#+TITLE: Simscape Model - Nano Active Stabilization System
:DRAWER:
#+LANGUAGE: en
#+EMAIL: dehaeze.thomas@gmail.com
#+AUTHOR: Dehaeze Thomas
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* Notes :noexport:
** Notes
Prefix is =nass=
The goals of this report are:
- [X] ([[file:~/Cloud/work-projects/ID31-NASS/matlab/nass-simscape/org/positioning_error.org][positioning_error]]): Explain how the NASS control is made (computation of the wanted position, measurement of the sample position, computation of the errors)
- [X] ([[file:~/Cloud/work-projects/ID31-NASS/matlab/nass-simscape/org/uncertainty_experiment.org][uncertainty_experiment]]): Effect of experimental conditions on the plant (payload mass, Ry position, Rz position, Rz velocity, etc...)
- [ ] Determination of the *optimal stiffness* for the hexapod actuators:
- [ ] [[file:~/Cloud/work-projects/ID31-NASS/matlab/nass-simscape/org/uncertainty_optimal_stiffness.org][uncertainty_optimal_stiffness]]
- [ ] [[file:~/Cloud/work-projects/ID31-NASS/matlab/nass-simscape/org/optimal_stiffness_disturbances.org][optimal_stiffness_disturbances]]
- [ ] [[file:~/Cloud/work-projects/ID31-NASS/documents/state-of-thesis-2020/index.org][state-of-thesis-2020]]
- [ ] [[file:/home/thomas/Cloud/meetings/group-meetings-me/2020-04-06-NASS-Design/2020-04-06-NASS-Design.org][group-meeting-optimal-stiffness]]
Should this be in this report? *This should be in chapter 2*
- [X] Explain why HAC-LAC strategy is nice (*It was already explained in uniaxial model*)
- [X] [[file:~/Cloud/work-projects/ID31-NASS/matlab/nass-simscape/org/control.org][different control architectures]]
- [X] [[file:~/Cloud/work-projects/ID31-NASS/matlab/stewart-simscape/org/control-vibration-isolation.org][hexapod - vibration isolation]]
- [X] How to apply/optimize IFF on an hexapod? ([[file:~/Cloud/work-projects/ID31-NASS/matlab/nass-simscape/org/control_active_damping.org][control_active_damping]], [[file:~/Cloud/work-projects/ID31-NASS/matlab/stewart-simscape/org/control-active-damping.org][active damping for stewart platforms]])
- [X] ([[file:~/Cloud/research/matlab/decoupling-strategies/svd-control.org][decoupling-strategies]]): Decoupling strategies for HAC? (maybe also in previous report)
*Will be in chapter 2*
- [X] Validation of the concept using simulations:
- [X] Find where this simulation in OL/CL is made (maybe for the conference?)
It was re-made for micro-station validation. Will just have to do the same simulation but with nano-hexapod in closed-loop
- Tomography experiment (maybe also Ty scans)
- Open VS Closed loop results
- *Conclusion*: concept validation
nano hexapod architecture with APA
decentralized IFF + centralized HAC
- In this section simple control (in the frame of the struts)
- Justify future used control architecture (control in the frame of the struts? Need to check what was done in ID31 tests)
- Table that compares different approaches (specify performances in different DoF, same plans on the diagonal, etc...)
- Literature review about Stewart platform control?
