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Worked on sections 2 and 3

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@ -23,7 +23,7 @@
#+LATEX_HEADER: \usepackage[USenglish]{babel}
#+LATEX_HEADER_EXTRA: \usepackage{tikz}
#+LATEX_HEADER_EXTRA: \usetikzlibrary{shapes.misc}
#+LATEX_HEADER_EXTRA: \usetikzlibrary{shapes.misc,arrows,arrows.meta}
#+LATEX_HEADER: \setcounter{footnote}{1}
#+LATEX_HEADER: \input{config.tex}
@ -60,9 +60,6 @@
** Establish the importance of the research topic :ignore:
# Active Damping + Rotating System
Controller Poles are shown by black crosses (
\begin{tikzpicture} \node[cross out, draw=black, minimum size=1ex, line width=2pt, inner sep=0pt, outer sep=0pt] at (0, 0){}; \end{tikzpicture}
).
** Applications of active damping :ignore:
# Link to previous paper / tomography
@ -78,7 +75,7 @@ Controller Poles are shown by black crosses (
** Describe the paper itself / the problem which is addressed :ignore:
Due to gyroscopic effects, the guaranteed robustness properties of Integral Force Feedback do not hold.
Either the control architecture can be slightly modfied or mechanical changes in the system can be performed.
Either the control architecture can be slightly modified or mechanical changes in the system can be performed.
** Introduce Each part of the paper :ignore:
@ -86,30 +83,31 @@ This paper has been published
The Matlab code that was use to obtain the results are available in cite:dehaeze20_activ_dampin_rotat_posit_platf.
* Dynamics of Rotating Positioning Platforms
<<sec:dynamics>>
** Model of a Rotating Positioning Platform
# Introduce the fact that we need a simple system representing the rotating aspect
To study how the rotation of positioning platforms does affect the use of force feedback, a simple model is developed.
In order to study how the rotation of a positioning platforms does affect the use of force feedback, a simple model of an X-Y positioning stage on top of a rotating stage is developed.
# Simplest system where gyroscopic forces can be studied
It represents an X-Y positioning stage on top of a Rotating Stage and is schematically represented in Figure ref:fig:system.
# Explain the frames (inertial frame x,y, rotating frame u,v)
Two frames of reference are used:
- $(\vec{i}_x, \vec{i}_y, \vec{i}_z)$ is an inertial frame
- $(\vec{i}_u, \vec{i}_v, \vec{i}_w)$ is a frame fixed on the Rotating Stage with its origin align with the rotation axis
The model is schematically represented in Figure ref:fig:system and forms the simplest system where gyroscopic forces can be studied.
# Present the system, parameters, assumptions (small displacements, perfect spindle)
The rotating stage is supposed to be ideal, meaning it is infinitely rigid and induces a rotation $\theta(t) = \Omega t$ where $\Omega$ is the rotational speed in $\si{\radian\per\second}$.
The rotating stage is supposed to be ideal, meaning it induces a perfect rotation $\theta(t) = \Omega t$ where $\Omega$ is the rotational speed in $\si{\radian\per\second}$.
# X-Y Stage
The parallel X-Y positioning stage consists of two orthogonal actuators represented by the three following elements in parallel:
- A spring with a stiffness $k$ in $\si{\newton\per\meter}$
- A dashpot with a damping coefficient $c$ in $\si{\newton\per\meter\second}$
- An ideal force source $F_u, F_v$ in $\si{\newton}$
The parallel X-Y positioning stage consists of two orthogonal actuators represented by three elements in parallel:
- a spring with a stiffness $k$ in $\si{\newton\per\meter}$
- a dashpot with a damping coefficient $c$ in $\si{\newton\per\meter\second}$
- an ideal force source $F_u, F_v$
# Payload
The X-Y stage is supporting a payload with a payload with a mass $m$ in $\si{\kilo\gram}$.
The position of the payload is represented by $(d_u, d_v)$ expressed in the rotating frame $(\vec{i}_u, \vec{i}_v)$.
A payload with a mass $m$ in $\si{\kilo\gram}$ is mounted on the rotating X-Y stage.
# Explain the frames (inertial frame x,y, rotating frame u,v)
Two reference frames are used:
- an inertial frame $(\vec{i}_x, \vec{i}_y, \vec{i}_z)$
- a uniform rotating frame $(\vec{i}_u, \vec{i}_v, \vec{i}_w)$ rigidly fixed on top of the rotating stage. $\vec{i}_w$ is aligned with the rotation axis
The position of the payload is represented by $(d_u, d_v)$ expressed in the rotating frame.
#+name: fig:system
#+caption: Schematic of the studied System
@ -122,7 +120,9 @@ To obtain of equation of motion for the system represented in Figure ref:fig:sys
\begin{equation}
\frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}_i} \right) + \frac{\partial D}{\partial \dot{q}_i} - \frac{\partial L}{\partial q_i} = Q_i
\end{equation}
with $L = T - V$ is the Lagrangian, $D$ is the dissipation function, and $Q_i$ is the generalized force associated with the generalized variable $[q_1\ q_2] = [d_u\ d_v]$:
with $L = T - V$ the Lagrangian, $D$ the dissipation function, and $Q_i$ the generalized force associated with the generalized variable $\begin{bmatrix}q_1 & q_2\end{bmatrix} = \begin{bmatrix}d_u & d_v\end{bmatrix}$.
The constant rotation in the $(\vec{i}_x, \vec{i}_y)$ plane is here disregarded as it is imposed by the rotating stage.
