509 lines
18 KiB
Org Mode
509 lines
18 KiB
Org Mode
#+TITLE: Matlab Functions used for the NASS Project
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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 no
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#+PROPERTY: header-args:matlab+ :mkdirp yes
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#+PROPERTY: header-args:shell :eval no-export
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#+PROPERTY: header-args:latex :headers '("\\usepackage{tikz}" "\\usepackage{import}" "\\import{$HOME/Cloud/thesis/latex/}{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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* Functions
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<<sec:functions>>
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** computePsdDispl
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:PROPERTIES:
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:header-args:matlab+: :tangle ../src/computePsdDispl.m
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:header-args:matlab+: :comments none :mkdirp yes :eval no
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:END:
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<<sec:computePsdDispl>>
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This Matlab function is accessible [[file:../src/computePsdDispl.m][here]].
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#+begin_src matlab
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function [psd_object] = computePsdDispl(sys_data, t_init, n_av)
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i_init = find(sys_data.time > t_init, 1);
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han_win = hanning(ceil(length(sys_data.Dx(i_init:end, :))/n_av));
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Fs = 1/sys_data.time(2);
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[pdx, f] = pwelch(sys_data.Dx(i_init:end, :), han_win, [], [], Fs);
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[pdy, ~] = pwelch(sys_data.Dy(i_init:end, :), han_win, [], [], Fs);
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[pdz, ~] = pwelch(sys_data.Dz(i_init:end, :), han_win, [], [], Fs);
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[prx, ~] = pwelch(sys_data.Rx(i_init:end, :), han_win, [], [], Fs);
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[pry, ~] = pwelch(sys_data.Ry(i_init:end, :), han_win, [], [], Fs);
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[prz, ~] = pwelch(sys_data.Rz(i_init:end, :), han_win, [], [], Fs);
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psd_object = struct(...
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'f', f, ...
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'dx', pdx, ...
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'dy', pdy, ...
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'dz', pdz, ...
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'rx', prx, ...
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'ry', pry, ...
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'rz', prz);
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end
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#+end_src
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** computeSetpoint
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:PROPERTIES:
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:header-args:matlab+: :tangle ../src/computeSetpoint.m
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:header-args:matlab+: :comments none :mkdirp yes :eval no
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:END:
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<<sec:computeSetpoint>>
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This Matlab function is accessible [[file:../src/computeSetpoint.m][here]].
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#+begin_src matlab
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function setpoint = computeSetpoint(ty, ry, rz)
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%%
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setpoint = zeros(6, 1);
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%% Ty
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Ty = [1 0 0 0 ;
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0 1 0 ty ;
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0 0 1 0 ;
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0 0 0 1 ];
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% Tyinv = [1 0 0 0 ;
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% 0 1 0 -ty ;
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% 0 0 1 0 ;
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% 0 0 0 1 ];
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%% Ry
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Ry = [ cos(ry) 0 sin(ry) 0 ;
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0 1 0 0 ;
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-sin(ry) 0 cos(ry) 0 ;
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0 0 0 1 ];
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% TMry = Ty*Ry*Tyinv;
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%% Rz
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Rz = [cos(rz) -sin(rz) 0 0 ;
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sin(rz) cos(rz) 0 0 ;
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0 0 1 0 ;
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0 0 0 1 ];
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% TMrz = Ty*TMry*Rz*TMry'*Tyinv;
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%% All stages
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% TM = TMrz*TMry*Ty;
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TM = Ty*Ry*Rz;
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[thetax, thetay, thetaz] = RM2angle(TM(1:3, 1:3));
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setpoint(1:3) = TM(1:3, 4);
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setpoint(4:6) = [thetax, thetay, thetaz];
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%% Custom Functions
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function [thetax, thetay, thetaz] = RM2angle(R)
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if abs(abs(R(3, 1)) - 1) > 1e-6 % R31 != 1 and R31 != -1
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thetay = -asin(R(3, 1));
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thetax = atan2(R(3, 2)/cos(thetay), R(3, 3)/cos(thetay));
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thetaz = atan2(R(2, 1)/cos(thetay), R(1, 1)/cos(thetay));
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else
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thetaz = 0;
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if abs(R(3, 1)+1) < 1e-6 % R31 = -1
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thetay = pi/2;
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thetax = thetaz + atan2(R(1, 2), R(1, 3));
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else
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thetay = -pi/2;
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thetax = -thetaz + atan2(-R(1, 2), -R(1, 3));
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end
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end
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end
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end
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#+end_src
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** converErrorBasis
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:PROPERTIES:
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:header-args:matlab+: :tangle ../src/converErrorBasis.m
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:header-args:matlab+: :comments none :mkdirp yes :eval no
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:END:
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<<sec:converErrorBasis>>
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This Matlab function is accessible [[file:../src/converErrorBasis.m][here]].
