Add HAC-LAC Tracking control simulations
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docs/figs/hybrid_control_Ex.pdf
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docs/figs/hybrid_control_Ex.png
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docs/figs/hybrid_control_Kl_loop_gain.pdf
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docs/figs/hybrid_control_Kl_loop_gain.png
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docs/figs/hybrid_control_Kl_plant_diagonal.pdf
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docs/figs/hybrid_control_Kl_plant_diagonal.png
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docs/figs/hybrid_control_Kl_plant_off_diagonal.pdf
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docs/figs/hybrid_control_Kl_plant_off_diagonal.png
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docs/figs/hybrid_control_Kx_loop_gain.pdf
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docs/figs/hybrid_control_Kx_loop_gain.png
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docs/figs/hybrid_control_Kx_plant.pdf
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docs/figs/hybrid_control_Kx_plant.png
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@ -772,6 +772,7 @@ The root locus is shown in figure [[fig:root_locus_dvf_rot_stiffness]].
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#+begin_important
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#+begin_important
|
||||||
Joint stiffness does increase the resonance frequencies of the system but does not change the attainable damping when using relative motion sensors.
|
Joint stiffness does increase the resonance frequencies of the system but does not change the attainable damping when using relative motion sensors.
|
||||||
#+end_important
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#+end_important
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||||||
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|
||||||
* Compliance and Transmissibility Comparison
|
* Compliance and Transmissibility Comparison
|
||||||
** Introduction :ignore:
|
** Introduction :ignore:
|
||||||
** Matlab Init :noexport:ignore:
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** Matlab Init :noexport:ignore:
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@ -1364,7 +1364,7 @@ The rotation point of the ground is located at the origin of frame $\{A\}$.
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:END:
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:END:
|
||||||
#+begin_src matlab
|
#+begin_src matlab
|
||||||
arguments
|
arguments
|
||||||
args.type char {mustBeMember(args.type, {'open-loop', 'iff', 'dvf', 'hac-iff', 'hac-dvf', 'ref-track-L', 'ref-track-X'})} = 'open-loop'
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args.type char {mustBeMember(args.type, {'open-loop', 'iff', 'dvf', 'hac-iff', 'hac-dvf', 'ref-track-L', 'ref-track-X', 'ref-track-hac-dvf'})} = 'open-loop'
|
||||||
end
|
end
|
||||||
#+end_src
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#+end_src
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||||||
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||||||
@ -1396,5 +1396,7 @@ The rotation point of the ground is located at the origin of frame $\{A\}$.
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controller.type = 5;
|
controller.type = 5;
|
||||||
case 'ref-track-X'
|
case 'ref-track-X'
|
||||||
controller.type = 6;
|
controller.type = 6;
|
||||||
|
case 'ref-track-hac-dvf'
|
||||||
|
controller.type = 7;
|
||||||
end
|
end
|
||||||
#+end_src
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#+end_src
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@ -121,7 +121,7 @@ Then, a diagonal (decentralized) controller $\bm{K}_\mathcal{L}$ is used such th
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#+end_src
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#+end_src
|
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|
|
||||||
** Identification of the plant
|
** Identification of the plant
|
||||||
Let's identify the transfer function from $\bm{\tau}$ to $\bm{L}$.
|
Let's identify the transfer function from $\bm{\tau}$ to $\bm{\mathcal{L}}$.
|
||||||
#+begin_src matlab
|
#+begin_src matlab
|
||||||
%% Name of the Simulink File
|
%% Name of the Simulink File
|
||||||
mdl = 'stewart_platform_model';
|
mdl = 'stewart_platform_model';
|
||||||
@ -231,7 +231,6 @@ We see that the plant is decoupled at low frequency which indicate that decentra
|
|||||||
** Controller Design
|
** Controller Design
|
||||||
The controller consists of:
|
The controller consists of:
|
||||||
- A pure integrator
|
- A pure integrator
|
||||||
- A lead around the crossover frequency to increase the phase margin
|
|
||||||
- A low pass filter with a cut-off frequency 3 times the crossover to increase the gain margin
|
- A low pass filter with a cut-off frequency 3 times the crossover to increase the gain margin
|
||||||
|
|
||||||
The obtained loop gains corresponding to the diagonal elements are shown in Figure [[fig:loop_gain_decentralized_L]].
|
The obtained loop gains corresponding to the diagonal elements are shown in Figure [[fig:loop_gain_decentralized_L]].
|
||||||
@ -428,6 +427,10 @@ However, as $\mathcal{X}$ is not directly measured, it is possible that importan
|
|||||||
simulinkproject('../');
|
simulinkproject('../');
|
||||||
#+end_src
|
#+end_src
|
||||||
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|
||||||
|
#+begin_src matlab
|
||||||
|
open('stewart_platform_model.slx');
|
||||||
|
#+end_src
|
||||||
|
|
||||||
** Control Schematic
|
** Control Schematic
|
||||||
The centralized controller takes the position error $\bm{\epsilon}_\mathcal{X}$ as an inputs and generate actuator forces $\bm{\tau}$ (see Figure [[fig:centralized_reference_tracking]]).
|
The centralized controller takes the position error $\bm{\epsilon}_\mathcal{X}$ as an inputs and generate actuator forces $\bm{\tau}$ (see Figure [[fig:centralized_reference_tracking]]).
|
||||||
The signals are:
|
The signals are:
|
||||||
@ -492,7 +495,7 @@ We can think of two ways to make the plant more diagonal that are described in s
|
|||||||
#+end_src
|
#+end_src
|
||||||
|
|
||||||
** Identification of the plant
|
** Identification of the plant
|
||||||
Let's identify the transfer function from $\bm{\tau}$ to $\bm{L}$.
