445 lines
17 KiB
Org Mode
445 lines
17 KiB
Org Mode
#+TITLE: Kinematics of the station
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:DRAWER:
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#+STARTUP: overview
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#+PROPERTY: header-args:matlab :session *MATLAB*
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#+PROPERTY: header-args:matlab+ :comments org
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#+PROPERTY: header-args:matlab+ :results none
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#+PROPERTY: header-args:matlab+ :exports both
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#+PROPERTY: header-args:matlab+ :eval no-export
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#+PROPERTY: header-args:matlab+ :output-dir figs
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#+PROPERTY: header-args:matlab+ :tangle no
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#+PROPERTY: header-args:matlab+ :mkdirp yes
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#+PROPERTY: header-args:shell :eval no-export
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#+PROPERTY: header-args:latex :headers '("\\usepackage{tikz}" "\\usepackage{import}" "\\import{$HOME/Cloud/thesis/latex/org/}{config.tex}")
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#+PROPERTY: header-args:latex+ :imagemagick t :fit yes
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#+PROPERTY: header-args:latex+ :iminoptions -scale 100% -density 150
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#+PROPERTY: header-args:latex+ :imoutoptions -quality 100
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#+PROPERTY: header-args:latex+ :results raw replace :buffer no
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#+PROPERTY: header-args:latex+ :eval no-export
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#+PROPERTY: header-args:latex+ :exports both
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#+PROPERTY: header-args:latex+ :mkdirp yes
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#+PROPERTY: header-args:latex+ :output-dir figs
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:END:
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* Introduction :ignore:
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In this document, we discuss the way the motion of each stage is defined.
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* Micro Hexapod
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** How the Symetrie Hexapod is controlled on the micro station
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For the Micro-Hexapod, the convention for the angles are defined in =MAN_A_Software API_4.0.150918_EN.pdf= on page 13 (section 2.4 - Rotation Vectors):
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#+begin_quote
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The *Euler type II convention* is used to express the rotation vector.
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This convention is mainly used in the aeronautics field (standard ISO 1151 concerning flight mechanics).
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This convention uses the concepts of rotation of vehicles (ship, car and plane).
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Generally, we consider that the main movement of the vehicle is following the X-axis and the Z-axis is parallel to the axis of gravity (at the initial position).
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The roll rotation is around the X-axis, the pitch is around the Y-axis and yaw is the rotation around the Z-axis.
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*The order of rotation is: Rx, Ry and then Rz.*
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In most case, rotations are related to a reference with fixed axis; thus we say the rotations are around fixed axes.
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The combination of these three rotations enables to write a rotation matrix.
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This writing is unique and equal to:
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\[ \bm{R} = \bm{R}_z(\gamma) \cdot \bm{R}_y(\beta) \cdot \bm{R}_x(\alpha) \]
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The Euler type II convention corresponding to the *succession of rotations with respect to fixed axes*: first around X0, then Y0 and Z0.
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This is equivalent to the succession of rotations with respect to mobile axes: first around Z0, then Y1' and X2'.
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#+end_quote
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More generally on the Control of the Micro-Hexapod:
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#+begin_quote
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Note that for all control modes, *the rotation center coincides with Object coordinate system origin*.
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Moreover, the movements are controlled with *translation components at first* (Tx, Ty, Tz) *then rotation components* (Rx, Ry, Rz).
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#+end_quote
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Thus, it does the translations and then the rotation around the new translated frame.
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** Control of the Micro-Hexapod using Simscape
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*** Introduction :ignore:
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We can think of two main ways to position the Micro-Hexapod using Simscape.
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The first one is to use only one Bushing Joint between the base and the mobile platform.
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The advantage is that it is very easy to impose the wanted displacement, however, we loose the dynamical properties of the Hexapod.
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The second way is to specify the wanted length of the legs of the Hexapod in order to have the wanted position of the mobile platform.
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This require a little bit more of mathematical derivations but this is the chosen solution.
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*** Using Bushing Joint
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In the documentation of the Bushing Joint (=doc "Bushing Joint"=) that is used to position the Hexapods, it is mention that the following frame is positioned with respect to the base frame in a way shown in figure [[fig:bushing_joint_transform]].
