From 6522211e01cede3708cbe65109fad63c5897a61a Mon Sep 17 00:00:00 2001 From: Thomas Dehaeze Date: Fri, 17 Apr 2020 14:11:34 +0200 Subject: [PATCH] Add analysis about virtual mass addition --- docs/control_virtual_mass.html | 251 ++ docs/figs/virtual_mass_L_primary_plant_L.pdf | Bin 0 -> 601762 bytes docs/figs/virtual_mass_L_primary_plant_L.png | Bin 0 -> 172457 bytes docs/figs/virtual_mass_L_primary_plant_X.pdf | Bin 0 -> 1840486 bytes docs/figs/virtual_mass_L_primary_plant_X.png | Bin 0 -> 262823 bytes docs/figs/virtual_mass_X_primary_plant_L.pdf | Bin 0 -> 602731 bytes docs/figs/virtual_mass_X_primary_plant_L.png | Bin 0 -> 172580 bytes docs/figs/virtual_mass_X_primary_plant_X.pdf | Bin 0 -> 1851086 bytes docs/figs/virtual_mass_X_primary_plant_X.png | Bin 0 -> 260882 bytes docs/figs/virtual_mass_loop_gain_L.pdf | Bin 0 -> 90851 bytes docs/figs/virtual_mass_loop_gain_L.png | Bin 0 -> 168462 bytes docs/figs/virtual_mass_loop_gain_X.pdf | 2911 ++++++++++++++++ docs/figs/virtual_mass_loop_gain_X.png | Bin 0 -> 225413 bytes docs/figs/virtual_mass_plant_L.pdf | 1533 +++++++++ docs/figs/virtual_mass_plant_L.png | Bin 0 -> 153432 bytes docs/figs/virtual_mass_plant_X.pdf | 3255 ++++++++++++++++++ docs/figs/virtual_mass_plant_X.png | Bin 0 -> 253426 bytes docs/optimal_stiffness_control.html | 54 +- mat/conf_log.mat | Bin 225 -> 223 bytes mat/conf_simscape.mat | Bin 208 -> 207 bytes mat/controller.mat | Bin 232 -> 232 bytes mat/nass_disturbances.mat | Bin 774585 -> 50897 bytes mat/nass_references.mat | Bin 2485365 -> 1472462 bytes org/control_virtual_mass.org | 707 ++++ org/optimal_stiffness_control.org | 19 +- 25 files changed, 8708 insertions(+), 22 deletions(-) create mode 100644 docs/control_virtual_mass.html create mode 100644 docs/figs/virtual_mass_L_primary_plant_L.pdf create mode 100644 docs/figs/virtual_mass_L_primary_plant_L.png create mode 100644 docs/figs/virtual_mass_L_primary_plant_X.pdf create mode 100644 docs/figs/virtual_mass_L_primary_plant_X.png create mode 100644 docs/figs/virtual_mass_X_primary_plant_L.pdf create mode 100644 docs/figs/virtual_mass_X_primary_plant_L.png create mode 100644 docs/figs/virtual_mass_X_primary_plant_X.pdf create mode 100644 docs/figs/virtual_mass_X_primary_plant_X.png create mode 100644 docs/figs/virtual_mass_loop_gain_L.pdf create mode 100644 docs/figs/virtual_mass_loop_gain_L.png create mode 100644 docs/figs/virtual_mass_loop_gain_X.pdf create mode 100644 docs/figs/virtual_mass_loop_gain_X.png create mode 100644 docs/figs/virtual_mass_plant_L.pdf create mode 100644 docs/figs/virtual_mass_plant_L.png create mode 100644 docs/figs/virtual_mass_plant_X.pdf create mode 100644 docs/figs/virtual_mass_plant_X.png create mode 100644 org/control_virtual_mass.org diff --git a/docs/control_virtual_mass.html b/docs/control_virtual_mass.html new file mode 100644 index 0000000..987dcc1 --- /dev/null +++ b/docs/control_virtual_mass.html @@ -0,0 +1,251 @@ + + + + + + + + +Decentralize control to add virtual mass + + + + + + + + + + + + +
+ UP + | + HOME +
+

Decentralize control to add virtual mass

+ + +
+

1 Initialization

+
+
+
initializeGround();
+initializeGranite();
+initializeTy();
+initializeRy();
+initializeRz();
+initializeMicroHexapod();
+initializeAxisc();
+initializeMirror();
+
+initializeSimscapeConfiguration();
+initializeDisturbances('enable', false);
+initializeLoggingConfiguration('log', 'none');
+
+initializeController('type', 'hac-dvf');
+
+
+ +

+We set the stiffness of the payload fixation: +

+
+
Kp = 1e8; % [N/m]
+
+
+
+
+ +
+

2 Identification

+
+
+
+

2.1 Identification of the transfer function from \(\tau\) to \(d\mathcal{L}\)

+
+
+
K = tf(zeros(6));
+Kdvf = tf(zeros(6));
+
+
+ +

+We identify the system for the following payload masses: +

+
+
Ms = [1, 10, 50];
+
+
+ +

+The nano-hexapod has the following leg’s stiffness and damping. +

+
+
initializeNanoHexapod('k', 1e5, 'c', 2e2);
+
+
+
+
+ +
+

2.2 Identification of the Primary plant without virtual add of mass

+
+
+
+

3 Adding Virtual Mass in the Leg’s Space

+
+
+
+

3.1 Plant

+
+ +
+

virtual_mass_plant_L.png +

+

Figure 1: Transfer function from \(\tau_i\) to \(d\mathcal{L}_i\) for three payload masses

