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<h1 class="title">Matlab Functions used for the NASS Project</h1>
<div id="table-of-contents">
<h2>Table of Contents</h2>
<div id="text-table-of-contents">
<ul>
<li><a href="#orgea7e3d7">1. computePsdDispl</a></li>
<li><a href="#org5e769e7">2. computeSetpoint</a></li>
<li><a href="#org0239a56">3. converErrorBasis</a></li>
<li><a href="#orgdc168b9">4. computeReferencePose</a></li>
<li><a href="#org493ab7f">5. Compute the Sample Position Error w.r.t. the NASS</a></li>
</ul>
</div>
</div>
<div id="outline-container-orgea7e3d7" class="outline-2">
<h2 id="orgea7e3d7"><span class="section-number-2">1</span> computePsdDispl</h2>
<div class="outline-text-2" id="text-1">
<p>
<a id="org2cc2589"></a>
</p>
<p>
This Matlab function is accessible <a href="../src/computePsdDispl.m">here</a>.
</p>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-keyword">function</span> <span class="org-variable-name">[psd_object]</span> = <span class="org-function-name">computePsdDispl</span>(<span class="org-variable-name">sys_data</span>, <span class="org-variable-name">t_init</span>, <span class="org-variable-name">n_av</span>)
i_init = find(sys_data.time <span class="org-type">&gt;</span> t_init, 1);
han_win = hanning(ceil(length(sys_data.Dx(i_init<span class="org-type">:</span>end, <span class="org-type">:</span>))<span class="org-type">/</span>n_av));
Fs = 1<span class="org-type">/</span>sys_data.time(2);
[pdx, f] = pwelch(sys_data.Dx(i_init<span class="org-type">:</span>end, <span class="org-type">:</span>), han_win, [], [], Fs);
[pdy, <span class="org-type">~</span>] = pwelch(sys_data.Dy(i_init<span class="org-type">:</span>end, <span class="org-type">:</span>), han_win, [], [], Fs);
[pdz, <span class="org-type">~</span>] = pwelch(sys_data.Dz(i_init<span class="org-type">:</span>end, <span class="org-type">:</span>), han_win, [], [], Fs);
[prx, <span class="org-type">~</span>] = pwelch(sys_data.Rx(i_init<span class="org-type">:</span>end, <span class="org-type">:</span>), han_win, [], [], Fs);
[pry, <span class="org-type">~</span>] = pwelch(sys_data.Ry(i_init<span class="org-type">:</span>end, <span class="org-type">:</span>), han_win, [], [], Fs);
[prz, <span class="org-type">~</span>] = pwelch(sys_data.Rz(i_init<span class="org-type">:</span>end, <span class="org-type">:</span>), han_win, [], [], Fs);
psd_object = struct(...
<span class="org-string">'f'</span>, f, ...
<span class="org-string">'dx'</span>, pdx, ...
<span class="org-string">'dy'</span>, pdy, ...
<span class="org-string">'dz'</span>, pdz, ...
<span class="org-string">'rx'</span>, prx, ...
<span class="org-string">'ry'</span>, pry, ...
<span class="org-string">'rz'</span>, prz);
<span class="org-keyword">end</span>
</pre>
</div>
</div>
</div>
<div id="outline-container-org5e769e7" class="outline-2">
<h2 id="org5e769e7"><span class="section-number-2">2</span> computeSetpoint</h2>
<div class="outline-text-2" id="text-2">
<p>
<a id="orge6ebb0b"></a>
</p>
<p>
This Matlab function is accessible <a href="../src/computeSetpoint.m">here</a>.
