1857 lines
67 KiB
Org Mode
1857 lines
67 KiB
Org Mode
#+TITLE: Finite Element Model with Simscape
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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="https://research.tdehaeze.xyz/css/style.css"/>
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#+HTML_HEAD: <script type="text/javascript" src="https://research.tdehaeze.xyz/js/script.js"></script>
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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/tikz/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 results
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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, Finite Element Models (FEM) of parts of the Nano-Hexapod are developed and integrated into Simscape for dynamical analysis.
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- Section [[sec:APA300ML]]:
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A super-element of the Amplified Piezoelectric Actuator APA300ML used for the NASS is exported using Ansys and imported in Simscape.
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The static and dynamical properties of the APA300ML are then estimated using the Simscape model.
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- Section [[sec:flexor_ID16]]:
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A first geometry of a Flexible joint is modelled and its characteristics are identified from the Stiffness matrix as well as from the Simscape model.
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- Section [[sec:flexor_025]]:
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An optimized flexible joint is developed for the Nano-Hexapod and is then imported in a Simscape model.
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- Section [[sec:integral_force_feedback]]:
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- Section [[sec:strut_fem]]:
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A super element of a complete strut is exported.
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* APA300ML
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:PROPERTIES:
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:header-args:matlab+: :tangle matlab/APA300ML.m
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:END:
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<<sec:APA300ML>>
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** Introduction :ignore:
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In this section, the Amplified Piezoelectric Actuator APA300ML ([[file:doc/APA300ML.pdf][doc]]) is modeled using a Finite Element Software.
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Then a /super element/ is exported and imported in Simscape where its dynamic is studied.
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A 3D view of the Amplified Piezoelectric Actuator (APA300ML) is shown in Figure [[fig:apa300ml_ansys]].
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The remote point used are also shown in this figure.
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#+name: fig:apa300ml_ansys
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#+caption: Ansys FEM of the APA300ML
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[[file:figs/apa300ml_ansys.jpg]]
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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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addpath('matlab/');
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addpath('matlab/APA300ML/');
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#+end_src
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#+begin_src matlab :eval no
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addpath('APA300ML/');
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#+end_src
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#+begin_src matlab
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open('APA300ML.slx');
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#+end_src
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** Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates
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We first extract the stiffness and mass matrices.
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#+begin_src matlab
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K = readmatrix('mat_K.CSV');
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M = readmatrix('mat_M.CSV');
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#+end_src
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#+begin_src matlab :exports results :results value table replace :tangle no
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data2orgtable(K(1:10, 1:10), {}, {}, ' %.1g ');
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#+end_src
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#+caption: First 10x10 elements of the Stiffness matrix
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#+RESULTS:
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| 200000000.0 | 30000.0 | -20000.0 | -70.0 | 300000.0 | 40.0 | 10000000.0 | 10000.0 | -6000.0 | 30.0 |
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| 30000.0 | 30000000.0 | 2000.0 | -200000.0 | 60.0 | -10.0 | 4000.0 | 2000000.0 | -500.0 | 9000.0 |
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| -20000.0 | 2000.0 | 7000000.0 | -10.0 | -30.0 | 10.0 | 6000.0 | 900.0 | -500000.0 | 3 |
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| -70.0 | -200000.0 | -10.0 | 1000.0 | -0.1 | 0.08 | -20.0 | -9000.0 | 3 | -30.0 |
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| 300000.0 | 60.0 | -30.0 | -0.1 | 900.0 | 0.1 | 30000.0 | 20.0 | -10.0 | 0.06 |
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| 40.0 | -10.0 | 10.0 | 0.08 | 0.1 | 10000.0 | 20.0 | 9 | -5 | 0.03 |
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| 10000000.0 | 4000.0 | 6000.0 | -20.0 | 30000.0 | 20.0 | 200000000.0 | 10000.0 | 9000.0 | 50.0 |
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| 10000.0 | 2000000.0 | 900.0 | -9000.0 | 20.0 | 9 | 10000.0 | 30000000.0 | -500.0 | 200000.0 |
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| -6000.0 | -500.0 | -500000.0 | 3 | -10.0 | -5 | 9000.0 | -500.0 | 7000000.0 | -2 |
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| 30.0 | 9000.0 | 3 | -30.0 | 0.06 | 0.03 | 50.0 | 200000.0 | -2 | 1000.0 |
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#+begin_src matlab :exports results :results value table replace :tangle no
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data2orgtable(M(1:10, 1:10), {}, {}, ' %.1g ');
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#+end_src
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#+caption: First 10x10 elements of the Mass matrix
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#+RESULTS:
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| 0.01 | -2e-06 | 1e-06 | 6e-09 | 5e-05 | -5e-09 | -0.0005 | -7e-07 | 6e-07 | -3e-09 |
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| -2e-06 | 0.01 | 8e-07 | -2e-05 | -8e-09 | 2e-09 | -9e-07 | -0.0002 | 1e-08 | -9e-07 |
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| 1e-06 | 8e-07 | 0.009 | 5e-10 | 1e-09 | -1e-09 | -5e-07 | 3e-08 | 6e-05 | 1e-10 |
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| 6e-09 | -2e-05 | 5e-10 | 3e-07 | 2e-11 | -3e-12 | 3e-09 | 9e-07 | -4e-10 | 3e-09 |
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| 5e-05 | -8e-09 | 1e-09 | 2e-11 | 6e-07 | -4e-11 | -1e-06 | -2e-09 | 1e-09 | -8e-12 |
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| -5e-09 | 2e-09 | -1e-09 | -3e-12 | -4e-11 | 1e-07 | -2e-09 | -1e-09 | -4e-10 | -5e-12 |
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| -0.0005 | -9e-07 | -5e-07 | 3e-09 | -1e-06 | -2e-09 | 0.01 | 1e-07 | -3e-07 | -2e-08 |
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| -7e-07 | -0.0002 | 3e-08 | 9e-07 | -2e-09 | -1e-09 | 1e-07 | 0.01 | -4e-07 | 2e-05 |
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| 6e-07 | 1e-08 | 6e-05 | -4e-10 | 1e-09 | -4e-10 | -3e-07 | -4e-07 | 0.009 | -2e-10 |
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| -3e-09 | -9e-07 | 1e-10 | 3e-09 | -8e-12 | -5e-12 | -2e-08 | 2e-05 | -2e-10 | 3e-07 |
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Then, we extract the coordinates of the interface nodes.
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#+begin_src matlab
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[int_xyz, int_i, n_xyz, n_i, nodes] = extractNodes('out_nodes_3D.txt');
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#+end_src
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#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
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data2orgtable([[1:length(int_i)]', int_i, int_xyz], {}, {'Node i', 'Node Number', 'x [m]', 'y [m]', 'z [m]'}, ' %f ');
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#+end_src
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#+caption: Coordinates of the interface nodes
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#+RESULTS:
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| Node i | Node Number | x [m] | y [m] | z [m] |
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|--------+-------------+---------+-------+--------|
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| 1.0 | 697783.0 | 0.0 | 0.0 | -0.015 |
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| 2.0 | 697784.0 | 0.0 | 0.0 | 0.015 |
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| 3.0 | 697785.0 | -0.0325 | 0.0 | 0.0 |
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| 4.0 | 697786.0 | -0.0125 | 0.0 | 0.0 |
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| 5.0 | 697787.0 | -0.0075 | 0.0 | 0.0 |
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| 6.0 | 697788.0 | 0.0125 | 0.0 | 0.0 |
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| 7.0 | 697789.0 | 0.0325 | 0.0 | 0.0 |
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#+begin_src matlab :exports results :results value table replace :tangle no
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data2orgtable([length(n_i); length(int_i); size(M,1) - 6*length(int_i); size(M,1)], {'Total number of Nodes', 'Number of interface Nodes', 'Number of Modes', 'Size of M and K matrices'}, {}, ' %.0f ');
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#+end_src
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#+caption: Some extracted parameters of the FEM
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#+RESULTS:
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| Total number of Nodes | 7 |
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| Number of interface Nodes | 7 |
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| Number of Modes | 120 |
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| Size of M and K matrices | 162 |
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Using =K=, =M= and =int_xyz=, we can now use the =Reduced Order Flexible Solid= simscape block.
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** Piezoelectric parameters
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In order to make the conversion from applied voltage to generated force or from the strain to the generated voltage, we need to defined some parameters corresponding to the piezoelectric material:
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#+begin_src matlab
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d33 = 300e-12; % Strain constant [m/V]
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n = 80; % Number of layers per stack
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eT = 1.6e-8; % Permittivity under constant stress [F/m]
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sD = 1e-11; % Compliance under constant electric displacement [m2/N]
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ka = 235e6; % Stack stiffness [N/m]
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C = 5e-6; % Stack capactiance [F]
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#+end_src
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The ratio of the developed force to applied voltage is:
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#+name: eq:piezo_voltage_to_force
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\begin{equation}
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F_a = g_a V_a, \quad g_a = d_{33} n k_a
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\end{equation}
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where:
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- $F_a$: developed force in [N]
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- $n$: number of layers of the actuator stack
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- $d_{33}$: strain constant in [m/V]
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- $k_a$: actuator stack stiffness in [N/m]
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- $V_a$: applied voltage in [V]
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If we take the numerical values, we obtain:
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#+begin_src matlab :results replace value
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d33*n*ka % [N/V]
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#+end_src
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#+RESULTS:
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: 5.64
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From cite:fleming14_desig_model_contr_nanop_system (page 123), the relation between relative displacement of the sensor stack and generated voltage is:
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#+name: eq:piezo_strain_to_voltage
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\begin{equation}
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V_s = \frac{d_{33}}{\epsilon^T s^D n} \Delta h
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\end{equation}
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where:
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- $V_s$: measured voltage in [V]
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- $d_{33}$: strain constant in [m/V]
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- $\epsilon^T$: permittivity under constant stress in [F/m]
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- $s^D$: elastic compliance under constant electric displacement in [m^2/N]
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- $n$: number of layers of the sensor stack
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- $\Delta h$: relative displacement in [m]
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If we take the numerical values, we obtain:
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#+begin_src matlab :results replace value
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1e-6*d33/(eT*sD*n) % [V/um]
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#+end_src
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#+RESULTS:
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: 23.438
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** Simscape Model
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The flexible element is imported using the =Reduced Order Flexible Solid= simscape block.
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Let's say we use two stacks as a force sensor and one stack as an actuator:
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- A =Relative Motion Sensor= block is added between the nodes A and C
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- An =Internal Force= block is added between the remote points E and B
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The interface nodes are shown in Figure [[fig:apa300ml_ansys]].
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One mass is fixed at one end of the piezo-electric stack actuator (remove point F), the other end is fixed to the world frame (remote point G).
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** Identification of the APA Characteristics
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*** Stiffness
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#+begin_src matlab :exports none
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m = 0.001;
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#+end_src
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The transfer function from vertical external force to the relative vertical displacement is identified.
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#+begin_src matlab :exports none
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%% Name of the Simulink File
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mdl = 'APA300ML';
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%% Input/Output definition
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clear io; io_i = 1;
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io(io_i) = linio([mdl, '/Fd'], 1, 'openinput'); io_i = io_i + 1;
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io(io_i) = linio([mdl, '/z'], 1, 'openoutput'); io_i = io_i + 1;
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G = linearize(mdl, io);
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#+end_src
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The inverse of its DC gain is the axial stiffness of the APA:
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#+begin_src matlab :results replace value
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1e-6/dcgain(G) % [N/um]
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#+end_src
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#+RESULTS:
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: 1.753
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The specified stiffness in the datasheet is $k = 1.8\, [N/\mu m]$.