+
+
+
+ +
+

3.2 Controller Design

+
+
+
Kdvf = 10*s^2/(1+s/2/pi/500)^2*eye(6);
+
+
+ + +
+

virtual_mass_loop_gain_L.png +

+

Figure 2: Loop Gain for the addition of virtual mass in the leg’s space

+
+
+
+ +
+

3.3 Identification of the Primary Plant

+
+ +
+

virtual_mass_L_primary_plant_X.png +

+

Figure 3: Comparison of the transfer function from \(\mathcal{F}_{x,y,z}\) to \(\mathcal{X}_{x,y,z}\) with and without the virtual addition of mass in the leg’s space

+
+ + +
+

virtual_mass_L_primary_plant_L.png +

+

Figure 4: Comparison of the transfer function from \(\tau_i\) to \(\mathcal{L}_{i}\) with and without the virtual addition of mass in the leg’s space

+
+
+
+
+ +
+

4 Adding Virtual Mass in the Task Space

+
+
+
+

4.1 Plant

+
+

+Let’s look at the transfer function from \(\bm{\mathcal{F}}\) to \(d\bm{\mathcal{X}}\): +\[ \frac{d\bm{\mathcal{L}}}{\bm{\mathcal{F}}} = \bm{J}^{-1} \frac{d\bm{\mathcal{L}}}{\bm{\tau}} \bm{J}^{-T} \] +

+ + +
+

virtual_mass_plant_X.png +

+

Figure 5: Dynamics from \(\mathcal{F}_{x,y,z}\) to \(\mathcal{X}_{x,y,z}\) used for virtual mass addition in the task space

+
+
+
+ +
+

4.2 Controller Design

+
+
+
KmX = (s^2*1/(1+s/2/pi/500)^2*diag([1 1 50 0 0 0]));
+
+
+ + +
+

virtual_mass_loop_gain_X.png +

+

Figure 6: Loop gain for virtual mass addition in the task space

+
+ +
+
Kdvf = inv(nano_hexapod.J')*KmX*inv(nano_hexapod.J);
+
+
+
+
+ +
+

4.3 Identification of the Primary Plant

+
+ +
+

virtual_mass_X_primary_plant_X.png +

+

Figure 7: Comparison of the transfer function from \(\mathcal{F}_{x,y,z}\) to \(\mathcal{X}_{x,y,z}\) with and without the virtual addition of mass in the task space

+
+ + +
+

virtual_mass_X_primary_plant_L.png +

+

Figure 8: Comparison of the transfer function from \(\tau_i\) to \(\mathcal{L}_{i}\) with and without the virtual addition of mass in the task space

+
+
+
+
+
+
+

Author: Dehaeze Thomas

+

Created: 2020-04-17 ven. 14:10

+
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    2. 1.4. Effect of the Low Authority Control on the Primary Plant
    3. 1.5. Effect of the Low Authority Control on the Sensibility to Disturbances
    4. -
    5. 1.6. Conclusion
    6. +
    7. 1.6. Conclusion
    8. 1.3. Controller Design
    9. 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zNk{bcm^#$vjwNYwa}^4#yB{lfDvVC{PG{|kiPff6;e1b4#^h)aV|~2ExsCwk4FsDW zz7##KH0i+Dp5tp5B620l28PxaJpzLWwB}M_)STkRT@0T~pb@vZTx!-zm%VKAmyRE3M7$3D?R215#awh!~& z+ix~K3<*$Xe1M5FW94^3L$MqUpk>EXAWQ-~kN-`_-dAw){iv3Zq5-zKi$ufZ*uVZ< zF&6AbJb$+0_ate69XDgGV04K}(ja!}{)~%cQz`JKLc>fQtK;VD*aRmu_c2L=R*5

    10. 2. Primary Control in the leg space @@ -52,7 +52,7 @@
    11. 2.3. Sensibility to Disturbances and Noise Budget
    12. 2.4. Simulations
    13. 2.5. Results
    14. -
    15. 2.6. Conclusion
    16. +
    17. 2.6. Conclusion
    18. 3. Primary Control in the task space @@ -64,7 +64,7 @@
    19. 3.3. Simulation
    20. -
    21. 3.4. Conclusion
    22. +
    23. 3.4. Conclusion
    24. @@ -207,6 +207,12 @@ Then, we compute the transfer function from forces applied by the actuators \(\b The obtained dynamics is shown in Figure 5.

      +
      +

      +A zero with a positive real part is introduced in the transfer function from \(\mathcal{F}_y\) to \(\mathcal{X}_y\) after Decentralized Direct Velocity Feedback is applied. +

      + +

      And we compute the transfer function from actuator forces \(\bm{\tau}\) to position error of each leg \(\bm{\epsilon}_\mathcal{L}\): @@ -237,9 +243,15 @@ The coupling does not change a lot with DVF.

      The coupling in the space of the legs \(\bm{G}_\mathcal{L}\) are shown in Figure 8. -The magnitude of the coupling around the resonance of the nano-hexapod (where the coupling is the highest) is considerably reduced when DVF is applied.

