through joints. +The typical configuration consists of a universal joint at one end and a spherical joint at the other, providing the necessary degrees of freedom[fn:nhexa_1]. #+name: fig:nhexa_stewart_architecture #+caption: Schematical representation of the Stewart platform architecture. @@ -679,10 +679,6 @@ This is summarized in Figure ref:fig:nhexa_stewart_notations. ** Kinematic Analysis <> -**** Introduction :ignore: - -The kinematic analysis of the Stewart platform involves understanding the geometric relationships between the platform position/orientation and the actuator lengths without considering the forces involved. - **** Loop Closure The foundation of the kinematic analysis lies in the geometric constraints imposed by each strut, which can be expressed using loop closure equations. @@ -692,7 +688,7 @@ For each strut $i$ (illustrated in Figure ref:fig:nhexa_stewart_loop_closure), t {}^A\bm{P}_B = {}^A\bm{a}_i + l_i{}^A\hat{\bm{s}}_i - \underbrace{{}^B\bm{b}_i}_{{}^A\bm{R}_B {}^B\bm{b}_i} \quad \text{for } i=1 \text{ to } 6 \end{equation} -This equation links the pose variables ${}^A\bm{P}$ and ${}^A\bm{R}_B$, the position vectors describing the known geometry of the base and the moving platform, $\bm{a}_i$ and $\bm{b}_i$, and the strut vector $l_i {}^A\hat{\bm{s}}_i$: +This equation links the pose[fn:nhexa_2] variables ${}^A\bm{P}$ and ${}^A\bm{R}_B$, the position vectors describing the known geometry of the base and the moving platform, $\bm{a}_i$ and $\bm{b}_i$, and the strut vector $l_i {}^A\hat{\bm{s}}_i$: #+name: fig:nhexa_stewart_loop_closure #+caption: Notations to compute the kinematic loop closure @@ -729,11 +725,11 @@ This approximation, which is developed in subsequent sections through the Jacobi The Jacobian matrix plays a central role in analyzing the Stewart platform's behavior, providing a linear mapping between the platform and actuator velocities. While the previously derived kinematic relationships are essential for position analysis, the Jacobian enables velocity analysis and forms the foundation for both static and dynamic studies. -**** Jacobian Computation - Velocity Loop Closure +**** Jacobian Computation As discussed in Section ref:ssec:nhexa_stewart_platform_kinematics, the strut lengths $\bm{\mathcal{L}}$ and the platform pose $\bm{\mathcal{X}}$ are related through a system of nonlinear algebraic equations representing the kinematic constraints imposed by the struts. -By taking the time derivative of the position loop close eqref:eq:nhexa_loop_closure, the /velocity loop closure/ is obtained eqref:eq:nhexa_loop_closure_velocity. +By taking the time derivative of the position loop close eqref:eq:nhexa_loop_closure, equation eqref:eq:nhexa_loop_closure_velocity[fn:nhexa_3] is obtained. \begin{equation}\label{eq:nhexa_loop_closure_velocity} {}^A\bm{v}_p + {}^A \dot{\bm{R}}_B {}^B\bm{b}_i + {}^A\bm{R}_B \underbrace{{}^B\dot{\bm{b}_i}}_{=0} = \dot{l}_i {}^A\hat{\bm{s}}_i + l_i {}^A\dot{\hat{\bm{s}}}_i + \underbrace{{}^A\dot{\bm{a}}_i}_{=0} @@ -791,10 +787,10 @@ While this approximation offers limited value for inverse kinematics, which can **** Range validity of the approximate inverse kinematics The accuracy of the Jacobian-based forward kinematics solution was estimated by a simple 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. +For a series of platform positions, 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 eqref:eq:nhexa_forward_kinematics_approximate, from which the error between the estimated and true poses can be calculated, both in terms of position $\epsilon_D$ and orientation $\epsilon_R$. -The estimation errors in the $x$, $y$, and $z$ directions are shown in Figure ref:fig:nhexa_forward_kinematics_approximate_errors. +For motion strokes from $1\,\mu m$ to $10\,mm$, the errors are estimated for all direction of motion, and the worst case errors 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. Since the maximum required stroke of the nano-hexapod ($\approx 100\,\mu m$) is three orders of magnitude smaller than its overall size ($\approx 100\,mm$), the Jacobian matrix can be considered constant throughout the workspace. @@ -802,42 +798,63 @@ It can be computed once at the rest position and used for both forward and inver #+begin_src matlab %% Estimate the errors associated with approximate forward kinematics using the Jacobian matrix -stewart = initializeSimplifiedNanoHexapod('H', 100e-3, 'MO_B', 0); +stewart = initializeSimplifiedNanoHexapod('H', 100e-3); -Xrs = logspace(-6, -1, 100); % Wanted X translation of the mobile platform [m] +Xrs = logspace(-6, -1, 10); % Wanted X translation of the mobile platform [m] +phis = linspace(-pi, pi, 100); % Tested azimutal angles [rad] +thetas = linspace(0, pi, 100); % Tested polar angles [rad] % Compute the strut exact length for each X-position -Ls_exact = zeros(6, length(Xrs)); +Xrs_errors = zeros(1, length(Xrs)); % Maximum distance error [m] +Rrs_errors = zeros(1, length(Xrs)); % Maximum angular error [rad] 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); + Xrs_error_min = 0; + Rrs_error_min = 0; + for theta = thetas + for phi = phis + ix = [sin(theta)*cos(phi); sin(theta)*sin(phi);cos(theta)]; % Unit vector for the displacement direction + [~, L_exact] = inverseKinematics(stewart, 'AP', Xrs(i)*ix); % Compute exact strut length for the wanted position + Xrs_approx = inv(stewart.geometry.J)*L_exact; % Approximate the position using the Jacobian + Xrs_error = norm(Xrs(i)*ix - Xrs_approx(1:3), 2); % Compute the position estimation error + Rrs_error = norm(Xrs_approx(4:6), 2); % Compute the angular estimation error + if Xrs_error > Xrs_error_min + Xrs_error_min = Xrs_error; + end + if Rrs_error > Rrs_error_min + Rrs_error_min = Rrs_error; + end + end + end + Xrs_errors(i) = Xrs_error_min; + Rrs_errors(i) = Rrs_error_min; end #+end_src #+begin_src matlab :exports none :results none %% Errors associated with the use of the Jacobian matrix to solve the forward kinematic problem figure; +yyaxis left 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'); +plot(1e6*Xrs, 1e9*Xrs_errors, 'DisplayName', '$\epsilon_D$'); +plot(1e6*Xrs, 1e6*Xrs, '--', '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'); +xlabel('Motion Stroke'); 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({'$1nm$', '$10nm$', '$100nm$', '$1\mu m$', '$10\mu m$'}); + +yyaxis right +plot(1e6*Xrs, 1e9*Rrs_errors, 'DisplayName', '$\epsilon_R$'); +set(gca, 'YScale', 'log'); +ylim([1, 1e4]); +yticks([1, 10, 100, 1000, 10000]); +yticklabels({'$1$nrad', '$10$nrad', '$100$nrad', '$1\mu$rad', '$10\mu$rad'}); #+end_src #+begin_src matlab :tangle no :exports results :results file replace @@ -845,7 +862,7 @@ exportFig('figs/nhexa_forward_kinematics_approximate_errors.pdf', 'width', 'wide #+end_src #+name: fig:nhexa_forward_kinematics_approximate_errors -#+caption: Errors associated with the use of the Jacobian matrix to solve the forward kinematic problem. A Stewart platform with a height of $100\,mm$ was used to perform this analysis +#+caption: Errors associated with the use of the Jacobian matrix to solve the forward kinematic problem. A Stewart platform with a height of $100\,mm$ was used to perform this analysis. $\epsilon_D$ corresponds to the distance between the true positioin and the estimated position. $\epsilon_R$ corresponds to the angular motion between the true orientation and the estimated orientation. #+RESULTS: [[file:figs/nhexa_forward_kinematics_approximate_errors.png]] @@ -911,7 +928,7 @@ These relationships reveal that the overall platform