Update paper bibliography
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@ -54,7 +54,7 @@
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* Introduction
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<<sec:introduction>>
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*** Establish the importance of the research topic :ignore:
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# Active Damping + Rotating Systems
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# Active Damping + Rotating System
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*** Applications of active damping :ignore:
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# Link to previous paper / tomography
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@ -93,7 +93,7 @@ Consider the rotating X-Y stage of Figure [[fig:rotating_xy_platform]].
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#+name: fig:rotating_xy_platform
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#+caption: Figure caption
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#+attr_latex: :scale 1
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[[file:figs/rotating_xy_platform.pdf]]
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[[file:figs/system.pdf]]
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#+name: fig:cedrat_xy25xs
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@ -109,7 +109,6 @@ Let's express the kinetic energy $T$ and the potential energy $V$ of the mass $m
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Dissipation function $R$
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Kinetic energy $T$
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Potential energy $V$
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#+name: eq:energy_inertial_frame
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\begin{subequations}
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\begin{align}
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T & = \frac{1}{2} m \left( \left( \dot{u} - \Omega v \right)^2 + \left( \dot{v} + \Omega u \right)^2 \right) \\
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@ -119,7 +118,6 @@ Potential energy $V$
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\end{subequations}
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The Lagrangian is the kinetic energy minus the potential energy:
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#+name: eq:lagrangian_inertial_frame
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\begin{equation}
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L = T - V
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\end{equation}
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@ -128,12 +126,8 @@ From the Lagrange's equations of the second kind eqref:eq:lagrange_second_kind,
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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 $Q_i$ is the generalized force associated with the generalized variable $q_i$ ($Q_1 = F_u$ and $Q_2 = F_v$).
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\begin{equation}
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\frac{d}{dt} \left( \frac{\partial T}{\partial \dot{q}_i} \right) - \frac{\partial T}{\partial q_i} + \frac{\partial R}{\partial \dot{q}_i} - \frac{\partial V}{\partial q_i} = Q_i
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\end{equation}
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with $Q_i$ is the generalized force associated with the generalized variable $q_i$ ($F_u$ and $F_v$).
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\begin{subequations}
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\begin{align}
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@ -143,12 +137,16 @@ with $Q_i$ is the generalized force associated with the generalized variable $q_
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\end{subequations}
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# Explain Gyroscopic effects
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- Coriolis Forces: coupling
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- Centrifugal forces: negative stiffness
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Without the coupling terms, each equation is the equation of a one degree of freedom mass-spring system with mass $m$ and stiffness $k- m\dot{\theta}^2$.
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Thus, the term $- m\dot{\theta}^2$ acts like a negative stiffness (due to *centrifugal forces*).
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** Transfer Functions in the Laplace domain
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# Laplace Domain
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\begin{subequations}
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\begin{align}
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u &= \frac{ms^2 + cs + k - m \Omega^2}{\left( m s^2 + cs + k - m \Omega^2 \right)^2 + \left( 2 m \Omega s \right)^2} F_u + \frac{2 m \Omega s}{\left( m s^2 + cs + k - m \Omega^2 \right)^2 + \left( 2 m \Omega s \right)^2} F_v \\
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@ -180,50 +178,10 @@ With:
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\end{align}
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\end{subequations}
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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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- $\xi$ damping ratio
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#+name: eq:lagrange_second_kind
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\begin{equation}
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\frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}_j} \right) = \frac{\partial L}{\partial q_j}
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\end{equation}
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#+name: eq:eom_mixed
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\begin{subequations}
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\begin{align}
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m\ddot{x} + kx = F_u \cos{\theta} - F_v \sin{\theta}\\
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m\ddot{y} + ky = F_u \sin{\theta} + F_v \cos{\theta}
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\end{align}
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\end{subequations}
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Performing the change coordinates from $(x, y)$ to $(d_x, d_y, \theta)$:
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\begin{subequations}
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\begin{align}
