diff --git a/Figures/coupling_heavy.pdf b/Figures/coupling_heavy.pdf index 6458900..ec149d6 100644 Binary files a/Figures/coupling_heavy.pdf and b/Figures/coupling_heavy.pdf differ diff --git a/Figures/coupling_heavy.png b/Figures/coupling_heavy.png index d098e97..1f091be 100644 Binary files a/Figures/coupling_heavy.png and b/Figures/coupling_heavy.png differ diff --git a/Figures/coupling_heavy.svg b/Figures/coupling_heavy.svg index ee2dc3e..5e8ff74 100644 --- a/Figures/coupling_heavy.svg +++ b/Figures/coupling_heavy.svg @@ -1,183 +1,215 @@ - + - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + 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b/Figures/coupling_simscape_heavy.svg @@ -0,0 +1,614 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + 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+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/rotating_frame.html b/rotating_frame.html index 94cae6e..f42b558 100644 --- a/rotating_frame.html +++ b/rotating_frame.html @@ -3,7 +3,7 @@ "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> - + Control in a rotating frame @@ -275,75 +275,75 @@ for the JavaScript code in this tag.

Table of Contents

-
-

1 Introduction

+
+

1 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.

-In section 2, a simple system composed of a spindle and a translation stage is defined and the equations of motion are written. +In section 2, 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.

-Then, in section 3, two different control approach are compared where: +Then, in section 3, two different control approach are compared where:

  • the measurement is made in the fixed frame
    • @@ -351,27 +351,27 @@ Then, in section 3, two different control approach are

-In section 4, the analytical study will be validated using a multi body model of the studied system. +In section 4, the analytical study will be validated using a multi body model of the studied system.

-Finally, in section 5, the control strategies are implemented using Simulink and Simscape and compared. +Finally, in section 5, the control strategies are implemented using Simulink and Simscape and compared.

-
-

2 System

+
+

2 System

- +

-
-

2.1 System description

+
+

2.1 System description

-The system consists of one 2 degree of freedom translation stage on top of a spindle (figure 1). +The system consists of one 2 degree of freedom translation stage on top of a spindle (figure 1).

@@ -384,7 +384,7 @@ The measurement is either the \(x-y\) displacement of the object located on top

-
+

rotating_frame_2dof.png

Figure 1: Schematic of the mecanical system

@@ -418,19 +418,19 @@ Indices \(u\) and \(v\) corresponds to signals in the rotating reference frame (
-
-

2.2 Equations

+
+

2.2 Equations

- -Based on the figure 1, we can write the equations of motion of the system. + +Based on the figure 1, 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 \(m\):

\begin{align} -\label{orgae08e6a} +\label{org97ab84a} T & = \frac{1}{2} m \left( \dot{x}^2 + \dot{y}^2 \right) \\ V & = \frac{1}{2} k \left( x^2 + y^2 \right) \end{align} @@ -439,7 +439,7 @@ V & = \frac{1}{2} k \left( x^2 + y^2 \right) The Lagrangian is the kinetic energy minus the potential energy.

\begin{equation} -\label{org30a9182} +\label{org5b05ded} L = T-V = \frac{1}{2} m \left( \dot{x}^2 + \dot{y}^2 \right) - \frac{1}{2} k \left( x^2 + y^2 \right) \end{equation} @@ -448,7 +448,7 @@ L = T-V = \frac{1}{2} m \left( \dot{x}^2 + \dot{y}^2 \right) - \frac{1}{2} k \le The partial derivatives of the Lagrangian with respect to the variables \((x, y)\) are:

\begin{align*} -\label{orgbf13b01} +\label{orgf971d6e} \frac{\partial L}{\partial x} & = -kx \\ \frac{\partial L}{\partial y} & = -ky \\ \frac{d}{dt}\frac{\partial L}{\partial \dot{x}} & = m\ddot{x} \\ @@ -518,11 +518,11 @@ We can then subtract and add the previous equations to obtain the following equa

\begin{equation} -\label{orgb7e67f4} +\label{orge8fa8fd} 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} \begin{equation} -\label{orgda77d65} +\label{orge386db1} 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} @@ -548,8 +548,8 @@ The resulting effect of those forces should then be higher when using softer act
-
-

2.3 Numerical Values for the NASS

+
+

2.3 Numerical Values for the NASS

Let's define the parameters for the NASS. @@ -612,8 +612,8 @@ Let's define the parameters for the NASS.

-
-

2.4 Euler and Coriolis forces - Numerical Result

+
+

2.4 Euler and Coriolis forces - Numerical Result

First we will determine the value for Euler and Coriolis forces during regular experiment. @@ -624,10 +624,10 @@ First we will determine the value for Euler and Coriolis forces during regular e

-The obtained values are displayed in table 1. +The obtained values are displayed in table 1.

- +
@@ -661,22 +661,22 @@ The obtained values are displayed in table 1. -
-

2.5 Negative Spring Effect - Numerical Result

+
+

2.5 Negative Spring Effect - Numerical Result

The negative stiffness due to the rotation is equal to \(-m{\omega_0}^2\).

