From a3cf7fc5373b97958b5eb5348f0343a85dad8390 Mon Sep 17 00:00:00 2001 From: Thomas Dehaeze Date: Fri, 23 Apr 2021 00:00:49 +0200 Subject: [PATCH] First analysis of the nano-hexapod --- ...o_hexapod_cartesian_plant_encoder_comp.pdf | 1955 +++++++++++++++ ...o_hexapod_cartesian_plant_encoder_comp.png | Bin 0 -> 189385 bytes ...o_hexapod_comp_cartesian_plants_plates.pdf | 1932 ++++++++++++++ ...o_hexapod_comp_cartesian_plants_plates.png | Bin 0 -> 149515 bytes ...o_hexapod_comp_cartesian_plants_struts.pdf | Bin 0 -> 218390 bytes ...o_hexapod_comp_cartesian_plants_struts.png | Bin 0 -> 167189 bytes .../nano_hexapod_decentralized_schematic.pdf | Bin 0 -> 65520 bytes .../nano_hexapod_decentralized_schematic.png | Bin 0 -> 6265 bytes docs/figs/nano_hexapod_diagonal_plant_cok.pdf | 2172 ++++++++++++++++ docs/figs/nano_hexapod_diagonal_plant_cok.png | Bin 0 -> 231246 bytes docs/figs/nano_hexapod_effect_encoder.pdf | Bin 0 -> 191744 bytes docs/figs/nano_hexapod_effect_encoder.png | Bin 0 -> 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zbCGxwbDF`w@t?~PoL%Tkzm`kIDYa>bq!Rzx_>Iirxoe)fZ?^5}kOW`In#-bY)3XaN zV!hBmbbC)FK1MJqC$qTh>jiA^dp`SiFXTCGZSSSo&;3EOlCWz9hzHlAu&ue7meMva z5#&BZg(4Z}5r4+EETZ z;%3vlD$)+iO=0nAKXbZ6BkJ?_UN<(9Pw?6>u_7+gAIBy9_}m*SSjmwrt#kJC-!o?0 zwoboeyeVB+QBM~!!EU=f4Vkb>fvjebI>SaYpp?Y9PNg{)*9o8ykiU zg!1|E{%1y?T&*f$sp8z)na!BE8};|t4$jyAz6^ez> #+end_src @@ -15,64 +19,789 @@ simulinkproject('../'); #+end_src -#+begin_src matlab +#+begin_src matlab :eval no +addpath('nano_hexapod'); +open('nano_hexapod/nano_hexapod.slx') +#+end_src + +#+begin_src matlab :tangle no addpath('matlab/nano_hexapod'); open('matlab/nano_hexapod/nano_hexapod.slx') #+end_src -** Nano Hexapod object -The nano-hexapod can be initialized and configured using the =initializeNanoHexapodFinal= function. +** Nano Hexapod - Configuration +<> + +The nano-hexapod can be initialized and configured using the =initializeNanoHexapodFinal= function ([[sec:initializeNanoHexapodFinal][link]]). +#+begin_src matlab +n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... + 'flex_top_type', '3dof', ... + 'motion_sensor_type', 'struts', ... + 'actuator_type', '2dof', ... + 'MO_B', 150e-3); +#+end_src + +We initialize the identification parameters. +#+begin_src matlab +%% Options for Linearized +options = linearizeOptions; +options.SampleTime = 0; + +%% Name of the Simulink File +mdl = 'nano_hexapod'; + +%% Input/Output definition +clear io; io_i = 1; +io(io_i) = linio([mdl, '/F'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs +io(io_i) = linio([mdl, '/D'], 1, 'openoutput'); io_i = io_i + 1; % Relative Motion Outputs +#+end_src + +** Effect of encoders on the decentralized plant +<> + +We here wish to compare the plant from actuators to the encoders when the encoders are either fixed on the struts or on the plates. + +Identify the plant when the encoders are on the struts: #+begin_src matlab n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... 'flex_top_type', '3dof', ... 'motion_sensor_type', 'struts', ... 'actuator_type', '2dof'); + +Gs = linearize(mdl, io, 0.0, options); +Gs.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'}; +Gs.OutputName = {'D1', 'D2', 'D3', 'D4', 'D5', 'D6'}; #+end_src -** Jacobian from Solidworks Model - -Center of joints a_i with respect to {F}: +And identify the plant when the encoders are fixed on the plates: #+begin_src matlab -Fa = [[-86.05, -74.78, 22.49], - [86.05, -74.78, 22.49], - [107.79, -37.13, 22.49], - [21.74, 111.91, 22.49], - [-21.74, 111.91, 22.49], - [-107.79, -37.13, 22.49]]'*1e-3; +n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... + 'flex_top_type', '3dof', ... + 'motion_sensor_type', 'plates', ... + 'actuator_type', '2dof'); + +Gp = linearize(mdl, io, 0.0, options); +Gp.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'}; +Gp.OutputName = {'D1', 'D2', 'D3', 'D4', 'D5', 'D6'}; #+end_src -Center of joints a_i with respect to {M}: +The obtained plants are compared in Figure [[fig:nano_hexapod_effect_encoder]]. +#+begin_src matlab :exports none +freqs = logspace(0, 3, 1000); + +figure; +tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +set(gca,'ColorOrderIndex',1); +plot(freqs, abs(squeeze(freqresp(Gs('D1', 'F1'), freqs, 'Hz'))), '-', 'DisplayName', 'Struts'); +for i = 1:6 + set(gca,'ColorOrderIndex',1); + plot(freqs, abs(squeeze(freqresp(Gs(['D', num2str(i)], ['F', num2str(i)]), freqs, 'Hz'))), '-', ... + 'HandleVisibility', 'off'); +end +set(gca,'ColorOrderIndex',2); +plot(freqs, abs(squeeze(freqresp(Gp('D1', 'F1'), freqs, 'Hz'))), '-', 'DisplayName', 'Plates'); +for i = 1:6 + set(gca,'ColorOrderIndex',2); + plot(freqs, abs(squeeze(freqresp(Gp(['D', num2str(i)], ['F', num2str(i)]), freqs, 'Hz'))), '-', ... + 'HandleVisibility', 'off'); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]); +legend('location', 'northeast'); +ylim([1e-11, 1e-6]); + +ax2 = nexttile; +hold on; +for i = 1:6 + set(gca,'ColorOrderIndex',1); + plot(freqs, 180/pi*angle(squeeze(freqresp(Gs(['D', num2str(i)], ['F', num2str(i)]), freqs, 'Hz'))), '-'); +end +for i = 1:6 + set(gca,'ColorOrderIndex',2); + plot(freqs, 180/pi*angle(squeeze(freqresp(Gp(['D', num2str(i)], ['F', num2str(i)]), freqs, 'Hz'))), '-'); +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)]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/nano_hexapod_effect_encoder.pdf', 'width', 'full', 'height', 'tall'); +#+end_src + +#+name: fig:nano_hexapod_effect_encoder +#+caption: Comparison of the plants from actuator to associated encoder when the encoders are either fixed to the struts or to the plates +#+RESULTS: +[[file:figs/nano_hexapod_effect_encoder.png]] + +#+begin_question +Why do we have zeros at 400Hz and 800Hz when the encoders are fixed on the struts? +#+end_question + +** Effect of APA flexibility +<> + +First identify the plant for APA represented by 2DoF system: #+begin_src matlab -Mb = [[-28.47, -106.25, -22.50], - [28.47, -106.25, -22.50], - [106.25, 28.47, -22.50], - [77.78, 77.78, -22.50], - [-77.78, 77.78, -22.50], - [-106.25, 28.47, -22.50]]'*1e-3; +n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... + 'flex_top_type', '3dof', ... + 'motion_sensor_type', 'struts', ... + 'actuator_type', '2dof', ... + 'actuator_Ga', 2); + +Gs = linearize(mdl, io, 0.0, options); +Gs.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'}; +Gs.OutputName = {'D1', 'D2', 'D3', 'D4', 'D5', 'D6'}; #+end_src +First identify the plant for APA represented by a flexible element: #+begin_src matlab -Fb = Mb + [0; 0; 95e-3]; +n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... + 'flex_top_type', '3dof', ... + 'motion_sensor_type', 'struts', ... + 'actuator_type', 'flexible'); -si = Fb - Fa; -si = si./vecnorm(si); +Gf = linearize(mdl, io, 0.0, options); +Gf.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'}; +Gf.OutputName = {'D1', 'D2', 'D3', 'D4', 'D5', 'D6'}; #+end_src +#+begin_src matlab :exports none +freqs = logspace(0, 3, 1000); + +figure; +tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +set(gca,'ColorOrderIndex',1); +plot(freqs, abs(squeeze(freqresp(Gs('D1', 'F1'), freqs, 'Hz'))), '-', 'DisplayName', '2DoF'); +for i = 1:6 + set(gca,'ColorOrderIndex',1); + plot(freqs, abs(squeeze(freqresp(Gs(['D', num2str(i)], ['F', num2str(i)]), freqs, 'Hz'))), '-', ... + 'HandleVisibility', 'off'); +end +set(gca,'ColorOrderIndex',2); +plot(freqs, abs(squeeze(freqresp(Gf('D1', 'F1'), freqs, 'Hz'))), '-', 'DisplayName', 'Flexible'); +for i = 1:6 + set(gca,'ColorOrderIndex',2); + plot(freqs, abs(squeeze(freqresp(Gf(['D', num2str(i)], ['F', num2str(i)]), freqs, 'Hz'))), '-', ... + 'HandleVisibility', 'off'); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]); +legend('location', 'northeast'); +ylim([1e-11, 1e-6]); + +ax2 = nexttile; +hold on; +for i = 1:6 + set(gca,'ColorOrderIndex',1); + plot(freqs, 180/pi*angle(squeeze(freqresp(Gs(['D', num2str(i)], ['F', num2str(i)]), freqs, 'Hz'))), '-'); +end +for i = 1:6 + set(gca,'ColorOrderIndex',2); + plot(freqs, 180/pi*angle(squeeze(freqresp(Gf(['D', num2str(i)], ['F', num2str(i)]), freqs, 'Hz'))), '-'); +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)]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/nano_hexapod_effect_flexible_apa.pdf', 'width', 'full', 'height', 'tall'); +#+end_src + +#+name: fig:nano_hexapod_effect_flexible_apa +#+caption: Comparison of the plants from actuator to associated strut encoder when the APA are modelled with a 2DoF system of with a flexible one +#+RESULTS: +[[file:figs/nano_hexapod_effect_flexible_apa.png]] + +#+begin_important +The first resonance is strange when using the flexible APA model (Figure [[fig:nano_hexapod_effect_flexible_apa]]). +Otherwise, the 2DoF model matches quite well the flexible model considering its simplicity. +#+end_important + +** Nano Hexapod - Number of DoF +<> + +In this section, we wish to see how the configuration of each element changes the number of the states of the obtained system. + +The most minimalist model is the following: #+begin_src matlab -ki = ones(1,6); -bi = Fb; +n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '3dof', ... + 'flex_top_type', '2dof', ... + 'motion_sensor_type', 'struts', ... + 'actuator_type', '2dof'); +G = linearize(mdl, io, 0.0, options); #+end_src +#+begin_src matlab :results value replace :exports results +sprintf('There are %i states.', length(G.statename)) +#+end_src + +#+RESULTS: +: There are 24 states. + +These states are summarized on table [[tab:num_states_nano_hexapod]]. + +#+name: tab:num_states_nano_hexapod +#+caption: Number of states for the minimalist model +#+attr_latex: :environment tabularx :width 0.3\linewidth :align cc +#+attr_latex: :center t :booktabs t :float t +| Element | States | +|-----------+--------| +| Struts | 2*6 | +| Top Plate | 12 | +|-----------+--------| +| Total: | 24 | +#+TBLFM: @>$2=vsum(@I..