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x & = d_u \cos{\theta} - d_v \sin{\theta}\\
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y & = d_u \sin{\theta} + d_v \cos{\theta}
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\end{align}
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\end{subequations}
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Gives
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#+name: eq:oem_coupled
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\begin{subequations}
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\begin{align}
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m \ddot{d_u} + (k - m\dot{\theta}^2) d_u &= F_u + 2 m\dot{d_v}\dot{\theta} + m d_v\ddot{\theta} \label{eq:du_coupled} \\
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m \ddot{d_v} + (k \underbrace{-\ m\dot{\theta}^2}_{\text{Centrif.}}) d_v &= F_v \underbrace{-\ 2 m\dot{d_u}\dot{\theta}}_{\text{Coriolis}} \underbrace{-\ m d_u\ddot{\theta}}_{\text{Euler}} \label{eq:dv_coupled}
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\end{align}
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\end{subequations}
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We obtain two differential equations that are coupled through:
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- *Euler forces*: $m d_v \ddot{\theta}$
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- *Coriolis forces*: $2 m \dot{d_v} \dot{\theta}$
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Without the coupling terms, each equation is the equation of a one degree of freedom mass-spring system with mass $m$ and stiffness $k- m\dot{\theta}^2$.
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Thus, the term $- m\dot{\theta}^2$ acts like a negative stiffness (due to *centrifugal forces*).
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** Constant Rotating Speed
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To simplify, let's consider a constant rotating speed $\dot{\theta} = \Omega$ and thus $\ddot{\theta} = 0$.
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@ -285,18 +243,73 @@ The magnitude of the coupling terms are increasing with the rotation speed.
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* Integral Force Feedback
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** Control Schematic
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Force Sensors are added in series with the actuators as shown in Figure [[fig:system_iff]].
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# Reference to IFF control
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#+name: fig:system_iff
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#+caption: System with Force Sensors in Series with the Actuators. Decentralized Integral Force Feedback is used
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#+attr_latex: :scale 1
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[[file:figs/system_iff.pdf]]
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** Equations
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The sensed forces are equal to:
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\begin{equation}
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\begin{bmatrix} f_{u} \\ f_{v} \end{bmatrix} =
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\begin{bmatrix} F_u \\ F_v \end{bmatrix} - (c s + k)
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\begin{bmatrix} d_u \\ d_v \end{bmatrix}
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\end{equation}
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Which then gives:
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\begin{equation}
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\begin{bmatrix} f_{u} \\ f_{v} \end{bmatrix} =
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\bm{G}_{f}
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\begin{bmatrix} F_u \\ F_v \end{bmatrix}
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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_{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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\end{bmatrix}
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\begin{bmatrix} F_u \\ F_v \end{bmatrix}
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\end{equation}
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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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** Plant Dynamics
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# General explanation for the Root Locus Plot
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# MIMO root locus: gain is simultaneously increased for both decentralized controllers
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# Explain the circles, crosses and black crosses (poles of the controller)
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#+name: fig:root_locus_pure_iff
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#+caption: Figure caption
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#+caption: Root Locus
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#+attr_latex: :scale 1
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[[file:figs/root_locus_pure_iff.pdf]]
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** Physical Interpretation
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* Integral Force Feedback with Low Pass Filters
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At low frequency, the gain is very large and thus no force is transmitted between the payload and the rotating stage.
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This means that at low frequency, the system is decoupled (the force sensor removed) and thus the system is unstable.
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* Integral Force Feedback with High Pass Filters
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** Modification of the Control Low
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# Reference to Preumont where its done
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# Explain why it is usually done and why it is done here: the problem is the high gain at low frequency => high pass filter
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** Close Loop Analysis
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#+name: fig:loop_gain_modified_iff
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#+caption: Figure caption
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@ -308,10 +321,18 @@ The magnitude of the coupling terms are increasing with the rotation speed.
