#+TITLE: Active Damping of Rotating Platforms using Integral Force Feedback
:DRAWER:
#+AUTHOR:
#+SUBTITLE:  ISMA-USD 2020, September 7-9, 2020
#+EMAIL:     dehaeze.thomas@gmail.com
#+DATE:

#+LATEX_HEADER_EXTRA: \author[shortname]{Thomas Dehaeze \inst{1,3} \and Christophe Collette \inst{1,2}}
#+LATEX_HEADER_EXTRA: \institute[shortinst]{\inst{1} Precision Mechatronics Laboratory, University of Liege, Belgium \and %
#+LATEX_HEADER_EXTRA:                       \inst{2} BEAMS Department, Free University of Brussels, Belgium \and %
#+LATEX_HEADER_EXTRA:                       \inst{3} European Synchrotron Radiation Facility, Grenoble, France}

#+LATEX_HEADER_EXTRA: \titlegraphic{\includegraphics[height=1.5cm]{figs/logo_pml.png} \hspace{5em} %
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#+LATEX_HEADER_EXTRA:       \includegraphics[height=1.5cm]{figs/logo_isma.png}}

# © 2020 Dehaeze Thomas This work is protected by copyright and, subject to the exceptions expressly laid down by law, may not be communicated to the public in whole or in part, reproduced, processed or stored in a automatized database, in any way whatsoever, without the express prior written consent of the author.

#+DESCRIPTION:
#+KEYWORDS:
#+LANGUAGE:  en

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:END:

* Dynamics of Rotating Platforms
** Model of a Rotating Positioning Platform

#+caption: Schematic of the studied System
#+attr_latex: :width 0.7\linewidth
[[file:figs/system.pdf]]

** Equations of Motion - Lagrangian Formalism

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*}

\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}

#+attr_latex: :options []{blue}{}
#+begin_cbox
#+begin_center
Centrifugal forces $\Longleftrightarrow$ Negative Stiffness

Coriolis Forces $\Longleftrightarrow$ Coupling
#+end_center
#+end_cbox

** Transfer Function Matrix the Laplace domain

\vspace{2em}

#+caption: Campbell Diagram : Evolution of the complex and real parts of the system's poles as a function of the rotational speed $\Omega$
#+attr_latex: :environment subfigure :width 0.49\linewidth :align c
| file:figs/campbell_diagram_real.pdf     | file:figs/campbell_diagram_imag.pdf          |
| <<fig:campbell_diagram_real>> Real Part | <<fig:campbell_diagram_imag>> Imaginary Part |

* Decentralized Integral Force Feedback
** Force Sensors and Decentralized IFF Control Architecture

#+caption: System with added Force Sensor in series with the actuators, $K_F(s) = g \cdot \frac{1}{s}$
#+attr_latex: :width 0.7\linewidth
[[file:figs/system_iff.pdf]]

** IFF Plant Dynamics

#+caption: Bode plot of the dynamics from force actuator to force sensor for several rotational speeds $\Omega$
#+attr_latex: :width \linewidth
[[file:figs/plant_iff_compare_rotating_speed.pdf]]

** Decentralized IFF with Pure Integrators

#+caption: Root Locus for Decentralized IFF for several rotating speeds $\Omega$
#+attr_latex: :width 0.7\linewidth
[[file:figs/root_locus_pure_iff.pdf]]

\vspace{-1em}

#+attr_latex: :options []{blue}{}
#+begin_cbox
\centering For $\Omega > 0$, the closed loop system is unstable
#+end_cbox

* Integral Force Feedback with High Pass Filter
** 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}

#+attr_latex: :options [b]{0.45\linewidth}
#+begin_minipage
#+caption: Loop Gain
#+attr_latex: :width \linewidth :float nil
[[file:figs/loop_gain_modified_iff.pdf]]
#+end_minipage
\hfill
#+attr_latex: :options [b]{0.5\linewidth}
#+begin_minipage
#+caption: Root Locus
#+attr_latex: :width \linewidth :float nil
[[file:figs/root_locus_modified_iff.pdf]]
#+end_minipage


