Initial Commit

css/htmlize.css

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js/jquery.stickytableheaders.min.js

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js/readtheorg.js

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+$(function() {
+    $('.note').before("<p class='admonition-title note'>Note</p>");
+    $('.seealso').before("<p class='admonition-title seealso'>See also</p>");
+    $('.warning').before("<p class='admonition-title warning'>Warning</p>");
+    $('.caution').before("<p class='admonition-title caution'>Caution</p>");
+    $('.attention').before("<p class='admonition-title attention'>Attention</p>");
+    $('.tip').before("<p class='admonition-title tip'>Tip</p>");
+    $('.important').before("<p class='admonition-title important'>Important</p>");
+    $('.hint').before("<p class='admonition-title hint'>Hint</p>");
+    $('.error').before("<p class='admonition-title error'>Error</p>");
+    $('.danger').before("<p class='admonition-title danger'>Danger</p>");
+});
+
+$( document ).ready(function() {
+
+    // Shift nav in mobile when clicking the menu.
+    $(document).on('click', "[data-toggle='wy-nav-top']", function() {
+      $("[data-toggle='wy-nav-shift']").toggleClass("shift");
+      $("[data-toggle='rst-versions']").toggleClass("shift");
+    });
+    // Close menu when you click a link.
+    $(document).on('click', ".wy-menu-vertical .current ul li a", function() {
+      $("[data-toggle='wy-nav-shift']").removeClass("shift");
+      $("[data-toggle='rst-versions']").toggleClass("shift");
+    });
+    $(document).on('click', "[data-toggle='rst-current-version']", function() {
+      $("[data-toggle='rst-versions']").toggleClass("shift-up");
+    });
+    // Make tables responsive
+    $("table.docutils:not(.field-list)").wrap("<div class='wy-table-responsive'></div>");
+});
+
+$( document ).ready(function() {
+    $('#text-table-of-contents ul').first().addClass('nav');
+                                        // ScrollSpy also requires that we use
+                                        // a Bootstrap nav component.
+    $('body').scrollspy({target: '#text-table-of-contents'});
+
+    // add sticky table headers
+    $('table').stickyTableHeaders();
+
+    // set the height of tableOfContents
+    var $postamble = $('#postamble');
+    var $tableOfContents = $('#table-of-contents');
+    $tableOfContents.css({paddingBottom: $postamble.outerHeight()});
+
+    // add TOC button
+    var toggleSidebar = $('<div id="toggle-sidebar"><a href="#table-of-contents"><h2>Table of Contents</h2></a></div>');
+    $('#content').prepend(toggleSidebar);
+
+    // add close button when sidebar showed in mobile screen
+    var closeBtn = $('<a class="close-sidebar" href="#">Close</a>');
+    var tocTitle = $('#table-of-contents').find('h2');
+    tocTitle.append(closeBtn);
+});
+
+window.SphinxRtdTheme = (function (jquery) {
+    var stickyNav = (function () {
+        var navBar,
+            win,
+            stickyNavCssClass = 'stickynav',
+            applyStickNav = function () {
+                if (navBar.height() <= win.height()) {
+                    navBar.addClass(stickyNavCssClass);
+                } else {
+                    navBar.removeClass(stickyNavCssClass);
+                }
+            },
+            enable = function () {
+                applyStickNav();
+                win.on('resize', applyStickNav);
+            },
+            init = function () {
+                navBar = jquery('nav.wy-nav-side:first');