stiffness and compliance ch - The geometric configuration embodied in the Jacobian matrix $\bm{J}$ This geometric dependency means that the platform's stiffness varies throughout its workspace, as the Jacobian matrix changes with the platform's position and orientation. -For the NASS application, where the workspace is small compared to the platform dimensions, these variations can be considered minimal. +For the NASS application, where the workspace is small compared to the platform dimensions, these variations can be considered negligible. However, the initial geometric configuration significantly affects the overall stiffness characteristics. The relationship between maximum stroke and stiffness presents another important design consideration. As both parameters are influenced by the geometric configuration, their optimization involves inherent trade-offs that must be carefully balanced based on the application requirements. @@ -920,7 +937,6 @@ The optimization of this configuration to achieve the desired stiffness while ha ** Dynamical Analysis <> -The dynamic behavior of a Stewart platform can be analyzed using various approaches depending on the desired level of model fidelity. For initial analysis, a simplified representation of the system has been developed. This model assumes perfectly rigid bodies for both the platform and base, connected by massless struts through ideal joints that exhibit neither friction nor compliance. @@ -992,7 +1008,7 @@ Through this multi-body modeling approach, each component model (including joint The analysis is structured as follows. First, the multi-body model is developed, and the geometric parameters, inertial properties, and actuator characteristics are established (Section ref:ssec:nhexa_model_def). -The model was then validated through comparison with the analytical equations in a simplified configuration (Section ref:ssec:nhexa_model_validation). +The model is then validated through comparison with the analytical equations in a simplified configuration (Section ref:ssec:nhexa_model_validation). Finally, the validated model is employed to analyze the nano-hexapod dynamics, from which insights for the control system design are derived (Section ref:ssec:nhexa_model_dynamics). ** Matlab Init :noexport:ignore: @@ -1220,7 +1236,7 @@ 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], ... +plot(freqs, abs(squeeze(freqresp(G_simscape(i,1), freqs, 'Hz'))), 'color', [colors(i,:), 0.5], 'linewidth', 2.5, ... 'DisplayName', sprintf('$l_%i/f_1$ - Multi-Body', i)) end for i = 1:6 @@ -1237,7 +1253,7 @@ 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]); + plot(freqs, 180/pi*angle(squeeze(freqresp(G_simscape(i,1), freqs, 'Hz'))), 'color', [colors(i,:),0.5], 'linewidth', 2.5); end for i = 1:6 plot(freqs, 180/pi*angle(squeeze(freqresp(G_analytical(i,1), freqs, 'Hz'))), '--', 'color', colors(i,:)); @@ -2171,6 +2187,7 @@ Lmimo = squeeze(freqresp(G_hac_struts*Khac, freqs, 'Hz')); for i_f = 2:length(freqs) Ldet(:, i_f) = eig(squeeze(Lmimo(:,:,i_f))); end +mod_margin = min(min(abs(Ldet + ones(size(Ldet))))); figure; hold on; @@ -2183,6 +2200,8 @@ plot(real(squeeze(Ldet(i,:))), -imag(squeeze(Ldet(i,:))), ... 'HandleVisibility', 'off'); end plot(-1, 0, 'kx', 'HandleVisibility', 'off'); +patch(-1 + mod_margin*cos([0:0.1:2*pi+0.1]), mod_margin*sin([0:0.1:2*pi+0.1]), colors(5,:), 'linestyle', '--', 'EdgeColor','black', 'FaceAlpha', 0.5, 'HandleVisibility', 'off'); +text(-1,0.1, 'Robustness', 'FontSize', 8, 'horizontalalignment', 'center') hold off; set(gca, 'XScale', 'lin'); set(gca, 'YScale', 'lin'); xlabel('Real Part'); ylabel('Imaginary Part'); @@ -2196,12 +2215,16 @@ exportFig('figs/nhexa_decentralized_hac_iff_root_locus.pdf', 'width', 'half', 'h #+begin_src matlab :exports none :results