From c6a4e5343e6b8813aa175c73ea4f7678a9bab32c Mon Sep 17 00:00:00 2001 From: Thomas Dehaeze Date: Tue, 17 Dec 2019 18:03:21 +0100 Subject: [PATCH] Important change for the implementation of motion input Now we provide the first and second derivatives. This permits to not have any motion error. Also, many experiments (tomography, ty-scans) are simulated --- experiment_tomography/index.html | 527 +++++++++++++++--- experiment_tomography/index.org | 521 +++++++++-------- experiment_tomography/mat/experiment.mat | Bin 4813358 -> 4346458 bytes .../matlab/sim_nano_station_tomo.slx | Bin 32011 -> 32851 bytes figs/exp_tomo_dist_rot.png | Bin 0 -> 46639 bytes figs/exp_tomo_dist_trans.png | Bin 81372 -> 54926 bytes figs/exp_tomo_offset_rot.png | Bin 0 -> 46249 bytes figs/exp_tomo_offset_trans.png | Bin 0 -> 57614 bytes figs/exp_tomo_without_dist_rot.png | Bin 0 -> 43866 bytes figs/exp_tomo_without_dist_trans.png | Bin 56938 -> 41473 bytes figs/exp_ty_scan_rot.png | Bin 0 -> 41465 bytes figs/exp_ty_scan_trans.png | Bin 0 -> 46191 bytes mat/conf_simscape.mat | Bin 13406 -> 13441 bytes mat/nass_disturbances.mat | Bin 777342 -> 50787 bytes mat/nass_references.mat | Bin 930532 -> 2482560 bytes simscape_subsystems/index.org | 112 ++-- simscape_subsystems/nass_references.slx | Bin 31209 -> 31455 bytes simscape_subsystems/spindle_D.slx | Bin 31023 -> 31187 bytes simscape_subsystems/tilt_stage_D.slx | Bin 40872 -> 41037 bytes simscape_subsystems/translation_stage_D.slx | Bin 38189 -> 38345 bytes src/initializeReferences.m | 112 ++-- 21 files changed, 876 insertions(+), 396 deletions(-) create mode 100644 figs/exp_tomo_dist_rot.png create mode 100644 figs/exp_tomo_offset_rot.png create mode 100644 figs/exp_tomo_offset_trans.png create mode 100644 figs/exp_tomo_without_dist_rot.png create mode 100644 figs/exp_ty_scan_rot.png create mode 100644 figs/exp_ty_scan_trans.png diff --git a/experiment_tomography/index.html b/experiment_tomography/index.html index 029e490..38021e4 100644 --- a/experiment_tomography/index.html +++ b/experiment_tomography/index.html @@ -3,7 +3,7 @@ "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> - + Tomography Experiment @@ -258,16 +258,70 @@ for the JavaScript code in this tag.

Table of Contents

-
-

1 Simscape Model

+

+The goal here is to simulate some scientific experiments with the tuned Simscape model when no control is applied to the nano-hexapod. +

+ +

+This has several goals: +

+ + +

+The document in organized as follow: +

+ + +
+

1 Simscape Model

+

+ +

+

The simulink file to do tomography experiments is sim_nano_station_tomo.slx.

@@ -277,11 +331,11 @@ The simulink file to do tomography experiments is sim_nano_station_tomo.sl

-We load the shared simulink configuration and we set a small StopTime. +We load the shared simulink configuration and we set the StopTime.

load('mat/conf_simscape.mat');
-set_param(conf_simscape, 'StopTime', '10');
+set_param(conf_simscape, 'StopTime', '5');
 
@@ -313,9 +367,231 @@ All stage is set to its zero position except the Spindle which is rotating at 60
-
-

2 Tomography Experiment with no disturbances

+
+

2 Tomography Experiment with no disturbances

+

+ +

+
+
+

2.1 Simulation Setup

+
+

+And we initialize the disturbances to be equal to zero. +

+
+
opts = struct(...
+    'Dwx', false, ... % Ground Motion - X direction
+    'Dwy', false, ... % Ground Motion - Y direction
+    'Dwz', false, ... % Ground Motion - Z direction
+    'Fty_x', false, ... % Translation Stage - X direction
+    'Fty_z', false, ... % Translation Stage - Z direction
+    'Frz_z', false  ... % Spindle - Z direction
+);
+initDisturbances(opts);
+
+
+ +

+We simulate the model. +

+
+
sim('sim_nano_station_tomo');
+
+
+ +

+And we save the obtained data. +

+
+
tomo_align_no_dist = struct('t', t, 'MTr', MTr);
+save('experiment_tomography/mat/experiment.mat', 'tomo_align_no_dist', '-append');
+
+
+
+
+ +
+

2.2 Analysis

+
+
+
load('experiment_tomography/mat/experiment.mat', 'tomo_align_no_dist');
+t = tomo_align_no_dist.t;
+MTr = tomo_align_no_dist.MTr;
+
+
+ +
+
Edx = squeeze(MTr(1, 4, :));
+Edy = squeeze(MTr(2, 4, :));
+Edz = squeeze(MTr(3, 4, :));
+% The angles obtained are u-v-w Euler angles (rotations in the moving frame)
+Ery = atan2( squeeze(MTr(1, 3, :)),           squeeze(sqrt(MTr(1, 1, :).^2 + MTr(1, 2, :).^2)));
+Erx = atan2(-squeeze(MTr(2, 3, :))./cos(Ery), squeeze(MTr(3, 3, :))./cos(Ery));
+Erz = atan2(-squeeze(MTr(1, 2, :))./cos(Ery), squeeze(MTr(1, 1, :))./cos(Ery));
+
+
+ + +
+

exp_tomo_without_dist_trans.png +

+

Figure 1: X-Y-Z translation of the sample w.r.t. granite when performing tomography experiment with no disturbances (png, pdf)

