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+ UP + | + HOME +
+

Study of the Flexible Joints

+ + +

+In this document is studied the effect of the mechanical behavior of the flexible joints that are located the extremities of each nano-hexapod’s legs. +

+ +

+Ideally, we want the x and y rotations to be free and all the translations to be blocked. +However, this is never the case and be have to consider: +

+
    +
  • Finite x and y rotational stiffnesses (Section 1)
  • +
  • Translation stiffness in the direction of the legs (Section 2)
  • +
+ +

+This may impose some limitations, also, the goal is to specify the required joints stiffnesses. +

+ +
+

1 Rotational Stiffness

+
+

+ +

+

+In this section, we wish to study the effect of the rotation flexibility of the nano-hexapod joints. +

+
+ +
+

1.1 Initialization

+
+

+Let’s initialize all the stages with default parameters. +

+
+
initializeGround();
+initializeGranite();
+initializeTy();
+initializeRy();
+initializeRz();
+initializeMicroHexapod();
+initializeAxisc();
+initializeMirror();
+
+
+ +

+Let’s consider the heaviest mass which should we the most problematic with it comes to the flexible joints. +

+
+
initializeSample('mass', 50, 'freq', 200*ones(6,1));
+initializeReferences('Rz_type', 'rotating-not-filtered', 'Rz_period', 60);
+
+
+
+
+ +
+

1.2 Realistic Rotational Stiffness Values

+
+

+Let’s compare the ideal case (zero stiffness in rotation and infinite stiffness in translation) with a more realistic case: +

+
    +
  • \(K_{\theta, \phi} = 15\,[Nm/rad]\) stiffness in flexion
  • +
  • \(K_{\psi} = 20\,[Nm/rad]\) stiffness in torsion
  • +
+ +
+
Kf_M = 15*ones(6,1);
+Kf_F = 15*ones(6,1);
+Kt_M = 20*ones(6,1);
+Kt_F = 20*ones(6,1);
+
+
+ +

+The stiffness and damping of the nano-hexapod’s legs are: +

+
+
k_opt = 1e5; % [N/m]
+c_opt = 2e2; % [N/(m/s)]
+
+
+ +

+This corresponds to the optimal identified stiffness. +

+
+ +
+

1.2.1 Direct Velocity Feedback

+
+

+We identify the dynamics from actuators force \(\tau_i\) to relative motion sensors \(d\mathcal{L}_i\) with and without considering the flexible joint stiffness. +

+ +

+The obtained dynamics are shown in Figure 1. +It is shown that the adding of stiffness for the flexible joints does increase a little bit the frequencies of the mass suspension modes. It stiffen the structure. +

+ + +
+

flex_joint_rot_dvf.png +

+

Figure 1: Dynamics from actuators force \(\tau_i\) to relative motion sensors \(d\mathcal{L}_i\) with (blue) and without (red) considering the flexible joint stiffness

+
+
+
+ +
+

1.2.2 Primary Plant

+
+

+Let’s now identify the dynamics from \(\bm{\tau}^\prime\) to \(\bm{\epsilon}_{\mathcal{X}_n}\) (for the primary controller designed in the frame of the legs). +

+ +

+The dynamics is compare with and without the joint flexibility in Figure 2. +The plant dynamics is not found to be changing significantly. +

+ + +
+

flex_joints_rot_primary_plant_L.png +

+

Figure 2: Dynamics from \(\bm{\tau}^\prime_i\) to \(\bm{\epsilon}_{\mathcal{X}_n,i}\) with perfect joints (dashed) and with flexible joints (solid)

+
+
+
+ +
+

1.2.3 Conclusion

+
+
+

+Considering realistic flexible joint rotational stiffness for the nano-hexapod does not seems to impose any limitation on the DVF control nor on the primary control. +

+ +

+It only increases a little bit the suspension modes of the sample on top of the nano-hexapod. +

+ +
+
+
+
+ +
+

1.3 Parametric Study

+
+

+We wish now to see what is the impact of the rotation stiffness of the flexible joints on the dynamics. +This will help to determine the requirements on the joint’s stiffness. +

+ +

+Let’s consider the following rotational stiffness of the flexible joints: +

+
+
Ks = [1, 5, 10, 50, 100]; % [Nm/rad]
+
+
+ +

+We also consider here a nano-hexapod with the identified optimal actuator stiffness. +

+
+ +
+

1.3.1 Direct Velocity Feedback

+
+

+The dynamics from the actuators to the relative displacement sensor in each leg is identified and shown in Figure 3. +

+ +

+The corresponding Root Locus plot is shown in Figure 4. +

+ +

+It is shown that the rotational stiffness of the flexible joints does indeed change a little the dynamics, but critical damping is stiff achievable with Direct Velocity Feedback. +

+ + +
+

flex_joints_rot_study_dvf.png +

+

Figure 3: Dynamics from \(\tau_i\) to \(d\mathcal{L}_i\) for all the considered Rotation Stiffnesses

+
+ + +
+

flex_joints_rot_study_dvf_root_locus.png +

+

Figure 4: Root Locus for all the considered Rotation Stiffnesses

+
+
+
+ +
+

1.3.2 Primary Control

+
+

+The dynamics from \(\bm{\tau}^\prime\) to \(\bm{\epsilon}_{\mathcal{X}_n}\) (for the primary controller designed in the frame of the legs) is shown in Figure 5. +