*In chapter 2: Special section about MIMO control, complementary filters, etc...*
** Outline
*** Control Kinematics
- Explain how the position error can be expressed in the frame of the nano-hexapod
- ([[file:~/Cloud/work-projects/ID31-NASS/matlab/nass-simscape/org/positioning_error.org][positioning_error]]): Explain how the NASS control is made (computation of the wanted position, measurement of the sample position, computation of the errors)
- Control architecture, block diagram
*** LAC
- How to apply/optimize IFF on an hexapod? ([[file:~/Cloud/work-projects/ID31-NASS/matlab/nass-simscape/org/control_active_damping.org][control_active_damping]], [[file:~/Cloud/work-projects/ID31-NASS/matlab/stewart-simscape/org/control-active-damping.org][active damping for stewart platforms]])
- Robustness to payload mass
- Root Locus
- Damping optimization
*** HAC
- ([[file:~/Cloud/work-projects/ID31-NASS/matlab/nass-simscape/org/uncertainty_experiment.org][uncertainty_experiment]]): Effect of experimental conditions on the plant (payload mass, Ry position, Rz position, Rz velocity, etc...)
- Determination of the *optimal stiffness* for the hexapod actuators:
- [ ] [[file:~/Cloud/work-projects/ID31-NASS/matlab/nass-simscape/org/uncertainty_optimal_stiffness.org][uncertainty_optimal_stiffness]]
- [ ] [[file:~/Cloud/work-projects/ID31-NASS/matlab/nass-simscape/org/optimal_stiffness_disturbances.org][optimal_stiffness_disturbances]]
- [ ] [[file:~/Cloud/work-projects/ID31-NASS/documents/state-of-thesis-2020/index.org][state-of-thesis-2020]]
- [ ] [[file:/home/thomas/Cloud/meetings/group-meetings-me/2020-04-06-NASS-Design/2020-04-06-NASS-Design.org][group-meeting-optimal-stiffness]]
- Effect of micro-station compliance
- Effect of IFF
- Effect of payload mass
- Decoupled plant
- Controller design
*** Simulations
- Take into account disturbances, metrology sensor noise. Maybe say here that we don't take in account other noise sources as they will be optimized latter (detail design phase)
- Tomography + lateral scans (same as what was done in open loop [[file:~/Cloud/work-projects/ID31-NASS/phd-thesis-chapters/A4-simscape-micro-station/simscape-micro-station.org::*Simulation of Scientific Experiments][here]])
- Validation of concept
** DONE Old Outline
CLOSED: [2024-11-07 Thu 16:19]
*** Introduction :ignore:
Discussion of:
- Transformation matrices / control architecture (computation of the position error in the frame of the nano-hexapod)
- Control of parallel architectures
- Control in the frame of struts or cartesian?
- Effect of rotation on IFF? => APA
- HAC-LAC
- New noise budgeting?
*** Control Kinematics
- Explain how the position error can be expressed in the frame of the nano-hexapod
- block diagram
- Explain how to go from external metrology to the frame of the nano-hexapod
*** High Authority Control - Low Authority Control (HAC-LAC)
- general idea
- case for parallel manipulator: decentralized LAC + centralized HAC
*** Decoupling Strategies for parallel manipulators
[[file:~/Cloud/research/matlab/decoupling-strategies/svd-control.org::+TITLE: Diagonal control using the SVD and the Jacobian Matrix][study]]
- Jacobian matrices, CoK, CoM, ...
- Discussion of cubic architecture
- SVD, Modal, ...
*** Decentralized Integral Force Feedback (LAC)
- Root Locus
- Damping optimization
*** Decoupled Dynamics
- Centralized HAC
- Control in the frame of the struts
- Effect of IFF
*** Centralized Position Controller (HAC)
- Decoupled plant
- Controller design
*** Time domain simulations
Goal: validation of the concept
- Take into account disturbances, sensor noise, etc...
