Last talk update
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@ -64,53 +64,30 @@
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\fontsize{8pt}{7.2}\selectfont
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:END:
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* Dynamics of Rotating Positioning Platforms
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* Dynamics of Rotating Platforms
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** Model of a Rotating Positioning Platform
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*** Column :BMCOL:
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:PROPERTIES:
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:BEAMER_col: 0.55
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:END:
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#+caption: Schematic of the studied System
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#+attr_latex: :width \linewidth
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#+attr_latex: :width 0.7\linewidth
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[[file:figs/system.pdf]]
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*** Column :BMCOL:
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:PROPERTIES:
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:BEAMER_col: 0.45
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:END:
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Simplest model to study the *gyroscopic effects* on Decentralized IFF
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\vspace{1em}
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Assumptions:
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- Perfect Rotating Stage
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- $\dot{\theta}(t) = \Omega = \text{cst}$
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- Small displacements
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- Position of the mass described by $[d_u\ d_v]$
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\vspace{1em}
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Two frames:
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- Inertial frame $(\vec{i}_x, \vec{i}_y, \vec{i}_z)$
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- Uniform rotating frame $(\vec{i}_u, \vec{i}_v, \vec{i}_w)$
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** Equations of Motion - Lagrangian Formalism
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\vspace{-1em}
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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 $L = T - V$ the Lagrangian, $D$ the dissipation function, and $Q_i$ the generalized force associated with the generalized variable.
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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), \quad V = \frac{1}{2} k \left( {d_u}^2 + {d_v}^2 \right) \\
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D &= \frac{1}{2} c \left( \dot{d}_u{}^2 + \dot{d}_v{}^2 \right), \quad Q_1 = F_u, \quad Q_2 = F_v
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\end{align*}
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\vspace{-1em}
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\vspace{1em}
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Equations of motion:
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\begin{align*}
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m \ddot{d}_u + c \dot{d}_u + ( k - m \Omega^2 ) d_u &= F_u + 2 m \Omega \dot{d}_v \\
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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}}
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\end{align*}
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\vspace{1em}
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#+attr_latex: :options []{blue}{}
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#+begin_cbox
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#+begin_center
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@ -121,73 +98,25 @@ Coriolis Forces $\Longleftrightarrow$ Coupling
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#+end_cbox
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** Transfer Function Matrix the Laplace domain
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\vspace{-1em}
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\begin{equation*}
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{\scriptsize \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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\begin{equation*}
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{\scriptsize \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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\vspace{2em}
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#+caption: Campbell Diagram : Evolution of the complex and real parts of the system's poles as a function of the rotational speed $\Omega$
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#+attr_latex: :environment subfigure :width 0.4\linewidth :align c
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#+attr_latex: :environment subfigure :width 0.49\linewidth :align c
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| file:figs/campbell_diagram_real.pdf | file:figs/campbell_diagram_imag.pdf |
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| <<fig:campbell_diagram_real>> Real Part | <<fig:campbell_diagram_imag>> Imaginary Part |
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** Bode Plots of the System's Dynamics
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* Decentralized Integral Force Feedback
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** Force Sensors and Decentralized IFF Control Architecture
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#+caption: Bode Plots for $\bm{G}_d$ for several rotational speed $\Omega$
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#+attr_latex: :environment subfigure :width 0.45\linewidth :align c
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| file:figs/plant_compare_rotating_speed_direct.pdf | file:figs/plant_compare_rotating_speed_coupling.pdf |
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| <<fig:plant_compare_rotating_speed_direct>> Direct Terms $d_u/F_u$, $d_v/F_v$ | <<fig:plant_compare_rotating_speed_coupling>> Coupling Terms $d_v/F_u$, $-d_u/F_v$ |
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For all the numerical analysis, $\omega_0 = \SI{1}{\radian\per\second}$, $k = \SI{1}{\newton\per\meter}$ and $\xi = 0.025 = \SI{2.5}{\percent}$.