+ +

+It is shown that the rotational stiffness of the flexible joints have very little impact on the dynamics. +

+ + +
+

flex_joints_rot_study_primary_plant.png +

+

Figure 5: Diagonal elements of the transfer function matrix from \(\bm{\tau}^\prime\) to \(\bm{\epsilon}_{\mathcal{X}_n}\) for the considered rotational stiffnesses

+
+
+
+ +
+

1.3.3 Conclusion

+
+
+

+The rotational stiffness of the flexible joint does not significantly change the dynamics. +

+ +
+
+
+
+
+ +
+

2 Translation Stiffness

+
+

+ +

+

+Let’s know consider a flexibility in translation of the flexible joint, in the axis of the legs. +

+
+ +
+

2.1 Realistic Translation Stiffness Values

+
+

+We choose realistic values of the axial stiffness of the joints: +\[ K_a = 60\,[N/\mu m] \] +

+ +
+
Kz_F = 60e6*ones(6,1); % [N/m]
+Kz_M = 60e6*ones(6,1); % [N/m]
+Cz_F = 1*ones(6,1); % [N/(m/s)]
+Cz_M = 1*ones(6,1); % [N/(m/s)]
+
+
+
+ +
+

2.1.1 Initialization

+
+

+Let’s initialize all the stages with default parameters. +

+
+
initializeGround();
+initializeGranite();
+initializeTy();
+initializeRy();
+initializeRz();
+initializeMicroHexapod();
+initializeAxisc();
+initializeMirror();
+
+
+ +

+Let’s consider the heaviest mass which should we the most problematic with it comes to the flexible joints. +

+
+
initializeSample('mass', 50, 'freq', 200*ones(6,1));
+initializeReferences('Rz_type', 'rotating-not-filtered', 'Rz_period', 60);
+
+
+
+
+ +
+

2.1.2 Direct Velocity Feedback

+
+

+The dynamics from actuators force \(\tau_i\) to relative motion sensors \(d\mathcal{L}_i\) with and without considering the flexible joint stiffness are identified. +

+ +

+The obtained dynamics are shown in Figure 6. +

+ + +
+

flex_joint_trans_dvf.png +

+

Figure 6: Dynamics from actuators force \(\tau_i\) to relative motion sensors \(d\mathcal{L}_i\) with (blue) and without (red) considering the flexible joint axis stiffness

+
+
+
+ +
+

2.1.3 Primary Plant

+
+
+
Kdvf = 5e3*s/(1+s/2/pi/1e3)*eye(6);
+
+
+ +

+Let’s now identify the dynamics from \(\bm{\tau}^\prime\) to \(\bm{\epsilon}_{\mathcal{X}_n}\) (for the primary controller designed in the frame of the legs). +

+ +

+The dynamics is compare with and without the joint flexibility in Figure 7. +

+ + +
+

flex_joints_trans_primary_plant_L.png +

+

Figure 7: Dynamics from \(\bm{\tau}^\prime_i\) to \(\bm{\epsilon}_{\mathcal{X}_n,i}\) with infinite axis stiffnes (solid) and with realistic axial stiffness (dashed)

+
+
+
+
+ +
+

2.2 Parametric study

+
+

+We wish now to see what is the impact of the axial stiffness of the flexible joints on the dynamics. +

+ +

+Let’s consider the following values for the axial stiffness: +

+
+
Kzs = [1e4, 1e5, 1e6, 1e7, 1e8, 1e9]; % [N/m]
+
+
+ +

+We also consider here a nano-hexapod with the identified optimal actuator stiffness (\(k = 10^5\,[N/m]\)). +

+
+ +
+

2.2.1 Direct Velocity Feedback

+
+

+The dynamics from the actuators to the relative displacement sensor in each leg is identified and shown in Figure 8. +

+ +

+It is shown that the axial stiffness of the flexible joints does have a huge impact on the dynamics. +

+ +

+If the axial stiffness of the flexible joints is \(K_a > 10^7\,[N/m]\) (here \(100\) times higher than the actuator stiffness), then the change of dynamics stays reasonably small. +

+ +

+This is more clear by looking at the root locus (Figures 9 and 10). +It can be seen that very little active damping can be achieve for rotational joint axial stiffnesses below \(10^7\,[N/m]\). +

+ + +
+

flex_joints_trans_study_dvf.png +

+

Figure 8: Dynamics from \(\tau_i\) to \(d\mathcal{L}_i\) for all the considered axis Stiffnesses

+
+ + +
+

flex_joints_trans_study_dvf_root_locus.png +

+

Figure 9: Root Locus for all the considered axial Stiffnesses

+
+ +
+
xlim([-1e3, 0]);
+ylim([0, 1e3]);
+
+
+ + +
+

flex_joints_trans_study_root_locus_unzoom.png +

+

Figure 10: Root Locus (unzoom) for all the considered axial Stiffnesses

+
+
+
+ +
+

2.2.2 Primary Control

+
+

+The dynamics from \(\bm{\tau}^\prime\) to \(\bm{\epsilon}_{\mathcal{X}_n}\) (for the primary controller designed in the frame of the legs) is shown in Figure 11. +