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#+attr_latex: :scale 1
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[[file:figs/root_locus_modified_iff_bis.pdf]]
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** Optimal Cut-Off Frequency
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#+name: fig:root_locus_wi_modified_iff
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#+caption: Figure caption
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#+attr_latex: :scale 1
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[[file:figs/root_locus_wi_modified_iff.pdf]]
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[[file:figs/root_locus_wi_modified_iff_bis.pdf]]
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#+name: fig:mod_iff_damping_wi
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#+caption: Figure caption
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#+attr_latex: :scale 1
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[[file:figs/mod_iff_damping_wi.pdf]]
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* Integral Force Feedback with Parallel Springs
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@ -1,4 +1,4 @@
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% Created 2020-06-22 lun. 13:27
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% Created 2020-06-22 lun. 17:38
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% Intended LaTeX compiler: pdflatex
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\documentclass{ISMA_USD2020}
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\usepackage[utf8]{inputenc}
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@ -43,14 +43,14 @@
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}
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\section{Introduction}
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\label{sec:orgd20252d}
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\label{sec:org67e0a4e}
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\label{sec:introduction}
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\cite{dehaeze18_sampl_stabil_for_tomog_exper}
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\section{System Under Study}
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\label{sec:orgacbe1ae}
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\label{sec:org85bcde2}
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\subsection{Rotating Positioning Platform}
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\label{sec:org07e4fc8}
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\label{sec:org4959a5e}
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Consider the rotating X-Y stage of Figure \ref{fig:rotating_xy_platform}.
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\begin{itemize}
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@ -63,7 +63,7 @@ Consider the rotating X-Y stage of Figure \ref{fig:rotating_xy_platform}.
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\begin{figure}[htbp]
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\centering
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\includegraphics[scale=1]{figs/rotating_xy_platform.pdf}
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\includegraphics[scale=1]{figs/system.pdf}
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\caption{\label{fig:rotating_xy_platform}Figure caption}
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\end{figure}
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@ -75,7 +75,7 @@ Consider the rotating X-Y stage of Figure \ref{fig:rotating_xy_platform}.
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\end{figure}
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\subsection{Equation of Motion}
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\label{sec:orgac1a52a}
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\label{sec:orgdb109d9}
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The system has two degrees of freedom and is thus fully described by the generalized coordinates \(u\) and \(v\).
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Let's express the kinetic energy \(T\) and the potential energy \(V\) of the mass \(m\) (neglecting the rotational energy):
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@ -84,7 +84,6 @@ Dissipation function \(R\)
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Kinetic energy \(T\)
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Potential energy \(V\)
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\begin{subequations}
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\label{eq:energy_inertial_frame}
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\begin{align}
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T & = \frac{1}{2} m \left( \left( \dot{u} - \Omega v \right)^2 + \left( \dot{v} + \Omega u \right)^2 \right) \\
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R & = \frac{1}{2} c \left( \dot{u}^2 + \dot{v}^2 \right) \\
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@ -94,7 +93,6 @@ Potential energy \(V\)
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The Lagrangian is the kinetic energy minus the potential energy:
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\begin{equation}
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\label{eq:lagrangian_inertial_frame}
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L = T - V
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\end{equation}
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@ -102,12 +100,8 @@ From the Lagrange's equations of the second kind \eqref{eq:lagrange_second_kind}
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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 \(Q_i\) is the generalized force associated with the generalized variable \(q_i\) (\(Q_1 = F_u\) and \(Q_2 = F_v\)).
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\begin{equation}
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\frac{d}{dt} \left( \frac{\partial T}{\partial \dot{q}_i} \right) - \frac{\partial T}{\partial q_i} + \frac{\partial R}{\partial \dot{q}_i} - \frac{\partial V}{\partial q_i} = Q_i
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\end{equation}
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with \(Q_i\) is the generalized force associated with the generalized variable \(q_i\) (\(F_u\) and \(F_v\)).