</p>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-keyword">function</span> <span class="org-variable-name">setpoint</span> = <span class="org-function-name">computeSetpoint</span>(<span class="org-variable-name">ty</span>, <span class="org-variable-name">ry</span>, <span class="org-variable-name">rz</span>)
<span class="org-matlab-cellbreak"><span class="org-comment">%%</span></span>
setpoint = zeros(<span class="org-variable-name">6</span>, 1);
<span class="org-matlab-cellbreak"><span class="org-comment">%% Ty</span></span>
Ty = [1 0 0 0 ;
0 1 0 ty ;
0 0 1 0 ;
0 0 0 1 ];
<span class="org-comment">% Tyinv = [1 0 0 0 ;</span>
<span class="org-comment">% 0 1 0 -ty ;</span>
<span class="org-comment">% 0 0 1 0 ;</span>
<span class="org-comment">% 0 0 0 1 ];</span>
<span class="org-matlab-cellbreak"><span class="org-comment">%% Ry</span></span>
Ry = [ cos(ry) 0 sin(ry) 0 ;
0 1 0 0 ;
<span class="org-type">-</span>sin(ry) 0 cos(ry) 0 ;
0 0 0 1 ];
<span class="org-comment">% TMry = Ty*Ry*Tyinv;</span>
<span class="org-matlab-cellbreak"><span class="org-comment">%% Rz</span></span>
Rz = [cos(rz) <span class="org-type">-</span>sin(rz) 0 0 ;
sin(rz) cos(rz) 0 0 ;
0 0 1 0 ;
0 0 0 1 ];
<span class="org-comment">% TMrz = Ty*TMry*Rz*TMry'*Tyinv;</span>
<span class="org-matlab-cellbreak"><span class="org-comment">%% All stages</span></span>
<span class="org-comment">% </span><span class="org-comment"><span class="org-constant">TM </span></span><span class="org-comment">= TMrz*TMry*Ty;</span>
TM = Ty<span class="org-type">*</span>Ry<span class="org-type">*</span>Rz;
[thetax, thetay, thetaz] = RM2angle(TM(1<span class="org-type">:</span>3, 1<span class="org-type">:</span>3));
setpoint<span class="org-type">(1:3) </span>= TM(1<span class="org-type">:</span>3, 4);
setpoint<span class="org-type">(4:6) </span>= [thetax, thetay, thetaz];
<span class="org-matlab-cellbreak"><span class="org-comment">%% Custom Functions</span></span>
<span class="org-keyword">function</span> <span class="org-variable-name">[thetax, thetay, thetaz]</span> = <span class="org-function-name">RM2angle</span>(<span class="org-variable-name">R</span>)
<span class="org-keyword">if</span> abs(abs(R(3, 1)) <span class="org-type">-</span> 1) <span class="org-type">&gt;</span> 1e<span class="org-type">-</span>6 <span class="org-comment">% R31 != 1 and R31 != -1</span>
thetay = <span class="org-type">-</span>asin(R(3, 1));
thetax = atan2(R(3, 2)<span class="org-type">/</span>cos(thetay), R(3, 3)<span class="org-type">/</span>cos(thetay));
thetaz = atan2(R(2, 1)<span class="org-type">/</span>cos(thetay), R(1, 1)<span class="org-type">/</span>cos(thetay));
<span class="org-keyword">else</span>
thetaz = 0;
<span class="org-keyword">if</span> abs(R(3, 1)<span class="org-type">+</span>1) <span class="org-type">&lt;</span> 1e<span class="org-type">-</span>6 <span class="org-comment">% R31 = -1</span>
thetay = <span class="org-constant">pi</span><span class="org-type">/</span>2;
thetax = thetaz <span class="org-type">+</span> atan2(R(1, 2), R(1, 3));
<span class="org-keyword">else</span>
thetay = <span class="org-type">-</span><span class="org-constant">pi</span><span class="org-type">/</span>2;
thetax = <span class="org-type">-</span>thetaz <span class="org-type">+</span> atan2(<span class="org-type">-</span>R(1, 2), <span class="org-type">-</span>R(1, 3));
<span class="org-keyword">end</span>
<span class="org-keyword">end</span>
<span class="org-keyword">end</span>
<span class="org-keyword">end</span>
</pre>
</div>
</div>
</div>
<div id="outline-container-org0239a56" class="outline-2">
<h2 id="org0239a56"><span class="section-number-2">3</span> converErrorBasis</h2>
<div class="outline-text-2" id="text-3">
<p>
<a id="org8c2ffe5"></a>
</p>
<p>
This Matlab function is accessible <a href="../src/converErrorBasis.m">here</a>.