+ + +
+

flex_joints_trans_study_primary_plant.png +

+

Figure 11: Diagonal elements of the transfer function matrix from \(\bm{\tau}^\prime\) to \(\bm{\epsilon}_{\mathcal{X}_n}\) for the considered axial stiffnesses

+
+
+
+
+ +
+

2.3 Conclusion

+
+
+

+The axial stiffness of the flexible joints should be maximized. +

+ +

+For the considered actuator stiffness \(k = 10^5\,[N/m]\), the axial stiffness of the rotational joints should ideally be above \(10^7\,[N/m]\). +

+ +

+This is a reasonable stiffness value for such joints. +

+ +

+We may interpolate the results and say that the axial joint stiffness should be 100 times higher than the actuator stiffness, but this should be confirmed with further analysis. +

+ +
+
+
+
+
+
+

Author: Dehaeze Thomas

+

Created: 2020-05-05 mar. 10:34

+
+ + diff --git a/org/flexible_joints_study.org b/org/flexible_joints_study.org index 43a7629..047df6e 100644 --- a/org/flexible_joints_study.org +++ b/org/flexible_joints_study.org @@ -75,17 +75,19 @@ Let's compare the ideal case (zero stiffness in rotation and infinite stiffness #+begin_src matlab Kf_M = 15*ones(6,1); - Kt_M = 20*ones(6,1); Kf_F = 15*ones(6,1); + Kt_M = 20*ones(6,1); Kt_F = 20*ones(6,1); #+end_src The stiffness and damping of the nano-hexapod's legs are: #+begin_src matlab - k = 1e5; % [N/m] - c = 2e2; % [N/(m/s)] + k_opt = 1e5; % [N/m] + c_opt = 2e2; % [N/(m/s)] #+end_src +This corresponds to the optimal identified stiffness. + *** Direct Velocity Feedback We identify the dynamics from actuators force $\tau_i$ to relative motion sensors $d\mathcal{L}_i$ with and without considering the flexible joint stiffness. @@ -103,7 +105,7 @@ It is shown that the adding of stiffness for the flexible joints does increase a #+end_src #+begin_src matlab :exports none - initializeNanoHexapod('k', k, 'c', c, ... + initializeNanoHexapod('k', k_opt, 'c', c_opt, ... 'type_F', 'universal_p', ... 'type_M', 'spherical_p'); @@ -113,7 +115,7 @@ It is shown that the adding of stiffness for the flexible joints does increase a #+end_src #+begin_src matlab :exports none - initializeNanoHexapod('k', k, 'c', c, ... + initializeNanoHexapod('k', k_opt, 'c', c_opt, ... 'type_F', 'universal', ... 'type_M', 'spherical', ... 'Kf_M', Kf_M, ... @@ -156,7 +158,7 @@ It is shown that the adding of stiffness for the flexible joints does increase a #+end_src #+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/flex_joint_rot_dvf.pdf', 'width', 'full', 'height', 'full') + exportFig('figs/flex_joint_rot_dvf.pdf', 'width', 'full', 'height', 'full'); #+end_src #+name: fig:flex_joint_rot_dvf @@ -187,23 +189,20 @@ The plant dynamics is not found to be changing significantly. #+end_src #+begin_src matlab :exports none - initializeNanoHexapod('k', k, 'c', c, ... + initializeNanoHexapod('k', k_opt, 'c', c_opt, ... 'type_F', 'universal_p', ... 'type_M', 'spherical_p'); - G_p = linearize(mdl, io); - G_p.InputName = {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'}; - G_p.OutputName = {'Ex', 'Ey', 'Ez', 'Erx', 'Ery', 'Erz'}; + G = linearize(mdl, io); + G.InputName = {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'}; + G.OutputName = {'Ex', 'Ey', 'Ez', 'Erx', 'Ery', 'Erz'}; - Gx_p = -G_p*inv(nano_hexapod.kinematics.J'); - Gx_p.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'}; - - Gl_p = -nano_hexapod.kinematics.J*G_p; + Gl_p = -nano_hexapod.kinematics.J*G; Gl_p.OutputName = {'E1', 'E2', 'E3', 'E4', 'E5', 'E6'}; #+end_src #+begin_src matlab :exports none - initializeNanoHexapod('k', k, 'c', c, ... + initializeNanoHexapod('k', k_opt, 'c', c_opt, ... 'type_F', 'universal', ... 'type_M', 'spherical', ... 'Kf_M', Kf_M, ... @@ -215,9 +214,6 @@ The plant dynamics is not found to be changing significantly. G.InputName = {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'}; G.OutputName = {'Ex', 'Ey', 'Ez', 'Erx', 'Ery', 'Erz'}; - Gx = -G*inv(nano_hexapod.kinematics.J'); - Gx.