9 changed files with 378 additions and 148 deletions
--- a/index.html
+++ b/index.html
@ -3,7 +3,7 @@
 "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
 <html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
 <head>
-<!-- 2020-10-08 jeu. 10:41 -->
+<!-- 2020-08-28 ven. 12:27 -->
 <meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
 <title>Active Damping of Rotating Platforms using Integral Force Feedback</title>
 <meta name="generator" content="Org mode" />
@ -36,57 +36,47 @@ The results reveal that, despite their different implementations, both modified
 </p>
 </blockquote>

-<div id="outline-container-orgf5de8e3" class="outline-2">
-<h2 id="orgf5de8e3">Paper (<a href="paper/dehaeze20_activ_dampin_rotat_platf_integ_force_feedb.pdf">link</a>)</h2>
-<div class="outline-text-2" id="text-orgf5de8e3">
+<div id="outline-container-org8a3e56b" class="outline-2">
+<h2 id="org8a3e56b">Paper (<a href="paper/dehaeze20_activ_dampin_rotat_platf_integ_force_feedb.pdf">link</a>)</h2>
+<div class="outline-text-2" id="text-org8a3e56b">
 <p>
 The paper has been created using <a href="https://orgmode.org/">Org Mode</a> (generating <a href="https://www.latex-project.org/">LaTeX</a> code) under <a href="https://www.gnu.org/software/emacs/">Emacs</a>.
 </p>
 </div>
 </div>

-<div id="outline-container-orgb09588e" class="outline-2">
-<h2 id="orgb09588e">Matlab Scripts (<a href="matlab/index.html">link</a>)</h2>
-<div class="outline-text-2" id="text-orgb09588e">
+<div id="outline-container-org9a11af3" class="outline-2">
+<h2 id="org9a11af3">Matlab Scripts (<a href="matlab/index.html">link</a>)</h2>
+<div class="outline-text-2" id="text-org9a11af3">
 <p>
 The Matlab scripts that permits to obtain all the results presented in the paper are accessible <a href="matlab/index.html">here</a>.
 </p>
 </div>
 </div>

-<div id="outline-container-orgb266de5" class="outline-2">
-<h2 id="orgb266de5">Figures (<a href="tikz/index.html">link</a>)</h2>
-<div class="outline-text-2" id="text-orgb266de5">
+<div id="outline-container-org542f0c3" class="outline-2">
+<h2 id="org542f0c3">Figures (<a href="tikz/index.html">link</a>)</h2>
+<div class="outline-text-2" id="text-org542f0c3">
 <p>
 All the figures in the paper are generated using either <a href="https://sourceforge.net/projects/pgf/">TikZ</a> or <a href="https://inkscape.org/">Inkscape</a>. The code snippets that was used to generate the figures are accessible <a href="tikz/index.html">here</a>.
 </p>
 </div>
 </div>