+
+ + +
+

exp_tomo_without_dist_rot.png +

+

Figure 2: X-Y-Z rotations of the sample w.r.t. granite when performing tomography experiment with no disturbances (png, pdf)

+
+
+
+ +
+

2.3 Conclusion

+
+
+

+When everything is aligned, the resulting error motion is very small (nm range) and is quite negligible with respect to the error when disturbances are included. +This residual error motion probably comes from a small misalignment somewhere. +

+ +
+
+
+
+ +
+

3 Tomography Experiment with included perturbations

+
+

+ +

+
+
+

3.1 Simulation Setup

+
+

+We now activate the disturbances. +

+
+
opts = struct(...
+    'Dwx', true, ... % Ground Motion - X direction
+    'Dwy', true, ... % Ground Motion - Y direction
+    'Dwz', true, ... % Ground Motion - Z direction
+    'Fty_x', true, ... % Translation Stage - X direction
+    'Fty_z', true, ... % Translation Stage - Z direction
+    'Frz_z', true  ... % Spindle - Z direction
+);
+initDisturbances(opts);
+
+
+ +

+We simulate the model. +

+
+
sim('sim_nano_station_tomo');
+
+
+ +

+And we save the obtained data. +

+
+
tomo_align_dist = struct('t', t, 'MTr', MTr);
+save('experiment_tomography/mat/experiment.mat', 'tomo_align_dist', '-append');
+
+
+
+
+ +
+

3.2 Analysis

+
+
+
load('experiment_tomography/mat/experiment.mat', 'tomo_align_dist');
+t = tomo_align_dist.t;
+MTr = tomo_align_dist.MTr;
+
+
+ +
+
Edx = squeeze(MTr(1, 4, :));
+Edy = squeeze(MTr(2, 4, :));
+Edz = squeeze(MTr(3, 4, :));
+% The angles obtained are u-v-w Euler angles (rotations in the moving frame)
+Ery = atan2( squeeze(MTr(1, 3, :)),           squeeze(sqrt(MTr(1, 1, :).^2 + MTr(1, 2, :).^2)));
+Erx = atan2(-squeeze(MTr(2, 3, :))./cos(Ery), squeeze(MTr(3, 3, :))./cos(Ery));
+Erz = atan2(-squeeze(MTr(1, 2, :))./cos(Ery), squeeze(MTr(1, 1, :))./cos(Ery));
+
+
+ + +
+

exp_tomo_dist_trans.png +

+

Figure 3: X-Y-Z translation of the sample w.r.t. the granite when performing tomography experiment with disturbances (png, pdf)

+
+ + +
+

exp_tomo_dist_rot.png +

+

Figure 4: X-Y-Z rotations of the sample w.r.t. the granite when performing tomography experiment with disturbances (png, pdf)

+
+
+
+ +
+

3.3 Conclusion

+
+
+

+Error motion is what expected from the disturbance measurements. +

+ +
+
+
+
+ +
+

4 Tomography when the micro-hexapod is not centered

+
+

+ +

+
+
+

4.1 Simulation Setup

+
+

+We first set the wanted translation of the Micro Hexapod. +

+
+
P_micro_hexapod = [0.01; 0; 0]; % [m]
+
+
+ +

+We initialize the reference path. +

+
+
initializeReferences(struct('Dh_pos', [P_micro_hexapod; 0; 0; 0], 'Rz_type', 'rotating', 'Rz_period', 1));
+
+
+ +

+We initialize the stages. +

+
+
initializeMicroHexapod(struct('AP', P_micro_hexapod));
+
+
+

And we initialize the disturbances to zero.

@@ -332,88 +608,199 @@ initDisturbances(opts
-
-
sim('sim_nano_station_tomo')
-
-
- -
-
Dsm_without_dist = Dsm;
-
-
- -
-
figure;
-hold on;
-plot(Dsm_without_dist.x.Time, Dsm_without_dist.x.Data, 'DisplayName', 'x')
-plot(Dsm_without_dist.y.Time, Dsm_without_dist.y.Data, 'DisplayName', 'y')
-plot(Dsm_without_dist.z.Time, Dsm_without_dist.z.Data, 'DisplayName', 'z')
-hold off;
-xlim([2, inf]);
-legend('location', 'northeast');
-
-
- - -
-

exp_tomo_without_dist_trans.png +

+We simulate the model.

-

Figure 1: X-Y-Z translation of the sample w.r.t. granite when performing tomography experiment with no disturbances (png, pdf)

+
+
sim('sim_nano_station_tomo');
+
-

-Rotations. -Think of the good way to plot these rotations with respect to time. +And we save the obtained data.

+
+
tomo_not_align = struct('t', t, 'MTr', MTr);
+save('experiment_tomography/mat/experiment.mat', 'tomo_not_align', '-append');
+
+
-
-

3 With Perturbations

-
+
+

4.2 Analysis

+
+
+
load('experiment_tomography/mat/experiment.mat', 'tomo_not_align');
+t = tomo_not_align.t;
+MTr = tomo_not_align.MTr;
+
+
+ +
+
Edx = squeeze(MTr(1, 4, :));
+Edy = squeeze(MTr(2, 4, :));
+Edz = squeeze(MTr(3, 4, :));
+% The angles obtained are u-v-w Euler angles (rotations in the moving frame)
+Ery = atan2( squeeze(MTr(1, 3, :)),           squeeze(sqrt(MTr(1, 1, :).^2 + MTr(1, 2, :).^2)));
+Erx = atan2(-squeeze(MTr(2, 3, :))./cos(Ery), squeeze(MTr(3, 3, :))./cos(Ery));
+Erz = atan2(-squeeze(MTr(1, 2, :))./cos(Ery), squeeze(MTr(1, 1, :))./cos(Ery));
+
+
+ + +
+

exp_tomo_offset_trans.png +

+

Figure 5: X-Y-Z translation of the sample w.r.t. granite when performing tomography experiment with no disturbances (png, pdf)