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\begin{subequations}
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\begin{align}
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@ -116,6 +110,18 @@ with \(Q_i\) is the generalized force associated with the generalized variable \
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\end{align}
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\end{subequations}
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\begin{itemize}
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\item Coriolis Forces: coupling
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\item Centrifugal forces: negative stiffness
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\end{itemize}
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Without the coupling terms, each equation is the equation of a one degree of freedom mass-spring system with mass \(m\) and stiffness \(k- m\dot{\theta}^2\).
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Thus, the term \(- m\dot{\theta}^2\) acts like a negative stiffness (due to \textbf{centrifugal forces}).
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\subsection{Transfer Functions in the Laplace domain}
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\label{sec:orgfcd3def}
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\begin{subequations}
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\begin{align}
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u &= \frac{ms^2 + cs + k - m \Omega^2}{\left( m s^2 + cs + k - m \Omega^2 \right)^2 + \left( 2 m \Omega s \right)^2} F_u + \frac{2 m \Omega s}{\left( m s^2 + cs + k - m \Omega^2 \right)^2 + \left( 2 m \Omega s \right)^2} F_v \\
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@ -146,56 +152,14 @@ With:
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\end{align}
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\end{subequations}
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\begin{itemize}
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\item \(\omega_0 = \sqrt{\frac{k}{m}}\): Natural frequency of the mass-spring system in \(\si{\radian/\s}\)
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\item \(\xi\) damping ratio
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\end{itemize}
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\begin{equation}
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\label{eq:lagrange_second_kind}
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\frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}_j} \right) = \frac{\partial L}{\partial q_j}
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\end{equation}
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\begin{subequations}
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\label{eq:eom_mixed}
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\begin{align}
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m\ddot{x} + kx = F_u \cos{\theta} - F_v \sin{\theta}\\
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m\ddot{y} + ky = F_u \sin{\theta} + F_v \cos{\theta}
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\end{align}
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\end{subequations}
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Performing the change coordinates from \((x, y)\) to \((d_x, d_y, \theta)\):
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\begin{subequations}
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\begin{align}
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x & = d_u \cos{\theta} - d_v \sin{\theta}\\
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y & = d_u \sin{\theta} + d_v \cos{\theta}
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\end{align}
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\end{subequations}
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Gives
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\begin{subequations}
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\label{eq:oem_coupled}
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\begin{align}
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m \ddot{d_u} + (k - m\dot{\theta}^2) d_u &= F_u + 2 m\dot{d_v}\dot{\theta} + m d_v\ddot{\theta} \label{eq:du_coupled} \\
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m \ddot{d_v} + (k \underbrace{-\ m\dot{\theta}^2}_{\text{Centrif.}}) d_v &= F_v \underbrace{-\ 2 m\dot{d_u}\dot{\theta}}_{\text{Coriolis}} \underbrace{-\ m d_u\ddot{\theta}}_{\text{Euler}} \label{eq:dv_coupled}
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\end{align}
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\end{subequations}
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We obtain two differential equations that are coupled through:
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\begin{itemize}
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\item \textbf{Euler forces}: \(m d_v \ddot{\theta}\)
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\item \textbf{Coriolis forces}: \(2 m \dot{d_v} \dot{\theta}\)
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\end{itemize}
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Without the coupling terms, each equation is the equation of a one degree of freedom mass-spring system with mass \(m\) and stiffness \(k- m\dot{\theta}^2\).
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Thus, the term \(- m\dot{\theta}^2\) acts like a negative stiffness (due to \textbf{centrifugal forces}).
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\subsection{Constant Rotating Speed}
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\label{sec:org47aaeee}
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\label{sec:org81c7074}
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To simplify, let's consider a constant rotating speed \(\dot{\theta} = \Omega\) and thus \(\ddot{\theta} = 0\).