- Tomography + lateral scans (same as what was done in open loop [[file:~/Cloud/work-projects/ID31-NASS/phd-thesis-chapters/A4-simscape-micro-station/simscape-micro-station.org::*Simulation of Scientific Experiments][here]])
** DONE [#A] Merge the micro-station model with the nano-hexapod model
CLOSED: [2025-02-12 Wed 12:10] SCHEDULED: <2025-02-12 Wed>
- [X] *Start from the Simscape model of the ID31 tests*
=/home/thomas/Cloud/work-projects/ID31-NASS/phd-thesis-chapters/C5-test-bench-id31/matlab/nass_model_id31.slx=
- [X] Remove LION metrology to have perfect measurement
- [X] Remove nano-hexapod model and add simplified model
- [ ] Add "cylindrical" payloads (configurable in mass)
** DONE [#B] Add payload configurable subsystem
CLOSED: [2025-02-12 Wed 14:17] SCHEDULED: <2025-02-12 Wed>
** DONE [#A] Verify formulas to have the errors in the frame of the nano-hexapod and in the frame of the granite
CLOSED: [2025-02-17 Mon 10:35] SCHEDULED: <2025-02-17 Mon>
Errors in the frame of the nano-hexapod:
\begin{equation}\label{eq:nass_transformation_error}
\bm{T}_{\text{error}} = \bm{T}_{\mu\text{-station}}^{-1} \cdot \bm{T}_{\text{sample}}
\end{equation}
Errors in the frame of the granite:
WTe(1:3, 4, i) = WTr(1:3, 4, i) - WTm(1:3, 4, i);
WTe(1:3, 1:3, i) = WTr(1:3, 1:3, end)*WTm(1:3, 1:3, end)';
** DONE [#A] Fix IFF and HAC controllers
CLOSED: [2025-02-17 Mon 16:00] SCHEDULED: <2025-02-17 Mon>
** DONE [#A] Compute all figures
CLOSED: [2025-02-17 Mon 18:26] SCHEDULED: <2025-02-17 Mon>
** DONE [#B] Discuss the necessity of estimated Rz?
CLOSED: [2025-02-17 Mon 18:26]
One big advantage of doing the control in the cartesian plane, is that we don't need the estimation of nano-hexapod Rz, therefore we don't need the encoders anymore!
Maybe this should be done *here*.
Here it can be reminded when doing the control in the cartesian frame.
** DONE [#B] Determine which .mat files are used and which are not
CLOSED: [2025-02-17 Mon 23:04]
#+begin_src matlab :eval no :tangle no
load("nass_model_conf_log.mat");
load("nass_model_conf_simscape.mat");
dist = load("nass_model_disturbances.mat");
load("nass_model_references.mat");
load("nass_model_controller.mat");
load("nass_model_stages.mat");
J_L_to_X = inv(nano_hexapod.geometry.J);
#+end_src
- [ ] matlab/mat/conf_log.mat
- [ ] matlab/mat/conf_simscape.mat
- [ ] matlab/mat/conf_simulink.mat
- [ ] matlab/mat/nano_hexapod.mat
- [ ] matlab/mat/nass_disturbances.mat
- [X] matlab/mat/nass_model_conf_log.mat
- [X] matlab/mat/nass_model_conf_simscape.mat
- [X] matlab/mat/nass_model_controller.mat
- [X] matlab/mat/nass_model_disturbances.mat
- [X] matlab/mat/nass_model_references.mat
- [X] matlab/mat/nass_model_stages.mat
- [ ] matlab/mat/nass_references.mat
- [ ] matlab/mat/nass_stages.mat
** TODO [#B] Check if things are compatible to results of uniaxial model
** DONE [#C] Check if it would be interesting to show soft/stiff nano-hexapod plants
CLOSED: [2025-02-17 Mon 18:26]
- [ ] Would we see u-station dynamics with very stiff nano-hexapod?
- [ ] Would rotation be difficult to handle with soft nano-hexapod?