+
+ + +
+

exp_tomo_offset_rot.png +

+

Figure 6: X-Y-Z rotations of the sample w.r.t. granite when performing tomography experiment with no disturbances (png, pdf)

+
+
+
+ +
+

4.3 Conclusion

+
+
+

+The main motions are translations in the X direction of the mobile platform (corresponds to the eccentricity of the micro-hexapod) and rotations along the rotating Y axis. +

+ +
+
+
+
+ +
+

5 Raster Scans with the translation stage

+
+

+ +

+
+
+

5.1 Simulation Setup

+
+

+We set the reference path. +

+
+
initializeReferences(struct('Dy_type', 'triangular', 'Dy_amplitude', 10e-3, 'Dy_period', 1));
+
+
+ +

+We initialize the stages. +

+
+
initializeGround();
+initializeGranite();
+initializeTy();
+initializeRy();
+initializeRz();
+initializeMicroHexapod();
+initializeAxisc();
+initializeMirror();
+initializeNanoHexapod(struct('actuator', 'piezo'));
+initializeSample(struct('mass', 1));
+
+
+ +

+And we initialize the disturbances to zero. +

opts = struct(...
-    'Dwx', true, ... % Ground Motion - X direction
-    'Dwy', true, ... % Ground Motion - Y direction
-    'Dwz', true, ... % Ground Motion - Z direction
-    'Fty_x', true, ... % Translation Stage - X direction
-    'Fty_z', true, ... % Translation Stage - Z direction
-    'Frz_z', true  ... % Spindle - Z direction
+    'Dwx', false, ... % Ground Motion - X direction
+    'Dwy', false, ... % Ground Motion - Y direction
+    'Dwz', false, ... % Ground Motion - Z direction
+    'Fty_x', false, ... % Translation Stage - X direction
+    'Fty_z', false, ... % Translation Stage - Z direction
+    'Frz_z', false  ... % Spindle - Z direction
 );
 initDisturbances(opts);
 
-
-
sim('sim_nano_station_tomo')
-
-
- -
-
figure;
-hold on;
-plot(Dsm.x.Time, Dsm.x.Data, 'DisplayName', 'x')
-plot(Dsm.y.Time, Dsm.y.Data, 'DisplayName', 'y')
-plot(Dsm.z.Time, Dsm.z.Data, 'DisplayName', 'z')
-hold off;
-xlim([2, inf]);
-legend('location', 'northeast');
-
-
- - -
-

exp_tomo_dist_trans.png +

+We simulate the model.

-

Figure 2: X-Y-Z translation of the sample w.r.t. the granite when performing tomography experiment with disturbances (png, pdf)

+
+
sim('sim_nano_station_tomo');
+
+
+ +

+And we save the obtained data. +

+
+
ty_scan = struct('t', t, 'MTr', MTr);
+save('experiment_tomography/mat/experiment.mat', 'ty_scan', '-append');
+
+
+
+
+ +
+

5.2 Analysis

+
+
+
load('experiment_tomography/mat/experiment.mat', 'ty_scan');
+t = ty_scan.t;
+MTr = ty_scan.MTr;
+
+
+ +
+
Edx = squeeze(MTr(1, 4, :));
+Edy = squeeze(MTr(2, 4, :));
+Edz = squeeze(MTr(3, 4, :));
+% The angles obtained are u-v-w Euler angles (rotations in the moving frame)
+Ery = atan2( squeeze(MTr(1, 3, :)),           squeeze(sqrt(MTr(1, 1, :).^2 + MTr(1, 2, :).^2)));
+Erx = atan2(-squeeze(MTr(2, 3, :))./cos(Ery), squeeze(MTr(3, 3, :))./cos(Ery));
+Erz = atan2(-squeeze(MTr(1, 2, :))./cos(Ery), squeeze(MTr(1, 1, :))./cos(Ery));
+
+
+ + +
+

exp_ty_scan_trans.png +

+

Figure 7: X-Y-Z translation of the sample w.r.t. granite when performing tomography experiment with no disturbances (png, pdf)

+
+ + +
+

exp_ty_scan_rot.png +

+

Figure 8: X-Y-Z rotations of the sample w.r.t. granite when performing tomography experiment with no disturbances (png, pdf)

+
+
+
+ +
+

5.3 Conclusion

+
+
+

+This is logic that the main error moving is translation along the Y axis and rotation along the X axis. +In order to reduce the errors, we can make a smoother reference path for the translation stage. +

