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tdehaeze committed 2019-01-21 23:33:23 +01:00
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height:100%;
margin-left:300px;
/* margin:auto; */
max-width:800px;
max-width:1200px;
min-height:100%;
padding:1.618em 3.236em;
}
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"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
<head>
<!-- 2019-01-18 ven. 17:19 -->
<!-- 2019-01-21 lun. 22:53 -->
<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
<meta name="viewport" content="width=device-width, initial-scale=1" />
<title>Rotating Frame</title>
@@ -249,7 +249,7 @@ for the JavaScript code in this tag.
</div>
<div id="postamble" class="status">
<p class="author">Author: Thomas Dehaeze</p>
<p class="date">Created: 2019-01-18 ven. 17:19</p>
<p class="date">Created: 2019-01-21 lun. 22:53</p>
<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
</div>
</body>
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#+TITLE: Rotating Frame
- [[file:rotating_frame.html][Rotating Frame]]
- [[file:rotating_frame.org][Rotating Frame]]
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#+LaTeX_HEADER: \newcommand{\authorLastName}{Dehaeze}
#+LaTeX_HEADER: \newcommand{\authorEmail}{dehaeze.thomas@gmail.com}
#+PROPERTY: header-args:matlab :session *MATLAB*
#+PROPERTY: header-args:matlab+ :tangle rotating_frame.m
#+PROPERTY: header-args:matlab+ :comments org
#+PROPERTY: header-args:matlab+ :exports both
#+PROPERTY: header-args:matlab+ :eval no-export
#+PROPERTY: header-args:matlab+ :output-dir Figures
* Goal of this note
The control objective is to stabilize the position of a rotating object with respect to a non-rotating frame.
* Introduction
The objective of this note it to highlight some control problems that arises when controlling the position of an object using actuators that are rotating with respect to a fixed reference frame.
The actuators are also rotating with the object.
In section [[sec:system]], a simple system composed of a spindle and a translation stage is defined and the equations of motion are written.
The rotation induces some coupling between the actuators and their displacement, and modifies the dynamics of the system.
This is studied using the equations, and some numerical computations are used to compare the use of voice coil and piezoelectric actuators.
We want to compare the two different approach:
Then, in section [[sec:control_strategies]], two different control approach are compared where:
- the measurement is made in the fixed frame
- the measurement is made in the rotating frame
In section [[sec:simscape]], the analytical study will be validated using a multi body model of the studied system.
Finally, in section [[sec:control]], the control strategies are implemented using Simulink and Simscape and compared.
* System
:PROPERTIES:
:HEADER-ARGS:matlab+: :tangle system_numerical_analysis.m
:END:
<<sec:system>>
** System description
The system consists of one 2 degree of freedom translation stage on top of a spindle (figure [[fig:rotating_frame_2dof]]).
The control inputs are the forces applied in the translation stage ($F_u$ and $F_v$). As the translation stage is rotating around the Z axis due to the spindle, the forces are applied along $u$ and $v$.
The control inputs are the forces applied by the actuators of the translation stage ($F_u$ and $F_v$).
As the translation stage is rotating around the Z axis due to the spindle, the forces are applied along $u$ and $v$.
The measurement is either the $x-y$ displacement of the object located on top of the translation stage or the $u-v$ displacement of the actuators.
The measurement is either the $x-y$ displacement of the object located on top of the translation stage or the $u-v$ displacement of the sample with respect to a fixed reference frame.
#+name: fig:rotating_frame_2dof
#+caption: Schematic of the mecanical system
@@ -57,7 +66,6 @@ Indices $u$ and $v$ corresponds to signals in the rotating reference frame ($\ve
** Equations
<<sec:equations>>
Based on the figure [[fig:rotating_frame_2dof]], we can write the equations of motion of the system.