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\begin{equation}
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@ -250,27 +214,74 @@ The magnitude of the coupling terms are increasing with the rotation speed.
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\end{figure}
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\section{Integral Force Feedback}
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\label{sec:org78c2eab}
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\label{sec:orgc6c1b99}
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\subsection{Control Schematic}
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\label{sec:org6a00238}
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\label{sec:orgb93b297}
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Force Sensors are added in series with the actuators as shown in Figure \ref{fig:system_iff}.
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\begin{figure}[htbp]
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\centering
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\includegraphics[scale=1]{figs/system_iff.pdf}
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\caption{\label{fig:system_iff}System with Force Sensors in Series with the Actuators. Decentralized Integral Force Feedback is used}
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\end{figure}
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\subsection{Equations}
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\label{sec:org5480f1b}
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\label{sec:org4072ea4}
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The sensed forces are equal to:
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\begin{equation}
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\begin{bmatrix} f_{u} \\ f_{v} \end{bmatrix} =
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\begin{bmatrix} F_u \\ F_v \end{bmatrix} - (c s + k)
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\begin{bmatrix} d_u \\ d_v \end{bmatrix}
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\end{equation}
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Which then gives:
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\begin{equation}
|
||||
\begin{bmatrix} f_{u} \\ f_{v} \end{bmatrix} =
|
||||
\bm{G}_{f}
|
||||
\begin{bmatrix} F_u \\ F_v \end{bmatrix}
|
||||
\end{equation}
|
||||
|
||||
\begin{equation}
|
||||
\begin{bmatrix} f_{u} \\ f_{v} \end{bmatrix} =
|
||||
\frac{1}{G_{fp}}
|
||||
\begin{bmatrix}
|
||||
G_{fz} & -G_{fc} \\
|
||||
G_{fc} & G_{fz}
|
||||
\end{bmatrix}
|
||||
\begin{bmatrix} F_u \\ F_v \end{bmatrix}
|
||||
\end{equation}
|
||||
|
||||
\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}
|
||||
|
||||
|
||||
\subsection{Plant Dynamics}
|
||||
\label{sec:orgbb0952e}
|
||||
\label{sec:org0250ac0}
|
||||
|
||||
\begin{figure}[htbp]
|
||||
\centering
|
||||
\includegraphics[scale=1]{figs/root_locus_pure_iff.pdf}
|
||||
\caption{\label{fig:root_locus_pure_iff}Figure caption}
|
||||
\caption{\label{fig:root_locus_pure_iff}Root Locus}
|
||||
\end{figure}
|
||||
|
||||
\subsection{Physical Interpretation}
|
||||
\label{sec:orgdb25e2c}
|
||||
\label{sec:orgb2d79d2}
|
||||
|
||||
\section{Integral Force Feedback with Low Pass Filters}
|
||||
\label{sec:org2985d35}
|
||||
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.
|
||||
|
||||
\section{Integral Force Feedback with High Pass Filters}
|
||||
\label{sec:orgabf7a6a}
|
||||
\subsection{Modification of the Control Low}
|
||||
\label{sec:org4766bd6}
|
||||
|
||||
|
||||
\subsection{Close Loop Analysis}
|
||||
\label{sec:org4c639fd}
|
||||
|
||||
\begin{figure}[htbp]
|
||||
\centering
|
||||
@ -284,14 +295,24 @@ The magnitude of the coupling terms are increasing with the rotation speed.