@II) + +If we add axial stiffness on the top joints, we should add 2 states for each struts. +#+begin_src matlab +n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... + 'flex_top_type', '2dof', ... + 'motion_sensor_type', 'struts', ... + 'actuator_type', '2dof'); +G = linearize(mdl, io, 0.0, options); +#+end_src + +#+begin_src matlab :results value replace :exports results +sprintf('There are %i states.', length(G.statename)) +#+end_src + +#+RESULTS: +: There are 36 states. + +If we add torsional stiffness on the bottom joints, we should again add 2 states for each struts. +#+begin_src matlab +n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... + 'flex_top_type', '3dof', ... + 'motion_sensor_type', 'struts', ... + 'actuator_type', '2dof'); +G = linearize(mdl, io, 0.0, options); +#+end_src + +#+begin_src matlab :results value replace :exports results +sprintf('There are %i states.', length(G.statename)) +#+end_src + +#+RESULTS: +: There are 48 states. + +Finally, if we add axial stiffness on the bottom joint, we should add 2 states for each struts. +#+begin_src matlab +n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... + 'flex_top_type', '4dof', ... + 'motion_sensor_type', 'struts', ... + 'actuator_type', '2dof'); +G = linearize(mdl, io, 0.0, options); +#+end_src + +#+begin_src matlab :results value replace :exports results +sprintf('There are %i states.', length(G.statename)) +#+end_src + +#+RESULTS: +: There are 60 states. + +#+begin_important +Obtained number of states is very comprehensible. +Depending on the physical effects we want to model, we therefore know how many states are added when configuring the model. +#+end_important + +** Direct Velocity Feedback Plant +<> + +The transfer function from actuator forces $\tau_i$ to the encoder measurements $\mathcal{L}_i$ is now identified both when the encoders are fixed to the struts. +#+begin_src matlab +%% Options for Linearized +options = linearizeOptions; +options.SampleTime = 0; + +%% Name of the Simulink File +mdl = 'nano_hexapod'; + +%% Input/Output definition +clear io; io_i = 1; +io(io_i) = linio([mdl, '/F'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs +io(io_i) = linio([mdl, '/D'], 1, 'openoutput'); io_i = io_i + 1; % Force Sensors + +n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... + 'flex_top_type', '3dof', ... + 'motion_sensor_type', 'struts', ... + 'actuator_type', '2dof'); + +Gdvf = linearize(mdl, io, 0.0, options); +#+end_src + +The DC gain from actuator to relative motion sensor should be equal to (for the 2dof APA): +\[ \frac{1}{k + k_a + kk_a/k_e} \] + +Which is equal to: +#+begin_src matlab :results value replace :exports results :tangle no +sprintf('DCgain = %.2e [m/N]', 1/(n_hexapod.actuator.k(1) + n_hexapod.actuator.ka(1) + n_hexapod.actuator.k(1)*n_hexapod.actuator.ka(1)/n_hexapod.actuator.ke(1))) +#+end_src + +#+RESULTS: +: DCgain = 1.87e-08 [m/N] + +Let's verify that by looking at the DC gain of the $6 \times 6$ DVF plant in Table [[tab:dvf_dc_gain]]. +#+begin_src matlab :results value replace :exports results :tangle no +dcgain(Gdvf) +#+end_src + +#+name: tab:dvf_dc_gain +#+caption: DC gain of the DVF plant +#+attr_latex: :environment tabularx :width 0.9\linewidth :align cccccc +#+attr_latex: :center t :booktabs t :float t +#+RESULTS: +| 1.8617e-08 | -1.0408e-10 | 1.3034e-10 | 3.2559e-11 | -1.1188e-10 | 9.0385e-11 | +| -5.1839e-11 | 1.8593e-08 | -4.4868e-11 | 8.016e-11 | 4.3558e-11 | -1.1164e-10 | +| 5.1963e-12 | -6.9001e-12 | 1.8564e-08 | 3.0844e-11 | 4.0097e-11 | -3.4387e-11 | +| 1.9359e-11 | 1.7432e-10 | -5.0928e-11 | 1.855e-08 | 1.6406e-10 | 4.5757e-12 | +| -2.1185e-11 | 2.1724e-10 | 1.5333e-12 | -8.802e-11 | 1.8803e-08 | -4.6946e-11 | +| -1.1728e-11 | -5.7682e-11 | 1.6213e-10 | 2.1934e-12 | -1.6237e-10 | 1.8715e-08 | + +And the bode plot of the DVF plant is shown in Figure [[fig:nano_hexapod_struts_2dof_dvf_plant]]. +#+begin_src matlab :exports none +freqs = 5*logspace(0, 2, 1000); + +figure; +tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +plot(freqs, abs(squeeze(freqresp(Gdvf(1,1), freqs, 'Hz'))), '-', ... + 'DisplayName', '$\mathcal{L}_{i}/\tau_i$') +for i = 1:6 + set(gca,'ColorOrderIndex',1); + plot(freqs, abs(squeeze(freqresp(Gdvf(i,i), freqs, 'Hz'))), '-', ... + 'HandleVisibility', 'off'); +end +% Off diagonal terms +plot(freqs, abs(squeeze(freqresp(Gdvf(1, 2), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ... + 'DisplayName', '$F_{m,i}/\tau_j \quad i \neq j$'); +for i = 1:5 + for j = i+1:6 + plot(freqs, abs(squeeze(freqresp(Gdvf(i, j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ... + 'HandleVisibility', 'off'); + end +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]); +legend('location', 'southeast'); + +ax2 = nexttile; +hold on; +for i = 1:6 + set(gca,'ColorOrderIndex',1); + plot(freqs, 180/pi*angle(squeeze(freqresp(Gdvf(i,i), freqs, 'Hz'))), '-'); +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)]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/nano_hexapod_struts_2dof_dvf_plant.pdf', 'width', 'full', 'height', 'tall'); +#+end_src + +#+name: fig:nano_hexapod_struts_2dof_dvf_plant +#+caption: Bode plot of the transfer functions from actuator forces $\tau_i$ to relative motion sensors attached to the struts $\mathcal{L}_i$. Diagonal terms are shown in blue, and off-diagonal terms in black. +#+RESULTS: +[[file:figs/nano_hexapod_struts_2dof_dvf_plant.png]] + +** Integral Force Feedback Plant +<> + +The transfer function from actuators to force sensors is identified. +#+begin_src matlab +%% Options for Linearized +options = linearizeOptions; +options.SampleTime = 0; + +%% Name of the Simulink File +mdl = 'nano_hexapod'; + +%% Input/Output definition +clear io; io_i = 1; +io(io_i) = linio([mdl, '/F'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs +io(io_i) = linio([mdl, '/Fm'], 1, 'openoutput'); io_i = io_i + 1; % Force Sensors + +n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... + 'flex_top_type', '3dof', ... + 'motion_sensor_type', 'struts', ... + 'actuator_type', '2dof'); + +Giff = linearize(mdl, io, 0.0, options); +#+end_src + +This is corresponding to the dynamics for the Integral Force Feedback (IFF) control strategy. + +The bode plot is shown in Figure [[fig:nano_hexapod_struts_2dof_iff_plant]]. +#+begin_src matlab :exports none +freqs = 5*logspace(0, 2, 1000); + +figure; +tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +plot(freqs, abs(squeeze(freqresp(Giff(1,1), freqs, 'Hz'))), '-', ... + 'DisplayName', '$F_{m,i}/\tau_i$') +for i = 1:6 + set(gca,'ColorOrderIndex',1); + plot(freqs, abs(squeeze(freqresp(Giff(i,i), freqs, 'Hz'))), '-', ... + 'HandleVisibility', 'off'); +end +% Off diagonal terms +plot(freqs, abs(squeeze(freqresp(Giff(1, 2), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ... + 'DisplayName', '$F_{m,i}/\tau_j \quad i \neq j$'); +for i = 1:5 + for j = i+1:6 + plot(freqs, abs(squeeze(freqresp(Giff(i, j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ... + 'HandleVisibility', 'off'); + end +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude [N/N]'); set(gca, 'XTickLabel',[]); +legend('location', 'southeast'); + +ax2 = nexttile; +hold on; +for i = 1:6 + set(gca,'ColorOrderIndex',1); + plot(freqs, 180/pi*angle(squeeze(freqresp(Giff(i,i), freqs, 'Hz'))), '-'); +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)]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/nano_hexapod_struts_2dof_iff_plant.pdf', 'width', 'full', 'height', 'tall'); +#+end_src + +#+name: fig:nano_hexapod_struts_2dof_iff_plant +#+caption: Bode plot of the transfer functions from actuator forces $\tau_i$ to force sensors $F_{m,i}$. Diagonal terms are shown in blue, and off-diagonal terms in black. +#+RESULTS: +[[file:figs/nano_hexapod_struts_2dof_iff_plant.png]] + +** Decentralized Plant - Cartesian coordinates +<> +*** Introduction :ignore: +Consider the plant shown in Figure [[fig:nano_hexapod_decentralized_schematic]] with: +- $\tau$ the 6 input forces (APA) +- $d\mathcal{L}$ the relative motion sensor outputs (encoders) +- $\mathcal{X}$ the motion of the top platform measured with "perfect" 6-dof sensor +- $J_a$ and $J_s$ the Jacobians for the actuators and sensors + +#+begin_src latex :file nano_hexapod_decentralized_schematic.pdf +\begin{tikzpicture} + % Blocs + \node[block={2.0cm}{2.0cm}] (P) {Plant}; + \coordinate[] (inputF) at (P.west); + \coordinate[] (outputL) at ($(P.south east)!0.8!