#+name: eq:energy_functions_lagrange
\begin{subequations}
\begin{align}
@ -137,18 +137,27 @@ Substituting equations eqref:eq:energy_functions_lagrange into eqref:eq:lagrangi
#+name: eq:eom_coupled
\begin{subequations}
\begin{align}
m \ddot{d}_u + c \dot{d}_u + ( k - m \Omega ) d_u &= F_u + 2 m \Omega \dot{d}_v \\
m \ddot{d}_v + c \dot{d}_v + ( k \underbrace{-\,m \Omega}_{\text{Centrif.}} ) d_v &= F_v \underbrace{-\,2 m \Omega \dot{d}_u}_{\text{Coriolis}}
m \ddot{d}_u + c \dot{d}_u + ( k - m \Omega^2 ) d_u &= F_u + 2 m \Omega \dot{d}_v \\
m \ddot{d}_v + c \dot{d}_v + ( k \underbrace{-\,m \Omega^2}_{\text{Centrif.}} ) d_v &= F_v \underbrace{-\,2 m \Omega \dot{d}_u}_{\text{Coriolis}}
\end{align}
\end{subequations}
# Explain Gyroscopic effects
The rotation of the XY positioning platform induces two Gyroscopic effects:
- Coriolis Forces: that adds coupling between the two orthogonal controlled directions
- Centrifugal forces: that can been seen as negative stiffness
The constant rotation of the system induces two Gyroscopic effects:
- Centrifugal forces: that can been seen as added negative stiffness along $\vec{i}_u$ and $\vec{i}_v$
- Coriolis Forces: that couples the motion in the two orthogonal directions
One can verify that without rotation ($\Omega = 0$) the system becomes equivalent as to two uncoupled one degree of freedom mass-spring-damper systems:
#+name: eq:oem_no_rotation
\begin{subequations}
\begin{align}
m \ddot{d}_u + c \dot{d}_u + k d_u &= F_u \\
m \ddot{d}_v + c \dot{d}_v + k d_v &= F_v
\end{align}
\end{subequations}
** Transfer Functions in the Laplace domain
To study the dynamics of the system, the differential equations of motions eqref:eq:eom_coupled are transformed in the Laplace domain and the transfer functions from $[F_u,\ F_v]$ to $[d_u,\ d_v]$ are obtained:
To study the dynamics of the system, the differential equations of motions eqref:eq:eom_coupled are transformed in the Laplace domain and the transfer function matrix from $\begin{bmatrix}F_u & F_v\end{bmatrix}$ to $\begin{bmatrix}d_u & d_v\end{bmatrix}$ is obtained:
#+name: eq:Gd_mimo_tf
\begin{equation}
\begin{bmatrix} d_u \\ d_v \end{bmatrix} = \bm{G}_d \begin{bmatrix} F_u \\ F_v \end{bmatrix}
@ -163,19 +172,8 @@ with $\bm{G}_d$ a $2 \times 2$ transfer function matrix
\end{bmatrix}
\end{equation}
One can verify that without rotation ($\Omega = 0$) the system becomes equivalent as to two uncoupled one degree of freedom mass-spring-damper systems:
#+name: eq:oem_no_rotation
\begin{subequations}
\begin{align}
d_u &= \frac{1}{m s^2 + cs + k} F_u \\
d_v &= \frac{1}{m s^2 + cs + k} F_v
\end{align}
\end{subequations}
** Change of Variables / Parameters for the study
# Change of variables
In order to make this study less dependent on the system parameters, the following change of variable is performed:
To simply the analysis, the following change of variable is performed:
- $\omega_0 = \sqrt{\frac{k}{m}}$: Undamped natural frequency of the mass-spring system in $\si{\radian/\s}$
- $\xi = \frac{c}{2 \sqrt{k m}}$: Damping ratio
@ -191,14 +189,12 @@ The transfer function matrix eqref:eq:Gd_m_k_c becomes equal to
\end{equation}
# Parameters
During the rest of this study, the following parameters are used for numerical analysis
- $\omega_0 = \SI{1}{\radian\per\second}$, $\xi = 0.025 = \SI{2.5}{\percent}$
- $k = \SI{1}{N/m}$, $m = \SI{1}{kg}$, $c = \SI{0.05}{\newton\per\meter\second}$
For all the numerical analysis in this study, $\omega_0 = \SI{1}{\radian\per\second}$, $k = \SI{1}{\newton\per\meter}$ and $\xi = 0.025 = \SI{2.5}{\percent}$.
# Say that these parameters are not realist but will be used to draw conclusions "relatively"
Even tough no system with such parameters will be encountered in practice, conclusions will be drawn relative to these parameters such that they can be generalized to any other parameter.
** System Dynamics and Campbell Diagram
# Campbell Diagram
# Poles computation
The poles of $\bm{G}_d$ are the complex solutions $p$ of
\begin{equation}
\left( \frac{p^2}{{\omega_0}^2} + 2 \xi \frac{p}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{p}{\omega_0} \right)^2 = 0
@ -213,12 +209,13 @@ Supposing small damping ($\xi \ll 1$), two pairs of complex conjugate poles are
\end{align}
\end{subequations}
# Campbell Diagram
The real part and complex part of these two pairs of complex conjugate poles are represented in Figure ref:fig:campbell_diagram as a function of the rotational speed $\Omega$.
As the rotational speed increases, $p_{+}$ goes to higher frequencies and $p_{-}$ to lower frequencies.
The system becomes unstable for $\Omega > \omega_0$ as the real part of $p_{-}$ is positive.
Physically, the negative stiffness term $-m\Omega^2$ induced by centrifugal forces exceeds the spring stiffness $k$.
As the rotation speed increases, $p_{+}$ goes to higher frequencies and $p_{-}$ to lower frequencies.
When the rotational speed $\Omega$ reaches $\omega_0$, the real part $p_{-}$ becomes positive rendering the system unstable.
Physically, the negative stiffness term induced by centrifugal forces exceeds the spring stiffness.
Thus, stiff positioning platforms should be used when working at high rotational speeds.
In the rest of this study, rotational speeds smaller than the undamped natural frequency of the system are used ($\Omega < \omega_0$).
#+name: fig:campbell_diagram
#+caption: Campbell Diagram : Evolution of the complex and real parts of the system's poles as a function of the rotational speed $\Omega$
@ -227,41 +224,43 @@ Thus, stiff positioning platforms should be used when working at high rotational
| <<fig:campbell_diagram_real>> Real Part | <<fig:campbell_diagram_imag>> Imaginary Part |
# Bode Plots for different ratio W/w0
Looking at the transfer function matrix $\bm{G}_d$ eqref:eq:Gd_w0_xi_k, one can see it has two distinct terms that can be studied separately:
- the direct (diagonal) terms (Figure ref:fig:plant_compare_rotating_speed_direct)
- the coupling (off-diagonal) terms (Figure ref:fig:plant_compare_rotating_speed_coupling)
The bode plot of the direct and coupling terms are shown in Figure ref:fig:plant_compare_rotating_speed for several rotational speed $\Omega$.
# Describe the dynamics: without rotation
Without rotation, the dynamics of the direct terms is equivalent to the dynamics of a one degree of freedom mass spring damper system and the coupling terms are null.
As the rotational speed increases, the pair of complex conjugate poles is separated into two pairs of complex conjugate poles, one going to lower frequencies and the other to higher frequencies.
When the
Looking at the transfer function matrix $\bm{G}_d$ in Eq. eqref:eq:Gd_w0_xi_k, one can see that the two diagonal (direct) terms are equal and the two off-diagonal (coupling) terms are opposite.
The bode plot of these two distinct terms are shown in Figure ref:fig:plant_compare_rotating_speed for several rotational speeds $\Omega$.
# Rapid Analysis of the dynamics
It is confirmed that the two pairs of complex conjugate poles are further separated as $\Omega$ increases.
For $\Omega > \omega_0$, the low frequency complex conjugate poles $p_{-}$ becomes unstable.
#+name: fig:plant_compare_rotating_speed
#+caption: Bode Plots for $\bm{G}_d$ for several rotational speed $\Omega$
#+attr_latex: :environment subfigure :width 0.45\linewidth :align c
| file:figs/plant_compare_rotating_speed_direct.pdf | file:figs/plant_compare_rotating_speed_coupling.pdf |
| <<fig:plant_compare_rotating_speed_direct>> Direct Terms $d_u/F_u$, $d_v/F_v$ | <<fig:plant_compare_rotating_speed_coupling>> Coupling Terms $d_v/F_u$, $d_u/F_v$ |
In the rest of this study, rotational speeds smaller than the undamped natural frequency of the system are used ($\Omega < \omega_0$).
| file:figs/plant_compare_rotating_speed_direct.pdf | file:figs/plant_compare_rotating_speed_coupling.pdf |
| <<fig:plant_compare_rotating_speed_direct>> Direct Terms $d_u/F_u$, $d_v/F_v$ | <<fig:plant_compare_rotating_speed_coupling>> Coupling Terms $d_v/F_u$, $-d_u/F_v$ |
* Decentralized Integral Force Feedback
<<sec:iff>>
** Force Sensors and Control Architecture
In order to apply Decentralized Integral Force Feedback to the system, force sensors are added in series of the two actuators (Figure ref:fig:system_iff).
