From 42a0f82b8e0ad2611a61fc45073fbd2fb970d591 Mon Sep 17 00:00:00 2001 From: Thomas Dehaeze Date: Wed, 1 Jul 2020 10:03:07 +0200 Subject: [PATCH] Worked on sections 2 and 3 --- paper/paper.org | 268 +++++++++++++++++++++++++----------------------- paper/paper.pdf | Bin 1217586 -> 1182750 bytes paper/paper.tex | 252 +++++++++++++++++++++++++-------------------- 3 files changed, 277 insertions(+), 243 deletions(-) diff --git a/paper/paper.org b/paper/paper.org index 3edd175..5b3e667 100644 --- a/paper/paper.org +++ b/paper/paper.org @@ -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 +<> ** 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 | <> Real Part | <> 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 | -| <> Direct Terms $d_u/F_u$, $d_v/F_v$ | <> 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 | +| <> Direct Terms $d_u/F_u$, $d_v/F_v$ | <> Coupling Terms $d_v/F_u$, $-d_u/F_v$ | * Decentralized Integral Force Feedback +<> ** 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 +<> ** 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 +<> ** 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: | <> Three values of $k_p$ | <> $k_p = 5 m \Omega^2$, optimal damping is shown | * Comparison of the Proposed Active Damping Techniques for Rotating Positioning Stages +<> ** 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 | | <> 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 <> diff --git a/paper/paper.pdf b/paper/paper.pdf index 2f9b3543ce9eee1787ebbf92d75ed773d82b4a38..ebdc693ab0cad762eddef083dd1a051842539b49 100644 GIT binary patch delta 341516 zcmZs?Q;;uA^d#7}ZQHhO+cs|7_t&;<+qP}n?%TGt-+v}1Hg;p4>SRRKOFdNOIhom5 zE|Z%sRWAjdMC5=D!or(|of6Wu{HkQAM+A#VodpB=Q2+A>*D$ zv4>3%4=6S`#QWId-F_kqW0YP3;g7&Wv^?U3&8bGHGJC!4}Ub8diN3E1@Tdjand|qc3m3wXVXh7Nr$U?kry9S3)50hKBJ$_#IA%s?bZja zyu(nI%h1?Whb~zMN*Q!4t!2?@lX7K}Us8UJzfs;OqE~^ai<=;tqa`$JDXW?P;Pon+ zE*fW?R$WKm_a=R>UwQ)QI%93|)m8FY#$?Hdy}L)iis%?;Ko$naa1>cAAkr_LK4iz9 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