-<div id="outline-container-org26afb06" class="outline-2">
-<h2 id="org26afb06"><span class="section-number-2">1</span> Talk (<a href="talk/talk.pdf">link</a>)</h2>
-<div class="outline-text-2" id="text-1">
-<iframe width="720"
- height="540"
- src="https://www.youtube.com/embed/F9j2-ge2FPE"
- frameborder="0" allowfullscreen> </iframe>
-</div>
-</div>
-
-<div id="outline-container-orgd36ede9" class="outline-2">
-<h2 id="orgd36ede9">Cite this paper</h2>
-<div class="outline-text-2" id="text-orgd36ede9">
+<div id="outline-container-org0f7ddc7" class="outline-2">
+<h2 id="org0f7ddc7">Cite this paper</h2>
+<div class="outline-text-2" id="text-org0f7ddc7">
 <p>
 To cite this paper use the following bibtex code.
 </p>
 <div class="org-src-container">
-<pre class="src src-bibtex"><span class="org-function-name">@inproceedings</span>{<span class="org-constant">dehaeze20_activ_dampin_rotat_platf_integ_force_feedb</span>,
-  <span class="org-variable-name">author</span>          = {Dehaeze, T. and Collette, C.},
-  <span class="org-variable-name">title</span>           = {Active Damping of Rotating Platforms using Integral Force
+<pre class="src src-bibtex">@inproceedings{dehaeze20_activ_dampin_rotat_platf_integ_force_feedb,
+  author          = {Dehaeze, T. and Collette, C.},
+  title           = {Active Damping of Rotating Platforms using Integral Force
                  Feedback},
-  <span class="org-variable-name">booktitle</span>       = {Proceedings of the International Conference on Modal
+  booktitle       = {Proceedings of the International Conference on Modal
                  Analysis Noise and Vibration Engineering (ISMA)},
-  <span class="org-variable-name">year</span>            = 2020,
+  year            = 2020,
 }
 </pre>
 </div>
--- a/index.org
+++ b/index.org
@ -40,15 +40,6 @@ The Matlab scripts that permits to obtain all the results presented in the paper
 :END:
 All the figures in the paper are generated using either [[https://sourceforge.net/projects/pgf/][TikZ]] or [[https://inkscape.org/][Inkscape]]. The code snippets that was used to generate the figures are accessible [[file:tikz/index.org][here]].

-* Talk ([[file:talk/talk.pdf][link]])
-
-#+begin_export html
-    <iframe width="720"
-     height="540"
-     src="https://www.youtube.com/embed/F9j2-ge2FPE"
-     frameborder="0" allowfullscreen> </iframe>
-#+end_export
-
 * Cite this paper
 :PROPERTIES:
 :UNNUMBERED: t
--- a/paper/figs/loop_gain_modified_iff.svg
+++ b/paper/figs/loop_gain_modified_iff.svg
--- a/talk/figs/iff_vs_passive.pdf
+++ b/talk/figs/iff_vs_passive.pdf
--- a/talk/figs/iff_vs_passive.png
+++ b/talk/figs/iff_vs_passive.png
--- a/talk/figs/plant_iff_compare_rotating_speed.pdf
+++ b/talk/figs/plant_iff_compare_rotating_speed.pdf
--- a/talk/talk.org
+++ b/talk/talk.org
@ -64,30 +64,53 @@
 \fontsize{8pt}{7.2}\selectfont
 :END:

-* Dynamics of Rotating Platforms
+* Dynamics of Rotating Positioning Platforms
 ** Model of a Rotating Positioning Platform
+*** Column                                                           :BMCOL:
+:PROPERTIES:
+:BEAMER_col: 0.55
+:END:

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

-** Equations of Motion - Lagrangian Formalism
+*** Column                                                           :BMCOL:
+:PROPERTIES:
+:BEAMER_col: 0.45
+:END:

-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*}
+Simplest model to study the *gyroscopic effects* on Decentralized IFF

 \vspace{1em}

-Equations of motion:
+Assumptions:
+- Perfect Rotating Stage
+- $\dot{\theta}(t) = \Omega = \text{cst}$
+- Small displacements
+- Position of the mass described by $[d_u\ d_v]$
+
+\vspace{1em}
+
+Two frames:
+- Inertial frame $(\vec{i}_x, \vec{i}_y, \vec{i}_z)$
+- Uniform rotating frame $(\vec{i}_u, \vec{i}_v, \vec{i}_w)$
+
+** Equations of Motion - Lagrangian Formalism
+\vspace{-1em}
+\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 $L = T - V$ the Lagrangian, $D$ the dissipation function, and $Q_i$ the generalized force associated with the generalized variable.
+\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), \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}
 \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
@ -98,25 +121,73 @@ Coriolis Forces $\Longleftrightarrow$ Coupling
 #+end_cbox

 ** Transfer Function Matrix the Laplace domain
-
-\vspace{2em}
+\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*}

 #+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
+#+attr_latex: :environment subfigure :width 0.4\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
+** Bode Plots of the System's Dynamics