# Description of the control architecture
In order to apply Decentralized Integral Force Feedback to the system, force sensors are added in series with the two actuators (Figure ref:fig:system_iff).
Two identical controllers $K_F$ are added to feedback each of the sensed forces to its collocated actuator.
The control diagram is shown in Figure ref:fig:control_diagram_iff.
# Reference to IFF control
# Explain what "decentralized" means
# => we consider the system has two SISO systems for the control
# Say that we will use the same controllers for the two directions
# Decentralized aspect + SISO approach
#+attr_latex: :options [t]{0.55\linewidth}
#+begin_minipage
#+name: fig:system_iff
#+caption: System with Force Sensors in Series with the Actuators. Decentralized Integral Force Feedback is used
#+attr_latex: :scale 1
#+caption: System with added Force Sensor in series with the actuators
#+attr_latex: :scale 1 :float nil
[[file:figs/system_iff.pdf]]
#+end_minipage
\hfill
#+attr_latex: :options [t]{0.40\linewidth}
#+begin_minipage
#+name: fig:control_diagram_iff
#+caption: Control Diagram for decentralized IFF
#+attr_latex: :scale 1 :float nil
[[file:figs/control_diagram_iff.pdf]]
#+end_minipage
** Plant Dynamics
The forces measured by the force sensors are equal to:
@ -286,11 +285,7 @@ with $\bm{G}_f$ a $2 \times 2$ transfer function matrix
\end{bmatrix}
\end{equation}
# Explain the two real zeros => change of gain but not of phase
# The alternating poles and zeros properties of collocated IFF holds
# but additional real zeros are added
The zeros of the diagonal terms of $\bm{G}_f$ are equal to (neglecting the damping)
The zeros of the diagonal terms are equal to (neglecting the damping for simplicity)
\begin{subequations}
\begin{align}
z_c &= \pm j \omega_0 \sqrt{\frac{1}{2} \sqrt{8 \frac{\Omega^2}{{\omega_0}^2} + 1} + \frac{\Omega^2}{{\omega_0}^2} + \frac{1}{2} } \label{eq:iff_zero_cc} \\
@ -301,73 +296,72 @@ The zeros of the diagonal terms of $\bm{G}_f$ are equal to (neglecting the dampi
The frequency of the two complex conjugate zeros $z_c$ eqref:eq:iff_zero_cc is between the frequency of the two pairs of complex conjugate poles $p_{-}$ and $p_{+}$ eqref:eq:pole_values.
This is the expected behavior of a collocated pair of actuator and sensor.
However, the two real zeros $z_c$ induces an increase of +2 of the slope without change of phase (Figure [[fig:plant_iff_compare_rotating_speed]]).
This represents non-minimum phase behavior.
# Explain physically why the real zeros
However for non-null rotational speeds, two real zeros $z_r$ eqref:eq:iff_zero_real appear in the diagonal terms which represent a non-minimum phase behavior.
This can be seen in the Bode plot of the diagonal terms (Figure ref:fig:plant_iff_compare_rotating_speed) where the magnitude experiences an increase of its slope without any change of phase.
# Show that the low frequency gain is no longer zero
The low frequency gain, for $\Omega < \omega_0$, is no longer zero:
The low frequency gain of $\bm{G}_f$ is no longer zero, and increases with the rotational speed $\Omega$
#+name: low_freq_gain_iff_plan
\begin{equation}
\bm{G}_{f0} = \lim_{\omega \to 0} \left| \bm{G}_f (j\omega) \right| = \begin{bmatrix}
\lim_{\omega \to 0} \left| \bm{G}_f (j\omega) \right| = \begin{bmatrix}
\frac{- \Omega^2}{{\omega_0}^2 - \Omega^2} & 0 \\
0 & \frac{- \Omega^2}{{\omega_0}^2 - \Omega^2}
\end{bmatrix}
\end{equation}
It increases with the rotational speed $\Omega$.
# Explain why do we have this low frequency gain
This low frequency gain can be explained as follows: a constant force induces a small displacement of the mass, which then increases the centrifugal forces measured by the force sensors.
# Another way to explain this low frequency gain is to model the centrifugal forces by a negative stiffness $k_p = -m \Omega^2$ in parallel with both the actuator and force sensor as in Figure ref:fig:system_parallel_springs.
#+name: fig:plant_iff_compare_rotating_speed
#+caption: Bode plot of $\bm{G}_f$ for several rotational speeds $\Omega$
#+caption: Bode plot of the diagonal terms of $\bm{G}_f$ for several rotational speeds $\Omega$
#+attr_latex: :scale 1
[[file:figs/plant_iff_compare_rotating_speed.pdf]]
** Decentralized Integral Force Feedback
** Decentralized Integral Force Feedback with Pure Integrators
The two IFF controllers $K_F$ are pure integrators
\begin{equation}
K_F(s) = g \cdot \frac{1}{s}
\bm{K}_F(s) = \begin{bmatrix} K_F(s) & 0 \\ 0 & K_F(s) \end{bmatrix}, \quad K_F(s) = g \cdot \frac{1}{s}
\end{equation}
# Problem of zero with a positive real part
Also, as one zero has a positive real part, the IFF control is no more unconditionally stable.
This is due to the fact that the zeros of the plant are the poles of the closed loop system with an infinite gain.
Thus, for some finite IFF gain, one pole will have a positive real part and the system will be unstable.
where $g$ is a scalar value representing the gain of the controller.
# General explanation for the Root Locus Plot
# MIMO root locus: gain is simultaneously increased for both decentralized controllers
# Explain the circles, crosses and black crosses (poles of the controller)
# transmission zeros
In order to see how the controller affects the poles of the closed loop system, the Root Locus is constructed as follows.
The poles of the closed-loop system are drawn in the complex plane as the gain $g$ varies from $0$ to $\infty$ for the two controllers simultaneously.
The closed-loop poles start at the open-loop poles (shown by $\tikz[baseline=-0.6ex] \node[cross out, draw=black, minimum size=1ex, line width=2pt, inner sep=0pt, outer sep=0pt] at (0, 0){};$) for $g = 0$ and coincide with the transmission zeros (shown by $\tikz[baseline=-0.6ex] \draw[line width=2pt, inner sep=0pt, outer sep=0pt] (0,0) circle[radius=3pt];$) as $g \to \infty$.
The direction of increasing gains is shown by the arrows $\tikz[baseline=-0.6ex] \draw[-{Stealth[round]},line width=2pt] (0,0) -- (0.3,0);$.
#+name: fig:root_locus_pure_iff
#+caption: Root Locus for the Decentralized Integral Force Feedback
#+attr_latex: :scale 1
[[file:figs/root_locus_pure_iff.pdf]]
# IFF is usually known for its guaranteed stability (add reference)
# This is not the case anymore due to gyroscopic effects
# IFF is usually known for its guaranteed stability (add reference) which is not the case anymore due to gyroscopic effects
Whereas collocated IFF is known for its guaranteed stability, which is the case here for $\Omega = 0$, this property is lost as soon as the rotational speed in non-null due to gyroscopic effects.