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#+name: fig:bushing_joint_transform
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#+caption: Joint Transformation Sequence for the Bushing Joint
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[[file:figs/bushing_joint_transform.png]]
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Basically, it performs the translations, and then the rotation along the X, Y and Z axis of the moving frame.
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The three rotations that we define thus corresponds to the Euler U-V-W angles.
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We should have the *same behavior* for the Micro-Hexapod on Simscape (same inputs at least).
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However, the Bushing Joint makes rotations around mobiles axes (X, Y' and then Z'') and not fixed axes (X, Y and Z).
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*** Using Inverse Kinematics and Leg Actuators
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Here, we can use the Inverse Kinematic of the Hexapod to determine the length of each leg in order to obtain some defined translation and rotation of the mobile platform.
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The advantages are:
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- we can position the Hexapod as we want by specifying a rotation matrix
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- the hexapod keeps its full flexibility as we don't specify any wanted displacements, only leg's rest position
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However:
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- even though the rest position of each leg (the position where the stiffness force is zero) is set correctly, the hexapod will we deflected due to gravity
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Thus, for this simulation, we *remove the gravity*.
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**** Theory
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For inverse kinematic analysis, it is assumed that the position ${}^A\bm{P}$ and orientation of the moving platform ${}^A\bm{R}_B$ are given and the problem is to obtain the joint variables, namely, $\bm{L} = [l_1, l_2, \dots, l_6]^T$.
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From the geometry of the manipulator, the loop closure for each limb, $i = 1, 2, \dots, 6$ can be written as
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\begin{align*}
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l_i {}^A\hat{\bm{s}}_i &= {}^A\bm{A} + {}^A\bm{b}_i - {}^A\bm{a}_i \\
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&= {}^A\bm{A} + {}^A\bm{R}_b {}^B\bm{b}_i - {}^A\bm{a}_i
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\end{align*}
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To obtain the length of each actuator and eliminate $\hat{\bm{s}}_i$, it is sufficient to dot multiply each side by itself:
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\begin{equation}
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l_i^2 \left[ {}^A\hat{\bm{s}}_i^T {}^A\hat{\bm{s}}_i \right] = \left[ {}^A\bm{P} + {}^A\bm{R}_B {}^B\bm{b}_i - {}^A\bm{a}_i \right]^T \left[ {}^A\bm{P} + {}^A\bm{R}_B {}^B\bm{b}_i - {}^A\bm{a}_i \right]
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\end{equation}
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Hence, for $i = 1, 2, \dots, 6$, each limb length can be uniquely determined by:
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\begin{equation}
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l_i = \sqrt{{}^A\bm{P}^T {}^A\bm{P} + {}^B\bm{b}_i^T {}^B\bm{b}_i + {}^A\bm{a}_i^T {}^A\bm{a}_i - 2 {}^A\bm{P}^T {}^A\bm{a}_i + 2 {}^A\bm{P}^T \left[{}^A\bm{R}_B {}^B\bm{b}_i\right] - 2 \left[{}^A\bm{R}_B {}^B\bm{b}_i\right]^T {}^A\bm{a}_i}
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\end{equation}
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If the position and orientation of the moving platform lie in the feasible workspace of the manipulator, one unique solution to the limb length is determined by the above equation.
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Otherwise, when the limbs' lengths derived yield complex numbers, then the position or orientation of the moving platform is not reachable.
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**** Matlab Init :noexport:ignore:
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#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
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<<matlab-dir>>
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#+end_src
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#+begin_src matlab :exports none :results silent :noweb yes
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<<matlab-init>>
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#+end_src
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#+begin_src matlab :tangle no
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simulinkproject('../');
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#+end_src
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**** Matlab Implementation
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We open the Simulink file.
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#+begin_src matlab
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open('nass_model.slx')
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#+end_src
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We load the configuration and set a small =StopTime=.
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#+begin_src matlab
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load('mat/conf_simulink.mat');
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set_param(conf_simulink, 'StopTime', '0.1');
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#+end_src
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We define the wanted position/orientation of the Hexapod under study.