+ +

Author: Dehaeze Thomas

-

Created: 2019-12-13 ven. 19:06

+

Created: 2019-12-17 mar. 18:01

Validate

diff --git a/experiment_tomography/index.org b/experiment_tomography/index.org index 9280bcb..336b5ff 100644 --- a/experiment_tomography/index.org +++ b/experiment_tomography/index.org @@ -42,6 +42,20 @@ :END: * Introduction :ignore: +The goal here is to simulate some scientific experiments with the tuned Simscape model when no control is applied to the nano-hexapod. + +This has several goals: +- Validate the model +- Estimate the expected error motion for the experiments +- Estimate the stroke that we may need for the nano-hexapod + +The document in organized as follow: +- In section [[sec:simscape_model]] the Simscape model is initialized +- In section [[sec:tomo_no_dist]] a tomography experiment is performed where the sample is aligned with the rotation axis. No disturbance is included +- In section [[sec:tomo_dist]], the same is done but with disturbance included +- In section [[sec:tomo_hexa_trans]] the micro-hexapod translate the sample such that its center of mass is no longer aligned with the rotation axis. No disturbance is included +- In section [[sec:ty_scans]], scans with the translation stage are simulated with no perturbation included + * Matlab Init :noexport:ignore: #+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) <> @@ -56,6 +70,8 @@ #+end_src * Simscape Model +<> + The simulink file to do tomography experiments is =sim_nano_station_tomo.slx=. #+begin_src matlab open('experiment_tomography/matlab/sim_nano_station_tomo.slx') @@ -81,14 +97,17 @@ We first initialize all the stages. initializeSample(struct('mass', 1)); #+end_src -* Tomography Experiment with no disturbances -** Simulation Setup We initialize the reference path for all the stages. All stage is set to its zero position except the Spindle which is rotating at 60rpm. #+begin_src matlab initializeReferences(struct('Rz_type', 'rotating', 'Rz_period', 1)); #+end_src +* Tomography Experiment with no disturbances +<> +** Introduction :ignore: + +** Simulation Setup And we initialize the disturbances to be equal to zero. #+begin_src matlab opts = struct(... @@ -109,11 +128,17 @@ We simulate the model. And we save the obtained data. #+begin_src matlab - MTr_alig_no_dist = MTr; - save('experiment_tomography/mat/experiment.mat', 'MTr_alig_no_dist', '-append'); + tomo_align_no_dist = struct('t', t, 'MTr', MTr); + save('experiment_tomography/mat/experiment.mat', 'tomo_align_no_dist', '-append'); #+end_src ** Analysis +#+begin_src matlab + load('experiment_tomography/mat/experiment.mat', 'tomo_align_no_dist'); + t = tomo_align_no_dist.t; + MTr = tomo_align_no_dist.MTr; +#+end_src + #+begin_src matlab Edx = squeeze(MTr(1, 4, :)); Edy = squeeze(MTr(2, 4, :)); @@ -126,18 +151,26 @@ And we save the obtained data. #+begin_src matlab :exports none figure; - hold on; + ax1 = subplot(1, 3, 1); plot(t, Edx, 'DisplayName', '$\epsilon_{x}$') - plot(t, Edy, 'DisplayName', '$\epsilon_{y}$') - plot(t, Edz, 'DisplayName', '$\epsilon_{z}$') - hold off; - xlabel('Time [s]'); ylabel('Displacement [m]'); - xlim([2, inf]); + ylabel('Displacement [m]'); legend('location', 'northeast'); + + ax2 = subplot(1, 3, 2); + plot(t, Edy, 'DisplayName', '$\epsilon_{y}$') + xlabel('Time [s]'); + legend('location', 'northeast'); + + ax3 = subplot(1, 3, 3); + plot(t, Edz, 'DisplayName', '$\epsilon_{z}$') + legend('location', 'northeast'); + + linkaxes([ax1,ax2,ax3],'x'); + xlim([1, inf]); #+end_src #+HEADER: :tangle no :exports results :results none :noweb yes -#+begin_src matlab :var filepath="figs/exp_tomo_without_dist_trans.pdf" :var figsize="wide-normal" :post pdf2svg(file=*this*, ext="png") +#+begin_src matlab :var filepath="figs/exp_tomo_without_dist_trans.pdf" :var figsize="full-normal" :post pdf2svg(file=*this*, ext="png") <> #+end_src @@ -147,18 +180,26 @@ And we save the obtained data. #+begin_src matlab :exports none figure; - hold on; + ax1 = subplot(1, 3, 1); plot(t, Erx, 'DisplayName', '$\epsilon_{\theta x}$') - plot(t, Ery, 'DisplayName', '$\epsilon_{\theta y}$') - plot(t, Erz, 'DisplayName', '$\epsilon_{\theta z}$') - hold off; - xlabel('Time [s]'); ylabel('Rotation [rad]'); - xlim([2, inf]); + ylabel('Rotation [rad]'); legend('location', 'northeast'); + + ax2 = subplot(1, 3, 2); + plot(t, Ery, 'DisplayName', '$\epsilon_{\theta y}$') + xlabel('Time [s]'); + legend('location', 'northeast'); + + ax3 = subplot(1, 3, 3); + plot(t, Erz, 'DisplayName', '$\epsilon_{\theta z}$') + legend('location', 'northeast'); + + linkaxes([ax1,ax2,ax3],'x'); + xlim([1, inf]); #+end_src #+HEADER: :tangle no :exports results :results none :noweb yes -#+begin_src matlab :var filepath="figs/exp_tomo_without_dist_rot.pdf" :var figsize="wide-normal" :post pdf2svg(file=*this*, ext="png") +#+begin_src matlab :var filepath="figs/exp_tomo_without_dist_rot.