Let's express the kinetic energy $T$ and the potential energy $V$ of the mass $m$:
@@ -131,97 +139,271 @@ Which is equivalent to:
We can then subtract and add the previous equations to obtain the following equations:
#+begin_important
\begin{align*}
m \ddot{d_u} + (k - m\dot{\theta}^2) d_u &= F_u + 2 m\dot{d_v}\dot{\theta} + m d_v\ddot{\theta} \\
m \ddot{d_v} + (k - m\dot{\theta}^2) d_v &= F_v - 2 m\dot{d_u}\dot{\theta} - m d_u\ddot{\theta} \\
\end{align*}
#+NAME: eq:du_coupled
\begin{equation}
m \ddot{d_u} + (k - m\dot{\theta}^2) d_u = F_u + 2 m\dot{d_v}\dot{\theta} + m d_v\ddot{\theta}
\end{equation}
#+NAME: eq:dv_coupled
\begin{equation}
m \ddot{d_v} + (k - m\dot{\theta}^2) d_v = F_v - 2 m\dot{d_u}\dot{\theta} - m d_u\ddot{\theta}
\end{equation}
#+end_important
** TODO Analysis
We obtain two differential equations that are coupled through:
- *Euler forces*: $m d_v \ddot{\theta}$
- *Coriolis forces*: $2 m \dot{d_v} \dot{\theta}$
Without the coupling terms, each equation is the equation of a one degree of freedom mass-spring system with mass $m$ and stiffness $k-d_u m\dot{\theta}^2$.
Thus, the term $-d_u m\dot{\theta}^2$ acts like a negative stiffness (due to *centrifugal forces*).
Without the coupling terms, each equation is the equation of a one degree of freedom mass-spring system with mass $m$ and stiffness $k- m\dot{\theta}^2$.
Thus, the term $- m\dot{\theta}^2$ acts like a negative stiffness (due to *centrifugal forces*).
*** Stiff actuators
Let's say we use stiff actuators such that $m \ddot{d_u} + (k - m\dot{\theta}^2) d_u \approx k d_u$.
The forces induced by the rotating reference frame are independent of the stiffness of the actuator.
The resulting effect of those forces should then be higher when using softer actuators.
Let's suppose that $F_u + 2 m\dot{d_v}\dot{\theta} + m d_v\ddot{\theta} \approx F_u$.
Then we obtain $d_u = \frac{F_u}{k}$ that we can re inject in the other equation to obtain:
\[ m \ddot{d_v} + (k - m\dot{\theta}^2) d_v &= F_v - 2 m\frac{\dot{F_u}}{k}\dot{\theta} - m \frac{F_u}{k}\ddot{\theta} \]
*** Negative Stiffness
If $\max{\dot{\theta}} \ll \sqrt{\frac{k}{m}}$, then the negative spring effect is negligible and $k - m\dot{\theta}^2 \approx k$.
* Analytical Computation of forces for the NASS
For the NASS, the Euler forces should be less of a problem as $\ddot{\theta}$ should be very small when conducting an experiment.
** Parameters
** Numerical Values for the NASS
#+begin_src matlab :exports none :results silent :noweb yes
<<matlab-init>>
#+end_src
Let's define the parameters for the NASS.
#+begin_src matlab :exports code :results silent
mlight = 35; % [kg]
mheavy = 85; % [kg]
#+begin_src matlab :exports none :results silent
mlight = 35; % Mass for light sample [kg]
mheavy = 85; % Mass for heavy sample [kg]
wlight = 2*pi; % [rad/s]
wheavy = 2*pi/60; % [rad/s]
wlight = 2*pi; % Max rot. speed for light sample [rad/s]
wheavy = 2*pi/60; % Max rot. speed for heavy sample [rad/s]
wdot = 1; % [rad/s2]
kvc = 1e3; % Voice Coil Stiffness [N/m]
kpz = 1e8; % Piezo Stiffness [N/m]
d = 0.1; % [m]
ddot = 0.2; % [m/s]
wdot = 1; % Maximum rotation acceleration [rad/s2]
d = 0.01; % Maximum excentricity from rotational axis [m]
ddot = 0.2; % Maximum Horizontal speed [m/s]
save('./mat/parameters.mat');
#+end_src
** Euler and Coriolis forces
#+begin_src matlab :results table :exports results
labels = {'Light sample mass [kg]', ...
'Heavy sample mass [kg]', ...
'Max rot. speed - light [rpm]', ...
'Max rot. speed - heavy [rpm]', ...