|
Let's identify the transfer function from $\bm{\tau}$ to $\bm{\mathcal{X}}$.
|
||||||
#+begin_src matlab
|
#+begin_src matlab
|
||||||
%% Name of the Simulink File
|
%% Name of the Simulink File
|
||||||
mdl = 'stewart_platform_model';
|
mdl = 'stewart_platform_model';
|
||||||
@ -650,7 +653,6 @@ Thus $J \cdot G(\omega = 0) = J \cdot \frac{\delta\bm{\mathcal{X}}}{\delta\bm{\t
|
|||||||
*** Controller Design
|
*** Controller Design
|
||||||
The controller consists of:
|
The controller consists of:
|
||||||
- A pure integrator
|
- A pure integrator
|
||||||
- A lead around the crossover frequency to increase the phase margin
|
|
||||||
- A low pass filter with a cut-off frequency 3 times the crossover to increase the gain margin
|
- A low pass filter with a cut-off frequency 3 times the crossover to increase the gain margin
|
||||||
|
|
||||||
The obtained loop gains corresponding to the diagonal elements are shown in Figure [[fig:loop_gain_centralized_L]].
|
The obtained loop gains corresponding to the diagonal elements are shown in Figure [[fig:loop_gain_centralized_L]].
|
||||||
@ -877,7 +879,6 @@ This control architecture can also give a dynamically decoupled plant if the Cen
|
|||||||
*** Controller Design
|
*** Controller Design
|
||||||
The controller consists of:
|
The controller consists of:
|
||||||
- A pure integrator
|
- A pure integrator
|
||||||
- A lead around the crossover frequency to increase the phase margin
|
|
||||||
- A low pass filter with a cut-off frequency 3 times the crossover to increase the gain margin
|
- A low pass filter with a cut-off frequency 3 times the crossover to increase the gain margin
|
||||||
|
|
||||||
The obtained loop gains corresponding to the diagonal elements are shown in Figure [[fig:loop_gain_centralized_X]].
|
The obtained loop gains corresponding to the diagonal elements are shown in Figure [[fig:loop_gain_centralized_X]].
|
||||||
@ -1221,6 +1222,7 @@ Thus, this method should be quite robust against parameter variation (e.g. the p
|
|||||||
|
|
||||||
* Hybrid Control Architecture - HAC-LAC with relative DVF
|
* Hybrid Control Architecture - HAC-LAC with relative DVF
|
||||||
<<sec:hybrid>>
|
<<sec:hybrid>>
|
||||||
|
** Introduction :ignore:
|
||||||
** Matlab Init :noexport:
|
** Matlab Init :noexport:
|
||||||
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
|
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
|
||||||
<<matlab-dir>>
|
<<matlab-dir>>
|
||||||
@ -1234,7 +1236,26 @@ Thus, this method should be quite robust against parameter variation (e.g. the p
|
|||||||
simulinkproject('../');
|
simulinkproject('../');
|
||||||
#+end_src
|
#+end_src
|
||||||
|
|
||||||
|
#+begin_src matlab
|
||||||
|
open('stewart_platform_model.slx');
|
||||||
|
#+end_src
|
||||||
|
|
||||||
** Control Schematic
|
** Control Schematic
|
||||||
|
Let's consider the control schematic shown in Figure [[fig:hybrid_reference_tracking]].
|
||||||
|
|
||||||
|
The first loop containing $\bm{K}_\mathcal{L}$ is a Decentralized Direct (Relative) Velocity Feedback.
|
||||||
|
|
||||||
|
A reference $\bm{r}_\mathcal{L}$ is computed using the inverse kinematics and corresponds to the wanted motion of each leg.
|
||||||
|
The actual length of each leg $\bm{\mathcal{L}}$ is subtracted and then passed trough the controller $\bm{K}_\mathcal{L}$.
|
||||||
|
|
||||||
|
The controller is a diagonal controller with pure derivative action on the diagonal.
|
||||||
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|
||||||
|
The effect of this loop is:
|
||||||
|
- it adds damping to the system (the force applied in each actuator is proportional to the relative velocity of the strut)
|
||||||
|
- it however does not go "against" the reference path $\bm{r}_\mathcal{X}$ thanks to the use of the inverse kinematics
|
||||||
|
|
||||||
|
Then, the second loop containing $\bm{K}_\mathcal{X}$ is designed based on the already damped plant (represented by the gray area).
|
||||||
|
This second loop is responsible for the reference tracking.