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*** Resonance Frequency
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The resonance frequency is specified to be between 650Hz and 840Hz.
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This is also the case for the FEM model (Figure [[fig:apa300ml_resonance]]).
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#+begin_src matlab :exports none
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freqs = logspace(2, 4, 5000);
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figure;
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hold on;
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plot(freqs, abs(squeeze(freqresp(G, freqs, 'Hz'))));
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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xlabel('Frequency [Hz]'); ylabel('Amplitude');
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hold off;
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#+end_src
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#+begin_src matlab :tangle no :exports results :results file replace
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exportFig('figs/apa300ml_resonance.pdf', 'width', 'wide', 'height', 'normal');
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#+end_src
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#+name: fig:apa300ml_resonance
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#+caption: First resonance is around 800Hz
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#+RESULTS:
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[[file:figs/apa300ml_resonance.png]]
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*** Amplification factor
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The amplification factor is the ratio of the vertical displacement to the stack displacement.
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#+begin_src matlab :exports none
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%% Name of the Simulink File
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mdl = 'APA300ML';
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%% Input/Output definition
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clear io; io_i = 1;
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io(io_i) = linio([mdl, '/F'], 1, 'openinput'); io_i = io_i + 1;
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io(io_i) = linio([mdl, '/z'], 1, 'openoutput'); io_i = io_i + 1;
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io(io_i) = linio([mdl, '/d'], 1, 'openoutput'); io_i = io_i + 1;
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G = linearize(mdl, io);
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#+end_src
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The ratio of the two displacement is computed from the FEM model.
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#+begin_src matlab :results replace value
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abs(dcgain(G(1,1))./dcgain(G(2,1)))
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#+end_src
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#+RESULTS:
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: 5.0749
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This is actually correct and approximately corresponds to the ratio of the piezo height and length:
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#+begin_src matlab :results replace value
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75/15
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#+end_src
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#+RESULTS:
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: 5
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*** Stroke
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Estimation of the actuator stroke:
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\[ \Delta H = A n \Delta L \]
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with:
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- $\Delta H$ Axial Stroke of the APA
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- $A$ Amplification factor (5 for the APA300ML)
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- $n$ Number of stack used
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- $\Delta L$ Stroke of the stack (0.1% of its length)
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#+begin_src matlab :results replace value
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1e6 * 5 * 3 * 20e-3 * 0.1e-2
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#+end_src
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#+RESULTS:
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: 300
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This is exactly the specified stroke in the data-sheet.
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** Identification of the Dynamics from actuator to replace displacement
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We first set the mass to be approximately zero.
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#+begin_src matlab :exports none
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m = 0.01;
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#+end_src
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The dynamics is identified from the applied force to the measured relative displacement.
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#+begin_src matlab :exports none
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%% Name of the Simulink File
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mdl = 'APA300ML';
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%% Input/Output definition
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clear io; io_i = 1;
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io(io_i) = linio([mdl, '/F'], 1, 'openinput'); io_i = io_i + 1;
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io(io_i) = linio([mdl, '/z'], 1, 'openoutput'); io_i = io_i + 1;
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Gh = -linearize(mdl, io);
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#+end_src
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The same dynamics is identified for a payload mass of 10Kg.
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#+begin_src matlab
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m = 10;
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#+end_src
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#+begin_src matlab :exports none
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%% Name of the Simulink File
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mdl = 'APA300ML';
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%% Input/Output definition
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clear io; io_i = 1;
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io(io_i) = linio([mdl, '/F'], 1, 'openinput'); io_i = io_i + 1;
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io(io_i) = linio([mdl, '/z'], 1, 'openoutput'); io_i = io_i + 1;
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Ghm = -linearize(mdl, io);
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#+end_src
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#+begin_src matlab :exports none
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freqs = logspace(0, 4, 5000);
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figure;
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tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
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ax1 = nexttile([2,1]);
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hold on;
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plot(freqs, abs(squeeze(freqresp(Gh, freqs, 'Hz'))), '-');
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plot(freqs, abs(squeeze(freqresp(Ghm, freqs, 'Hz'))), '-');
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Amplitude'); set(gca, 'XTickLabel',[]);
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hold off;
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ax2 = nexttile;
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hold on;
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plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gh, freqs, 'Hz')))), '-', ...
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'DisplayName', '$m = 0kg$');
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plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Ghm, freqs, 'Hz')))), '-', ...
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'DisplayName', '$m = 10kg$');
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
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yticks(-360:90:360);
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ylim([-360 0]);
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xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
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hold off;
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linkaxes([ax1,ax2],'x');
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xlim([freqs(1), freqs(end)]);
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legend('location', 'southwest');
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#+end_src
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#+begin_src matlab :tangle no :exports results :results file replace
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exportFig('figs/apa300ml_plant_dynamics.pdf', 'width', 'wide', 'height', 'tall');
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#+end_src
|
|
|
|
#+name: fig:apa300ml_plant_dynamics
|
|
#+caption: Transfer function from forces applied by the stack to the axial displacement of the APA
|
|
#+RESULTS:
|
|
[[file:figs/apa300ml_plant_dynamics.png]]
|
|
|
|
The root locus corresponding to Direct Velocity Feedback with a mass of 10kg is shown in Figure [[fig:apa300ml_dvf_root_locus]].
|
|
#+begin_src matlab :exports none
|
|
figure;
|
|
|
|
gains = logspace(0, 5, 500);
|
|
|
|
hold on;
|
|
plot(real(pole(Ghm)), imag(pole(G)), 'kx');
|
|
plot(real(tzero(Ghm)), imag(tzero(G)), 'ko');
|
|
for k = 1:length(gains)
|
|
cl_poles = pole(feedback(Ghm, gains(k)*s));
|
|
plot(real(cl_poles), imag(cl_poles), 'k.');
|
|
end
|
|
hold off;
|
|
axis square;
|
|
xlim([-500, 10]); ylim([0, 510]);
|
|
|
|
xlabel('Real Part'); ylabel('Imaginary Part');
|
|
#+end_src
|
|
|
|
#+begin_src matlab :tangle no :exports results :results file replace
|
|
exportFig('figs/apa300ml_dvf_root_locus.pdf', 'width', 'wide', 'height', 'tall');
|
|
#+end_src
|
|
|
|
#+name: fig:apa300ml_dvf_root_locus
|
|
#+caption: Root Locus for Direct Velocity Feedback
|
|
#+RESULTS:
|
|
[[file:figs/apa300ml_dvf_root_locus.png]]
|
|
|
|
** Identification of the Dynamics from actuator to force sensor
|
|
Let's use 2 stacks as a force sensor and 1 stack as force actuator.
|
|
|
|
The transfer function from actuator voltage to sensor voltage is identified and shown in Figure [[fig:apa300ml_iff_plant]].
|
|
#+begin_src matlab :exports none
|
|
m = 10;
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
%% Name of the Simulink File
|
|
mdl = 'APA300ML';
|
|
|
|
%% Input/Output definition
|
|
clear io; io_i = 1;
|
|
io(io_i) = linio([mdl, '/Va'], 1, 'openinput'); io_i = io_i + 1;
|
|
io(io_i) = linio([mdl, '/Vs'], 1, 'openoutput'); io_i = io_i + 1;
|
|
|
|
Giff = -linearize(mdl, io);
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(0, 4, 5000);
|
|
|
|
figure;
|
|
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
|
|
|
|
ax1 = nexttile([2,1]);
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(Giff, freqs, 'Hz'))), '-');
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude'); set(gca, 'XTickLabel',[]);
|
|
hold off;
|
|
|
|
ax2 = nexttile;
|
|
hold on;
|
|
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Giff, freqs, 'Hz')))), '-');
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
|
yticks(-360:90:360);
|
|
ylim([-180 180]);
|
|
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
|
|
hold off;
|
|
linkaxes([ax1,ax2],'x');
|
|
xlim([freqs(1), freqs(end)]);
|
|
#+end_src
|
|
|
|
#+begin_src matlab :tangle no :exports results :results file replace
|
|
exportFig('figs/apa300ml_iff_plant.pdf', 'width', 'wide', 'height', 'tall');
|
|
#+end_src
|
|
|
|
#+name: fig:apa300ml_iff_plant
|
|
#+caption: Transfer function from actuator to force sensor
|
|
#+RESULTS:
|
|
[[file:figs/apa300ml_iff_plant.png]]
|
|
|
|
For root locus corresponding to IFF is shown in Figure [[fig:apa300ml_iff_root_locus]].
|
|
#+begin_src matlab :exports none
|
|
figure;
|
|
|
|
gains = logspace(0, 5, 500);
|
|
|
|
hold on;
|
|
plot(real(pole(Giff)), imag(pole(Giff)), 'kx');
|
|
plot(real(tzero(Giff)), imag(tzero(Giff)), 'ko');
|
|
for k = 1:length(gains)
|
|
cl_poles = pole(feedback(Giff, gains(k)/s));
|
|
plot(real(cl_poles), imag(cl_poles), 'k.');
|
|
end
|
|
hold off;
|
|
axis square;
|
|
xlim([-500, 10]); ylim([0, 510]);
|
|
|
|
xlabel('Real Part'); ylabel('Imaginary Part');
|
|
#+end_src
|
|
|
|
#+begin_src matlab :tangle no :exports results :results file replace
|
|
exportFig('figs/apa300ml_iff_root_locus.pdf', 'width', 'wide', 'height', 'tall');
|
|
#+end_src
|
|
|
|
#+name: fig:apa300ml_iff_root_locus
|
|
#+caption: Root Locus for IFF
|
|
#+RESULTS:
|
|
[[file:figs/apa300ml_iff_root_locus.png]]
|
|
|
|
** Identification for a simpler model
|
|
The goal in this section is to identify the parameters of a simple APA model from the FEM.
|
|
This can be useful is a lower order model is to be used for simulations.
|
|
|
|
The presented model is based on cite:souleille18_concep_activ_mount_space_applic.
|
|
|
|
The model represents the Amplified Piezo Actuator (APA) from Cedrat-Technologies (Figure [[fig:souleille18_model_piezo]]).
|
|
The parameters are shown in the table below.
|
|
|
|
#+name: fig:souleille18_model_piezo
|
|
#+caption: Picture of an APA100M from Cedrat Technologies. Simplified model of a one DoF payload mounted on such isolator
|
|
[[file:./figs/souleille18_model_piezo.png]]
|
|
|
|
#+caption:Parameters used for the model of the APA 100M
|
|
| | Meaning |
|
|
|-------+----------------------------------------------------------------|
|
|
| $k_e$ | Stiffness used to adjust the pole of the isolator |
|
|
| $k_1$ | Stiffness of the metallic suspension when the stack is removed |
|
|
| $k_a$ | Stiffness of the actuator |
|
|
| $c_1$ | Added viscous damping |
|
|
|
|
The goal is to determine $k_e$, $k_a$ and $k_1$ so that the simplified model fits the FEM model.