none %% Loop gain for the Decentralized HAC_IFF +i_fb = find(abs(squeeze(freqresp(G_hac_struts(1,1)*Khac(1,1), freqs, 'Hz')))<1, 1); + figure; tiledlayout(3, 1, 'TileSpacing', 'compact', 'Padding', 'None'); ax1 = nexttile([2,1]); hold on; +patch([freqs(1:i_fb), freqs(i_fb), freqs(1)], [abs(squeeze(freqresp(G_hac_struts(1,1)*Khac(1,1), [freqs(1:i_fb)], 'Hz'))); 1; 1], colors(5,:), 'EdgeColor','none', 'FaceAlpha', 0.5) plot(freqs, abs(squeeze(freqresp(G_hac_struts(1,1)*Khac(1,1), freqs, 'Hz')))); +text(1.2,2.5,{'Disturbance', 'rejection'}, 'FontSize', 8) hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); ylabel('Loop Gain'); set(gca, 'XTickLabel',[]); @@ -2226,7 +2249,7 @@ exportFig('figs/nhexa_decentralized_hac_iff_loop_gain.pdf', 'width', 'half', 'he #+end_src #+name: fig:nhexa_decentralized_hac_iff_results -#+caption: Decentralized HAC-IFF +#+caption: Decentralized HAC-IFF. Loop gain (\subref{fig:nhexa_decentralized_hac_iff_loop_gain}) is used for the design of the controller and to estimate the disturbance rejection performances. Characteristic Loci (\subref{fig:nhexa_decentralized_hac_iff_root_locus}) is used to verify the stability and robustness of the feedback loop. #+attr_latex: :options [htbp] #+begin_figure #+attr_latex: :caption \subcaption{\label{fig:nhexa_decentralized_hac_iff_loop_gain}Loop Gain} @@ -2469,10 +2492,10 @@ function [nano_hexapod] = initializeSimplifiedNanoHexapod(args) args.H (1,1) double {mustBeNumeric, mustBePositive} = 95e-3 % Height of the nano-hexapod [m] args.MO_B (1,1) double {mustBeNumeric} = 150e-3 % Height of {B} w.r.t. {M} [m] %% generateGeneralConfiguration - args.FH (1,1) double {mustBeNumeric, mustBePositive} = 15e-3 % Height of fixed joints [m] + args.FH (1,1) double {mustBeNumeric, mustBeNonnegative} = 15e-3 % Height of fixed joints [m] args.FR (1,1) double {mustBeNumeric, mustBePositive} = 120e-3 % Radius of fixed joints [m] args.FTh (6,1) double {mustBeNumeric} = [220, 320, 340, 80, 100, 200]*(pi/180) % Angles of fixed joints [rad] - args.MH (1,1) double {mustBeNumeric, mustBePositive} = 15e-3 % Height of mobile joints [m] + args.MH (1,1) double {mustBeNumeric, mustBeNonnegative} = 15e-3 % Height of mobile joints [m] args.MR (1,1) double {mustBeNumeric, mustBePositive} = 110e-3 % Radius of mobile joints [m] args.MTh (6,1) double {mustBeNumeric} = [255, 285, 15, 45, 135, 165]*(pi/180) % Angles of fixed joints [rad] %% Actuators @@ -2713,10 +2736,10 @@ function [stewart] = generateGeneralConfiguration(stewart, args) arguments stewart - args.FH (1,1) double {mustBeNumeric, mustBePositive} = 15e-3 + args.FH (1,1) double {mustBeNumeric, mustBeNonnegative} = 15e-3 args.FR (1,1) double {mustBeNumeric, mustBePositive} = 115e-3; args.FTh (6,1) double {mustBeNumeric} = [-10, 10, 120-10, 120+10, 240-10, 240+10]*(pi/180); - args.MH (1,1) double {mustBeNumeric, mustBePositive} = 15e-3 + args.MH (1,1) double {mustBeNumeric, mustBeNonnegative} = 15e-3 args.MR (1,1) double {mustBeNumeric, mustBePositive} = 90e-3; args.MTh (6,1) double {mustBeNumeric} = [-60+10, 60-10, 60+10, 180-10, 180+10, -60-10]*(pi/180); end @@ -3718,6 +3741,69 @@ if isfield(stewart.kinematics, 'C') end #+end_src +*** =generateCubicConfiguration=: Generate a Cubic Configuration +#+begin_src matlab :tangle matlab/src/generateCubicConfiguration.m :comments none :mkdirp yes :eval no +function [stewart] = generateCubicConfiguration(stewart, args) +% generateCubicConfiguration - Generate a Cubic Configuration +% +% Syntax: [stewart] = generateCubicConfiguration(stewart, args) +% +% Inputs: +% - stewart - A structure with the following fields +% - geometry.H [1x1] - Total height of the platform [m] +% - args - Can have the following fields: +% - Hc [1x1] - Height of the "useful" part of the cube [m] +% - FOc [1x1] - Height