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") <> #+end_src @@ -166,7 +207,17 @@ And we save the obtained data. #+CAPTION: X-Y-Z rotations of the sample w.r.t. granite when performing tomography experiment with no disturbances ([[./figs/exp_tomo_without_dist_rot.png][png]], [[./figs/exp_tomo_without_dist_rot.pdf][pdf]]) [[file:figs/exp_tomo_without_dist_rot.png]] -* With Perturbations +** Conclusion + +#+begin_important + When everything is aligned, the resulting error motion is very small (nm range) and is quite negligible with respect to the error when disturbances are included. + This residual error motion probably comes from a small misalignment somewhere. +#+end_important + +* Tomography Experiment with included perturbations +<> +** Introduction :ignore: + ** Simulation Setup We now activate the disturbances. #+begin_src matlab @@ -188,11 +239,17 @@ We simulate the model. And we save the obtained data. #+begin_src matlab - MTr_alig_dist = MTr; - save('experiment_tomography/mat/experiment.mat', 'MTr_alig_dist', '-append'); + tomo_align_dist = struct('t', t, 'MTr', MTr); + save('experiment_tomography/mat/experiment.mat', 'tomo_align_dist', '-append'); #+end_src ** Analysis +#+begin_src matlab + load('experiment_tomography/mat/experiment.mat', 'tomo_align_dist'); + t = tomo_align_dist.t; + MTr = tomo_align_dist.MTr; +#+end_src + #+begin_src matlab Edx = squeeze(MTr(1, 4, :)); Edy = squeeze(MTr(2, 4, :)); @@ -205,18 +262,26 @@ And we save the obtained data. #+begin_src matlab :exports none figure; - hold on; + ax1 = subplot(1, 3, 1); plot(t, Edx, 'DisplayName', '$\epsilon_{x}$') - plot(t, Edy, 'DisplayName', '$\epsilon_{y}$') - plot(t, Edz, 'DisplayName', '$\epsilon_{z}$') - hold off; - xlabel('Time [s]'); ylabel('Displacement [m]'); - xlim([2, inf]); + ylabel('Displacement [m]'); legend('location', 'northeast'); + + ax2 = subplot(1, 3, 2); + plot(t, Edy, 'DisplayName', '$\epsilon_{y}$') + xlabel('Time [s]'); + legend('location', 'northeast'); + + ax3 = subplot(1, 3, 3); + plot(t, Edz, 'DisplayName', '$\epsilon_{z}$') + legend('location', 'northeast'); + + linkaxes([ax1,ax2,ax3],'x'); + xlim([1, inf]); #+end_src #+HEADER: :tangle no :exports results :results none :noweb yes -#+begin_src matlab :var filepath="figs/exp_tomo_dist_trans.pdf" :var figsize="wide-normal" :post pdf2svg(file=*this*, ext="png") +#+begin_src matlab :var filepath="figs/exp_tomo_dist_trans.pdf" :var figsize="full-normal" :post pdf2svg(file=*this*, ext="png") <> #+end_src @@ -226,18 +291,26 @@ And we save the obtained data. #+begin_src matlab :exports none figure; - hold on; + ax1 = subplot(1, 3, 1); plot(t, Erx, 'DisplayName', '$\epsilon_{\theta x}$') - plot(t, Ery, 'DisplayName', '$\epsilon_{\theta y}$') - plot(t, Erz, 'DisplayName', '$\epsilon_{\theta z}$') - hold off; - xlabel('Time [s]'); ylabel('Rotation [rad]'); - xlim([2, inf]); + ylabel('Rotation [rad]'); legend('location', 'northeast'); + + ax2 = subplot(1, 3, 2); + plot(t, Ery, 'DisplayName', '$\epsilon_{\theta y}$') + xlabel('Time [s]'); + legend('location', 'northeast'); + + ax3 = subplot(1, 3, 3); + plot(t, Erz, 'DisplayName', '$\epsilon_{\theta z}$') + legend('location', 'northeast'); + + linkaxes([ax1,ax2,ax3],'x'); + xlim([1, inf]); #+end_src #+HEADER: :tangle no :exports results :results none :noweb yes -#+begin_src matlab :var filepath="figs/exp_tomo_dist_rot.pdf" :var figsize="wide-normal" :post pdf2svg(file=*this*, ext="png") +#+begin_src matlab :var filepath="figs/exp_tomo_dist_rot.pdf" :var figsize="full-normal" :post pdf2svg(file=*this*, ext="png") <> #+end_src @@ -245,7 +318,16 @@ And we save the obtained data. #+CAPTION: X-Y-Z rotations of the sample w.r.t. the granite when performing tomography experiment with disturbances ([[./figs/exp_tomo_dist_rot.png][png]], [[./figs/exp_tomo_dist_rot.pdf][pdf]]) [[file:figs/exp_tomo_dist_rot.png]] -* Tomography +** Conclusion + +#+begin_important + Error motion is what expected from the disturbance measurements. +#+end_important + +* Tomography when the micro-hexapod is not centered +<> +** Introduction :ignore: + ** Simulation Setup We first set the wanted translation of the Micro Hexapod. #+begin_src matlab @@ -282,11 +364,17 @@ We simulate the model. And we save the obtained data. #+begin_src matlab - MTr_not_alig = MTr; - save('experiment_tomography/mat/experiment.mat', 'MTr_not_alig', '-append'); + tomo_not_align = struct('t', t, 'MTr', MTr); + save('experiment_tomography/mat/experiment.mat', 'tomo_not_align', '-append'); #+end_src ** Analysis +#+begin_src matlab + load('experiment_tomography/mat/experiment.mat', 'tomo_not_align'); + t = tomo_not_align.t; + MTr = tomo_not_align.MTr; +#+end_src + #+begin_src matlab Edx = squeeze(MTr(1, 4, :)); Edy = squeeze(MTr(2, 4, :)); @@ -299,18 +387,26 @@ And we save the obtained data. #+begin_src matlab :exports none figure; - hold on; + ax1 = subplot(1, 3, 1); plot(t, Edx, 'DisplayName', '$\epsilon_{x}$') - plot(t, Edy, 'DisplayName', '$\epsilon_{y}$') - plot(t, Edz, 'DisplayName', '$\epsilon_{z}$') - hold off; - xlabel('Time [s]'); ylabel('Displacement [m]'); - xlim([2, inf]); + ylabel('Displacement [m]'); legend('location', 'northeast'); + + ax2 = subplot(1, 3, 2); + plot(t, Edy, 'DisplayName', '$\epsilon_{y}$') + xlabel('Time [s]'); + legend('location', 'northeast'); + + ax3 = subplot(1, 3, 3); + plot(t, Edz, 'DisplayName', '$\epsilon_{z}$') + legend('location', 'northeast'); + + linkaxes([ax1,ax2,ax3],'x'); + xlim([1, inf]); #+end_src #+HEADER: :tangle no :exports results :results none :noweb yes -#+begin_src matlab :var filepath="figs/exp_tomo_offset_trans.pdf" :var figsize="wide-normal" :post pdf2svg(file=*this*, ext="png") +#+begin_src matlab :var filepath="figs/exp_tomo_offset_trans.pdf" :var figsize="full-normal" :post pdf2svg(file=*this*, ext="png") <> #+end_src @@ -320,18 +416,26 @@ And we save the obtained data. #+begin_src matlab :exports none figure; - hold on; + ax1 = subplot(1, 3, 1); plot(t, Erx, 'DisplayName', '$\epsilon_{\theta x}$') - plot(t, Ery, 'DisplayName', '$\epsilon_{\theta y}$') - plot(t, Erz, 'DisplayName', '$\epsilon_{\theta z}$') - hold off; - xlabel('Time [s]'); ylabel('Rotation [rad]'); - xlim([2, inf]); + ylabel('Rotation [rad]'); legend('location', 'northeast'); + + ax2 = subplot(1, 3, 2); + plot(t, Ery, 'DisplayName', '$\epsilon_{\theta y}$') + xlabel('Time [s]'); + legend('location', 'northeast'); + + ax3 = subplot(1, 3, 3); + plot(t, Erz, 'DisplayName', '$\epsilon_{\theta z}$') + legend('location', 'northeast'); + + linkaxes([ax1,ax2,ax3],'x'); + xlim([1, inf]); #+end_src #+HEADER: :tangle no :exports results :results none :noweb yes -#+begin_src matlab :var filepath="figs/exp_tomo_offset_rot.pdf" :var figsize="wide-normal" :post pdf2svg(file=*this*, ext="png") +#+begin_src matlab :var filepath="figs/exp_tomo_offset_rot.pdf" :var figsize="full-normal" :post pdf2svg(file=*this*, ext="png") <> #+end_src @@ -339,245 +443,138 @@ And we save the obtained data. #+CAPTION: X-Y-Z rotations of the sample w.r.t. granite when performing tomography experiment with no disturbances ([[./figs/exp_tomo_offset_rot.png][png]], [[./figs/exp_tomo_offset_rot.pdf][pdf]]) [[file:figs/exp_tomo_offset_rot.png]] -* TODO Tests on the transformation from reference to wanted position :noexport: -:PROPERTIES: -:header-args:matlab+: :eval no -:END: +** Conclusion + +#+begin_important + The main motions are translations in the X direction of the mobile platform (corresponds to the eccentricity of the micro-hexapod) and rotations along the rotating Y axis. +#+end_important + +* Raster Scans with the translation stage +<> ** Introduction :ignore: -#+begin_quote -Rx = [1 0 0; - 0 cos(t) -sin(t); - 0 sin(t) cos(t)]; -Ry = [ cos(t) 0 sin(t); - 0 1 0; - -sin(t) 0 cos(t)]; - -Rz = [cos(t) -sin(t) 0; - sin(t) cos(t) 0; - 0 0 1]; -#+end_quote - -Let's define the following frames: -- $\{W\}$ the frame that is *fixed to the granite* and its origin at the theoretical meeting point between the X-ray and the spindle axis. -- $\{S\}$ the frame *attached to the sample* (in reality attached to the top platform of the nano-hexapod) with its origin at 175mm above the top platform of the nano-hexapod. - Its origin is $O_S$. -- $\{T\}$ the theoretical wanted frame that correspond to the wanted pose of the frame $\{S\}$. - $\{T\}$ is computed from the wanted position of each stage. It is thus theoretical and does not correspond to a real position. - The origin of $T$ is $O_T$ and is the wanted position of the sample. - -Thus: -- the *measurement* of the position of the sample corresponds to ${}^W O_S = \begin{bmatrix} {}^WP_{x,m} & {}^WP_{y,m} & {}^WP_{z,m} \end{bmatrix}^T$ in translation and to $\theta_m {}^W\bm{s}_m = \theta_m \cdot \begin{bmatrix} {}^Ws_{x,m} & {}^Ws_{y,m} & {}^Ws_{z,m} \end{bmatrix}^T$ in rotations -- the *wanted position* of the sample expressed w.r.t. the granite is ${}^W O_T = \begin{bmatrix} {}^WP_{x,r} & {}^WP_{y,r} & {}^WP_{z,r} \end{bmatrix}^T$ in translation and to $\theta_r {}^W\bm{s}_r = \theta_r \cdot \begin{bmatrix} {}^Ws_{x,r} & {}^Ws_{y,r} & {}^Ws_{z,r} \end{bmatrix}^T$ in rotations - -** Wanted Position of the Sample with respect to the Granite -Let's define the wanted position of each stage. +** Simulation Setup +We set the reference path. #+begin_src matlab - Ty = 0; % [m] - Ry = 3*pi/180; % [rad] - Rz = 180*pi/180; % [rad] - - % Hexapod (first consider only translations) - Thx = 0; % [m] - Thy = 0; % [m] - Thz = 0; % [m] + initializeReferences(struct('Dy_type', 'triangular', 'Dy_amplitude', 10e-3, 'Dy_period', 1)); #+end_src -Now, we compute the