</p>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-keyword">function</span> <span class="org-variable-name">error_nass</span> = <span class="org-function-name">convertErrorBasis</span>(<span class="org-variable-name">pos</span>, <span class="org-variable-name">setpoint</span>, <span class="org-variable-name">ty</span>, <span class="org-variable-name">ry</span>, <span class="org-variable-name">rz</span>)
<span class="org-comment">% convertErrorBasis -</span>
<span class="org-comment">%</span>
<span class="org-comment">% Syntax: convertErrorBasis(p_error, ty, ry, rz)</span>
<span class="org-comment">%</span>
<span class="org-comment">% Inputs:</span>
<span class="org-comment">% - p_error - Position error of the sample w.r.t. the granite [m, rad]</span>
<span class="org-comment">% - ty - Measured translation of the Ty stage [m]</span>
<span class="org-comment">% - ry - Measured rotation of the Ry stage [rad]</span>
<span class="org-comment">% - rz - Measured rotation of the Rz stage [rad]</span>
<span class="org-comment">%</span>
<span class="org-comment">% Outputs:</span>
<span class="org-comment">% - P_nass - Position error of the sample w.r.t. the NASS base [m]</span>
<span class="org-comment">% - R_nass - Rotation error of the sample w.r.t. the NASS base [rad]</span>
<span class="org-comment">%</span>
<span class="org-comment">% Example:</span>
<span class="org-comment">%</span>
<span class="org-matlab-cellbreak"><span class="org-comment">%% If line vector =&gt; column vector</span></span>
<span class="org-keyword">if</span> size(pos, 2) <span class="org-type">==</span> 6
pos = pos<span class="org-type">'</span>;
<span class="org-keyword">end</span>
<span class="org-keyword">if</span> size(setpoint, 2) <span class="org-type">==</span> 6
setpoint = setpoint<span class="org-type">'</span>;
<span class="org-keyword">end</span>
<span class="org-matlab-cellbreak"><span class="org-comment">%% Position of the sample in the frame fixed to the Granite</span></span>
P_granite = [pos(1<span class="org-type">:</span>3); 1]; <span class="org-comment">% Position [m]</span>
R_granite = [setpoint(1<span class="org-type">:</span>3); 1]; <span class="org-comment">% Reference [m]</span>
<span class="org-matlab-cellbreak"><span class="org-comment">%% Transformation matrices of the stages</span></span>
<span class="org-comment">% T-y</span>
TMty = [1 0 0 0 ;
0 1 0 ty ;
0 0 1 0 ;
0 0 0 1 ];
<span class="org-comment">% R-y</span>
TMry = [ cos(ry) 0 sin(ry) 0 ;
0 1 0 0 ;
<span class="org-type">-</span>sin(ry) 0 cos(ry) 0 ;
0 0 0 1 ];
<span class="org-comment">% R-z</span>
TMrz = [cos(rz) <span class="org-type">-</span>sin(rz) 0 0 ;
sin(rz) cos(rz) 0 0 ;
0 0 1 0 ;
0 0 0 1 ];
<span class="org-matlab-cellbreak"><span class="org-comment">%% Compute Point coordinates in the new reference fixed to the NASS base</span></span>
<span class="org-comment">% P_nass = TMrz*TMry*TMty*P_granite;</span>
P_nass = TMrz<span class="org-type">\</span>TMry<span class="org-type">\</span>TMty<span class="org-type">\</span>P_granite;
R_nass = TMrz<span class="org-type">\</span>TMry<span class="org-type">\</span>TMty<span class="org-type">\</span>R_granite;
dx = R_nass(1)<span class="org-type">-</span>P_nass(1);
dy = R_nass(2)<span class="org-type">-</span>P_nass(2);
dz = R_nass(3)<span class="org-type">-</span>P_nass(3);
<span class="org-matlab-cellbreak"><span class="org-comment">%% Compute new basis vectors linked to the NASS base</span></span>