|
||||
\caption{\label{fig:root_locus_modified_iff}Figure caption}
|
||||
\end{figure}
|
||||
|
||||
\subsection{Optimal Cut-Off Frequency}
|
||||
\label{sec:orge829a45}
|
||||
|
||||
\begin{figure}[htbp]
|
||||
\centering
|
||||
\includegraphics[scale=1]{figs/root_locus_wi_modified_iff.pdf}
|
||||
\includegraphics[scale=1]{figs/root_locus_wi_modified_iff_bis.pdf}
|
||||
\caption{\label{fig:root_locus_wi_modified_iff}Figure caption}
|
||||
\end{figure}
|
||||
|
||||
|
||||
\begin{figure}[htbp]
|
||||
\centering
|
||||
\includegraphics[scale=1]{figs/mod_iff_damping_wi.pdf}
|
||||
\caption{\label{fig:mod_iff_damping_wi}Figure caption}
|
||||
\end{figure}
|
||||
|
||||
\section{Integral Force Feedback with Parallel Springs}
|
||||
\label{sec:orga4142a5}
|
||||
\label{sec:orgd96ea25}
|
||||
|
||||
\begin{figure}[htbp]
|
||||
\centering
|
||||
@ -330,7 +351,7 @@ The magnitude of the coupling terms are increasing with the rotation speed.
|
||||
\end{figure}
|
||||
|
||||
\section{Direct Velocity Feedback}
|
||||
\label{sec:org6a1be4f}
|
||||
\label{sec:org027d051}
|
||||
|
||||
\begin{figure}[htbp]
|
||||
\centering
|
||||
@ -339,7 +360,7 @@ The magnitude of the coupling terms are increasing with the rotation speed.
|
||||
\end{figure}
|
||||
|
||||
\section{Comparison of the Proposed Active Damping Techniques}
|
||||
\label{sec:orga9658c0}
|
||||
\label{sec:org1eaa959}
|
||||
|
||||
\begin{figure}[htbp]
|
||||
\centering
|
||||
@ -360,12 +381,12 @@ The magnitude of the coupling terms are increasing with the rotation speed.
|
||||
\end{figure}
|
||||
|
||||
\section{Conclusion}
|
||||
\label{sec:orgcdf948f}
|
||||
\label{sec:org1b2b4ae}
|
||||
\label{sec:conclusion}
|
||||
|
||||
|
||||
\section*{Acknowledgment}
|
||||
\label{sec:org6c21e13}
|
||||
\label{sec:org2ae16a5}
|
||||
|
||||
\bibliography{ref.bib}
|
||||
\end{document}
|
||||
|
@ -16,3 +16,33 @@
|
||||
venue = {Paris, France},
|
||||
tags = {nass},
|
||||
}
|
||||
|
||||
@book{skogestad07_multiv_feedb_contr,
|
||||
author = {Skogestad, Sigurd and Postlethwaite, Ian},
|
||||
title = {Multivariable Feedback Control: Analysis and Design},
|
||||
year = {2007},
|
||||
publisher = {John Wiley},
|
||||
isbn = {9780470011683},
|
||||
}
|
||||
|
||||
@book{preumont18_vibrat_contr_activ_struc_fourt_edition,
|
||||
author = {Andre Preumont},
|
||||
title = {Vibration Control of Active Structures - Fourth Edition},
|
||||
year = {2018},
|
||||
publisher = {Springer International Publishing},
|
||||
url = {https://doi.org/10.1007/978-3-319-72296-2},
|
||||
doi = {10.1007/978-3-319-72296-2},
|
||||
pages = {nil},
|
||||
series = {Solid Mechanics and Its Applications},
|
||||
}
|
||||
|
||||
@article{souleille18_concep_activ_mount_space_applic,
|
||||
author = {Souleille, Adrien and Lampert, Thibault and Lafarga, V and Hellegouarch, Sylvain and Rondineau, Alan and Rodrigues, Gon{\c{c}}alo and Collette, Christophe},
|
||||
title = {A Concept of Active Mount for Space Applications},
|
||||
journal = {CEAS Space Journal},
|
||||
volume = {10},
|
||||
number = {2},
|
||||
pages = {157--165},
|
||||
year = {2018},
|
||||
publisher = {Springer},
|
||||
}
|
||||
|
Loading…
Reference in New Issue
Block a user