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#+begin_src matlab
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tx = 0.05; % [rad]
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ty = 0.1; % [rad]
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tz = 0.02; % [rad]
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Rx = [1 0 0;
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0 cos(tx) -sin(tx);
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0 sin(tx) cos(tx)];
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Ry = [ cos(ty) 0 sin(ty);
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0 1 0;
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-sin(ty) 0 cos(ty)];
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Rz = [cos(tz) -sin(tz) 0;
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sin(tz) cos(tz) 0;
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0 0 1];
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ARB = Rz*Ry*Rx;
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AP = [0.1; 0.005; 0.01]; % [m]
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#+end_src
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#+begin_src matlab
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initializeSimscapeConfiguration('gravity', false);
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initializeGround('type', 'none');
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initializeGranite('type', 'none');
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initializeTy('type', 'none');
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initializeRy('type', 'none');
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initializeRz('type', 'none');
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initializeMicroHexapod('type', 'rigid', 'AP', AP, 'ARB', ARB);
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initializeAxisc('type', 'none');
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initializeMirror('type', 'none');
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initializeNanoHexapod('type', 'none');
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initializeSample('type', 'none');
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initializeLoggingConfiguration('log', 'all');
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#+end_src
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We run the simulation.
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#+begin_src matlab
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sim('nass_model');
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#+end_src
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And we verify that we indeed succeed to go to the wanted position.
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#+begin_src matlab :results table replace
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[simout.Dhm.x.Data(end) ; simout.Dhm.y.Data(end) ; simout.Dhm.z.Data(end)] - AP
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#+end_src
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#+RESULTS:
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| 8.4655e-16 |
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| 1.5586e-15 |
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| -2.1337e-16 |
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#+begin_src matlab :results table replace
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simout.Dhm.R.Data(:, :, end)-ARB
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#+end_src
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#+RESULTS:
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| -1.1102e-16 | -1.36e-15 | 4.2744e-15 |
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| 1.0651e-15 | 6.6613e-16 | 5.1278e-15 |
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| -4.2882e-15 | -4.9336e-15 | 1.1102e-16 |
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* TODO Tests on the transformation from reference to wanted position :noexport:
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:PROPERTIES:
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:header-args:matlab+: :eval no
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:END:
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** Introduction :ignore:
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#+begin_quote
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Rx = [1 0 0;
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0 cos(t) -sin(t);
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0 sin(t) cos(t)];
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Ry = [ cos(t) 0 sin(t);
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0 1 0;
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-sin(t) 0 cos(t)];
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Rz = [cos(t) -sin(t) 0;
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sin(t) cos(t) 0;
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0 0 1];
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#+end_quote
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Let's define the following frames:
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- $\{W\}$ the frame that is *fixed to the granite* and its origin at the theoretical meeting point between the X-ray and the spindle axis.
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- $\{S\}$ the frame *attached to the sample* (in reality attached to the top platform of the nano-hexapod) with its origin at 175mm above the top platform of the nano-hexapod.
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Its origin is $O_S$.
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- $\{T\}$ the theoretical wanted frame that correspond to the wanted pose of the frame $\{S\}$.
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$\{T\}$ is computed from the wanted position of each stage. It is thus theoretical and does not correspond to a real position.
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The origin of $T$ is $O_T$ and is the wanted position of the sample.
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Thus:
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- the *measurement* of the position of the sample corresponds to ${}^W O_S = \begin{bmatrix} {}^WP_{x,m} & {}^WP_{y,m} & {}^WP_{z,m} \end{bmatrix}^T$ in translation and to $\theta_m {}^W\bm{s}_m = \theta_m \cdot \begin{bmatrix} {}^Ws_{x,m} & {}^Ws_{y,m} & {}^Ws_{z,m} \end{bmatrix}^T$ in rotations
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- the *wanted position* of the sample expressed w.r.t. the granite is ${}^W O_T = \begin{bmatrix} {}^WP_{x,r} & {}^WP_{y,r} & {}^WP_{z,r} \end{bmatrix}^T$ in translation and to $\theta_r {}^W\bm{s}_r = \theta_r \cdot \begin{bmatrix} {}^Ws_{x,r} & {}^Ws_{y,r} & {}^Ws_{z,r} \end{bmatrix}^T$ in rotations
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** Wanted Position of the Sample with respect to the Granite
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Let's define the wanted position of each stage.