<span class="org-comment">% ux_nass = TMrz*TMry*TMty*[1; 0; 0; 0];</span>
<span class="org-comment">% ux_nass = ux_nass(1:3);</span>
<span class="org-comment">% uy_nass = TMrz*TMry*TMty*[0; 1; 0; 0];</span>
<span class="org-comment">% uy_nass = uy_nass(1:3);</span>
<span class="org-comment">% uz_nass = TMrz*TMry*TMty*[0; 0; 1; 0];</span>
<span class="org-comment">% uz_nass = uz_nass(1:3);</span>
ux_nass = TMrz<span class="org-type">\</span>TMry<span class="org-type">\</span>TMty<span class="org-type">\</span>[1; 0; 0; 0];
ux_nass = ux_nass(1<span class="org-type">:</span>3);
uy_nass = TMrz<span class="org-type">\</span>TMry<span class="org-type">\</span>TMty<span class="org-type">\</span>[0; 1; 0; 0];
uy_nass = uy_nass(1<span class="org-type">:</span>3);
uz_nass = TMrz<span class="org-type">\</span>TMry<span class="org-type">\</span>TMty<span class="org-type">\</span>[0; 0; 1; 0];
uz_nass = uz_nass(1<span class="org-type">:</span>3);
<span class="org-matlab-cellbreak"><span class="org-comment">%% Rotations error w.r.t. granite Frame</span></span>
<span class="org-comment">% Rotations error w.r.t. granite Frame</span>
rx_nass = pos(4);
ry_nass = pos(5);
rz_nass = pos(6);
<span class="org-comment">% Rotation matrices of the Sample w.r.t. the Granite</span>
Mrx_error = [1 0 0 ;
0 cos(<span class="org-type">-</span>rx_nass) <span class="org-type">-</span>sin(<span class="org-type">-</span>rx_nass) ;
0 sin(<span class="org-type">-</span>rx_nass) cos(<span class="org-type">-</span>rx_nass)];
Mry_error = [ cos(<span class="org-type">-</span>ry_nass) 0 sin(<span class="org-type">-</span>ry_nass) ;
0 1 0 ;
<span class="org-type">-</span>sin(<span class="org-type">-</span>ry_nass) 0 cos(<span class="org-type">-</span>ry_nass)];
Mrz_error = [cos(<span class="org-type">-</span>rz_nass) <span class="org-type">-</span>sin(<span class="org-type">-</span>rz_nass) 0 ;
sin(<span class="org-type">-</span>rz_nass) cos(<span class="org-type">-</span>rz_nass) 0 ;
0 0 1];
<span class="org-comment">% Rotation matrix of the Sample w.r.t. the Granite</span>
Mr_error = Mrz_error<span class="org-type">*</span>Mry_error<span class="org-type">*</span>Mrx_error;
<span class="org-matlab-cellbreak"><span class="org-comment">%% Use matrix to solve</span></span>
R = Mr_error<span class="org-type">/</span>[ux_nass, uy_nass, uz_nass]; <span class="org-comment">% Rotation matrix from NASS base to Sample</span>
[thetax, thetay, thetaz] = RM2angle(R);
error_nass = [dx; dy; dz; thetax; thetay; thetaz];
<span class="org-matlab-cellbreak"><span class="org-comment">%% Custom Functions</span></span>
<span class="org-keyword">function</span> <span class="org-variable-name">[thetax, thetay, thetaz]</span> = <span class="org-function-name">RM2angle</span>(<span class="org-variable-name">R</span>)
<span class="org-keyword">if</span> abs(abs(R(3, 1)) <span class="org-type">-</span> 1) <span class="org-type">&gt;</span> 1e<span class="org-type">-</span>6 <span class="org-comment">% R31 != 1 and R31 != -1</span>
thetay = <span class="org-type">-</span>asin(R(3, 1));
<span class="org-comment">% thetaybis = pi-thetay;</span>
thetax = atan2(R(3, 2)<span class="org-type">/</span>cos(thetay), R(3, 3)<span class="org-type">/</span>cos(thetay));
<span class="org-comment">% thetaxbis = atan2(R(3, 2)/cos(thetaybis), R(3, 3)/cos(thetaybis));</span>
thetaz = atan2(R(2, 1)<span class="org-type">/</span>cos(thetay), R(1, 1)<span class="org-type">/</span>cos(thetay));