+                win    = jquery(window);
+            };
+        jquery(init);
+        return {
+            enable : enable
+        };
+    }());
+    return {
+        StickyNav : stickyNav
+    };
+}($));

rotating_frame.org

@@ -0,0 +1,280 @@
+#+TITLE: Control in a rotating frame
+#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="css/htmlize.css"/>
+#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="css/readtheorg.css"/>
+#+HTML_HEAD: <script src="js/jquery.min.js"></script>
+#+HTML_HEAD: <script src="js/bootstrap.min.js"></script>
+#+HTML_HEAD: <script type="text/javascript" src="js/jquery.stickytableheaders.min.js"></script>
+#+HTML_HEAD: <script type="text/javascript" src="js/readtheorg.js"></script>
+#+LATEX_CLASS: cleanreport
+#+LaTeX_CLASS_OPTIONS: [tocnp, secbreak, minted]
+#+STARTUP: overview
+#+LaTeX_HEADER: \usepackage{svg}
+#+LaTeX_HEADER: \newcommand{\authorFirstName}{Thomas}
+#+LaTeX_HEADER: \newcommand{\authorLastName}{Dehaeze}
+#+LaTeX_HEADER: \newcommand{\authorEmail}{dehaeze.thomas@gmail.com}
+#+PROPERTY: header-args:matlab  :session *MATLAB*
+#+PROPERTY: header-args:matlab+ :tangle rotating_frame.m
+#+PROPERTY: header-args:matlab+ :comments org
+#+PROPERTY: header-args:matlab+ :exports both
+#+PROPERTY: header-args:matlab+ :eval no-export
+#+PROPERTY: header-args:matlab+ :output-dir Figures
+
+* Goal of this note
+The control objective is to stabilize the position of a rotating object with respect to a non-rotating frame.
+
+The actuators are also rotating with the object.
+
+We want to compare the two different approach:
+- the measurement is made in the fixed frame
+- the measurement is made in the rotating frame
+
+* System
+  <<sec:system>>
+** System description
+The system consists of one 2 degree of freedom translation stage on top of a spindle (figure [[fig:rotating_frame_2dof]]).
+
+The control inputs are the forces applied in the translation stage ($F_u$ and $F_v$). As the translation stage is rotating around the Z axis due to the spindle, the forces are applied along $u$ and $v$.
+
+The measurement is either the $x-y$ displacement of the object located on top of the translation stage or the $u-v$ displacement of the actuators.
+
+#+name: fig:rotating_frame_2dof
+#+caption: Schematic of the mecanical system
+[[./Figures/rotating_frame_2dof.png]]
+
+In the following block diagram:
+- $G$ is the transfer function from the forces applied in the actuators to the measurement
+- $K$ is the controller to design
+- $J$ is a Jacobian matrix usually used to change the reference frame
+
+Indices $x$ and $y$ corresponds to signals in the fixed reference frame (along $\vec{i}_x$ and $\vec{i}_y$):
+- $D_x$ is the measured position of the sample
+- $r_x$ is the reference signal which corresponds to the wanted $D_x$
+- $\epsilon_x$ is the position error
+
+Indices $u$ and $v$ corresponds to signals in the rotating reference frame ($\vec{i}_u$ and $\vec{i}_v$):
+- $F_u$ and $F_v$ are forces applied by the actuators
+- $\epsilon_u$ and $\epsilon_v$ are position error of the sample along $\vec{i}_u$ and $\vec{i}_v$
+
+** Equations
+  <<sec:equations>>
+
+Based on the figure [[fig:rotating_frame_2dof]], we can write the equations of motion of the system.
+
+Let's express the kinetic energy $T$ and the potential energy $V$ of the mass $m$:
+#+name: eq:energy_inertial_frame
+\begin{align}
+T & = \frac{1}{2} m \left( \dot{x}^2 + \dot{y}^2 \right) \\
+V & = \frac{1}{2} k \left( x^2 + y^2 \right)
+\end{align}
+
+The Lagrangian is the kinetic energy minus the potential energy.