'Voice Coil Stiffness [N/m]', ...
'Piezo Stiffness [N/m]', ...
'Max rot. acceleration [rad/s2]', ...
'Max mass excentricity [m]', ...
'Max Horizontal speed [m/s]'};
data = [mlight, mheavy, 60*wlight/2/pi, 60*wheavy/2/pi, kvc, kpz, wdot, d, ddot];
data2orgtable(data', labels, {}, ' %.1e ')
#+end_src
#+RESULTS:
| Light sample mass [kg] | 3.5e+01 |
| Heavy sample mass [kg] | 8.5e+01 |
| Max rot. speed - light [rpm] | 6.0e+01 |
| Max rot. speed - heavy [rpm] | 1.0e+00 |
| Voice Coil Stiffness [N/m] | 1.0e+03 |
| Piezo Stiffness [N/m] | 1.0e+08 |
| Max rot. acceleration [rad/s2] | 1.0e+00 |
| Max mass excentricity [m] | 1.0e-02 |
| Max Horizontal speed [m/s] | 2.0e-01 |
** Euler and Coriolis forces - Numerical Result
First we will determine the value for Euler and Coriolis forces during regular experiment.
- *Euler forces*: $m d_v \ddot{\theta}$
- *Coriolis forces*: $2 m \dot{d_v} \dot{\theta}$
#+begin_src matlab :exports none :results silent
Felight = mlight*d*wdot;
Feheavy = mheavy*d*wdot;
Fclight = 2*mlight*d*wlight;
Fcheavy = 2*mheavy*d*wheavy;
Fclight = 2*mlight*ddot*wlight;
Fcheavy = 2*mheavy*ddot*wheavy;
#+end_src
We then compute the corresponding values of the Coriolis and Euler forces, and the obtained values are displayed in table [[tab:euler_coriolis]].
The obtained values are displayed in table [[tab:euler_coriolis]].
#+begin_src matlab :results value table :exports results :post addhdr(*this*)
ans = sprintf(' | Light | Heavy | \n Coriolis | %.1f N | %.1f N | \n Euler | %.1f N | %.1f N', Fclight, Fcheavy, Felight, Feheavy)
data = [Fclight, Fcheavy ;
Felight, Feheavy];
data2orgtable(data, {'Coriolis', 'Euler'}, {'Light', 'Heavy'}, ' %.1fN ')
#+end_src
#+NAME: tab:euler_coriolis
#+CAPTION: Euler and Coriolis forces for the NASS
#+RESULTS:
| | Light | Heavy |
|----------+--------+-------|
| Coriolis | 44.0 N | 1.8 N |
| Euler | 3.5 N | 8.5 N |
| | Light | Heavy |
|----------+-------+-------|
| Coriolis | 88.0N | 3.6N |
| Euler | 0.4N | 0.8N |
** Negative Spring Effect - Numerical Result
The negative stiffness due to the rotation is equal to $-m{\omega_0}^2$.
** Negative Spring Effect
#+begin_src matlab :exports none :results silent
Klight = mlight*d*wdot^2;
Kheavy = mheavy*d*wdot^2;
Klight = mlight*wlight^2;
Kheavy = mheavy*wheavy^2;
#+end_src
The values for the negative spring effect are displayed in table [[tab:negative_spring]].
This is definitely negligible when using piezoelectric actuators. It may not be the case when using voice coil actuators.
#+begin_src matlab :results value table :exports results :post addhdr(*this*)
ans = sprintf(' | Light | Heavy | \n Neg. Spring | %.1f N/m | %.1f N/m', Klight, Kheavy)
data = [Klight, Kheavy];
data2orgtable(data, {'Neg. Spring'}, {'Light', 'Heavy'}, ' %.1f[N/m] ')
#+end_src
#+NAME: tab:negative_spring
#+CAPTION: Negative Spring effect
#+RESULTS:
| | Light | Heavy |
|-------------+---------+---------|
| Neg. Spring | 3.5 N/m | 8.5 N/m |
| | Light | Heavy |
|-------------+-------------+----------|
| Neg. Spring | 1381.7[N/m] | 0.9[N/m] |
** Limitations due to coupling
To simplify, we consider a constant rotating speed $\dot{\theta} = {\omega_0}$ and thus $\ddot{\theta} = 0$.