This can be seen in the Root Locus (Figure ref:fig:root_locus_pure_iff) where the pole corresponding to the controller is bounded to the right half plane implying closed-loop system instability.
# Physical Interpretation
# Physical Interpretation ?
# This instability can be explained by the gyroscopic effects.
# At low frequency, the gain is very large and thus no force is transmitted to the payload.
# This means that at low frequency, the system is decoupled (the force sensor removed) and thus the system is unstable.
At low frequency, the gain is very large and thus no force is transmitted between the payload and the rotating stage.
This means that at low frequency, the system is decoupled (the force sensor removed) and thus the system is unstable.
# Introduce next two sections where either:
# - IFF is modified to deal with this low frequency behavior
# - physical system is modified
# Introduce next two sections
Two system modifications are proposed in the next sections to deal with this stability problem.
Either the control law can be change (Section ref:sec:iff_hpf) or the mechanical system slightly modified (Section ref:sec:iff_kp).
* Integral Force Feedback with High Pass Filters
<<sec:iff_hpf>>
** Modification of the Control Low
# Reference to Preumont where its done
# Equation with the new control law
# Equivalent as to add a HFP or to slightly move the pole to the left
#+NAME: eq:IFF_LHF
\begin{equation}
K_{F}(s) = g \cdot \frac{1}{s} \cdot \underbrace{\frac{s/\omega_i}{1 + s/\omega_i}}_{\text{HPF}} = g \cdot \frac{1}{s + \omega_i}
\end{equation}
# Explain why it is usually done and why it is done here: the problem is the high gain at low frequency => high pass filter
@ -377,26 +371,37 @@ This means that at low frequency, the system is decoupled (the force sensor remo
# Explain that now the low frequency loop gain does not reach a gain more than 1 (if g not so high)
#+name: fig:loop_gain_modified_iff
#+caption: Bode Plot of the Loop Gain for IFF with and without the HPF, $\Omega = 0.1 \omega_0$
#+caption: Bode Plot of the Loop Gain for IFF with and without the HPF with $\omega_i = 0.1 \omega_0$, $g = 2$ and $\Omega = 0.1 \omega_0$
#+attr_latex: :scale 1
[[file:figs/loop_gain_modified_iff.pdf]]
# Not the system can be stable for small values of g
# Actually, the system becomes unstable for g > ... => it has been verified
# Explain how the root locus changes (the pole corresponding to the controller is moved to the left)
# Explain that it is stable for small values of $g$ but at some point, the system goes unstable
# Explain what is the maximum value of the gain
As shown in Figure ref:fig:root_locus_modified_iff, the poles of the closed loop system are stable for $g < g_\text{max}$
\begin{equation}
g_\text{max} = \omega_i \left( \frac{{\omega_0}^2}{\Omega^2} - 1 \right) \label{eq:iff_gmax}
g_{\text{max}} = \omega_i \left( \frac{{\omega_0}^2}{\Omega^2} - 1 \right)
\end{equation}
# Small rotational speeds allows to increase the control gain
# Large wi allows more gain but less damping
# Say that this corresponds as to have a low frequency gain of the loop gain less thank 1
#+name: fig:root_locus_modified_iff
#+caption: Root Locus for IFF with and without the HPF, $\Omega = 0.1 \omega_0$
#+attr_latex: :scale 1
[[file:figs/root_locus_modified_iff.pdf]]
** Optimal Cut-Off Frequency
** Optimal Control Parameters
# Controller: two parameters: gain and wi
Two parameters can be tuned for the controller eqref:eq:IFF_LHF, the gain $g$ and the frequency of the pole $\omega_i$.
# Try few wi
Root Locus plots for several $\omega_i$ are shown in Figure ref:fig:root_locus_wi_modified_iff.
# Small wi seems to allow more damping to be added
# but the gain is limited to small values
@ -409,6 +414,7 @@ This means that at low frequency, the system is decoupled (the force sensor remo
[[file:figs/root_locus_wi_modified_iff.pdf]]
# Study this trade-off
The optimal values of $\omega_i$ and $g$ may be considered as the values for which the closed-loop poles are equally damped.
# Explain how the figure is obtained
@ -422,17 +428,31 @@ This means that at low frequency, the system is decoupled (the force sensor remo
[[file:figs/mod_iff_damping_wi.pdf]]
* Integral Force Feedback with Parallel Springs
<<sec:iff_kp>>
** Stiffness in Parallel with the Force Sensor
# Zeros = remove force sensor
# We want to have stable zeros => add stiffnesses in parallel
Stiffness can be added in parallel to the force sensor to counteract the negative stiffness due to centrifugal forces.
If the added stiffness is higher than the maximum negative stiffness, then the poles of the IFF damped system will stay in the (stable) right half-plane.
#+name: fig:system_parallel_springs
#+caption: System with added springs in parallel with the actuators
#+caption:
#+attr_latex: :scale 1
[[file:figs/system_parallel_springs.pdf]]
# Sensed Force
The forces measured by the force sensors are equal to:
#+name: eq:measured_force_kp
\begin{equation}
\begin{bmatrix} f_{u} \\ f_{v} \end{bmatrix} =
\begin{bmatrix} F_u \\ F_v \end{bmatrix} - (c s + k_a)
\begin{bmatrix} d_u \\ d_v \end{bmatrix}
\end{equation}
# Maybe add the fact that this is equivalent to amplified piezo for instance
# Add reference to cite:souleille18_concep_activ_mount_space_applic
This could represent a system where
** Plant Dynamics
@ -517,28 +537,26 @@ The overall stiffness $k$ stays constant:
| <<fig:root_locus_iff_kps>> Three values of $k_p$ | <<fig:root_locus_opt_gain_iff_kp>> $k_p = 5 m \Omega^2$, optimal damping is shown |
* Comparison of the Proposed Active Damping Techniques for Rotating Positioning Stages
<<sec:comparison>>
** Physical Comparison
** Attainable Damping
# Both techniques provides very good amount of damping
#+name: fig:comp_root_locus
#+caption: Root Locus for the three proposed decentralized active damping techniques: IFF with HFP, IFF with parallel springs, and relative DVF, $\Omega = 0.1 \omega_0$
#+attr_latex: :scale 1
[[file:figs/comp_root_locus.pdf]]