-#+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
+#+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                                |
+| <<fig:plant_compare_rotating_speed_direct>> Direct Terms $d_u/F_u$, $d_v/F_v$ | <<fig:plant_compare_rotating_speed_coupling>> Coupling Terms $d_v/F_u$, $-d_u/F_v$ |
+
+For all the numerical analysis, $\omega_0 = \SI{1}{\radian\per\second}$, $k = \SI{1}{\newton\per\meter}$ and $\xi = 0.025 = \SI{2.5}{\percent}$.
+
+* Problem with the Decentralized Integral Force Feedback
+** Force Sensors and Control Architecture
+\vspace{-1em}
+*** Column                                                           :BMCOL:
+:PROPERTIES:
+:BEAMER_col: 0.6
+:END:
+
+#+caption: System with added Force Sensor in series with the actuators
+#+attr_latex: :width \linewidth
 [[file:figs/system_iff.pdf]]

-** IFF Plant Dynamics
+*** Column                                                           :BMCOL:
+:PROPERTIES:
+:BEAMER_col: 0.4
+:END:

-#+caption: Bode plot of the dynamics from force actuator to force sensor for several rotational speeds $\Omega$
+#+caption: Control Diagram for decentralized IFF
 #+attr_latex: :width \linewidth
+[[file:figs/control_diagram_iff.pdf]]
+
+\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*}
+
+** 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*}
+
+#+caption: Bode plot of the diagonal terms of $\bm{G}_f$ for several rotational speeds $\Omega$
+#+attr_latex: :width 0.9\linewidth
 [[file:figs/plant_iff_compare_rotating_speed.pdf]]

 ** Decentralized IFF with Pure Integrators
@ -132,15 +203,15 @@ Coriolis Forces $\Longleftrightarrow$ Coupling
 \centering For $\Omega > 0$, the closed loop system is unstable
 #+end_cbox

-* Integral Force Feedback with High Pass Filter
-** Modification of the Control Law
+* Modification of the control law: Add High-Pass Filter
+** Modification of the Control Low
+
+\vspace{-1em}

 \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
@ -155,6 +226,14 @@ Coriolis Forces $\Longleftrightarrow$ Coupling
 [[file:figs/root_locus_modified_iff.pdf]]
 #+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*}
+

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

@ -193,31 +272,56 @@ Coriolis Forces $\Longleftrightarrow$ Coupling
 #+attr_latex: :width \linewidth
 [[file:figs/mod_iff_damping_wi.pdf]]

-* Integral Force Feedback with Parallel Springs
+* Modification of the Mechanical System: Parallel Stiffness
 ** Stiffness in Parallel with the Force Sensor
+\vspace{-1em}
+*** Column                                                           :BMCOL:
+:PROPERTIES:
+:BEAMER_col: 0.6
+:END:

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

+*** Column                                                           :BMCOL:
+:PROPERTIES:
+:BEAMER_col: 0.4
+:END:
+
+#+attr_latex: :options [Intuitive Idea]{blue}{}
+#+begin_cbox
+  $k_p$ 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
+
 ** Effect of the Parallel Stiffness on the Plant Dynamics

 #+attr_latex: :options [b]{0.42\linewidth}
 #+begin_minipage
-#+caption: Bode Plot of $f_u/F_u$
+#+caption: Bode Plot of $f_u/F_u$ for $k_p = 0$, $k_p < m \Omega^2$ and $k_p > m \Omega^2$, $\Omega = 0.1 \omega_0$
 #+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
+#+caption: Root Locus for IFF without parallel spring, with parallel springs with stiffness $k_p < m \Omega^2$ and $k_p > m \Omega^2$, $\Omega = 0.1 \omega_0$
 #+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*.
@ -225,9 +329,10 @@ If $k_p > m \Omega^2$, the poles of the closed-loop system stay in the (stable)