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#+begin_src matlab
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function error_nass = convertErrorBasis(pos, setpoint, ty, ry, rz)
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% convertErrorBasis -
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%
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% Syntax: convertErrorBasis(p_error, ty, ry, rz)
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%
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% Inputs:
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% - p_error - Position error of the sample w.r.t. the granite [m, rad]
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% - ty - Measured translation of the Ty stage [m]
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% - ry - Measured rotation of the Ry stage [rad]
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% - rz - Measured rotation of the Rz stage [rad]
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%
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% Outputs:
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% - P_nass - Position error of the sample w.r.t. the NASS base [m]
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% - R_nass - Rotation error of the sample w.r.t. the NASS base [rad]
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%
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% Example:
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%
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%% If line vector => column vector
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if size(pos, 2) == 6
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pos = pos';
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end
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if size(setpoint, 2) == 6
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setpoint = setpoint';
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end
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%% Position of the sample in the frame fixed to the Granite
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P_granite = [pos(1:3); 1]; % Position [m]
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R_granite = [setpoint(1:3); 1]; % Reference [m]
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%% Transformation matrices of the stages
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% T-y
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TMty = [1 0 0 0 ;
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0 1 0 ty ;
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0 0 1 0 ;
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0 0 0 1 ];
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% R-y
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TMry = [ cos(ry) 0 sin(ry) 0 ;
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0 1 0 0 ;
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-sin(ry) 0 cos(ry) 0 ;
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0 0 0 1 ];
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% R-z
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TMrz = [cos(rz) -sin(rz) 0 0 ;
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sin(rz) cos(rz) 0 0 ;
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0 0 1 0 ;
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0 0 0 1 ];
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%% Compute Point coordinates in the new reference fixed to the NASS base
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% P_nass = TMrz*TMry*TMty*P_granite;
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P_nass = TMrz\TMry\TMty\P_granite;
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R_nass = TMrz\TMry\TMty\R_granite;
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dx = R_nass(1)-P_nass(1);
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dy = R_nass(2)-P_nass(2);
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dz = R_nass(3)-P_nass(3);
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%% Compute new basis vectors linked to the NASS base
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% ux_nass = TMrz*TMry*TMty*[1; 0; 0; 0];
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% ux_nass = ux_nass(1:3);
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% uy_nass = TMrz*TMry*TMty*[0; 1; 0; 0];
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% uy_nass = uy_nass(1:3);
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% uz_nass = TMrz*TMry*TMty*[0; 0; 1; 0];
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% uz_nass = uz_nass(1:3);
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ux_nass = TMrz\TMry\TMty\[1; 0; 0; 0];
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ux_nass = ux_nass(1:3);
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uy_nass = TMrz\TMry\TMty\[0; 1; 0; 0];
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uy_nass = uy_nass(1:3);
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uz_nass = TMrz\TMry\TMty\[0; 0; 1; 0];
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uz_nass = uz_nass(1:3);
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%% Rotations error w.r.t. granite Frame
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% Rotations error w.r.t. granite Frame
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rx_nass = pos(4);
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ry_nass = pos(5);
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rz_nass = pos(6);
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% Rotation matrices of the Sample w.r.t. the Granite
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Mrx_error = [1 0 0 ;
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0 cos(-rx_nass) -sin(-rx_nass) ;
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0 sin(-rx_nass) cos(-rx_nass)];
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Mry_error = [ cos(-ry_nass) 0 sin(-ry_nass) ;
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0 1 0 ;
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-sin(-ry_nass) 0 cos(-ry_nass)];
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Mrz_error = [cos(-rz_nass) -sin(-rz_nass) 0 ;
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sin(-rz_nass) cos(-rz_nass) 0 ;
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0 0 1];
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% Rotation matrix of the Sample w.r.t. the Granite
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Mr_error = Mrz_error*Mry_error*Mrx_error;
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%% Use matrix to solve
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R = Mr_error/[ux_nass, uy_nass, uz_nass]; % Rotation matrix from NASS base to Sample
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[thetax, thetay, thetaz] = RM2angle(R);
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error_nass = [dx; dy; dz; thetax; thetay; thetaz];
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%% Custom Functions
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function [thetax, thetay, thetaz] = RM2angle(R)
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if abs(abs(R(3, 1)) - 1) > 1e-6 % R31 != 1 and R31 != -1
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thetay = -asin(R(3, 1));
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% thetaybis = pi-thetay;
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thetax = atan2(R(3, 2)/cos(thetay), R(3, 3)/cos(thetay));
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% thetaxbis = atan2(R(3, 2)/cos(thetaybis), R(3, 3)/cos(thetaybis));
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thetaz = atan2(R(2, 1)/cos(thetay), R(1, 1)/cos(thetay));
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% thetazbis = atan2(R(2, 1)/cos(thetaybis), R(1, 1)/cos(thetaybis));
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else
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thetaz = 0;
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if abs(R(3, 1)+1) < 1e-6 % R31 = -1
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thetay = pi/2;
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thetax = thetaz + atan2(R(1, 2), R(1, 3));
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else
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thetay = -pi/2;
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thetax = -thetaz + atan2(-R(1, 2), -R(1, 3));
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end
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end
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end
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end
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#+end_src
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** generateDiagPidControl
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:PROPERTIES:
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:header-args:matlab+: :tangle ../src/generateDiagPidControl.m
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:header-args:matlab+: :comments none :mkdirp yes :eval no
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:END:
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<<sec:generateDiagPidControl>>
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This Matlab function is accessible [[file:../src/generateDiagPidControl.m][here]].