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* Problem with the Decentralized Integral Force Feedback
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** Force Sensors and Control Architecture
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\vspace{-1em}
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*** Column :BMCOL:
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:PROPERTIES:
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:BEAMER_col: 0.6
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:END:
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#+caption: System with added Force Sensor in series with the actuators
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#+attr_latex: :width \linewidth
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#+caption: System with added Force Sensor in series with the actuators, $K_F(s) = g \cdot \frac{1}{s}$
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#+attr_latex: :width 0.7\linewidth
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[[file:figs/system_iff.pdf]]
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*** Column :BMCOL:
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:PROPERTIES:
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:BEAMER_col: 0.4
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:END:
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** IFF Plant Dynamics
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#+caption: Control Diagram for decentralized IFF
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#+caption: Bode plot of the dynamics from force actuator to force sensor for several rotational speeds $\Omega$
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#+attr_latex: :width \linewidth
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[[file:figs/control_diagram_iff.pdf]]
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\begin{equation*}
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\bm{K}_F(s) = \begin{bmatrix} K_F(s) & 0 \\ 0 & K_F(s) \end{bmatrix}
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\end{equation*}
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\begin{equation*}
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K_F(s) = g \cdot \frac{1}{s}
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\end{equation*}
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** Plant Dynamics
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\vspace{-1em}
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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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#+caption: Bode plot of the diagonal terms of $\bm{G}_f$ for several rotational speeds $\Omega$
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#+attr_latex: :width 0.9\linewidth
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[[file:figs/plant_iff_compare_rotating_speed.pdf]]
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** Decentralized IFF with Pure Integrators
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@ -203,15 +132,15 @@ For all the numerical analysis, $\omega_0 = \SI{1}{\radian\per\second}$, $k = \S
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\centering For $\Omega > 0$, the closed loop system is unstable
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#+end_cbox
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* Modification of the control law: Add High-Pass Filter
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** Modification of the Control Low
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\vspace{-1em}
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* Integral Force Feedback with High Pass Filter
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** Modification of the Control Law
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\begin{equation*}
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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}
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\end{equation*}
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\vspace{1em}
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#+attr_latex: :options [b]{0.45\linewidth}
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#+begin_minipage
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#+caption: Loop Gain
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@ -226,14 +155,6 @@ For all the numerical analysis, $\omega_0 = \SI{1}{\radian\per\second}$, $k = \S
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[[file:figs/root_locus_modified_iff.pdf]]
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#+end_minipage
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\vspace{-1em}
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\begin{align*}
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\text{Added HPF} &\Longleftrightarrow \text{limit the low frequency gain} \\
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&\Longleftrightarrow \text{shift the pole to the left along the real axis} \\
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&\Longrightarrow \text{stable system for small values of the gain}
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\end{align*}
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** Effect of $\omega_i$ on the attainable damping
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@ -272,56 +193,31 @@ For all the numerical analysis, $\omega_0 = \SI{1}{\radian\per\second}$, $k = \S
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#+attr_latex: :width \linewidth
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[[file:figs/mod_iff_damping_wi.pdf]]
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* Modification of the Mechanical System: Parallel Stiffness
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* Integral Force Feedback with Parallel Springs
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** Stiffness in Parallel with the Force Sensor
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\vspace{-1em}
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*** Column :BMCOL:
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:PROPERTIES:
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:BEAMER_col: 0.6
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:END:
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#+caption: Studied system with additional springs in parallel with the actuators and force sensors
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#+attr_latex: :width \linewidth
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#+caption: System with additional springs in parallel with the actuators and force sensors
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#+attr_latex: :width 0.65\linewidth
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[[file:figs/system_parallel_springs.pdf]]
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*** Column :BMCOL:
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:PROPERTIES:
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:BEAMER_col: 0.4
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:END:
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#+attr_latex: :options [Intuitive Idea]{blue}{}
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#+begin_cbox
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$k_p$ is used to counteract the negative stiffness $-m\Omega^2$ when high control gains are used.
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#+end_cbox
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\vspace{-2em}
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\begin{align*}
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k_p &= \alpha k \\
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k_a &= (1 - \alpha) k
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\end{align*}
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with $0 < \alpha < 1$.