From equations [[eq:du_coupled]] and [[eq:dv_coupled]], we obtain:
\begin{align*}
(m s^2 + (k - m{\omega_0}^2)) d_u &= F_u + 2 m {\omega_0} s d_v \\
(m s^2 + (k - m{\omega_0}^2)) d_v &= F_v - 2 m {\omega_0} s d_u \\
\end{align*}
From second equation:
\[ d_v = \frac{1}{m s^2 + (k - m{\omega_0}^2)} F_v - \frac{2 m {\omega_0} s}{m s^2 + (k - m{\omega_0}^2)} d_u \]
And we re-inject $d_v$ into the first equation:
\begin{equation*}
(m s^2 + (k - m{\omega_0}^2)) d_u = F_u + \frac{2 m {\omega_0} s}{m s^2 + (k - m{\omega_0}^2)} F_v - \frac{(2 m {\omega_0} s)^2}{m s^2 + (k - m{\omega_0}^2)} d_u
\end{equation*}
\begin{equation*}
\frac{(m s^2 + (k - m{\omega_0}^2))^2 + (2 m {\omega_0} s)^2}{m s^2 + (k - m{\omega_0}^2)} d_u = F_u + \frac{2 m {\omega_0} s}{m s^2 + (k - m{\omega_0}^2)} F_v
\end{equation*}
Finally we obtain $d_u$ function of $F_u$ and $F_v$.
\[ d_u = \frac{m s^2 + (k - m{\omega_0}^2)}{(m s^2 + (k - m{\omega_0}^2))^2 + (2 m {\omega_0} s)^2} F_u + \frac{2 m {\omega_0} s}{(m s^2 + (k - m{\omega_0}^2))^2 + (2 m {\omega_0} s)^2} F_v \]
Similarly we can obtain $d_v$ function of $F_u$ and $F_v$:
\[ d_v = \frac{m s^2 + (k - m{\omega_0}^2)}{(m s^2 + (k - m{\omega_0}^2))^2 + (2 m {\omega_0} s)^2} F_v - \frac{2 m {\omega_0} s}{(m s^2 + (k - m{\omega_0}^2))^2 + (2 m {\omega_0} s)^2} F_u \]
The two previous equations can be written in a matrix form:
#+begin_important
\begin{equation}
\begin{bmatrix} d_u \\ d_v \end{bmatrix} =
\frac{1}{(m s^2 + (k - m{\omega_0}^2))^2 + (2 m {\omega_0} s)^2}
\begin{bmatrix}
ms^2 + (k-m{\omega_0}^2) & 2 m \omega_0 s \\
-2 m \omega_0 s & ms^2 + (k-m{\omega_0}^2) \\
\end{bmatrix}
\begin{bmatrix} F_u \\ F_v \end{bmatrix}
\end{equation}
#+end_important
Then, coupling is negligible if $|-m \omega^2 + (k - m{\omega_0}^2)| \gg |2 m {\omega_0} \omega|$.
*** Numerical Analysis
We plot on the same graph $\frac{|-m \omega^2 + (k - m {\omega_0}^2)|}{|2 m \omega_0 \omega|}$ for the voice coil and the piezo:
- with the light sample (figure [[fig:coupling_light]]).
- with the heavy sample (figure [[fig:coupling_heavy]]).