(P.north east)$); + \coordinate[] (outputX) at ($(P.south east)!0.2!(P.north east)$); + + \node[block, left= of inputF] (Ja) {$\bm{J}^{-T}_a$}; + \node[block, right= of outputL] (Js) {$\bm{J}^{-1}_s$}; + + % Connections and labels + \draw[->] ($(Ja.west)+(-1,0)$) -- (Ja.west) node[above left]{$\bm{\mathcal{F}}$}; + \draw[->] (Ja.east) -- (inputF) node[above left]{$\bm{\tau}$}; + \draw[->] (outputL) -- (Js.west) node[above left]{$d\bm{\mathcal{L}}$}; + \draw[->] (Js.east) -- ++(1, 0) node[above left]{$d\bm{\mathcal{X}}$}; + \draw[->] (outputX.east) -- ++(1, 0) node[above left]{$\bm{\mathcal{X}}$}; +\end{tikzpicture} +#+end_src + +#+name: fig:nano_hexapod_decentralized_schematic +#+caption: Plant in the cartesian Frame +#+RESULTS: +[[file:figs/nano_hexapod_decentralized_schematic.png]] + +*** Verification of the Sensor Jacobian +The "perfect" sensor output $\mathcal{X}$ is used to verify that the sensor Jacobian is working correctly both when the encoders are fixed to the struts and to the plates. + +Let's then identify the plant for both configuration, and compare the transfer functions from $\mathcal{F}$ to $d\mathcal{X}$ and to $\mathcal{X}$. +#+begin_src matlab +%% Options for Linearized +options = linearizeOptions; +options.SampleTime = 0; + +%% Name of the Simulink File +mdl = 'nano_hexapod'; + +%% Input/Output definition +clear io; io_i = 1; +io(io_i) = linio([mdl, '/F'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs +io(io_i) = linio([mdl, '/D'], 1, 'openoutput'); io_i = io_i + 1; % Relative Motion Outputs +io(io_i) = linio([mdl, '/X'], 1, 'openoutput'); io_i = io_i + 1; % Relative Motion Outputs +#+end_src + +Start when the encoders are fixed on the struts. +#+begin_src matlab +n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... + 'flex_top_type', '3dof', ... + 'motion_sensor_type', 'struts', ... + 'actuator_type', '2dof', ... + 'MO_B', 150e-3); +Gs = linearize(mdl, io, 0.0, options); +Gs.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'}; +Gs.OutputName = {'D1', 'D2', 'D3', 'D4', 'D5', 'D6', ... + 'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'}; + +% Cartesian plant using the Jacobians +Gsc = inv(n_hexapod.geometry.Js)*Gs({'D1', 'D2', 'D3', 'D4', 'D5', 'D6'}, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'})*inv(n_hexapod.geometry.J)'; +% Cartesian plant using the perfect sensor +Gsp = -Gs({'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'}, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'})*inv(n_hexapod.geometry.J)'; +#+end_src + +The diagonal elements of the plant are shown in Figure [[fig:nano_hexapod_comp_cartesian_plants_struts]]. +#+begin_src matlab :exports none +freqs = 5*logspace(0, 2, 1000); + +labels = {'$D_x/\mathcal{F}_x$', '$D_y/\mathcal{F}_y$', '$D_z/\mathcal{F}_z$', ... + '$R_x/\mathcal{M}_x$', '$R_y/\mathcal{M}_y$', '$R_z/\mathcal{M}_z$'}; + +figure; +tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +for i = 1:6 + plot(freqs, abs(squeeze(freqresp(Gsc(i,i), freqs, 'Hz'))), '-', ... + 'DisplayName', labels{i}); +end +set(gca,'ColorOrderIndex',1); +for i = 1:6 + plot(freqs, abs(squeeze(freqresp(Gsp(i,i), freqs, 'Hz'))), '--', ... + 'HandleVisibility', 'off'); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]); +legend('location', 'southwest'); +ylim([1e-11, 1e-4]); + +ax2 = nexttile; +hold on; +for i = 1:6 + plot(freqs, 180/pi*angle(squeeze(freqresp(Gsc(i,i), freqs, 'Hz'))), '-'); +end +set(gca,'ColorOrderIndex',1); +for i = 1:6 + plot(freqs, 180/pi*angle(squeeze(freqresp(Gsp(i,i), freqs, 'Hz'))), '--'); +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)]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/nano_hexapod_comp_cartesian_plants_struts.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:nano_hexapod_comp_cartesian_plants_struts +#+caption: Bode plot of the diagonal elements of the decentralized (cartesian) plant when using the sensor Jacobian (solid) and when using "perfect" 6dof sensor (dashed). The encoders are fixed on the struts. +#+RESULTS: +[[file:figs/nano_hexapod_comp_cartesian_plants_struts.png]] + +The same if performed when the encoders are fixed to the plates. +#+begin_src matlab +n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... + 'flex_top_type', '3dof', ... + 'motion_sensor_type', 'plates', ... + 'actuator_type', '2dof', ... + 'MO_B', 150e-3); +Gp = linearize(mdl, io, 0.0, options); +Gp.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'}; +Gp.OutputName = {'D1', 'D2', 'D3', 'D4', 'D5', 'D6', ... + 'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'}; + +% Cartesian plant using the Jacobians +Gpc = inv(n_hexapod.geometry.Js)*Gp({'D1', 'D2', 'D3', 'D4', 'D5', 'D6'}, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'})*inv(n_hexapod.geometry.J)'; +% Cartesian plant using the perfect sensor +Gpp = -Gp({'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'}, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'})*inv(n_hexapod.geometry.J)'; +#+end_src + +The obtained bode plots are shown in Figure [[fig:nano_hexapod_comp_cartesian_plants_plates]]. +#+begin_src matlab :exports none +freqs = 5*logspace(0, 2, 1000); + +labels = {'$D_x/\mathcal{F}_x$', '$D_y/\mathcal{F}_y$', '$D_z/\mathcal{F}_z$', ... + '$R_x/\mathcal{M}_x$', '$R_y/\mathcal{M}_y$', '$R_z/\mathcal{M}_z$'}; + +figure; +tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +for i = 1:6 + plot(freqs, abs(squeeze(freqresp(Gpc(i,i), freqs, 'Hz'))), '-', ... + 'DisplayName', labels{i}); +end +set(gca,'ColorOrderIndex',1); +for i = 1:6 + plot(freqs, abs(squeeze(freqresp(Gpp(i,i), freqs, 'Hz'))), '--', ... + 'HandleVisibility', 'off'); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]); +legend('location', 'southwest'); +ylim([1e-11, 1e-4]); + +ax2 = nexttile; +hold on; +for i = 1:6 + plot(freqs, 180/pi*angle(squeeze(freqresp(Gpc(i,i), freqs, 'Hz'))), '-'); +end +set(gca,'ColorOrderIndex',1); +for i = 1:6 + plot(freqs, 180/pi*angle(squeeze(freqresp(Gpp(i,i), freqs, 'Hz'))), '--'); +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)]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/nano_hexapod_comp_cartesian_plants_plates.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:nano_hexapod_comp_cartesian_plants_plates +#+caption: Bode plot of the diagonal elements of the decentralized (cartesian) plant when using the sensor Jacobian (solid) and when using "perfect" 6dof sensor (dashed). The encoders are fixed on the plates. +#+RESULTS: +[[file:figs/nano_hexapod_comp_cartesian_plants_plates.png]] + +#+begin_important +The Jacobian for the encoders is working properly both when the encoders are fixed to the plates or to the struts. + +However, then the encoders are fixed to the struts, there is a mismatch between the estimated motion and the measured motion above 100Hz due to a complex conjugate zero. +#+end_important + +*** Comparison of the decentralized plants + +The decentralized plants are now compared whether the encoders are fixed on the struts or on the plates in Figure [[fig:nano_hexapod_cartesian_plant_encoder_comp]]. +#+begin_src matlab :exports none +freqs = 5*logspace(0, 2, 1000); + +labels = {'$D_x/\mathcal{F}_x$', '$D_y/\mathcal{F}_y$', '$D_z/\mathcal{F}_z$', ... + '$R_x/\mathcal{M}_x$', '$R_y/\mathcal{M}_y$', '$R_z/\mathcal{M}_z$'}; + +figure; +tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +for i = 1:6 + plot(freqs, abs(squeeze(freqresp(Gsc(i,i), freqs, 'Hz'))), '-', ... + 'DisplayName', labels{i}); +end +set(gca,'ColorOrderIndex',1); +for i = 1:6 + plot(freqs, abs(squeeze(freqresp(Gpc(i,i), freqs, 'Hz'))), '--', ... + 'HandleVisibility', 'off'); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]); +legend('location', 'southwest'); +ylim([1e-11, 1e-4]); + +ax2 = nexttile; +hold on; +for i = 1:6 + plot(freqs, 180/pi*angle(squeeze(freqresp(Gsc(i,i), freqs, 'Hz'))), '-'); +end +set(gca,'ColorOrderIndex',1); +for i = 1:6 + plot(freqs, 180/pi*angle(squeeze(freqresp(Gpc(i,i), freqs, 'Hz'))), '--'); +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)]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/nano_hexapod_cartesian_plant_encoder_comp.pdf', 'width', 'full', 'height', 'tall'); +#+end_src + +#+name: fig:nano_hexapod_cartesian_plant_encoder_comp +#+caption: Bode plot of the "cartesian" plant (transfer function from $\mathcal{F}$ to $d\mathcal{X}$) when the encoders are fixed on the struts (solid) and on the plates (dashed) +#+RESULTS: +[[file:figs/nano_hexapod_cartesian_plant_encoder_comp.png]] + +** Decentralized Plant - Decoupling at the Center of Stiffness +<> + +*** Center of Stiffness +<> + +Let's define some parameters: +#+begin_src matlab +si = n_hexapod.geometry.si; % Orientation of struts +bi = n_hexapod.geometry.Fb; % Location of bi w.r.t. {F} +ki = ones(1,6); % Normalized strut stiffness +#+end_src + +In order to find is the Center of Stiffness (CoK) exists, we have to verify is the following is diagonal: #+begin_src matlab :results value replace :exports both ki.