|
|
|
|
\[ \alpha = \frac{x_1}{f}(\omega=0) = \frac{\frac{k_e}{k_e + k_a}}{k_1 + \frac{k_e k_a}{k_e + k_a}} \]
|
|
\[ \beta = \frac{x_1}{F}(\omega=0) = \frac{1}{k_1 + \frac{k_e k_a}{k_e + k_a}} \]
|
|
|
|
If we can fix $k_a$, we can determine $k_e$ and $k_1$ with:
|
|
\[ k_e = \frac{k_a}{\frac{\beta}{\alpha} - 1} \]
|
|
\[ k_1 = \frac{1}{\beta} - \frac{k_e k_a}{k_e + k_a} \]
|
|
|
|
#+begin_src matlab :exports none
|
|
m = 10;
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
%% Name of the Simulink File
|
|
mdl = 'APA300ML';
|
|
|
|
%% Input/Output definition
|
|
clear io; io_i = 1;
|
|
io(io_i) = linio([mdl, '/Fd'], 1, 'openinput'); io_i = io_i + 1; % External Vertical Force [N]
|
|
io(io_i) = linio([mdl, '/w'], 1, 'openinput'); io_i = io_i + 1; % Base Motion [m]
|
|
io(io_i) = linio([mdl, '/Fa'], 1, 'openinput'); io_i = io_i + 1; % Actuator Force [N]
|
|
io(io_i) = linio([mdl, '/z'], 1, 'openoutput'); io_i = io_i + 1; % Vertical Displacement [m]
|
|
io(io_i) = linio([mdl, '/Vs'], 1, 'openoutput'); io_i = io_i + 1; % Force Sensor [V]
|
|
io(io_i) = linio([mdl, '/d'], 1, 'openoutput'); io_i = io_i + 1; % Stack Displacement [m]
|
|
|
|
G = linearize(mdl, io);
|
|
|
|
G.InputName = {'Fd', 'w', 'Fa'};
|
|
G.OutputName = {'y', 'Fs', 'd'};
|
|
#+end_src
|
|
|
|
From the identified dynamics, compute $\alpha$ and $\beta$
|
|
#+begin_src matlab
|
|
alpha = abs(dcgain(G('y', 'Fa')));
|
|
beta = abs(dcgain(G('y', 'Fd')));
|
|
#+end_src
|
|
|
|
$k_a$ is estimated using the following formula:
|
|
#+begin_src matlab
|
|
ka = 0.8/abs(dcgain(G('y', 'Fa')));
|
|
#+end_src
|
|
The factor can be adjusted to better match the curves.
|
|
|
|
Then $k_e$ and $k_1$ are computed.
|
|
#+begin_src matlab
|
|
ke = ka/(beta/alpha - 1);
|
|
k1 = 1/beta - ke*ka/(ke + ka);
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
|
|
data2orgtable(1e-6*[ka; ke; k1], {'ka', 'ke', 'k1'}, {'Value [N/um]'}, ' %.1f ');
|
|
#+end_src
|
|
|
|
#+RESULTS:
|
|
| | Value [N/um] |
|
|
|----+--------------|
|
|
| ka | 40.5 |
|
|
| ke | 1.5 |
|
|
| k1 | 0.4 |
|
|
|
|
The damping in the system is adjusted to match the FEM model if necessary.
|
|
#+begin_src matlab
|
|
c1 = 1e2;
|
|
#+end_src
|
|
|
|
The analytical model of the simpler system is defined below:
|
|
#+begin_src matlab
|
|
Ga = 1/(m*s^2 + k1 + c1*s + ke*ka/(ke + ka)) * ...
|
|
[ 1 , k1 + c1*s + ke*ka/(ke + ka) , ke/(ke + ka) ;
|
|
-ke*ka/(ke + ka), ke*ka/(ke + ka)*m*s^2 , -ke/(ke + ka)*(m*s^2 + c1*s + k1)];
|
|
|
|
Ga.InputName = {'Fd', 'w', 'Fa'};
|
|
Ga.OutputName = {'y', 'Fs'};
|
|
#+end_src
|
|
|
|
And the DC gain is adjusted for the force sensor:
|
|
#+begin_src matlab
|
|
F_gain = dcgain(G('Fs', 'Fd'))/dcgain(Ga('Fs', 'Fd'));
|
|
#+end_src
|
|
|
|
The dynamics of the FEM model and the simpler model are compared in Figure [[fig:apa300ml_comp_simpler_model]].
|
|
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(0, 5, 1000);
|
|
|
|
figure;
|
|
tiledlayout(2, 3, 'TileSpacing', 'None', 'Padding', 'None');
|
|
|
|
ax1 = nexttile;
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G( 'y', 'w'), freqs, 'Hz'))));
|
|
plot(freqs, abs(squeeze(freqresp(Ga('y', 'w'), freqs, 'Hz'))));
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
set(gca, 'XTickLabel',[]);
|
|
ylabel('$x_1/w$ [m/m]');
|
|
ylim([1e-6, 1e2]);
|
|
|
|
ax2 = nexttile;
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G( 'y', 'Fa'), freqs, 'Hz'))));
|
|
plot(freqs, abs(squeeze(freqresp(Ga('y', 'Fa'), freqs, 'Hz'))));
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
set(gca, 'XTickLabel',[]);
|
|
ylabel('$x_1/f$ [m/N]');
|
|
ylim([1e-14, 1e-6]);
|
|
|
|
ax3 = nexttile;
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G( 'y', 'Fd'), freqs, 'Hz'))));
|
|
plot(freqs, abs(squeeze(freqresp(Ga('y', 'Fd'), freqs, 'Hz'))));
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
set(gca, 'XTickLabel',[]);
|
|
ylabel('$x_1/F$ [m/N]');
|
|
ylim([1e-14, 1e-4]);
|
|
|
|
ax4 = nexttile;
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G( 'Fs', 'w'), freqs, 'Hz'))));
|
|
plot(freqs, abs(squeeze(freqresp(F_gain*Ga('Fs', 'w'), freqs, 'Hz'))));
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
xlabel('Frequency [Hz]');
|
|
ylabel('$F_s/w$ [m/m]');
|
|
ylim([1e2, 1e8]);
|
|
|
|
ax5 = nexttile;
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G( 'Fs', 'Fa'), freqs, 'Hz'))));
|
|
plot(freqs, abs(squeeze(freqresp(F_gain*Ga('Fs', 'Fa'), freqs, 'Hz'))));
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
xlabel('Frequency [Hz]');
|
|
ylabel('$F_s/f$ [m/N]');
|
|
ylim([1e-4, 1e1]);
|
|
|
|
ax6 = nexttile;
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G( 'Fs', 'Fd'), freqs, 'Hz'))));
|
|
plot(freqs, abs(squeeze(freqresp(F_gain*Ga('Fs', 'Fd'), freqs, 'Hz'))));
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
xlabel('Frequency [Hz]');
|
|
ylabel('$F_s/F$ [m/N]');
|
|
ylim([1e-7, 1e2]);
|
|
|
|
linkaxes([ax1,ax2,ax3,ax4,ax5,ax6],'x');
|
|
#+end_src
|
|
|
|
#+begin_src matlab :tangle no :exports results :results file replace
|
|
exportFig('figs/apa300ml_comp_simpler_model.pdf', 'width', 'full', 'height', 'full');
|
|
#+end_src
|
|
|
|
#+name: fig:apa300ml_comp_simpler_model
|
|
#+caption: Comparison of the Dynamics between the FEM model and the simplified one
|
|
#+RESULTS:
|
|
[[file:figs/apa300ml_comp_simpler_model.png]]
|
|
|
|
The simplified model has also been implemented in Simscape.
|
|
|
|
The dynamics of the Simscape simplified model is identified and compared with the FEM one in Figure [[fig:apa300ml_comp_simpler_simscape]].
|
|
#+begin_src matlab :exports none
|
|
%% Name of the Simulink File
|
|
mdl = 'APA300ML_simplified';
|
|
|
|
%% Input/Output definition
|
|
clear io; io_i = 1;
|
|
io(io_i) = linio([mdl, '/Fd'], 1, 'openinput'); io_i = io_i + 1; % External Vertical Force [N]
|
|
io(io_i) = linio([mdl, '/w'], 1, 'openinput'); io_i = io_i + 1; % Base Motion [m]
|
|
io(io_i) = linio([mdl, '/Fa'], 1, 'openinput'); io_i = io_i + 1; % Actuator Force [N]
|
|
io(io_i) = linio([mdl, '/y'], 1, 'openoutput'); io_i = io_i + 1; % Vertical Displacement [m]
|
|
io(io_i) = linio([mdl, '/Fs'], 1, 'openoutput'); io_i = io_i + 1; % Force Sensor [V]
|
|
|
|
Gs = linearize(mdl, io);
|
|
|
|
Gs.InputName = {'Fd', 'w', 'Fa'};
|
|
Gs.OutputName = {'y', 'Fs'};
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(0, 5, 1000);
|
|
|
|
figure;
|
|
tiledlayout(2, 3, 'TileSpacing', 'None', 'Padding', 'None');
|
|
|
|
ax1 = nexttile;
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G( 'y', 'w'), freqs, 'Hz'))));
|
|
plot(freqs, abs(squeeze(freqresp(Gs('y', 'w'), freqs, 'Hz'))));
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
set(gca, 'XTickLabel',[]);
|
|
ylabel('$x_1/w$ [m/m]');
|
|
ylim([1e-6, 1e2]);
|
|
|
|
ax2 = nexttile;
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G( 'y', 'Fa'), freqs, 'Hz'))));
|
|
plot(freqs, abs(squeeze(freqresp(Gs('y', 'Fa'), freqs, 'Hz'))));
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
set(gca, 'XTickLabel',[]);
|
|
ylabel('$x_1/f$ [m/N]');
|
|
ylim([1e-14, 1e-6]);
|
|
|
|
ax3 = nexttile;
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G( 'y', 'Fd'), freqs, 'Hz'))));
|
|
plot(freqs, abs(squeeze(freqresp(Gs('y', 'Fd'), freqs, 'Hz'))));
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
set(gca, 'XTickLabel',[]);
|
|
ylabel('$x_1/F$ [m/N]');
|
|
ylim([1e-14, 1e-4]);
|
|
|
|
ax4 = nexttile;
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G( 'Fs', 'w'), freqs, 'Hz'))));
|
|
plot(freqs, abs(squeeze(freqresp(F_gain*Gs('Fs', 'w'), freqs, 'Hz'))));
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
xlabel('Frequency [Hz]');
|
|
ylabel('$F_s/w$ [m/m]');
|
|
ylim([1e2, 1e8]);
|
|
|
|
ax5 = nexttile;
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G( 'Fs', 'Fa'), freqs, 'Hz'))));
|
|
plot(freqs, abs(squeeze(freqresp(F_gain*Gs('Fs', 'Fa'), freqs, 'Hz'))));
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
xlabel('Frequency [Hz]');
|
|
ylabel('$F_s/f$ [m/N]');
|
|
ylim([1e-4, 1e1]);
|
|
|
|
ax6 = nexttile;
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G( 'Fs', 'Fd'), freqs, 'Hz'))));
|
|
plot(freqs, abs(squeeze(freqresp(F_gain*Gs('Fs', 'Fd'), freqs, 'Hz'))));
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
xlabel('Frequency [Hz]');
|
|
ylabel('$F_s/F$ [m/N]');
|
|
ylim([1e-7, 1e2]);
|
|
|
|
linkaxes([ax1,ax2,ax3,ax4,ax5,ax6],'x');
|
|
#+end_src
|
|
|
|
#+begin_src matlab :tangle no :exports results :results file replace
|
|
exportFig('figs/apa300ml_comp_simpler_simscape.pdf', 'width', 'full', 'height', 'full');
|
|
#+end_src
|
|
|
|
#+name: fig:apa300ml_comp_simpler_simscape
|
|
#+caption: Comparison of the Dynamics between the FEM model and the simplified simscape model
|
|
#+RESULTS:
|
|
[[file:figs/apa300ml_comp_simpler_simscape.png]]
|
|
|
|
* First Flexible Joint Geometry
|
|
:PROPERTIES:
|
|
:header-args:matlab+: :tangle matlab/flexor_ID16.m
|
|
:END:
|
|
<<sec:flexor_ID16>>
|
|
** Introduction :ignore:
|
|
The studied flexor is shown in Figure [[fig:flexor_id16_screenshot]].
