Continue flexible joint study
This commit is contained in:
@@ -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
|
||||
<<sec:trans_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
|
||||
|
||||
Reference in New Issue
Block a user