** Transmissibility and Compliance
# IFF with HPF and IFF with kp give very similar results
# Both techniques provides very good amount of damping
# IFF with HPF and IFF with kp give very similar results!
# IFF degrades the compliance at low frequency (add reference)
# Relative DVF degrades the transmissibility at high frequency
# Passive Damping degrades the transmissibility at high frequency
# The roll-off is -1 instead of -2
#+name: fig:comp_active_damping
@ -547,16 +565,6 @@ The overall stiffness $k$ stays constant:
| file:figs/comp_compliance.pdf | file:figs/comp_transmissibility.pdf |
| <<fig:comp_compliance>> Transmissibility | <<fig:comp_transmissibility>> Compliance |
# #+name: fig:comp_compliance
# #+caption: Figure caption
# #+attr_latex: :scale 1
# [[file:figs/comp_compliance.pdf]]
# #+name: fig:comp_transmissibility
# #+caption: Figure caption
# #+attr_latex: :scale 1
# [[file:figs/comp_transmissibility.pdf]]
* Conclusion
<<sec:conclusion>>

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@ -1,4 +1,4 @@
% Created 2020-06-29 lun. 10:22
% Created 2020-07-01 mer. 10:02
% Intended LaTeX compiler: pdflatex
\documentclass{ISMA_USD2020}
\usepackage[utf8]{inputenc}
@ -34,7 +34,7 @@
\affil[3] {European Synchrotron Radiation Facility \NewLineAffil Grenoble, France e-mail: \textbf{thomas.dehaeze@esrf.fr}}
\bibliographystyle{IEEEtran}
\usepackage{tikz}
\usetikzlibrary{shapes.misc}
\usetikzlibrary{shapes.misc,arrows,arrows.meta}
\date{}
\title{Active Damping of Rotating Positioning Platforms using Force Feedback}
\hypersetup{
@ -53,41 +53,40 @@
}
\section{Introduction}
\label{sec:org8a431d3}
\label{sec:org38db2d2}
\label{sec:introduction}
Controller Poles are shown by black crosses (
\begin{tikzpicture} \node[cross out, draw=black, minimum size=1ex, line width=2pt, inner sep=0pt, outer sep=0pt] at (0, 0){}; \end{tikzpicture}
).
Due to gyroscopic effects, the guaranteed robustness properties of Integral Force Feedback do not hold.
Either the control architecture can be slightly modfied or mechanical changes in the system can be performed.
Either the control architecture can be slightly modified or mechanical changes in the system can be performed.
This paper has been published
The Matlab code that was use to obtain the results are available in \cite{dehaeze20_activ_dampin_rotat_posit_platf}.
\section{Dynamics of Rotating Positioning Platforms}
\label{sec:org6c19606}
\label{sec:orgd0419a7}
\label{sec:dynamics}
\subsection{Model of a Rotating Positioning Platform}
\label{sec:orga59b20f}
To study how the rotation of positioning platforms does affect the use of force feedback, a simple model is developed.
\label{sec:org4c3a41a}
In order to study how the rotation of a positioning platforms does affect the use of force feedback, a simple model of an X-Y positioning stage on top of a rotating stage is developed.
It represents an X-Y positioning stage on top of a Rotating Stage and is schematically represented in Figure \ref{fig:system}.
The model is schematically represented in Figure \ref{fig:system} and forms the simplest system where gyroscopic forces can be studied.
Two frames of reference are used:
The rotating stage is supposed to be ideal, meaning it induces a perfect rotation \(\theta(t) = \Omega t\) where \(\Omega\) is the rotational speed in \(\si{\radian\per\second}\).
The parallel X-Y positioning stage consists of two orthogonal actuators represented by three elements in parallel:
\begin{itemize}
\item \((\vec{i}_x, \vec{i}_y, \vec{i}_z)\) is an inertial frame
\item \((\vec{i}_u, \vec{i}_v, \vec{i}_w)\) is a frame fixed on the Rotating Stage with its origin align with the rotation axis
\item a spring with a stiffness \(k\) in \(\si{\newton\per\meter}\)
\item a dashpot with a damping coefficient \(c\) in \(\si{\newton\per\meter\second}\)
\item an ideal force source \(F_u, F_v\)
\end{itemize}
The rotating stage is supposed to be ideal, meaning it is infinitely rigid and induces a rotation \(\theta(t) = \Omega t\) where \(\Omega\) is the rotational speed in \(\si{\radian\per\second}\).
A payload with a mass \(m\) in \(\si{\kilo\gram}\) is mounted on the rotating X-Y stage.
The parallel X-Y positioning stage consists of two orthogonal actuators represented by the three following elements in parallel:
Two reference frames are used:
\begin{itemize}
\item A spring with a stiffness \(k\) in \(\si{\newton\per\meter}\)
\item A dashpot with a damping coefficient \(c\) in \(\si{\newton\per\meter\second}\)
\item An ideal force source \(F_u, F_v\) in \(\si{\newton}\)
\item an inertial frame \((\vec{i}_x, \vec{i}_y, \vec{i}_z)\)
\item a uniform rotating frame \((\vec{i}_u, \vec{i}_v, \vec{i}_w)\) rigidly fixed on top of the rotating stage. \(\vec{i}_w\) is aligned with the rotation axis
\end{itemize}
The X-Y stage is supporting a payload with a payload with a mass \(m\) in \(\si{\kilo\gram}\).
The position of the payload is represented by \((d_u, d_v)\) expressed in the rotating frame \((\vec{i}_u, \vec{i}_v)\).
The position of the payload is represented by \((d_u, d_v)\) expressed in the rotating frame.
\begin{figure}[htbp]
\centering
@ -96,13 +95,15 @@ The position of the payload is represented by \((d_u, d_v)\) expressed in the ro
\end{figure}
\subsection{Equations of Motion}
\label{sec:orgad9d82d}
\label{sec:orga2d4956}
To obtain of equation of motion for the system represented in Figure \ref{fig:system}, the Lagrangian equations are used:
\begin{equation}
\label{eq:lagrangian_equations}
\frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}_i} \right) + \frac{\partial D}{\partial \dot{q}_i} - \frac{\partial L}{\partial q_i} = Q_i
\end{equation}
with \(L = T - V\) is the Lagrangian, \(D\) is the dissipation function, and \(Q_i\) is the generalized force associated with the generalized variable \([q_1\ q_2] = [d_u\ d_v]\):
with \(L = T - V\) the Lagrangian, \(D\) the dissipation function, and \(Q_i\) the generalized force associated with the generalized variable \(\begin{bmatrix}q_1 & q_2\end{bmatrix} = \begin{bmatrix}d_u & d_v\end{bmatrix}\).
The constant rotation in the \((\vec{i}_x, \vec{i}_y)\) plane is here disregarded as it is imposed by the rotating stage.
