diff --git a/Figures/coupling_ration_light_heavy.pdf b/Figures/coupling_ration_light_heavy.pdf new file mode 100644 index 0000000..fa37b32 Binary files /dev/null and b/Figures/coupling_ration_light_heavy.pdf differ diff --git a/Figures/coupling_ration_light_heavy.png b/Figures/coupling_ration_light_heavy.png new file mode 100644 index 0000000..d4a272f Binary files /dev/null and b/Figures/coupling_ration_light_heavy.png differ diff --git a/Figures/coupling_ration_light_heavy.svg b/Figures/coupling_ration_light_heavy.svg new file mode 100644 index 0000000..e7c9ddc --- /dev/null +++ b/Figures/coupling_ration_light_heavy.svg @@ -0,0 +1,536 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/rotating_frame.html b/rotating_frame.html index f42b558..a2f1cd0 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{org97ab84a} +\label{org93a4d45} 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{org5b05ded} +\label{org19136da} 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{orgf971d6e} +\label{org4fc9f2b} \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{orge8fa8fd} +\label{orgf3ca0ca} 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{orge386db1} +\label{org5e2eb96} 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{orge8fa8fd} and \eqref{orge386db1}, we obtain: +From equations \eqref{orgf3ca0ca} and \eqref{org5e2eb96}, 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

@@ -968,17 +968,49 @@ 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 Parameter for the Simscape simulations

+
+
w = 2*pi; % Rotation speed [rad/s]
+
+theta_e = 0; % Static measurement error on the angle theta [rad]
+
+m = 5; % mass of the sample [kg]
+
+mTuv = 30;% Mass of the moving part of the Tuv stage [kg]
+kTuv = 1e8; % Stiffness of the Tuv stage [N/m]
+cTuv = 0; % Damping of the Tuv stage [N/(m/s)]
+
+
+ +
+
mlight = 5; % Mass for light sample [kg]
+mheavy = 55; % Mass for heavy sample [kg]
+
+wlight = 2*pi; % Max rot. speed for light sample [rad/s]
+wheavy = 2*pi/60; % Max rot. speed for heavy sample [rad/s]
+
+kvc = 1e3; % Voice Coil Stiffness [N/m]
+kpz = 1e8; % Piezo Stiffness [N/m]
+
+d = 0.01; % Maximum excentricity from rotational axis [m]
+
+
+
+
+ +
+

4.2 Identification in the rotating referenced frame

+

We initialize the inputs and outputs of the system to identify.

@@ -998,34 +1030,60 @@ io(3(4) = linio([mdl, '/dv'], 1, 'output');
-
-
-

4.1.1 Piezo and Voice coil

-

We start we identify the transfer functions at high speed with the light sample.

-
rot_speed = wlight;
-angle_e = 0;
-m = mlight;
+
w = wlight; % Rotation speed [rad/s]
+m = mlight; % mass of the sample [kg]
 
-k = kpz;
-c = 1e3;
+kTuv = kpz;
 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'};
+
+kTuv = kvc;
+Gvc_light = linearize(mdl, io, 0.1);
 Gvc_light.InputName  = {'Fu', 'Fv'};
 Gvc_light.OutputName = {'Du', 'Dv'};
+

+Then we identify the system with an heavy mass and low speed. +

+
+
w = wheavy; % Rotation speed [rad/s]
+m = mheavy; % mass of the sample [kg]
+
+kTuv = kpz;
+Gpz_heavy = linearize(mdl, io, 0.1);
+Gpz_heavy.InputName  = {'Fu', 'Fv'};
+Gpz_heavy.OutputName = {'Du', 'Dv'};
+
+kTuv = kvc;
+Gvc_heavy = linearize(mdl, io, 0.1);
+Gvc_heavy.InputName  = {'Fu', 'Fv'};
+Gvc_heavy.OutputName = {'Du', 'Dv'};
+
+
+ +

+Finally, we plot the coupling ratio in both case (figure 6). +We obtain the same result than the analytical case (figures 2 and 3). +

+ +
+

coupling_ration_light_heavy.png +

+
+ + + + + +

coupling_simscape_light.png @@ -1076,11 +1134,10 @@ Plot the ratio between the main transfer function and the coupling term:

-
-
-

4.1.2 Low rotation speed and High rotation speed

-
+
+

4.2.1 Low rotation speed and High rotation speed

+
rot_speed = 2*pi/60; angle_e = 0;
 G_low = linearize(mdl, io, 0.1);
@@ -1104,9 +1161,9 @@ bode(G_low, G_high
-
4.2 Identification in the fixed frame

-

+

+
4.3 Identification in the fixed frame

+

 

 Let's define some options as well as the inputs and outputs for linearization.
 

