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Update the dynamic study

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Thomas Dehaeze
2020-02-13 15:37:06 +01:00
parent fee3dd4ca3
commit 2f4af4914e
7 changed files with 96 additions and 49 deletions
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@@ -40,6 +40,8 @@
* Compare external forces and forces applied by the actuators
** Introduction :ignore:
In this section, we wish to compare the effect of forces/torques applied by the actuators with the effect of external forces/torques on the displacement of the mobile platform.
** Matlab Init :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
<<matlab-dir>>
@@ -58,6 +60,7 @@
#+end_src
** Comparison with fixed support
Let's generate a Stewart platform.
#+begin_src matlab
stewart = initializeStewartPlatform();
stewart = initializeFramesPositions(stewart, 'H', 90e-3, 'MO_B', 45e-3);
@@ -72,12 +75,14 @@
stewart = initializeInertialSensor(stewart, 'type', 'none');
#+end_src
We don't put any flexibility below the Stewart platform such that *its base is fixed to an inertial frame*.
We also don't put any payload on top of the Stewart platform.
#+begin_src matlab
ground = initializeGround('type', 'none');
payload = initializePayload('type', 'none');
#+end_src
Estimation of the transfer function from $\bm{\tau}$ to $\mathcal{\bm{X}}$:
The transfer function from actuator forces $\bm{\tau}$ to the relative displacement of the mobile platform $\mathcal{\bm{X}}$ is extracted.
#+begin_src matlab
%% Options for Linearized
options = linearizeOptions;
@@ -97,12 +102,13 @@ Estimation of the transfer function from $\bm{\tau}$ to $\mathcal{\bm{X}}$:
G.OutputName = {'Edx', 'Edy', 'Edz', 'Erx', 'Ery', 'Erz'};
#+end_src
Using the Jacobian matrix, we compute the transfer function from force/torques applied by the actuators on the frame $\{B\}$ fixed to the mobile platform:
#+begin_src matlab
Gc = minreal(G*inv(stewart.kinematics.J'));
Gc.InputName = {'Fnx', 'Fny', 'Fnz', 'Mnx', 'Mny', 'Mnz'};
#+end_src
Estimation of the transfer function from $\bm{\mathcal{F}}_{\text{ext}}$ to $\mathcal{\bm{X}}$:
We also extract the transfer function from external forces $\bm{\mathcal{F}}_{\text{ext}}$ on the frame $\{B\}$ fixed to the mobile platform to the relative displacement $\mathcal{\bm{X}}$ of $\{B\}$ with respect to frame $\{A\}$:
#+begin_src matlab
%% Input/Output definition
clear io; io_i = 1;
@@ -115,6 +121,8 @@ Estimation of the transfer function from $\bm{\mathcal{F}}_{\text{ext}}$ to $\ma
Gd.OutputName = {'Edx', 'Edy', 'Edz', 'Erx', 'Ery', 'Erz'};
#+end_src
The comparison of the two transfer functions is shown in Figure [[fig:comparison_Fext_F_fixed_base]].
#+begin_src matlab :exports none
freqs = logspace(1, 4, 1000);
@@ -126,7 +134,7 @@ Estimation of the transfer function from $\bm{\mathcal{F}}_{\text{ext}}$ to $\ma
plot(freqs, abs(squeeze(freqresp(Gd(1,1), freqs, 'Hz'))), '--');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [N/N]'); set(gca, 'XTickLabel',[]);
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
ax2 = subplot(2, 1, 2);
hold on;
@@ -137,26 +145,28 @@ Estimation of the transfer function from $\bm{\mathcal{F}}_{\text{ext}}$ to $\ma
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-180, 180]);
yticks([-180, -90, 0, 90, 180]);
legend({'$\mathcal{X}_{x}/\mathcal{F}_{x}$', '$\mathcal{X}_{x}/\mathcal{F}_{x,ext}$'});
linkaxes([ax1,ax2],'x');
#+end_src
#+header: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/comparison_Fext_F_fixed_base.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src
#+name: fig:comparison_Fext_F_fixed_base
#+caption: Comparison of the transfer functions from $\bm{\mathcal{F}}$ to $\mathcal{\bm{X}}$ and from $\bm{\mathcal{F}}_{\text{ext}}$ to $\mathcal{\bm{X}}$ ([[./figs/comparison_Fext_F_fixed_base.png][png]], [[./figs/comparison_Fext_F_fixed_base.pdf][pdf]])
[[file:figs/comparison_Fext_F_fixed_base.png]]
** Comparison with a flexible support
We redo the identification for when the Stewart platform is on a flexible support.