#+HEADER: :exports none :results silent
#+begin_src matlab
f = logspace(-1, 2, 1000);
figure;
hold on;
plot(f, abs(-mlight*(2*pi*f).^2 + kvc - mlight * wlight^2)./abs(2*mlight*wlight*2*pi*f), 'DisplayName', 'Voice Coil')
plot(f, abs(-mlight*(2*pi*f).^2 + kpz - mlight * wlight^2)./abs(2*mlight*wlight*2*pi*f), 'DisplayName', 'Piezo')
plot(f, ones(1, length(f)), 'k--', 'HandleVisibility', 'off')
set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
xlabel('Frequency [Hz]');
legend('Location', 'northeast');
hold off;
#+end_src
#+HEADER: :tangle no :exports results :results file :noweb yes
#+HEADER: :var filepath="Figures/coupling_light.png" :var figsize="normal-normal"
#+begin_src matlab
<<plt-matlab>>
#+end_src
#+NAME: fig:coupling_light
#+CAPTION: Relative Coupling for light mass and high rotation speed
#+RESULTS:
[[file:Figures/coupling_light.png]]
#+HEADER: :exports none :results silent
#+begin_src matlab
f = logspace(-1, 2, 1000);
figure;
hold on;
plot(f, abs(-mheavy*(2*pi*f).^2 + kvc - mheavy * wheavy^2)./abs(2*mheavy*wheavy*2*pi*f), 'DisplayName', 'Voice Coil')
plot(f, abs(-mheavy*(2*pi*f).^2 + kpz - mheavy * wheavy^2)./abs(2*mheavy*wheavy*2*pi*f), 'DisplayName', 'Piezo')
plot(f, ones(1, length(f)), 'k--', 'HandleVisibility', 'off')
set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
xlabel('Frequency [Hz]');
legend('Location', 'northeast');
hold off;
#+end_src
#+HEADER: :tangle no :exports results :results file :noweb yes
#+HEADER: :var filepath="Figures/coupling_heavy.png" :var figsize="normal-normal"
#+begin_src matlab
<<plt-matlab>>
#+end_src
#+NAME: fig:coupling_heavy
#+CAPTION: Relative Coupling for heavy mass and low rotation speed
#+RESULTS:
[[file:Figures/coupling_heavy.png]]
#+begin_important
Coupling is higher for actuators with small stiffness.
#+end_important
** Limitations due to negative stiffness effect
If $\max{\dot{\theta}} \ll \sqrt{\frac{k}{m}}$, then the negative spring effect is negligible and $k - m\dot{\theta}^2 \approx k$.
Let's estimate what is the maximum rotation speed for which the negative stiffness effect is still negligible ($\omega_\text{max} = 0.1 \sqrt{\frac{k}{m}}$). Results are shown table [[tab:negative_stiffness]].
#+begin_src matlab :results table :exports results :post addhdr(*this*)
data = 0.1*60*(1/2/pi)*[sqrt(kvc/mlight), sqrt(kpz/mlight);
sqrt(kvc/mheavy), sqrt(kpz/mheavy)];
data2orgtable(data, {'Light', 'Heavy'}, {'Voice Coil', 'Piezo'}, ' %.0f[rpm] ')
#+end_src
#+NAME: tab:negative_stiffness
#+CAPTION: Maximum rotation speed at which negative stiffness is negligible ($0.1\sqrt{\frac{k}{m}}$)
#+RESULTS:
| | Voice Coil | Piezo |
|-------+------------+-----------|
| Light | 5[rpm] | 1614[rpm] |
| Heavy | 3[rpm] | 1036[rpm] |
The negative spring effect is proportional to the rotational speed $\omega$.
The system dynamics will be much more affected when using soft actuator.
#+begin_important
Negative stiffness effect has very important effect when using soft actuators.
#+end_important
The system can even goes unstable when $m \omega^2 > k$, that is when the centrifugal forces are higher than the forces due to stiffness.
From this analysis, we can determine the lowest practical stiffness that is possible to use: $k_\text{min} = 10 m \omega^2$ (table [[tab:min_k]])
#+begin_src matlab :results table :exports results :post addhdr(*this*)
data = 10*[mlight*2*pi, mheavy*2*pi/60]
data2orgtable(data, {'k min [N/m]'}, {'Light', 'Heavy'}, ' %.0f ')
#+end_src
#+NAME: tab:min_k
#+CAPTION: Minimum possible stiffness
#+RESULTS:
| | Light | Heavy |
|-------------+-------+-------|
| k min [N/m] | 2199 | 89 |
* Control Strategies
<<sec:control_strategies>>
@@ -257,50 +439,264 @@ The corresponding block diagram is shown figure [[fig:control_measure_rotating_2
The loop gain is $L = G K$.