+#+name: eq:lagrangian_inertial_frame
+\begin{equation}
+L = T-V = \frac{1}{2} m \left( \dot{x}^2 + \dot{y}^2 \right) - \frac{1}{2} k \left( x^2 + y^2 \right)
+\end{equation}
+
+
+The partial derivatives of the Lagrangian with respect to the variables $(x, y)$ are:
+#+name: eq:inertial_frame_deriv
+\begin{align*}
+\frac{\partial L}{\partial x} & = -kx \\
+\frac{\partial L}{\partial y} & = -ky \\
+\frac{d}{dt}\frac{\partial L}{\partial \dot{x}} & = m\ddot{x} \\
+\frac{d}{dt}\frac{\partial L}{\partial \dot{y}} & = m\ddot{y}
+\end{align*}
+
+The external forces applied to the mass are:
+\begin{align*}
+F_{\text{ext}, x} &= F_u \cos{\theta} - F_v \sin{\theta}\\
+F_{\text{ext}, y} &= F_u \sin{\theta} + F_v \cos{\theta}
+\end{align*}
+
+By appling the Lagrangian equations, we obtain:
+\begin{align}
+m\ddot{x} + kx = F_u \cos{\theta} - F_v \sin{\theta}\\
+m\ddot{y} + ky = F_u \sin{\theta} + F_v \cos{\theta}
+\end{align}
+
+We then change coordinates from $(x, y)$ to $(d_x, d_y, \theta)$.
+\begin{align*}
+x & = d_u \cos{\theta} - d_v \sin{\theta}\\
+y & = d_u \sin{\theta} + d_v \cos{\theta}
+\end{align*}
+
+We obtain:
+\begin{align*}
+\ddot{x} & = \ddot{d_u} \cos{\theta} - 2\dot{d_u}\dot{\theta}\sin{\theta} - d_u\ddot{\theta}\sin{\theta} - d_u\dot{\theta}^2 \cos{\theta}
+           - \ddot{d_v} \sin{\theta} - 2\dot{d_v}\dot{\theta}\cos{\theta} - d_v\ddot{\theta}\cos{\theta} + d_v\dot{\theta}^2 \sin{\theta} \\
+\ddot{y} & = \ddot{d_u} \sin{\theta} + 2\dot{d_u}\dot{\theta}\cos{\theta} + d_u\ddot{\theta}\cos{\theta} - d_u\dot{\theta}^2 \sin{\theta}
+           + \ddot{d_v} \cos{\theta} - 2\dot{d_v}\dot{\theta}\sin{\theta} - d_v\ddot{\theta}\sin{\theta} - d_v\dot{\theta}^2 \cos{\theta} \\
+\end{align*}
+
+By injecting the previous result into the Lagrangian equation [[eq:lagrangian_eq_inertial]], we obtain:
+\begin{align*}
+ m \ddot{d_u} \cos{\theta} - 2m\dot{d_u}\dot{\theta}\sin{\theta} - m d_u\ddot{\theta}\sin{\theta} - m d_u\dot{\theta}^2 \cos{\theta}
+-m \ddot{d_v} \sin{\theta} - 2m\dot{d_v}\dot{\theta}\cos{\theta} - m d_v\ddot{\theta}\cos{\theta} + m d_v\dot{\theta}^2 \sin{\theta}
++ k d_u \cos{\theta} - k d_v \sin{\theta} = F_u \cos{\theta} - F_v \sin{\theta} \\
+  m \ddot{d_u} \sin{\theta} + 2m\dot{d_u}\dot{\theta}\cos{\theta} + m d_u\ddot{\theta}\cos{\theta} - m d_u\dot{\theta}^2 \sin{\theta}
++ m \ddot{d_v} \cos{\theta} - 2m\dot{d_v}\dot{\theta}\sin{\theta} - m d_v\ddot{\theta}\sin{\theta} - m d_v\dot{\theta}^2 \cos{\theta}
++ k d_u \sin{\theta} + k d_v \cos{\theta} = F_u \sin{\theta} + F_v \cos{\theta}
+\end{align*}
+
+Which is equivalent to:
+\begin{align*}
+ m \ddot{d_u} - 2m\dot{d_u}\dot{\theta}\frac{\sin{\theta}}{\cos{\theta}} - m d_u\ddot{\theta}\frac{\sin{\theta}}{\cos{\theta}} - m d_u\dot{\theta}^2
+-m \ddot{d_v} \frac{\sin{\theta}}{\cos{\theta}} - 2m\dot{d_v}\dot{\theta} - m d_v\ddot{\theta} + m d_v\dot{\theta}^2 \frac{\sin{\theta}}{\cos{\theta}}