      +
      +

      +The magnitude of the coupling between \(\tau_i\) and \(d\mathcal{L}_j\) (Figure 8) around the resonance of the nano-hexapod (where the coupling is the highest) is considerably reduced when DVF is applied. +

      + +
      + +
      +

      +Decentralized Direct Velocity Feedback is shown to increase the effect of stages vibrations at high frequency and to reduce the effect of ground motion and direct forces at low frequency. +

      +
      -
      -

      1.6 Conclusion

      +
      +

      1.6 Conclusion

      @@ -332,7 +350,7 @@ The controller for decentralized direct velocity feedback is the one designed in

      2.1 Plant in the leg space

      -We now loop at the transfer function matrix from \(\bm{\tau}^\prime\) to \(\bm{\epsilon}_{\mathcal{X}_n}\) for the design of \(\bm{K}_\mathcal{L}\). +We now look at the transfer function matrix from \(\bm{\tau}^\prime\) to \(\bm{\epsilon}_{\mathcal{X}_n}\) for the design of \(\bm{K}_\mathcal{L}\).

      @@ -400,17 +418,17 @@ Kl = 2e7 * eye(6) *

      Figure 12: Loop gain for the primary plant

      +

      +Finally, we include the Jacobian in the control and we ignore the measurement of the vertical rotation as for the real system. +

      load('mat/stages.mat', 'nano_hexapod');
       K = Kl*nano_hexapod.J*diag([1, 1, 1, 1, 1, 0]);
       
      +
      +
      -

      -Check the MIMO stability -

      -
      -

      2.3 Sensibility to Disturbances and Noise Budget

      @@ -519,8 +537,8 @@ Finally, the time domain position error signals are shown in Figure -

      2.6 Conclusion

      +
      +

      2.6 Conclusion

      @@ -614,8 +632,8 @@ Kx(6,6) = 5e4 * ...

      3.3 Simulation

      -
      -

      3.4 Conclusion

      +
      +

      3.4 Conclusion

      @@ -629,7 +647,7 @@ Kx(6,6) = 5e4 * ...

