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Thomas Dehaeze
2020-06-25 18:28:07 +02:00
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@@ -249,7 +249,7 @@ Stiff positioning platforms should be used if high rotational speeds or heavy pa
| <<fig:campbell_diagram_real>> Real Part | <<fig:campbell_diagram_imag>> Imaginary Part |
* Decentralized Integral Force Feedback
** Control Schematic
** System Schematic and Control Architecture
Force Sensors are added in series with the actuators as shown in Figure [[fig:system_iff]].
@@ -332,7 +332,7 @@ It increase with the rotational speed $\Omega$.
** Decentralized Integral Force Feedback
\begin{equation}
\bm{K}_F(s) = g \cdot \frac{1}{s}
K_F(s) = g \cdot \frac{1}{s}
\end{equation}
# Problem of zero with a positive real part
@@ -341,10 +341,9 @@ This is due to the fact that the zeros of the plant are the poles of the closed
Thus, for some finite IFF gain, one pole will have a positive real part and the system will be unstable.
# 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
#+name: fig:root_locus_pure_iff
#+caption: Root Locus for the Decentralized Integral Force Feedback
@@ -370,7 +369,7 @@ This means that at low frequency, the system is decoupled (the force sensor remo
# Equation with the new control law
\begin{equation}
\bm{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}
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}
@@ -429,19 +428,32 @@ This means that at low frequency, the system is decoupled (the force sensor remo
* Integral Force Feedback with Parallel Springs
** Stiffness in Parallel with the Force Sensor
# Zeros = remove force sensor
# We want to have stable zeros => add stiffnesses in parallel
#+name: fig:system_parallel_springs
#+caption: System with added springs $k_p$ in parallel with the actuators
#+caption: System with added springs in parallel with the actuators
#+attr_latex: :scale 1
[[file:figs/system_parallel_springs.pdf]]
# Maybe add the fact that this is equivalent to amplified piezo for instance
# Add reference to cite:souleille18_concep_activ_mount_space_applic
** Plant Dynamics
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}
\begin{align}
k_p &= \alpha k \\
k_a &= (1 - \alpha) k
\end{align}
\end{subequations}
The overall stiffness $k$ stays constant:
\begin{equation}
k = k_a + k_p
\end{equation}
# Equations: sensed force
\begin{equation}
\begin{bmatrix} f_u \\ f_v \end{bmatrix} =
@@ -459,30 +471,27 @@ This means that at low frequency, the system is decoupled (the force sensor remo
\begin{bmatrix} F_u \\ F_v \end{bmatrix}
\end{equation}
With:
\begin{align}
G_{kp} &= \left( \frac{s^2}{{\omega_0^\prime}^2} + 2\xi^\prime \frac{s}{{\omega_0^\prime}^2} + 1 - \frac{\Omega^2}{{\omega_0^\prime}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0^\prime}\frac{s}{\omega_0^\prime} \right)^2 \\
G_{kz} &= \left( \frac{s^2}{{\omega_0^\prime}^2} + \frac{k_p}{k + k_p} - \frac{\Omega^2}{{\omega_0^\prime}^2} \right) \left( \frac{s^2}{{\omega_0^\prime}^2} + 2\xi^\prime \frac{s}{{\omega_0^\prime}^2} + 1 - \frac{\Omega^2}{{\omega_0^\prime}^2} \right) + \left( 2 \frac{\Omega}{\omega_0^\prime}\frac{s}{\omega_0^\prime} \right)^2 \\
G_{kc} &= \left( 2 \xi^\prime \frac{s}{\omega_0^\prime} + \frac{k}{k + k_p} \right) \left( 2 \frac{\Omega}{\omega_0^\prime}\frac{s}{\omega_0^\prime} \right)
\end{align}
\begin{subequations}
\begin{align}
G_{kp} &= \left( \frac{s^2}{{\omega_0}^2} + 2\xi \frac{s}{\omega_0} + 1 - \frac{\Omega^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0}\frac{s}{\omega_0} \right)^2 \\
G_{kz} &= \left( \frac{s^2}{{\omega_0}^2} - \frac{\Omega^2}{{\omega_0}^2} + \alpha \right) \left( \frac{s^2}{{\omega_0}^2} + 2\xi \frac{s}{\omega_0} + 1 - \frac{\Omega^2}{{\omega_0}^2} \right) + \left( 2 \frac{\Omega}{\omega_0}\frac{s}{\omega_0} \right)^2 \\
G_{kc} &= \left( 2 \xi \frac{s}{\omega_0} + 1 - \alpha \right) \left( 2 \frac{\Omega}{\omega_0}\frac{s}{\omega_0} \right)
\end{align}
\end{subequations}
# New parameters
where:
- $\omega_0^\prime = \frac{k + k_p}{m}$
- $\xi^\prime = \frac{c}{2 \sqrt{(k + k_p) m}}$
# News terms with \alpha are added
# w0 and xi are the same as before => only the zeros are changing and not the poles.
** Effect of the Parallel Stiffness on the Plant Dynamics
# Negative Stiffness due to rotation => the stiffness should be larger than that
# TODO: Verify that
# For kp < negative stiffness => real zeros
# For kp > negative stiffness => complex conjugate zeros
# For kp < negative stiffness => real zeros => non-minimum phase
# For kp > negative stiffness => complex conjugate zeros => minimum phase
\begin{equation}
\frac{k_p}{k + k_p} - \frac{\Omega^2}{{\omega_0^\prime}^2} > 0
\end{equation}
Which is equivalent to
\begin{equation}
k_p > m \Omega^2
\begin{align}
\alpha > \frac{\Omega^2}{{\omega_0}^2} \\
\Leftrightarrow k_p > m \Omega^2
\end{align}
\end{equation}
#+name: fig:plant_iff_kp
@@ -490,7 +499,8 @@ Which is equivalent to
#+attr_latex: :scale 1
[[file:figs/plant_iff_kp.pdf]]
# Location of the zeros as a function of kp
# Location of the zeros as a function of kp => maybe to complex
# Do we talk about siso zeros of mimo (transmission zeros)?
# Try to show that we don't have anymore real zeros that was making the system non-minimum phase
@@ -504,36 +514,31 @@ Which is equivalent to
# For kp > m Omega => unconditionally stable
** Optimal Parallel Stiffness
# Explain that we have k = ka + kp constant in order to have the same resonance
# Attainable damping generally proportional to the distance between the poles and zeros (add reference, probably preumont)
# The zero is the poles of the system without the force sensors => w =
# The zero is the poles of the system without the force sensors => w0 = sqrt(kp/m) +/- Omega ?? => seems not true
# Thus, small kp is wanted: kp close to m Omega^2 should give the optimal damping but is not acceptable for robustness reasons
# Large Stiffness decreases the attainable damping
#+name: fig:root_locus_iff_kps
#+caption: Root Locus for IFF for several parallel spring stiffnesses $k_p$
#+attr_latex: :scale 1
[[file:figs/root_locus_iff_kps.pdf]]
# Example with kp = 5 m Omega
#+name: fig:root_locus_opt_gain_iff_kp
#+caption: Root Locus for IFF with $k_p = 5 m \Omega^2$. The poles of the system using the gain that yields the maximum damping ratio are shown by black crosses
#+attr_latex: :scale 1
[[file:figs/root_locus_opt_gain_iff_kp.pdf]]
#+name: fig:root_locus_iff_kps_opt
#+caption: Root Locus for IFF when parallel stiffness is used
#+attr_latex: :environment subfigure :width 0.49\linewidth :align c
| file:figs/root_locus_iff_kps.pdf | file:figs/root_locus_opt_gain_iff_kp.pdf |
| <<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 |
* Direct Velocity Feedback
** Control Schematic
** System Schematic and Control Architecture
# Basic Idea of DVF
# Equation with the control law
# Equation with the control law: pure gain
\begin{equation}
K_V(s) = g
\end{equation}
#+name: fig:system_dvf
#+caption: System with relative velocity sensors and with decentralized controllers $K_V$
@@ -556,7 +561,7 @@ Which is equivalent to
\begin{equation}
\begin{bmatrix} v_u \\ v_v \end{bmatrix} =
\frac{s}{k} \frac{1}{G_{vp}}
\frac{1}{k} \frac{1}{G_{vp}}
\begin{bmatrix}
G_{vz} & G_{vc} \\
-G_{vc} & G_{vz}
@@ -564,11 +569,13 @@ Which is equivalent to
\begin{bmatrix} F_u \\ F_v \end{bmatrix}
\end{equation}
With:
\begin{align}
G_{vp} &= \left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2 \\
G_{vz} &= \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \\
G_{vc} &= 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0}
\end{align}
\begin{subequations}
\begin{align}
G_{vp} &= \left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2 \\
G_{vz} &= s \left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right) \\
G_{vc} &= 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0}
\end{align}
\end{subequations}
# Show that the rotation have somehow less impact on the plant than for IFF

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@@ -1,4 +1,4 @@
% Created 2020-06-25 jeu. 10:07
% Created 2020-06-25 jeu. 17:24
% Intended LaTeX compiler: pdflatex
\documentclass{ISMA_USD2020}
\usepackage[utf8]{inputenc}
@@ -53,18 +53,20 @@
}
\section{Introduction}
\label{sec:org5780a8f}
\label{sec:org7ff5c70}
\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.