corresponding wanted translation and rotation of the sample with respect to the granite frame $\{W\}$. -This corresponds to ${}^WO_T$ and $\theta_m {}^Ws_m$. - -To do so, we have to define the homogeneous transformation for each stage. +We initialize the stages. #+begin_src matlab - % Translation Stage - Rty = [1 0 0 0; - 0 1 0 Ty; - 0 0 1 0; - 0 0 0 1]; - - % Tilt Stage - Pure rotating aligned with Ob - Rry = [ cos(Ry) 0 sin(Ry) 0; - 0 1 0 0; - -sin(Ry) 0 cos(Ry) 0; - 0 0 0 1]; - - % Spindle - Rotation along the Z axis - Rrz = [cos(Rz) -sin(Rz) 0 0 ; - sin(Rz) cos(Rz) 0 0 ; - 0 0 1 0 ; - 0 0 0 1 ]; - - % Micro-Hexapod (only rotations first) - Rh = [1 0 0 Thx ; - 0 1 0 Thy ; - 0 0 1 Thz ; - 0 0 0 1 ]; + initializeGround(); + initializeGranite(); + initializeTy(); + initializeRy(); + initializeRz(); + initializeMicroHexapod(); + initializeAxisc(); + initializeMirror(); + initializeNanoHexapod(struct('actuator', 'piezo')); + initializeSample(struct('mass', 1)); #+end_src -We combine the individual homogeneous transformations into one homogeneous transformation for all the station. +And we initialize the disturbances to zero. #+begin_src matlab - Ttot = Rty*Rry*Rrz*Rh; + opts = struct(... + 'Dwx', false, ... % Ground Motion - X direction + 'Dwy', false, ... % Ground Motion - Y direction + 'Dwz', false, ... % Ground Motion - Z direction + 'Fty_x', false, ... % Translation Stage - X direction + 'Fty_z', false, ... % Translation Stage - Z direction + 'Frz_z', false ... % Spindle - Z direction + ); + initDisturbances(opts); #+end_src -Using this homogeneous transformation, we can compute the wanted position and orientation of the sample with respect to the granite. - -Translation. +We simulate the model. #+begin_src matlab - WOr = Ttot*[0;0;0;1]; - WOr = WOr(1:3); + sim('sim_nano_station_tomo'); #+end_src -Rotation. +And we save the obtained data. #+begin_src matlab - thetar = acos((trace(Ttot(1:3, 1:3))-1)/2) - if thetar == 0 - WSr = [0; 0; 0]; - else - [V, D] = eig(Ttot(1:3, 1:3)); - WSr = thetar*V(:, abs(diag(D) - 1) < eps(1)); - end + ty_scan = struct('t', t, 'MTr', MTr); + save('experiment_tomography/mat/experiment.mat', 'ty_scan', '-append'); +#+end_src + +** Analysis +#+begin_src matlab + load('experiment_tomography/mat/experiment.mat', 'ty_scan'); + t = ty_scan.t; + MTr = ty_scan.MTr; #+end_src #+begin_src matlab - WPr = [WOr ; WSr]; + Edx = squeeze(MTr(1, 4, :)); + Edy = squeeze(MTr(2, 4, :)); + Edz = squeeze(MTr(3, 4, :)); + % The angles obtained are u-v-w Euler angles (rotations in the moving frame) + Ery = atan2( squeeze(MTr(1, 3, :)), squeeze(sqrt(MTr(1, 1, :).^2 + MTr(1, 2, :).^2))); + Erx = atan2(-squeeze(MTr(2, 3, :))./cos(Ery), squeeze(MTr(3, 3, :))./cos(Ery)); + Erz = atan2(-squeeze(MTr(1, 2, :))./cos(Ery), squeeze(MTr(1, 1, :))./cos(Ery)); #+end_src -** Measured Position of the Sample with respect to the Granite -The measurement of the position of the sample using the metrology system gives the position and orientation of the sample with respect to the granite. -#+begin_src matlab - % Measurements: Xm, Ym, Zm, Rx, Ry, Rz - Dxm = 0; % [m] - Dym = 0; % [m] - Dzm = 0; % [m] +#+begin_src matlab :exports none + figure; + ax1 = subplot(1, 3, 1); + plot(t, Edx, 'DisplayName', '$\epsilon_{x}$') + ylabel('Displacement [m]'); + legend('location', 'northeast'); - Rxm = 0*pi/180; % [rad] - Rym = 0*pi/180; % [rad] - Rzm = 180*pi/180; % [rad] + ax2 = subplot(1, 3, 2); + plot(t, Edy, 'DisplayName', '$\epsilon_{y}$') + xlabel('Time [s]'); + legend('location', 'northeast'); + + ax3 = subplot(1, 3, 3); + plot(t, Edz, 'DisplayName', '$\epsilon_{z}$') + legend('location', 'northeast'); + + linkaxes([ax1,ax2,ax3],'x'); + xlim([1, inf]); #+end_src -Let's compute the corresponding orientation using screw axis. -#+begin_src matlab - Trxm = [1 0 0; - 0 cos(Rxm) -sin(Rxm); - 0 sin(Rxm) cos(Rxm)]; - Trym = [ cos(Rym) 0 sin(Rym); - 0 1 0; - -sin(Rym) 0 cos(Rym)]; - Trzm = [cos(Rzm) -sin(Rzm) 0; - sin(Rzm) cos(Rzm) 0; - 0 0 1]; - - STw = [[ Trym*Trxm*Trzm , [Dxm; Dym; Dzm]]; 0 0 0 1]; +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/exp_ty_scan_trans.pdf" :var figsize="full-normal" :post pdf2svg(file=*this*, ext="png") + <> #+end_src -We then obtain the orientation measurement in the form of screw coordinate $\theta_m ({}^Ws_{x,m},\ {}^Ws_{y,m},\ {}^Ws_{z,m})^T$ where: -- $\theta_m = \cos^{-1} \frac{\text{Tr}(R) - 1}{2}$ -- ${}^W\bm{s}_m$ is the eigen vector of the rotation matrix $R$ corresponding to the eigen value $\lambda = 1$ +#+NAME: fig:exp_ty_scan_trans +#+CAPTION: X-Y-Z translation of the sample w.r.t. granite when performing tomography experiment with no disturbances ([[./figs/exp_ty_scan_trans.png][png]], [[./figs/exp_ty_scan_trans.pdf][pdf]]) +[[file:figs/exp_ty_scan_trans.png]] -#+begin_src matlab - thetam = acos((trace(STw(1:3, 