|
|
|
|
The stiffness and mass matrices representing the dynamics of the flexor are exported from a FEM.
|
|
It is then imported into Simscape.
|
|
|
|
A simplified model of the flexor is then developped.
|
|
|
|
#+name: fig:flexor_id16_screenshot
|
|
#+caption: Flexor studied
|
|
[[file:figs/flexor_id16_screenshot.png]]
|
|
|
|
** Matlab Init :noexport:ignore:
|
|
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
|
|
<<matlab-dir>>
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none :results silent :noweb yes
|
|
<<matlab-init>>
|
|
#+end_src
|
|
|
|
#+begin_src matlab :tangle no
|
|
addpath('matlab/');
|
|
addpath('matlab/flexor_ID16/');
|
|
#+end_src
|
|
|
|
#+begin_src matlab :eval no
|
|
addpath('flexor_ID16/');
|
|
#+end_src
|
|
|
|
#+begin_src matlab
|
|
open('flexor_ID16.slx');
|
|
#+end_src
|
|
|
|
** Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates
|
|
We first extract the stiffness and mass matrices.
|
|
#+begin_src matlab
|
|
K = extractMatrix('mat_K_6modes_2MDoF.matrix');
|
|
M = extractMatrix('mat_M_6modes_2MDoF.matrix');
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports results :results value table replace :tangle no
|
|
data2orgtable(K(1:10, 1:10), {}, {}, ' %.2e ');
|
|
#+end_src
|
|
|
|
#+caption: First 10x10 elements of the Stiffness matrix
|
|
#+RESULTS:
|
|
| 11200000.0 | 195.0 | 2220.0 | -0.719 | -265.0 | 1.59 | -11200000.0 | -213.0 | -2220.0 | 0.147 |
|
|
| 195.0 | 11400000.0 | 1290.0 | -148.0 | -0.188 | 2.41 | -212.0 | -11400000.0 | -1290.0 | 148.0 |
|
|
| 2220.0 | 1290.0 | 119000000.0 | 1.31 | 1.49 | 1.79 | -2220.0 | -1290.0 | -119000000.0 | -1.31 |
|
|
| -0.719 | -148.0 | 1.31 | 33.0 | 0.000488 | -0.000977 | 0.141 | 148.0 | -1.31 | -33.0 |
|
|
| -265.0 | -0.188 | 1.49 | 0.000488 | 33.0 | 0.00293 | 266.0 | 0.154 | -1.49 | 0.00026 |
|
|
| 1.59 | 2.41 | 1.79 | -0.000977 | 0.00293 | 236.0 | -1.32 | -2.55 | -1.79 | 0.000379 |
|
|
| -11200000.0 | -212.0 | -2220.0 | 0.141 | 266.0 | -1.32 | 11400000.0 | 24600.0 | 1640.0 | 120.0 |
|
|
| -213.0 | -11400000.0 | -1290.0 | 148.0 | 0.154 | -2.55 | 24600.0 | 11400000.0 | 1290.0 | -72.0 |
|
|
| -2220.0 | -1290.0 | -119000000.0 | -1.31 | -1.49 | -1.79 | 1640.0 | 1290.0 | 119000000.0 | 1.32 |
|
|
| 0.147 | 148.0 | -1.31 | -33.0 | 0.00026 | 0.000379 | 120.0 | -72.0 | 1.32 | 34.7 |
|
|
|
|
#+begin_src matlab :exports results :results value table replace :tangle no
|
|
data2orgtable(M(1:10, 1:10), {}, {}, ' %.1g ');
|
|
#+end_src
|
|
|
|
#+caption: First 10x10 elements of the Mass matrix
|
|
#+RESULTS:
|
|
| 0.02 | 1e-09 | -4e-08 | -1e-10 | 0.0002 | -3e-11 | 0.004 | 5e-08 | 7e-08 | 1e-10 |
|
|
| 1e-09 | 0.02 | -3e-07 | -0.0002 | -1e-10 | -2e-09 | 2e-08 | 0.004 | 3e-07 | 1e-05 |
|
|
| -4e-08 | -3e-07 | 0.02 | 7e-10 | -2e-09 | 1e-09 | 3e-07 | 7e-08 | 0.003 | 1e-09 |
|
|
| -1e-10 | -0.0002 | 7e-10 | 4e-06 | -1e-12 | -6e-13 | 2e-10 | -7e-06 | -8e-10 | -1e-09 |
|
|
| 0.0002 | -1e-10 | -2e-09 | -1e-12 | 3e-06 | 2e-13 | 9e-06 | 4e-11 | 2e-09 | -3e-13 |
|
|
| -3e-11 | -2e-09 | 1e-09 | -6e-13 | 2e-13 | 4e-07 | 8e-11 | 9e-10 | -1e-09 | 2e-12 |
|
|
| 0.004 | 2e-08 | 3e-07 | 2e-10 | 9e-06 | 8e-11 | 0.02 | -7e-08 | -3e-07 | -2e-10 |
|
|
| 5e-08 | 0.004 | 7e-08 | -7e-06 | 4e-11 | 9e-10 | -7e-08 | 0.01 | -4e-08 | 0.0002 |
|
|
| 7e-08 | 3e-07 | 0.003 | -8e-10 | 2e-09 | -1e-09 | -3e-07 | -4e-08 | 0.02 | -1e-09 |
|
|
| 1e-10 | 1e-05 | 1e-09 | -1e-09 | -3e-13 | 2e-12 | -2e-10 | 0.0002 | -1e-09 | 2e-06 |
|
|
|
|
Then, we extract the coordinates of the interface nodes.
|
|
#+begin_src matlab
|
|
[int_xyz, int_i, n_xyz, n_i, nodes] = extractNodes('out_nodes_3D.txt');
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
|
|
data2orgtable([[1:length(int_i)]', int_i, int_xyz], {}, {'Node i', 'Node Number', 'x [m]', 'y [m]', 'z [m]'}, ' %f ');
|
|
#+end_src
|
|
|
|
#+caption: Coordinates of the interface nodes
|
|
#+RESULTS:
|
|
| Node i | Node Number | x [m] | y [m] | z [m] |
|
|
|--------+-------------+-------+-------+-------|
|
|
| 1.0 | 181278.0 | 0.0 | 0.0 | 0.0 |
|
|
| 2.0 | 181279.0 | 0.0 | 0.0 | -0.0 |
|
|
|
|
#+begin_src matlab :exports results :results value table replace :tangle no
|
|
data2orgtable([length(n_i); length(int_i); size(M,1) - 6*length(int_i); size(M,1)], {'Total number of Nodes', 'Number of interface Nodes', 'Number of Modes', 'Size of M and K matrices'}, {}, ' %.0f ');
|
|
#+end_src
|
|
|
|
#+RESULTS:
|
|
| Total number of Nodes | 2 |
|
|
| Number of interface Nodes | 2 |
|
|
| Number of Modes | 6 |
|
|
| Size of M and K matrices | 18 |
|
|
|
|
Using =K=, =M= and =int_xyz=, we can use the =Reduced Order Flexible Solid= simscape block.
|
|
|
|
** Identification of the parameters using Simscape and looking at the Stiffness Matrix
|
|
The flexor is now imported into Simscape and its parameters are estimated using an identification.
|
|
|
|
#+begin_src matlab :exports none
|
|
m = 1;
|
|
#+end_src
|
|
|
|
The dynamics is identified from the applied force/torque to the measured displacement/rotation of the flexor.
|
|
#+begin_src matlab :exports none
|
|
%% Name of the Simulink File
|
|
mdl = 'flexor_ID16';
|
|
|
|
%% Input/Output definition
|
|
clear io; io_i = 1;
|
|
io(io_i) = linio([mdl, '/T'], 1, 'openinput'); io_i = io_i + 1;
|
|
io(io_i) = linio([mdl, '/D'], 1, 'openoutput'); io_i = io_i + 1;
|
|
|
|
G = linearize(mdl, io);
|
|
#+end_src
|
|
|
|
And we find the same parameters as the one estimated from the Stiffness matrix.
|
|
|
|
#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
|
|
data2orgtable([1e-6*K(3,3), K(4,4), K(5,5), K(6,6) ; 1e-6./dcgain(G(3,3)), 1./dcgain(G(4,4)), 1./dcgain(G(5,5)), 1./dcgain(G(6,6))]', {'Axial Stiffness Dz [N/um]', 'Bending Stiffness Rx [Nm/rad]', 'Bending Stiffness Ry [Nm/rad]', 'Torsion Stiffness Rz [Nm/rad]'}, {'*Caracteristic*', '*Value*', '*Identification*'}, ' %0.f ');
|
|
#+end_src
|
|
|
|
#+RESULTS:
|
|
| *Caracteristic* | *Value* | *Identification* |
|
|
|-------------------------------+---------+------------------|
|
|
| Axial Stiffness Dz [N/um] | 119 | 119 |
|
|
| Bending Stiffness Rx [Nm/rad] | 33 | 33 |
|
|
| Bending Stiffness Ry [Nm/rad] | 33 | 33 |
|
|
| Torsion Stiffness Rz [Nm/rad] | 236 | 236 |
|
|
|
|
** Simpler Model
|
|
Let's now model the flexible joint with a "perfect" Bushing joint as shown in Figure [[fig:flexible_joint_simscape]].
|
|
|
|
#+name: fig:flexible_joint_simscape
|
|
#+caption: Bushing Joint used to model the flexible joint
|
|
[[file:figs/flexible_joint_simscape.png]]
|
|
|
|
The parameters of the Bushing joint (stiffnesses) are estimated from the Stiffness matrix that was computed from the FEM.
|
|
#+begin_src matlab
|
|
Kx = K(1,1); % [N/m]
|
|
Ky = K(2,2); % [N/m]
|
|
Kz = K(3,3); % [N/m]
|
|
Krx = K(4,4); % [Nm/rad]
|
|
Kry = K(5,5); % [Nm/rad]
|
|
Krz = K(6,6); % [Nm/rad]
|
|
#+end_src
|
|
|
|
The dynamics from the applied force/torque to the measured displacement/rotation of the flexor is identified again for this simpler model.