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:orga8db619}
\label{sec:org029ca48}
\subsection{Studied Rotating Positioning Platform}
\label{sec:org70ddefe}
\label{sec:org7eec6c5}
Consider the rotating X-Y stage of Figure \ref{fig:system}.
\begin{itemize}
@@ -82,7 +84,7 @@ Consider the rotating X-Y stage of Figure \ref{fig:system}.
\end{figure}
\subsection{Equations of Motion}
\label{sec:org647b64d}
\label{sec:orgca181a2}
The system has two degrees of freedom and is thus fully described by the generalized coordinates \([q_1\ q_2] = [d_u\ d_v]\) (describing the position of the mass in the rotating frame).
Let's express the kinetic energy \(T\), the potential energy \(V\) of the mass \(m\) (neglecting the rotational energy) as well as the deceptive function \(R\):
@@ -117,7 +119,7 @@ The Gyroscopic effects can be seen from the two following terms:
\end{itemize}
\subsection{Transfer Functions in the Laplace domain}
\label{sec:org55c9228}
\label{sec:orge7e184a}
Using the Laplace transformation on the equations of motion \eqref{eq:eom_coupled}, the transfer functions from \([F_u,\ F_v]\) to \([d_u,\ d_v]\) are obtained:
\begin{subequations}
@@ -139,7 +141,7 @@ Without rotation \(\Omega = 0\) and the system corresponds to two uncoupled one
\end{subequations}
\subsection{Change of Variables / Parameters for the study}
\label{sec:orgb7d090c}
\label{sec:org3cdb1ab}
In order this study is more independent on the system parameters, the following change of variable is performed:
\begin{itemize}
@@ -171,8 +173,13 @@ With:
\(G_{dp}\) describes to poles of the system, \(G_{dz}\) the zeros of the diagonal terms and \(G_{dc}\) the coupling.
\begin{itemize}
\item \(k = \SI{1}{N/m}\), \(m = \SI{1}{kg}\), \(c = \SI{0.05}{\newton\per\meter\second}\)
\item \(\omega_0 = \SI{1}{\radian\per\second}\), \(\xi = 0.025\)
\end{itemize}
\subsection{System Dynamics and Campbell Diagram}
\label{sec:org24f5f5f}
\label{sec:org42dee20}
The bode plot of \(\bm{G}_d\) is shown in Figure \ref{fig:plant_compare_rotating_speed}.
\begin{figure}[htbp]
@@ -189,6 +196,14 @@ The bode plot of \(\bm{G}_d\) is shown in Figure \ref{fig:plant_compare_rotating
\end{figure}
The poles are the roots of \(G_{dp}\).
Two pairs of complex conjugate poles (supposing small damping \(\xi \approx 0\)):
\begin{subequations}
\begin{align}
p_1 &= \pm j (\omega_0 - \Omega) \\
p_2 &= \pm j (\omega_0 + \Omega)
\end{align}
\end{subequations}
When the rotation speed in non-null, the resonance frequency is split into two pairs of complex conjugate poles.
As the rotation speed increases, one of the two resonant frequency goes to lower frequencies as the other one goes to higher frequencies.
@@ -213,9 +228,9 @@ Stiff positioning platforms should be used if high rotational speeds or heavy pa
\end{figure}
\section{Decentralized Integral Force Feedback}
\label{sec:orgd957fd6}
\subsection{Control Schematic}
\label{sec:orgc01d8cf}
\label{sec:orgda38ad7}
\subsection{System Schematic and Control Architecture}
\label{sec:orgf08eb9d}
Force Sensors are added in series with the actuators as shown in Figure \ref{fig:system_iff}.