1:3))-1)/2); % [rad] - if thetam == 0 - WSm = [0; 0; 0]; - else - [V, D] = eig(STw(1:3, 1:3)); - WSm = thetam*V(:, abs(diag(D) - 1) < eps(1)); - end +#+begin_src matlab :exports none + figure; + ax1 = subplot(1, 3, 1); + plot(t, Erx, 'DisplayName', '$\epsilon_{\theta x}$') + ylabel('Rotation [rad]'); + legend('location', 'northeast'); + + ax2 = subplot(1, 3, 2); + plot(t, Ery, 'DisplayName', '$\epsilon_{\theta y}$') + xlabel('Time [s]'); + legend('location', 'northeast'); + + ax3 = subplot(1, 3, 3); + plot(t, Erz, 'DisplayName', '$\epsilon_{\theta z}$') + legend('location', 'northeast'); + + linkaxes([ax1,ax2,ax3],'x'); + xlim([1, inf]); #+end_src -#+begin_src matlab - WPm = [Dxm ; Dym ; Dzm ; WSm]; +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/exp_ty_scan_rot.pdf" :var figsize="full-normal" :post pdf2svg(file=*this*, ext="png") + <> #+end_src -** Positioning Error with respect to the Granite -The wanted position expressed with respect to the granite is ${}^WO_T$ and the measured position with respect to the granite is ${}^WO_S$, thus the *position error* expressed in $\{W\}$ is -\[ {}^W E = {}^W O_T - {}^W O_S \] -The same is true for rotations: -\[ \theta_\epsilon {}^W\bm{s}_\epsilon = \theta_r {}^W\bm{s}_r - \theta_m {}^W\bm{s}_m \] +#+NAME: fig:exp_ty_scan_rot +#+CAPTION: X-Y-Z rotations of the sample w.r.t. granite when performing tomography experiment with no disturbances ([[./figs/exp_ty_scan_rot.png][png]], [[./figs/exp_ty_scan_rot.pdf][pdf]]) +[[file:figs/exp_ty_scan_rot.png]] -#+begin_src matlab - WPe = WPr - WPm; -#+end_src +** Conclusion -#+begin_quote -Now we want to express this error in a frame attached to the *base of the nano-hexapod* with its origin at the same point where the Jacobian of the nano-hexapod is computed (175mm above the top platform + 90mm of total height of the nano-hexapod). - -Or maybe should we want to express this error with respect to the *top platform of the nano-hexapod*? -We are measuring the position of the top-platform, and we don't know exactly the position of the bottom platform. -We could compute the position of the bottom platform in two ways: -- from the encoders of each stage -- from the measurement of the nano-hexapod top platform + the internal metrology in the nano-hexapod (capacitive sensors e.g) - -A third option is to say that the maximum stroke of the nano-hexapod is so small that the error should no change to much by the change of base. -#+end_quote - -** Position Error Expressed in the Nano-Hexapod Frame -We now want the position error to be expressed in $\{S\}$ (the frame attach to the sample) for control: -\[ {}^S E = {}^S T_W \cdot {}^W E \] - -Thus we need to compute the homogeneous transformation ${}^ST_W$. -Fortunately, this homogeneous transformation can be computed from the measurement of the sample position and orientation with respect to the granite. -#+begin_src matlab - Trxm = [1 0 0; - 0 cos(Rxm) -sin(Rxm); - 0 sin(Rxm) cos(Rxm)]; - Trym = [ cos(Rym) 0 sin(Rym); - 0 1 0; - -sin(Rym) 0 cos(Rym)]; - Trzm = [cos(Rzm) -sin(Rzm) 0; - sin(Rzm) cos(Rzm) 0; - 0 0 1]; - - STw = [[ Trym*Trxm*Trzm , [Dxm; Dym; Dzm]]; 0 0 0 1]; -#+end_src - -Translation Error. -#+begin_src matlab - SEm = STw * [WPe(1:3); 0]; - SEm = SEm(1:3); -#+end_src - -Rotation Error. -#+begin_src matlab - SEr = STw * [WPe(4:6); 0]; - SEr = SEr(1:3); -#+end_src - -#+begin_src matlab - Etot = [SEm ; SEr] -#+end_src -** Another try -Let's denote: -- $\{W\}$ the initial fixed frame -- $\{R\}$ the reference frame corresponding to the wanted pose of the sample -- $\{M\}$ the frame corresponding to the measured pose of the sample - -We have then computed: -- ${}^WT_R$ -- ${}^WT_M$ - -We have: -\begin{align} - {}^MT_R &= {}^MT_W {}^WT_R \\ - &= {}^WT_M^t {}^WT_R -\end{align} - -#+begin_src matlab - MTr = STw'*Ttot; -#+end_src - -Position error: -#+begin_src matlab - MTr(1:3, 1:4)*[0; 0; 0; 1] -#+end_src - -Orientation error: -#+begin_src matlab - MTr(1:3, 1:3) -#+end_src - -** Verification -How can we verify that the computation is correct? -Options: -- Test with simscape multi-body - - Impose motion on each stage - - Measure the position error w.r.t. the NASS - - Compare with the computation +#+begin_important + This is logic that the main error moving is translation along the Y axis and rotation along the X axis. + In order to reduce the errors, we can make a smoother reference path for the translation stage. +#+end_important diff --git a/experiment_tomography/mat/experiment.mat b/experiment_tomography/mat/experiment.mat index 40023d0dc11032a3d8877ef2eebe350cb61ab9b7..6a61a91595649cc7823f4c6525dc959571da801b 100644 GIT binary patch literal 4346458 zcma%=LvSq&)2-i=Op04VuRbBna3aZEm z3K6iLr>vJ)^ea4|4*F)$J^F)(rx{D11d|I;5yS@{3WR0Z(=dF1G%dvc^Me%?wFGLb|{ 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