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#+begin_src matlab
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Ty = 0; % [m]
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Ry = 3*pi/180; % [rad]
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Rz = 180*pi/180; % [rad]
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% Hexapod (first consider only translations)
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Thx = 0; % [m]
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Thy = 0; % [m]
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Thz = 0; % [m]
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#+end_src
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Now, we compute the corresponding wanted translation and rotation of the sample with respect to the granite frame $\{W\}$.
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This corresponds to ${}^WO_T$ and $\theta_m {}^Ws_m$.
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To do so, we have to define the homogeneous transformation for each stage.
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#+begin_src matlab
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% Translation Stage
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Rty = [1 0 0 0;
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0 1 0 Ty;
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0 0 1 0;
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0 0 0 1];
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% Tilt Stage - Pure rotating aligned with Ob
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Rry = [ cos(Ry) 0 sin(Ry) 0;
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0 1 0 0;
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-sin(Ry) 0 cos(Ry) 0;
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0 0 0 1];
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% Spindle - Rotation along the Z axis
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Rrz = [cos(Rz) -sin(Rz) 0 0 ;
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sin(Rz) cos(Rz) 0 0 ;
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0 0 1 0 ;
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0 0 0 1 ];
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% Micro-Hexapod (only rotations first)
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Rh = [1 0 0 Thx ;
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0 1 0 Thy ;
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0 0 1 Thz ;
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0 0 0 1 ];
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#+end_src
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We combine the individual homogeneous transformations into one homogeneous transformation for all the station.
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#+begin_src matlab
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Ttot = Rty*Rry*Rrz*Rh;
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#+end_src
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Using this homogeneous transformation, we can compute the wanted position and orientation of the sample with respect to the granite.
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Translation.
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#+begin_src matlab
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WOr = Ttot*[0;0;0;1];
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WOr = WOr(1:3);
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#+end_src
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Rotation.
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#+begin_src matlab
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thetar = acos((trace(Ttot(1:3, 1:3))-1)/2)
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if thetar == 0
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WSr = [0; 0; 0];
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else
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[V, D] = eig(Ttot(1:3, 1:3));
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WSr = thetar*V(:, abs(diag(D) - 1) < eps(1));
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end
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#+end_src
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#+begin_src matlab
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WPr = [WOr ; WSr];
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#+end_src
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** Measured Position of the Sample with respect to the Granite
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The measurement of the position of the sample using the metrology system gives the position and orientation of the sample with respect to the granite.
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#+begin_src matlab
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% Measurements: Xm, Ym, Zm, Rx, Ry, Rz
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Dxm = 0; % [m]
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Dym = 0; % [m]
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Dzm = 0; % [m]
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Rxm = 0*pi/180; % [rad]
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Rym = 0*pi/180; % [rad]
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Rzm = 180*pi/180; % [rad]
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#+end_src
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Let's compute the corresponding orientation using screw axis.
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#+begin_src matlab
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Trxm = [1 0 0;
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0 cos(Rxm) -sin(Rxm);
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0 sin(Rxm) cos(Rxm)];
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Trym = [ cos(Rym) 0 sin(Rym);
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0 1 0;
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-sin(Rym) 0 cos(Rym)];
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Trzm = [cos(Rzm) -sin(Rzm) 0;
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sin(Rzm) cos(Rzm) 0;
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0 0 1];
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STw = [[ Trym*Trxm*Trzm , [Dxm; Dym; Dzm]]; 0 0 0 1];
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#+end_src
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We then obtain the orientation measurement in the form of screw coordinate $\theta_m ({}^Ws_{x,m},\ {}^Ws_{y,m},\ {}^Ws_{z,m})^T$ where:
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- $\theta_m = \cos^{-1} \frac{\text{Tr}(R) - 1}{2}$
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- ${}^W\bm{s}_m$ is the eigen vector of the rotation matrix $R$ corresponding to the eigen value $\lambda = 1$
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#+begin_src matlab
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thetam = acos((trace(STw(1:3, 1:3))-1)/2); % [rad]
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if thetam == 0
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WSm = [0; 0; 0];
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else
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[V, D] = eig(STw(1:3, 1:3));
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WSm = thetam*V(:, abs(diag(D) - 1) < eps(1));
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end
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#+end_src
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#+begin_src matlab
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WPm = [Dxm ; Dym ; Dzm ; WSm];
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#+end_src
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** Positioning Error with respect to the Granite
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The wanted position expressed with respect to the granite is ${}^WO_T$ and the measured position with respect to the granite is ${}^WO_S$, thus the *position error* expressed in $\{W\}$ is
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\[ {}^W E = {}^W O_T - {}^W O_S \]
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The same is true for rotations:
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\[ \theta_\epsilon {}^W\bm{s}_\epsilon = \theta_r {}^W\bm{s}_r - \theta_m {}^W\bm{s}_m \]
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#+begin_src matlab
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WPe = WPr - WPm;
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#+end_src
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#+begin_quote
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Now we want to express this error in a frame attached to the *base of the nano-hexapod* with its origin at the same point where the Jacobian of the nano-hexapod is computed (175mm above the top platform + 90mm of total height of the nano-hexapod).