|
|
#+begin_src matlab :exports none
|
|
%% Name of the Simulink File
|
|
mdl = 'flexor_ID16_simplified';
|
|
|
|
%% Input/Output definition
|
|
clear io; io_i = 1;
|
|
io(io_i) = linio([mdl, '/T'], 1, 'openinput'); io_i = io_i + 1;
|
|
io(io_i) = linio([mdl, '/D'], 1, 'openoutput'); io_i = io_i + 1;
|
|
|
|
Gs = linearize(mdl, io);
|
|
#+end_src
|
|
|
|
The two obtained dynamics are compared in Figure
|
|
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(0, 5, 1000);
|
|
|
|
figure;
|
|
tiledlayout(1, 2, 'TileSpacing', 'None', 'Padding', 'None');
|
|
|
|
ax1 = nexttile;
|
|
hold on;
|
|
set(gca,'ColorOrderIndex',1)
|
|
plot(freqs, abs(squeeze(freqresp(G(1,1), freqs, 'Hz'))), '-', 'DisplayName', '$D_x/F_x$');
|
|
set(gca,'ColorOrderIndex',1)
|
|
plot(freqs, abs(squeeze(freqresp(Gs(1,1), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
|
|
set(gca,'ColorOrderIndex',2)
|
|
plot(freqs, abs(squeeze(freqresp(G(2,2), freqs, 'Hz'))), '-', 'DisplayName', '$D_y/F_y$');
|
|
set(gca,'ColorOrderIndex',2)
|
|
plot(freqs, abs(squeeze(freqresp(Gs(2,2), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
|
|
set(gca,'ColorOrderIndex',3)
|
|
plot(freqs, abs(squeeze(freqresp(G(3,3), freqs, 'Hz'))), '-', 'DisplayName', '$D_z/F_z$');
|
|
set(gca,'ColorOrderIndex',3)
|
|
plot(freqs, abs(squeeze(freqresp(Gs(3,3), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
xlabel('Frequency [Hz]'); ylabel('Amplitude [m/N]');
|
|
hold off;
|
|
legend('location', 'southwest');
|
|
|
|
ax2 = nexttile;
|
|
hold on;
|
|
set(gca,'ColorOrderIndex',1)
|
|
plot(freqs, abs(squeeze(freqresp(G(4,4), freqs, 'Hz'))), '-', 'DisplayName', '$R_x/M_x$');
|
|
set(gca,'ColorOrderIndex',1)
|
|
plot(freqs, abs(squeeze(freqresp(Gs(4,4), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
|
|
set(gca,'ColorOrderIndex',2)
|
|
plot(freqs, abs(squeeze(freqresp(G(5,5), freqs, 'Hz'))), '-', 'DisplayName', '$R_y/M_y$');
|
|
set(gca,'ColorOrderIndex',2)
|
|
plot(freqs, abs(squeeze(freqresp(Gs(5,5), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
|
|
set(gca,'ColorOrderIndex',3)
|
|
plot(freqs, abs(squeeze(freqresp(G(6,6), freqs, 'Hz'))), '-', 'DisplayName', '$R_z/M_z$');
|
|
set(gca,'ColorOrderIndex',3)
|
|
plot(freqs, abs(squeeze(freqresp(Gs(6,6), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
xlabel('Frequency [Hz]'); ylabel('Amplitude [rad/Nm]');
|
|
hold off;
|
|
legend('location', 'southwest');
|
|
#+end_src
|
|
|
|
#+begin_src matlab :tangle no :exports results :results file replace
|
|
exportFig('figs/flexor_ID16_compare_bushing_joint.pdf', 'width', 'wide', 'height', 'normal');
|
|
#+end_src
|
|
|
|
#+name: fig:flexor_ID16_compare_bushing_joint
|
|
#+caption: Comparison of the Joint compliance between the FEM model and the simpler model
|
|
#+RESULTS:
|
|
[[file:figs/flexor_ID16_compare_bushing_joint.png]]
|
|
|
|
* Optimized Flexible Joint
|
|
:PROPERTIES:
|
|
:header-args:matlab+: :tangle matlab/flexor_025.m
|
|
:END:
|
|
<<sec:flexor_025>>
|
|
** Introduction :ignore:
|
|
|
|
The joint geometry has been optimized using Ansys to have lower bending stiffness while keeping a large axial stiffness.
|
|
|
|
The obtained geometry is shown in Figure [[fig:optimal_flexor]].
|
|
|
|
#+name: fig:optimal_flexor
|
|
#+caption: Flexor studied
|
|
[[file:figs/flexor_025_MDoF.jpg]]
|
|
|
|
** Matlab Init :noexport:ignore:
|
|
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
|
|
<<matlab-dir>>
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none :results silent :noweb yes
|
|
<<matlab-init>>
|
|
#+end_src
|
|
|
|
#+begin_src matlab :tangle no
|
|
addpath('matlab/');
|
|
addpath('matlab/flexor_025/');
|
|
#+end_src
|
|
|
|
#+begin_src matlab :eval no
|
|
addpath('flexor_025/');
|
|
#+end_src
|
|
|
|
#+begin_src matlab
|
|
open('flexor_025.slx');
|
|
#+end_src
|
|
|
|
** Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates
|
|
We first extract the stiffness and mass matrices.
|
|
#+begin_src matlab
|
|
K = readmatrix('mat_K.CSV');
|
|
M = readmatrix('mat_M.CSV');
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports results :results value table replace :tangle no
|
|
data2orgtable(K(1:10, 1:10), {}, {}, ' %.2e ');
|
|
#+end_src
|
|
|
|
#+caption: First 10x10 elements of the Stiffness matrix
|
|
#+RESULTS:
|
|
| 12700000.0 | -18.5 | -26.8 | 0.00162 | -4.63 | 64.0 | -12700000.0 | 18.3 | 26.7 | 0.00234 |
|
|
| -18.5 | 12700000.0 | -499.0 | -132.0 | 0.00414 | -0.495 | 18.4 | -12700000.0 | 499.0 | 132.0 |
|
|
| -26.8 | -499.0 | 94000000.0 | -470.0 | 0.00771 | -0.855 | 26.8 | 498.0 | -94000000.0 | 470.0 |
|
|
| 0.00162 | -132.0 | -470.0 | 4.83 | 2.61e-07 | 0.000123 | -0.00163 | 132.0 | 470.0 | -4.83 |
|
|
| -4.63 | 0.00414 | 0.00771 | 2.61e-07 | 4.83 | 4.43e-05 | 4.63 | -0.00413 | -0.00772 | -4.3e-07 |
|
|
| 64.0 | -0.495 | -0.855 | 0.000123 | 4.43e-05 | 260.0 | -64.0 | 0.495 | 0.855 | -0.000124 |
|
|
| -12700000.0 | 18.4 | 26.8 | -0.00163 | 4.63 | -64.0 | 12700000.0 | -18.2 | -26.7 | -0.00234 |
|
|
| 18.3 | -12700000.0 | 498.0 | 132.0 | -0.00413 | 0.495 | -18.2 | 12700000.0 | -498.0 | -132.0 |
|
|
| 26.7 | 499.0 | -94000000.0 | 470.0 | -0.00772 | 0.855 | -26.7 | -498.0 | 94000000.0 | -470.0 |
|
|
| 0.00234 | 132.0 | 470.0 | -4.83 | -4.3e-07 | -0.000124 | -0.00234 | -132.0 | -470.0 | 4.83 |
|
|
|
|
|
|
#+begin_src matlab :exports results :results value table replace :tangle no
|
|
data2orgtable(M(1:10, 1:10), {}, {}, ' %.1g ');
|
|
#+end_src
|
|
|
|
#+caption: First 10x10 elements of the Mass matrix
|
|
#+RESULTS:
|
|
| 0.006 | 8e-09 | -2e-08 | -1e-10 | 3e-05 | 3e-08 | 0.003 | -3e-09 | 9e-09 | 2e-12 |
|
|
| 8e-09 | 0.02 | 1e-07 | -3e-05 | 1e-11 | 6e-10 | 1e-08 | 0.003 | -5e-08 | 3e-09 |
|
|
| -2e-08 | 1e-07 | 0.01 | -6e-08 | -6e-11 | -8e-12 | -1e-07 | 1e-08 | 0.003 | -1e-08 |
|
|
| -1e-10 | -3e-05 | -6e-08 | 1e-06 | 7e-14 | 6e-13 | 1e-10 | 1e-06 | -1e-08 | 3e-10 |
|
|
| 3e-05 | 1e-11 | -6e-11 | 7e-14 | 2e-07 | 1e-10 | 3e-08 | -7e-12 | 6e-11 | -6e-16 |
|
|
| 3e-08 | 6e-10 | -8e-12 | 6e-13 | 1e-10 | 5e-07 | 1e-08 | -5e-10 | -1e-11 | 1e-13 |
|
|
| 0.003 | 1e-08 | -1e-07 | 1e-10 | 3e-08 | 1e-08 | 0.02 | -2e-08 | 1e-07 | -4e-12 |
|
|
| -3e-09 | 0.003 | 1e-08 | 1e-06 | -7e-12 | -5e-10 | -2e-08 | 0.006 | -8e-08 | 3e-05 |
|
|
| 9e-09 | -5e-08 | 0.003 | -1e-08 | 6e-11 | -1e-11 | 1e-07 | -8e-08 | 0.01 | -6e-08 |
|
|
| 2e-12 | 3e-09 | -1e-08 | 3e-10 | -6e-16 | 1e-13 | -4e-12 | 3e-05 | -6e-08 | 2e-07 |
|
|
|
|
|
|
Then, we extract the coordinates of the interface nodes.
|
|
#+begin_src matlab
|
|
[int_xyz, int_i, n_xyz, n_i, nodes] = extractNodes('out_nodes_3D.txt');
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports results :results value table replace :tangle no
|
|
data2orgtable([length(n_i); length(int_i); size(M,1) - 6*length(int_i); size(M,1)], {'Total number of Nodes', 'Number of interface Nodes', 'Number of Modes', 'Size of M and K matrices'}, {}, ' %.0f ');
|
|
#+end_src
|
|
|
|
#+RESULTS:
|
|
| Total number of Nodes | 2 |
|
|
| Number of interface Nodes | 2 |
|
|
| Number of Modes | 6 |
|
|
| Size of M and K matrices | 18 |
|
|
|
|
#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
|
|
data2orgtable([[1:length(int_i)]', int_i, int_xyz], {}, {'Node i', 'Node Number', 'x [m]', 'y [m]', 'z [m]'}, ' %f ');
|
|
#+end_src
|
|
|
|
#+caption: Coordinates of the interface nodes
|
|
#+RESULTS:
|
|
| Node i | Node Number | x [m] | y [m] | z [m] |
|
|
|--------+-------------+-------+-------+-------|
|
|
| 1.0 | 528875.0 | 0.0 | 0.0 | 0.0 |
|
|
| 2.0 | 528876.0 | 0.0 | 0.0 | -0.0 |
|
|
|
|
Using =K=, =M= and =int_xyz=, we can use the =Reduced Order Flexible Solid= simscape block.
|
|
|
|
** Identification of the parameters using Simscape
|
|
The flexor is now imported into Simscape and its parameters are estimated using an identification.
|
|
|
|
#+begin_src matlab :exports none
|
|
m = 1;
|
|
#+end_src
|
|
|
|
The dynamics is identified from the applied force/torque to the measured displacement/rotation of the flexor.