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#+begin_src matlab
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function [K] = generateDiagPidControl(G, fs)
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%%
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pid_opts = pidtuneOptions(...
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'PhaseMargin', 50, ...
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'DesignFocus', 'disturbance-rejection');
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%%
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K = tf(zeros(6));
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for i = 1:6
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input_name = G.InputName(i);
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output_name = G.OutputName(i);
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K(i, i) = tf(pidtune(minreal(G(output_name, input_name)), 'PIDF', 2*pi*fs, pid_opts));
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end
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K.InputName = G.OutputName;
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K.OutputName = G.InputName;
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end
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#+end_src
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** identifyPlant
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:PROPERTIES:
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:header-args:matlab+: :tangle ../src/identifyPlant.m
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:header-args:matlab+: :comments none :mkdirp yes :eval no
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:END:
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<<sec:identifyPlant>>
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This Matlab function is accessible [[file:../src/identifyPlant.m][here]].
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#+begin_src matlab
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function [sys] = identifyPlant(opts_param)
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%% Default values for opts
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opts = struct();
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%% Populate opts with input parameters
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if exist('opts_param','var')
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for opt = fieldnames(opts_param)'
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opts.(opt{1}) = opts_param.(opt{1});
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end
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end
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%% Options for Linearized
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options = linearizeOptions;
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options.SampleTime = 0;
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%% Name of the Simulink File
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mdl = 'sim_nano_station_id';
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%% Input/Output definition
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io(1) = linio([mdl, '/Fn'], 1, 'input'); % Cartesian forces applied by NASS
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io(2) = linio([mdl, '/Dw'], 1, 'input'); % Ground Motion
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io(3) = linio([mdl, '/Fs'], 1, 'input'); % External forces on the sample
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io(4) = linio([mdl, '/Fnl'], 1, 'input'); % Forces applied on the NASS's legs
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io(5) = linio([mdl, '/Fd'], 1, 'input'); % Disturbance Forces
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io(6) = linio([mdl, '/Dsm'], 1, 'output'); % Displacement of the sample
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io(7) = linio([mdl, '/Fnlm'], 1, 'output'); % Force sensor in NASS's legs
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io(8) = linio([mdl, '/Dnlm'], 1, 'output'); % Displacement of NASS's legs
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io(9) = linio([mdl, '/Es'], 1, 'output'); % Position Error w.r.t. NASS base
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io(10) = linio([mdl, '/Vlm'], 1, 'output'); % Measured absolute velocity of the legs
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%% Run the linearization
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G = linearize(mdl, io, options);
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G.InputName = {'Fnx', 'Fny', 'Fnz', 'Mnx', 'Mny', 'Mnz', ...
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'Dgx', 'Dgy', 'Dgz', ...
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'Fsx', 'Fsy', 'Fsz', 'Msx', 'Msy', 'Msz', ...
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'F1', 'F2', 'F3', 'F4', 'F5', 'F6', ...
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'Frzz', 'Ftyx', 'Ftyz'};
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G.OutputName = {'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz', ...
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'Fm1', 'Fm2', 'Fm3', 'Fm4', 'Fm5', 'Fm6', ...
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'Dm1', 'Dm2', 'Dm3', 'Dm4', 'Dm5', 'Dm6', ...