 ** 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]]
+#+caption: Root Locus for IFF when parallel stiffness $k_p$ is added, $\Omega = 0.1 \omega_0$
+#+attr_latex: :environment subfigure :width 0.49\linewidth :align c
+| file:figs/root_locus_iff_kps.pdf                                          | file:figs/root_locus_opt_gain_iff_kp.pdf                                                           |
+| <<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 |

 #+attr_latex: :options []{blue}{}
 #+begin_cbox
@ -235,7 +340,7 @@ If $k_p > m \Omega^2$, the poles of the closed-loop system stay in the (stable)
 Large parallel stiffness $k_p$ reduces the attainable damping.
 #+end_cbox

-* Comparison and Discussion
+* Comparison of the two Proposed Modifications
 ** Comparison of the Attainable Damping

 #+caption: Root Locus for the two proposed modifications of decentralized IFF, $\Omega = 0.1 \omega_0$
@ -249,14 +354,25 @@ Large parallel stiffness $k_p$ reduces the attainable damping.
 | file:figs/comp_compliance.pdf      | file:figs/comp_transmissibility.pdf            |
 | <<fig:comp_compliance>> Compliance | <<fig:comp_transmissibility>> Transmissibility |

-** @@latex:@@
+** Conclusion & Further work

-\vspace{8em}
-#+begin_center
-  \Huge Thank you!
-#+end_center
-\vspace{8em}
+The two proposed techniques gives almost identical results but are very different when it comes to their implementations

-Contact: [[mailto:dehaeze.thomas@gmail.com][dehaeze.thomas@gmail.com]]
-\vspace{1em}
-\small https://tdehaeze.github.io/dehaeze20_contr_stewa_platf/
+\vspace{2em}
+
+The best technique depends on the application
+
+\vspace{2em}
+
+#+attr_latex: :options {r}{0.45\textwidth}
+#+begin_wrapfigure
+\vspace{-1em}
+#+attr_latex: :width \linewidth
+[[file:figs/apa_schematic.pdf]]
+#+end_wrapfigure
+
+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
--- a/talk/talk.pdf
+++ b/talk/talk.pdf
--- a/talk/talk.tex
+++ b/talk/talk.tex
@ -1,4 +1,4 @@
-% Created 2020-08-24 lun. 18:19
+% Created 2020-07-29 mer. 15:27
 % Intended LaTeX compiler: pdflatex
 \documentclass[t, minted]{clean-beamer}
 \usepackage[utf8]{inputenc}
@ -52,7 +52,7 @@
 pdftitle={Active Damping of Rotating Platforms using Integral Force Feedback},
 pdfkeywords={},
 pdfsubject={},
- pdfcreator={Emacs 27.1.50 (Org mode 9.4)}, 
+ pdfcreator={Emacs 27.0.91 (Org mode 9.4)}, 
 pdflang={English}}
 \begin{document}

@ -62,32 +62,57 @@
 \end{frame}


-\section{Dynamics of Rotating Platforms}
-\label{sec:orge3e010f}
-\begin{frame}[label={sec:orgac66c89}]{Model of a Rotating Positioning Platform}
+\section{Dynamics of Rotating Positioning Platforms}
+\label{sec:orge7d05a8}
+\begin{frame}[label={sec:orgb466daa}]{Model of a Rotating Positioning Platform}
+\begin{columns}
+\begin{column}{0.55\columnwidth}
 \begin{figure}[htbp]
 \centering
-\includegraphics[width=0.7\linewidth]{figs/system.pdf}
+\includegraphics[width=\linewidth]{figs/system.pdf}
 \caption{Schematic of the studied System}
 \end{figure}
-\end{frame}
+\end{column}

-\begin{frame}[label={sec:org614e76f}]{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*}
+\begin{column}{0.45\columnwidth}
+Simplest model to study the \textbf{gyroscopic effects} on Decentralized IFF