*si*si' #+end_src #+RESULTS: -| 1.8977 | 2.4659e-17 | 1.3383e-18 | +| 1.8977 | 2.4659e-17 | 5.1838e-19 | | 2.4659e-17 | 1.8977 | -2.3143e-05 | -| 1.3383e-18 | -2.3143e-05 | 2.2046 | +| 5.1838e-19 | -2.3143e-05 | 2.2046 | +And we can find the location of the CoK with respect to {F}: #+begin_src matlab OkX = (ki.*cross(bi, si)*si')/(ki.*si*si'); Ok = [OkX(3,2);OkX(1,3);OkX(2,1)] @@ -89,33 +818,36 @@ ans = Ok The center of the cube is therefore 52.7mm above the bottom platform {F} frame. -Let's compute the Jacobian at this location: +Let's initialize the hexapod with frame {A} and {B} at the CoK: #+begin_src matlab -Jk = [si', cross(bi - Ok, si)']; +n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... + 'flex_top_type', '3dof', ... + 'motion_sensor_type', 'struts', ... + 'actuator_type', '2dof', ... + 'MO_B', Ok(3)-95e-3); #+end_src -And the stiffness matrix: +And the (normalized) stiffness matrix is computed as follows: #+begin_src matlab :results value replace -Jk'*diag(ki)*Jk +n_hexapod.geometry.J'*diag(ki)*n_hexapod.geometry.J #+end_src +#+name: tab:stiffness_matrix_cok +#+caption: Normalized Stiffness Matrix - Center of Stiffness +#+attr_latex: :environment tabularx :width 0.9\linewidth :align cccccc +#+attr_latex: :center t :booktabs t :float t #+RESULTS: -| 1.8977 | 0 | 0 | 0 | -3.4694e-18 | 2.5509e-06 | -| 0 | 1.8977 | -2.3143e-05 | 4.1751e-06 | 1.3878e-17 | 0 | -| 0 | -2.3143e-05 | 2.2046 | -5.0916e-11 | -3.4694e-18 | 0 | -| 0 | 4.1751e-06 | -5.0916e-11 | 0.012594 | 2.1684e-19 | 8.6736e-19 | -| -3.2704e-18 | 0 | -4.206e-18 | 3.9451e-19 | 0.012594 | -9.3183e-08 | -| 2.5509e-06 | 0 | 0 | 8.6736e-19 | -9.3183e-08 | 0.043362 | +| 1.8977 | 0 | 0 | 0 | -2.0817e-17 | -1.5311e-06 | +| 0 | 1.8977 | -2.3143e-05 | 4.175e-06 | 0 | 0 | +| 0 | -2.3143e-05 | 2.2046 | 4.7422e-06 | 0 | 0 | +| 0 | 4.175e-06 | 4.7422e-06 | 0.012594 | 2.1684e-19 | -8.6736e-19 | +| -1.8521e-17 | 0 | 0 | 0 | 0.012594 | -9.3183e-08 | +| -1.5311e-06 | -6.9389e-18 | 2.7756e-17 | -8.6736e-19 | -9.3183e-08 | 0.043362 | -** Jacobian for encoders on the plates +And we indeed obtain a diagonal stiffness matrix. -*Goal*: Compute the 6-DoF motion of the top platform (relative to the bottom platform) from the 6 encoder measurements. +*** Obtained plant -The main differences with the case where the encoders are parallel with the strut are: -- the encoder and ruler may not be parallel anymore as the top platform is moving -- the derivation of the Jacobian (if possible) will not be the same - -** Compare encoders on the struts and encoders on the platforms #+begin_src matlab %% Options for Linearized options = linearizeOptions; @@ -128,713 +860,92 @@ mdl = 'nano_hexapod'; clear io; io_i = 1; io(io_i) = linio([mdl, '/F'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs io(io_i) = linio([mdl, '/D'], 1, 'openoutput'); io_i = io_i + 1; % Relative Motion Outputs +io(io_i) = linio([mdl, '/X'], 1, 'openoutput'); io_i = io_i + 1; % Relative Motion Outputs #+end_src #+begin_src matlab -n_hexapod.motion_sensor = struct(); -n_hexapod.motion_sensor.type = 1; % 1: Leg / 2: Plate -#+end_src - -#+begin_src matlab -Gs = linearize(mdl, io, 0.0, options); -Gs.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'}; -Gs.OutputName = {'D1', 'D2', 'D3', 'D4', 'D5', 'D6'}; -#+end_src - -#+begin_src matlab -n_hexapod.motion_sensor = struct(); -n_hexapod.motion_sensor.type = 2; % 1: Leg / 2: Plate -#+end_src - -#+begin_src matlab -Gp = linearize(mdl, io, 0.0, options); -Gp.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'}; -Gp.OutputName = {'D1', 'D2', 'D3', 'D4', 'D5', 'D6'}; -#+end_src - -#+begin_src matlab :exports none -freqs = logspace(0, 3, 1000); - -figure; -tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); - -ax1 = nexttile([2,1]); -hold on; -for i = 1:6 - plot(freqs, abs(squeeze(freqresp(Gs(['D', num2str(i)], ['F', num2str(i)]), freqs, 'Hz')))); -end -for i = 1:5 - for j = i+1:6 - plot(freqs, abs(squeeze(freqresp(Gs(['D', num2str(i)], ['F', num2str(j)]), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2]); - end -end -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]); -ylim([1e-10, 1e-6]); - -ax2 = nexttile; -hold on; -for i = 1:6 - plot(freqs, 180/pi*angle(squeeze(freqresp(Gs(['D', num2str(i)], ['F', num2str(i)]), freqs, 'Hz')))); -end -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); -ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); -ylim([0, 180]); -yticks([-180, -90, 0, 90, 180]); - -linkaxes([ax1,ax2],'x'); -xlim([freqs(1), freqs(end)]); -#+end_src - -#+begin_src matlab :exports none -freqs = logspace(0, 3, 1000); - -figure; -tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); - -ax1 = nexttile([2,1]); -hold on; -for i = 1:6 - plot(freqs, abs(squeeze(freqresp(Gp(['D', num2str(i)], ['F', num2str(i)]), freqs, 'Hz')))); -end -for i = 1:5 - for j = i+1:6 - plot(freqs, abs(squeeze(freqresp(Gp(['D', num2str(i)], ['F', num2str(j)]), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2]); - end -end -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]); -ylim([1e-10, 1e-6]); - -ax2 = nexttile; -hold on; -for i = 1:6 - plot(freqs, 180/pi*angle(squeeze(freqresp(Gp(['D', num2str(i)], ['F', num2str(i)]), freqs, 'Hz')))); -end -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); -ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); -ylim([0, 180]); -yticks([-180, -90, 0, 90, 180]); - -linkaxes([ax1,ax2],'x'); -xlim([freqs(1), freqs(end)]); -#+end_src - -** Nano Hexapod -#+begin_src matlab -%% Options for Linearized -options = linearizeOptions; -options.SampleTime = 0; - -%% Name of the Simulink File -mdl = 'nano_hexapod'; - -%% Input/Output definition -clear io; io_i = 1; -io(io_i) = linio([mdl, '/F'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs -io(io_i) = linio([mdl, '/Fe'], 1, 'openinput'); io_i = io_i + 1; % External Force -io(io_i) = linio([mdl, '/D'], 1, 'openoutput'); io_i = io_i + 1; % Relative Motion Outputs -io(io_i) = linio([mdl, '/Fm'], 1, 'openoutput'); io_i = io_i + 1; % Force Sensors -io(io_i) = linio([mdl, '/X'], 1, 'openoutput'); io_i = io_i + 1; % Absolute Motion of top platform - -%% Run the linearization G = linearize(mdl, io, 0.0, options); -G.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6', ... - 'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'}; +G.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'}; G.OutputName = {'D1', 'D2', 'D3', 'D4', 'D5', 'D6', ... - 'Fm1', 'Fm2', 'Fm3', 'Fm4', 'Fm5', 'Fm6' ... 'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'}; + +% % Cartesian plant using the Jacobians +% Gsc = inv(n_hexapod.geometry.Js)*Gs({'D1', 'D2', 'D3', 'D4', 'D5', 'D6'}, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'})*inv(n_hexapod.geometry.J)'; +% % Cartesian plant using the perfect sensor +% Gsp = -Gs({'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'}, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'})*inv(n_hexapod.geometry.J)'; #+end_src -#+begin_src matlab :results output replace :exports results :tangle no -size(G) +#+begin_src matlab +Gc = inv(n_hexapod.geometry.J)*G({'D1', 'D2', 'D3', 'D4', 'D5', 'D6'}, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'})*inv(n_hexapod.geometry.J'); #+end_src -#+RESULTS: -: size(G) -: State-space model with 18 outputs, 12 inputs, and 60 states. - -Verify the number of DOF: -| Element | DoF | -|--------------------+-----| -| Bot Plate | 0 | -| Struts | 6*4 | -| Top Plate + Sample | 12 | -|--------------------+-----| -| Total: | 36 | -#+TBLFM: @>$2=vsum(@I..