206 lines
9.0 KiB
Org Mode
206 lines
9.0 KiB
Org Mode
#+TITLE: Kinematics of the station
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:DRAWER:
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#+STARTUP: overview
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#+LANGUAGE: en
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#+EMAIL: dehaeze.thomas@gmail.com
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#+AUTHOR: Dehaeze Thomas
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#+HTML_LINK_HOME: ../index.html
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#+HTML_LINK_UP: ../index.html
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#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="../css/htmlize.css"/>
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#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="../css/readtheorg.css"/>
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#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="../css/zenburn.css"/>
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#+HTML_HEAD: <script type="text/javascript" src="../js/jquery.min.js"></script>
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#+HTML_HEAD: <script type="text/javascript" src="../js/bootstrap.min.js"></script>
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#+HTML_HEAD: <script type="text/javascript" src="../js/jquery.stickytableheaders.min.js"></script>
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#+HTML_HEAD: <script type="text/javascript" src="../js/readtheorg.js"></script>
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#+HTML_MATHJAX: align: center tagside: right font: TeX
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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 matlab/modal_frf_coh.m
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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/}{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('kinematics/matlab/hexapod_tests.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_simscape.mat');
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set_param(conf_simscape, 'StopTime', '0.5');
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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 = 0.1; % [rad]
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ty = 0.2; % [rad]
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tz = 0.05; % [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 = [0.01; 0.02; 0.03]; % [m]
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hexapod = initializeMicroHexapod(struct('AP', AP, 'ARB', ARB));
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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('hexapod_tests')
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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.x.Data(end) ; simout.y.Data(end) ; simout.z.Data(end)] - AP
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#+end_src
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#+RESULTS:
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| 1.611e-10 |
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| -1.3748e-10 |
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| 8.4879e-11 |
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#+begin_src matlab :results table replace
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simout.R.Data(:, :, end)-ARB
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#+end_src
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#+RESULTS:
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| -1.2659e-10 | 6.5603e-11 | 6.2183e-10 |
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| 1.0354e-10 | -5.2439e-11 | -5.2425e-10 |
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| -5.9816e-10 | 5.532e-10 | -1.7737e-10 |
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