* Effect of the rotating Speed
* Multi Body Model - Simscape
:PROPERTIES:
:HEADER-ARGS:matlab+: :tangle simscape_analysis.m
:END:
<<sec:simscape>>
#+begin_src matlab :exports none :results silent :noweb yes
<<matlab-init>>
load('./mat/parameters.mat');
#+end_src
#+begin_src matlab :exports none :results silent
open rotating_frame.slx
#+end_src
** Identification in the rotating referenced frame
We initialize the inputs and outputs of the system to identify.
#+begin_src matlab :exports code :results silent
%% Options for Linearized
options = linearizeOptions;
options.SampleTime = 0;
%% Name of the Simulink File
mdl = 'rotating_frame';
%% Input/Output definition
io(1) = linio([mdl, '/fu'], 1, 'input');
io(2) = linio([mdl, '/fv'], 1, 'input');
io(3) = linio([mdl, '/du'], 1, 'output');
io(4) = linio([mdl, '/dv'], 1, 'output');
#+end_src
*** Piezo and Voice coil
We start we identify the transfer functions at high speed with the light sample.
#+begin_src matlab :exports code :results silent
rot_speed = wlight;
angle_e = 0;
m = mlight;
k = kpz;
c = 1e3;
Gpz_light = linearize(mdl, io, 0.1);
k = kvc;
c = 1e3;
Gvc_light = linearize(mdl, io, 0.1);
Gpz_light.InputName = {'Fu', 'Fv'};
Gpz_light.OutputName = {'Du', 'Dv'};
Gvc_light.InputName = {'Fu', 'Fv'};
Gvc_light.OutputName = {'Du', 'Dv'};
#+end_src
#+begin_src matlab :exports none :results silent
figure;
bode(Gpz_light, Gvc_light);
#+end_src
And then with the heavy sample.
#+begin_src matlab :exports code :results silent
rot_speed = wheavy;
angle_e = 0;
m = mheavy;
k = kpz;
c = 1e3;
Gpz_heavy = linearize(mdl, io, 0.1);
k = kvc;
c = 1e3;
Gvc_heavy = linearize(mdl, io, 0.1);
Gpz_heavy.InputName = {'Fu', 'Fv'};
Gpz_heavy.OutputName = {'Du', 'Dv'};
Gvc_heavy.InputName = {'Fu', 'Fv'};
Gvc_heavy.OutputName = {'Du', 'Dv'};
#+end_src
#+begin_src matlab :exports none :results silent
figure;
bode(Gpz_heavy, Gvc_heavy);
#+end_src
Plot the ratio between the main transfer function and the coupling term:
#+begin_src matlab :results silent :exports none
freqs = logspace(-2, 3, 1000);
figure;
hold on;
plot(freqs, abs(squeeze(freqresp(Gvc_light('Du', 'Fu'), freqs, 'Hz'))))./abs(squeeze(freqresp(Gvc_light('Dv', 'Fu'), freqs, 'Hz')));
plot(freqs, abs(squeeze(freqresp(Gpz_light('Du', 'Fu'), freqs, 'Hz'))))./abs(squeeze(freqresp(Gpz_light('Dv', 'Fu'), freqs, 'Hz')));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Coupling ratio');
legend({'Voice Coil', 'Piezoelectric'})
#+end_src
#+begin_src matlab :results silent :exports none
freqs = logspace(-2, 3, 1000);
figure;
hold on;
plot(freqs, abs(squeeze(freqresp(Gvc_heavy('Du', 'Fu'), freqs, 'Hz'))))./abs(squeeze(freqresp(Gvc_heavy('Dv', 'Fu'), freqs, 'Hz')));
plot(freqs, abs(squeeze(freqresp(Gpz_heavy('Du', 'Fu'), freqs, 'Hz'))))./abs(squeeze(freqresp(Gpz_heavy('Dv', 'Fu'), freqs, 'Hz')));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Coupling ratio');
legend({'Voice Coil', 'Piezoelectric'})
#+end_src
*** Low rotation speed and High rotation speed
#+begin_src matlab :exports code :results silent
rot_speed = 2*pi/60; angle_e = 0;
G_low = linearize(mdl, io, 0.1);
rot_speed = 2*pi; angle_e = 0;
G_high = linearize(mdl, io, 0.1);
G_low.InputName = {'Fu', 'Fv'};
G_low.OutputName = {'Du', 'Dv'};
G_high.InputName = {'Fu', 'Fv'};
G_high.OutputName = {'Du', 'Dv'};
#+end_src
#+begin_src matlab :results silent
figure;
bode(G_low, G_high);
#+end_src
** Identification in the fixed frame
Let's define some options as well as the inputs and outputs for linearization.