-The values for the negative spring effect are displayed in table 2. +The values for the negative spring effect are displayed in table 2.

This is definitely negligible when using piezoelectric actuators. It may not be the case when using voice coil actuators.

-
Table 1: Euler and Coriolis forces for the NASS
+
@@ -704,15 +704,15 @@ This is definitely negligible when using piezoelectric actuators. It may not be -
-

2.6 Limitations due to coupling

+
+

2.6 Limitations due to coupling

To simplify, we consider a constant rotating speed \(\dot{\theta} = {\omega_0}\) and thus \(\ddot{\theta} = 0\).

-From equations \eqref{orgb7e67f4} and \eqref{orgda77d65}, we obtain: +From equations \eqref{orge8fa8fd} and \eqref{orge386db1}, we obtain:

\begin{align*} (m s^2 + (k - m{\omega_0}^2)) d_u &= F_u + 2 m {\omega_0} s d_v \\ @@ -766,26 +766,26 @@ Then, coupling is negligible if \(|-m \omega^2 + (k - m{\omega_0}^2)| \gg |2 m {

-
-

2.6.1 Numerical Analysis

+
+

2.6.1 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 2).
    • -
  • with the heavy sample (figure 3).
    • +
  • with the light sample (figure 2).
    • +
  • with the heavy sample (figure 3).
-
+

coupling_light.png

Figure 2: Relative Coupling for light mass and high rotation speed

-
+

coupling_heavy.png

Figure 3: Relative Coupling for heavy mass and low rotation speed

@@ -801,17 +801,17 @@ Coupling is higher for actuators with small stiffness.
-
-

2.7 Limitations due to negative stiffness effect

+
+

2.7 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 3. +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 3.

-
Table 2: Negative Spring effect
+
@@ -860,10 +860,10 @@ The system can even goes unstable when \(m \omega^2 > k\), that is when the cent

-From this analysis, we can determine the lowest practical stiffness that is possible to use: \(k_\text{min} = 10 m \omega^2\) (table 4) +From this analysis, we can determine the lowest practical stiffness that is possible to use: \(k_\text{min} = 10 m \omega^2\) (table 4)

-
Table 3: Maximum rotation speed at which negative stiffness is negligible (\(0.1\sqrt{\frac{k}{m}}\))
+
@@ -892,15 +892,15 @@ From this analysis, we can determine the lowest practical stiffness that is poss -
-

3 Control Strategies

+
+

3 Control Strategies

- +

-
-

3.1 Measurement in the fixed reference frame

+
+

3.1 Measurement in the fixed reference frame

First, let's consider a measurement in the fixed referenced frame. @@ -923,11 +923,11 @@ Finally, the control low \(K\) links the position errors \([\epsilon_u, \epsilon

-The block diagram is shown on figure 4. +The block diagram is shown on figure 4.

-
+

control_measure_fixed_2dof.png

Figure 4: Control with a measure from fixed frame

@@ -943,19 +943,19 @@ One question we wish to answer is: is \(G(\theta) J(\theta) = G(\theta_0) J(\the
-
-

3.2 Measurement in the rotating frame

+
+

3.2 Measurement in the rotating frame

Let's consider that the measurement is made in the rotating reference frame.

-The corresponding block diagram is shown figure 5 +The corresponding block diagram is shown figure 5

-
+

control_measure_rotating_2dof.png

Figure 5: Control with a measure from rotating frame

@@ -968,16 +968,16 @@ The loop gain is \(L = G K\).
-
-

4 Multi Body Model - Simscape

+
+

4 Multi Body Model - Simscape

- +

-
-

4.1 Identification in the rotating referenced frame

+
+

4.1 Identification in the rotating referenced frame

We initialize the inputs and outputs of the system to identify. @@ -1000,8 +1000,8 @@ io(4

-
-

4.1.1 Piezo and Voice coil

+
+

4.1.1 Piezo and Voice coil

We start we identify the transfer functions at high speed with the light sample. @@ -1026,6 +1026,12 @@ Gvc_light.OutputName = { +

coupling_simscape_light.png +

+
+

And then with the heavy sample.

@@ -1049,13 +1055,31 @@ Gvc_heavy.OutputName = { +

coupling_simscape_heavy.png +

+
+

Plot the ratio between the main transfer function and the coupling term:

+ +
+

coupling_ration_simscape_light.png +

+
+ + +
+

coupling_ration_simscape_heavy.png +

-
-

4.1.2 Low rotation speed and High rotation speed

+
+ +
+

4.1.2 Low rotation speed and High rotation speed

rot_speed = 2*pi/60; angle_e = 0;
@@ -1080,8 +1104,8 @@ bode(G_low, G_high
-
4.2 Identification in the fixed frame

+

+
4.2 Identification in the fixed frame

 

 

 Let's define some options as well as the inputs and outputs for linearization.
@@ -1162,8 +1186,8 @@ bode(Ge)
 