|
||||||
|
|
||||||
#+begin_src latex :file hybrid_reference_tracking.pdf
|
#+begin_src latex :file hybrid_reference_tracking.pdf
|
||||||
\begin{tikzpicture}
|
\begin{tikzpicture}
|
||||||
@ -1260,8 +1281,8 @@ Thus, this method should be quite robust against parameter variation (e.g. the p
|
|||||||
% Connections and labels
|
% Connections and labels
|
||||||
\draw[<-] (subx.west)node[above left]{$\bm{r}_{\mathcal{X}}$} -- ++(-1.2, 0);
|
\draw[<-] (subx.west)node[above left]{$\bm{r}_{\mathcal{X}}$} -- ++(-1.2, 0);
|
||||||
\draw[->] ($(subx.west) + (-0.8, 0)$)node[branch]{} |- (invK.west);
|
\draw[->] ($(subx.west) + (-0.8, 0)$)node[branch]{} |- (invK.west);
|
||||||
\draw[->] (invK.east) -- (subl.west) node[above left]{$\bm{\epsilon}_\mathcal{L}$};
|
\draw[->] (invK.east) -- (subl.west) node[above left]{$\bm{r}_\mathcal{L}$};
|
||||||
\draw[->] (subl.south) -- (Kl.north);
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\draw[->] (subl.south) -- (Kl.north) node[above right]{$\bm{\epsilon}_\mathcal{L}$};
|
||||||
\draw[->] (Kl.south) -- (addF.north);
|
\draw[->] (Kl.south) -- (addF.north);
|
||||||
|
|
||||||
\draw[->] (subx.east) -- (Kx.west) node[above left]{$\bm{\epsilon}_\mathcal{X}$};
|
\draw[->] (subx.east) -- (Kx.west) node[above left]{$\bm{\epsilon}_\mathcal{X}$};
|
||||||
@ -1278,11 +1299,418 @@ Thus, this method should be quite robust against parameter variation (e.g. the p
|
|||||||
#+end_src
|
#+end_src
|
||||||
|
|
||||||
#+name: fig:hybrid_reference_tracking
|
#+name: fig:hybrid_reference_tracking
|
||||||
#+caption: Centralized Controller
|
#+caption: Hybrid Control Architecture
|
||||||
#+RESULTS:
|
#+RESULTS:
|
||||||
[[file:figs/hybrid_reference_tracking.png]]
|
[[file:figs/hybrid_reference_tracking.png]]
|
||||||
|
|
||||||
|
** Initialize the Stewart platform
|
||||||
|
#+begin_src matlab
|
||||||
|
stewart = initializeStewartPlatform();
|
||||||
|
stewart = initializeFramesPositions(stewart, 'H', 90e-3, 'MO_B', 45e-3);
|
||||||
|
stewart = generateGeneralConfiguration(stewart);
|
||||||
|
stewart = computeJointsPose(stewart);
|
||||||
|
stewart = initializeStrutDynamics(stewart);
|
||||||
|
stewart = initializeJointDynamics(stewart, 'type_F', 'universal_p', 'type_M', 'spherical_p');
|
||||||
|
stewart = initializeCylindricalPlatforms(stewart);
|
||||||
|
stewart = initializeCylindricalStruts(stewart);
|
||||||
|
stewart = computeJacobian(stewart);
|
||||||
|
stewart = initializeStewartPose(stewart);
|
||||||
|
stewart = initializeInertialSensor(stewart, 'type', 'accelerometer', 'freq', 5e3);
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+begin_src matlab
|
||||||
|
ground = initializeGround('type', 'rigid');
|
||||||
|
payload = initializePayload('type', 'none');
|
||||||
|
controller = initializeController('type', 'open-loop');
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+begin_src matlab
|
||||||
|
disturbances = initializeDisturbances();
|
||||||
|
references = initializeReferences(stewart);
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
** First Control Loop - $\bm{K}_\mathcal{L}$
|
||||||
|
*** Identification
|
||||||
|
Let's identify the transfer function from $\bm{\tau}$ to $\bm{L}$.
|
||||||
|
#+begin_src matlab
|
||||||
|
%% 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]
|
||||||
|
io(io_i) = linio([mdl, '/Stewart Platform'], 1, 'openoutput', [], 'dLm'); io_i = io_i + 1; % Relative Displacement Outputs [m]
|
||||||
|
|
||||||
|
%% Run the linearization
|
||||||
|
Gl = linearize(mdl, io);
|
||||||
|
Gl.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
|
||||||
|
Gl.OutputName = {'L1', 'L2', 'L3', 'L4', 'L5', 'L6'};
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
*** Obtained Plant
|
||||||
|
The diagonal elements of the plant are shown in Figure [[fig:hybrid_control_Kl_plant_diagonal]] while the off diagonal terms are shown in Figure [[fig:hybrid_control_Kl_plant_off_diagonal]].
|
||||||
|
|
||||||
|
#+begin_src matlab :exports none
|
||||||
|
freqs = logspace(1, 4, 1000);
|
||||||
|
|
||||||
|
figure;
|
||||||
|
|
||||||
|
ax1 = subplot(2, 1, 1);
|
||||||
|
hold on;
|
||||||
|
for i = 1:6
|
||||||
|
plot(freqs, abs(squeeze(freqresp(Gl(i, i), 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 = 1:6
|
||||||
|
plot(freqs, 180/pi*angle(squeeze(freqresp(Gl(i, i), freqs, 'Hz'))));
|
||||||
|
end
|
||||||
|
hold off;
|
||||||
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||||
|
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
||||||
|
ylim([-180, 180]);
|
||||||
|
yticks([-180, -90, 0, 90, 180]);
|
||||||
|
|
||||||
|
linkaxes([ax1,ax2],'x');
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+header: :tangle no :exports results :results none :noweb yes
|
||||||
|
#+begin_src matlab :var filepath="figs/hybrid_control_Kl_plant_diagonal.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
||||||
|
<<plt-matlab>>
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+name: fig:hybrid_control_Kl_plant_diagonal
|
||||||
|