#+begin_src matlab :exports code :results silent
%% Options for Linearized
options = linearizeOptions;
options.SampleTime = 0;
%% Name of the Simulink File
mdl = 'rotating_frame';
%% Input/Output definition
io(1) = linio([mdl, '/fx'], 1, 'input');
io(2) = linio([mdl, '/fy'], 1, 'input');
io(3) = linio([mdl, '/dx'], 1, 'output');
io(4) = linio([mdl, '/dy'], 1, 'output');
#+end_src
We then define the error estimation of the error and the rotational speed.
#+begin_src matlab :exports code :results silent
%% Run the linearization
angle_e = 0;
rot_speed = 0;
#+end_src
Finally, we run the linearization.
#+begin_src matlab :exports code :results silent
G = linearize(mdl, io, 0);
%% Input/Output names
G.InputName = {'Fx', 'Fy'};
G.OutputName = {'Dx', 'Dy'};
#+end_src
#+begin_src matlab :exports code :results silent
%% Run the linearization
angle_e = 0;
rot_speed = 2*pi;
Gr = linearize(mdl, io, 0);
%% Input/Output names
Gr.InputName = {'Fx', 'Fy'};
Gr.OutputName = {'Dx', 'Dy'};
#+end_src
#+begin_src matlab :exports code :results silent
%% Run the linearization
angle_e = 1*2*pi/180;
rot_speed = 2*pi;
Ge = linearize(mdl, io, 0);
%% Input/Output names
Ge.InputName = {'Fx', 'Fy'};
Ge.OutputName = {'Dx', 'Dy'};
#+end_src
#+begin_src matlab :exports code :results silent
figure;
bode(G);
% exportFig('G_x_y', 'wide-tall');
figure;
bode(Ge);
% exportFig('G_x_y_e', 'normal-normal');
#+end_src
** Identification from actuator forces to displacement in the fixed frame
#+begin_src matlab :exports code :results silent
%% Options for Linearized
options = linearizeOptions;
options.SampleTime = 0;
%% Name of the Simulink File
mdl = 'rotating_frame';
%% Input/Output definition
io(1) = linio([mdl, '/fu'], 1, 'input');
io(2) = linio([mdl, '/fv'], 1, 'input');
io(3) = linio([mdl, '/dx'], 1, 'output');
io(4) = linio([mdl, '/dy'], 1, 'output');
#+end_src
#+begin_src matlab :exports code :results silent
rot_speed = 2*pi;
angle_e = 0;
G = linearize(mdl, io, 0.0);
G.InputName = {'Fu', 'Fv'};
G.OutputName = {'Dx', 'Dy'};
#+end_src
#+begin_src matlab :exports code :results silent
rot_speed = 2*pi;
angle_e = 0;
G1 = linearize(mdl, io, 0.4);
G1.InputName = {'Fu', 'Fv'};
G1.OutputName = {'Dx', 'Dy'};
#+end_src
#+begin_src matlab :exports code :results silent
rot_speed = 2*pi;
angle_e = 0;
G2 = linearize(mdl, io, 0.8);
G2.InputName = {'Fu', 'Fv'};
G2.OutputName = {'Dx', 'Dy'};
#+end_src
#+begin_src matlab :exports code :results silent
figure;
bode(G, G1, G2);
exportFig('G_u_v_to_x_y', 'wide-tall');
#+end_src
** Effect of the rotating Speed
<<sec:effect_rot_speed>>
#+begin_src matlab :exports none :results silent :noweb yes
<<matlab-init>>
#+end_src
** TODO Use realistic parameters for the mass of the sample and stiffness of the X-Y stage
** TODO Check if the plant is changing a lot when we are not turning to when we are turning at the maximum speed (60rpm)
* Effect of the X-Y stage stiffness
*** TODO Use realistic parameters for the mass of the sample and stiffness of the X-Y stage
*** TODO Check if the plant is changing a lot when we are not turning to when we are turning at the maximum speed (60rpm)
** Effect of the X-Y stage stiffness
<<sec:effect_stiffness>>