 
-

-
4.3 Identification from actuator forces to displacement in the fixed frame

+

+
4.3 Identification from actuator forces to displacement in the fixed frame

 

 

 
%% Options for Linearized
@@ -1221,48 +1245,48 @@ exportFig('G_
 

 

 
-

-
4.4 Effect of the rotating Speed

+

+
4.4 Effect of the rotating Speed

 

 

-
+
 

 

 
-

-
4.4.1 TODO Use realistic parameters for the mass of the sample and stiffness of the X-Y stage

+

+
4.4.1 TODO Use realistic parameters for the mass of the sample and stiffness of the X-Y stage

 

-

-
4.4.2 TODO Check if the plant is changing a lot when we are not turning to when we are turning at the maximum speed (60rpm)

+

+
4.4.2 TODO Check if the plant is changing a lot when we are not turning to when we are turning at the maximum speed (60rpm)

 

 

-

-
4.5 Effect of the X-Y stage stiffness

+

+
4.5 Effect of the X-Y stage stiffness

 

 

-
+
 

 

-

-
4.5.1 TODO At full speed, check how the coupling changes with the stiffness of the actuators

+

+
4.5.1 TODO At full speed, check how the coupling changes with the stiffness of the actuators

 

 

 

-

-
5 Control Implementation

+

+
5 Control Implementation

 

 

-
+
 

 

-

-
5.1 Measurement in the fixed reference frame

+

+
5.1 Measurement in the fixed reference frame

 

 

 

 

 
Author: Thomas Dehaeze

-
Created: 2019-01-21 lun. 23:32

+
Created: 2019-01-21 lun. 23:44

 
Validate

 

 
diff --git a/rotating_frame.org b/rotating_frame.org
index c044f56..451b296 100644
--- a/rotating_frame.org
+++ b/rotating_frame.org
@@ -310,17 +310,18 @@ We plot on the same graph $\frac{|-m \omega^2 + (k - m {\omega_0}^2)|}{|2 m \ome
 
 #+HEADER: :exports none :results silent
 #+begin_src matlab
-  f = logspace(-1, 2, 1000);
+  f = logspace(-1, 3, 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')
+  hold off;
+  xlim([f(1), f(end)]);
   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
@@ -336,17 +337,16 @@ We plot on the same graph $\frac{|-m \omega^2 + (k - m {\omega_0}^2)|}{|2 m \ome
 
 #+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')
+  hold off;
+  xlim([f(1), f(end)]);
   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
@@ -498,6 +498,15 @@ We start we identify the transfer functions at high speed with the light sample.
   bode(Gpz_light, Gvc_light);
 #+end_src
 
+#+HEADER: :tangle no :exports results :results file :noweb yes
+#+HEADER: :var filepath="Figures/coupling_simscape_light.png" :var figsize="wide-tall"
+#+begin_src matlab
+  <>
+#+end_src
+
+#+RESULTS:
+[[file:Figures/coupling_simscape_light.png]]
+
 And then with the heavy sample.
 #+begin_src matlab :exports code :results silent
   rot_speed = wheavy;
@@ -523,33 +532,62 @@ And then with the heavy sample.
   bode(Gpz_heavy, Gvc_heavy);
 #+end_src
 
+#+HEADER: :tangle no :exports results :results file :noweb yes
+#+HEADER: :var filepath="Figures/coupling_simscape_heavy.png" :var figsize="wide-tall"
+#+begin_src matlab
+  <>
+#+end_src
+
+#+RESULTS:
+[[file:Figures/coupling_simscape_heavy.png]]
+
 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')));
+  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;
+  xlim([freqs(1), freqs(end)]);
   set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
   xlabel('Frequency [Hz]'); ylabel('Coupling ratio');
   legend({'Voice Coil', 'Piezoelectric'})
 #+end_src
 
+#+HEADER: :tangle no :exports results :results file :noweb yes
+#+HEADER: :var filepath="Figures/coupling_ration_simscape_light.png" :var figsize="wide-tall"
+#+begin_src matlab
+  <>
+#+end_src
+
+#+RESULTS:
+[[file:Figures/coupling_ration_simscape_light.png]]
+
 #+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')));
+  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;
+  xlim([freqs(1), freqs(end)]);
   set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
   xlabel('Frequency [Hz]'); ylabel('Coupling ratio');
   legend({'Voice Coil', 'Piezoelectric'})
 #+end_src
 
+#+HEADER: :tangle no :exports results :results file :noweb yes
+#+HEADER: :var filepath="Figures/coupling_ration_simscape_heavy.png" :var figsize="wide-tall"
+#+begin_src matlab
+  <>
+#+end_src
+
+#+RESULTS:
+[[file:Figures/coupling_ration_simscape_heavy.png]]
+
 *** Low rotation speed and High rotation speed
 #+begin_src matlab :exports code :results silent
   rot_speed = 2*pi/60; angle_e = 0;
Table 4: Minimum possible stiffness