@II) - -*** DVF Plant - -The DC gain from actuator to relative motion sensor should be equal to (for the 2dof APA): -\[ \frac{1}{k + k_a + kk_a/k_e} \] - #+begin_src matlab :results value replace :exports results :tangle no -1/(k + ka + k*ka/ke) +dcgain(Gc) #+end_src #+RESULTS: -: 1.8732e-08 +| 9.8602e-09 | 7.8692e-11 | 1.4426e-11 | -2.663e-10 | 2.8e-10 | -4.7559e-11 | +| 8.457e-11 | 9.8788e-09 | -2.4002e-11 | -2.9502e-11 | -1.3262e-10 | -8.7346e-11 | +| -4.3244e-11 | 2.4075e-13 | 8.4775e-09 | 1.1442e-11 | -2.5809e-10 | 2.8796e-11 | +| -1.8326e-09 | -9.318e-10 | 6.8188e-10 | 1.4697e-06 | 5.5936e-09 | 8.7632e-10 | +| 4.6906e-10 | 1.5911e-09 | 1.6989e-10 | -5.223e-09 | 1.4729e-06 | -2.6059e-10 | +| -6.5754e-11 | -3.0408e-12 | 5.394e-11 | -1.0917e-10 | 6.9479e-10 | 4.2979e-07 | -Which is almost the case: - -#+begin_src matlab :results value replace :exports results :tangle no -dcgain(G({'D1', 'D2', 'D3', 'D4', 'D5', 'D6'}, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'})) +As the rotations and translations have very different gains, we normalize each motion to one. +#+begin_src matlab +Gc = diag(1./diag(dcgain(Gc)))*Gc; #+end_src -#+RESULTS: -| -1.8676e-08 | 5.2396e-11 | -8.8935e-11 | 2.6392e-11 | 9.7629e-11 | -1.5609e-10 | -| -1.4754e-11 | -1.8725e-08 | 1.4348e-11 | -5.7189e-11 | -3.3327e-11 | 7.0799e-11 | -| -3.4924e-11 | -7.7445e-11 | -1.8607e-08 | -4.6578e-11 | -1.9592e-11 | 4.0299e-11 | -| 6.1699e-11 | -7.1962e-11 | 1.3374e-11 | -1.8623e-08 | -1.7898e-10 | 5.4541e-11 | -| 9.735e-11 | -1.0698e-10 | -1.6424e-11 | 2.3718e-11 | -1.8826e-08 | 8.387e-11 | -| -3.94e-11 | 2.6436e-11 | -1.1338e-10 | 5.0793e-11 | 1.2358e-10 | -1.8792e-08 | - -Other (off-diagonal) elements should be close to zero (probably limited to joint's stiffness). -*However, when setting joint's axial stiffness to infinity, I obtain better diagonal*. - +The diagonal and off-diagonal elements are shown in Figure [[fig:nano_hexapod_diagonal_plant_cok]], and we can see good decoupling at low frequency. #+begin_src matlab :exports none -Gdvf = minreal(G({'D1', 'D2', 'D3', 'D4', 'D5', 'D6'}, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'})); +freqs = logspace(0, 3, 1000); -freqs = 2*logspace(0, 2, 1000); +labels = {'$D_x/\mathcal{F}_x$', '$D_y/\mathcal{F}_y$', '$D_z/\mathcal{F}_z$', ... + '$R_x/\mathcal{M}_x$', '$R_y/\mathcal{M}_y$', '$R_z/\mathcal{M}_z$'}; figure; -tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); - -ax1 = nexttile([2,1]); hold on; for i = 1:6 - plot(freqs, abs(squeeze(freqresp(Gdvf(['D', num2str(i)], ['F', num2str(i)]), freqs, 'Hz')))); + plot(freqs, abs(squeeze(freqresp(Gc(i,i), freqs, 'Hz'))), '-', ... + 'DisplayName', labels{i}); end +plot(freqs, abs(squeeze(freqresp(Gc(1, 2), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ... + 'DisplayName', 'Off-Diagonal'); for i = 1:5 for j = i+1:6 - plot(freqs, abs(squeeze(freqresp(Gdvf(['D', num2str(i)], ['F', num2str(j)]), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2]); + plot(freqs, abs(squeeze(freqresp(Gc(i, j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ... + 'HandleVisibility', 'off'); end end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]); -ylim([1e-10, 1e-6]); - -ax2 = nexttile; -hold on; -for i = 1:6 - plot(freqs, 180/pi*angle(squeeze(freqresp(Gdvf(['D', num2str(i)], ['F', num2str(i)]), freqs, 'Hz')))); -end -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); -ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); -ylim([0, 180]); -yticks([-180, -90, 0, 90, 180]); - -linkaxes([ax1,ax2],'x'); -xlim([freqs(1), freqs(end)]); +xlabel('Frequency [Hz]'); ylabel('Amplitude'); +xlim([freqs(1), freqs(end)]); ylim([1e-8, 1e3]) +legend('location', 'southwest'); #+end_src #+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/nano_hexapod_struts_2dof_dvf_plant.pdf', 'width', 'wide', 'height', 'tall'); +exportFig('figs/nano_hexapod_diagonal_plant_cok.pdf', 'width', 'wide', 'height', 'tall'); #+end_src -#+name: fig:nano_hexapod_struts_2dof_dvf_plant -#+caption: IFF Plant +#+name: fig:nano_hexapod_diagonal_plant_cok +#+caption: Diagonal and off-diagonal elements of the (normalized) decentralized plant with the Jacobians estimated at the "center of stiffness" #+RESULTS: -[[file:figs/nano_hexapod_struts_2dof_dvf_plant.png]] - -*** IFF Plant - -#+begin_src matlab :exports none -freqs = 2*logspace(0, 2, 1000); - -figure; -tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); - -ax1 = nexttile([2,1]); -hold on; -for i = 1:6 - plot(freqs, abs(squeeze(freqresp(G(['Fm', num2str(i)], ['F', num2str(i)]), freqs, 'Hz')))); -end -for i = 1:5 - for j = i+1:6 - plot(freqs, abs(squeeze(freqresp(G(['Fm', num2str(i)], ['F', num2str(j)]), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2]); - end -end -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Amplitude [N/N]'); set(gca, 'XTickLabel',[]); -ylim([1e-4, 1e1]); - -ax2 = nexttile; -hold on; -for i = 1:6 - plot(freqs, 180/pi*angle(squeeze(freqresp(G(['Fm', num2str(i)], ['F', num2str(i)]), freqs, 'Hz')))); -end -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); -ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); -ylim([-180, 0]); -yticks([-180, -90, 0, 90, 180]); - -linkaxes([ax1,ax2],'x'); -xlim([freqs(1), freqs(end)]); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/nano_hexapod_struts_2dof_iff_plant.pdf', 'width', 'wide', 'height', 'tall'); -#+end_src - -#+name: fig:nano_hexapod_struts_2dof_iff_plant -#+caption: DVF Plant -#+RESULTS: -[[file:figs/nano_hexapod_struts_2dof_iff_plant.png]] - -** Decoupling at the CoK - -#+begin_src matlab -Gx = inv(Jk)*G({'D1', 'D2', 'D3', 'D4', 'D5', 'D6'}, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'})*inv(Jk'); -Gx.inputname = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'}; -Gx.outputname = {'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'}; -#+end_src - -#+begin_src matlab :results value replace :exports results :tangle no -dcgain(Gx) -#+end_src - -#+RESULTS: -| -9.8399e-09 | -8.0785e-11 | -1.5723e-11 | 2.6356e-10 | -4.9857e-10 | 5.2177e-11 | -| -8.7554e-11 | -9.8585e-09 | 2.6292e-11 | 2.0485e-10 | 2.0385e-10 | 8.2942e-11 | -| 4.3837e-11 | -5.5862e-14 | -8.4859e-09 | -9.0036e-12 | 2.9325e-10 | -3.104e-11 | -| 1.901e-09 | 5.3078e-10 | -7.5477e-10 | -1.4759e-06 | -7.7918e-09 | -7.491e-10 | -| -7.1795e-11 | -1.6333e-09 | -1.8359e-10 | 5.2707e-09 | -1.4794e-06 | 3.2304e-10 | -| 7.0322e-11 | 1.7713e-12 | -5.8019e-11 | 7.4838e-11 | -8.9677e-10 | -4.2938e-07 | - -#+begin_src matlab :exports none -freqs = 5*logspace(0, 2, 1000); - -figure; -tiledlayout(6, 6, 'TileSpacing', 'None', 'Padding', 'None'); - -for i = 1:6 - for j = 1:6 - nexttile; - plot(freqs, abs(squeeze(freqresp(Gx(i, j), freqs, 'Hz'))), 'k-'); - set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); - end -end -axs = findobj(gcf,'Type','axes','Visible','on'); -linkaxes(axs, 'xy'); -xlim([freqs(1), freqs(end)]); ylim([1e-10, 1e-4]); -#+end_src - -#+begin_src matlab :exports none -freqs = logspace(0, 2, 1000); - -figure; -tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); - -ax1 = nexttile([2,1]); -hold on; -for i = 1:6 - plot(freqs, abs(squeeze(freqresp(Gx(i,i), freqs, 'Hz'))), '-'); -end -for i = 1:5 - for j = i+1:6 - plot(freqs, abs(squeeze(freqresp(Gx(i, j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2]); - end -end -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]); - -ax2 = nexttile; -hold on; -for i = 1:6 - plot(freqs, 180/pi*angle(squeeze(freqresp(Gx(i,i), freqs, 'Hz'))), '-'); -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)]); -#+end_src - -** Jacobian -*** Compute Jacobian -#+begin_src matlab -Ai = [out.b1.Data(1,:) - out.b2.Data(1,:) - out.b3.Data(1,:) - out.b4.Data(1,:) - out.b5.Data(1,:) - out.b6.Data(1,:)]'; - - -Bi = [out.a1.Data(1,:) - out.a2.Data(1,:) - out.a3.Data(1,:) - out.a4.Data(1,:) - out.a5.Data(1,:) - out.a6.Data(1,:)]'; -#+end_src - -#+begin_src matlab -stewart = initializeStewartPlatform(); -stewart = initializeFramesPositions(stewart, 'H', 95e-3, 'MO_B', 150e-3); -stewart = generateGeneralConfiguration(stewart); -stewart.platform_F.Fa = Ai; -stewart.platform_M.Mb = Bi - [0;0;1]*stewart.geometry.H; -stewart = computeJointsPose(stewart); -stewart = initializeStewartPose(stewart, 'AP', [0;0;0], 'ARB', eye(3)); -#+end_src - -#+begin_src matlab -stewart = initializeCylindricalPlatforms(stewart); -stewart = initializeCylindricalStruts(stewart); -#+end_src - -#+begin_src matlab -stewart = initializeStrutDynamics(stewart, 'K', 1e6*ones(6,1), 'C', 1e2*ones(6,1)); -stewart = initializeAmplifiedStrutDynamics(stewart); -stewart = initializeJointDynamics(stewart); -#+end_src - -#+begin_src matlab -displayArchitecture(stewart, 'views', 'all'); -#+end_src - -#+begin_src matlab :results output replace :exports results -describeStewartPlatform(stewart) -#+end_src - -#+RESULTS: -#+begin_example -describeStewartPlatform(stewart) -GEOMETRY: -- The height between the fixed based and the top platform is 95 [mm]. -- Frame {A} is located 150 [mm] above the top platform. -- The initial length of the struts are: - 82.8, 82.8, 82.8, 82.8, 