Reworked the TF matrices / removed DVF
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paper/paper.org
@ -1,4 +1,4 @@
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#+TITLE: Decentralized Active Damping of Rotating Positioning Platforms
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#+TITLE: Active Damping of Rotating Positioning Platforms using Force Feedback
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:DRAWER:
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#+LATEX_CLASS: ISMA_USD2020
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#+OPTIONS: toc:nil
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@ -86,58 +86,54 @@ This paper has been published
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The Matlab code that was use to obtain the results are available in cite:dehaeze20_activ_dampin_rotat_posit_platf.
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* Dynamics of Rotating Positioning Platforms
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** Studied Rotating Positioning Platform
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# Introduce the fact that we need a simple system representing the rotating aspect.
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** Model of a Rotating Positioning Platform
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# Introduce the fact that we need a simple system representing the rotating aspect
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To study how the rotation of positioning platforms does affect the use of force feedback, a simple model is developed.
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# Simplest system where gyroscopic forces can be studied
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Consider the rotating X-Y stage of Figure [[fig:system]].
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# Present the system, parameters, assumptions
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It represents an X-Y positioning stage on top of a Rotating Stage and is schematically represented in Figure [[fig:system]].
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# Explain the frames (inertial frame x,y, rotating frame u,v)
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# iu, iv is linked to the rotating stage and supposed to be perfect
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Two frames of reference are used:
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- $(\vec{i}_x, \vec{i}_y, \vec{i}_z)$ is an inertial frame
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- $(\vec{i}_u, \vec{i}_v, \vec{i}_w)$ is a frame fixed on the Rotating Stage with its origin align with the rotation axis
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# Small displacements
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# Present the system, parameters, assumptions (small displacements, perfect spindle)
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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 $\si{\radian\per\second}$.
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# Constant rotational speed
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# X-Y Stage
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The parallel X-Y positioning stage consists of two orthogonal actuators represented by the three following elements in parallel:
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- A spring with a stiffness $k$ in $\si{\newton\per\meter}$
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- A dashpot with a damping coefficient $c$ in $\si{\newton\per\meter\second}$
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- An ideal force source $F_u, F_v$ in $\si{\newton}$
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- $k$: Actuator's Stiffness [N/m]
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- $m$: Payload's mass [kg]
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- $\Omega = \dot{\theta}$: rotation speed [rad/s]
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- $F_u$, $F_v$
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- $d_u$, $d_v$
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# Payload
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The X-Y stage is supporting a payload with a payload with a mass $m$ in $\si{\kilo\gram}$.
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The position of the payload is represented by $(d_u, d_v)$ expressed in the rotating frame $(\vec{i}_u, \vec{i}_v)$.
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#+name: fig:system
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#+caption: Schematic of the studied System
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#+attr_latex: :scale 1
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[[file:figs/system.pdf]]
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# #+name: fig:cedrat_xy25xs
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# #+caption: Figure caption
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# #+attr_latex: :width 0.5\linewidth
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# [[file:figs/cedrat_xy25xs.jpg]]
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** Equations of Motion
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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).
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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 $R$:
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To obtain of equation of motion for the system represented in Figure [[fig:system]], the Lagrangian equations are used:
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#+name: eq:lagrangian_equations
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\begin{equation}
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\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
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\end{equation}
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with $L = T - V$ is the Lagrangian, $D$ is the dissipation function, and $Q_i$ is the generalized force associated with the generalized variable $[q_1\ q_2] = [d_u\ d_v]$:
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#+name: eq:energy_functions_lagrange
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\begin{subequations}
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\begin{align}
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T & = \frac{1}{2} m \left( \left( \dot{d}_u - \Omega d_v \right)^2 + \left( \dot{d}_v + \Omega d_u \right)^2 \right) \\
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V & = \frac{1}{2} k \left( {d_u}^2 + {d_v}^2 \right) \\
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R & = \frac{1}{2} c \left( \dot{d}_u{}^2 + \dot{d}_v{}^2 \right)
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D & = \frac{1}{2} c \left( \dot{d}_u{}^2 + \dot{d}_v{}^2 \right) \\
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Q_1 &= F_u, \quad Q_2 = F_v
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\end{align}
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\end{subequations}
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The equations of motion are derived from the Lagrangian equation:
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#+name: eq:lagrangian_equations
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\begin{equation}
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\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
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\end{equation}
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with $L = T - V$ is the Lagrangian and $Q_i$ is the generalized force associated with the generalized variable $q_i$ ($Q_1 = F_u$ and $Q_2 = F_v$).