 \vspace{1em}

-Equations of motion:
+Assumptions:
+\begin{itemize}
+\item Perfect Rotating Stage
+\item \(\dot{\theta}(t) = \Omega = \text{cst}\)
+\item Small displacements
+\item Position of the mass described by \([d_u\ d_v]\)
+\end{itemize}
+
+\vspace{1em}
+
+Two frames:
+\begin{itemize}
+\item Inertial frame \((\vec{i}_x, \vec{i}_y, \vec{i}_z)\)
+\item Uniform rotating frame \((\vec{i}_u, \vec{i}_v, \vec{i}_w)\)
+\end{itemize}
+\end{column}
+\end{columns}
+\end{frame}
+
+\begin{frame}[label={sec:orgc029b67}]{Equations of Motion - Lagrangian Formalism}
+\vspace{-1em}
+\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 \(L = T - V\) the Lagrangian, \(D\) the dissipation function, and \(Q_i\) the generalized force associated with the generalized variable.
+\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), \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}
 \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
@ -97,16 +122,27 @@ Coriolis Forces \(\Longleftrightarrow\) Coupling
 \end{cbox}
 \end{frame}

-\begin{frame}[label={sec:org0a25ba1}]{Transfer Function Matrix the Laplace domain}
-\vspace{2em}
+\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{figure}[htbp]
-\begin{subfigure}[c]{0.49\linewidth}
+\begin{subfigure}[c]{0.4\linewidth}
 \includegraphics[width=\linewidth]{figs/campbell_diagram_real.pdf}
 \caption{\label{fig:campbell_diagram_real} Real Part}
 \end{subfigure}
 \hfill
-\begin{subfigure}[c]{0.49\linewidth}
+\begin{subfigure}[c]{0.4\linewidth}
 \includegraphics[width=\linewidth]{figs/campbell_diagram_imag.pdf}
 \caption{\label{fig:campbell_diagram_imag} Imaginary Part}
 \end{subfigure}
@ -116,25 +152,73 @@ Coriolis Forces \(\Longleftrightarrow\) Coupling
 \end{figure}
 \end{frame}

-\section{Decentralized Integral Force Feedback}
-\label{sec:org8107aa6}
-\begin{frame}[label={sec:org818bd83}]{Force Sensors and Decentralized IFF Control Architecture}
+\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}
 \begin{figure}[htbp]
 \centering
-\includegraphics[width=0.7\linewidth]{figs/system_iff.pdf}
-\caption{System with added Force Sensor in series with the actuators, \(K_F(s) = g \cdot \frac{1}{s}\)}
+\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 \(\Omega\)}
 \end{figure}
 \end{frame}

-\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{frame}[label={sec:orgb8d521c}]{Decentralized IFF with Pure Integrators}
 \begin{figure}[htbp]
 \centering
 \includegraphics[width=0.7\linewidth]{figs/root_locus_pure_iff.pdf}
@ -148,15 +232,15 @@ Coriolis Forces \(\Longleftrightarrow\) Coupling
 \end{cbox}
 \end{frame}

-\section{Integral Force Feedback with High Pass Filter}
-\label{sec:org789619d}
-\begin{frame}[label={sec:org2020343}]{Modification of the Control Law}
+\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}
+
 \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}
@ -170,10 +254,18 @@ Coriolis Forces \(\Longleftrightarrow\) Coupling
 \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:org38b9bd2}]{Effect of \(\omega_i\) on the attainable damping}
+\begin{frame}[label={sec:org4390eac}]{Effect of \(\omega_i\) on the attainable damping}
 \begin{figure}[htbp]
 \centering
 \includegraphics[width=\linewidth]{figs/root_locus_wi_modified_iff.pdf}
@ -199,7 +291,7 @@ small \(\omega_i\) \(\Longrightarrow\) reduces maximum gain \(g_\text{max}\)
 \end{columns}
 \end{frame}

-\begin{frame}[label={sec:orga2c2166}]{Optimal Control Parameters}
+\begin{frame}[label={sec:org84c72de}]{Optimal Control Parameters}
 \vspace{1em}

 \begin{figure}[htbp]
@ -209,22 +301,45 @@ small \(\omega_i\) \(\Longrightarrow\) reduces maximum gain \(g_\text{max}\)
 \end{figure}
 \end{frame}