** TODO At full speed, check how the coupling changes with the stiffness of the actuators
* Noweb Snippets :noexport:
:PROPERTIES:
:HEADER-ARGS:matlab+: :results none :exports none :tangle no
:HEADER-ARGS:emacs-lisp+: :tangle no :eval no-export
:END:
** Matlab Init
#+NAME: matlab-init
#+BEGIN_SRC matlab :results none :exports none
clear; close all; clc;
%% Add path with some functions
addpath('./src/');
%% Intialize Laplace variable
s = tf('s');
%% Initialize ans with org-babel
ans = 0;
#+END_SRC
** Matlab Export Figure
#+NAME: matlab-export-figure
#+BEGIN_SRC matlab :var fn="figure" ft="png" size="normal-normal" :exports none
exportFig(fn, size);
ans = [fn, ".", ft];
#+END_SRC
** Add hline to org table
#+name: addhdr
#+begin_src emacs-lisp :var tbl=""
(cons (car tbl) (cons 'hline (cdr tbl)))
#+end_src
*** TODO At full speed, check how the coupling changes with the stiffness of the actuators
* Control Implementation
<<sec:control>>
** Measurement in the fixed reference frame
rotating_frame_old.org
-54
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@@ -1,54 +0,0 @@
#+TITLE: Control in a rotating frame
#+AUTHOR: Dehaeze Thomas
#+EMAIL: dehaeze.thomas@gmail.com
#+DATE: {{{time(%Y-%m-%d)}}}
#+DESCRIPTION: Course about Optimal and Robust Control
#+KEYWORDS: control optimal robust lqr lqg hinfinity
#+LANGUAGE: en
#+STARTUP: beamer
#+LaTeX_CLASS: clean-beamer
#+LaTeX_CLASS_OPTIONS: [t]
#+OPTIONS: H:2
#+OPTIONS: num:t toc:t ::t |:t ^:{} -:t f:t *:t <:t
#+SELECT_TAGS: export
#+EXCLUDE_TAGS: noexport
#+COLUMNS: %20ITEM %13BEAMER_env(Env) %6BEAMER_envargs(Args) %4BEAMER_col(Col) %7BEAMER_extra(Extra)
#+latex_header: \AtBeginSection[]{\begin{frame}<beamer>\frametitle{Topic}\tableofcontents[currentsection]\end{frame}}
* Simscape Model
#+name: fig:simscape
#+caption: Screen of simscape multibody model
#+attr_latex: :float t :width 0.7\linewidth
[[./Figures/simscape.png]]
* Simscape Model
#+name: fig:simulink_ctrl
#+caption: Simulink Blocks
#+attr_latex: :float t :width 0.9\linewidth
[[./Figures/simulink_ctrl.png]]
* Identification in the rotating frame
The rotation speed increase the coupling between the rotating actuators and sensors (figure ref:fig:G_u_v).
#+name: fig:G_u_v
#+caption: Transfer function from forces to displacement in the rotating frame
#+attr_latex: :float t
[[./Figures/G_u_v.pdf]]
* Identification in the cartesian frame
#+name: fig:G_x_y
#+caption: Plant from force to displacement in the cartesian frame
#+attr_latex: :float t :width 0.8\linewidth
[[./Figures/G_x_y.pdf]]
* Identification in the cartesian frame
#+name: fig:G_x_y_e
#+caption: Plant from force to displacement in the cartesian frame with small angle estimation error
#+attr_latex: :float t :width 0.8\linewidth
[[./Figures/G_x_y_e.pdf]]
* Control result
#+name: fig:D_x_y
#+caption: Control result with a simple PID
#+attr_latex: :float t :width 0.8\linewidth
[[./Figures/D_x_y.pdf]]
rotating_frame_old.pdf
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