++ k d_u - k d_v \frac{\sin{\theta}}{\cos{\theta}} = F_u - F_v \frac{\sin{\theta}}{\cos{\theta}} \\
+  m \ddot{d_u} + 2m\dot{d_u}\dot{\theta}\frac{\cos{\theta}}{\sin{\theta}} + m d_u\ddot{\theta}\frac{\cos{\theta}}{\sin{\theta}} - m d_u\dot{\theta}^2
++ m \ddot{d_v} \frac{\cos{\theta}}{\sin{\theta}} - 2m\dot{d_v}\dot{\theta} - m d_v\ddot{\theta} - m d_v\dot{\theta}^2 \frac{\cos{\theta}}{\sin{\theta}}
++ k d_u + k d_v \frac{\cos{\theta}}{\sin{\theta}} = F_u + F_v \frac{\cos{\theta}}{\sin{\theta}}
+\end{align*}
+
+We can then subtract and add the previous equations to obtain the following equations:
+#+begin_important
+\begin{align*}
+ m \ddot{d_u} + (k - m\dot{\theta}^2) d_u &= F_u + 2 m\dot{d_v}\dot{\theta} + m d_v\ddot{\theta} \\
+ m \ddot{d_v} + (k - m\dot{\theta}^2) d_v &= F_v - 2 m\dot{d_u}\dot{\theta} - m d_u\ddot{\theta} \\
+\end{align*}
+#+end_important
+
+** Analysis
+We obtain two differential equations that are coupled through:
+- *Euler forces*: $m d_v \ddot{\theta}$
+- *Coriolis forces*: $2 m \dot{d_v} \dot{\theta}$
+
+Without the coupling terms, each equation is the equation of a one degree of freedom mass-spring system with mass $m$ and stiffness $k-d_u m\dot{\theta}^2$.
+Thus, the term $-d_u m\dot{\theta}^2$ acts like a negative stiffness (due to *centrifugal forces*).
+
+* Analytical Computation of forces for the NASS
+For the NASS, the Euler forces should be less of a problem as $\ddot{\theta}$ should be very small when conducting an experiment.
+
+First we will determine the value for Euler and Coriolis forces during regular experiment.
+
+** Euler and Coriolis forces
+
+Let's define the parameters for the NASS.
+#+begin_src matlab :exports code :results silent
+  mlight = 35; % [kg]
+  mheavy = 85; % [kg]
+
+  wlight = 2*pi; % [rad/s]
+  wheavy = 2*pi/60; % [rad/s]
+
+  wdot = 1; % [rad/s2]
+
+  d = 0.1; % [m]
+  ddot = 0.2; % [m/s]
+#+end_src
+
+#+begin_src matlab :exports none :results silent
+  Felight = mlight*d*wdot;
+  Feheavy = mheavy*d*wdot;
+
+  Fclight = 2*mlight*d*wlight;
+  Fcheavy = 2*mheavy*d*wheavy;
+#+end_src
+
+We then compute the corresponding values of the Coriolis and Euler forces, and the obtained values are displayed in table [[tab:euler_coriolis]].
+
+#+begin_src matlab :results value table :exports results :post addhdr(*this*)
+  ans = sprintf(' | Light | Heavy | \n Coriolis | %.1f N | %.1f N | \n Euler | %.1f N | %.1f N', Fclight, Fcheavy, Felight, Feheavy)
+#+end_src
+
+#+NAME: tab:euler_coriolis
+#+CAPTION: Euler and Coriolis forces for the NASS
+#+RESULTS:
+|          | Light  | Heavy |
+|----------+--------+-------|
+| Coriolis | 44.0 N | 1.8 N |
+| Euler    | 3.5 N  | 8.5 N |
+
+** Spring Softening Effect
+#+begin_src matlab :exports none :results silent
+  Klight = mlight*d*wdot^2;
+  Kheavy = mheavy*d*wdot^2;
+#+end_src
+
+The values for the spring softening effect are displayed in table [[tab:spring_softening]].