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Substituting equations eqref:eq:energy_functions_lagrange into eqref:eq:lagrangian_equations gives the two coupled differential equations:
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#+name: eq:eom_coupled
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\begin{subequations}
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\begin{align}
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@ -147,14 +143,12 @@ with $L = T - V$ is the Lagrangian and $Q_i$ is the generalized force associated
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\end{subequations}
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# Explain Gyroscopic effects
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The Gyroscopic effects can be seen from the two following terms:
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- Coriolis Forces: coupling
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- Centrifugal forces: negative stiffness
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The rotation of the XY positioning platform induces two Gyroscopic effects:
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- Coriolis Forces: that adds coupling between the two orthogonal controlled directions
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- Centrifugal forces: that can been seen as negative stiffness
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** Transfer Functions in the Laplace domain
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# Laplace Domain
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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:
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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:
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#+name: eq:oem_laplace_domain
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\begin{subequations}
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\begin{align}
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@ -163,8 +157,7 @@ Using the Laplace transformation on the equations of motion eqref:eq:eom_coupled
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\end{align}
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\end{subequations}
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Without rotation $\Omega = 0$ and the system corresponds to two uncoupled one degree of freedom mass-spring-damper systems:
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One can verify that without rotation ($\Omega = 0$) the system becomes equivalent as to two uncoupled one degree of freedom mass-spring-damper systems:
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#+name: eq:oem_no_rotation
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\begin{subequations}
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\begin{align}
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@ -176,33 +169,24 @@ Without rotation $\Omega = 0$ and the system corresponds to two uncoupled one de
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** Change of Variables / Parameters for the study
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# Change of variables
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In order this study is more independent on the system parameters, the following change of variable is performed:
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In order to make this study less dependent on the system parameters, the following change of variable is performed:
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- $\omega_0 = \sqrt{\frac{k}{m}}$: Natural frequency of the mass-spring system in $\si{\radian/\s}$
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- $\xi = \frac{c}{2 \sqrt{k m}}$: Damping ratio
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#+name: eq:tf_d
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\begin{equation}
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\begin{bmatrix} d_u \\ d_v \end{bmatrix} = \bm{G}_d \begin{bmatrix} F_u \\ F_v \end{bmatrix}
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\end{equation}
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Where $\bm{G}_d$ is a $2 \times 2$ transfer function matrix.
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#+name: eq:tf_d
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\begin{equation}
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\bm{G}_d = \frac{1}{k} \frac{1}{G_{dp}}
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\begin{bmatrix}
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G_{dz} & G_{dc} \\
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-G_{dc} & G_{dz}
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\end{bmatrix}
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\bm{G}_{d} =
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\frac{1}{k}
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\begin{bmatrix}
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\frac{\frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2}}{\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} & \frac{2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0}}{\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} \\
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\frac{- 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0}}{\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} & \frac{\frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2}}{\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}
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\end{bmatrix}
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\end{equation}
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With:
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\begin{subequations}
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\begin{align}
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G_{dp} &= \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 \\
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G_{dz} &= \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \\
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G_{dc} &= 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0}
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\end{align}
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\end{subequations}
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$G_{dp}$ describes to poles of the system, $G_{dz}$ the zeros of the diagonal terms and $G_{dc}$ the coupling.
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# Parameters
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- $k = \SI{1}{N/m}$, $m = \SI{1}{kg}$, $c = \SI{0.05}{\newton\per\meter\second}$
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@ -277,19 +261,11 @@ Re-injecting eqref:eq:tf_d into eqref:eq:measured_force yields:
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Where $\bm{G}_f$ is a $2 \times 2$ transfer function matrix.
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\begin{equation}
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\bm{G}_f =
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\frac{1}{G_{fp}}
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\begin{bmatrix}
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G_{fz} & -G_{fc} \\
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G_{fc} & G_{fz}
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\bm{G}_{f} = \begin{bmatrix}
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\frac{\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}{\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} & \frac{- \left( 2 \xi \frac{s}{\omega_0} + 1 \right) \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)}{\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} \\
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\frac{\left( 2 \xi \frac{s}{\omega_0} + 1 \right) \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)}{\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} & \frac{\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}{\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}
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\end{bmatrix}
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\end{equation}
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with:
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\begin{align}
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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 \\
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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 \\
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G_{fc} &= \left( 2 \xi \frac{s}{\omega_0} + 1 \right) \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)
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\end{align}
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# Explain the two real zeros => change of gain but not of phase
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# The alternating poles and zeros properties of collocated IFF holds
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@ -462,22 +438,12 @@ The overall stiffness $k$ stays constant:
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\end{equation}
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\begin{equation}
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\begin{bmatrix} f_u \\ f_v \end{bmatrix} =
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\frac{1}{G_{kp}}
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\bm{G}_k =
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\begin{bmatrix}
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G_{kz} & -G_{kc} \\
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G_{kc} & G_{kz}
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\frac{\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}{\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} & \frac{- \left( 2 \xi \frac{s}{\omega_0} + 1 - \alpha \right) \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)}{\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} \\
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\frac{\left( 2 \xi \frac{s}{\omega_0} + 1 - \alpha \right) \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)}{\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} & \frac{\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}{\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}
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\end{bmatrix}
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\begin{bmatrix} F_u \\ F_v \end{bmatrix}
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\end{equation}
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With:
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\begin{subequations}
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\begin{align}
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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 \\
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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 \\
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G_{kc} &= \left( 2 \xi \frac{s}{\omega_0} + 1 - \alpha \right) \left( 2 \frac{\Omega}{\omega_0}\frac{s}{\omega_0} \right)
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\end{align}
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\end{subequations}
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# News terms with \alpha are added
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# w0 and xi are the same as before => only the zeros are changing and not the poles.
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@ -530,7 +496,7 @@ With:
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| <<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 |
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* Direct Velocity Feedback
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* Direct Velocity Feedback :noexport:
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** System Schematic and Control Architecture
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# Basic Idea of DVF
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