\begin{subequations}
\label{eq:energy_functions_lagrange}
\begin{align}
@ -117,20 +118,29 @@ Substituting equations \eqref{eq:energy_functions_lagrange} into \eqref{eq:lagra
\begin{subequations}
\label{eq:eom_coupled}
\begin{align}
m \ddot{d}_u + c \dot{d}_u + ( k - m \Omega ) d_u &= F_u + 2 m \Omega \dot{d}_v \\
m \ddot{d}_v + c \dot{d}_v + ( k \underbrace{-\,m \Omega}_{\text{Centrif.}} ) d_v &= F_v \underbrace{-\,2 m \Omega \dot{d}_u}_{\text{Coriolis}}
m \ddot{d}_u + c \dot{d}_u + ( k - m \Omega^2 ) d_u &= F_u + 2 m \Omega \dot{d}_v \\
m \ddot{d}_v + c \dot{d}_v + ( k \underbrace{-\,m \Omega^2}_{\text{Centrif.}} ) d_v &= F_v \underbrace{-\,2 m \Omega \dot{d}_u}_{\text{Coriolis}}
\end{align}
\end{subequations}
The rotation of the XY positioning platform induces two Gyroscopic effects:
The constant rotation of the system induces two Gyroscopic effects:
\begin{itemize}
\item Coriolis Forces: that adds coupling between the two orthogonal controlled directions
\item Centrifugal forces: that can been seen as negative stiffness
\item Centrifugal forces: that can been seen as added negative stiffness along \(\vec{i}_u\) and \(\vec{i}_v\)
\item Coriolis Forces: that couples the motion in the two orthogonal directions
\end{itemize}
One can verify that without rotation (\(\Omega = 0\)) the system becomes equivalent as to two uncoupled one degree of freedom mass-spring-damper systems:
\begin{subequations}
\label{eq:oem_no_rotation}
\begin{align}
m \ddot{d}_u + c \dot{d}_u + k d_u &= F_u \\
m \ddot{d}_v + c \dot{d}_v + k d_v &= F_v
\end{align}
\end{subequations}
\subsection{Transfer Functions in the Laplace domain}
\label{sec:orgb80a7b8}
To study the dynamics of the system, the differential equations of motions \eqref{eq:eom_coupled} are transformed in the Laplace domain and the transfer functions from \([F_u,\ F_v]\) to \([d_u,\ d_v]\) are obtained:
\label{sec:org16be790}
To study the dynamics of the system, the differential equations of motions \eqref{eq:eom_coupled} are transformed in the Laplace domain and the transfer function matrix from \(\begin{bmatrix}F_u & F_v\end{bmatrix}\) to \(\begin{bmatrix}d_u & d_v\end{bmatrix}\) is obtained:
\begin{equation}
\label{eq:Gd_mimo_tf}
\begin{bmatrix} d_u \\ d_v \end{bmatrix} = \bm{G}_d \begin{bmatrix} F_u \\ F_v \end{bmatrix}
@ -145,19 +155,7 @@ with \(\bm{G}_d\) a \(2 \times 2\) transfer function matrix
\end{bmatrix}
\end{equation}
One can verify that without rotation (\(\Omega = 0\)) the system becomes equivalent as to two uncoupled one degree of freedom mass-spring-damper systems:
\begin{subequations}
\label{eq:oem_no_rotation}
\begin{align}
d_u &= \frac{1}{m s^2 + cs + k} F_u \\
d_v &= \frac{1}{m s^2 + cs + k} F_v
\end{align}
\end{subequations}
\subsection{Change of Variables / Parameters for the study}
\label{sec:org97136f3}
In order to make this study less dependent on the system parameters, the following change of variable is performed:
To simply the analysis, the following change of variable is performed:
\begin{itemize}
\item \(\omega_0 = \sqrt{\frac{k}{m}}\): Undamped natural frequency of the mass-spring system in \(\si{\radian/\s}\)
\item \(\xi = \frac{c}{2 \sqrt{k m}}\): Damping ratio
@ -174,14 +172,12 @@ The transfer function matrix \eqref{eq:Gd_m_k_c} becomes equal to
\end{bmatrix}
\end{equation}
During the rest of this study, the following parameters are used for numerical analysis
\begin{itemize}
\item \(\omega_0 = \SI{1}{\radian\per\second}\), \(\xi = 0.025 = \SI{2.5}{\percent}\)
\item \(k = \SI{1}{N/m}\), \(m = \SI{1}{kg}\), \(c = \SI{0.05}{\newton\per\meter\second}\)
\end{itemize}
For all the numerical analysis in this study, \(\omega_0 = \SI{1}{\radian\per\second}\), \(k = \SI{1}{\newton\per\meter}\) and \(\xi = 0.025 = \SI{2.5}{\percent}\).
Even tough no system with such parameters will be encountered in practice, conclusions will be drawn relative to these parameters such that they can be generalized to any other parameter.
\subsection{System Dynamics and Campbell Diagram}
\label{sec:orgf368845}
\label{sec:org0ed334c}
The poles of \(\bm{G}_d\) are the complex solutions \(p\) of
\begin{equation}
\left( \frac{p^2}{{\omega_0}^2} + 2 \xi \frac{p}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{p}{\omega_0} \right)^2 = 0
@ -197,11 +193,11 @@ Supposing small damping (\(\xi \ll 1\)), two pairs of complex conjugate poles ar
\end{subequations}
The real part and complex part of these two pairs of complex conjugate poles are represented in Figure \ref{fig:campbell_diagram} as a function of the rotational speed \(\Omega\).
As the rotational speed increases, \(p_{+}\) goes to higher frequencies and \(p_{-}\) to lower frequencies.
The system becomes unstable for \(\Omega > \omega_0\) as the real part of \(p_{-}\) is positive.
Physically, the negative stiffness term \(-m\Omega^2\) induced by centrifugal forces exceeds the spring stiffness \(k\).
As the rotation speed increases, \(p_{+}\) goes to higher frequencies and \(p_{-}\) to lower frequencies.
When the rotational speed \(\Omega\) reaches \(\omega_0\), the real part \(p_{-}\) becomes positive rendering the system unstable.
Physically, the negative stiffness term induced by centrifugal forces exceeds the spring stiffness.
Thus, stiff positioning platforms should be used when working at high rotational speeds.
In the rest of this study, rotational speeds smaller than the undamped natural frequency of the system are used (\(\Omega < \omega_0\)).
\begin{figure}[htbp]
\begin{subfigure}[c]{0.4\linewidth}
@ -216,17 +212,11 @@ Thus, stiff positioning platforms should be used when working at high rotational
\centering
\end{figure}
Looking at the transfer function matrix \(\bm{G}_d\) \eqref{eq:Gd_w0_xi_k}, one can see it has two distinct terms that can be studied separately:
\begin{itemize}
\item the direct (diagonal) terms (Figure \ref{fig:plant_compare_rotating_speed_direct})
\item the coupling (off-diagonal) terms (Figure \ref{fig:plant_compare_rotating_speed_coupling})
\end{itemize}
Looking at the transfer function matrix \(\bm{G}_d\) in Eq. \eqref{eq:Gd_w0_xi_k}, one can see that the two diagonal (direct) terms are equal and the two off-diagonal (coupling) terms are opposite.
The bode plot of these two distinct terms are shown in Figure \ref{fig:plant_compare_rotating_speed} for several rotational speeds \(\Omega\).
The bode plot of the direct and coupling terms are shown in Figure \ref{fig:plant_compare_rotating_speed} for several rotational speed \(\Omega\).
Without rotation, the dynamics of the direct terms is equivalent to the dynamics of a one degree of freedom mass spring damper system and the coupling terms are null.
As the rotational speed increases, the pair of complex conjugate poles is separated into two pairs of complex conjugate poles, one going to lower frequencies and the other to higher frequencies.
When the
It is confirmed that the two pairs of complex conjugate poles are further separated as \(\Omega\) increases.