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'Edx', 'Edy', 'Edz', 'Erx', 'Ery', 'Erz', ...
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'Vm1', 'Vm2', 'Vm3', 'Vm4', 'Vm5', 'Vm6'};
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%% Create the sub transfer functions
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minreal_tol = sqrt(eps);
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% From forces applied in the cartesian frame to displacement of the sample in the cartesian frame
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sys.G_cart = minreal(G({'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'}, {'Fnx', 'Fny', 'Fnz', 'Mnx', 'Mny', 'Mnz'}), minreal_tol, false);
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% From ground motion to Sample displacement
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sys.G_gm = minreal(G({'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'}, {'Dgx', 'Dgy', 'Dgz'}), minreal_tol, false);
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% From direct forces applied on the sample to displacement of the sample
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sys.G_fs = minreal(G({'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'}, {'Fsx', 'Fsy', 'Fsz', 'Msx', 'Msy', 'Msz'}), minreal_tol, false);
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% From forces applied on NASS's legs to force sensor in each leg
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sys.G_iff = minreal(G({'Fm1', 'Fm2', 'Fm3', 'Fm4', 'Fm5', 'Fm6'}, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'}), minreal_tol, false);
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% From forces applied on NASS's legs to displacement of each leg
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sys.G_dleg = minreal(G({'Dm1', 'Dm2', 'Dm3', 'Dm4', 'Dm5', 'Dm6'}, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'}), minreal_tol, false);
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% From forces/torques applied by the NASS to position error
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sys.G_plant = minreal(G({'Edx', 'Edy', 'Edz', 'Erx', 'Ery', 'Erz'}, {'Fnx', 'Fny', 'Fnz', 'Mnx', 'Mny', 'Mnz'}), minreal_tol, false);
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% From forces/torques applied by the NASS to velocity of the actuator
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sys.G_geoph = minreal(G({'Vm1', 'Vm2', 'Vm3', 'Vm4', 'Vm5', 'Vm6'}, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'}), minreal_tol, false);
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% From various disturbance forces to position error
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sys.G_dist = minreal(G({'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'}, {'Frzz', 'Ftyx', 'Ftyz'}), minreal_tol, false);
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%% We remove low frequency and high frequency dynamics that are usually unstable
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% using =freqsep= is risky as it may change the shape of the transfer functions
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% f_min = 0.1; % [Hz]
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% f_max = 1e4; % [Hz]
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% [~, sys.G_cart] = freqsep(freqsep(sys.G_cart, 2*pi*f_max), 2*pi*f_min);
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% [~, sys.G_gm] = freqsep(freqsep(sys.G_gm, 2*pi*f_max), 2*pi*f_min);
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% [~, sys.G_fs] = freqsep(freqsep(sys.G_fs, 2*pi*f_max), 2*pi*f_min);
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% [~, sys.G_iff] = freqsep(freqsep(sys.G_iff, 2*pi*f_max), 2*pi*f_min);
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% [~, sys.G_dleg] = freqsep(freqsep(sys.G_dleg, 2*pi*f_max), 2*pi*f_min);
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% [~, sys.G_plant] = freqsep(freqsep(sys.G_plant, 2*pi*f_max), 2*pi*f_min);
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|
|
|
%% We finally verify that the system is stable
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|
if ~isstable(sys.G_cart) || ~isstable(sys.G_gm) || ~isstable(sys.G_fs) || ~isstable(sys.G_iff) || ~isstable(sys.G_dleg) || ~isstable(sys.G_plant)
|
|
warning('One of the identified system is unstable');
|
|
end
|
|
end
|
|
#+end_src
|
|
|
|
** runSimulation
|
|
:PROPERTIES:
|
|
:header-args:matlab+: :tangle ../src/runSimulation.m
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|
:header-args:matlab+: :comments none :mkdirp yes :eval no
|
|
:END:
|
|
<<sec:runSimulation>>
|
|
|
|
This Matlab function is accessible [[file:../src/runSimulation.m][here]].