+This is definitely negligible when using piezoelectric actuators. It may not be the case when using voice coil actuators.
+
+#+begin_src matlab :results value table :exports results :post addhdr(*this*)
+  ans = sprintf(' | Light | Heavy | \n Spring Soft. | %.1f N/m | %.1f N/m', Klight, Kheavy)
+#+end_src
+
+#+NAME: tab:spring_softening
+#+CAPTION: Spring Softening effect
+#+RESULTS:
+|              | Light   | Heavy   |
+|--------------+---------+---------|
+| Spring Soft. | 3.5 N/m | 8.5 N/m |
+
+* Control Strategies
+  <<sec:control_strategies>>
+** Measurement in the fixed reference frame
+First, let's consider a measurement in the fixed referenced frame.
+
+The transfer function from actuator $[F_u, F_v]$ to sensor $[D_x, D_y]$ is then $G(\theta)$.
+
+Then the measurement is subtracted to the reference signal $[r_x, r_y]$ to obtain the position error in the fixed reference frame $[\epsilon_x, \epsilon_y]$.
+
+The position error $[\epsilon_x, \epsilon_y]$ is then express in the rotating frame corresponding to the actuators $[\epsilon_u, \epsilon_v]$.
+
+Finally, the control low $K$ links the position errors $[\epsilon_u, \epsilon_v]$ to the actuator forces $[F_u, F_v]$.
+
+The block diagram is shown on figure [[fig:control_measure_fixed_2dof]].
+
+#+name: fig:control_measure_fixed_2dof
+#+caption: Control with a measure from fixed frame
+[[./Figures/control_measure_fixed_2dof.png]]
+
+The loop gain is then $L = G(\theta) K J(\theta)$.
+
+*** QUESTION Is the loop gain is changing with the angle ?
+Is \[ G(\theta) J(\theta) = G(\theta_0) J(\theta_0) \] ?
+
+** Measurement in the rotating frame
+Let's consider that the measurement is in the rotating reference frame.
+
+The corresponding block diagram is shown figure [[fig:control_measure_rotating_2dof]]
+
+#+name: fig:control_measure_rotating_2dof
+#+caption: Control with a measure from rotating frame
+[[./Figures/control_measure_rotating_2dof.png]]
+
+The loop gain is $L = G K$.
+
+* Effect of the rotating Speed
+  <<sec:effect_rot_speed>>
+** TODO Use realistic parameters for the mass of the sample and stiffness of the X-Y stage
+
+** TODO Check if the plant is changing a lot when we are not turning to when we are turning at the maximum speed (60rpm)
+
+* Effect of the X-Y stage stiffness
+  <<sec:effect_stiffness>>
+** TODO At full speed, check how the coupling changes with the stiffness of the actuators
+* Noweb Snippets                                                   :noexport:
+  :PROPERTIES:
+  :HEADER-ARGS:matlab+: :results none :exports none :tangle no
+  :HEADER-ARGS:emacs-lisp+: :tangle no :eval no-export
+  :END:
+
+** Matlab Init
+#+NAME: matlab-init
+#+BEGIN_SRC matlab :results none :exports none
+  clear; close all; clc;
+  addpath('./src/');
+  ans = 0;
+#+END_SRC
+
+** Matlab Export Figure
+#+NAME: matlab-export-figure
+#+BEGIN_SRC matlab :var fn="figure" ft="png" size="normal-normal" :exports none
+  exportFig(fn, size);
+  ans = [fn, ".", ft];
+#+END_SRC
+
+** Add hline to org table
+#+name: addhdr
+#+begin_src emacs-lisp :var tbl=""
+(cons (car tbl) (cons 'hline (cdr tbl)))
+#+end_src