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'}; - Gl = -nano_hexapod.kinematics.J*G; Gl.OutputName = {'E1', 'E2', 'E3', 'E4', 'E5', 'E6'}; #+end_src @@ -283,7 +279,7 @@ This will help to determine the requirements on the joint's stiffness. Let's consider the following rotational stiffness of the flexible joints: #+begin_src matlab - Ks = [1, 10, 100]; % [Nm/rad] + Ks = [1, 5, 10, 50, 100]; % [Nm/rad] #+end_src We also consider here a nano-hexapod with the identified optimal actuator stiffness. @@ -307,28 +303,24 @@ It is shown that the rotational stiffness of the flexible joints does indeed cha %% 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 + io(io_i) = linio([mdl, '/Micro-Station'], 3, 'openoutput', [], 'Dnlm'); io_i = io_i + 1; % Relative Displacement Sensors - Gdvf_s = {zeros(length(Ks), 1)}; + G_dvf_s = {zeros(length(Ks), 1)}; for i = 1:length(Ks) - initializeNanoHexapod('k', k, 'c', c, ... + initializeNanoHexapod('k', k_opt, 'c', c_opt, ... 'type_F', 'universal', ... 'type_M', 'spherical', ... 'Kf_M', Ks(i), ... 'Kt_M', Ks(i), ... 'Kf_F', Ks(i), ... - 'Kt_F', Ks(i), ... - 'Cf_M', 0.2*sqrt(Ks(i)*1), ... - 'Ct_M', 0.2*sqrt(Ks(i)*1), ... - 'Cf_F', 0.2*sqrt(Ks(i)*1), ... - 'Ct_F', 0.2*sqrt(Ks(i)*1)); + 'Kt_F', Ks(i)); - Gdvf = linearize(mdl, io); - G_dvf.InputName = {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'}; - G_dvf.OutputName = {'Dnlm1', 'Dnlm2', 'Dnlm3', 'Dnlm4', 'Dnlm5', 'Dnlm6'}; + G = linearize(mdl, io); + G.InputName = {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'}; + G.OutputName = {'Dnlm1', 'Dnlm2', 'Dnlm3', 'Dnlm4', 'Dnlm5', 'Dnlm6'}; - Gdvf_s(i) = {Gdvf}; + G_dvf_s(i) = {G}; end #+end_src @@ -340,7 +332,7 @@ It is shown that the rotational stiffness of the flexible joints does indeed cha ax1 = subplot(2, 1, 1); hold on; for i = 1:length(Ks) - plot(freqs, abs(squeeze(freqresp(Gdvf_s{i}(1, 1), freqs, 'Hz')))); + plot(freqs, abs(squeeze(freqresp(G_dvf_s{i}(1, 1), freqs, 'Hz')))); end plot(freqs, abs(squeeze(freqresp(G_dvf_p(1, 1), freqs, 'Hz'))), 'k--'); hold off; @@ -350,7 +342,7 @@ It is shown that the rotational stiffness of the flexible joints does indeed cha ax2 = subplot(2, 1, 2); hold on; for i = 1:length(Ks) - plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gdvf_s{i}(1, 1), freqs, 'Hz')))), ... + plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_dvf_s{i}(1, 1), freqs, 'Hz')))), ... 'DisplayName', sprintf('$k = %.0g$ [N/m]', Ks(i))); end plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_dvf_p(1, 1), freqs, 'Hz')))), 'k--', ... @@ -366,7 +358,7 @@ It is shown that the rotational stiffness of the flexible joints does indeed cha #+end_src #+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/flex_joints_rot_study_dvf.pdf', 'width', 'full', 'height', 'full') + exportFig('figs/flex_joints_rot_study_dvf.pdf', 'width', 'full', 'height', 'full'); #+end_src #+name: fig:flex_joints_rot_study_dvf @@ -382,14 +374,14 @@ exportFig('figs/flex_joints_rot_study_dvf.pdf', 'width', 'full', 'height', 'full hold on; for i = 1:length(Ks) set(gca,'ColorOrderIndex',i); - plot(real(pole(Gdvf_s{i})), imag(pole(Gdvf_s{i})), 'x', ... + plot(real(pole(G_dvf_s{i})), imag(pole(G_dvf_s{i})), 'x', ... 'DisplayName', sprintf('$k = %.0g$ [N/m]', Ks(i))); set(gca,'ColorOrderIndex',i); - plot(real(tzero(Gdvf_s{i})), imag(tzero(Gdvf_s{i})), 'o', ... + plot(real(tzero(G_dvf_s{i})), imag(tzero(G_dvf_s{i})), 'o', ... 'HandleVisibility', 'off'); for k = 1:length(gains) set(gca,'ColorOrderIndex',i); - cl_poles = pole(feedback(Gdvf_s{i}, (gains(k)*s)*eye(6))); + cl_poles = pole(feedback(G_dvf_s{i}, (gains(k)*s)*eye(6))); plot(real(cl_poles), imag(cl_poles), '.', ... 