#+caption: Diagonal elements of the plant for the design of $\bm{K}_\mathcal{L}$ ([[./figs/hybrid_control_Kl_plant_diagonal.png][png]], [[./figs/hybrid_control_Kl_plant_diagonal.pdf][pdf]])
|
||||||
|
[[file:figs/hybrid_control_Kl_plant_diagonal.png]]
|
||||||
|
|
||||||
|
#+begin_src matlab :exports none
|
||||||
|
freqs = logspace(1, 4, 1000);
|
||||||
|
|
||||||
|
figure;
|
||||||
|
|
||||||
|
ax1 = subplot(2, 1, 1);
|
||||||
|
hold on;
|
||||||
|
for i = 1:5
|
||||||
|
for j = i+1:6
|
||||||
|
plot(freqs, abs(squeeze(freqresp(Gl(i, j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2]);
|
||||||
|
end
|
||||||
|
end
|
||||||
|
set(gca,'ColorOrderIndex',1);
|
||||||
|
plot(freqs, abs(squeeze(freqresp(Gl(1, 1), freqs, 'Hz'))));
|
||||||
|
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 = 1:5
|
||||||
|
for j = i+1:6
|
||||||
|
plot(freqs, 180/pi*angle(squeeze(freqresp(Gl(i, j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2]);
|
||||||
|
end
|
||||||
|
end
|
||||||
|
set(gca,'ColorOrderIndex',1);
|
||||||
|
plot(freqs, 180/pi*angle(squeeze(freqresp(Gl(1, 1), freqs, 'Hz'))));
|
||||||
|
hold off;
|
||||||
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||||
|
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
||||||
|
ylim([-180, 180]);
|
||||||
|
yticks([-180, -90, 0, 90, 180]);
|
||||||
|
|
||||||
|
linkaxes([ax1,ax2],'x');
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+header: :tangle no :exports results :results none :noweb yes
|
||||||
|
#+begin_src matlab :var filepath="figs/hybrid_control_Kl_plant_off_diagonal.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
||||||
|
<<plt-matlab>>
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+name: fig:hybrid_control_Kl_plant_off_diagonal
|
||||||
|
#+caption: Off-diagonal elements of the plant for the design of $\bm{K}_\mathcal{L}$ ([[./figs/hybrid_control_Kl_plant_off_diagonal.png][png]], [[./figs/hybrid_control_Kl_plant_off_diagonal.pdf][pdf]])
|
||||||
|
[[file:figs/hybrid_control_Kl_plant_off_diagonal.png]]
|
||||||
|
|
||||||
|
*** Controller Design
|
||||||
|
We apply a decentralized (diagonal) direct velocity feedback.
|
||||||
|
Thus, we apply a pure derivative action.
|
||||||
|
In order to make the controller realizable, we add a low pass filter at high frequency.
|
||||||
|
The gain of the controller is chosen such that the resonances are critically damped.
|
||||||
|
|
||||||
|
The obtain loop gain is shown in Figure [[fig:hybrid_control_Kl_loop_gain]].
|
||||||
|
|
||||||
|
#+begin_src matlab
|
||||||
|
Kl = 1e4 * s / (1 + s/2/pi/1e4) * eye(6);
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+begin_src matlab :exports none
|
||||||
|
freqs = logspace(1, 4, 1000);
|
||||||
|
|
||||||
|
figure;
|
||||||
|
|
||||||
|
ax1 = subplot(2, 1, 1);
|
||||||
|
hold on;
|
||||||
|
plot(freqs, abs(squeeze(freqresp(Gl(1, 1)*Kl(1,1), freqs, 'Hz'))));
|
||||||
|
hold off;
|
||||||
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||||
|
ylabel('Loop Gain'); set(gca, 'XTickLabel',[]);
|
||||||
|
|
||||||
|
ax2 = subplot(2, 1, 2);
|
||||||
|
hold on;
|
||||||
|
plot(freqs, 180/pi*angle(squeeze(freqresp(Gl(1, 1)*Kl(1,1), freqs, 'Hz'))));
|
||||||
|
hold off;
|
||||||
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||||
|
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
||||||
|
ylim([-180, 180]);
|
||||||
|
yticks([-180, -90, 0, 90, 180]);
|
||||||
|
|
||||||
|
linkaxes([ax1,ax2],'x');
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+header: :tangle no :exports results :results none :noweb yes
|
||||||
|
#+begin_src matlab :var filepath="figs/hybrid_control_Kl_loop_gain.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
||||||
|
<<plt-matlab>>
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+name: fig:hybrid_control_Kl_loop_gain
|
||||||
|
#+caption: Obtain Loop Gain for the DVF control loop ([[./figs/hybrid_control_Kl_loop_gain.png][png]], [[./figs/hybrid_control_Kl_loop_gain.pdf][pdf]])
|
||||||
|
[[file:figs/hybrid_control_Kl_loop_gain.png]]
|
||||||
|
|
||||||
|
** Second Control Loop - $\bm{K}_\mathcal{X}$
|
||||||
|
*** Identification
|
||||||
|
#+begin_src matlab
|
||||||
|
Kx = tf(zeros(6));
|
||||||
|
|
||||||
|
controller = initializeController('type', 'ref-track-hac-dvf');
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+begin_src matlab
|
||||||
|
%% Name of the Simulink File
|
||||||
|
mdl = 'stewart_platform_model';
|
||||||
|
|
||||||
|
%% Input/Output definition
|
||||||
|
clear io; io_i = 1;
|
||||||
|
io(io_i) = linio([mdl, '/Controller'], 1, 'input'); io_i = io_i + 1; % Actuator Force Inputs [N]
|
||||||
|
io(io_i) = linio([mdl, '/Relative Motion Sensor'], 1, 'openoutput'); io_i = io_i + 1; % Relative Displacement Outputs [m]
|
||||||
|
|
||||||
|
%% Run the linearization
|
||||||
|
G = linearize(mdl, io);
|
||||||
|
G.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
|
||||||
|
G.OutputName = {'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'};
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
*** Obtained Plant
|
||||||
|
We use the Jacobian matrix to apply forces in the cartesian frame.