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\vspace{1em}
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The overall stiffness $k = k_a + k_p = \text{cst}$ $\Longrightarrow$ the open-loop poles remains unchanged
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** Effect of the Parallel Stiffness on the Plant Dynamics
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#+attr_latex: :options [b]{0.42\linewidth}
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#+begin_minipage
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#+caption: Bode Plot of $f_u/F_u$ for $k_p = 0$, $k_p < m \Omega^2$ and $k_p > m \Omega^2$, $\Omega = 0.1 \omega_0$
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#+caption: Bode Plot of $f_u/F_u$
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#+attr_latex: :width \linewidth
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[[file:figs/plant_iff_kp.pdf]]
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#+end_minipage
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\hfill
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#+attr_latex: :options [b]{0.55\linewidth}
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#+begin_minipage
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#+caption: Root Locus for IFF without parallel spring, with parallel springs with stiffness $k_p < m \Omega^2$ and $k_p > m \Omega^2$, $\Omega = 0.1 \omega_0$
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#+caption: Root Locus for IFF
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#+attr_latex: :width \linewidth
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[[file:figs/root_locus_iff_kp.pdf]]
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#+end_minipage
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\vspace{1em}
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#+attr_latex: :options []{blue}{}
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#+begin_cbox
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If $k_p > m \Omega^2$, the poles of the closed-loop system stay in the (stable) right half-plane, and hence the *unconditional stability of IFF is recovered*.
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@ -329,10 +225,9 @@ If $k_p > m \Omega^2$, the poles of the closed-loop system stay in the (stable)
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** Optimal Parallel Stiffness
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#+caption: Root Locus for IFF when parallel stiffness $k_p$ is added, $\Omega = 0.1 \omega_0$
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#+attr_latex: :environment subfigure :width 0.49\linewidth :align c
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| file:figs/root_locus_iff_kps.pdf | file:figs/root_locus_opt_gain_iff_kp.pdf |
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| <<fig:root_locus_iff_kps>> Comparison of three parallel stiffnesses $k_p$ | <<fig:root_locus_opt_gain_iff_kp>> $k_p = 5 m \Omega^2$, optimal damping $\xi_\text{opt}$ is shown |
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#+caption: Root Locus for three parallel stiffnesses $k_p$
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#+attr_latex: :width 0.60\linewidth
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[[file:figs/root_locus_iff_kps.pdf]]
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#+attr_latex: :options []{blue}{}
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#+begin_cbox
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@ -340,7 +235,7 @@ If $k_p > m \Omega^2$, the poles of the closed-loop system stay in the (stable)
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Large parallel stiffness $k_p$ reduces the attainable damping.
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#+end_cbox
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* Comparison of the two Proposed Modifications
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* Comparison and Discussion
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** Comparison of the Attainable Damping
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#+caption: Root Locus for the two proposed modifications of decentralized IFF, $\Omega = 0.1 \omega_0$
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@ -354,25 +249,14 @@ Large parallel stiffness $k_p$ reduces the attainable damping.
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| file:figs/comp_compliance.pdf | file:figs/comp_transmissibility.pdf |
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| <<fig:comp_compliance>> Compliance | <<fig:comp_transmissibility>> Transmissibility |
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** Conclusion & Further work
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** @@latex:@@
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The two proposed techniques gives almost identical results but are very different when it comes to their implementations
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\vspace{8em}
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#+begin_center
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\Huge Thank you!