'HandleVisibility', 'off'); end @@ -403,7 +395,7 @@ exportFig('figs/flex_joints_rot_study_dvf.pdf', 'width', 'full', 'height', 'full #+end_src #+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/flex_joints_rot_study_dvf_root_locus.pdf', 'width', 'wide', 'height', 'tall') + exportFig('figs/flex_joints_rot_study_dvf_root_locus.pdf', 'width', 'wide', 'height', 'tall'); #+end_src #+name: fig:flex_joints_rot_study_dvf_root_locus @@ -412,7 +404,11 @@ exportFig('figs/flex_joints_rot_study_dvf_root_locus.pdf', 'width', 'wide', 'hei [[file:figs/flex_joints_rot_study_dvf_root_locus.png]] *** Primary Control -#+begin_src matlab +The dynamics from $\bm{\tau}^\prime$ to $\bm{\epsilon}_{\mathcal{X}_n}$ (for the primary controller designed in the frame of the legs) is shown in Figure [[fig:flex_joints_rot_study_primary_plant]]. + +It is shown that the rotational stiffness of the flexible joints have very little impact on the dynamics. + +#+begin_src matlab :exports none Kdvf = 5e3*s/(1+s/2/pi/1e3)*eye(6); #+end_src @@ -427,33 +423,24 @@ exportFig('figs/flex_joints_rot_study_dvf_root_locus.pdf', 'width', 'wide', 'hei load('mat/stages.mat', 'nano_hexapod'); - Gx_3dof_s = {zeros(length(Ks), 1)}; - Gl_3dof_s = {zeros(length(Ks), 1)}; + Gl_s = {zeros(length(Ks), 1)}; for i = 1:length(Ks) - initializeNanoHexapod('k', k, 'c', c, ... + initializeNanoHexapod('k', k_opt, 'c', c_opt, ... 'type_F', 'universal', ... 'type_M', 'spherical', ... 'Kf_M', Ks(i), ... - 'Kt_M', 20, ... + 'Kt_M', Ks(i), ... 'Kf_F', Ks(i), ... - 'Kt_F', 20, ... - 'Cf_M', 0, ... - 'Ct_M', 0, ... - 'Cf_F', 0, ... - 'Ct_F', 0); + 'Kt_F', Ks(i)); G = linearize(mdl, io); G.InputName = {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'}; G.OutputName = {'Ex', 'Ey', 'Ez', 'Erx', 'Ery', 'Erz'}; - Gx_3dof = -G*inv(nano_hexapod.kinematics.J'); - Gx_3dof.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'}; - Gx_3dof_s(i) = {Gx_3dof}; - - Gl_3dof = -nano_hexapod.kinematics.J*G; - Gl_3dof.OutputName = {'E1', 'E2', 'E3', 'E4', 'E5', 'E6'}; - Gl_3dof_s(i) = {Gl_3dof}; + Gl = -nano_hexapod.kinematics.J*G; + Gl.OutputName = {'E1', 'E2', 'E3', 'E4', 'E5', 'E6'}; + Gl_s(i) = {Gl}; end #+end_src @@ -465,7 +452,7 @@ exportFig('figs/flex_joints_rot_study_dvf_root_locus.pdf', 'width', 'wide', 'hei ax1 = subplot(2, 1, 1); hold on; for i = 1:length(Ks) - plot(freqs, abs(squeeze(freqresp(Gl_3dof_s{i}(1, 1), freqs, 'Hz')))); + plot(freqs, abs(squeeze(freqresp(Gl_s{i}(1, 1), freqs, 'Hz')))); end plot(freqs, abs(squeeze(freqresp(Gl_p(1, 1), freqs, 'Hz'))), 'k--'); hold off; @@ -475,7 +462,7 @@ exportFig('figs/flex_joints_rot_study_dvf_root_locus.pdf', 'width', 'wide', 'hei ax2 = subplot(2, 1, 2); hold on; for i = 1:length(Ks) - plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gl_3dof_s{i}(1, 1), freqs, 'Hz')))), ... + plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gl_s{i}(1, 1), freqs, 'Hz')))), ... 'DisplayName', sprintf('$k = %.0g$ [N/m]', Ks(i))); end plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gl_p(1, 1), freqs, 'Hz')))), 'k--', ... @@ -501,14 +488,14 @@ exportFig('figs/flex_joints_rot_study_dvf_root_locus.pdf', 'width', 'wide', 'hei *** Conclusion #+begin_important - + The rotational stiffness of the flexible joint does not significantly change the dynamics. #+end_important * Translation Stiffness <> ** Introduction :ignore: -Let's know consider a flexibility in translation of the flexible joint. +Let's know consider a flexibility in translation of the flexible joint, in the axis of the legs. ** Matlab Init :noexport:ignore: #+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) @@ -528,7 +515,19 @@ Let's know consider a flexibility in translation of the flexible joint. open('nass_model.slx') #+end_src -** Initialization +** Realistic Translation Stiffness Values +*** Introduction :ignore: +We choose realistic values of the axial stiffness of the joints: +\[ K_a = 60\,[N/\mu m] \] + +#+begin_src matlab + Kz_F = 60e6*ones(6,1); % [N/m] + Kz_M = 60e6*ones(6,1); % [N/m] + Cz_F = 1*ones(6,1); % [N/(m/s)] + Cz_M = 1*ones(6,1); % [N/(m/s)] +#+end_src + +*** Initialization Let's initialize all the stages with default parameters. #+begin_src matlab initializeGround(); @@ -560,13 +559,10 @@ Let's consider the heaviest mass which should we the most problematic with it co Kdvf = tf(zeros(6)); #+end_src -** Direct Velocity Feedback -#+begin_src matlab - Kz_F = 60e6*ones(6,1); % [N/m] - Kz_M = 60e6*ones(6,1); % [N/m] - Cz_F = 1e2*ones(6,1); % [N/m] - Cz_M = 1e2*ones(6,1); % [N/m] -#+end_src +*** Direct Velocity Feedback +The dynamics from actuators force $\tau_i$ to relative motion sensors $d\mathcal{L}_i$ with and without considering the flexible joint stiffness are identified. + +The obtained dynamics are shown in Figure [[fig:flex_joint_trans_dvf]]. #+begin_src matlab :exports none %% Name of the Simulink File @@ -579,7 +575,7 @@ Let's consider the heaviest mass which should we the most problematic with it co #+end_src #+begin_src matlab :exports none - initializeNanoHexapod('k', 1e5, 'c', 2e2, ... + initializeNanoHexapod('k', k_opt, 'c', c_opt, ... 