      Author: Dehaeze Thomas

      -

      Created: 2020-04-17 ven. 10:25

      +

      Created: 2020-04-17 ven. 14:10

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--git a/org/control_virtual_mass.org b/org/control_virtual_mass.org new file mode 100644 index 0000000..37c61dc --- /dev/null +++ b/org/control_virtual_mass.org @@ -0,0 +1,707 @@ +#+TITLE: Decentralize control to add virtual mass +#+SETUPFILE: ./setup/org-setup-file.org + +* Matlab Init :noexport:ignore: +#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) +<> +#+end_src + +#+begin_src matlab :exports none :results silent :noweb yes +<> +#+end_src + +#+begin_src matlab :tangle no + simulinkproject('../'); +#+end_src + +#+begin_src matlab + load('mat/conf_simulink.mat'); + open('nass_model.slx') +#+end_src + +* Initialization +#+begin_src matlab + initializeGround(); + initializeGranite(); + initializeTy(); + initializeRy(); + initializeRz(); + initializeMicroHexapod(); + initializeAxisc(); + initializeMirror(); + + initializeSimscapeConfiguration(); + initializeDisturbances('enable', false); + initializeLoggingConfiguration('log', 'none'); + + initializeController('type', 'hac-dvf'); +#+end_src + +We set the stiffness of the payload fixation: +#+begin_src matlab + Kp = 1e8; % [N/m] +#+end_src + +* Identification +** Identification of the transfer function from $\tau$ to $d\mathcal{L}$ +#+begin_src matlab + K = tf(zeros(6)); + Kdvf = tf(zeros(6)); +#+end_src + +We identify the system for the following payload masses: +#+begin_src matlab + Ms = [1, 10, 50]; +#+end_src + +#+begin_src matlab :exports none + Gm = {zeros(length(Ms), 1)}; +#+end_src + +The nano-hexapod has the following leg's stiffness and damping. +#+begin_src matlab + initializeNanoHexapod('k', 1e5, 'c', 2e2); +#+end_src + +#+begin_src matlab :exports none + %% Name of the Simulink File + mdl = 'nass_model'; + + %% Input/Output definition + clear io; io_i = 1; + io(io_i) = linio([mdl, '/Controller'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs + io(io_i) = linio([mdl, '/Micro-Station'], 3, 'openoutput', [], 'Dnlm'); io_i = io_i + 1; % Force Sensors +#+end_src + +#+begin_src matlab :exports none + for i = 1:length(Ms) + initializeSample('mass', Ms(i), 'freq', sqrt(Kp/Ms(i))/2/pi*ones(6,1)); + initializeReferences('Rz_type', 'rotating-not-filtered', 'Rz_period', Ms(i)); + + %% Run the linearization + G_dvf = linearize(mdl, io); + G_dvf.InputName = {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'}; + G_dvf.OutputName = {'Dnlm1', 'Dnlm2', 'Dnlm3', 'Dnlm4', 'Dnlm5', 'Dnlm6'}; + Gm(i) = {G_dvf}; + end +#+end_src + +** Identification of the Primary plant without virtual add of mass +#+begin_src matlab :exports none + G_x = {zeros(length(Ms), 1)}; + G_l = {zeros(length(Ms), 1)}; +#+end_src + +#+begin_src matlab :exports none + for i = 1:length(Ms) + initializeSample('mass', Ms(i), 'freq', sqrt(Kp/Ms(i))/2/pi*ones(6,1)); + initializeReferences('Rz_type', 'rotating-not-filtered', 'Rz_period', Ms(i)); + + %% Run the linearization + G = linearize(mdl, io); + G.InputName = {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'}; + G.OutputName = {'Ex', 'Ey', 'Ez', 'Erx', 'Ery', 'Erz'}; + + Gx = -G*inv(nano_hexapod.J'); + Gx.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'}; + G_x(i) = {Gx}; + + Gl = -nano_hexapod.J*G; + Gl.OutputName = {'E1', 'E2', 'E3', 'E4', 'E5', 'E6'}; + G_l(i) = {Gl}; + end +#+end_src + +* Adding Virtual Mass in the Leg's Space +** Plant +#+begin_src matlab :exports none + freqs = logspace(-1, 3, 1000); + + figure; + + ax1 = subplot(2, 1, 1); + hold on; + for i = 1:length(Ms) + plot(freqs, abs(squeeze(freqresp(Gm{i}(1, 1), freqs, 'Hz')))); + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]); + + ax2 = subplot(2, 1, 2); + hold on; + for i = 1:length(Ms) + plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gm{i}(1, 1), freqs, 'Hz')))), ... + 'DisplayName', sprintf('$m_p = %.0f$ [kg]', Ms(i))); + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); + ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); + ylim([-270, 90]); + yticks([-360:90:360]); + legend('location', 'northeast'); + + linkaxes([ax1,ax2],'x'); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/virtual_mass_plant_L.pdf', 'width', 'full', 'height', 'full') +#+end_src + +#+name: fig:virtual_mass_plant_L +#+caption: Transfer function from $\tau_i$ to $d\mathcal{L}_i$ for three payload masses +#+RESULTS: +[[file:figs/virtual_mass_plant_L.png]] + +** Controller Design +#+begin_src matlab + Kdvf = 10*s^2/(1+s/2/pi/500)^2*eye(6); +#+end_src + +#+begin_src matlab :exports none + for i = 1:length(Ms) + isstable(feedback(Gm{i}*Kdvf, eye(6), -1)) + end +#+end_src + +#+begin_src matlab :exports none + freqs = logspace(-1, 4, 1000); + + figure; + + ax1 = subplot(2, 1, 1); + hold on; + for i = 1:length(Ms) + plot(freqs, abs(squeeze(freqresp(Gm{i}(1, 1)*Kdvf(1,1), freqs, 'Hz')))); + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Loop Gain'); set(gca, 'XTickLabel',[]); + + ax2 = subplot(2, 1, 2); + hold on; + for i = 1:length(Ms) + plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gm{i}(1, 1)*Kdvf(1,1), freqs, 'Hz')))), ... + 'DisplayName', sprintf('$m_p = %.0f$ [kg]', Ms(i))); + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); + ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); + ylim([-180, 180]); + yticks([-360:90:360]); + legend('location', 'northeast'); + + linkaxes([ax1,ax2],'x'); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/virtual_mass_loop_gain_L.pdf', 'width', 'full', 'height', 'full') +#+end_src + +#+name: fig:virtual_mass_loop_gain_L +#+caption: Loop Gain for the addition of virtual mass in the leg's space +#+RESULTS: +[[file:figs/virtual_mass_loop_gain_L.png]] + +** Identification of the Primary Plant +#+begin_src matlab :exports none + %% Name of the Simulink File + mdl = 'nass_model'; + + %% Input/Output definition + clear io; io_i = 1; + io(io_i) = linio([mdl, '/Controller'], 1, 'input'); io_i = io_i + 1; % Actuator Inputs + io(io_i) = linio([mdl, '/Tracking Error'], 1, 'output', [], 'En'); io_i = io_i + 1; % Position Errror + + load('mat/stages.mat', 'nano_hexapod'); + + GmL_x = {zeros(length(Ms), 1)}; + GmL_l = {zeros(length(Ms), 1)}; + + for i = 1:length(Ms) + initializeSample('mass', Ms(i), 'freq', sqrt(Kp/Ms(i))/2/pi*ones(6,1)); + initializeReferences('Rz_type', 'rotating-not-filtered', 'Rz_period', Ms(i)); + + %% Run the linearization + G = linearize(mdl, io); + G.InputName = {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'}; + G.OutputName = {'Ex', 'Ey', 'Ez', 'Erx', 'Ery', 'Erz'}; + + Gx = -G*inv(nano_hexapod.J'); + Gx.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'}; + GmL_x(i) = {Gx}; + + Gl = -nano_hexapod.J*G; + Gl.OutputName = {'E1', 'E2', 'E3', 'E4', 'E5', 'E6'}; + GmL_l(i) = {Gl}; + end +#+end_src + +#+begin_src matlab :exports none + freqs = logspace(0, 3, 5000); + + figure; + + ax1 = subplot(2, 2, 1); + hold on; + for i = 1:length(Ms) + set(gca,'ColorOrderIndex',i); + plot(freqs, abs(squeeze(freqresp(G_x{i}(1, 1), freqs, 'Hz')))); + set(gca,'ColorOrderIndex',i); + plot(freqs, abs(squeeze(freqresp(G_x{i}(2, 2), freqs, 'Hz')))); + set(gca,'ColorOrderIndex',i); + plot(freqs, abs(squeeze(freqresp(GmL_x{i}(1, 1), freqs, 'Hz'))), '--'); + set(gca,'ColorOrderIndex',i); + plot(freqs, abs(squeeze(freqresp(GmL_x{i}(2, 2), freqs, 'Hz'))), '--'); + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]); + title('$\mathcal{X}_x/\mathcal{F}_x$, $\mathcal{X}_y/\mathcal{F}_y$') + + ax2 = subplot(2, 2, 2); + hold on; + for i = 1:length(Ms) + set(gca,'ColorOrderIndex',i); + plot(freqs, abs(squeeze(freqresp(G_x{i}(3, 3), freqs, 'Hz')))); + set(gca,'ColorOrderIndex',i); + plot(freqs, abs(squeeze(freqresp(GmL_x{i}(3, 3), freqs, 'Hz'))), '--'); + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]); + title('$\mathcal{X}_z/\mathcal{F}_z$') + + ax3 = subplot(2, 2, 3); + hold on; + for i = 1:length(Ms) + set(gca,'ColorOrderIndex',i); + plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_x{i}(1, 1), freqs, 'Hz'))))); + set(gca,'ColorOrderIndex',i); + plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_x{i}(2, 2), freqs, 'Hz'))))); + set(gca,'ColorOrderIndex',i); + plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(GmL_x{i}(1, 1), freqs, 'Hz')))), '--'); + set(gca,'ColorOrderIndex',i); + plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(GmL_x{i}(2, 2), freqs, 'Hz')))), '--'); + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); + ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); + ylim([-270, 90]); + yticks([-360:90:360]); + + ax4 = subplot(2, 2, 4); + hold on; + for i = 1:length(Ms) + set(gca,'ColorOrderIndex',i); + plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_x{i}(3, 3), freqs, 'Hz')))), ... + 'DisplayName', sprintf('$m_p = %.0f [kg]$', Ms(i))); + set(gca,'ColorOrderIndex',i); + plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(GmL_x{i}(3, 3), freqs, 'Hz')))), '--', ... + 'HandleVisibility', 'off'); + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); + ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); + ylim([-270, 90]); + yticks([-360:90:360]); + legend('location', 'southwest'); + + linkaxes([ax1,ax2,ax3,ax4],'x'); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/virtual_mass_L_primary_plant_X.pdf', 'width', 'full', 'height', 'full') +#+end_src + +#+name: fig:virtual_mass_L_primary_plant_X +#+caption: Comparison of the transfer function from $\mathcal{F}_{x,y,z}$ to $\mathcal{X}_{x,y,z}$ with and without the virtual addition of mass in the leg's space +#+RESULTS: +[[file:figs/virtual_mass_L_primary_plant_X.png]] + +#+begin_src matlab :exports none + freqs = logspace(0, 3, 5000); + + figure; + + ax1 = subplot(2, 1, 1); + hold on; + for i = 1:length(Ms) + set(gca,'ColorOrderIndex',i); + plot(freqs, abs(squeeze(freqresp(G_l{i}(1, 1), freqs, 'Hz')))); + set(gca,'ColorOrderIndex',i); + plot(freqs, abs(squeeze(freqresp(GmL_l{i}(1, 1), freqs, 