<span class="org-comment">% thetazbis = atan2(R(2, 1)/cos(thetaybis), R(1, 1)/cos(thetaybis));</span>
<span class="org-keyword">else</span>
thetaz = 0;
<span class="org-keyword">if</span> abs(R(3, 1)<span class="org-type">+</span>1) <span class="org-type">&lt;</span> 1e<span class="org-type">-</span>6 <span class="org-comment">% R31 = -1</span>
thetay = <span class="org-constant">pi</span><span class="org-type">/</span>2;
thetax = thetaz <span class="org-type">+</span> atan2(R(1, 2), R(1, 3));
<span class="org-keyword">else</span>
thetay = <span class="org-type">-</span><span class="org-constant">pi</span><span class="org-type">/</span>2;
thetax = <span class="org-type">-</span>thetaz <span class="org-type">+</span> atan2(<span class="org-type">-</span>R(1, 2), <span class="org-type">-</span>R(1, 3));
<span class="org-keyword">end</span>
<span class="org-keyword">end</span>
<span class="org-keyword">end</span>
<span class="org-keyword">end</span>
</pre>
</div>
</div>
</div>
<div id="outline-container-orgdc168b9" class="outline-2">
<h2 id="orgdc168b9"><span class="section-number-2">4</span> computeReferencePose</h2>
<div class="outline-text-2" id="text-4">
<p>
<a id="org98cbe6e"></a>
</p>
<p>
This Matlab function is accessible <a href="src/computeReferencePose.m">here</a>.
</p>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-keyword">function</span> <span class="org-variable-name">[WTr]</span> = <span class="org-function-name">computeReferencePose</span>(<span class="org-variable-name">Dy</span>, <span class="org-variable-name">Ry</span>, <span class="org-variable-name">Rz</span>, <span class="org-variable-name">Dh</span>, <span class="org-variable-name">Dn</span>)
<span class="org-comment">% computeReferencePose - Compute the homogeneous transformation matrix corresponding to the wanted pose of the sample</span>
<span class="org-comment">%</span>
<span class="org-comment">% Syntax: [WTr] = computeReferencePose(Dy, Ry, Rz, Dh, Dn)</span>
<span class="org-comment">%</span>
<span class="org-comment">% Inputs:</span>
<span class="org-comment">% - Dy - Reference of the Translation Stage [m]</span>
<span class="org-comment">% - Ry - Reference of the Tilt Stage [rad]</span>
<span class="org-comment">% - Rz - Reference of the Spindle [rad]</span>
<span class="org-comment">% - Dh - Reference of the Micro Hexapod (Pitch, Roll, Yaw angles) [m, m, m, rad, rad, rad]</span>
<span class="org-comment">% - Dn - Reference of the Nano Hexapod [m, m, m, rad, rad, rad]</span>
<span class="org-comment">%</span>
<span class="org-comment">% Outputs:</span>
<span class="org-comment">% - WTr -</span>
<span class="org-matlab-cellbreak"><span class="org-comment">%% Translation Stage</span></span>
Rty = [1 0 0 0;
0 1 0 Dy;
0 0 1 0;
0 0 0 1];
<span class="org-matlab-cellbreak"><span class="org-comment">%% Tilt Stage - Pure rotating aligned with Ob</span></span>
Rry = [ cos(Ry) 0 sin(Ry) 0;
0 1 0 0;
<span class="org-type">-</span>sin(Ry) 0 cos(Ry) 0;
0 0 0 1];
<span class="org-matlab-cellbreak"><span class="org-comment">%% Spindle - Rotation along the Z axis</span></span>
Rrz = [cos(Rz) <span class="org-type">-</span>sin(Rz) 0 0 ;
sin(Rz) cos(Rz) 0 0 ;
0 0 1 0 ;
0 0 0 1 ];
<span class="org-matlab-cellbreak"><span class="org-comment">%% Micro-Hexapod</span></span>
Rhx = [1 0 0;
0 cos(Dh(4)) <span class="org-type">-</span>sin(Dh(4));
0 sin(Dh(4)) cos(Dh(4))];
Rhy = [ cos(Dh(5)) 0 sin(Dh(5));
0 1 0;