of the center of the cube with respect to {F} [m] +% - FHa [1x1] - Height of the plane joining the points ai with respect to the frame {F} [m] +% - MHb [1x1] - Height of the plane joining the points bi with respect to the frame {M} [m] +% +% Outputs: + +% - stewart - updated Stewart structure with the added fields: +% - platform_F.Fa [3x6] - Its i'th column is the position vector of joint ai with respect to {F} +% - platform_M.Mb [3x6] - Its i'th column is the position vector of joint bi with respect to {M} + +arguments + stewart + args.Hc (1,1) double {mustBeNumeric, mustBePositive} = 60e-3 + args.FOc (1,1) double {mustBeNumeric} = 50e-3 + args.FHa (1,1) double {mustBeNumeric, mustBeNonnegative} = 15e-3 + args.MHb (1,1) double {mustBeNumeric, mustBeNonnegative} = 15e-3 +end + +assert(isfield(stewart.geometry, 'H'), 'stewart.geometry should have attribute H') +H = stewart.geometry.H; + +% We define the useful points of the cube with respect to the Cube's center. +% ${}^{C}C$ are the 6 vertices of the cubes expressed in a frame {C} which is located at the center of the cube and aligned with {F} and {M}. + +sx = [ 2; -1; -1]; +sy = [ 0; 1; -1]; +sz = [ 1; 1; 1]; + +R = [sx, sy, sz]./vecnorm([sx, sy, sz]); + +L = args.Hc*sqrt(3); + +Cc = R'*[[0;0;L],[L;0;L],[L;0;0],[L;L;0],[0;L;0],[0;L;L]] - [0;0;1.5*args.Hc]; + +CCf = [Cc(:,1), Cc(:,3), Cc(:,3), Cc(:,5), Cc(:,5), Cc(:,1)]; % CCf(:,i) corresponds to the bottom cube's vertice corresponding to the i'th leg +CCm = [Cc(:,2), Cc(:,2), Cc(:,4), Cc(:,4), Cc(:,6), Cc(:,6)]; % CCm(:,i) corresponds to the top cube's vertice corresponding to the i'th leg + +% We can compute the vector of each leg ${}^{C}\hat{\bm{s}}_{i}$ (unit vector from ${}^{C}C_{f}$ to ${}^{C}C_{m}$). + +CSi = (CCm - CCf)./vecnorm(CCm - CCf); + +% We now which to compute the position of the joints $a_{i}$ and $b_{i}$. +Fa = CCf + [0; 0; args.FOc] + ((args.FHa-(args.FOc-args.Hc/2))./CSi(3,:)).*CSi; +Mb = CCf + [0; 0; args.FOc-H] + ((H-args.MHb-(args.FOc-args.Hc/2))./CSi(3,:)).*CSi; + +stewart.platform_F.Fa = Fa; +stewart.platform_M.Mb = Mb; +#+end_src + * Footnotes -[fn:1]Different architecture exists, typically referred as "6-SPS" (Spherical, Prismatic, Spherical) or "6-UPS" 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za?!F(ouJt*cC>3T#ygJ?Y+j&=gWFM90PYd;tE2oq3y64ua_bbf?r-HSm|IwkHgznw z%+OOzS9CxKU$Z2Jp`EMA-B07%x4nZ1H>l!*)XoHqI+#<*9h3Ms&(tiwX4$?L``(&i@~=I^2LuY&>$b#~Cbu)EL#-j<%e*)dMhz|Yi@F}V5Du@>pT-~?F!=$|__C6XP;Q(a||UELYP z5?Cp@PLpU)?G$=Y{G&v$!d-**aZLI$o2ZVOiW zZTf~XVl9QJ!ZHNfZNkEH9}cosrd)8kx@J-&T{k%&oY+O^8+10QMeZ`7CoZwS+Hi0^ zgql9bUR>_lLQ~(oG|n<=QmOstTvt*^9c1|ENUzL{@a+v-ym&I>nS*Z&2AtMQG+Vc;qVCy3kYDbuqbLPVf`P0d8Baw diff --git a/simscape-nano-hexapod.tex b/simscape-nano-hexapod.tex index 9da2ae2..86b4408 100644 --- a/simscape-nano-hexapod.tex +++ b/simscape-nano-hexapod.tex @@ -1,4 +1,4 @@ -% Created 2025-02-19 Wed 19:45 +% Created 2025-03-28 Fri 14:30 % Intended LaTeX compiler: pdflatex \documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt} @@ -45,7 +45,7 @@ However, the development of the Nano Active Stabilization System (NASS) now requ To develop this model, the architecture of the active platform must first be determined. The selection of an appropriate architecture begins with a review of existing positioning stages that incorporate active platforms similar to NASS (Section \ref{ssec:nhexa_sample_stages}). -This review reveals two distinctive features of the NASS that set it apart from existing systems: the fact that the active platform is continuously rotating and its requirement to accommodate high payload masses. +This review reveals two distinctive features of the NASS that set it apart from existing systems: the fact that the active platform is continuously rotating and its requirement