@@ -225,8 +240,8 @@ Force Sensors are added in series with the actuators as shown in Figure \ref{fig
\caption{\label{fig:system_iff}System with Force Sensors in Series with the Actuators. Decentralized Integral Force Feedback is used}
\end{figure}
\subsection{Equations}
\label{sec:orge5896ec}
\subsection{Plant Dynamics}
\label{sec:orgf2f22c2}
The forces measured by the force sensors are equal to:
\begin{equation}
\label{eq:measured_force}
@@ -235,11 +250,12 @@ The forces measured by the force sensors are equal to:
\begin{bmatrix} d_u \\ d_v \end{bmatrix}
\end{equation}
Reinjecting \eqref{eq:tf_d} into \eqref{eq:measured_force} yields:
Re-injecting \eqref{eq:tf_d} into \eqref{eq:measured_force} yields:
\begin{equation}
\label{eq:tf_f}
\begin{bmatrix} f_{u} \\ f_{v} \end{bmatrix} = \bm{G}_{f} \begin{bmatrix} F_u \\ F_v \end{bmatrix}
\end{equation}
Where \(\bm{G}_f\) is a \(2 \times 2\) transfer function matrix.
\begin{equation}
\bm{G}_f =
@@ -249,24 +265,38 @@ Reinjecting \eqref{eq:tf_d} into \eqref{eq:measured_force} yields:
G_{fc} & G_{fz}
\end{bmatrix}
\end{equation}
with:
\begin{align}
G_{fp} &= \left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2 \\
G_{fz} &= \left( \frac{s^2}{{\omega_0}^2} - \frac{\Omega^2}{{\omega_0}^2} \right) \left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right) + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2 \\
G_{fc} &= \left( 2 \xi \frac{s}{\omega_0} + 1 \right) \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)
\end{align}
The zeros of the diagonal terms are the roots of \(G_{fz}\) (supposing small damping):
\begin{subequations}
\begin{align}
z_1 &= \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} } \\
z_2 &= \pm \omega_0 \sqrt{\frac{1}{2} \sqrt{8 \frac{\Omega^2}{{\omega_0}^2} + 1} - \frac{\Omega^2}{{\omega_0}^2} - \frac{1}{2} }
\end{align}
\end{subequations}
The frequency of the two complex conjugate zeros \(z_1\) is between the frequency of the two pairs of complex conjugate poles \(p_1\) and \(p_2\).
This is the expected behavior of a collocated pair of actuator and sensor.
However, the two real zeros \(z_2\) 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.
The low frequency gain, for \(\Omega < \omega_0\), is no longer zero:
\begin{equation}
\bm{G}_f =
\frac{1}{\left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2}
\begin{bmatrix}
\left( \frac{s^2}{{\omega_0}^2} - \frac{\Omega^2}{{\omega_0}^2} \right) \left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right) + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2 & -G_{fc} \\
G_{fc} & \left( \frac{s^2}{{\omega_0}^2} - \frac{\Omega^2}{{\omega_0}^2} \right) \left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right) + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2
\label{low_freq_gain_iff_plan}
\bm{G}_{f0} = \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}
\subsection{Plant Dynamics}
\label{sec:org0a22a10}
It increase with the rotational speed \(\Omega\).
\begin{figure}[htbp]
\centering
@@ -274,8 +304,16 @@ Reinjecting \eqref{eq:tf_d} into \eqref{eq:measured_force} yields:
\caption{\label{fig:plant_iff_compare_rotating_speed}Bode plot of \(\bm{G}_f\) for several rotational speeds \(\Omega\)}
\end{figure}
\subsection{Problems with Integral Force Feedback}
\label{sec:orgd432439}
\subsection{Decentralized Integral Force Feedback}
\label{sec:orge1a14b4}
\begin{equation}
\bm{K}_F(s) = g \cdot \frac{1}{s}
\end{equation}
Also, as one zero has a positive real part, the \textbf{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.
\begin{figure}[htbp]
\centering
@@ -287,23 +325,26 @@ At low frequency, the gain is very large and thus no force is transmitted betwee
This means that at low frequency, the system is decoupled (the force sensor removed) and thus the system is unstable.