** DONE [#C] Why not plant with very stiff actuators?
CLOSED: [2025-02-17 Mon 18:26]
- [ ] Check if it is confirms that having very stiff actuators is bad
Not much better decoupling: 10Hz of bandwidth achievable, but may have worst sensitivity to disturbances
#+begin_src matlab
%% Identify the IFF plant dynamics using the Simscape model
% Initialize each Simscape model elements
initializeGround();
initializeGranite();
initializeTy();
initializeRy();
initializeRz();
initializeMicroHexapod();
initializeSimplifiedNanoHexapod();
% Initial Simscape Configuration
initializeSimscapeConfiguration('gravity', false);
initializeDisturbances('enable', false);
initializeLoggingConfiguration('log', 'none');
initializeController('type', 'open-loop');
initializeReferences();
% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Controller'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs [N]
io(io_i) = linio([mdl, '/NASS'], 3, 'openoutput', [], 'fn'); io_i = io_i + 1; % Force Sensors [N]
initializeSimplifiedNanoHexapod('actuator_k', 1e8, 'actuator_kp', 0, 'actuator_c', 1e2);
initializeSample('type', 'cylindrical', 'm', 1);
G_m1_iff_pz = linearize(mdl, io);
G_m1_iff_pz.InputName = {'f1', 'f2', 'f3', 'f4', 'f5', 'f6'};
G_m1_iff_pz.OutputName = {'fn1', 'fn2', 'fn3', 'fn4', 'fn5', 'fn6'};
#+end_src
#+begin_src matlab :exports none :results none
%% IFF Plant - Without parallel stiffness
f = logspace(0,4,1000);
figure;
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
for i = 1:5
for j = i+1:6
plot(f, abs(squeeze(freqresp(G_m1_iff_pz(i,j), f, 'Hz'))), 'color', [0, 0, 0, 0.2], ...
'HandleVisibility', 'off');
end
end
plot(f, abs(squeeze(freqresp(G_m1_iff_pz(1,1), f, 'Hz'))), 'color', colors(1,:), ...
'DisplayName', '$f_{ni}/f_i$ - $k_p = 0$')
for i = 2:6
plot(f, abs(squeeze(freqresp(G_m1_iff_pz(i,i), f, 'Hz'))), 'color', colors(1,:), ...
'HandleVisibility', 'off');
end
plot(f, abs(squeeze(freqresp(G_m1_iff_pz(1,2), f, 'Hz'))), 'color', [0, 0, 0, 0.2], ...
'DisplayName', '$f_{ni}/f_j$ - $k_p = 0$')
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [N/N]'); set(gca, 'XTickLabel',[]);
ylim([1e-4, 1e1]);
leg = legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 1);
leg.ItemTokenSize(1) = 15;
ax2 = nexttile;
hold on;
for i = 1:6
plot(f, 180/pi*unwrap(angle(squeeze(freqresp(G_m1_iff_pz(i,i), f, 'Hz')))), 'color', colors(1,:));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-20, 200]);
yticks([0:45:180]);
linkaxes([ax1,ax2],'x');
xlim([f(1), f(end)]);
#+end_src
#+begin_src matlab
%% Identify the IFF plant dynamics using the Simscape model
% Initialize each Simscape model elements
initializeGround();
initializeGranite();
initializeTy();
initializeRy();
initializeRz();
initializeMicroHexapod();
initializeSimplifiedNanoHexapod('actuator_k', 1e8, 'actuator_kp', 0, 'actuator_c', 1e2);
initializeSample('type', 'cylindrical', 'm', 1);
% Initial Simscape Configuration
initializeSimscapeConfiguration('gravity', false);
initializeDisturbances('enable', false);
initializeLoggingConfiguration('log', 'none');
initializeController('type', 'open-loop');
initializeReferences();
% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Controller'], 1, 'input'); io_i = io_i + 1; % Actuator Inputs [N]
io(io_i) = linio([mdl, '/Tracking Error'], 1, 'openoutput', [], 'EdL'); io_i = io_i + 1; % Strut errors [m]
%% Identify HAC Plant without using IFF
initializeSample('type', 'cylindrical', 'm', 1);
G_m1_pz = linearize(mdl, io);
G_m1_pz.InputName = {'f1', 'f2', 'f3', 'f4', 'f5', 'f6'};
G_m1_pz.OutputName = {'l1', 'l2', 'l3', 'l4', 'l5', 'l6'};
initializeSample('type', 'cylindrical', 'm', 25);
G_m25_pz = linearize(mdl, io);
G_m25_pz.InputName = {'f1', 'f2', 'f3', 'f4', 'f5', 'f6'};
G_m25_pz.OutputName = {'l1', 'l2', 'l3', 'l4', 'l5', 'l6'};
initializeSample('type', 'cylindrical', 'm', 50);
G_m50_pz = linearize(mdl, io);
G_m50_pz.InputName = {'f1', 'f2', 'f3', 'f4', 'f5', 'f6'};
G_m50_pz.OutputName = {'l1', 'l2', 'l3', 'l4', 'l5', 'l6'};
#+end_src
#+begin_src matlab :exports none :results none
figure;
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
plot(freqs, abs(squeeze(freqresp(G_m1_pz(1,1), freqs, 'Hz'))), 'color', colors(1,:), ...