We now add a flexible support under the Stewart platform.
#+begin_src matlab
ground = initializeGround('type', 'flexible');
#+end_src
Estimation of the transfer function from $\bm{\tau}$ to $\mathcal{\bm{X}}$:
And we perform again the identification.
#+begin_src matlab
%% Options for Linearized
options = linearizeOptions;
options.SampleTime = 0;
%% Name of the Simulink File
mdl = 'stewart_platform_model';
%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Controller'], 1, 'openinput'); io_i = io_i + 1; % Actuator Force Inputs [N]
@@ -166,15 +176,10 @@ Estimation of the transfer function from $\bm{\tau}$ to $\mathcal{\bm{X}}$:
G = linearize(mdl, io, options);
G.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
G.OutputName = {'Edx', 'Edy', 'Edz', 'Erx', 'Ery', 'Erz'};
#+end_src
#+begin_src matlab
Gc = minreal(G*inv(stewart.kinematics.J'));
Gc.InputName = {'Fnx', 'Fny', 'Fnz', 'Mnx', 'Mny', 'Mnz'};
#+end_src
Estimation of the transfer function from $\bm{\mathcal{F}}_{\text{ext}}$ to $\mathcal{\bm{X}}$:
#+begin_src matlab
%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Disturbances'], 1, 'openinput', [], 'F_ext'); io_i = io_i + 1; % External forces/torques applied on {B}
@@ -186,6 +191,8 @@ Estimation of the transfer function from $\bm{\mathcal{F}}_{\text{ext}}$ to $\ma
Gd.OutputName = {'Edx', 'Edy', 'Edz', 'Erx', 'Ery', 'Erz'};
#+end_src
The comparison between the obtained transfer functions is shown in Figure [[fig:comparison_Fext_F_flexible_base]].
#+begin_src matlab :exports none
freqs = logspace(1, 4, 1000);
@@ -197,7 +204,7 @@ Estimation of the transfer function from $\bm{\mathcal{F}}_{\text{ext}}$ to $\ma
plot(freqs, abs(squeeze(freqresp(Gd(1,1), freqs, 'Hz'))), '--');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [N/N]'); set(gca, 'XTickLabel',[]);
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
ax2 = subplot(2, 1, 2);
hold on;
@@ -208,10 +215,20 @@ Estimation of the transfer function from $\bm{\mathcal{F}}_{\text{ext}}$ to $\ma
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-180, 180]);
yticks([-180, -90, 0, 90, 180]);
legend({'$\mathcal{X}_{x}/\mathcal{F}_{x}$', '$\mathcal{X}_{x}/\mathcal{F}_{x,ext}$'});
linkaxes([ax1,ax2],'x');
#+end_src
#+header: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/comparison_Fext_F_flexible_base.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src
#+name: fig:comparison_Fext_F_flexible_base
#+caption: Comparison of the transfer functions from $\bm{\mathcal{F}}$ to $\mathcal{\bm{X}}$ and from $\bm{\mathcal{F}}_{\text{ext}}$ to $\mathcal{\bm{X}}$ ([[./figs/comparison_Fext_F_flexible_base.png][png]], [[./figs/comparison_Fext_F_flexible_base.pdf][pdf]])
[[file:figs/comparison_Fext_F_flexible_base.png]]
** Conclusion
#+begin_important
The transfer function from forces/torques applied by the actuators on the payload $\bm{\mathcal{F}} = \bm{J}^T \bm{\tau}$ to the pose of the mobile platform $\bm{\mathcal{X}}$ is the same as the transfer function from external forces/torques to $\bm{\mathcal{X}}$ as long as the Stewart platform's base is fixed.
@@ -219,6 +236,8 @@ The transfer function from forces/torques applied by the actuators on the payloa
* Comparison of the static transfer function and the Compliance matrix
** Introduction :ignore:
In this section, we see how the Compliance matrix of the Stewart platform is linked to the static relation between $\mathcal{\bm{F}}$ to $\mathcal{\bm{X}}$.
** Matlab Init :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
<<matlab-dir>>
@@ -252,6 +271,7 @@ Initialization of the Stewart platform.
stewart = initializeInertialSensor(stewart, 'type', 'none');
#+end_src
No flexibility below the Stewart platform and no payload.
#+begin_src matlab
ground = initializeGround('type', 'none');
payload = initializePayload('type', 'none');