diff --git a/paper/figs/loop_gain_modified_iff.svg b/paper/figs/loop_gain_modified_iff.svg index 10e1c7c..c3c0a5e 100644 Binary files a/paper/figs/loop_gain_modified_iff.svg and b/paper/figs/loop_gain_modified_iff.svg differ diff --git a/talk/figs/iff_vs_passive.pdf b/talk/figs/iff_vs_passive.pdf new file mode 100644 index 0000000..84a08bd Binary files /dev/null and b/talk/figs/iff_vs_passive.pdf differ diff --git a/talk/figs/iff_vs_passive.png b/talk/figs/iff_vs_passive.png new file mode 100644 index 0000000..fa4a76b Binary files /dev/null and b/talk/figs/iff_vs_passive.png differ diff --git a/talk/figs/plant_iff_compare_rotating_speed.pdf b/talk/figs/plant_iff_compare_rotating_speed.pdf index 5420a74..ca00621 100644 Binary files a/talk/figs/plant_iff_compare_rotating_speed.pdf and b/talk/figs/plant_iff_compare_rotating_speed.pdf differ diff --git a/talk/talk.org b/talk/talk.org index e4d833d..2f83311 100644 --- a/talk/talk.org +++ b/talk/talk.org @@ -64,53 +64,30 @@ \fontsize{8pt}{7.2}\selectfont :END: -* Dynamics of Rotating Positioning Platforms +* Dynamics of Rotating Platforms ** Model of a Rotating Positioning Platform -*** Column :BMCOL: -:PROPERTIES: -:BEAMER_col: 0.55 -:END: #+caption: Schematic of the studied System -#+attr_latex: :width \linewidth +#+attr_latex: :width 0.7\linewidth [[file:figs/system.pdf]] -*** Column :BMCOL: -:PROPERTIES: -:BEAMER_col: 0.45 -:END: - -Simplest model to study the *gyroscopic effects* on Decentralized IFF - -\vspace{1em} - -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} + +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 $L = T - V$ 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} + +\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 @@ -121,73 +98,25 @@ Coriolis Forces $\Longleftrightarrow$ Coupling #+end_cbox ** 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*} + +\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 0.4\linewidth :align c +#+attr_latex: :environment subfigure :width 0.49\linewidth :align c | file:figs/campbell_diagram_real.pdf | file:figs/campbell_diagram_imag.pdf | | <> Real Part | <> Imaginary Part | -** Bode Plots of the System's Dynamics +* Decentralized Integral Force Feedback +** Force Sensors and Decentralized IFF Control Architecture -#+caption: Bode Plots for $\bm{G}_d$ for several rotational speed $\Omega$ -#+attr_latex: :environment subfigure :width 0.45\linewidth :align c -| file:figs/plant_compare_rotating_speed_direct.pdf | file:figs/plant_compare_rotating_speed_coupling.pdf | -| <> Direct Terms $d_u/F_u$, $d_v/F_v$ | <> Coupling Terms $d_v/F_u$, $-d_u/F_v$ | - -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 +#+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]] -*** Column :BMCOL: -:PROPERTIES: -:BEAMER_col: 0.4 -:END: +** IFF Plant Dynamics -#+caption: Control Diagram for decentralized IFF +#+caption: Bode plot of the dynamics from force actuator to force sensor for several rotational speeds $\Omega$ #+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 @@ -203,15 +132,15 @@ For all the numerical analysis, $\omega_0 = \SI{1}{\radian\per\second}$, $k = \S \centering For $\Omega > 0$, the closed loop system is unstable #+end_cbox -* Modification of the control law: Add High-Pass Filter -** Modification of the Control Low - -\vspace{-1em} +* 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 @@ -226,14 +155,6 @@ For all the numerical analysis, $\omega_0 = \SI{1}{\radian\per\second}$, $k = \S [[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 @@ -272,56 +193,31 @@ For all the numerical analysis, $\omega_0 = \SI{1}{\radian\per\second}$, $k = \S #+attr_latex: :width \linewidth [[file:figs/mod_iff_damping_wi.pdf]] -* Modification of the Mechanical System: Parallel Stiffness +* Integral Force Feedback with Parallel Springs ** Stiffness in Parallel with the Force Sensor -\vspace{-1em} -*** Column :BMCOL: -:PROPERTIES: -:BEAMER_col: 0.6 -:END: -#+caption: Studied system with additional springs in parallel with the actuators and force sensors -#+attr_latex: :width \linewidth +#+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]] -*** 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 $f_u/F_u$ for $k_p = 0$, $k_p < m \Omega^2$ and $k_p > m \Omega^2$, $\Omega = 0.1 \omega_0$ +#+caption: Bode Plot of $f_u/F_u$ #+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 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 #+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*. @@ -329,10 +225,9 @@ If $k_p > m \Omega^2$, the poles of the closed-loop system stay in the (stable) ** Optimal Parallel Stiffness -#+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 | -| <> Comparison of three parallel stiffnesses $k_p$ | <> $k_p = 5 m \Omega^2$, optimal damping $\xi_\text{opt}$ is shown | +#+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 @@ -340,7 +235,7 @@ 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 of the two Proposed Modifications +* Comparison and Discussion ** Comparison of the Attainable Damping #+caption: Root Locus for the two proposed modifications of decentralized IFF, $\Omega = 0.1 \omega_0$ @@ -354,25 +249,14 @@ Large parallel stiffness $k_p$ reduces the attainable damping. | file:figs/comp_compliance.pdf | file:figs/comp_transmissibility.pdf | | <> Compliance | <> Transmissibility | -** Conclusion & Further work +** @@latex:@@ -The two proposed techniques gives almost identical results but are very different when it comes to their implementations +\vspace{8em} +#+begin_center + \Huge Thank you! +#+end_center +\vspace{8em} -\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 +Contact: [[mailto:dehaeze.thomas@gmail.com][dehaeze.thomas@gmail.com]] +\vspace{1em} +\small https://tdehaeze.github.io/dehaeze20_contr_stewa_platf/ diff --git a/talk/talk.pdf b/talk/talk.pdf index 8bf114e..e665fc6 100644 Binary files a/talk/talk.pdf and b/talk/talk.pdf differ diff --git a/talk/talk.tex b/talk/talk.tex index a6c5e47..615fb83 100644 --- a/talk/talk.tex +++ b/talk/talk.tex @@ -1,4 +1,4 @@ -% Created 2020-07-29 mer. 15:27 +% Created 2020-08-24 lun. 18:19 % 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 27.0.91 (Org mode 9.4)}, + pdfcreator={Emacs 27.1.50 (Org mode 9.4)}, pdflang={English}} \begin{document} @@ -62,57 +62,32 @@ \end{frame} -\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} +\section{Dynamics of Rotating Platforms} +\label{sec:orge3e010f} +\begin{frame}[label={sec:orgac66c89}]{Model of a Rotating Positioning Platform} \begin{figure}[htbp] \centering -\includegraphics[width=\linewidth]{figs/system.pdf} +\includegraphics[width=0.7\linewidth]{figs/system.pdf} \caption{Schematic of the studied System} \end{figure} -\end{column} - -\begin{column}{0.45\columnwidth} -Simplest model to study the \textbf{gyroscopic effects} on Decentralized IFF - -\vspace{1em} - -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{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*} -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} + +\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}