\section{Integral Force Feedback with High Pass Filters}
\label{sec:org2e1883a}
\label{sec:org7f551f8}
\subsection{Modification of the Control Low}
\label{sec:org218110f}
\label{sec:org4c39c6d}
\begin{equation}
\bm{K}_{F}(s) = \frac{1}{s} \underbrace{\frac{s/\omega_i}{1 + s/\omega_i}}_{\text{HPF}} = \frac{1}{s + \omega_i}
\bm{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:org03090fc}
\label{sec:org2673698}
\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}
\end{figure}
\begin{equation}
g_\text{max} = \omega_i \left( \frac{{\omega_0}^2}{\Omega^2} - 1 \right) \label{eq:iff_gmax}
\end{equation}
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/root_locus_modified_iff.pdf}
@@ -311,7 +352,7 @@ This means that at low frequency, the system is decoupled (the force sensor remo
\end{figure}
\subsection{Optimal Cut-Off Frequency}
\label{sec:org6ba4e55}
\label{sec:org7d8a789}
\begin{figure}[htbp]
\centering
@@ -326,16 +367,31 @@ 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:org90ec20f}
\label{sec:org9bc19d0}
\subsection{Stiffness in Parallel with the Force Sensor}
\label{sec:org60d6640}
\label{sec:orgdfd59fa}
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/system_parallel_springs.pdf}
\caption{\label{fig:system_parallel_springs}System with added springs \(k_p\) in parallel with the actuators}
\caption{\label{fig:system_parallel_springs}System with added springs in parallel with the actuators}
\end{figure}
\subsection{Plant Dynamics}
\label{sec:org70fc8fa}
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}
\begin{align}
k_p &= \alpha k \\
k_a &= (1 - \alpha) k
\end{align}
\end{subequations}
The overall stiffness \(k\) stays constant:
\begin{equation}
k = k_a + k_p
\end{equation}
\begin{equation}
\begin{bmatrix} f_u \\ f_v \end{bmatrix} =
\bm{G}_k
@@ -352,27 +408,21 @@ This means that at low frequency, the system is decoupled (the force sensor remo
\begin{bmatrix} F_u \\ F_v \end{bmatrix}
\end{equation}
With:
\begin{align}
G_{kp} &= \left( \frac{s^2}{{\omega_0^\prime}^2} + 2\xi^\prime \frac{s}{{\omega_0^\prime}^2} + 1 - \frac{\Omega^2}{{\omega_0^\prime}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0^\prime}\frac{s}{\omega_0^\prime} \right)^2 \\
G_{kz} &= \left( \frac{s^2}{{\omega_0^\prime}^2} + \frac{k_p}{k + k_p} - \frac{\Omega^2}{{\omega_0^\prime}^2} \right) \left( \frac{s^2}{{\omega_0^\prime}^2} + 2\xi^\prime \frac{s}{{\omega_0^\prime}^2} + 1 - \frac{\Omega^2}{{\omega_0^\prime}^2} \right) + \left( 2 \frac{\Omega}{\omega_0^\prime}\frac{s}{\omega_0^\prime} \right)^2 \\
G_{kc} &= \left( 2 \xi^\prime \frac{s}{\omega_0^\prime} + \frac{k}{k + k_p} \right) \left( 2 \frac{\Omega}{\omega_0^\prime}\frac{s}{\omega_0^\prime} \right)
\end{align}
where:
\begin{itemize}
\item \(\omega_0^\prime = \frac{k + k_p}{m}\)
\item \(\xi^\prime = \frac{c}{2 \sqrt{(k + k_p) m}}\)
\end{itemize}
\begin{subequations}
\begin{align}
G_{kp} &= \left( \frac{s^2}{{\omega_0}^2} + 2\xi \frac{s}{\omega_0} + 1 - \frac{\Omega^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0}\frac{s}{\omega_0} \right)^2 \\
G_{kz} &= \left( \frac{s^2}{{\omega_0}^2} - \frac{\Omega^2}{{\omega_0}^2} + \alpha \right) \left( \frac{s^2}{{\omega_0}^2} + 2\xi \frac{s}{\omega_0} + 1 - \frac{\Omega^2}{{\omega_0}^2} \right) + \left( 2 \frac{\Omega}{\omega_0}\frac{s}{\omega_0} \right)^2 \\
G_{kc} &= \left( 2 \xi \frac{s}{\omega_0} + 1 - \alpha \right) \left( 2 \frac{\Omega}{\omega_0}\frac{s}{\omega_0} \right)
\end{align}
\end{subequations}
\subsection{Effect of the Parallel Stiffness on the Plant Dynamics}
\label{sec:org3ec34fe}
\label{sec:orge20adc9}
\begin{equation}