'type_F', 'universal_3dof', ... 'type_M', 'spherical_3dof', ... 'Kz_F', Kz_F, ... @@ -593,7 +589,7 @@ Let's consider the heaviest mass which should we the most problematic with it co #+end_src #+begin_src matlab :exports none - initializeNanoHexapod('k', 1e5, 'c', 2e2, ... + initializeNanoHexapod('k', k_opt, 'c', c_opt, ... 'type_F', 'universal', ... 'type_M', 'spherical'); @@ -632,19 +628,23 @@ Let's consider the heaviest mass which should we the most problematic with it co #+end_src #+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/flex_joint_trans_dvf.pdf', 'width', 'full', 'height', 'full') + exportFig('figs/flex_joint_trans_dvf.pdf', 'width', 'full', 'height', 'full'); #+end_src #+name: fig:flex_joint_trans_dvf -#+caption: +#+caption: Dynamics from actuators force $\tau_i$ to relative motion sensors $d\mathcal{L}_i$ with (blue) and without (red) considering the flexible joint axis stiffness #+RESULTS: [[file:figs/flex_joint_trans_dvf.png]] -** Primary Plant +*** Primary Plant #+begin_src matlab Kdvf = 5e3*s/(1+s/2/pi/1e3)*eye(6); #+end_src +Let's now identify the dynamics from $\bm{\tau}^\prime$ to $\bm{\epsilon}_{\mathcal{X}_n}$ (for the primary controller designed in the frame of the legs). + +The dynamics is compare with and without the joint flexibility in Figure [[fig:flex_joints_trans_primary_plant_L]]. + #+begin_src matlab :exports none %% Name of the Simulink File mdl = 'nass_model'; @@ -658,7 +658,7 @@ exportFig('figs/flex_joint_trans_dvf.pdf', 'width', 'full', 'height', 'full') #+end_src #+begin_src matlab :exports none - initializeNanoHexapod('k', 1e5, 'c', 2e2, ... + initializeNanoHexapod('k', k_opt, 'c', c_opt, ... 'type_F', 'universal_3dof', ... 'type_M', 'spherical_3dof', ... 'Kz_F', Kz_F, ... @@ -666,19 +666,16 @@ exportFig('figs/flex_joint_trans_dvf.pdf', 'width', 'full', 'height', 'full') 'Cz_F', Cz_F, ... 'Cz_M', Cz_M); - G_3dof = linearize(mdl, io); - G_3dof.InputName = {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'}; - G_3dof.OutputName = {'Ex', 'Ey', 'Ez', 'Erx', 'Ery', 'Erz'}; + G = linearize(mdl, io); + G.InputName = {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'}; + G.OutputName = {'Ex', 'Ey', 'Ez', 'Erx', 'Ery', 'Erz'}; - Gx_3dof = -G_3dof*inv(nano_hexapod.kinematics.J'); - Gx_3dof.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'}; - - Gl_3dof = -nano_hexapod.kinematics.J*G_3dof; + Gl_3dof = -nano_hexapod.kinematics.J*G; Gl_3dof.OutputName = {'E1', 'E2', 'E3', 'E4', 'E5', 'E6'}; #+end_src #+begin_src matlab :exports none - initializeNanoHexapod('k', 1e5, 'c', 2e2, ... + initializeNanoHexapod('k', k_opt, 'c', c_opt, ... 'type_F', 'universal', ... 'type_M', 'spherical'); @@ -686,9 +683,6 @@ exportFig('figs/flex_joint_trans_dvf.pdf', 'width', 'full', 'height', 'full') G.InputName = {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'}; G.OutputName = {'Ex', 'Ey', 'Ez', 'Erx', 'Ery', 'Erz'}; - Gx = -G*inv(nano_hexapod.kinematics.J'); - Gx.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'}; - Gl = -nano_hexapod.kinematics.J*G; Gl.OutputName = {'E1', 'E2', 'E3', 'E4', 'E5', 'E6'}; #+end_src @@ -733,21 +727,40 @@ exportFig('figs/flex_joint_trans_dvf.pdf', 'width', 'full', 'height', 'full') #+end_src #+begin_src matlab :tangle no :exports results :results file replace - exportFig('figs/flex_joints_trans_primary_plant_L.pdf', 'width', 'full', 'height', 'full') + exportFig('figs/flex_joints_trans_primary_plant_L.pdf', 'width', 'full', 'height', 'full'); #+end_src #+name: fig:flex_joints_trans_primary_plant_L -#+caption: +#+caption: Dynamics from $\bm{\tau}^\prime_i$ to $\bm{\epsilon}_{\mathcal{X}_n,i}$ with infinite axis stiffnes (solid) and with realistic axial stiffness (dashed) #+RESULTS: [[file:figs/flex_joints_trans_primary_plant_L.png]] ** Parametric study *** Introduction :ignore: +We wish now to see what is the impact of the *axial* stiffness of the flexible joints on the dynamics. + +Let's consider the following values for the axial stiffness: #+begin_src matlab Kzs = [1e4, 1e5, 1e6, 1e7, 1e8, 1e9]; % [N/m] #+end_src +We also consider here a nano-hexapod with the identified optimal actuator stiffness ($k = 10^5\,[N/m]$). + +#+begin_src matlab :exports none + K = tf(zeros(6)); + Kdvf = tf(zeros(6)); +#+end_src + *** Direct Velocity Feedback +The dynamics from the actuators to the relative displacement sensor in each leg is identified and shown in Figure [[fig:flex_joints_trans_study_dvf]]. + +It is shown that the axial stiffness of the flexible joints does have a huge impact on the dynamics. + +If the axial stiffness of the flexible joints is $K_a > 10^7\,[N/m]$ (here $100$ times higher than the actuator stiffness), then the change of dynamics stays reasonably small. + +This is more clear by looking at the root locus (Figures [[fig:flex_joints_trans_study_dvf_root_locus]] and [[fig:flex_joints_trans_study_root_locus_unzoom]]). +It can be seen that very little active damping can be achieve for rotational joint axial stiffnesses below $10^7\,[N/m]$. + #+begin_src matlab :exports none %% Name of the Simulink File mdl = 'nass_model'; @@ -756,25 +769,23 @@ exportFig('figs/flex_joint_trans_dvf.pdf', 'width', 'full', 'height', 'full') 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 - Gdvf_3dof_s = {zeros(length(Kzs), 1)}; + G_dvf_3dof_s = {zeros(length(Kzs), 1)}; for i = 1:length(Kzs) - initializeNanoHexapod('k', 1e5, 'c', 2e2, ... + initializeNanoHexapod('k', k_opt, 'c', c_opt, ... 'type_F', 'universal_3dof', ... 'type_M', 'spherical_3dof', ... 'Kz_F', Kzs(i), ... 'Kz_M', Kzs(i), ... - 'Cz_F', 0.2*sqrt(Kzs(i)*10), ... - 'Cz_M', 0.2*sqrt(Kzs(i)*10)); + 'Cz_F', 0.05*sqrt(Kzs(i)*10), ... + 'Cz_M', 0.05*sqrt(Kzs(i)*10)); G = linearize(mdl, io); G.InputName = {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'}; G.OutputName = {'Dnlm1', 'Dnlm2', 'Dnlm3', 'Dnlm4', 'Dnlm5', 'Dnlm6'}; - - Gdvf_3dof_s(i) = {G}; + + G_dvf_3dof_s(i) = {G}; end #+end_src @@ -786,9 +797,9 @@ exportFig('figs/flex_joint_trans_dvf.pdf', 'width', 'full', 'height', 'full') ax1 = subplot(2, 1, 1); hold on; for i = 1:length(Kzs) - plot(freqs, abs(squeeze(freqresp(Gdvf_3dof_s{i}(1, 1), freqs, 'Hz')))); + plot(freqs, abs(squeeze(freqresp(G_dvf_3dof_s{i}(1, 1), freqs, 'Hz')))); end - plot(freqs, abs(squeeze(freqresp(Gdvf(1, 1), freqs, 'Hz'))), 'k--'); + plot(freqs, abs(squeeze(freqresp(G_dvf(1, 1), freqs, 'Hz'))), 'k--'); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]); @@ -796,23 +807,85 @@ exportFig('figs/flex_joint_trans_dvf.pdf', 'width', 'full', 'height', 'full') ax2 = subplot(2, 1, 2); hold on; for i = 1:length(Kzs) - plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gdvf_3dof_s{i}(1, 1), freqs, 'Hz')))), ... - 'DisplayName', sprintf('$k = %.0g$ [N/m]', Kzs(i))); - plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gdvf(1, 1), freqs, 'Hz')))), 'k--', ... - 'DisplayName', 'Perfect Joint'); + plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_dvf_3dof_s{i}(1, 1), freqs, 'Hz')))), ... + 'DisplayName', sprintf('$k = %.0g$ [N/m]', Kzs(i))); end + plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_dvf(1, 1), freqs, 'Hz')))), 'k--', ... + 'DisplayName', 'Perfect Joint'); 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'); + legend('location', 'southwest'); linkaxes([ax1,ax2],'x'); #+end_src -*** Primary Control +#+begin_src matlab :tangle no :exports results :results file replace + exportFig('figs/flex_joints_trans_study_dvf.pdf', 'width', 'full', 'height', 'full'); +#+end_src + +#+name: fig:flex_joints_trans_study_dvf +#+caption: Dynamics from $\tau_i$ to $d\mathcal{L}_i$ for all the considered axis Stiffnesses +#+RESULTS: +[[file:figs/flex_joints_trans_study_dvf.png]] + +#+begin_src