|
||||||
|
#+begin_src matlab
|
||||||
|
Gx = G*inv(stewart.kinematics.J');
|
||||||
|
Gx.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
The obtained plant is shown in Figure [[fig:hybrid_control_Kx_plant]].
|
||||||
|
#+begin_src matlab :exports none
|
||||||
|
freqs = logspace(1, 4, 1000);
|
||||||
|
|
||||||
|
labels = {'$D_x/\mathcal{F}_x$', '$D_y/\mathcal{F}_y$', '$D_z/\mathcal{F}_z$', '$R_x/\mathcal{M}_x$', '$R_y/\mathcal{M}_y$', '$R_z/\mathcal{M}_z$'};
|
||||||
|
|
||||||
|
figure;
|
||||||
|
|
||||||
|
ax1 = subplot(2, 2, 1);
|
||||||
|
hold on;
|
||||||
|
for i = 1:6
|
||||||
|
plot(freqs, abs(squeeze(freqresp(Gx(i, i), freqs, 'Hz'))));
|
||||||
|
end
|
||||||
|
hold off;
|
||||||
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||||
|
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
|
||||||
|
title('Diagonal elements of the Plant');
|
||||||
|
|
||||||
|
ax2 = subplot(2, 2, 3);
|
||||||
|
hold on;
|
||||||
|
for i = 1:6
|
||||||
|
plot(freqs, 180/pi*angle(squeeze(freqresp(Gx(i, i), freqs, 'Hz'))), 'DisplayName', labels{i});
|
||||||
|
end
|
||||||
|
hold off;
|
||||||
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||||
|
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
||||||
|
ylim([-180, 180]);
|
||||||
|
yticks([-180, -90, 0, 90, 180]);
|
||||||
|
legend();
|
||||||
|
|
||||||
|
ax3 = subplot(2, 2, 2);
|
||||||
|
hold on;
|
||||||
|
for i = 1:5
|
||||||
|
for j = i+1:6
|
||||||
|
plot(freqs, abs(squeeze(freqresp(Gx(i, j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2]);
|
||||||
|
end
|
||||||
|
end
|
||||||
|
set(gca,'ColorOrderIndex',1);
|
||||||
|
plot(freqs, abs(squeeze(freqresp(Gx(1, 1), freqs, 'Hz'))));
|
||||||
|
hold off;
|
||||||
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||||
|
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
|
||||||
|
title('Off-Diagonal elements of the Plant');
|
||||||
|
|
||||||
|
ax4 = subplot(2, 2, 4);
|
||||||
|
hold on;
|
||||||
|
for i = 1:5
|
||||||
|
for j = i+1:6
|
||||||
|
plot(freqs, 180/pi*angle(squeeze(freqresp(Gx(i, j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2]);
|
||||||
|
end
|
||||||
|
end
|
||||||
|
set(gca,'ColorOrderIndex',1);
|
||||||
|
plot(freqs, 180/pi*angle(squeeze(freqresp(Gx(1, 1), freqs, 'Hz'))));
|
||||||
|
hold off;
|
||||||
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||||
|
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
||||||
|
ylim([-180, 180]);
|
||||||
|
yticks([-180, -90, 0, 90, 180]);
|
||||||
|
|
||||||
|
linkaxes([ax1,ax2,ax3,ax4],'x');
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+header: :tangle no :exports results :results none :noweb yes
|
||||||
|
#+begin_src matlab :var filepath="figs/hybrid_control_Kx_plant.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
||||||
|
<<plt-matlab>>
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+name: fig:hybrid_control_Kx_plant
|
||||||
|
#+caption: Diagonal and Off-diagonal elements of the plant for the design of $\bm{K}_\mathcal{L}$ ([[./figs/hybrid_control_Kx_plant.png][png]], [[./figs/hybrid_control_Kx_plant.pdf][pdf]])
|
||||||
|
[[file:figs/hybrid_control_Kx_plant.png]]
|
||||||
|
|
||||||
|
*** Controller Design
|
||||||
|
The controller consists of:
|
||||||
|
- A pure integrator
|
||||||
|
- A low pass filter with a cut-off frequency 3 times the crossover to increase the gain margin
|
||||||
|
|
||||||
|
#+begin_src matlab
|
||||||
|
wc = 2*pi*200; % Bandwidth Bandwidth [rad/s]
|
||||||
|
|
||||||
|
h = 3; % Lead parameter
|
||||||
|
|
||||||
|
Kx = (1/h) * (1 + s/wc*h)/(1 + s/wc/h) * wc/s * ((s/wc/2 + 1)/(s/wc/2));
|
||||||
|
|
||||||
|
% Normalization of the gain of have a loop gain of 1 at frequency wc
|
||||||
|
Kx = Kx.*diag(1./diag(abs(freqresp(Gx*Kx, wc))));
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+begin_src matlab :exports none
|
||||||
|
freqs = logspace(1, 3, 1000);
|
||||||
|
|
||||||
|
figure;
|
||||||
|
|
||||||
|
ax1 = subplot(2, 1, 1);
|
||||||
|
hold on;
|
||||||
|
for i = 1:6
|
||||||
|
plot(freqs, abs(squeeze(freqresp(Kx(i,i)*Gx(i, i), 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 = 1:6
|
||||||
|
plot(freqs, 180/pi*angle(squeeze(freqresp(Kx(i,i)*Gx(i, i), freqs, 'Hz'))));
|
||||||
|
end
|
||||||
|
hold off;
|
||||||
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||||
|
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
||||||
|
ylim([-180, 180]);
|
||||||
|
yticks([-180, -90, 0, 90, 180]);
|
||||||
|
|
||||||
|
linkaxes([ax1,ax2],'x');
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+header: :tangle no :exports results :results none :noweb yes
|
||||||
|
#+begin_src matlab :var filepath="figs/hybrid_control_Kx_loop_gain.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
||||||
|
<<plt-matlab>>
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+name: fig:hybrid_control_Kx_loop_gain
|
||||||
|
#+caption: Obtained Loop Gain for the controller $\bm{K}_\mathcal{X}$ ([[./figs/hybrid_control_Kx_loop_gain.png][png]], [[./figs/hybrid_control_Kx_loop_gain.pdf][pdf]])
|
||||||
|
[[file:figs/hybrid_control_Kx_loop_gain.png]]
|
||||||
|
|
||||||
|
Then we include the Jacobian in the controller matrix.