For \(\Omega > \omega_0\), the low frequency complex conjugate poles \(p_{-}\) becomes unstable.
\begin{figure}[htbp]
\begin{subfigure}[c]{0.45\linewidth}
@ -235,28 +225,37 @@ When the
\end{subfigure}
\begin{subfigure}[c]{0.45\linewidth}
\includegraphics[width=\linewidth]{figs/plant_compare_rotating_speed_coupling.pdf}
\caption{\label{fig:plant_compare_rotating_speed_coupling} Coupling Terms \(d_v/F_u\), \(d_u/F_v\)}
\caption{\label{fig:plant_compare_rotating_speed_coupling} Coupling Terms \(d_v/F_u\), \(-d_u/F_v\)}
\end{subfigure}
\caption{\label{fig:plant_compare_rotating_speed}Bode Plots for \(\bm{G}_d\) for several rotational speed \(\Omega\)}
\centering
\end{figure}
In the rest of this study, rotational speeds smaller than the undamped natural frequency of the system are used (\(\Omega < \omega_0\)).
\section{Decentralized Integral Force Feedback}
\label{sec:orge7b2b3c}
\label{sec:org284335d}
\label{sec:iff}
\subsection{Force Sensors and Control Architecture}
\label{sec:org2b4254d}
In order to apply Decentralized Integral Force Feedback to the system, force sensors are added in series of the two actuators (Figure \ref{fig:system_iff}).
\label{sec:org5b8858f}
In order to apply Decentralized Integral Force Feedback to the system, force sensors are added in series with the two actuators (Figure \ref{fig:system_iff}).
Two identical controllers \(K_F\) are added to feedback each of the sensed forces to its collocated actuator.
The control diagram is shown in Figure \ref{fig:control_diagram_iff}.
\begin{figure}[htbp]
\centering
\begin{minipage}[t]{0.55\linewidth}
\begin{center}
\includegraphics[scale=1]{figs/system_iff.pdf}
\caption{\label{fig:system_iff}System with Force Sensors in Series with the Actuators. Decentralized Integral Force Feedback is used}
\end{figure}
\captionof{figure}{\label{fig:system_iff}System with added Force Sensor in series with the actuators}
\end{center}
\end{minipage}
\hfill
\begin{minipage}[t]{0.40\linewidth}
\begin{center}
\includegraphics[scale=1]{figs/control_diagram_iff.pdf}
\captionof{figure}{\label{fig:control_diagram_iff}Control Diagram for decentralized IFF}
\end{center}
\end{minipage}
\subsection{Plant Dynamics}
\label{sec:org59a4f35}
\label{sec:orge8dea8f}
The forces measured by the force sensors are equal to:
\begin{equation}
\label{eq:measured_force}
@ -279,7 +278,7 @@ with \(\bm{G}_f\) a \(2 \times 2\) transfer function matrix
\end{bmatrix}
\end{equation}
The zeros of the diagonal terms of \(\bm{G}_f\) are equal to (neglecting the damping)
The zeros of the diagonal terms are equal to (neglecting the damping for simplicity)
\begin{subequations}
\begin{align}
z_c &= \pm j \omega_0 \sqrt{\frac{1}{2} \sqrt{8 \frac{\Omega^2}{{\omega_0}^2} + 1} + \frac{\Omega^2}{{\omega_0}^2} + \frac{1}{2} } \label{eq:iff_zero_cc} \\
@ -290,37 +289,38 @@ The zeros of the diagonal terms of \(\bm{G}_f\) are equal to (neglecting the dam
The frequency of the two complex conjugate zeros \(z_c\) \eqref{eq:iff_zero_cc} is between the frequency of the two pairs of complex conjugate poles \(p_{-}\) and \(p_{+}\) \eqref{eq:pole_values}.
This is the expected behavior of a collocated pair of actuator and sensor.
However, the two real zeros \(z_c\) induces an increase of +2 of the slope without change of phase (Figure \ref{fig:plant_iff_compare_rotating_speed}).
This represents non-minimum phase behavior.
However for non-null rotational speeds, two real zeros \(z_r\) \eqref{eq:iff_zero_real} appear in the diagonal terms which represent a non-minimum phase behavior.
This can be seen in the Bode plot of the diagonal terms (Figure \ref{fig:plant_iff_compare_rotating_speed}) where the magnitude experiences an increase of its slope without any change of phase.
The low frequency gain, for \(\Omega < \omega_0\), is no longer zero:
The low frequency gain of \(\bm{G}_f\) is no longer zero, and increases with the rotational speed \(\Omega\)
\begin{equation}
\label{low_freq_gain_iff_plan}
\bm{G}_{f0} = \lim_{\omega \to 0} \left| \bm{G}_f (j\omega) \right| = \begin{bmatrix}
\lim_{\omega \to 0} \left| \bm{G}_f (j\omega) \right| = \begin{bmatrix}
\frac{- \Omega^2}{{\omega_0}^2 - \Omega^2} & 0 \\
0 & \frac{- \Omega^2}{{\omega_0}^2 - \Omega^2}
\end{bmatrix}
\end{equation}
It increases with the rotational speed \(\Omega\).
This low frequency gain can be explained as follows: a constant force induces a small displacement of the mass, which then increases the centrifugal forces measured by the force sensors.
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/plant_iff_compare_rotating_speed.pdf}
\caption{\label{fig:plant_iff_compare_rotating_speed}Bode plot of \(\bm{G}_f\) for several rotational speeds \(\Omega\)}
\caption{\label{fig:plant_iff_compare_rotating_speed}Bode plot of the diagonal terms of \(\bm{G}_f\) for several rotational speeds \(\Omega\)}
\end{figure}
\subsection{Decentralized Integral Force Feedback}
\label{sec:orgf040b7e}
\subsection{Decentralized Integral Force Feedback with Pure Integrators}
\label{sec:org8280bcd}
The two IFF controllers \(K_F\) are pure integrators
\begin{equation}
K_F(s) = g \cdot \frac{1}{s}
\bm{K}_F(s) = \begin{bmatrix} K_F(s) & 0 \\ 0 & K_F(s) \end{bmatrix}, \quad K_F(s) = g \cdot \frac{1}{s}
\end{equation}
where \(g\) is a scalar value representing the gain of the controller.
Also, as one zero has a positive real part, the IFF control is no more unconditionally stable.
This is due to the fact that the zeros of the plant are the poles of the closed loop system with an infinite gain.
Thus, for some finite IFF gain, one pole will have a positive real part and the system will be unstable.
In order to see how the controller affects the poles of the closed loop system, the Root Locus is constructed as follows.
The poles of the closed-loop system are drawn in the complex plane as the gain \(g\) varies from \(0\) to \(\infty\) for the two controllers simultaneously.
The closed-loop poles start at the open-loop poles (shown by \(\tikz[baseline=-0.6ex] \node[cross out, draw=black, minimum size=1ex, line width=2pt, inner sep=0pt, outer sep=0pt] at (0, 0){};\)) for \(g = 0\) and coincide with the transmission zeros (shown by \(\tikz[baseline=-0.6ex] \draw[line width=2pt, inner sep=0pt, outer sep=0pt] (0,0) circle[radius=3pt];\)) as \(g \to \infty\).
The direction of increasing gains is shown by the arrows \(\tikz[baseline=-0.6ex] \draw[-{Stealth[round]},line width=2pt] (0,0) -- (0.3,0);\).
\begin{figure}[htbp]
\centering
@ -328,38 +328,49 @@ Thus, for some finite IFF gain, one pole will have a positive real part and the
\caption{\label{fig:root_locus_pure_iff}Root Locus for the Decentralized Integral Force Feedback}
\end{figure}
At low frequency, the gain is very large and thus no force is transmitted between the payload and the rotating stage.