** Effect of $\omega_i$ on the attainable damping

#+caption: Root Locus for several HPF cut-off frequencies $\omega_i$, $\Omega = 0.1 \omega_0$
#+attr_latex: :width \linewidth
[[file:figs/root_locus_wi_modified_iff.pdf]]

\vspace{-2em}

*** Column                                                           :BMCOL:
:PROPERTIES:
:BEAMER_col: 0.3
:END:

\begin{equation*}
  g_{\text{max}} = \omega_i \left( \frac{{\omega_0}^2}{\Omega^2} - 1 \right)
\end{equation*}

*** Column                                                           :BMCOL:
:PROPERTIES:
:BEAMER_col: 0.7
:END:

#+attr_latex: :options []{blue}{}
#+begin_cbox
  small $\omega_i$ $\Longrightarrow$ increase maximum damping

  small $\omega_i$ $\Longrightarrow$ reduces maximum gain $g_\text{max}$
#+end_cbox

** Optimal Control Parameters

\vspace{1em}

#+caption: Attainable damping ratio $\xi_\text{cl}$ as a function of the ratio $\omega_i/\omega_0$. Corresponding control gain $g_\text{opt}$ and $g_\text{max}$ are also shown
#+attr_latex: :width \linewidth
[[file:figs/mod_iff_damping_wi.pdf]]

* Integral Force Feedback with Parallel Springs
** Stiffness in Parallel with the Force Sensor

#+caption: System with additional springs in parallel with the actuators and force sensors
#+attr_latex: :width 0.65\linewidth
[[file:figs/system_parallel_springs.pdf]]

** Effect of the Parallel Stiffness on the Plant Dynamics

#+attr_latex: :options [b]{0.42\linewidth}
#+begin_minipage
#+caption: Bode Plot of $f_u/F_u$
#+attr_latex: :width \linewidth
[[file:figs/plant_iff_kp.pdf]]
#+end_minipage
\hfill
#+attr_latex: :options [b]{0.55\linewidth}
#+begin_minipage
#+caption: Root Locus for IFF
#+attr_latex: :width \linewidth
[[file:figs/root_locus_iff_kp.pdf]]
#+end_minipage

\vspace{1em}

#+attr_latex: :options []{blue}{}
#+begin_cbox
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*.
#+end_cbox

** Optimal Parallel Stiffness

#+caption: Root Locus for three parallel stiffnesses $k_p$
#+attr_latex: :width 0.60\linewidth
[[file:figs/root_locus_iff_kps.pdf]]

#+attr_latex: :options []{blue}{}
#+begin_cbox
\centering
Large parallel stiffness $k_p$ reduces the attainable damping.
#+end_cbox

* Comparison and Discussion
** Comparison of the Attainable Damping

#+caption: Root Locus for the two proposed modifications of decentralized IFF, $\Omega = 0.1 \omega_0$
#+attr_latex: :width 0.7\linewidth
[[file:figs/comp_root_locus.pdf]]

** Comparison Transmissibility and Compliance

#+caption: Comparison of the two proposed Active Damping Techniques, $\Omega = 0.1 \omega_0$
#+attr_latex: :environment subfigure :width 0.49\linewidth :align c
| file:figs/comp_compliance.pdf      | file:figs/comp_transmissibility.pdf            |
| <<fig:comp_compliance>> Compliance | <<fig:comp_transmissibility>> Transmissibility |

** @@latex:@@

\vspace{8em}
#+begin_center
  \Huge Thank you!
#+end_center
\vspace{8em}

Contact: [[mailto:dehaeze.thomas@gmail.com][dehaeze.thomas@gmail.com]]
\vspace{1em}
\small https://tdehaeze.github.io/dehaeze20_contr_stewa_platf/