'Hz'))), '--'); + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]); + + ax2 = subplot(2, 1, 2); + hold on; + for i = 1:length(Ms) + set(gca,'ColorOrderIndex',i); + plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_l{i}(1, 1), freqs, 'Hz')))), ... + 'DisplayName', sprintf('$m_p = %.0f [kg]$', Ms(i))); + set(gca,'ColorOrderIndex',i); + plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(GmL_l{i}(1, 1), freqs, 'Hz')))), '--', ... + 'HandleVisibility', 'off'); + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); + ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); + ylim([-270, 90]); + yticks([-360:90:360]); + legend('location', 'southwest'); + + linkaxes([ax1,ax2],'x'); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/virtual_mass_L_primary_plant_L.pdf', 'width', 'full', 'height', 'full') +#+end_src + +#+name: fig:virtual_mass_L_primary_plant_L +#+caption: Comparison of the transfer function from $\tau_i$ to $\mathcal{L}_{i}$ with and without the virtual addition of mass in the leg's space +#+RESULTS: +[[file:figs/virtual_mass_L_primary_plant_L.png]] + +* Adding Virtual Mass in the Task Space +** Plant +Let's look at the transfer function from $\bm{\mathcal{F}}$ to $d\bm{\mathcal{X}}$: +\[ \frac{d\bm{\mathcal{L}}}{\bm{\mathcal{F}}} = \bm{J}^{-1} \frac{d\bm{\mathcal{L}}}{\bm{\tau}} \bm{J}^{-T} \] + +#+begin_src matlab :exports none + load('mat/stages.mat', 'nano_hexapod'); + + GmX = {zeros(length(Ms), 1)}; + for i = 1:length(Ms) + GmX(i) = {inv(nano_hexapod.J) * Gm{i} * inv(nano_hexapod.J')}; + end +#+end_src + +#+begin_src matlab :exports none + freqs = logspace(-1, 3, 1000); + + figure; + + ax1 = subplot(2, 2, 1); + hold on; + for i = 1:length(Ms) + set(gca,'ColorOrderIndex',i); + plot(freqs, abs(squeeze(freqresp(GmX{i}(1, 1), freqs, 'Hz')))); + set(gca,'ColorOrderIndex',i); + plot(freqs, abs(squeeze(freqresp(GmX{i}(2, 2), freqs, 'Hz')))); + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]); + + ax2 = subplot(2, 2, 3); + hold on; + for i = 1:length(Ms) + set(gca,'ColorOrderIndex',i); + plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(GmX{i}(1, 1), freqs, 'Hz')))), ... + 'DisplayName', sprintf('$m_p = %.0f$ [kg]', Ms(i))); + set(gca,'ColorOrderIndex',i); + plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(GmX{i}(2, 2), freqs, 'Hz')))), ... + 'HandleVisibility', 'off'); + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); + ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); + ylim([-270, 90]); + yticks([-360:90:360]); + legend('location', 'northeast'); + + ax1 = subplot(2, 2, 2); + hold on; + for i = 1:length(Ms) + set(gca,'ColorOrderIndex',i); + plot(freqs, abs(squeeze(freqresp(GmX{i}(3, 3), freqs, 'Hz')))); + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]); + + ax2 = subplot(2, 2, 4); + hold on; + for i = 1:length(Ms) + set(gca,'ColorOrderIndex',i); + plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(GmX{i}(3, 3), freqs, 'Hz')))), ... + 'DisplayName', sprintf('$m_p = %.0f$ [kg]', Ms(i))); + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); + ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); + ylim([-270, 90]); + yticks([-360:90:360]); + legend('location', 'northeast'); + + linkaxes([ax1,ax2],'x'); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/virtual_mass_plant_X.pdf', 'width', 'full', 'height', 'full') +#+end_src + +#+name: fig:virtual_mass_plant_X +#+caption: Dynamics from $\mathcal{F}_{x,y,z}$ to $\mathcal{X}_{x,y,z}$ used for virtual mass addition in the task space +#+RESULTS: +[[file:figs/virtual_mass_plant_X.png]] + +** Controller Design +#+begin_src matlab + KmX = (s^2*1/(1+s/2/pi/500)^2*diag([1 1 50 0 0 0])); +#+end_src + +#+begin_src matlab :exports none + for i = 1:length(Ms) + isstable(feedback(GmX{i}*KmX, eye(6), -1)) + end +#+end_src + +#+begin_src matlab :exports none + freqs = logspace(-1, 3, 1000); + + figure; + + ax1 = subplot(2, 2, 1); + hold on; + for i = 1:length(Ms) + LmX = GmX{i}*KmX; + set(gca,'ColorOrderIndex',i); + plot(freqs, abs(squeeze(freqresp(LmX(1, 1), freqs, 'Hz')))); + set(gca,'ColorOrderIndex',i); + plot(freqs, abs(squeeze(freqresp(LmX(2, 2), freqs, 'Hz')))); + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]); + + ax2 = subplot(2, 2, 3); + hold on; + for i = 1:length(Ms) + LmX = GmX{i}*KmX; + set(gca,'ColorOrderIndex',i); + plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(LmX(1, 1), freqs, 'Hz')))), ... + 'DisplayName', sprintf('$m_p = %.0f$ [kg]', Ms(i))); + set(gca,'ColorOrderIndex',i); + plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(LmX(2, 2), freqs, 'Hz')))), ... + 'HandleVisibility', 'off'); + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); + ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); + ylim([-270, 90]); + yticks([-360:90:360]); + legend('location', 'northeast'); + + ax1 = subplot(2, 2, 2); + hold on; + for i = 1:length(Ms) + LmX = GmX{i}*KmX; + set(gca,'ColorOrderIndex',i); + plot(freqs, abs(squeeze(freqresp(LmX(3, 3), freqs, 'Hz')))); + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]); + + ax2 = subplot(2, 2, 4); + hold on; + for i = 1:length(Ms) + LmX = GmX{i}*KmX; + set(gca,'ColorOrderIndex',i); + plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(LmX(3, 3), freqs, 'Hz')))), ... + 'DisplayName', sprintf('$m_p = %.0f$ [kg]', Ms(i))); + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); + ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); + ylim([-270, 90]); + yticks([-360:90:360]); + legend('location', 'northeast'); + + linkaxes([ax1,ax2],'x'); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/virtual_mass_loop_gain_X.pdf', 'width', 'full', 'height', 'full') +#+end_src + +#+name: fig:virtual_mass_loop_gain_X +#+caption: Loop gain for virtual mass addition in the task space +#+RESULTS: +[[file:figs/virtual_mass_loop_gain_X.png]] + +#+begin_src matlab + Kdvf = inv(nano_hexapod.J')*KmX*inv(nano_hexapod.J); +#+end_src + +#+begin_src matlab :exports none + for i = 1:length(Ms) + isstable(feedback(Gm{i}*Kdvf, eye(6), -1)) + end +#+end_src + +** Identification of the Primary Plant +#+begin_src matlab :exports none + %% Name of the Simulink File + mdl = 'nass_model'; + + %% Input/Output definition + clear io; io_i = 1; + io(io_i) = linio([mdl, '/Controller'], 1, 'input'); io_i = io_i + 1; % Actuator Inputs + io(io_i) = linio([mdl, '/Tracking Error'], 1, 'output', [], 'En'); io_i = io_i + 1; % Position Errror + + load('mat/stages.mat', 'nano_hexapod'); + + GmX_x = {zeros(length(Ms), 1)}; + GmX_l = {zeros(length(Ms), 1)}; + + for i = 1:length(Ms) + initializeSample('mass', Ms(i), 'freq', sqrt(Kp/Ms(i))/2/pi*ones(6,1)); + initializeReferences('Rz_type', 'rotating-not-filtered', 'Rz_period', Ms(i)); + + %% Run the linearization + G = linearize(mdl, io); + G.InputName = {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'}; + G.OutputName = {'Ex', 'Ey', 'Ez', 'Erx', 'Ery', 'Erz'}; + + Gx = -G*inv(nano_hexapod.J'); + Gx.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'}; + GmX_x(i) = {Gx}; + + Gl = -nano_hexapod.J*G; + Gl.OutputName = {'E1', 'E2', 'E3', 'E4', 'E5', 'E6'}; + GmX_l(i) = {Gl}; + end +#+end_src + +#+begin_src matlab :exports none + freqs = logspace(0, 3, 5000); + + figure; + + ax1 = subplot(2, 2, 1); + hold on; + for i = 1:length(Ms) + set(gca,'ColorOrderIndex',i); + plot(freqs, abs(squeeze(freqresp(G_x{i}(1, 1), freqs, 'Hz')))); + set(gca,'ColorOrderIndex',i); + plot(freqs, abs(squeeze(freqresp(G_x{i}(2, 2), freqs, 'Hz')))); + set(gca,'ColorOrderIndex',i); + plot(freqs, abs(squeeze(freqresp(GmX_x{i}(1, 1), freqs, 'Hz'))), '--'); + set(gca,'ColorOrderIndex',i); + plot(freqs, abs(squeeze(freqresp(GmX_x{i}(2, 2), freqs, 'Hz'))), '--'); + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]); + title('$\mathcal{X}_x/\mathcal{F}_x$, $\mathcal{X}_y/\mathcal{F}_y$') + + ax2 = subplot(2, 2, 2); + hold on; + for i = 1:length(Ms) + set(gca,'ColorOrderIndex',i); + plot(freqs, abs(squeeze(freqresp(G_x{i}(3, 3), freqs, 'Hz')))); + set(gca,'ColorOrderIndex',i); + plot(freqs, abs(squeeze(freqresp(GmX_x{i}(3, 3), freqs, 'Hz'))), '--'); + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]); + title('$\mathcal{X}_z/\mathcal{F}_z$') + + ax3 = subplot(2, 2, 3); + hold on; + for i = 1:length(Ms) + set(gca,'ColorOrderIndex',i); + plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_x{i}(1, 1), freqs, 'Hz'))))); + set(gca,'ColorOrderIndex',i); + plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_x{i}(2, 2), freqs, 'Hz'))))); + set(gca,'ColorOrderIndex',i); + plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(GmX_x{i}(1, 1), freqs, 'Hz')))), '--'); + set(gca,'ColorOrderIndex',i); + plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(GmX_x{i}(2, 2), freqs, 'Hz')))), '--'); + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); + ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); + ylim([-270, 90]); + yticks([-360:90:360]); + + ax4 = subplot(2, 2, 4); + hold on; + for i = 1:length(Ms) + set(gca,'ColorOrderIndex',i); + plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_x{i}(3, 3), freqs, 'Hz')))), ... + 'DisplayName', sprintf('$m_p = %.0f [kg]$', Ms(i))); + set(gca,'ColorOrderIndex',i); + plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(GmX_x{i}(3, 3), freqs, 'Hz')))), '--', ... + 'HandleVisibility', 'off'); + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); + ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); + ylim([-270, 90]); + yticks([-360:90:360]); + legend('location', 'southwest'); + + linkaxes([ax1,ax2,ax3,ax4],'x'); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/virtual_mass_X_primary_plant_X.pdf', 'width', 'full', 'height', 'full') +#+end_src + +#+name: fig:virtual_mass_X_primary_plant_X +#+caption: Comparison of the transfer function from $\mathcal{F}_{x,y,z}$ to $\mathcal{X}_{x,y,z}$ with and without the virtual addition of mass in the task space +#+RESULTS: +[[file:figs/virtual_mass_X_primary_plant_X.png]] + +#+begin_src matlab :exports none + freqs = logspace(0, 3, 5000); + + figure; + + ax1 = subplot(2, 1, 1); + hold on; + for i = 1:length(Ms) + set(gca,'ColorOrderIndex',i); + plot(freqs, abs(squeeze(freqresp(G_l{i}(1, 1), freqs, 'Hz')))); + set(gca,'ColorOrderIndex',i); + plot(freqs, abs(squeeze(freqresp(GmX_l{i}(1, 1), freqs, 'Hz'))), '--'); + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]); + + ax2 = subplot(2, 1, 2); + hold on; + for i = 1:length(Ms) + set(gca,'ColorOrderIndex',i); + plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_l{i}(1, 1), freqs, 'Hz')))), ... + 'DisplayName', sprintf('$m_p = %.0f [kg]$', Ms(i))); + set(gca,'ColorOrderIndex',i); + plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(GmX_l{i}(1, 1), freqs, 'Hz')))), '--', ... + 'HandleVisibility', 'off'); + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); + ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); + ylim([-270, 90]); + yticks([-360:90:360]); + legend('location', 'southwest'); + + linkaxes([ax1,ax2],'x'); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/virtual_mass_X_primary_plant_L.pdf', 'width', 'full', 'height', 'full') +#+end_src + +#+name: fig:virtual_mass_X_primary_plant_L +#+caption: Comparison of the transfer function from $\tau_i$ to $\mathcal{L}_{i}$ with and without the virtual addition of mass in the task space +#+RESULTS: +[[file:figs/virtual_mass_X_primary_plant_L.png]] diff --git a/org/optimal_stiffness_control.org b/org/optimal_stiffness_control.org index 893271e..160cc87 100644 --- a/org/optimal_stiffness_control.org +++ b/org/optimal_stiffness_control.org @@ -233,6 +233,9 @@ Then, we compute the transfer function from forces applied by the actuators $\bm \[ \bm{G}_\mathcal{X}(s) = \frac{\bm{\epsilon}_{\mathcal{X}_n}}{\bm{\mathcal{F}}} = \bm{G}(s) \bm{J}^{-T} \] The obtained dynamics is shown in Figure [[fig:opt_stiff_primary_plant_damped_X]]. +#+begin_important + A zero with a positive real part is introduced in the transfer function from $\mathcal{F}_y$ to $\mathcal{X}_y$ after Decentralized Direct Velocity Feedback is applied. +#+end_important And we compute the transfer function from actuator forces $\bm{\tau}$ to position error of each leg $\bm{\epsilon}_\mathcal{L}$: \[ \bm{G}_\mathcal{L} = \frac{\bm{\epsilon}_\mathcal{L}}{\bm{\tau}} = \bm{J} \bm{G}(s) \] @@ -449,7 +452,10 @@ The coupling does not change a lot with DVF. The coupling in the space of the legs $\bm{G}_\mathcal{L}$ are shown in Figure [[fig:opt_stiff_primary_plant_damped_coupling_L]]. -The magnitude of the coupling around the resonance of the nano-hexapod (where the coupling is the highest) is considerably reduced when DVF is applied. + +#+begin_important +The magnitude of the coupling between $\tau_i$ and $d\mathcal{L}_j$ (Figure [[fig:opt_stiff_primary_plant_damped_coupling_L]]) around the resonance of the nano-hexapod (where the coupling is the highest) is considerably reduced when DVF is applied. +#+end_important #+begin_src matlab :exports none freqs = logspace(0, 3, 1000); @@ -653,6 +659,11 @@ exportFig('figs/opt_stiff_sensibility_dist_dvf.pdf', 'width', 'full', 'height', #+RESULTS: [[file:figs/opt_stiff_sensibility_dist_dvf.png]] +*** Conclusion :ignore: +#+begin_important + Decentralized Direct Velocity Feedback is shown to increase the effect of stages vibrations at high frequency and to reduce the effect of ground motion and direct forces at low frequency. +#+end_important + ** Conclusion #+begin_important @@ -672,7 +683,7 @@ In this section we implement the control architecture shown in Figure [[fig:cont The controller for decentralized direct velocity feedback is the one designed in Section [[sec:lac_dvf]]. ** Plant in the leg space -We now loop at the transfer function matrix from $\bm{\tau}^\prime$ to $\bm{\epsilon}_{\mathcal{X}_n}$ for the design of $\bm{K}_\mathcal{L}$. +We now look at the transfer function matrix from $\bm{\tau}^\prime$ to $\bm{\epsilon}_{\mathcal{X}_n}$ for the design of $\bm{K}_\mathcal{L}$. The diagonal elements of the transfer function matrix from $\bm{\tau}^\prime$ to $\bm{\epsilon}_{\mathcal{X}_n}$ for the three considered masses are shown in Figure [[fig:opt_stiff_primary_plant_L]]. @@ -793,7 +804,7 @@ The loop gain is shown in Figure [[fig:opt_stiff_primary_loop_gain_L]]. end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); - ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]); + ylabel('Loop Gain'); set(gca, 'XTickLabel',[]); ax2 = subplot(2, 1, 2); hold on; @@ -821,12 +832,12 @@ exportFig('figs/opt_stiff_primary_loop_gain_L.pdf', 'width', 'full', 'height', ' #+RESULTS: [[file:figs/opt_stiff_primary_loop_gain_L.png]] +Finally, we include the Jacobian in the control and we ignore the measurement of the vertical rotation as for the real system. #+begin_src matlab load('mat/stages.mat', 'nano_hexapod'); K = Kl*nano_hexapod.J*diag([1, 1, 1, 1, 1, 0]); #+end_src -Check the MIMO stability #+begin_src matlab :exports none for i = 1:length(Ms) isstable(feedback(nano_hexapod.J\Gm_l{i}*K, eye(6), -1))

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