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#+end_center
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\vspace{8em}
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\vspace{2em}
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The best technique depends on the application
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\vspace{2em}
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#+attr_latex: :options {r}{0.45\textwidth}
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#+begin_wrapfigure
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\vspace{-1em}
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#+attr_latex: :width \linewidth
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[[file:figs/apa_schematic.pdf]]
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#+end_wrapfigure
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Amplified Piezoelectric Actuators are a nice way to have an actuator, a force sensors and a parallel stiffness in a compact manner
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\vspace{2em}
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Will be tested on the nano-hexapod
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Contact: [[mailto:dehaeze.thomas@gmail.com][dehaeze.thomas@gmail.com]]
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\vspace{1em}
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\small https://tdehaeze.github.io/dehaeze20_contr_stewa_platf/
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@ -1,4 +1,4 @@
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% Created 2020-07-29 mer. 15:27
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% Created 2020-08-24 lun. 18:19
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% Intended LaTeX compiler: pdflatex
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\documentclass[t, minted]{clean-beamer}
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\usepackage[utf8]{inputenc}
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@ -52,7 +52,7 @@
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pdftitle={Active Damping of Rotating Platforms using Integral Force Feedback},
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pdfkeywords={},
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pdfsubject={},
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pdfcreator={Emacs 27.0.91 (Org mode 9.4)},
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pdfcreator={Emacs 27.1.50 (Org mode 9.4)},
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pdflang={English}}
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\begin{document}
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@ -62,57 +62,32 @@
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\end{frame}
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\section{Dynamics of Rotating Positioning Platforms}
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\label{sec:orge7d05a8}
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\begin{frame}[label={sec:orgb466daa}]{Model of a Rotating Positioning Platform}
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\begin{columns}
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\begin{column}{0.55\columnwidth}
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\section{Dynamics of Rotating Platforms}
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\label{sec:orge3e010f}
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\begin{frame}[label={sec:orgac66c89}]{Model of a Rotating Positioning Platform}
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\begin{figure}[htbp]
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\centering
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\includegraphics[width=\linewidth]{figs/system.pdf}
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\includegraphics[width=0.7\linewidth]{figs/system.pdf}
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\caption{Schematic of the studied System}
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\end{figure}
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\end{column}