to accommodate variable payload masses. In existing systems, the sample mass is typically negligible compared to the stage mass, whereas in NASS, the sample mass significantly influences the system's dynamic behavior. These distinctive requirements drive the selection of the active platform architecture. @@ -53,7 +53,7 @@ In Section \ref{ssec:nhexa_active_platforms}, different active platform configur \section{Sample Stages with Active Control} \label{ssec:nhexa_sample_stages} -The positioning of samples with respect to to X-ray beam, that can be focused to sizes below 100 nanometers, presents significant challenges, because mechanical positioning systems are typically limited to micron-scale accuracy. +The positioning of samples with respect to X-ray beam, that can be focused to sizes below 100 nanometers, presents significant challenges, because mechanical positioning systems are typically limited to micron-scale accuracy. To overcome this limitation, external metrology systems have been implemented to measure sample positions with nanometer accuracy, enabling real-time feedback control for sample stabilization. A review of existing sample stages with active vibration control reveals various approaches to implementing such feedback systems. @@ -179,7 +179,7 @@ It must accommodate payloads up to 50 kg while maintaining high dynamical perfor For light samples, the typical design strategy of maximizing actuator stiffness works well because resonance frequencies in the kilohertz range can be achieved, enabling control bandwidths up to 100 Hz. However, achieving such resonance frequencies with a 50 kg payload would require unrealistic stiffness values of approximately \(2000\,N/\mu m\). This limitation necessitates alternative control approaches, and the High Authority Control/Low Authority Control (HAC-LAC) strategy is proposed to address this challenge. -Consequently, the design must incorporate force sensors for active damping. +To this purpose, the design includes force sensors for active damping. Compliant mechanisms must also be used to eliminate friction and backlash, which would otherwise compromise the nano-positioning capabilities. Two primary categories of positioning platform architectures are considered: serial and parallel mechanisms. @@ -257,7 +257,7 @@ These theoretical foundations form the basis for subsequent design decisions and \label{ssec:nhexa_stewart_platform_architecture} The Stewart platform consists of two rigid platforms connected by six parallel struts (Figure \ref{fig:nhexa_stewart_architecture}). -Each strut incorporates an active prismatic joint that allows for controlled length variation, with its ends attached to the fixed and mobile platforms through joints. +Each strut is modelled with an active prismatic joint that allows for controlled length variation, with its ends attached to the fixed and mobile platforms through joints. The typical configuration consists of a universal joint at one end and a spherical joint at the other, providing the necessary degrees of freedom\footnote{Different architecture exists, typically referred as ``6-SPS'' (Spherical, Prismatic, Spherical) or ``6-UPS'' (Universal, Prismatic, Spherical)}. \begin{figure}[htbp] @@ -291,7 +291,6 @@ This is summarized in Figure \ref{fig:nhexa_stewart_notations}. \section{Kinematic Analysis} \label{ssec:nhexa_stewart_platform_kinematics} -The kinematic analysis of the Stewart platform involves understanding the geometric relationships between the platform position/orientation and the actuator lengths without considering the forces involved. \paragraph{Loop Closure} The foundation of the kinematic analysis lies in the geometric constraints imposed by each strut, which can be expressed using loop closure equations. @@ -301,7 +300,7 @@ For each strut \(i\) (illustrated in Figure \ref{fig:nhexa_stewart_loop_closure} {}^A\bm{P}_B = {}^A\bm{a}_i + l_i{}^A\hat{\bm{s}}_i - \underbrace{{}^B\bm{b}_i}_{{}^A\bm{R}_B {}^B\bm{b}_i} \quad \text{for } i=1 \text{ to } 6 \end{equation} -This equation links the pose variables \({}^A\bm{P}\) and \({}^A\bm{R}_B\), the position vectors describing the known geometry of the base and the moving platform, \(\bm{a}_i\) and \(\bm{b}_i\), and the strut vector \(l_i {}^A\hat{\bm{s}}_i\): +This equation links the pose\footnote{The \emph{pose} represents the position and orientation of an object} variables \({}^A\bm{P}\) and \({}^A\bm{R}_B\), the position vectors describing the known geometry of the base and the moving platform, \(\bm{a}_i\) and \(\bm{b}_i\), and the strut vector \(l_i {}^A\hat{\bm{s}}_i\): \begin{figure}[htbp] \centering @@ -336,11 +335,11 @@ This approximation, which is developed in subsequent sections through the Jacobi \label{ssec:nhexa_stewart_platform_jacobian} The Jacobian matrix plays a central role in analyzing the Stewart platform's behavior, providing a linear mapping between the platform and actuator velocities. While the previously derived kinematic relationships are essential for position analysis, the Jacobian enables velocity analysis and forms the foundation for both static and dynamic studies. -\paragraph{Jacobian Computation - Velocity Loop Closure} +\paragraph{Jacobian Computation} As discussed in Section \ref{ssec:nhexa_stewart_platform_kinematics}, the strut lengths \(\bm{\mathcal{L}}\) and the platform pose \(\bm{\mathcal{X}}\) are related through a system of nonlinear algebraic equations representing the kinematic constraints imposed by the struts. -By taking the time derivative of the position loop close \eqref{eq:nhexa_loop_closure}, the \emph{velocity loop closure} is obtained \eqref{eq:nhexa_loop_closure_velocity}. +By taking the time derivative of the position loop close \eqref{eq:nhexa_loop_closure}, equation \eqref{eq:nhexa_loop_closure_velocity}\footnote{Such equation is called the \emph{velocity loop closure}} is obtained. \begin{equation}\label{eq:nhexa_loop_closure_velocity} {}^A\bm{v}_p + {}^A \dot{\bm{R}}_B {}^B\bm{b}_i + {}^A\bm{R}_B \underbrace{{}^B\dot{\bm{b}_i}}_{=0} = \dot{l}_i {}^A\hat{\bm{s}}_i + l_i {}^A\dot{\hat{\bm{s}}}_i + \underbrace{{}^A\dot{\bm{a}}_i}_{=0} @@ -400,10 +399,10 @@ While this approximation offers limited value for inverse kinematics, which can \paragraph{Range validity of the approximate inverse kinematics} The accuracy of the Jacobian-based forward kinematics solution was estimated by a simple 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. +For a series of platform positions, 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 \eqref{eq:nhexa_forward_kinematics_approximate}, from which the error between the estimated and true poses can be calculated, both in terms of position \(\epsilon_D\) and orientation \(\epsilon_R\). -The estimation errors in the \(x\), \(y\), and \(z\) directions are shown in Figure \ref{fig:nhexa_forward_kinematics_approximate_errors}. +For motion strokes from \(1\,\mu m\) to \(10\,mm\), the errors are estimated for all direction of motion, and the worst case errors 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. Since the maximum required stroke of the nano-hexapod (\(\approx 100\,\mu m\)) is three orders of magnitude smaller than its overall size (\(\approx 100\,mm\)), the Jacobian matrix can be considered constant throughout the workspace. @@ -412,7 +411,7 @@ It can be computed once at the rest position and used for both forward and inver \begin{figure}[htbp] \centering \includegraphics[scale=1]{figs/nhexa_forward_kinematics_approximate_errors.png} -\caption{\label{fig:nhexa_forward_kinematics_approximate_errors}Errors