From 651e67f3d1aa45778ee23e54f005c27baf2b4daf Mon Sep 17 00:00:00 2001 From: Thomas Dehaeze Date: Fri, 26 Jun 2020 11:57:54 +0200 Subject: [PATCH] Reworked the TF matrices / removed DVF --- paper/paper.org | 130 ++++++++++++++++++------------------------------ 1 file changed, 48 insertions(+), 82 deletions(-) diff --git a/paper/paper.org b/paper/paper.org index 378aab3..0f5f99b 100644 --- a/paper/paper.org +++ b/paper/paper.org @@ -1,4 +1,4 @@ -#+TITLE: Decentralized Active Damping of Rotating Positioning Platforms +#+TITLE: Active Damping of Rotating Positioning Platforms using Force Feedback :DRAWER: #+LATEX_CLASS: ISMA_USD2020 #+OPTIONS: toc:nil @@ -86,58 +86,54 @@ 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 -** Studied Rotating Positioning Platform -# Introduce the fact that we need a simple system representing the rotating aspect. - +** 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. # Simplest system where gyroscopic forces can be studied -Consider the rotating X-Y stage of Figure [[fig:system]]. - -# Present the system, parameters, assumptions +It represents an X-Y positioning stage on top of a Rotating Stage and is schematically represented in Figure [[fig:system]]. # Explain the frames (inertial frame x,y, rotating frame u,v) -# iu, iv is linked to the rotating stage and supposed to be perfect +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 -# Small displacements +# 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 $\si{\radian\per\second}$. -# Constant rotational speed +# 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 $\si{\newton}$ -- $k$: Actuator's Stiffness [N/m] -- $m$: Payload's mass [kg] -- $\Omega = \dot{\theta}$: rotation speed [rad/s] -- $F_u$, $F_v$ -- $d_u$, $d_v$ +# 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 $(\vec{i}_u, \vec{i}_v)$. #+name: fig:system #+caption: Schematic of the studied System #+attr_latex: :scale 1 [[file:figs/system.pdf]] -# #+name: fig:cedrat_xy25xs -# #+caption: Figure caption -# #+attr_latex: :width 0.5\linewidth -# [[file:figs/cedrat_xy25xs.jpg]] - ** Equations of Motion -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 $R$: +To obtain of equation of motion for the system represented in Figure [[fig:system]], the Lagrangian equations are used: +#+name: eq:lagrangian_equations +\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 $[q_1\ q_2] = [d_u\ d_v]$: #+name: eq:energy_functions_lagrange \begin{subequations} \begin{align} 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) \\ V & = \frac{1}{2} k \left( {d_u}^2 + {d_v}^2 \right) \\ - R & = \frac{1}{2} c \left( \dot{d}_u{}^2 + \dot{d}_v{}^2 \right) + D & = \frac{1}{2} c \left( \dot{d}_u{}^2 + \dot{d}_v{}^2 \right) \\ + Q_1 &= F_u, \quad Q_2 = F_v \end{align} \end{subequations} -The equations of motion are derived from the Lagrangian equation: -#+name: eq:lagrangian_equations -\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 and $Q_i$ is the generalized force associated with the generalized variable $q_i$ ($Q_1 = F_u$ and $Q_2 = F_v$). - +Substituting equations eqref:eq:energy_functions_lagrange into eqref:eq:lagrangian_equations gives the two coupled differential equations: #+name: eq:eom_coupled \begin{subequations} \begin{align} @@ -147,14 +143,12 @@ with $L = T - V$ is the Lagrangian and $Q_i$ is the generalized force associated \end{subequations} # Explain Gyroscopic effects -The Gyroscopic effects can be seen from the two following terms: -- Coriolis Forces: coupling -- Centrifugal forces: negative stiffness +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 ** Transfer Functions in the Laplace domain - -# Laplace Domain -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: +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: #+name: eq:oem_laplace_domain \begin{subequations} \begin{align} @@ -163,8 +157,7 @@ Using the Laplace transformation on the equations of motion eqref:eq:eom_coupled \end{align} \end{subequations} - -Without rotation $\Omega = 0$ and the system corresponds to two uncoupled one degree of freedom mass-spring-damper systems: +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} @@ -176,33 +169,24 @@ Without rotation $\Omega = 0$ and the system corresponds to two uncoupled one de ** Change of Variables / Parameters for the study # Change of variables -In order this study is more independent on the system parameters, the following change of variable is performed: +In order to make this study less dependent on the system parameters, the following change of variable is performed: - $\omega_0 = \sqrt{\frac{k}{m}}$: Natural frequency of the mass-spring system in $\si{\radian/\s}$ - $\xi = \frac{c}{2 \sqrt{k m}}$: Damping ratio - + #+name: eq:tf_d \begin{equation} \begin{bmatrix} d_u \\ d_v \end{bmatrix} = \bm{G}_d \begin{bmatrix} F_u \\ F_v \end{bmatrix} \end{equation} -Where $\bm{G}_d$ is a $2 \times 2$ transfer function matrix. +#+name: eq:tf_d \begin{equation} -\bm{G}_d = \frac{1}{k} \frac{1}{G_{dp}} -\begin{bmatrix} - G_{dz} & G_{dc} \\ - -G_{dc} & G_{dz} -\end{bmatrix} +\bm{G}_{d} = + \frac{1}{k} + \begin{bmatrix} + \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} \\ + \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} + \end{bmatrix} \end{equation} -With: -\begin{subequations} - \begin{align} - 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 \\ - G_{dz} &= \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \\ - G_{dc} &= 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} - \end{align} -\end{subequations} - -$G_{dp}$ describes to poles of the system, $G_{dz}$ the zeros of the diagonal terms and $G_{dc}$ the coupling. # Parameters - $k = \SI{1}{N/m}$, $m = \SI{1}{kg}$, $c = \SI{0.05}{\newton\per\meter\second}$ @@ -277,19 +261,11 @@ Re-injecting eqref:eq:tf_d into eqref:eq:measured_force yields: Where $\bm{G}_f$ is a $2 \times 2$ transfer function matrix. \begin{equation} -\bm{G}_f = -\frac{1}{G_{fp}} -\begin{bmatrix} - G_{fz} & -G_{fc} \\ - G_{fc} & G_{fz} + \bm{G}_{f} = \begin{bmatrix} + \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} \\ + \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} \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} # Explain the two real zeros => change of gain but not of phase # The alternating poles and zeros properties of collocated IFF holds @@ -462,22 +438,12 @@ The overall stiffness $k$ stays constant: \end{equation} \begin{equation} -\begin{bmatrix} f_u \\ f_v \end{bmatrix} = -\frac{1}{G_{kp}} +\bm{G}_k = \begin{bmatrix} - G_{kz} & -G_{kc} \\ - G_{kc} & G_{kz} + \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} \\ + \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} \end{bmatrix} -\begin{bmatrix} F_u \\ F_v \end{bmatrix} \end{equation} -With: -\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} # News terms with \alpha are added # w0 and xi are the same as before => only the zeros are changing and not the poles. @@ -530,7 +496,7 @@ With: | <> Three values of $k_p$ | <> $k_p = 5 m \Omega^2$, optimal damping is shown | -* Direct Velocity Feedback +* Direct Velocity Feedback :noexport: ** System Schematic and Control Architecture # Basic Idea of DVF