<span class="org-type">-</span>sin(Dh(5)) 0 cos(Dh(5))];
Rhz = [cos(Dh(6)) <span class="org-type">-</span>sin(Dh(6)) 0;
sin(Dh(6)) cos(Dh(6)) 0;
0 0 1];
Rh = [1 0 0 Dh(1) ;
0 1 0 Dh(2) ;
0 0 1 Dh(3) ;
0 0 0 1 ];
Rh(1<span class="org-type">:</span>3, 1<span class="org-type">:</span>3) = Rhz<span class="org-type">*</span>Rhy<span class="org-type">*</span>Rhx;
<span class="org-matlab-cellbreak"><span class="org-comment">%% Nano-Hexapod</span></span>
Rnx = [1 0 0;
0 cos(Dn(4)) <span class="org-type">-</span>sin(Dn(4));
0 sin(Dn(4)) cos(Dn(4))];
Rny = [ cos(Dn(5)) 0 sin(Dn(5));
0 1 0;
<span class="org-type">-</span>sin(Dn(5)) 0 cos(Dn(5))];
Rnz = [cos(Dn(6)) <span class="org-type">-</span>sin(Dn(6)) 0;
sin(Dn(6)) cos(Dn(6)) 0;
0 0 1];
Rn = [1 0 0 Dn(1) ;
0 1 0 Dn(2) ;
0 0 1 Dn(3) ;
0 0 0 1 ];
Rn(1<span class="org-type">:</span>3, 1<span class="org-type">:</span>3) = Rnz<span class="org-type">*</span>Rny<span class="org-type">*</span>Rnx;
<span class="org-matlab-cellbreak"><span class="org-comment">%% Total Homogeneous transformation</span></span>
WTr = Rty<span class="org-type">*</span>Rry<span class="org-type">*</span>Rrz<span class="org-type">*</span>Rh<span class="org-type">*</span>Rn;
<span class="org-keyword">end</span>
</pre>
</div>
</div>
</div>
<div id="outline-container-org493ab7f" class="outline-2">
<h2 id="org493ab7f"><span class="section-number-2">5</span> Compute the Sample Position Error w.r.t. the NASS</h2>
<div class="outline-text-2" id="text-5">
<p>
<a id="org6dcd4fb"></a>
</p>
<p>
This Matlab function is accessible <a href="src/computeSampleError.m">here</a>.
</p>
<div class="org-src-container">
<pre class="src src-matlab"><span class="org-keyword">function</span> <span class="org-variable-name">[MTr]</span> = <span class="org-function-name">computeSampleError</span>(<span class="org-variable-name">WTm</span>, <span class="org-variable-name">WTr</span>)
<span class="org-comment">% computeSampleError -</span>
<span class="org-comment">%</span>
<span class="org-comment">% Syntax: [MTr] = computeSampleError(WTm, WTr)</span>
<span class="org-comment">%</span>
<span class="org-comment">% Inputs:</span>
<span class="org-comment">% - WTm - Homoegeneous transformation that represent the</span>
<span class="org-comment">% wanted pose of the sample with respect to the granite</span>
<span class="org-comment">% - WTr - Homoegeneous transformation that represent the</span>
<span class="org-comment">% measured pose of the sample with respect to the granite</span>
<span class="org-comment">%</span>
<span class="org-comment">% Outputs:</span>
<span class="org-comment">% - MTr - Homoegeneous transformation that represent the</span>
<span class="org-comment">% wanted pose of the sample expressed in a frame</span>
<span class="org-comment">% attached to the top platform of the nano-hexapod</span>
MTr = zeros(4,4);
MTr = [WTm(1<span class="org-type">:</span>3,1<span class="org-type">:</span>3)<span class="org-type">'</span>, <span class="org-type">-</span>WTm(1<span class="org-type">:</span>3,1<span class="org-type">:</span>3)<span class="org-type">'*</span>WTm(1<span class="org-type">:</span>3,4) ; 0 0 0 1]<span class="org-type">*</span>WTr;
<span class="org-keyword">end</span>
</pre>
</div>
</div>
</div>
</div>
<div id="postamble" class="status">
<p class="author">Author: Dehaeze Thomas</p>
<p class="date">Created: 2020-02-25 mar. 18:20</p>
</div>
</body>
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