@@ -1186,9 +1243,9 @@ bode(Ge)
 

 
-

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

-

+

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

+

 

 
%% Options for Linearized
 options = linearizeOptions;
@@ -1245,48 +1302,48 @@ exportFig('G_
 

 

 
-

-
4.4 Effect of the rotating Speed

-

+

+
4.5 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

+

+
4.5.1 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)

+

+
4.5.2 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

-

+

+
4.6 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

+

+
4.6.1 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. 23:44

+
Created: 2019-01-23 mer. 15:21

 
Validate

 

 
diff --git a/rotating_frame.org b/rotating_frame.org
index 451b296..b1b8ab9 100644
--- a/rotating_frame.org
+++ b/rotating_frame.org
@@ -454,7 +454,34 @@ The loop gain is $L = G K$.
   open rotating_frame.slx
 #+end_src
 
+** Parameter for the Simscape simulations
+#+begin_src matlab :exports code :results silent
+  w = 2*pi; % Rotation speed [rad/s]
+
+  theta_e = 0; % Static measurement error on the angle theta [rad]
+
+  m = 5; % mass of the sample [kg]
+
+  mTuv = 30;% Mass of the moving part of the Tuv stage [kg]
+  kTuv = 1e8; % Stiffness of the Tuv stage [N/m]
+  cTuv = 0; % Damping of the Tuv stage [N/(m/s)]
+#+end_src
+
+#+begin_src matlab :exports code :results silent
+  mlight = 5; % Mass for light sample [kg]
+  mheavy = 55; % Mass for heavy sample [kg]
+
+  wlight = 2*pi; % Max rot. speed for light sample [rad/s]
+  wheavy = 2*pi/60; % Max rot. speed for heavy sample [rad/s]
+
+  kvc = 1e3; % Voice Coil Stiffness [N/m]
+  kpz = 1e8; % Piezo Stiffness [N/m]
+
+  d = 0.01; % Maximum excentricity from rotational axis [m]
+#+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
@@ -472,27 +499,72 @@ We initialize the inputs and outputs of the system to identify.
   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;
+  w = wlight; % Rotation speed [rad/s]
+  m = mlight; % mass of the sample [kg]
 
-  k = kpz;
-  c = 1e3;
+  kTuv = kpz;
   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'};
+
+  kTuv = kvc;
+  Gvc_light = linearize(mdl, io, 0.1);
   Gvc_light.InputName  = {'Fu', 'Fv'};
   Gvc_light.OutputName = {'Du', 'Dv'};
 #+end_src
 
+Then we identify the system with an heavy mass and low speed.
+#+begin_src matlab :exports code :results silent
+  w = wheavy; % Rotation speed [rad/s]
+  m = mheavy; % mass of the sample [kg]
+
+  kTuv = kpz;
+  Gpz_heavy = linearize(mdl, io, 0.1);
+  Gpz_heavy.InputName  = {'Fu', 'Fv'};
+  Gpz_heavy.OutputName = {'Du', 'Dv'};
+
+  kTuv = kvc;
+  Gvc_heavy = linearize(mdl, io, 0.1);
+  Gvc_heavy.InputName  = {'Fu', 'Fv'};
+  Gvc_heavy.OutputName = {'Du', 'Dv'};
+#+end_src
+
+Finally, we plot the coupling ratio in both case (figure [[fig:coupling_ration_light_heavy]]).
+We obtain the same result than the analytical case (figures [[fig:coupling_light]] and [[fig:coupling_heavy]]).
+#+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'))));
+  set(gca,'ColorOrderIndex',1);
+  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({'light - VC', 'light - PZ', 'heavy - VC',  'heavy - PZ'})
+#+end_src
+
+#+HEADER: :tangle no :exports results :results file :noweb yes
+#+HEADER: :var filepath="Figures/coupling_ration_light_heavy.png" :var figsize="wide-tall"
+#+begin_src matlab
+  <>
+#+end_src
+
+#+NAME: fig:coupling_ration_light_heavy
+#+RESULTS:
+[[file:Figures/coupling_ration_light_heavy.png]]
+
+
+
+
+
+
 #+begin_src matlab :exports none :results silent
   figure;
   bode(Gpz_light, Gvc_light);
diff --git a/rotating_frame.slx b/rotating_frame.slx
index 0bf0008..276eeed 100644
Binary files a/rotating_frame.slx and b/rotating_frame.slx differ
Table 4: Minimum possible stiffness