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index.html
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index.html
@ -3,7 +3,7 @@
|
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"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
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<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
|
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<head>
|
||||
<!-- 2020-08-28 ven. 12:27 -->
|
||||
<!-- 2020-10-08 jeu. 10:41 -->
|
||||
<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" />
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@ -36,47 +36,57 @@ The results reveal that, despite their different implementations, both modified
|
||||
</p>
|
||||
</blockquote>
|
||||
|
||||
<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">
|
||||
<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">
|
||||
<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-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">
|
||||
<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">
|
||||
<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-org542f0c3" class="outline-2">
|
||||
<h2 id="org542f0c3">Figures (<a href="tikz/index.html">link</a>)</h2>
|
||||
<div class="outline-text-2" id="text-org542f0c3">
|
||||
<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">
|
||||
<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-org0f7ddc7" class="outline-2">
|
||||
<h2 id="org0f7ddc7">Cite this paper</h2>
|
||||
<div class="outline-text-2" id="text-org0f7ddc7">
|
||||
<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">
|
||||
<p>
|
||||
To cite this paper use the following bibtex code.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<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
|
||||
<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
|
||||
Feedback},
|
||||
booktitle = {Proceedings of the International Conference on Modal
|
||||
<span class="org-variable-name">booktitle</span> = {Proceedings of the International Conference on Modal
|
||||
Analysis Noise and Vibration Engineering (ISMA)},
|
||||
year = 2020,
|
||||
<span class="org-variable-name">year</span> = 2020,
|
||||
}
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
@ -40,6 +40,15 @@ 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
|
||||
|
||||
Binary file not shown.
|
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talk/figs/iff_vs_passive.pdf
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talk/figs/iff_vs_passive.pdf
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talk/figs/iff_vs_passive.png
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talk/figs/iff_vs_passive.png
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204
talk/talk.org
204
talk/talk.org
@ -64,53 +64,30 @@
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||||
\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 |
|
||||
| <<fig:campbell_diagram_real>> Real Part | <<fig:campbell_diagram_imag>> 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 |
|
||||
| <<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
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||||
#+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 |
|
||||
| <<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 |
|
||||
#+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 |
|
||||
| <<fig:comp_compliance>> Compliance | <<fig:comp_transmissibility>> 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/
|
||||
|
||||
BIN
talk/talk.pdf
BIN
talk/talk.pdf
Binary file not shown.
261
talk/talk.tex
261
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}
|
||||
|
||||
Loading…
Reference in New Issue
Block a user