|
|
|
|
#+begin_src matlab
|
|
function [] = runSimulation(sys_name, sys_mass, ctrl_type, act_damp)
|
|
%% Load the controller and save it for the simulation
|
|
if strcmp(ctrl_type, 'cl') && strcmp(act_damp, 'none')
|
|
K_obj = load('./mat/K_fb.mat');
|
|
K = K_obj.(sprintf('K_%s_%s', sys_mass, sys_name)); %#ok
|
|
save('./mat/controllers.mat', 'K');
|
|
elseif strcmp(ctrl_type, 'cl') && strcmp(act_damp, 'iff')
|
|
K_obj = load('./mat/K_fb_iff.mat');
|
|
K = K_obj.(sprintf('K_%s_%s_iff', sys_mass, sys_name)); %#ok
|
|
save('./mat/controllers.mat', 'K');
|
|
elseif strcmp(ctrl_type, 'ol')
|
|
K = tf(zeros(6)); %#ok
|
|
save('./mat/controllers.mat', 'K');
|
|
else
|
|
error('ctrl_type should be cl or ol');
|
|
end
|
|
|
|
%% Active Damping
|
|
if strcmp(act_damp, 'iff')
|
|
K_iff_crit = load('./mat/K_iff_crit.mat');
|
|
K_iff = K_iff_crit.(sprintf('K_iff_%s_%s', sys_mass, sys_name)); %#ok
|
|
save('./mat/controllers.mat', 'K_iff', '-append');
|
|
elseif strcmp(act_damp, 'none')
|
|
K_iff = tf(zeros(6)); %#ok
|
|
save('./mat/controllers.mat', 'K_iff', '-append');
|
|
end
|
|
|
|
%%
|
|
if strcmp(sys_name, 'pz')
|
|
initializeNanoHexapod(struct('actuator', 'piezo'));
|
|
elseif strcmp(sys_name, 'vc')
|
|
initializeNanoHexapod(struct('actuator', 'lorentz'));
|
|
else
|
|
error('sys_name should be pz or vc');
|
|
end
|
|
|
|
if strcmp(sys_mass, 'light')
|
|
initializeSample(struct('mass', 1));
|
|
elseif strcmp(sys_mass, 'heavy')
|
|
initializeSample(struct('mass', 50));
|
|
else
|
|
error('sys_mass should be light or heavy');
|
|
end
|
|
|
|
%% Run the simulation
|
|
sim('sim_nano_station_ctrl.slx');
|
|
|
|
%% Split the Dsample matrix into vectors
|
|
[Dx, Dy, Dz, Rx, Ry, Rz] = matSplit(Es.Data, 1); %#ok
|
|
time = Dsample.Time; %#ok
|
|
|
|
%% Save the result
|
|
filename = sprintf('sim_%s_%s_%s_%s', sys_mass, sys_name, ctrl_type, act_damp);
|
|
save(sprintf('./mat/%s.mat', filename), ...
|
|
'time', 'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz', 'K');
|
|
end
|
|
#+end_src
|
|
|
|
** Inverse Kinematics of the Hexapod
|
|
:PROPERTIES:
|
|
:header-args:matlab+: :tangle ../src/inverseKinematicsHexapod.m
|
|
:header-args:matlab+: :comments none :mkdirp yes :eval no
|
|
:END:
|
|
<<sec:inverseKinematicsHexapod>>
|
|
|
|
This Matlab function is accessible [[file:src/inverseKinematicsHexapod.m][here]].
|
|
|
|
#+begin_src matlab
|
|
function [L] = inverseKinematicsHexapod(hexapod, AP, ARB)
|
|
% inverseKinematicsHexapod - Compute the initial position of each leg to have the wanted Hexapod's position
|
|
%
|
|
% Syntax: inverseKinematicsHexapod(hexapod, AP, ARB)
|
|
%
|
|
% Inputs:
|
|
% - hexapod - Hexapod object containing the geometry of the hexapod
|
|
% - AP - Position vector of point OB expressed in frame {A} in [m]
|
|
% - ARB - Rotation Matrix expressed in frame {A}
|
|
|
|
% Wanted Length of the hexapod's legs [m]
|
|
L = zeros(6, 1);
|
|
|
|
for i = 1:length(L)
|
|
Bbi = hexapod.pos_top_tranform(i, :)' - 1e-3*[0 ; 0 ; hexapod.TP.thickness+hexapod.Leg.sphere.top+hexapod.SP.thickness.top+hexapod.jacobian]; % [m]
|
|
Aai = hexapod.pos_base(i, :)' + 1e-3*[0 ; 0 ; hexapod.BP.thickness+hexapod.Leg.sphere.bottom+hexapod.SP.thickness.bottom-hexapod.h-hexapod.jacobian]; % [m]
|
|
|
|
L(i) = sqrt(AP'*AP + Bbi'*Bbi + Aai'*Aai - 2*AP'*Aai + 2*AP'*(ARB*Bbi) - 2*(ARB*Bbi)'*Aai);
|
|
end
|
|
end
|
|
#+end_src
|