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\begin{column}{0.45\columnwidth}
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Simplest model to study the \textbf{gyroscopic effects} on Decentralized IFF
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\vspace{1em}
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Assumptions:
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\begin{itemize}
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\item Perfect Rotating Stage
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\item \(\dot{\theta}(t) = \Omega = \text{cst}\)
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\item Small displacements
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\item Position of the mass described by \([d_u\ d_v]\)
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\end{itemize}
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\vspace{1em}
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Two frames:
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\begin{itemize}
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\item Inertial frame \((\vec{i}_x, \vec{i}_y, \vec{i}_z)\)
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\item Uniform rotating frame \((\vec{i}_u, \vec{i}_v, \vec{i}_w)\)
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\end{itemize}
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\end{column}
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\end{columns}
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\end{frame}
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\begin{frame}[label={sec:orgc029b67}]{Equations of Motion - Lagrangian Formalism}
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\vspace{-1em}
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\begin{frame}[label={sec:org614e76f}]{Equations of Motion - Lagrangian Formalism}
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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 \(L = T - V\) the Lagrangian, \(D\) the dissipation function, and \(Q_i\) the generalized force associated with the generalized variable.
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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), \quad V = \frac{1}{2} k \left( {d_u}^2 + {d_v}^2 \right) \\
|
||||
D &= \frac{1}{2} c \left( \dot{d}_u{}^2 + \dot{d}_v{}^2 \right), \quad Q_1 = F_u, \quad Q_2 = F_v
|
||||
\end{align*}
|
||||
\vspace{-1em}
|
||||
|
||||
\vspace{1em}
|
||||
|
||||
Equations of motion:
|
||||
\begin{align*}
|
||||
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*}
|
||||
|
||||
\vspace{1em}
|
||||
|
||||
\begin{cbox}[]{blue}{}
|
||||
\begin{center}
|
||||
Centrifugal forces \(\Longleftrightarrow\) Negative Stiffness
|
||||
@ -122,27 +97,16 @@ Coriolis Forces \(\Longleftrightarrow\) Coupling
|
||||
\end{cbox}
|
||||
\end{frame}
|
||||
|
||||
\begin{frame}[label={sec:org9fc6840}]{Transfer Function Matrix the Laplace domain}
|
||||
\vspace{-1em}
|
||||
\begin{equation*}
|
||||
{\scriptsize \begin{bmatrix} d_u \\ d_v \end{bmatrix} = \bm{G}_d \begin{bmatrix} F_u \\ F_v \end{bmatrix}}
|
||||
\end{equation*}
|
||||
\begin{equation*}
|
||||
{\scriptsize \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*}
|
||||
\begin{frame}[label={sec:org0a25ba1}]{Transfer Function Matrix the Laplace domain}
|
||||
\vspace{2em}
|
||||
|
||||
\begin{figure}[htbp]
|
||||
\begin{subfigure}[c]{0.4\linewidth}
|
||||
\begin{subfigure}[c]{0.49\linewidth}
|
||||
\includegraphics[width=\linewidth]{figs/campbell_diagram_real.pdf}
|
||||
\caption{\label{fig:campbell_diagram_real} Real Part}
|
||||
\end{subfigure}
|
||||
\hfill
|
||||
\begin{subfigure}[c]{0.4\linewidth}
|
||||
\begin{subfigure}[c]{0.49\linewidth}
|
||||
\includegraphics[width=\linewidth]{figs/campbell_diagram_imag.pdf}
|
||||
\caption{\label{fig:campbell_diagram_imag} Imaginary Part}
|
||||
\end{subfigure}
|
||||
@ -152,73 +116,25 @@ Coriolis Forces \(\Longleftrightarrow\) Coupling
|
||||
\end{figure}
|
||||
\end{frame}
|
||||
|
||||
\begin{frame}[label={sec:orge87dc7b}]{Bode Plots of the System's Dynamics}
|
||||
\begin{figure}[htbp]
|
||||
\begin{subfigure}[c]{0.45\linewidth}
|
||||