matlab :exports none + figure; + + gains = logspace(2, 5, 300); + + hold on; + for i = 1:length(Kzs) + set(gca,'ColorOrderIndex',i); + plot(real(pole(G_dvf_3dof_s{i})), imag(pole(G_dvf_3dof_s{i})), 'x', ... + 'DisplayName', sprintf('$k = %.0g$ [N/m]', Kzs(i))); + set(gca,'ColorOrderIndex',i); + plot(real(tzero(G_dvf_3dof_s{i})), imag(tzero(G_dvf_3dof_s{i})), 'o', ... + 'HandleVisibility', 'off'); + for k = 1:length(gains) + set(gca,'ColorOrderIndex',i); + cl_poles = pole(feedback(G_dvf_3dof_s{i}, (gains(k)*s)*eye(6))); + plot(real(cl_poles), imag(cl_poles), '.', ... + 'HandleVisibility', 'off'); + end + end + hold off; + axis square; + xlim([-140, 10]); ylim([0, 150]); + + xlabel('Real Part'); ylabel('Imaginary Part'); + legend('location', 'northwest'); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace + exportFig('figs/flex_joints_trans_study_dvf_root_locus.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:flex_joints_trans_study_dvf_root_locus +#+caption: Root Locus for all the considered axial Stiffnesses +#+RESULTS: +[[file:figs/flex_joints_trans_study_dvf_root_locus.png]] + #+begin_src matlab + xlim([-1e3, 0]); + ylim([0, 1e3]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace + exportFig('figs/flex_joints_trans_study_root_locus_unzoom.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:flex_joints_trans_study_root_locus_unzoom +#+caption: Root Locus (unzoom) for all the considered axial Stiffnesses +#+RESULTS: +[[file:figs/flex_joints_trans_study_root_locus_unzoom.png]] + +*** Primary Control +The dynamics from $\bm{\tau}^\prime$ to $\bm{\epsilon}_{\mathcal{X}_n}$ (for the primary controller designed in the frame of the legs) is shown in Figure [[fig:flex_joints_trans_study_primary_plant]]. + +#+begin_src matlab :exports none Kdvf = 5e3*s/(1+s/2/pi/1e3)*eye(6); #+end_src @@ -826,29 +899,22 @@ exportFig('figs/flex_joint_trans_dvf.pdf', 'width', 'full', 'height', 'full') io(io_i) = linio([mdl, '/Tracking Error'], 1, 'output', [], 'En'); io_i = io_i + 1; % Position Errror load('mat/stages.mat', 'nano_hexapod'); -#+end_src -#+begin_src matlab :exports none - Gx_3dof_s = {zeros(length(Kzs), 1)}; Gl_3dof_s = {zeros(length(Kzs), 1)}; for i = 1:length(Kzs) - initializeNanoHexapod('k', 1e5, 'c', 2e2, ... + initializeNanoHexapod('k', k_opt, 'c', c_opt, ... 'type_F', 'universal_3dof', ... 'type_M', 'spherical_3dof', ... 'Kz_F', Kzs(i), ... 'Kz_M', Kzs(i), ... - 'Cz_F', 0.2*sqrt(Kzs(i)*10), ... - 'Cz_M', 0.2*sqrt(Kzs(i)*10)); + 'Cz_F', 0.05*sqrt(Kzs(i)*10), ... + 'Cz_M', 0.05*sqrt(Kzs(i)*10)); G = linearize(mdl, io); G.InputName = {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'}; G.OutputName = {'Ex', 'Ey', 'Ez', 'Erx', 'Ery', 'Erz'}; - Gx_3dof = -G*inv(nano_hexapod.kinematics.J'); - Gx_3dof.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'}; - Gx_3dof_s(i) = {Gx_3dof}; - Gl_3dof = -nano_hexapod.kinematics.J*G; Gl_3dof.OutputName = {'E1', 'E2', 'E3', 'E4', 'E5', 'E6'}; Gl_3dof_s(i) = {Gl_3dof}; @@ -885,7 +951,22 @@ exportFig('figs/flex_joint_trans_dvf.pdf', 'width', 'full', 'height', 'full') linkaxes([ax1,ax2],'x'); #+end_src +#+begin_src matlab :tangle no :exports results :results file replace + exportFig('figs/flex_joints_trans_study_primary_plant.pdf', 'width', 'full', 'height', 'full'); +#+end_src + +#+name: fig:flex_joints_trans_study_primary_plant +#+caption: Diagonal elements of the transfer function matrix from $\bm{\tau}^\prime$ to $\bm{\epsilon}_{\mathcal{X}_n}$ for the considered axial stiffnesses +#+RESULTS: +[[file:figs/flex_joints_trans_study_primary_plant.png]] + ** Conclusion #+begin_important + The axial stiffness of the flexible joints should be maximized. + For the considered actuator stiffness $k = 10^5\,[N/m]$, the axial stiffness of the rotational joints should ideally be above $10^7\,[N/m]$. + + This is a reasonable stiffness value for such joints. + + We may interpolate the results and say that the axial joint stiffness should be 100 times higher than the actuator stiffness, but this should be confirmed with further analysis. #+end_important