-\section{Integral Force Feedback with Parallel Springs}
-\label{sec:org7a61560}
-\begin{frame}[label={sec:orga6be4f6}]{Stiffness in Parallel with the Force Sensor}
+\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}
 \begin{figure}[htbp]
 \centering
-\includegraphics[width=0.65\linewidth]{figs/system_parallel_springs.pdf}
-\caption{System with additional springs in parallel with the actuators and force sensors}
+\includegraphics[width=\linewidth]{figs/system_parallel_springs.pdf}
+\caption{Studied 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:org497c282}]{Effect of the Parallel Stiffness on the Plant Dynamics}
+\begin{frame}[label={sec:org223db59}]{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 \(f_u/F_u\)}
+\caption{Bode Plot of \(f_u/F_u\) for \(k_p = 0\), \(k_p < m \Omega^2\) and \(k_p > m \Omega^2\), \(\Omega = 0.1 \omega_0\)}
 \end{figure}
 \end{minipage}
 \hfill
@ -232,22 +347,29 @@ small \(\omega_i\) \(\Longrightarrow\) reduces maximum gain \(g_\text{max}\)
 \begin{figure}[htbp]
 \centering
 \includegraphics[width=\linewidth]{figs/root_locus_iff_kp.pdf}
-\caption{Root Locus for IFF}
+\caption{Root Locus for IFF without parallel spring, with parallel springs with stiffness \(k_p < m \Omega^2\) and \(k_p > m \Omega^2\), \(\Omega = 0.1 \omega_0\)}
 \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:orgd7eccc9}]{Optimal Parallel Stiffness}
+\begin{frame}[label={sec:org8be51fd}]{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}{}
@ -256,9 +378,9 @@ Large parallel stiffness \(k_p\) reduces the attainable damping.
 \end{cbox}
 \end{frame}

-\section{Comparison and Discussion}
-\label{sec:org79d4359}
-\begin{frame}[label={sec:org7dda174}]{Comparison of the Attainable Damping}
+\section{Comparison of the two Proposed Modifications}
+\label{sec:orge227508}
+\begin{frame}[label={sec:org783f2c4}]{Comparison of the Attainable Damping}
 \begin{figure}[htbp]
 \centering
 \includegraphics[width=0.7\linewidth]{figs/comp_root_locus.pdf}
@ -266,7 +388,7 @@ Large parallel stiffness \(k_p\) reduces the attainable damping.
 \end{figure}
 \end{frame}

-\begin{frame}[label={sec:org9effb95}]{Comparison Transmissibility and Compliance}
+\begin{frame}[label={sec:orgdd42828}]{Comparison Transmissibility and Compliance}
 \begin{figure}[htbp]
 \begin{subfigure}[c]{0.49\linewidth}
 \includegraphics[width=\linewidth]{figs/comp_compliance.pdf}
@ -283,15 +405,26 @@ Large parallel stiffness \(k_p\) reduces the attainable damping.
 \end{figure}
 \end{frame}

-\begin{frame}[label={sec:org46c3efd}]{}
-\vspace{8em}
-\begin{center}
-\Huge Thank you!
-\end{center}
-\vspace{8em}
+\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

-Contact: \href{mailto:dehaeze.thomas@gmail.com}{dehaeze.thomas@gmail.com}
-\vspace{1em}
-\small \url{https://tdehaeze.github.io/dehaeze20\_contr\_stewa\_platf/}
+\vspace{2em}
+
+The best technique depends on the application
+
+\vspace{2em}
+
+\begin{wrapfigure}{r}{0.45\textwidth}
+\vspace{-1em}
+\begin{center}
+\includegraphics[width=\linewidth]{figs/apa_schematic.pdf}
+\end{center}
+\end{wrapfigure}
+
+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
 \end{frame}
 \end{document}