associated with the use of the Jacobian matrix to solve the forward kinematic problem. A Stewart platform with a height of \(100\,mm\) was used to perform this analysis} +\caption{\label{fig:nhexa_forward_kinematics_approximate_errors}Errors associated with the use of the Jacobian matrix to solve the forward kinematic problem. A Stewart platform with a height of \(100\,mm\) was used to perform this analysis. \(\epsilon_D\) corresponds to the distance between the true positioin and the estimated position. \(\epsilon_R\) corresponds to the angular motion between the true orientation and the estimated orientation.} \end{figure} \paragraph{Static Forces} @@ -481,7 +480,7 @@ These relationships reveal that the overall platform stiffness and compliance ch \end{itemize} This geometric dependency means that the platform's stiffness varies throughout its workspace, as the Jacobian matrix changes with the platform's position and orientation. -For the NASS application, where the workspace is small compared to the platform dimensions, these variations can be considered minimal. +For the NASS application, where the workspace is small compared to the platform dimensions, these variations can be considered negligible. However, the initial geometric configuration significantly affects the overall stiffness characteristics. The relationship between maximum stroke and stiffness presents another important design consideration. As both parameters are influenced by the geometric configuration, their optimization involves inherent trade-offs that must be carefully balanced based on the application requirements. @@ -490,7 +489,6 @@ The optimization of this configuration to achieve the desired stiffness while ha \section{Dynamical Analysis} \label{ssec:nhexa_stewart_platform_dynamics} -The dynamic behavior of a Stewart platform can be analyzed using various approaches depending on the desired level of model fidelity. For initial analysis, a simplified representation of the system has been developed. This model assumes perfectly rigid bodies for both the platform and base, connected by massless struts through ideal joints that exhibit neither friction nor compliance. @@ -555,7 +553,7 @@ Through this multi-body modeling approach, each component model (including joint The analysis is structured as follows. First, the multi-body model is developed, and the geometric parameters, inertial properties, and actuator characteristics are established (Section \ref{ssec:nhexa_model_def}). -The model was then validated through comparison with the analytical equations in a simplified configuration (Section \ref{ssec:nhexa_model_validation}). +The model is then validated through comparison with the analytical equations in a simplified configuration (Section \ref{ssec:nhexa_model_validation}). Finally, the validated model is employed to analyze the nano-hexapod dynamics, from which insights for the control system design are derived (Section \ref{ssec:nhexa_model_dynamics}). \section{Model Definition} \label{ssec:nhexa_model_def} @@ -970,7 +968,7 @@ Additionally, the distance of the loci from the \(-1\) point provides informatio \end{center} \subcaption{\label{fig:nhexa_decentralized_hac_iff_root_locus}Characteristic Loci} \end{subfigure} -\caption{\label{fig:nhexa_decentralized_hac_iff_results}Decentralized HAC-IFF} +\caption{\label{fig:nhexa_decentralized_hac_iff_results}Decentralized HAC-IFF. Loop gain (\subref{fig:nhexa_decentralized_hac_iff_loop_gain}) is used for the design of the controller and to estimate the disturbance rejection performances. Characteristic Loci (\subref{fig:nhexa_decentralized_hac_iff_root_locus}) is used to verify the stability and robustness of the feedback loop.} \end{figure} \section*{Conclusion}