|
|
#+begin_src matlab :exports none
|
|
%% Name of the Simulink File
|
|
mdl = 'flexor_025';
|
|
|
|
%% Input/Output definition
|
|
clear io; io_i = 1;
|
|
io(io_i) = linio([mdl, '/T'], 1, 'openinput'); io_i = io_i + 1;
|
|
io(io_i) = linio([mdl, '/D'], 1, 'openoutput'); io_i = io_i + 1;
|
|
|
|
G = linearize(mdl, io);
|
|
#+end_src
|
|
|
|
And we find the same parameters as the one estimated from the Stiffness matrix.
|
|
|
|
#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
|
|
data2orgtable([1e-6*K(3,3), K(4,4), K(5,5), K(6,6) ; 1e-6./dcgain(G(3,3)), 1./dcgain(G(4,4)), 1./dcgain(G(5,5)), 1./dcgain(G(6,6))]', {'Axial Stiffness Dz [N/um]', 'Bending Stiffness Rx [Nm/rad]', 'Bending Stiffness Ry [Nm/rad]', 'Torsion Stiffness Rz [Nm/rad]'}, {'*Caracteristic*', '*Value*', '*Identification*'}, ' %.1f ');
|
|
#+end_src
|
|
|
|
#+RESULTS:
|
|
| *Caracteristic* | *Value* | *Identification* |
|
|
|-------------------------------+---------+------------------|
|
|
| Axial Stiffness Dz [N/um] | 94.0 | 93.9 |
|
|
| Bending Stiffness Rx [Nm/rad] | 4.8 | 4.8 |
|
|
| Bending Stiffness Ry [Nm/rad] | 4.8 | 4.8 |
|
|
| Torsion Stiffness Rz [Nm/rad] | 260.2 | 260.2 |
|
|
|
|
** Simpler Model
|
|
Let's now model the flexible joint with a "perfect" Bushing joint as shown in Figure [[fig:flexible_joint_simscape]].
|
|
|
|
#+name: fig:flexible_joint_simscape
|
|
#+caption: Bushing Joint used to model the flexible joint
|
|
[[file:figs/flexible_joint_simscape.png]]
|
|
|
|
The parameters of the Bushing joint (stiffnesses) are estimated from the Stiffness matrix that was computed from the FEM.
|
|
#+begin_src matlab
|
|
Kx = K(1,1); % [N/m]
|
|
Ky = K(2,2); % [N/m]
|
|
Kz = K(3,3); % [N/m]
|
|
Krx = K(4,4); % [Nm/rad]
|
|
Kry = K(5,5); % [Nm/rad]
|
|
Krz = K(6,6); % [Nm/rad]
|
|
#+end_src
|
|
|
|
The dynamics from the applied force/torque to the measured displacement/rotation of the flexor is identified again for this simpler model.
|
|
#+begin_src matlab :exports none
|
|
%% Name of the Simulink File
|
|
mdl = 'flexor_025_simplified';
|
|
|
|
%% Input/Output definition
|
|
clear io; io_i = 1;
|
|
io(io_i) = linio([mdl, '/T'], 1, 'openinput'); io_i = io_i + 1;
|
|
io(io_i) = linio([mdl, '/D'], 1, 'openoutput'); io_i = io_i + 1;
|
|
|
|
Gs = linearize(mdl, io);
|
|
#+end_src
|
|
|
|
The two obtained dynamics are compared in Figure
|
|
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(0, 5, 1000);
|
|
|
|
figure;
|
|
tiledlayout(1, 2, 'TileSpacing', 'None', 'Padding', 'None');
|
|
|
|
ax1 = nexttile;
|
|
hold on;
|
|
set(gca,'ColorOrderIndex',1)
|
|
plot(freqs, abs(squeeze(freqresp(G(1,1), freqs, 'Hz'))), '-', 'DisplayName', '$D_x/F_x$');
|
|
set(gca,'ColorOrderIndex',1)
|
|
plot(freqs, abs(squeeze(freqresp(Gs(1,1), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
|
|
set(gca,'ColorOrderIndex',2)
|
|
plot(freqs, abs(squeeze(freqresp(G(2,2), freqs, 'Hz'))), '-', 'DisplayName', '$D_y/F_y$');
|
|
set(gca,'ColorOrderIndex',2)
|
|
plot(freqs, abs(squeeze(freqresp(Gs(2,2), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
|
|
set(gca,'ColorOrderIndex',3)
|
|
plot(freqs, abs(squeeze(freqresp(G(3,3), freqs, 'Hz'))), '-', 'DisplayName', '$D_z/F_z$');
|
|
set(gca,'ColorOrderIndex',3)
|
|
plot(freqs, abs(squeeze(freqresp(Gs(3,3), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
xlabel('Frequency [Hz]'); ylabel('Amplitude [m/N]');
|
|
hold off;
|
|
legend('location', 'southwest');
|
|
|
|
ax2 = nexttile;
|
|
hold on;
|
|
set(gca,'ColorOrderIndex',1)
|
|
plot(freqs, abs(squeeze(freqresp(G(4,4), freqs, 'Hz'))), '-', 'DisplayName', '$R_x/M_x$');
|
|
set(gca,'ColorOrderIndex',1)
|
|
plot(freqs, abs(squeeze(freqresp(Gs(4,4), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
|
|
set(gca,'ColorOrderIndex',2)
|
|
plot(freqs, abs(squeeze(freqresp(G(5,5), freqs, 'Hz'))), '-', 'DisplayName', '$R_y/M_y$');
|
|
set(gca,'ColorOrderIndex',2)
|
|
plot(freqs, abs(squeeze(freqresp(Gs(5,5), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
|
|
set(gca,'ColorOrderIndex',3)
|
|
plot(freqs, abs(squeeze(freqresp(G(6,6), freqs, 'Hz'))), '-', 'DisplayName', '$R_z/M_z$');
|
|
set(gca,'ColorOrderIndex',3)
|
|
plot(freqs, abs(squeeze(freqresp(Gs(6,6), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
xlabel('Frequency [Hz]'); ylabel('Amplitude [rad/Nm]');
|
|
hold off;
|
|
legend('location', 'southwest');
|
|
#+end_src
|
|
|
|
#+begin_src matlab :tangle no :exports results :results file replace
|
|
exportFig('figs/flexor_ID16_compare_bushing_joint.pdf', 'width', 'wide', 'height', 'normal');
|
|
#+end_src
|
|
|
|
#+name: fig:flexor_ID16_compare_bushing_joint
|
|
#+caption: Comparison of the Joint compliance between the FEM model and the simpler model
|
|
#+RESULTS:
|
|
[[file:figs/flexor_ID16_compare_bushing_joint.png]]
|
|
|
|
* Force Sensor and Integral Force Feedback with an Amplified Piezoelectric Actuator
|
|
<<sec:integral_force_feedback>>
|
|
** Introduction :ignore:
|
|
In this section, we try to replicate the results obtained in cite:souleille18_concep_activ_mount_space_applic.
|
|
|
|
** Matlab Init :noexport:ignore:
|
|
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
|
|
<<matlab-dir>>
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none :results silent :noweb yes
|
|
<<matlab-init>>
|
|
#+end_src
|
|
|
|
#+begin_src matlab
|
|
addpath('./data/piezo_amplified_IFF/');
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
open('piezo_amplified_IFF');
|
|
#+end_src
|
|
|
|
** Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates
|
|
We first extract the stiffness and mass matrices.
|
|
#+begin_src matlab
|
|
K = extractMatrix('piezo_amplified_IFF_K.txt');
|
|
M = extractMatrix('piezo_amplified_IFF_M.txt');
|
|
#+end_src
|
|
|
|
Then, we extract the coordinates of the interface nodes.
|
|
#+begin_src matlab
|
|
[int_xyz, int_i, n_xyz, n_i, nodes] = extractNodes('piezo_amplified_IFF.txt');
|
|
#+end_src
|
|
|
|
** IFF Plant
|
|
The transfer function from the force actuator to the force sensor is identified and shown in Figure [[fig:piezo_amplified_iff_plant]].
|
|
|
|
#+begin_src matlab
|
|
Kiff = tf(0);
|
|
#+end_src
|
|
|
|
#+begin_src matlab
|
|
m = 0;
|
|
#+end_src
|
|
|
|
#+begin_src matlab
|
|
%% Name of the Simulink File
|
|
mdl = 'piezo_amplified_IFF';
|
|
|
|
%% Input/Output definition
|
|
clear io; io_i = 1;
|
|
io(io_i) = linio([mdl, '/Kiff'], 1, 'openinput'); io_i = io_i + 1;
|
|
io(io_i) = linio([mdl, '/G'], 1, 'openoutput'); io_i = io_i + 1;
|
|
|
|
Gf = linearize(mdl, io);
|
|
#+end_src
|
|
|
|
#+begin_src matlab
|
|
m = 10;
|
|
#+end_src
|
|
|
|
#+begin_src matlab
|
|
Gfm = linearize(mdl, io);
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(1, 5, 1000);
|
|
|
|
figure;
|
|
|
|
ax1 = subplot(2,1,1);
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(Gf, freqs, 'Hz'))), '-');
|
|
plot(freqs, abs(squeeze(freqresp(Gfm, freqs, 'Hz'))), '-');
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude'); set(gca, 'XTickLabel',[]);
|
|
hold off;
|
|
|
|
ax2 = subplot(2,1,2);
|
|
hold on;
|
|
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gf, freqs, 'Hz')))), '-', ...
|
|
'DisplayName', '$m = 0kg$');
|
|
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gfm, freqs, 'Hz')))), '-', ...
|
|
'DisplayName', '$m = 10kg$');
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
|
yticks(-360:90:360);
|
|
ylim([-390 30]);
|
|
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
|
|
hold off;
|
|
linkaxes([ax1,ax2],'x');
|
|
xlim([freqs(1), freqs(end)]);
|
|
legend('location', 'southwest');
|
|
#+end_src
|
|
|
|
#+begin_src matlab :tangle no :exports results :results file replace
|
|
exportFig('figs/piezo_amplified_iff_plant.pdf', 'width', 'full', 'height', 'full');
|
|
#+end_src
|
|
|
|
#+name: fig:piezo_amplified_iff_plant
|
|
#+caption: IFF Plant
|
|
#+RESULTS:
|
|
[[file:figs/piezo_amplified_iff_plant.png]]
|
|
|
|
** IFF controller
|
|
The controller is defined and the loop gain is shown in Figure [[fig:piezo_amplified_iff_loop_gain]].
|
|
#+begin_src matlab
|
|
Kiff = -1e12/s;
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(1, 5, 1000);
|
|
|
|
figure;
|
|
|
|
ax1 = subplot(2,1,1);
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(Gf*Kiff, freqs, 'Hz'))), '-');
|
|
plot(freqs, abs(squeeze(freqresp(Gfm*Kiff, freqs, 'Hz'))), '-');
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude'); set(gca, 'XTickLabel',[]);
|
|
hold off;
|
|
|
|
ax2 = subplot(2,1,2);
|
|
hold on;
|
|
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gf*Kiff, freqs, 'Hz')))), '-', ...
|
|
'DisplayName', '$m = 0kg$');
|
|
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gfm*Kiff, freqs, 'Hz')))), '-', ...