|
||||||
|
#+begin_src matlab
|
||||||
|
Kx = inv(stewart.kinematics.J')*Kx;
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
** Simulations
|
||||||
|
We specify the reference path to follow.
|
||||||
|
#+begin_src matlab
|
||||||
|
t = linspace(0, 10, 10000);
|
||||||
|
|
||||||
|
r = zeros(6, length(t));
|
||||||
|
|
||||||
|
r(1, :) = 5e-3*sin(2*pi*t);
|
||||||
|
|
||||||
|
references = initializeReferences(stewart, 't', t, 'r', r);
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
We run the simulation and we save the results.
|
||||||
|
#+begin_src matlab
|
||||||
|
sim('stewart_platform_model')
|
||||||
|
simout_H = simout;
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
The obtained position error is shown in Figure [[fig:hybrid_control_Ex]].
|
||||||
|
|
||||||
|
#+begin_src matlab :exports none
|
||||||
|
figure;
|
||||||
|
subplot(2, 3, 1);
|
||||||
|
hold on;
|
||||||
|
plot(simout_H.r.r.Time, squeeze(simout_H.r.r.Data(1, 1, :))-simout_H.x.Xr.Data(:, 1))
|
||||||
|
hold off;
|
||||||
|
xlabel('Time [s]');
|
||||||
|
ylabel('Dx [m]');
|
||||||
|
|
||||||
|
subplot(2, 3, 2);
|
||||||
|
hold on;
|
||||||
|
plot(simout_H.r.r.Time, squeeze(simout_H.r.r.Data(2, 1, :))-simout_H.x.Xr.Data(:, 2))
|
||||||
|
hold off;
|
||||||
|
xlabel('Time [s]');
|
||||||
|
ylabel('Dy [m]');
|
||||||
|
|
||||||
|
subplot(2, 3, 3);
|
||||||
|
hold on;
|
||||||
|
plot(simout_H.r.r.Time, squeeze(simout_H.r.r.Data(3, 1, :))-simout_H.x.Xr.Data(:, 3))
|
||||||
|
hold off;
|
||||||
|
xlabel('Time [s]');
|
||||||
|
ylabel('Dz [m]');
|
||||||
|
|
||||||
|
subplot(2, 3, 4);
|
||||||
|
hold on;
|
||||||
|
plot(simout_H.r.r.Time, squeeze(simout_H.r.r.Data(4, 1, :))-simout_H.x.Xr.Data(:, 4))
|
||||||
|
hold off;
|
||||||
|
xlabel('Time [s]');
|
||||||
|
ylabel('Rx [rad]');
|
||||||
|
|
||||||
|
subplot(2, 3, 5);
|
||||||
|
hold on;
|
||||||
|
plot(simout_H.r.r.Time, squeeze(simout_H.r.r.Data(5, 1, :))-simout_H.x.Xr.Data(:, 5))
|
||||||
|
hold off;
|
||||||
|
xlabel('Time [s]');
|
||||||
|
ylabel('Ry [rad]');
|
||||||
|
|
||||||
|
subplot(2, 3, 6);
|
||||||
|
hold on;
|
||||||
|
plot(simout_H.r.r.Time, squeeze(simout_H.r.r.Data(6, 1, :))-simout_H.x.Xr.Data(:, 6))
|
||||||
|
hold off;
|
||||||
|
xlabel('Time [s]');
|
||||||
|
ylabel('Rz [rad]');
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+header: :tangle no :exports results :results none :noweb yes
|
||||||
|
#+begin_src matlab :var filepath="figs/hybrid_control_Ex.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
||||||
|
<<plt-matlab>>
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+name: fig:hybrid_control_Ex
|
||||||
|
#+caption: Obtained position error $\bm{\epsilon}_\mathcal{X}$ ([[./figs/hybrid_control_Ex.png][png]], [[./figs/hybrid_control_Ex.pdf][pdf]])
|
||||||
|
[[file:figs/hybrid_control_Ex.png]]
|
||||||
|
|
||||||
|
** Conclusion
|
||||||
|
|
||||||
* Position Error computation
|
* Position Error computation
|
||||||
<<sec:compute_pose_error>>
|
<<sec:compute_pose_error>>
|
||||||
@ -1371,3 +1799,112 @@ We have:
|
|||||||
|
|
||||||
[Edx, Edy, Edz, Erx, Ery, Erz]
|
[Edx, Edy, Edz, Erx, Ery, Erz]
|
||||||
#+end_src
|
#+end_src
|
||||||
|
|
||||||
|
* Tests
|
||||||
|
#+begin_src matlab
|
||||||
|
Dxr = 0.1;
|
||||||
|
Dyr = 0.1;
|
||||||
|
Dzr = 0;
|
||||||
|
Rxr = 0;
|
||||||
|
Ryr = 0;
|
||||||
|
Rzr = 0.1;
|
||||||
|
|
||||||
|
Dxm = 0.1;
|
||||||
|
Dym = 0.1;
|
||||||
|
Dzm = 0.1;
|
||||||
|
Rxm = 0.1;
|
||||||
|
Rym = 0.1;
|
||||||
|
Rzm = 0.1;
|
||||||
|
|
||||||
|
[Edx, Edy, Edz, Erx, Ery, Erz] = computePosErrorTest(Dxr, Dyr, Dzr, Rxr, Ryr, Rzr, Dxm, Dym, Dzm, Rxm, Rym, Rzm)
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
* Function Name
|
||||||
|
:PROPERTIES:
|
||||||
|
:header-args:matlab+: :tangle ../src/computePosErrorTest.m
|
||||||
|
:header-args:matlab+: :comments none :mkdirp yes :eval no
|
||||||
|
:END:
|
||||||
|
|
||||||
|
#+begin_src matlab
|
||||||
|
function [Edx, Edy, Edz, Erx, Ery, Erz] = computePosErrorTest(Dxr, Dyr, Dzr, Rxr, Ryr, Rzr, Dxm, Dym, Dzm, Rxm, Rym, Rzm)
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
Let's note:
|
||||||
|
- $\{W\}$ the fixed measurement frame (corresponding to the metrology frame / the frame where the wanted displacement are expressed).