82.8, 82.8 [mm] - -ACTUATORS: -- The actuators are mechanicaly amplified. -- The vertical stiffness and damping contribution of the piezoelectric stack is: - ka = 2e+07 [N/m] ca = 1e+01 [N/(m/s)] -- Vertical stiffness when the piezoelectric stack is removed is: - kr = 5e+06 [N/m] cr = 1e+01 [N/(m/s)] - -JOINTS: -- The joints on the fixed based are universal joints -- The joints on the mobile based are spherical joints -- The position of the joints on the fixed based with respect to {F} are (in [mm]): - -86.2 -74.7 22.3 - 86.3 -74.6 22.3 - 108 -37.3 22.3 - 21.5 112 22.3 - -21.5 112 22.3 - -108 -37.5 22.2 -- The position of the joints on the mobile based with respect to {M} are (in [mm]): - -28.4 -106 -22.4 - 28.5 -106 -22.5 - 106 28.5 -22.5 - 77.8 77.8 -22.5 - -77.8 77.8 -22.5 - -106 28.4 -22.5 -#+end_example - -#+begin_src matlab -stewart = computeJacobian(stewart); -#+end_src - -*** Verify Jacobian -#+begin_src matlab -load('./mat/nano_hexapod_object.mat', 'stewart'); -J = stewart.kinematics.J; -#+end_src - -Transformation of motion: -#+begin_src matlab -G1 = -inv(J)*G({'D1', 'D2', 'D3', 'D4', 'D5', 'D6'}, {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'}); -G2 = G({'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'}, {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'}); -#+end_src - -#+begin_src matlab :exports none -freqs = logspace(1, 3, 1000); - -figure; -tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); - -ax1 = nexttile([2,1]); -hold on; -for i = 1:6 - plot(freqs, abs(squeeze(freqresp(G1(i,i), freqs, 'Hz'))), '-'); -end -set(gca,'ColorOrderIndex',1); -for i = 1:6 - plot(freqs, abs(squeeze(freqresp(G2(i,i), freqs, 'Hz'))), '--'); -end -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]); - -ax2 = nexttile; -hold on; -for i = 1:6 - plot(freqs, 180/pi*angle(squeeze(freqresp(G1(i,i), freqs, 'Hz'))), '-'); -end -set(gca,'ColorOrderIndex',1); -for i = 1:6 - plot(freqs, 180/pi*angle(squeeze(freqresp(G2(i,i), freqs, 'Hz'))), '--'); -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'); -#+end_src - -Transformation of Forces and Torques -#+begin_src matlab -G1 = G({'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'}, {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'}); -G2 = G({'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'}, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'})*inv(J'); -#+end_src - -*Not working because the forces applied on the APA are not really forces applied on the top platform (reduced by a factor ~ka/ke)* - -#+begin_src matlab :exports none -freqs = logspace(1, 3, 1000); - -figure; -tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); - -ax1 = nexttile([2,1]); -hold on; -for i = 1:6 - plot(freqs, abs(squeeze(freqresp(G1(i,i), freqs, 'Hz'))), '-'); -end -set(gca,'ColorOrderIndex',1); -for i = 1:6 - plot(freqs, abs(squeeze(freqresp(G2(i,i), freqs, 'Hz'))), '--'); -end -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]); - -ax2 = nexttile; -hold on; -for i = 1:6 - plot(freqs, 180/pi*angle(squeeze(freqresp(G1(i,i), freqs, 'Hz'))), '-'); -end -set(gca,'ColorOrderIndex',1); -for i = 1:6 - plot(freqs, 180/pi*angle(squeeze(freqresp(G2(i,i), freqs, 'Hz'))), '--'); -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'); -#+end_src - -*** Plant from APA forces to sample's displacement -#+begin_src matlab -Gx = G({'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'}, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'})*inv(J'); -Gx.inputname = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'}; -#+end_src - -#+begin_src matlab :exports none -freqs = 5*logspace(0, 2, 1000); - -figure; -tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); - -ax1 = nexttile([2,1]); -hold on; -for i = 1:6 - plot(freqs, abs(squeeze(freqresp(Gx(i,i), freqs, 'Hz'))), '-'); -end -for i = 1:5 - for j = i+1:6 - plot(freqs, abs(squeeze(freqresp(Gx(i, j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2]); - end -end -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]); - -ax2 = nexttile; -hold on; -for i = 1:6 - plot(freqs, 180/pi*angle(squeeze(freqresp(Gx(i,i), freqs, 'Hz'))), '-'); -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)]); -#+end_src - -#+begin_src matlab :exports none -freqs = 5*logspace(0, 2, 1000); - -figure; -tiledlayout(6, 6, 'TileSpacing', 'None', 'Padding', 'None'); - -for i = 1:6 - for j = 1:6 - nexttile; - plot(freqs, abs(squeeze(freqresp(Gx(i, j), freqs, 'Hz'))), 'k-'); - set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); - end -end -axs = findobj(gcf,'Type','axes','Visible','on'); -linkaxes(axs, 'xy'); -xlim([freqs(1), freqs(end)]); ylim([1e-10, 1e-4]); -#+end_src - -#+begin_src matlab :exports none -freqs = 5*logspace(0, 2, 1000); - -figure; -hold on; -for i = 1:3 - plot(freqs, abs(squeeze(freqresp(Gx(i, i), freqs, 'Hz'))), 'k-'); -end -for i = 1:3 - for j = i+1:3 - plot(freqs, abs(squeeze(freqresp(Gx(i, j), freqs, 'Hz'))), 'r-'); - set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); - end -end -hold off; -xlim([freqs(1), freqs(end)]); -ylim([1e-10, 1e-4]); -#+end_src - -#+begin_src matlab :exports none -freqs = 5*logspace(0, 2, 1000); - -figure; -hold on; -for i = 4:6 - plot(freqs, abs(squeeze(freqresp(Gx(i, i), freqs, 'Hz'))), 'k-'); -end -for i = 4:6 - for j = i+1:6 - plot(freqs, abs(squeeze(freqresp(Gx(i, j), freqs, 'Hz'))), 'r-'); - set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); - end -end -hold off; -xlim([freqs(1), freqs(end)]); -ylim([1e-10, 1e-4]); -#+end_src - -*** Plant in the Cartesian frame -Transfer function from forces/torque to displacement/rotation: -#+begin_src matlab -GJ = inv(J)*G({'D1', 'D2', 'D3', 'D4', 'D5', 'D6'}, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'})*inv(J'); -GJ.inputname = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'}; -GJ.outputname = {'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'}; -#+end_src - -#+begin_src matlab :exports none -freqs = logspace(1, 3, 1000); - -figure; -tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); - -ax1 = nexttile([2,1]); -hold on; -for i = 1:6 - plot(freqs, abs(squeeze(freqresp(GJ(i,i), freqs, 'Hz'))), '-'); -end -for i = 1:5 - for j = i+1:6 - plot(freqs, abs(squeeze(freqresp(GJ(i, j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2]); - end -end -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]); - -ax2 = nexttile; -hold on; -for i = 1:6 - plot(freqs, 180/pi*angle(squeeze(freqresp(GJ(i,i), freqs, 'Hz'))), '-'); -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'); -#+end_src - -*** Plant in the frame of the legs => Verification of Jacobian for displacements -Transfer Function in the frame of the legs -#+begin_src matlab -G1 = -J*G({'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'}, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'}); -G1.outputname = {'D1', 'D2', 'D3', 'D4', 'D5', 'D6'}; - -G2 = G({'D1', 'D2', 'D3', 'D4', 'D5', 'D6'}, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'}); -#+end_src +[[file:figs/nano_hexapod_diagonal_plant_cok.png]] #+begin_important -It is working fine (until internal resonance of strut). +The Jacobian matrices can be used to decoupled the plant at low frequency. #+end_important -#+begin_src matlab :exports none -freqs = logspace(1, 3, 1000); - -figure; -tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); - -ax1 = nexttile([2,1]); -hold on; -for i = 1:6 - plot(freqs, abs(squeeze(freqresp(G1(i,i), freqs, 'Hz'))), '-'); -end -set(gca,'ColorOrderIndex',1); -for i = 1:6 - plot(freqs, abs(squeeze(freqresp(G2(i,i), freqs, 'Hz'))), '--'); -end -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]); - -ax2 = nexttile; -hold on; -for i = 1:6 - plot(freqs, 180/pi*angle(squeeze(freqresp(G1(i,i), freqs, 'Hz'))), '-'); -end -set(gca,'ColorOrderIndex',1); -for i = 1:6 - plot(freqs, 180/pi*angle(squeeze(freqresp(G2(i,i), freqs, 'Hz'))), '--'); -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'); -#+end_src - ** Stiffness matrix +*** Introduction :ignore: + +The stiffness matrix of the nano-hexapod describes its induced static displacement/rotation when a force/torque is applied on its top platform. +The location of the applied force/torque and the expressed displacement/rotation can be defined as wanted. +Such location (or frame) is then used for the computation of the Jacobian which in turns is used to compute the stiffness matrix. + +*** Compute the theoretical stiffness of the nano-hexapod + Neglecting stiffness of the joints, we have: \[ K = J^t \mathcal{K} J \] where $\mathcal{K}$ is a diagonal 6x6 matrix with axial stiffness of the struts on the diagonal. @@ -844,55 +955,199 @@ Let's note the axial stiffness of the APA: Them axial stiffness of the struts $k_s$: \[ k_s = \frac{k_z k_{\text{APA}}}{k_z + 2 