\frac{k_p}{k + k_p} - \frac{\Omega^2}{{\omega_0^\prime}^2} > 0
\end{equation}
Which is equivalent to
\begin{equation}
k_p > m \Omega^2
\begin{align}
\alpha > \frac{\Omega^2}{{\omega_0}^2} \\
\Leftrightarrow k_p > m \Omega^2
\end{align}
\end{equation}
\begin{figure}[htbp]
@@ -388,26 +438,25 @@ Which is equivalent to
\end{figure}
\subsection{Optimal Parallel Stiffness}
\label{sec:org9c47159}
\label{sec:orgb14b5c2}
\begin{figure}[htbp]
\begin{subfigure}[c]{0.49\linewidth}
\includegraphics[width=\linewidth]{figs/root_locus_iff_kps.pdf}
\caption{\label{fig:root_locus_iff_kps} Three values of \(k_p\)}
\end{subfigure}
\begin{subfigure}[c]{0.49\linewidth}
\includegraphics[width=\linewidth]{figs/root_locus_opt_gain_iff_kp.pdf}
\caption{\label{fig:root_locus_opt_gain_iff_kp} \(k_p = 5 m \Omega^2\), optimal damping is shown}
\end{subfigure}
\caption{\label{fig:root_locus_iff_kps_opt}Root Locus for IFF when parallel stiffness is used}
\centering
\includegraphics[scale=1]{figs/root_locus_iff_kps.pdf}
\caption{\label{fig:root_locus_iff_kps}Root Locus for IFF for several parallel spring stiffnesses \(k_p\)}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/root_locus_opt_gain_iff_kp.pdf}
\caption{\label{fig:root_locus_opt_gain_iff_kp}Root Locus for IFF with \(k_p = 5 m \Omega^2\). The poles of the system using the gain that yields the maximum damping ratio are shown by black crosses}
\end{figure}
\section{Direct Velocity Feedback}
\label{sec:org5cb3076}
\subsection{Control Schematic}
\label{sec:orgaaa522f}
\label{sec:org7628683}
\subsection{System Schematic and Control Architecture}
\label{sec:org6953bc7}
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/system_dvf.pdf}
@@ -415,7 +464,7 @@ Which is equivalent to
\end{figure}
\subsection{Equations}
\label{sec:orge0a4555}
\label{sec:orgfab42cd}
\begin{equation}
\begin{bmatrix} v_u \\ v_v \end{bmatrix} =
@@ -425,7 +474,7 @@ Which is equivalent to
\begin{equation}
\begin{bmatrix} v_u \\ v_v \end{bmatrix} =
\frac{s}{k} \frac{1}{G_{vp}}
\frac{1}{k} \frac{1}{G_{vp}}
\begin{bmatrix}
G_{vz} & G_{vc} \\
-G_{vc} & G_{vz}
@@ -433,15 +482,17 @@ Which is equivalent to
\begin{bmatrix} F_u \\ F_v \end{bmatrix}
\end{equation}
With:
\begin{align}
G_{vp} &= \left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2 \\
G_{vz} &= \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \\
G_{vc} &= 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0}
\end{align}
\begin{subequations}
\begin{align}
G_{vp} &= \left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2 \\
G_{vz} &= s \left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right) \\
G_{vc} &= 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0}
\end{align}
\end{subequations}
\subsection{Relative Direct Velocity Feedback}
\label{sec:org5401110}
\label{sec:org5cf0d6b}
\begin{figure}[htbp]
\centering
@@ -450,14 +501,14 @@ With:
\end{figure}
\section{Comparison of the Proposed Active Damping Techniques for Rotating Positioning Stages}
\label{sec:org4cbf163}
\label{sec:orga2b60a8}
\subsection{Physical Comparison}
\label{sec:org5eba275}
\label{sec:orgb69316c}
\subsection{Attainable Damping}
\label{sec:org44635e0}
\label{sec:org5c23e13}
\begin{figure}[htbp]
\centering
@@ -467,7 +518,7 @@ With:
\subsection{Transmissibility and Compliance}
\label{sec:org58e9594}
\label{sec:org723fb63}
\begin{figure}[htbp]
@@ -484,11 +535,11 @@ With:
\end{figure}
\section{Conclusion}
\label{sec:org292b448}
\label{sec:orgd52b568}
\label{sec:conclusion}
\section*{Acknowledgment}
\label{sec:orgff7af07}
\label{sec:orge4d73e9}
\bibliography{ref.bib}
\end{document}