|
|
'DisplayName', '$m = 10kg$');
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
|
yticks(-360:90:360);
|
|
ylim([-180 180]);
|
|
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
|
|
hold off;
|
|
linkaxes([ax1,ax2],'x');
|
|
xlim([freqs(1), freqs(end)]);
|
|
legend('location', 'southwest');
|
|
#+end_src
|
|
|
|
#+begin_src matlab :tangle no :exports results :results file replace
|
|
exportFig('figs/piezo_amplified_iff_loop_gain.pdf', 'width', 'full', 'height', 'full');
|
|
#+end_src
|
|
|
|
#+name: fig:piezo_amplified_iff_loop_gain
|
|
#+caption: IFF Loop Gain
|
|
#+RESULTS:
|
|
[[file:figs/piezo_amplified_iff_loop_gain.png]]
|
|
|
|
** Closed Loop System
|
|
#+begin_src matlab
|
|
m = 10;
|
|
#+end_src
|
|
|
|
#+begin_src matlab
|
|
Kiff = -1e12/s;
|
|
#+end_src
|
|
|
|
#+begin_src matlab
|
|
%% Name of the Simulink File
|
|
mdl = 'piezo_amplified_IFF';
|
|
|
|
%% Input/Output definition
|
|
clear io; io_i = 1;
|
|
io(io_i) = linio([mdl, '/Dw'], 1, 'openinput'); io_i = io_i + 1;
|
|
io(io_i) = linio([mdl, '/F'], 1, 'openinput'); io_i = io_i + 1;
|
|
io(io_i) = linio([mdl, '/Fd'], 1, 'openinput'); io_i = io_i + 1;
|
|
io(io_i) = linio([mdl, '/d'], 1, 'openoutput'); io_i = io_i + 1;
|
|
io(io_i) = linio([mdl, '/G'], 1, 'output'); io_i = io_i + 1;
|
|
|
|
Giff = linearize(mdl, io);
|
|
Giff.InputName = {'w', 'f', 'F'};
|
|
Giff.OutputName = {'x1', 'Fs'};
|
|
#+end_src
|
|
|
|
#+begin_src matlab
|
|
Kiff = tf(0);
|
|
#+end_src
|
|
|
|
#+begin_src matlab
|
|
G = linearize(mdl, io);
|
|
G.InputName = {'w', 'f', 'F'};
|
|
G.OutputName = {'x1', 'Fs'};
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(1, 3, 1000);
|
|
|
|
figure;
|
|
|
|
ax1 = subplot(2, 3, 1);
|
|
title('$\displaystyle \frac{x_1}{w}$')
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G('x1', 'w'), freqs, 'Hz'))));
|
|
plot(freqs, abs(squeeze(freqresp(Giff('x1', 'w'), freqs, 'Hz'))));
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/m]');xlabel('Frequency [Hz]');
|
|
|
|
ax2 = subplot(2, 3, 2);
|
|
title('$\displaystyle \frac{x_1}{f}$')
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G('x1', 'f'), freqs, 'Hz'))));
|
|
plot(freqs, abs(squeeze(freqresp(Giff('x1', 'f'), freqs, 'Hz'))));
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/N]');xlabel('Frequency [Hz]');
|
|
|
|
ax3 = subplot(2, 3, 3);
|
|
title('$\displaystyle \frac{x_1}{F}$')
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G('x1', 'F'), freqs, 'Hz'))));
|
|
plot(freqs, abs(squeeze(freqresp(Giff('x1', 'F'), freqs, 'Hz'))));
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/N]');xlabel('Frequency [Hz]');
|
|
|
|
ax4 = subplot(2, 3, 4);
|
|
title('$\displaystyle \frac{F_s}{w}$')
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G('Fs', 'w'), freqs, 'Hz'))));
|
|
plot(freqs, abs(squeeze(freqresp(Giff('Fs', 'w'), freqs, 'Hz'))));
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/m]');xlabel('Frequency [Hz]');
|
|
|
|
ax5 = subplot(2, 3, 5);
|
|
title('$\displaystyle \frac{F_s}{f}$')
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G('Fs', 'f'), freqs, 'Hz'))));
|
|
plot(freqs, abs(squeeze(freqresp(Giff('Fs', 'f'), freqs, 'Hz'))));
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/N]');xlabel('Frequency [Hz]');
|
|
|
|
ax6 = subplot(2, 3, 6);
|
|
title('$\displaystyle \frac{F_s}{F}$')
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G('Fs', 'F'), freqs, 'Hz'))));
|
|
plot(freqs, abs(squeeze(freqresp(Giff('Fs', 'F'), freqs, 'Hz'))));
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
|
|
|
|
linkaxes([ax1,ax2,ax3,ax4,ax5,ax6],'x');
|
|
#+end_src
|
|
|
|
#+begin_src matlab :tangle no :exports results :results file replace
|
|
exportFig('figs/piezo_amplified_iff_comp.pdf', 'width', 1500, 'height', 'full');
|
|
#+end_src
|
|
|
|
#+name: fig:piezo_amplified_iff_comp
|
|
#+caption: OL and CL transfer functions
|
|
#+RESULTS:
|
|
[[file:figs/piezo_amplified_iff_comp.png]]
|
|
|
|
|
|
#+name: fig:souleille18_results
|
|
#+caption: Results obtained in cite:souleille18_concep_activ_mount_space_applic
|
|
[[file:figs/souleille18_results.png]]
|
|
|
|
* Complete Strut with Encoder
|
|
<<sec:strut_fem>>
|
|
** Introduction
|
|
|
|
#+name: fig:strut_encoder_points3
|
|
#+caption: Interface points
|
|
[[file:data/strut_encoder/points3.jpg]]
|
|
|
|
Flexible joints have 0.25mm width.
|
|
|
|
** Matlab Init :noexport:ignore:
|
|
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
|
|
<<matlab-dir>>
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none :results silent :noweb yes
|
|
<<matlab-init>>
|
|
#+end_src
|
|
|
|
#+begin_src matlab
|
|
addpath('./data/strut_encoder/');
|
|
#+end_src
|
|
|
|
#+begin_src matlab
|
|
open('strut_encoder.slx');
|
|
#+end_src
|
|
|
|
** Import Mass Matrix, Stiffness Matrix, and Interface Nodes Coordinates
|
|
We first extract the stiffness and mass matrices.
|
|
#+begin_src matlab
|
|
K = readmatrix('mat_K.CSV');
|
|
M = readmatrix('mat_M.CSV');
|
|
#+end_src
|
|
|
|
Then, we extract the coordinates of the interface nodes.
|
|
#+begin_src matlab
|
|
[int_xyz, int_i, n_xyz, n_i, nodes] = extractNodes('out_nodes_3D.txt');
|
|
#+end_src
|
|
|
|
#+begin_src matlab
|
|
save('./mat/strut_encoder.mat', 'int_xyz', 'int_i', 'n_xyz', 'n_i', 'nodes', 'M', 'K');
|
|
#+end_src
|
|
|
|
** Output parameters
|
|
#+begin_src matlab
|
|
load('./mat/strut_encoder.mat', 'int_xyz', 'int_i', 'n_xyz', 'n_i', 'nodes', 'M', 'K');
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports results :results value table replace :tangle no
|
|
data2orgtable([length(n_i); length(int_i); size(M,1) - 6*length(int_i); size(M,1)], {'Total number of Nodes', 'Number of interface Nodes', 'Number of Modes', 'Size of M and K matrices'}, {}, ' %.0f ');
|
|
#+end_src
|
|
|
|
#+RESULTS:
|
|
| Total number of Nodes | 8 |
|
|
| Number of interface Nodes | 8 |
|
|
| Number of Modes | 6 |
|
|
| Size of M and K matrices | 54 |
|
|
|
|
#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
|
|
data2orgtable([[1:length(int_i)]', int_i, int_xyz], {}, {'Node i', 'Node Number', 'x [m]', 'y [m]', 'z [m]'}, ' %f ');
|
|
#+end_src
|
|
|
|
#+caption: Coordinates of the interface nodes
|
|
#+RESULTS:
|
|
| Node i | Node Number | x [m] | y [m] | z [m] |
|
|
|--------+-------------+---------+--------+----------|
|
|
| 1.0 | 504411.0 | 0.0 | 0.0 | 0.0405 |
|
|
| 2.0 | 504412.0 | 0.0 | 0.0 | -0.0405 |
|
|
| 3.0 | 504413.0 | -0.0325 | 0.0 | 0.0 |
|
|
| 4.0 | 504414.0 | -0.0125 | 0.0 | 0.0 |
|
|
| 5.0 | 504415.0 | -0.0075 | 0.0 | 0.0 |
|
|
| 6.0 | 504416.0 | 0.0325 | 0.0 | 0.0 |
|
|
| 7.0 | 504417.0 | 0.004 | 0.0145 | -0.00175 |
|
|
| 8.0 | 504418.0 | 0.004 | 0.0166 | -0.00175 |
|
|
|
|
#+begin_src matlab :exports results :results value table replace :tangle no
|
|
data2orgtable(K(1:10, 1:10), {}, {}, ' %.1g ');
|
|
#+end_src
|
|
|
|
#+caption: First 10x10 elements of the Stiffness matrix
|
|
#+RESULTS:
|
|
| 2000000.0 | 1000000.0 | -3000000.0 | -400.0 | 300.0 | 200.0 | -30.0 | 2000.0 | -10000.0 | 0.3 |
|
|
| 1000000.0 | 4000000.0 | -8000000.0 | -900.0 | 400.0 | -50.0 | -6000.0 | 10000.0 | -20000.0 | 3 |
|
|
| -3000000.0 | -8000000.0 | 20000000.0 | 2000.0 | -900.0 | 200.0 | -10000.0 | 20000.0 | -300000.0 | 7 |
|
|
| -400.0 | -900.0 | 2000.0 | 5 | -0.1 | 0.05 | 1 | -3 | 6 | -0.0007 |
|
|
| 300.0 | 400.0 | -900.0 | -0.1 | 5 | 0.04 | -0.1 | 0.5 | -3 | 0.0001 |
|
|
| 200.0 | -50.0 | 200.0 | 0.05 | 0.04 | 300.0 | 4 | -0.01 | -1 | 3e-05 |
|
|
| -30.0 | -6000.0 | -10000.0 | 1 | -0.1 | 4 | 3000000.0 | -1000000.0 | -2000000.0 | -300.0 |
|
|
| 2000.0 | 10000.0 | 20000.0 | -3 | 0.5 | -0.01 | -1000000.0 | 6000000.0 | 7000000.0 | 1000.0 |
|
|
| -10000.0 | -20000.0 | -300000.0 | 6 | -3 | -1 | -2000000.0 | 7000000.0 | 20000000.0 | 2000.0 |
|
|
| 0.3 | 3 | 7 | -0.0007 | 0.0001 | 3e-05 | -300.0 | 1000.0 | 2000.0 | 5 |
|
|
|
|
|
|
#+begin_src matlab :exports results :results value table replace :tangle no
|
|
data2orgtable(M(1:10, 1:10), {}, {}, ' %.1g ');
|
|
#+end_src
|
|
|
|
#+caption: First 10x10 elements of the Mass matrix
|
|
#+RESULTS:
|
|
| 0.04 | -0.005 | 0.007 | 2e-06 | 0.0001 | -5e-07 | -1e-05 | -9e-07 | 8e-05 | -5e-10 |
|
|
| -0.005 | 0.03 | 0.02 | -0.0001 | 1e-06 | -3e-07 | 3e-05 | -0.0001 | 8e-05 | -3e-08 |
|
|
| 0.007 | 0.02 | 0.08 | -6e-06 | -5e-06 | -7e-07 | 4e-05 | -0.0001 | 0.0005 | -3e-08 |
|
|
| 2e-06 | -0.0001 | -6e-06 | 2e-06 | -4e-10 | 2e-11 | -8e-09 | 3e-08 | -2e-08 | 6e-12 |
|
|
| 0.0001 | 1e-06 | -5e-06 | -4e-10 | 3e-06 | 2e-10 | -3e-09 | 3e-09 | -7e-09 | 6e-13 |
|
|
| -5e-07 | -3e-07 | -7e-07 | 2e-11 | 2e-10 | 5e-07 | -2e-08 | 5e-09 | -5e-09 | 1e-12 |
|
|
| -1e-05 | 3e-05 | 4e-05 | -8e-09 | -3e-09 | -2e-08 | 0.04 | 0.004 | 0.003 | 1e-06 |
|
|
| -9e-07 | -0.0001 | -0.0001 | 3e-08 | 3e-09 | 5e-09 | 0.004 | 0.02 | -0.02 | 0.0001 |
|
|
| 8e-05 | 8e-05 | 0.0005 | -2e-08 | -7e-09 | -5e-09 | 0.003 | -0.02 | 0.08 | -5e-06 |
|
|
| -5e-10 | -3e-08 | -3e-08 | 6e-12 | 6e-13 | 1e-12 | 1e-06 | 0.0001 | -5e-06 | 2e-06 |
|
|
|
|
Using =K=, =M= and =int_xyz=, we can use the =Reduced Order Flexible Solid= simscape block.