k_{\text{APA}}} \] +with $k_z$ the axial stiffness of the flexible joints. + +#+begin_src matlab +n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... + 'flex_top_type', '4dof', ... + 'motion_sensor_type', 'struts', ... + 'actuator_type', '2dof'); +#+end_src + +#+begin_src matlab +k = n_hexapod.actuator.k(1); +ke = n_hexapod.actuator.ke(1); +ka = n_hexapod.actuator.ka(1); +kz = n_hexapod.flex_top.kz(1); +#+end_src #+begin_src matlab kAPA = k + ke*ka/(ke + ka); +#+end_src + +#+begin_src matlab :results value replace :exports results :tangle no +sprintf('kAPA = %.3e [N/m]', kAPA) +#+end_src + +#+RESULTS: +: kAPA = 1.799e+06 [N/m] + +#+begin_src matlab ks = kz*kAPA/(kz + 2*kAPA); #+end_src +#+begin_src matlab :results value replace :exports results :tangle no +sprintf('ks = %.3e [N/m]', ks) +#+end_src + +#+RESULTS: +: ks = 1.737e+06 [N/m] + +We can see that the axial stiffness of the flexible joint as little impact on the total axial stiffness of the struts. + +Let's now compute the stiffness matrix corresponding to an hexapod with perfect joints and the above computed axial stiffness: #+begin_src matlab -Ks = J'*(ks*eye(6))*J +Ks = n_hexapod.geometry.J'*(ks*eye(6))*n_hexapod.geometry.J; +#+end_src + +And the compliance matrix can be computed as the inverse of the stiffness matrix: +#+begin_src matlab +C = inv(Ks); #+end_src #+begin_src matlab :results value replace :exports results :tangle no -inv(Ks) +ans = C #+end_src +#+name: tab:compliance_matrix_perfect_joints +#+caption: Compliance Matrix - Perfect Joints +#+attr_latex: :environment tabularx :width 0.9\linewidth :align cccccc +#+attr_latex: :center t :booktabs t :float t #+RESULTS: -| 1.9937e-06 | 7.7027e-12 | -1.5885e-11 | -1.6811e-10 | 8.7902e-06 | 3.0476e-11 | -| 7.7027e-12 | 1.9937e-06 | 2.7221e-11 | -8.7901e-06 | 6.5081e-12 | 2.8619e-11 | -| -1.5885e-11 | 2.7221e-11 | 2.6114e-07 | -1.3209e-10 | -6.1725e-11 | 4.0402e-11 | -| -1.6811e-10 | -8.7901e-06 | -1.3209e-10 | 4.5712e-05 | -5.7781e-10 | -4.5817e-10 | -| 8.7902e-06 | 6.5081e-12 | -6.1725e-11 | -5.7781e-10 | 4.5712e-05 | 5.1628e-10 | -| 3.0476e-11 | 2.8619e-11 | 4.0402e-11 | -4.5817e-10 | 5.1628e-10 | 1.3277e-05 | +| 1.9938e-06 | -2.3138e-22 | 3.3403e-23 | 1.0202e-21 | 8.7906e-06 | 2.9603e-11 | +| -3.1875e-23 | 1.9938e-06 | 2.2094e-11 | -8.7909e-06 | -1.6576e-22 | -3.5622e-28 | +| 6.6811e-23 | 2.2094e-11 | 2.6115e-07 | -9.8337e-11 | 3.4744e-22 | 7.4663e-28 | +| 1.4054e-22 | -8.7909e-06 | -9.8337e-11 | 4.5715e-05 | 7.3086e-22 | 1.5706e-27 | +| 8.7906e-06 | -1.0202e-21 | 1.7371e-22 | 4.498e-21 | 4.5714e-05 | 9.8237e-11 | +| 2.9603e-11 | -1.9261e-22 | -1.7611e-27 | 8.4925e-22 | 9.8237e-11 | 1.3277e-05 | + +*** Comparison with Simscape Model +#+begin_src matlab +%% Options for Linearized +options = linearizeOptions; +options.SampleTime = 0; + +%% Name of the Simulink File +mdl = 'nano_hexapod'; + +%% Input/Output definition +clear io; io_i = 1; +io(io_i) = linio([mdl, '/Fe'], 1, 'openinput'); io_i = io_i + 1; +io(io_i) = linio([mdl, '/X'], 1, 'openoutput'); io_i = io_i + 1; + +G = linearize(mdl, io, 0.0, options); +#+end_src #+begin_src matlab :results value replace :exports results :tangle no -dcgain(G({'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'}, {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'})) +dcgain(G) #+end_src +#+name: tab:compliance_matrix_simscape_joints +#+caption: Compliance Matrix - Estimated from Simscape +#+attr_latex: :environment tabularx :width 0.9\linewidth :align cccccc +#+attr_latex: :center t :booktabs t :float t #+RESULTS: -| 1.9869e-06 | 2.043e-08 | 1.6444e-09 | -4.094e-08 | 8.7579e-06 | -3.5002e-09 | -| 3.746e-09 | 1.9841e-06 | -5.5274e-09 | -8.7257e-06 | -5.5082e-08 | -7.6978e-09 | -| -3.2056e-09 | -6.4295e-11 | 2.6079e-07 | 3.4103e-10 | -9.3873e-09 | 9.6146e-10 | -| -1.1677e-08 | -8.736e-06 | 2.4322e-08 | 4.5343e-05 | 2.52e-07 | 2.5932e-08 | -| 8.7439e-06 | 8.441e-08 | 5.9144e-09 | -1.6879e-07 | 4.546e-05 | -9.8986e-09 | -| 3.3453e-09 | 4.508e-10 | 1.8718e-09 | -2.7294e-09 | 2.8776e-08 | 1.3193e-05 | - -We can see that we have almost the same stiffness: - -#+begin_src matlab :results value replace :exports results :tangle no -inv(Ks)./dcgain(G({'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'}, {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'})) -#+end_src - -#+RESULTS: -| 1.0034 | 0.00037703 | -0.0096597 | 0.0041063 | 1.0037 | -0.0087071 | -| 0.0020562 | 1.0048 | -0.0049247 | 1.0074 | -0.00011815 | -0.0037178 | -| 0.0049552 | -0.42338 | 1.0013 | -0.38734 | 0.0065754 | 0.042021 | -| 0.014397 | 1.0062 | -0.005431 | 1.0081 | -0.0022929 | -0.017668 | -| 1.0053 | 7.7101e-05 | -0.010436 | 0.0034233 | 1.0055 | -0.052157 | -| 0.0091102 | 0.063485 | 0.021584 | 0.16786 | 0.017941 | 1.0063 | +| 1.9863e-06 | 2.4683e-08 | 2.2859e-09 | -6.4768e-08 | 8.7307e-06 | 3.2754e-10 | +| -3.693e-09 | 1.9663e-06 | -7.4503e-09 | -8.615e-06 | -6.7926e-08 | -2.0193e-08 | +| -3.5525e-09 | -8.8671e-10 | 2.6042e-07 | 4.7504e-09 | -9.9677e-09 | 1.7242e-10 | +| 2.1133e-08 | -8.6554e-06 | 3.2841e-08 | 4.4849e-05 | 3.0873e-07 | 8.1525e-08 | +| 8.7375e-06 | 9.9165e-08 | 8.135e-09 | -2.5208e-07 | 4.5331e-05 | 3.0602e-09 | +| 6.1611e-09 | 6.1733e-09 | 2.6778e-09 | -3.3188e-08 | 3.3887e-08 | 1.3212e-05 | * To-order :noexport: + +** Verify why unstable strut +#+begin_src matlab :results value replace +n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '3dof', ... + 'flex_top_type', '2dof', ... + 'motion_sensor_type', 'plates', ... + 'actuator_type', 'flexible'); +isstable(linearize(mdl, io, 0.0, options)) +#+end_src + +#+RESULTS: +: 1 + +#+begin_src matlab :results value replace +n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '3dof', ... + 'flex_top_type', '2dof', ... + 'motion_sensor_type', 'struts', ... + 'actuator_type', 'flexible'); +isstable(linearize(mdl, io, 0.0, options)) +#+end_src + +#+RESULTS: +: 0 + +#+begin_src matlab :results value replace +n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... + 'flex_top_type', '3dof', ... + 'motion_sensor_type', 'plates', ... + 'actuator_type', 'flexible'); +isstable(linearize(mdl, io, 0.0, options)) +#+end_src + +#+RESULTS: +: 0 + +#+begin_src matlab :results value replace +n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... + 'flex_top_type', '3dof', ... + 'motion_sensor_type', 'struts', ... + 'actuator_type', 'flexible'); +isstable(linearize(mdl, io, 0.0, options)) +#+end_src + +#+RESULTS: +: 0 + + +** Verify why unstable strut +#+begin_src matlab :results value replace +n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '3dof', ... + 'flex_top_type', '2dof', ... + 'motion_sensor_type', 'plates', ... + 'actuator_type', '2dof'); +isstable(linearize(mdl, io, 0.0, options)) +#+end_src + +#+RESULTS: +: 1 + +#+begin_src matlab :results value replace +n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '3dof', ... + 'flex_top_type', '2dof', ... + 'motion_sensor_type', 'struts', ... + 'actuator_type', '2dof'); +isstable(linearize(mdl, io, 0.0, options)) +#+end_src + +#+RESULTS: +: 1 + +#+begin_src matlab :results value replace +n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... + 'flex_top_type', '3dof', ... + 'motion_sensor_type', 'plates', ... + 'actuator_type', '2dof'); +isstable(linearize(mdl, io, 0.0, options)) +#+end_src + +#+RESULTS: +: 1 + +#+begin_src matlab :results value replace +n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... + 'flex_top_type', '3dof', ... + 'motion_sensor_type', 'struts', ... + 'actuator_type', '2dof'); +isstable(linearize(mdl, io, 0.0, options)) +#+end_src + +#+RESULTS: +: 1 + + ** APA300ML structure #+begin_src matlab @@ -1601,7 +1856,7 @@ linkaxes([ax3,ax4],'y'); :UNNUMBERED: t :END: #+begin_src matlab - function [nano_hexapod] = initializeNanoHexapodFinal(args) +function [nano_hexapod] = initializeNanoHexapodFinal(args) #+end_src ** Optional Parameters @@ -1609,47 +1864,49 @@ linkaxes([ax3,ax4],'y'); :UNNUMBERED: t :END: #+begin_src matlab - arguments - %% Bottom Flexible Joints - args.flex_bot_type char {mustBeMember(args.flex_bot_type,{'2dof', '3dof', '4dof', 'flexible'})} = '4dof' - args.flex_bot_kRx (6,1) double {mustBeNumeric} = ones(6,1)*5 % X bending stiffness [Nm/rad] - args.flex_bot_kRy (6,1) double {mustBeNumeric} = ones(6,1)*5 % Y bending stiffness [Nm/rad] - args.flex_bot_kRz (6,1) double {mustBeNumeric} = ones(6,1)*260 % Torsionnal stiffness [Nm/rad] - args.flex_bot_kz (6,1) double {mustBeNumeric} = ones(6,1)*1e8 % Axial Stiffness [N/m] - args.flex_bot_cRx (6,1) double {mustBeNumeric} = ones(6,1)*0.1 % X bending Damping [Nm/(rad/s)] - args.flex_bot_cRy (6,1) double {mustBeNumeric} = ones(6,1)*0.1 % Y bending Damping [Nm/(rad/s)] - args.flex_bot_cRz (6,1) double {mustBeNumeric} = ones(6,1)*0.1 % Torsionnal Damping [Nm/(rad/s)] - args.flex_bot_cz (6,1) double {mustBeNumeric} = ones(6,1)*1e2 % Axial Damping [N/(m/s)] - %% Top Flexible Joints - args.flex_top_type char {mustBeMember(args.flex_top_type,{'2dof', '3dof', '4dof', 'flexible'})} = '4dof' - args.flex_top_kRx (6,1) double {mustBeNumeric} = ones(6,1)*5 % X bending stiffness [Nm/rad] - args.flex_top_kRy (6,1) double {mustBeNumeric} = ones(6,1)*5 % Y bending stiffness [Nm/rad] - args.flex_top_kRz (6,1) double {mustBeNumeric} = ones(6,1)*260 % Torsionnal stiffness [Nm/rad] - args.flex_top_kz (6,1) double {mustBeNumeric} = ones(6,1)*1e8 % Axial Stiffness [N/m] - args.flex_top_cRx (6,1) double {mustBeNumeric} = ones(6,1)*0.1 % X bending Damping [Nm/(rad/s)] - args.flex_top_cRy (6,1) double {mustBeNumeric} = ones(6,1)*0.1 % Y bending Damping [Nm/(rad/s)] - args.flex_top_cRz (6,1) double {mustBeNumeric} = ones(6,1)*0.1 % Torsionnal Damping [Nm/(rad/s)] - args.flex_top_cz (6,1) double {mustBeNumeric} = ones(6,1)*1e2 % Axial Damping [N/(m/s)] - %% Relative Motion Sensor - args.motion_sensor_type char {mustBeMember(args.motion_sensor_type,{'struts', 'plates'})} = 'struts' - %% Actuators - args.actuator_type char {mustBeMember(args.actuator_type,{'2dof', 'flexible frame', 'flexible'})} = 'flexible' - args.actuator_Ga (6,1) double {mustBeNumeric} = ones(6,1)*1 % Actuator gain [N/V] - args.actuator_Gs (6,1) double {mustBeNumeric} = ones(6,1)*1 % Sensor gain [V/m] - % For 2DoF - args.actuator_k (6,1) double {mustBeNumeric} = ones(6,1)*0.35e6 % [N/m] - args.actuator_ke (6,1) double {mustBeNumeric} = ones(6,1)*1.5e6 % [N/m] - args.actuator_ka (6,1) double {mustBeNumeric} = ones(6,1)*43e6 % [N/m] - args.actuator_c (6,1) double {mustBeNumeric} = ones(6,1)*3e1 % [N/(m/s)] - args.actuator_ce (6,1) double {mustBeNumeric} = ones(6,1)*1e1 % [N/(m/s)] - args.actuator_ca (6,1) double {mustBeNumeric} = ones(6,1)*1e1 % [N/(m/s)] - args.actuator_Leq (6,1) double {mustBeNumeric} = ones(6,1)*0.056 % [m] - % For Flexible Frame - args.actuator_ks (6,1) double {mustBeNumeric} = ones(6,1)*235e6 % Stiffness of one stack [N/m] - args.actuator_cs (6,1) double {mustBeNumeric} = ones(6,1)*1e1 % Stiffness of one stack [N/m] - % For Flexible - args.actuator_xi (1,1) double {mustBeNumeric} = 0.01 % Sensor gain [V/m] - end +arguments + %% Bottom Flexible Joints + args.flex_bot_type char {mustBeMember(args.flex_bot_type,{'2dof', '3dof', '4dof', 'flexible'})} = '4dof' + args.flex_bot_kRx (6,1) double {mustBeNumeric} = ones(6,1)*5 % X bending stiffness [Nm/rad] + args.flex_bot_kRy (6,1) double {mustBeNumeric} = ones(6,1)*5 % Y bending stiffness [Nm/rad] + args.flex_bot_kRz (6,1) double {mustBeNumeric} = ones(6,1)*260 % Torsionnal stiffness [Nm/rad] + args.flex_bot_kz (6,1) double {mustBeNumeric} = ones(6,1)*1e8 % Axial Stiffness [N/m] + args.flex_bot_cRx (6,1) double {mustBeNumeric} = ones(6,1)*0.1 % X bending Damping [Nm/(rad/s)] + args.flex_bot_cRy (6,1) double {mustBeNumeric} = ones(6,1)*0.1 % Y bending Damping [Nm/(rad/s)] + args.flex_bot_cRz (6,1) double {mustBeNumeric} = ones(6,1)*0.1 % Torsionnal Damping [Nm/(rad/s)] + args.flex_bot_cz (6,1) double {mustBeNumeric} = ones(6,1)*1e2 % Axial Damping [N/(m/s)] + %% Top Flexible Joints + args.flex_top_type char {mustBeMember(args.flex_top_type,{'2dof', '3dof', '4dof', 'flexible'})} = '4dof' + args.flex_top_kRx (6,1) double {mustBeNumeric} = ones(6,1)*5 % X bending stiffness [Nm/rad] + args.flex_top_kRy (6,1) double {mustBeNumeric} = ones(6,1)*5 % Y bending stiffness [Nm/rad] + args.flex_top_kRz (6,1) double {mustBeNumeric} = ones(6,1)*260 % Torsionnal stiffness [Nm/rad] + args.flex_top_kz (6,1) double {mustBeNumeric} = ones(6,1)*1e8 % Axial Stiffness [N/m] + args.flex_top_cRx (6,1) double {mustBeNumeric} = ones(6,1)*0.1 % X bending Damping [Nm/(rad/s)] + args.flex_top_cRy (6,1) double {mustBeNumeric} = ones(6,1)*0.1 % Y bending Damping [Nm/(rad/s)] + args.flex_top_cRz (6,1) double {mustBeNumeric} = ones(6,1)*0.1 % Torsionnal Damping [Nm/(rad/s)] + args.flex_top_cz (6,1) double {mustBeNumeric} = ones(6,1)*1e2 % Axial Damping [N/(m/s)] + %% Jacobian - Location of frame {A} and {B} + args.MO_B (1,1) double {mustBeNumeric} = 150e-3 % Height of {B} w.r.t. {M} [m] + %% Relative Motion Sensor + args.motion_sensor_type char {mustBeMember(args.motion_sensor_type,{'struts', 'plates'})} = 'struts' + %% Actuators + args.actuator_type char {mustBeMember(args.actuator_type,{'2dof', 'flexible frame', 'flexible'})} = 'flexible' + args.actuator_Ga (6,1) double {mustBeNumeric} = ones(6,1)*1 % Actuator gain [N/V] + args.actuator_Gs (6,1) double {mustBeNumeric} = ones(6,1)*1 % Sensor gain [V/m] + % For 2DoF + args.actuator_k (6,1) double {mustBeNumeric} = ones(6,1)*0.35e6 % [N/m] + args.actuator_ke (6,1) double {mustBeNumeric} = ones(6,1)*1.5e6 % [N/m] + args.actuator_ka (6,1) double {mustBeNumeric} = ones(6,1)*43e6 % [N/m] + args.actuator_c (6,1) double {mustBeNumeric} = ones(6,1)*3e1 % [N/(m/s)] + args.actuator_ce (6,1) double {mustBeNumeric} = ones(6,1)*1e1 % [N/(m/s)] + args.actuator_ca (6,1) double {mustBeNumeric} = ones(6,1)*1e1 % [N/(m/s)] + args.actuator_Leq (6,1) double {mustBeNumeric} = ones(6,1)*0.056 % [m] + % For Flexible Frame + args.actuator_ks (6,1) double {mustBeNumeric} = ones(6,1)*235e6 % Stiffness of one stack [N/m] + args.actuator_cs (6,1) double {mustBeNumeric} = ones(6,1)*1e1 % Stiffness of one stack [N/m] + % For Flexible + args.actuator_xi (1,1) double {mustBeNumeric} = 0.01 % Damping Ratio +end #+end_src ** Nano Hexapod Object @@ -1781,9 +2038,9 @@ switch args.actuator_type nano_hexapod.actuator.M = readmatrix('APA300ML_b_mat_M.CSV'); % Mass Matrix nano_hexapod.actuator.P = extractNodes('APA300ML_b_out_nodes_3D.txt'); % Node coordinates [m] case 'flexible' - nano_hexapod.actuator.K = readmatrix('APA300ML_full_mat_K.CSV'); % Stiffness Matrix - nano_hexapod.actuator.M = readmatrix('APA300ML_full_mat_M.CSV'); % Mass Matrix - nano_hexapod.actuator.P = extractNodes('APA300ML_full_out_nodes_3D.txt'); % Node coordiantes [m] + nano_hexapod.actuator.K = readmatrix('full_APA300ML_K.CSV'); % Stiffness Matrix + nano_hexapod.actuator.M = readmatrix('full_APA300ML_M.CSV'); % Mass Matrix + nano_hexapod.actuator.P = extractNodes('full_APA300ML_out_nodes_3D.txt'); % Node coordiantes [m] end nano_hexapod.actuator.xi = args.actuator_xi; % Damping ratio @@ -1792,10 +2049,97 @@ nano_hexapod.actuator.ks = args.actuator_ks; % Stiffness of one stack [N/m] nano_hexapod.actuator.cs = args.actuator_cs; % Damping of one stack [N/m] #+end_src +** Geometry +:PROPERTIES: +:UNNUMBERED: t +:END: + +#+begin_src matlab +nano_hexapod.geometry = struct(); +#+end_src + +Center of joints $a_i$ with respect to {F}: +#+begin_src matlab +Fa = [[-86.05, -74.78, 22.49], + [ 86.05, -74.78, 22.49], + [ 107.79, -37.13, 22.49], + [ 21.74, 111.91, 22.49], + [-21.74, 111.91, 22.49], + [-107.79, -37.13, 22.49]]'*1e-3; % Ai w.r.t. {F} [m] +#+end_src + +Center of joints $b_i$ with respect to {M}: +#+begin_src matlab +Mb = [[-28.47, -106.25, -22.50], + [ 28.47, -106.25, -22.50], + [ 106.25, 28.47, -22.50], + [ 77.78, 77.78, -22.50], + [-77.78, 77.78, -22.50], + [-106.25, 28.47, -22.50]]'*1e-3; % Bi w.r.t. {M} [m] +#+end_src + +Now compute the positions $b_i$ with respect to {F}: +#+begin_src matlab +Fb = Mb + [0; 0; 95e-3]; % Bi w.r.t. {F} [m] +#+end_src + +The unit vector representing the orientation of the struts can then be computed: +#+begin_src matlab +si = Fb - Fa; +si = si./vecnorm(si); % Normalize +#+end_src + +Location of encoder measurement points when fixed on the plates: +#+begin_src matlab +Fc = [[-29.362, -105.765, 52.605] + [ 29.362, -105.765, 52.605] + [ 106.276, 27.454, 52.605] + [ 76.914, 78.31, 52.605] + [-76.914, 78.31, 52.605] + [-106.276, 27.454, 52.605]]'*1e-3; % Meas pos w.r.t. {F} +Mc = Fc - [0; 0; 95e-3]; % Meas pos w.r.t. {M} +#+end_src + +#+begin_src matlab +nano_hexapod.geometry.Fa = Fa; +nano_hexapod.geometry.Fb = Fb; +nano_hexapod.geometry.Fc = Fc; +nano_hexapod.geometry.Mb = Mb; +nano_hexapod.geometry.Mc = Mc; +nano_hexapod.geometry.si = si; +nano_hexapod.geometry.MO_B = args.MO_B; +#+end_src + +** Jacobian for Actuators +:PROPERTIES: +:UNNUMBERED: t +:END: + +#+begin_src matlab +Bb = Mb - [0; 0; args.MO_B]; + +nano_hexapod.geometry.J = [nano_hexapod.geometry.si', cross(Bb, nano_hexapod.geometry.si)']; +#+end_src + +** Jacobian for Sensors +:PROPERTIES: +:UNNUMBERED: t +:END: +#+begin_src matlab +switch args.motion_sensor_type + case 'struts' + nano_hexapod.geometry.Js = nano_hexapod.geometry.J; + case 'plates' + Bc = Mc - [0; 0; args.MO_B]; + nano_hexapod.geometry.Js = [nano_hexapod.geometry.si', cross(Bc, nano_hexapod.geometry.si)']; +end +#+end_src + ** Save the Structure :PROPERTIES: :UNNUMBERED: t :END: + #+begin_src matlab if nargout == 0 save('./mat/stages.mat', 'nano_hexapod', '-append');