\includegraphics[width=\linewidth]{figs/plant_compare_rotating_speed_direct.pdf}
|
||||
\caption{\label{fig:plant_compare_rotating_speed_direct} Direct Terms \(d_u/F_u\), \(d_v/F_v\)}
|
||||
\end{subfigure}
|
||||
\hfill
|
||||
\begin{subfigure}[c]{0.45\linewidth}
|
||||
\includegraphics[width=\linewidth]{figs/plant_compare_rotating_speed_coupling.pdf}
|
||||
\caption{\label{fig:plant_compare_rotating_speed_coupling} Coupling Terms \(d_v/F_u\), \(-d_u/F_v\)}
|
||||
\end{subfigure}
|
||||
\hfill
|
||||
\caption{Bode Plots for \(\bm{G}_d\) for several rotational speed \(\Omega\)}
|
||||
\centering
|
||||
\end{figure}
|
||||
|
||||
For all the numerical analysis, \(\omega_0 = \SI{1}{\radian\per\second}\), \(k = \SI{1}{\newton\per\meter}\) and \(\xi = 0.025 = \SI{2.5}{\percent}\).
|
||||
\end{frame}
|
||||
|
||||
\section{Problem with the Decentralized Integral Force Feedback}
|
||||
\label{sec:org59c3dbc}
|
||||
\begin{frame}[label={sec:org6faf2d3}]{Force Sensors and Control Architecture}
|
||||
\vspace{-1em}
|
||||
\begin{columns}
|
||||
\begin{column}{0.6\columnwidth}
|
||||
\section{Decentralized Integral Force Feedback}
|
||||
\label{sec:org8107aa6}
|
||||
\begin{frame}[label={sec:org818bd83}]{Force Sensors and Decentralized IFF Control Architecture}
|
||||
\begin{figure}[htbp]
|
||||
\centering
|
||||
\includegraphics[width=\linewidth]{figs/system_iff.pdf}
|
||||
\caption{System with added Force Sensor in series with the actuators}
|
||||
\end{figure}
|
||||
\end{column}
|
||||
|
||||
\begin{column}{0.4\columnwidth}
|
||||
\begin{figure}[htbp]
|
||||
\centering
|
||||
\includegraphics[width=\linewidth]{figs/control_diagram_iff.pdf}
|
||||
\caption{Control Diagram for decentralized IFF}
|
||||
\end{figure}
|
||||
|
||||
\begin{equation*}
|
||||
\bm{K}_F(s) = \begin{bmatrix} K_F(s) & 0 \\ 0 & K_F(s) \end{bmatrix}
|
||||
\end{equation*}
|
||||
|
||||
\begin{equation*}
|
||||
K_F(s) = g \cdot \frac{1}{s}
|
||||
\end{equation*}
|
||||
\end{column}
|
||||
\end{columns}
|
||||
\end{frame}
|
||||
|
||||
\begin{frame}[label={sec:org387ca99}]{Plant Dynamics}
|
||||
\vspace{-1em}
|
||||
|
||||
\begin{equation*}
|
||||
\begin{bmatrix} f_{u} \\ f_{v} \end{bmatrix} =
|
||||
\begin{bmatrix} F_u \\ F_v \end{bmatrix} - (c s + k)
|
||||
\begin{bmatrix} d_u \\ d_v \end{bmatrix}
|
||||
\end{equation*}
|
||||
|
||||
\begin{figure}[htbp]
|
||||
\centering
|
||||
\includegraphics[width=0.9\linewidth]{figs/plant_iff_compare_rotating_speed.pdf}
|
||||
\caption{Bode plot of the diagonal terms of \(\bm{G}_f\) for several rotational speeds \(\Omega\)}
|
||||
\includegraphics[width=0.7\linewidth]{figs/system_iff.pdf}
|
||||
\caption{System with added Force Sensor in series with the actuators, \(K_F(s) = g \cdot \frac{1}{s}\)}
|
||||
\end{figure}
|
||||
\end{frame}
|
||||
|
||||
\begin{frame}[label={sec:orgb8d521c}]{Decentralized IFF with Pure Integrators}
|
||||
\begin{frame}[label={sec:org159eb7e}]{IFF Plant Dynamics}
|
||||
\begin{figure}[htbp]
|
||||
\centering
|
||||
\includegraphics[width=\linewidth]{figs/plant_iff_compare_rotating_speed.pdf}
|
||||
\caption{Bode plot of the dynamics from force actuator to force sensor for several rotational speeds \(\Omega\)}
|
||||
\end{figure}
|
||||
\end{frame}
|
||||
|
||||
\begin{frame}[label={sec:orge6035da}]{Decentralized IFF with Pure Integrators}
|
||||
\begin{figure}[htbp]
|
||||
\centering
|
||||
\includegraphics[width=0.7\linewidth]{figs/root_locus_pure_iff.pdf}
|
||||
@ -232,15 +148,15 @@ For all the numerical analysis, \(\omega_0 = \SI{1}{\radian\per\second}\), \(k =
|
||||
\end{cbox}
|
||||
\end{frame}
|
||||
|
||||
\section{Modification of the control law: Add High-Pass Filter}
|
||||
\label{sec:orgc733c30}
|
||||
\begin{frame}[label={sec:orga459f5e}]{Modification of the Control Low}
|
||||
\vspace{-1em}
|
||||
|
||||
\section{Integral Force Feedback with High Pass Filter}
|
||||
\label{sec:org789619d}
|
||||
\begin{frame}[label={sec:org2020343}]{Modification of the Control Law}
|
||||
\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*}
|
||||
|
||||
\vspace{1em}
|
||||
|
||||
\begin{minipage}[b]{0.45\linewidth}
|
||||
\begin{center}
|
||||
\includegraphics[width=\linewidth]{figs/loop_gain_modified_iff.pdf}