|
||||||
|
The center of the frame if $O_W$
|
||||||
|
- $\{M\}$ is the frame fixed to the measured elements.
|
||||||
|
$O_M$ is the point where the pose of the element is measured
|
||||||
|
- $\{R\}$ is a virtual frame corresponding to the wanted pose of the element.
|
||||||
|
$O_R$ is the origin of this frame where the we want to position the point $O_M$ of the element.
|
||||||
|
- $\{V\}$ is a frame which its axes are aligned with $\{W\}$ and its origin $O_V$ is coincident with the $O_M$
|
||||||
|
|
||||||
|
We measure the position and orientation (pose) of the element represented by the frame $\{M\}$ with respect to frame $\{W\}$.
|
||||||
|
Thus we can compute the Homogeneous transformation matrix ${}^WT_M$.
|
||||||
|
#+begin_src matlab
|
||||||
|
%% Measured Pose
|
||||||
|
WTm = zeros(4,4);
|
||||||
|
|
||||||
|
WTm(1:3, 1:3) = [cos(Rzm) -sin(Rzm) 0;
|
||||||
|
sin(Rzm) cos(Rzm) 0;
|
||||||
|
0 0 1] * ...
|
||||||
|
[cos(Rym) 0 sin(Rym);
|
||||||
|
0 1 0;
|
||||||
|
-sin(Rym) 0 cos(Rym)] * ...
|
||||||
|
[1 0 0;
|
||||||
|
0 cos(Rxm) -sin(Rxm);
|
||||||
|
0 sin(Rxm) cos(Rxm)];
|
||||||
|
WTm(1:4, 4) = [Dxm ; Dym ; Dzm; 1];
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
We can also compute the Homogeneous transformation matrix ${}^WT_R$ corresponding to the transformation required to go from fixed frame $\{W\}$ to the wanted frame $\{R\}$.
|
||||||
|
#+begin_src matlab
|
||||||
|
%% Reference Pose
|
||||||
|
WTr = zeros(4,4);
|
||||||
|
|
||||||
|
WTr(1:3, 1:3) = [cos(Rzr) -sin(Rzr) 0;
|
||||||
|
sin(Rzr) cos(Rzr) 0;
|
||||||
|
0 0 1] * ...
|
||||||
|
[cos(Ryr) 0 sin(Ryr);
|
||||||
|
0 1 0;
|
||||||
|
-sin(Ryr) 0 cos(Ryr)] * ...
|
||||||
|
[1 0 0;
|
||||||
|
0 cos(Rxr) -sin(Rxr);
|
||||||
|
0 sin(Rxr) cos(Rxr)];
|
||||||
|
WTr(1:4, 4) = [Dxr ; Dyr ; Dzr; 1];
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
We can also compute ${}^WT_V$.
|
||||||
|
#+begin_src matlab
|
||||||
|
WTv = eye(4);
|
||||||
|
WTv(1:3, 4) = WTm(1:3, 4);
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
Now we want to express ${}^MT_R$ which corresponds to the transformation required to go to wanted position expressed in the frame of the measured element.