This means that at low frequency, the system is decoupled (the force sensor removed) and thus the system is unstable.
Whereas collocated IFF is known for its guaranteed stability, which is the case here for \(\Omega = 0\), this property is lost as soon as the rotational speed in non-null due to gyroscopic effects.
This can be seen in the Root Locus (Figure \ref{fig:root_locus_pure_iff}) where the pole corresponding to the controller is bounded to the right half plane implying closed-loop system instability.
Two system modifications are proposed in the next sections to deal with this stability problem.
Either the control law can be change (Section \ref{sec:iff_hpf}) or the mechanical system slightly modified (Section \ref{sec:iff_kp}).
\section{Integral Force Feedback with High Pass Filters}
\label{sec:org5533f47}
\label{sec:org6cc7c03}
\label{sec:iff_hpf}
\subsection{Modification of the Control Low}
\label{sec:orge3f4cc0}
\label{sec:org9575a34}
\begin{equation}
\label{eq:IFF_LHF}
K_{F}(s) = g \cdot \frac{1}{s} \cdot \underbrace{\frac{s/\omega_i}{1 + s/\omega_i}}_{\text{HPF}} = g \cdot \frac{1}{s + \omega_i}
\end{equation}
\subsection{Feedback Analysis}
\label{sec:orgd0fed6b}
\label{sec:orgbc7c7f2}
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/loop_gain_modified_iff.pdf}
\caption{\label{fig:loop_gain_modified_iff}Bode Plot of the Loop Gain for IFF with and without the HPF, \(\Omega = 0.1 \omega_0\)}
\caption{\label{fig:loop_gain_modified_iff}Bode Plot of the Loop Gain for IFF with and without the HPF with \(\omega_i = 0.1 \omega_0\), \(g = 2\) and \(\Omega = 0.1 \omega_0\)}
\end{figure}
As shown in Figure \ref{fig:root_locus_modified_iff}, the poles of the closed loop system are stable for \(g < g_\text{max}\)
\begin{equation}
g_\text{max} = \omega_i \left( \frac{{\omega_0}^2}{\Omega^2} - 1 \right) \label{eq:iff_gmax}
g_{\text{max}} = \omega_i \left( \frac{{\omega_0}^2}{\Omega^2} - 1 \right)
\end{equation}
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/root_locus_modified_iff.pdf}
\caption{\label{fig:root_locus_modified_iff}Root Locus for IFF with and without the HPF, \(\Omega = 0.1 \omega_0\)}
\end{figure}
\subsection{Optimal Cut-Off Frequency}
\label{sec:org4740973}
\subsection{Optimal Control Parameters}
\label{sec:org6e31c47}
Two parameters can be tuned for the controller \eqref{eq:IFF_LHF}, the gain \(g\) and the frequency of the pole \(\omega_i\).
Root Locus plots for several \(\omega_i\) are shown in Figure \ref{fig:root_locus_wi_modified_iff}.
\begin{figure}[htbp]
\centering
@ -367,6 +378,8 @@ This means that at low frequency, the system is decoupled (the force sensor remo
\caption{\label{fig:root_locus_wi_modified_iff}Root Locus for several HPF cut-off frequencies \(\omega_i\), \(\Omega = 0.1 \omega_0\)}
\end{figure}
The optimal values of \(\omega_i\) and \(g\) may be considered as the values for which the closed-loop poles are equally damped.
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/mod_iff_damping_wi.pdf}
@ -374,17 +387,32 @@ This means that at low frequency, the system is decoupled (the force sensor remo
\end{figure}
\section{Integral Force Feedback with Parallel Springs}
\label{sec:org1b53815}
\label{sec:org8681b34}
\label{sec:iff_kp}
\subsection{Stiffness in Parallel with the Force Sensor}
\label{sec:org3a8c426}
\label{sec:org0dff3a7}
Stiffness can be added in parallel to the force sensor to counteract the negative stiffness due to centrifugal forces.
If the added stiffness is higher than the maximum negative stiffness, then the poles of the IFF damped system will stay in the (stable) right half-plane.
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/system_parallel_springs.pdf}
\caption{\label{fig:system_parallel_springs}System with added springs in parallel with the actuators}
\label{fig:system_parallel_springs}
\end{figure}
The forces measured by the force sensors are equal to:
\begin{equation}
\label{eq:measured_force_kp}
\begin{bmatrix} f_{u} \\ f_{v} \end{bmatrix} =
\begin{bmatrix} F_u \\ F_v \end{bmatrix} - (c s + k_a)
\begin{bmatrix} d_u \\ d_v \end{bmatrix}
\end{equation}
This could represent a system where
\subsection{Plant Dynamics}
\label{sec:orgf26a6f4}
\label{sec:orgc00a18e}
We define an adimensional parameter \(\alpha\), \(0 \le \alpha < 1\), that describes the proportion of the stiffness in parallel with the actuator and force sensor:
\begin{subequations}
@ -416,7 +444,7 @@ The overall stiffness \(k\) stays constant:
\end{equation}
\subsection{Effect of the Parallel Stiffness on the Plant Dynamics}
\label{sec:org6a55282}
\label{sec:orgdbd366f}
\begin{equation}
\begin{aligned}
\alpha > \frac{\Omega^2}{{\omega_0}^2} \\
@ -437,7 +465,7 @@ The overall stiffness \(k\) stays constant:
\end{figure}
\subsection{Optimal Parallel Stiffness}
\label{sec:org358dd73}
\label{sec:org8a2fab9}
\begin{figure}[htbp]
\begin{subfigure}[c]{0.49\linewidth}
\includegraphics[width=\linewidth]{figs/root_locus_iff_kps.pdf}
@ -452,14 +480,15 @@ The overall stiffness \(k\) stays constant:
\end{figure}
\section{Comparison of the Proposed Active Damping Techniques for Rotating Positioning Stages}
\label{sec:org3cc6699}
\label{sec:orgf0c901c}
\label{sec:comparison}
\subsection{Physical Comparison}
\label{sec:orgc34b986}
\label{sec:orgb81fedd}
\subsection{Attainable Damping}
\label{sec:org993a1d7}
\label{sec:org9975e71}
\begin{figure}[htbp]
\centering
@ -467,11 +496,8 @@ The overall stiffness \(k\) stays constant:
\caption{\label{fig:comp_root_locus}Root Locus for the three proposed decentralized active damping techniques: IFF with HFP, IFF with parallel springs, and relative DVF, \(\Omega = 0.1 \omega_0\)}
\end{figure}
\subsection{Transmissibility and Compliance}
\label{sec:org0674052}
\label{sec:org0a87dc5}
\begin{figure}[htbp]
\begin{subfigure}[c]{0.45\linewidth}
\includegraphics[width=\linewidth]{figs/comp_compliance.pdf}
@ -486,11 +512,11 @@ The overall stiffness \(k\) stays constant:
\end{figure}
\section{Conclusion}
\label{sec:orgba18ca5}
\label{sec:orgcf2c965}
\label{sec:conclusion}
\section*{Acknowledgment}
\label{sec:org4c68ce2}
\label{sec:orga6bcde3}
\bibliography{ref.bib}
\end{document}