|
||||
@ -254,18 +170,10 @@ For all the numerical analysis, \(\omega_0 = \SI{1}{\radian\per\second}\), \(k =
|
||||
\captionof{figure}{Root Locus}
|
||||
\end{center}
|
||||
\end{minipage}
|
||||
|
||||
\vspace{-1em}
|
||||
|
||||
\begin{align*}
|
||||
\text{Added HPF} &\Longleftrightarrow \text{limit the low frequency gain} \\
|
||||
&\Longleftrightarrow \text{shift the pole to the left along the real axis} \\
|
||||
&\Longrightarrow \text{stable system for small values of the gain}
|
||||
\end{align*}
|
||||
\end{frame}
|
||||
|
||||
|
||||
\begin{frame}[label={sec:org4390eac}]{Effect of \(\omega_i\) on the attainable damping}
|
||||
\begin{frame}[label={sec:org38b9bd2}]{Effect of \(\omega_i\) on the attainable damping}
|
||||
\begin{figure}[htbp]
|
||||
\centering
|
||||
\includegraphics[width=\linewidth]{figs/root_locus_wi_modified_iff.pdf}
|
||||
@ -291,7 +199,7 @@ small \(\omega_i\) \(\Longrightarrow\) reduces maximum gain \(g_\text{max}\)
|
||||
\end{columns}
|
||||
\end{frame}
|
||||
|
||||
\begin{frame}[label={sec:org84c72de}]{Optimal Control Parameters}
|
||||
\begin{frame}[label={sec:orga2c2166}]{Optimal Control Parameters}
|
||||
\vspace{1em}
|
||||
|
||||
\begin{figure}[htbp]
|
||||
@ -301,45 +209,22 @@ small \(\omega_i\) \(\Longrightarrow\) reduces maximum gain \(g_\text{max}\)
|
||||
\end{figure}
|
||||
\end{frame}
|
||||
|
||||
\section{Modification of the Mechanical System: Parallel Stiffness}
|
||||
\label{sec:orgfa77c9b}
|
||||
\begin{frame}[label={sec:org4d07a64}]{Stiffness in Parallel with the Force Sensor}
|
||||
\vspace{-1em}
|
||||
\begin{columns}
|
||||
\begin{column}{0.6\columnwidth}
|
||||
\section{Integral Force Feedback with Parallel Springs}
|
||||
\label{sec:org7a61560}
|
||||
\begin{frame}[label={sec:orga6be4f6}]{Stiffness in Parallel with the Force Sensor}
|
||||
\begin{figure}[htbp]
|
||||
\centering
|
||||
\includegraphics[width=\linewidth]{figs/system_parallel_springs.pdf}
|
||||
\caption{Studied system with additional springs in parallel with the actuators and force sensors}
|
||||
\includegraphics[width=0.65\linewidth]{figs/system_parallel_springs.pdf}
|
||||
\caption{System with additional springs in parallel with the actuators and force sensors}
|
||||
\end{figure}
|
||||
\end{column}
|
||||
|
||||
\begin{column}{0.4\columnwidth}
|
||||
\begin{cbox}[Intuitive Idea]{blue}{}
|
||||
\(k_p\) is used to counteract the negative stiffness \(-m\Omega^2\) when high control gains are used.
|
||||
\end{cbox}
|
||||
|
||||
\vspace{-2em}
|
||||
|
||||
\begin{align*}
|
||||
k_p &= \alpha k \\
|
||||
k_a &= (1 - \alpha) k
|
||||
\end{align*}
|
||||
with \(0 < \alpha < 1\).
|
||||
|
||||
\vspace{1em}
|
||||
|
||||
The overall stiffness \(k = k_a + k_p = \text{cst}\) \(\Longrightarrow\) the open-loop poles remains unchanged
|
||||
\end{column}
|
||||
\end{columns}
|
||||
\end{frame}
|
||||
|
||||
\begin{frame}[label={sec:org223db59}]{Effect of the Parallel Stiffness on the Plant Dynamics}
|
||||
\begin{frame}[label={sec:org497c282}]{Effect of the Parallel Stiffness on the Plant Dynamics}
|
||||
\begin{minipage}[b]{0.42\linewidth}
|
||||
\begin{figure}[htbp]
|
||||
\centering
|
||||
\includegraphics[width=\linewidth]{figs/plant_iff_kp.pdf}
|
||||
\caption{Bode Plot of \(f_u/F_u\) for \(k_p = 0\), \(k_p < m \Omega^2\) and \(k_p > m \Omega^2\), \(\Omega = 0.1 \omega_0\)}
|
||||
\caption{Bode Plot of \(f_u/F_u\)}
|
||||
\end{figure}
|
||||
\end{minipage}
|
||||
\hfill
|
||||
@ -347,29 +232,22 @@ The overall stiffness \(k = k_a + k_p = \text{cst}\) \(\Longrightarrow\) the ope
|
||||
\begin{figure}[htbp]
|
||||
\centering
|
||||
\includegraphics[width=\linewidth]{figs/root_locus_iff_kp.pdf}
|
||||
\caption{Root Locus for IFF without parallel spring, with parallel springs with stiffness \(k_p < m \Omega^2\) and \(k_p > m \Omega^2\), \(\Omega = 0.1 \omega_0\)}
|
||||
\caption{Root Locus for IFF}
|
||||
\end{figure}
|
||||
\end{minipage}
|
||||
|
||||
\vspace{1em}
|
||||
|
||||
\begin{cbox}[]{blue}{}
|
||||
If \(k_p > m \Omega^2\), the poles of the closed-loop system stay in the (stable) right half-plane, and hence the \textbf{unconditional stability of IFF is recovered}.