|
|
|
|
|
|
|
|
|
|
** Piezoelectric parameters
|
|
Parameters for the APA300ML:
|
|
|
|
#+begin_src matlab
|
|
d33 = 3e-10; % Strain constant [m/V]
|
|
n = 80; % Number of layers per stack
|
|
eT = 1.6e-8; % Permittivity under constant stress [F/m]
|
|
sD = 2e-11; % Elastic compliance under constant electric displacement [m2/N]
|
|
ka = 235e6; % Stack stiffness [N/m]
|
|
C = 5e-6; % Stack capactiance [F]
|
|
#+end_src
|
|
|
|
#+begin_src matlab
|
|
na = 2; % Number of stacks used as actuator
|
|
ns = 1; % Number of stacks used as force sensor
|
|
#+end_src
|
|
|
|
** Identification of the Dynamics
|
|
#+begin_src matlab :exports none
|
|
m = 0.01;
|
|
#+end_src
|
|
|
|
The dynamics is identified from the applied force to the measured relative displacement.
|
|
#+begin_src matlab :exports none
|
|
%% Name of the Simulink File
|
|
mdl = 'strut_encoder';
|
|
|
|
%% Input/Output definition
|
|
clear io; io_i = 1;
|
|
io(io_i) = linio([mdl, '/Fa'], 1, 'openinput'); io_i = io_i + 1;
|
|
io(io_i) = linio([mdl, '/L'], 1, 'openoutput'); io_i = io_i + 1;
|
|
|
|
Gh = -linearize(mdl, io);
|
|
#+end_src
|
|
|
|
The same dynamics is identified for a payload mass of 10Kg.
|
|
#+begin_src matlab
|
|
m = 10;
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
Ghm = -linearize(mdl, io);
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(0, 4, 5000);
|
|
|
|
figure;
|
|
tiledlayout(2, 1, 'TileSpacing', 'None', 'Padding', 'None');
|
|
|
|
% ax1 = subplot(2,1,1);
|
|
ax1 = nexttile;
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(Gh, freqs, 'Hz'))), '-');
|
|
plot(freqs, abs(squeeze(freqresp(Ghm, freqs, 'Hz'))), '-');
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude'); set(gca, 'XTickLabel',[]);
|
|
axis padded 'auto x'
|
|
hold off;
|
|
|
|
ax2 = nexttile;
|
|
% ax2 = subplot(2,1,2);
|
|
hold on;
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(Gh, freqs, 'Hz'))), '-', ...
|
|
'DisplayName', '$m = 0kg$');
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(Ghm, freqs, 'Hz'))), '-', ...
|
|
'DisplayName', '$m = 10kg$');
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
|
yticks(-360:90:360);
|
|
% ylim([-360 0]);
|
|
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
|
|
hold off;
|
|
axis padded 'auto x'
|
|
|
|
% linkaxes([ax1,ax2],'x');
|
|
% xlim([freqs(1), freqs(end)]);
|
|
% legend('location', 'southwest');
|
|
#+end_src
|
|
|
|
* To Order :noexport:
|
|
** TODO Identification of the stiffness properties of the APA300ML
|
|
*** APA Alone
|
|
#+begin_src matlab :exports none
|
|
m = 10;
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
%% Name of the Simulink File
|
|
mdl = 'APA300ML_characterisation';
|
|
|
|
%% Input/Output definition
|
|
clear io; io_i = 1;
|
|
io(io_i) = linio([mdl, '/Fd'], 1, 'openinput'); io_i = io_i + 1; % External Vertical Force [N]
|
|
io(io_i) = linio([mdl, '/D'], 1, 'openoutput'); io_i = io_i + 1; % Displacement/Rotation [m]
|
|
|
|
G = linearize(mdl, io);
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
|
|
data2orgtable([1e-6./dcgain(G(1,1)), 1e-6./dcgain(G(2,2)), 1e-6./dcgain(G(3,3)), 1./dcgain(G(4,4)), 1./dcgain(G(5,5)), 1./dcgain(G(6,6))]', {'Kx [N/um]', 'Ky [N/um]', 'Kz [N/um]', 'Rx [Nm/rad]', 'Ry [Nm/rad]', 'Rz [Nm/rad]'}, {'*Caracteristics*', '*Value*'}, ' %.1f ');
|
|
#+end_src
|
|
|
|
#+RESULTS:
|
|
| *Caracteristics* | *Value* |
|
|
|------------------+---------|
|
|
| Kx [N/um] | 0.8 |
|
|
| Ky [N/um] | 1.6 |
|
|
| Kz [N/um] | 1.8 |
|
|
| Rx [Nm/rad] | 71.4 |
|
|
| Ry [Nm/rad] | 148.2 |
|
|
| Rz [Nm/rad] | 4241.8 |
|
|
|
|
*** See how the global stiffness is changing with the flexible joints
|
|
#+begin_src matlab
|
|
flex = load('./mat/flexor_ID16.mat', 'int_xyz', 'int_i', 'n_xyz', 'n_i', 'nodes', 'M', 'K');
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
%% Name of the Simulink File
|
|
mdl = 'APA300ML_characterisation_with_joints';
|
|
|
|
%% Input/Output definition
|
|
clear io; io_i = 1;
|
|
io(io_i) = linio([mdl, '/Fd'], 1, 'openinput'); io_i = io_i + 1; % External Vertical Force [N]
|
|
io(io_i) = linio([mdl, '/D'], 1, 'openoutput'); io_i = io_i + 1; % Displacement/Rotation [m]
|
|
|
|
Gf = linearize(mdl, io);
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
|
|
data2orgtable([1e-6./dcgain(Gf(1,1)), 1e-6./dcgain(Gf(2,2)), 1e-6./dcgain(Gf(3,3)), 1./dcgain(Gf(4,4)), 1./dcgain(Gf(5,5)), 1./dcgain(Gf(6,6))]', {'Kx [N/um]', 'Ky [N/um]', 'Kz [N/um]', 'Rx [Nm/rad]', 'Ry [Nm/rad]', 'Rz [Nm/rad]'}, {'*Caracteristic*', '*Value*'}, ' %.1f ');
|
|
#+end_src
|
|
|
|
#+RESULTS:
|
|
| *Caracteristic* | *Value* |
|
|
|-----------------+---------|
|
|
| Kx [N/um] | 0.0 |
|
|
| Ky [N/um] | 0.0 |
|
|
| Kz [N/um] | 1.8 |
|
|
| Rx [Nm/rad] | 722.9 |
|
|
| Ry [Nm/rad] | 129.6 |
|
|
| Rz [Nm/rad] | 115.3 |
|
|
|
|
|
|
#+begin_src matlab
|
|
freqs = logspace(-2, 5, 1000);
|
|
|
|
figure;
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G(2,2), freqs, 'Hz'))), '-', 'DisplayName', 'APA');
|
|
plot(freqs, abs(squeeze(freqresp(Gf(2,2), freqs, 'Hz'))), '-', 'DisplayName', 'Flex');
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
xlabel('Frequency [Hz]'); ylabel('Amplitude [m/N]');
|
|
hold off;
|
|
legend('location', 'northeast');
|
|
#+end_src
|
|
|
|
#+begin_src matlab
|
|
freqs = logspace(-2, 5, 1000);
|
|
|
|
figure;
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G(3,3), freqs, 'Hz'))), '-', 'DisplayName', 'APA');
|
|
plot(freqs, abs(squeeze(freqresp(Gf(3,3), freqs, 'Hz'))), '-', 'DisplayName', 'Flex');
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
xlabel('Frequency [Hz]'); ylabel('Amplitude [m/N]');
|
|
hold off;
|
|
legend('location', 'northeast');
|
|
#+end_src
|
|
|
|
** TODO Effect of APA300ML in the flexibility of the leg
|
|
#+begin_src matlab :exports none
|
|
m = 10;
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
%% Name of the Simulink File
|
|
mdl = 'APA300ML_flex_joints';
|
|
|
|
%% Input/Output definition
|
|
clear io; io_i = 1;
|
|
io(io_i) = linio([mdl, '/Fd'], 1, 'openinput'); io_i = io_i + 1; % External Vertical Force [N]
|
|
io(io_i) = linio([mdl, '/D'], 1, 'openoutput'); io_i = io_i + 1; % Displacement/Rotation [m]
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
G = linearize(mdl, io);
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
Gf = linearize(mdl, io);
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
|
|
data2orgtable([[1e-6./dcgain(G(1,1)), 1e-6./dcgain(G(2,2)), 1e-6./dcgain(G(3,3)), 1./dcgain(G(4,4)), 1./dcgain(G(5,5)), 1./dcgain(G(6,6))]', [1e-6./dcgain(Gf(1,1)), 1e-6./dcgain(Gf(2,2)), 1e-6./dcgain(Gf(3,3)), 1./dcgain(Gf(4,4)), 1./dcgain(Gf(5,5)), 1./dcgain(Gf(6,6))]'], {'Kx [N/um]', 'Ky [N/um]', 'Kz [N/um]', 'Rx [Nm/rad]', 'Ry [Nm/rad]', 'Rz [Nm/rad]'}, {'*Caracteristic*', '*Rigid APA*', '*Flexible APA*'}, ' %.3f ');
|
|
#+end_src
|
|
|
|
|
|
#+RESULTS:
|
|
| *Caracteristic* | *Rigid APA* | *Flexible APA* |
|
|
|-----------------+-------------+----------------|
|
|
| Kx [N/um] | 0.018 | 0.019 |
|
|
| Ky [N/um] | 0.018 | 0.018 |
|
|
| Kz [N/um] | 60.0 | 2.647 |
|
|
| Rx [Nm/rad] | 16.705 | 557.682 |
|
|
| Ry [Nm/rad] | 16.535 | 185.939 |
|
|
| Rz [Nm/rad] | 118.0 | 114.803 |
|
|
|
|
* Bibliography :ignore:
|
|
bibliographystyle:unsrt
|
|
bibliography:ref.bib
|