|
||||||
|
This homogeneous transformation can be computed from the previously computed matrices:
|
||||||
|
\[ {}^MT_R = ({{}^WT_M}^{-1}) {}^WT_R \]
|
||||||
|
|
||||||
|
#+begin_src matlab
|
||||||
|
%% Wanted pose expressed in a frame corresponding to the actual measured pose
|
||||||
|
MTr = [WTm(1:3,1:3)', -WTm(1:3,1:3)'*WTm(1:3,4) ; 0 0 0 1]*WTr;
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
Now we want to express ${}^VT_R$:
|
||||||
|
\[ {}^VT_R = ({{}^WT_V}^{-1}) {}^WT_R \]
|
||||||
|
#+begin_src matlab
|
||||||
|
%% Wanted pose expressed in a frame coincident with the actual position but with no rotation
|
||||||
|
VTr = [WTv(1:3,1:3)', -WTv(1:3,1:3)'*WTv(1:3,4) ; 0 0 0 1] * WTr;
|
||||||
|
#+end_src
|
||||||
|
|
||||||
|
#+begin_src matlab
|
||||||
|
%% Extract Translations and Rotations from the Homogeneous Matrix
|
||||||
|
T = MTr;
|
||||||
|
Edx = T(1, 4);
|
||||||
|
Edy = T(2, 4);
|
||||||
|
Edz = T(3, 4);
|
||||||
|
|
||||||
|
% The angles obtained are u-v-w Euler angles (rotations in the moving frame)
|
||||||
|
Ery = atan2( T(1, 3), sqrt(T(1, 1)^2 + T(1, 2)^2));
|
||||||
|
Erx = atan2(-T(2, 3)/cos(Ery), T(3, 3)/cos(Ery));
|
||||||
|
Erz = atan2(-T(1, 2)/cos(Ery), T(1, 1)/cos(Ery));
|
||||||
|
end
|
||||||
|
#+end_src
|
||||||
|
50
src/computePosErrorTest.m
Normal file
@ -0,0 +1,50 @@
|
|||||||
|
function [Edx, Edy, Edz, Erx, Ery, Erz] = computePosErrorTest(Dxr, Dyr, Dzr, Rxr, Ryr, Rzr, Dxm, Dym, Dzm, Rxm, Rym, Rzm)
|
||||||
|
|
||||||
|
%% Measured Pose
|
||||||
|
WTm = zeros(4,4);
|
||||||
|
|
||||||
|
WTm(1:3, 1:3) = [cos(Rzm) -sin(Rzm) 0;
|
||||||
|
sin(Rzm) cos(Rzm) 0;
|
||||||
|
0 0 1] * ...
|
||||||
|
[cos(Rym) 0 sin(Rym);
|
||||||
|
0 1 0;
|
||||||
|
-sin(Rym) 0 cos(Rym)] * ...
|
||||||
|
[1 0 0;
|
||||||
|
0 cos(Rxm) -sin(Rxm);
|
||||||
|
0 sin(Rxm) cos(Rxm)];
|
||||||
|
WTm(1:4, 4) = [Dxm ; Dym ; Dzm; 1];
|
||||||
|
|
||||||
|
%% Reference Pose
|
||||||
|
WTr = zeros(4,4);
|
||||||
|
|
||||||
|
WTr(1:3, 1:3) = [cos(Rzr) -sin(Rzr) 0;
|
||||||
|
sin(Rzr) cos(Rzr) 0;
|
||||||
|
0 0 1] * ...
|
||||||
|
[cos(Ryr) 0 sin(Ryr);
|
||||||
|
0 1 0;
|
||||||
|
-sin(Ryr) 0 cos(Ryr)] * ...
|
||||||
|
[1 0 0;
|
||||||
|
0 cos(Rxr) -sin(Rxr);
|
||||||
|
0 sin(Rxr) cos(Rxr)];
|
||||||
|
WTr(1:4, 4) = [Dxr ; Dyr ; Dzr; 1];
|
||||||
|
|
||||||
|
WTv = eye(4);
|
||||||
|
WTv(1:3, 4) = WTm(1:3, 4);
|
||||||
|
|
||||||
|
%% Wanted pose expressed in a frame corresponding to the actual measured pose
|
||||||
|
MTr = [WTm(1:3,1:3)', -WTm(1:3,1:3)'*WTm(1:3,4) ; 0 0 0 1]*WTr;
|
||||||
|
|
||||||
|
%% Wanted pose expressed in a frame coincident with the actual position but with no rotation
|
||||||
|
VTr = [WTv(1:3,1:3)', -WTv(1:3,1:3)'*WTv(1:3,4) ; 0 0 0 1] * WTr;
|
||||||
|
|
||||||
|
%% Extract Translations and Rotations from the Homogeneous Matrix
|
||||||
|
T = MTr;
|
||||||
|
Edx = T(1, 4);
|
||||||
|
Edy = T(2, 4);
|
||||||
|
Edz = T(3, 4);
|
||||||
|
|
||||||
|
% The angles obtained are u-v-w Euler angles (rotations in the moving frame)
|
||||||
|
Ery = atan2( T(1, 3), sqrt(T(1, 1)^2 + T(1, 2)^2));
|
||||||
|
Erx = atan2(-T(2, 3)/cos(Ery), T(3, 3)/cos(Ery));
|
||||||
|
Erz = atan2(-T(1, 2)/cos(Ery), T(1, 1)/cos(Ery));
|
||||||
|
end
|
@ -7,7 +7,7 @@ function [controller] = initializeController(args)
|
|||||||
% - args - Can have the following fields:
|
% - args - Can have the following fields:
|
||||||
|
|
||||||
arguments
|
arguments
|
||||||
args.type char {mustBeMember(args.type, {'open-loop', 'iff', 'dvf', 'hac-iff', 'hac-dvf', 'ref-track-L', 'ref-track-X'})} = 'open-loop'
|
args.type char {mustBeMember(args.type, {'open-loop', 'iff', 'dvf', 'hac-iff', 'hac-dvf', 'ref-track-L', 'ref-track-X', 'ref-track-hac-dvf'})} = 'open-loop'
|
||||||
end
|
end
|
||||||
|
|
||||||
controller = struct();
|
controller = struct();
|
||||||
@ -27,4 +27,6 @@ switch args.type
|
|||||||
controller.type = 5;
|
controller.type = 5;
|
||||||
case 'ref-track-X'
|
case 'ref-track-X'
|
||||||
controller.type = 6;
|
controller.type = 6;
|
||||||
|
case 'ref-track-hac-dvf'
|
||||||
|
controller.type = 7;
|
||||||
end
|
end
|
||||||
|