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Or maybe should we want to express this error with respect to the *top platform of the nano-hexapod*?
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We are measuring the position of the top-platform, and we don't know exactly the position of the bottom platform.
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We could compute the position of the bottom platform in two ways:
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- from the encoders of each stage
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- from the measurement of the nano-hexapod top platform + the internal metrology in the nano-hexapod (capacitive sensors e.g)
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A third option is to say that the maximum stroke of the nano-hexapod is so small that the error should no change to much by the change of base.
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#+end_quote
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** Position Error Expressed in the Nano-Hexapod Frame
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We now want the position error to be expressed in $\{S\}$ (the frame attach to the sample) for control:
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\[ {}^S E = {}^S T_W \cdot {}^W E \]
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Thus we need to compute the homogeneous transformation ${}^ST_W$.
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Fortunately, this homogeneous transformation can be computed from the measurement of the sample position and orientation with respect to the granite.
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#+begin_src matlab
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Trxm = [1 0 0;
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0 cos(Rxm) -sin(Rxm);
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0 sin(Rxm) cos(Rxm)];
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Trym = [ cos(Rym) 0 sin(Rym);
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0 1 0;
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-sin(Rym) 0 cos(Rym)];
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Trzm = [cos(Rzm) -sin(Rzm) 0;
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sin(Rzm) cos(Rzm) 0;
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0 0 1];
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STw = [[ Trym*Trxm*Trzm , [Dxm; Dym; Dzm]]; 0 0 0 1];
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#+end_src
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Translation Error.
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#+begin_src matlab
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SEm = STw * [WPe(1:3); 0];
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SEm = SEm(1:3);
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#+end_src
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Rotation Error.
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#+begin_src matlab
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SEr = STw * [WPe(4:6); 0];
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SEr = SEr(1:3);
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#+end_src
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#+begin_src matlab
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Etot = [SEm ; SEr]
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#+end_src
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** Another try
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Let's denote:
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- $\{W\}$ the initial fixed frame
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- $\{R\}$ the reference frame corresponding to the wanted pose of the sample
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- $\{M\}$ the frame corresponding to the measured pose of the sample
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We have then computed:
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- ${}^WT_R$
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- ${}^WT_M$
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We have:
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\begin{align}
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{}^MT_R &= {}^MT_W {}^WT_R \\
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&= {}^WT_M^t {}^WT_R
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\end{align}
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|
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#+begin_src matlab
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MTr = STw'*Ttot;
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#+end_src
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|
|
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Position error:
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|
#+begin_src matlab
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|
MTr(1:3, 1:4)*[0; 0; 0; 1]
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|
#+end_src
|
|
|
|
Orientation error:
|
|
#+begin_src matlab
|
|
MTr(1:3, 1:3)
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|
#+end_src
|
|
|
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** Verification
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|
How can we verify that the computation is correct?
|
|
Options:
|
|
- Test with simscape multi-body
|
|
- Impose motion on each stage
|
|
- Measure the position error w.r.t. the NASS
|
|
- Compare with the computation
|