|
||||
\end{cbox}
|
||||
\end{frame}
|
||||
|
||||
\begin{frame}[label={sec:org8be51fd}]{Optimal Parallel Stiffness}
|
||||
\begin{frame}[label={sec:orgd7eccc9}]{Optimal Parallel Stiffness}
|
||||
\begin{figure}[htbp]
|
||||
\begin{subfigure}[c]{0.49\linewidth}
|
||||
\includegraphics[width=\linewidth]{figs/root_locus_iff_kps.pdf}
|
||||
\caption{\label{fig:root_locus_iff_kps} Comparison of three parallel stiffnesses \(k_p\)}
|
||||
\end{subfigure}
|
||||
\hfill
|
||||
\begin{subfigure}[c]{0.49\linewidth}
|
||||
\includegraphics[width=\linewidth]{figs/root_locus_opt_gain_iff_kp.pdf}
|
||||
\caption{\label{fig:root_locus_opt_gain_iff_kp} \(k_p = 5 m \Omega^2\), optimal damping \(\xi_\text{opt}\) is shown}
|
||||
\end{subfigure}
|
||||
\hfill
|
||||
\caption{Root Locus for IFF when parallel stiffness \(k_p\) is added, \(\Omega = 0.1 \omega_0\)}
|
||||
\centering
|
||||
\includegraphics[width=0.60\linewidth]{figs/root_locus_iff_kps.pdf}
|
||||
\caption{Root Locus for three parallel stiffnesses \(k_p\)}
|
||||
\end{figure}
|
||||
|
||||
\begin{cbox}[]{blue}{}
|
||||
@ -378,9 +256,9 @@ Large parallel stiffness \(k_p\) reduces the attainable damping.
|
||||
\end{cbox}
|
||||
\end{frame}
|
||||
|
||||
\section{Comparison of the two Proposed Modifications}
|
||||
\label{sec:orge227508}
|
||||
\begin{frame}[label={sec:org783f2c4}]{Comparison of the Attainable Damping}
|
||||
\section{Comparison and Discussion}
|
||||
\label{sec:org79d4359}
|
||||
\begin{frame}[label={sec:org7dda174}]{Comparison of the Attainable Damping}
|
||||
\begin{figure}[htbp]
|
||||
\centering
|
||||
\includegraphics[width=0.7\linewidth]{figs/comp_root_locus.pdf}
|
||||
@ -388,7 +266,7 @@ Large parallel stiffness \(k_p\) reduces the attainable damping.
|
||||
\end{figure}
|
||||
\end{frame}
|
||||
|
||||
\begin{frame}[label={sec:orgdd42828}]{Comparison Transmissibility and Compliance}
|
||||
\begin{frame}[label={sec:org9effb95}]{Comparison Transmissibility and Compliance}
|
||||
\begin{figure}[htbp]
|
||||
\begin{subfigure}[c]{0.49\linewidth}
|
||||
\includegraphics[width=\linewidth]{figs/comp_compliance.pdf}
|
||||
@ -405,26 +283,15 @@ Large parallel stiffness \(k_p\) reduces the attainable damping.
|
||||
\end{figure}
|
||||
\end{frame}
|
||||
|
||||
\begin{frame}[label={sec:org5db221d}]{Conclusion \& Further work}
|
||||
The two proposed techniques gives almost identical results but are very different when it comes to their implementations
|
||||
|
||||
\vspace{2em}
|
||||
|
||||
The best technique depends on the application
|
||||
|
||||
\vspace{2em}
|
||||
|
||||
\begin{wrapfigure}{r}{0.45\textwidth}
|
||||
\vspace{-1em}
|
||||
\begin{frame}[label={sec:org46c3efd}]{}
|
||||
\vspace{8em}
|
||||
\begin{center}
|
||||
\includegraphics[width=\linewidth]{figs/apa_schematic.pdf}
|
||||
\Huge Thank you!
|
||||
\end{center}
|
||||
\end{wrapfigure}
|
||||
\vspace{8em}
|
||||
|
||||
Amplified Piezoelectric Actuators are a nice way to have an actuator, a force sensors and a parallel stiffness in a compact manner
|
||||
|
||||
\vspace{2em}
|
||||
|
||||
Will be tested on the nano-hexapod
|
||||
Contact: \href{mailto:dehaeze.thomas@gmail.com}{dehaeze.thomas@gmail.com}
|
||||
\vspace{1em}
|
||||
\small \url{https://tdehaeze.github.io/dehaeze20\_contr\_stewa\_platf/}
|
||||
\end{frame}
|
||||
\end{document}
|
||||
|
||||
Loading…
Reference in New Issue
Block a user