'DisplayName', '$f_{ni}/f_i$ - 1kg')
plot(freqs, abs(squeeze(freqresp(G_m25_pz(1,1), freqs, 'Hz'))), 'color', colors(2,:), ...
'DisplayName', '$f_{ni}/f_i$ - 25kg')
plot(freqs, abs(squeeze(freqresp(G_m50_pz(1,1), freqs, 'Hz'))), 'color', colors(3,:), ...
'DisplayName', '$f_{ni}/f_i$ - 50kg')
for i = 1:5
for j = i+1:6
plot(freqs, abs(squeeze(freqresp(G_m1_pz(i,j), freqs, 'Hz'))), 'color', [colors(1,:), 0.2], ...
'HandleVisibility', 'off');
plot(freqs, abs(squeeze(freqresp(G_m25_pz(i,j), freqs, 'Hz'))), 'color', [colors(2,:), 0.2], ...
'HandleVisibility', 'off');
plot(freqs, abs(squeeze(freqresp(G_m50_pz(i,j), freqs, 'Hz'))), 'color', [colors(3,:), 0.2], ...
'HandleVisibility', 'off');
end
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
% ylim([1e-5, 1e1]);
leg = legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 1);
leg.ItemTokenSize(1) = 15;
ax2 = nexttile;
hold on;
for i = 1:6
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_m1_pz(i,i), freqs, 'Hz')))), 'color', colors(1,:));
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_m25_pz(i,i), freqs, 'Hz')))), 'color', colors(2,:));
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_m50_pz(i,i), freqs, 'Hz')))), 'color', colors(3,:));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-200, 20]);
yticks([-180:45:180]);
linkaxes([ax1,ax2],'x');
xlim([freqs(1), freqs(end)]);
#+end_src
Compare with Hexapod alone:
#+begin_src matlab
%% Identify the IFF plant dynamics using the Simscape model
% Initialize each Simscape model elements
initializeGround('type', 'rigid');
initializeGranite('type', 'rigid');
initializeTy('type', 'rigid');
initializeRy('type', 'rigid');
initializeRz('type', 'rigid');
initializeMicroHexapod('type', 'rigid');
initializeSimplifiedNanoHexapod('actuator_k', 1e8, 'actuator_kp', 0, 'actuator_c', 1e2);
initializeSample('type', 'cylindrical', 'm', 25);
% Initial Simscape Configuration
initializeSimscapeConfiguration('gravity', false);
initializeDisturbances('enable', false);
initializeLoggingConfiguration('log', 'none');
initializeController('type', 'open-loop');
initializeReferences();
% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Controller'], 1, 'input'); io_i = io_i + 1; % Actuator Inputs [N]
io(io_i) = linio([mdl, '/Tracking Error'], 1, 'openoutput', [], 'EdL'); io_i = io_i + 1; % Strut errors [m]
%% Identify HAC Plant without using IFF
G_m25_pz_rigid = linearize(mdl, io);
G_m25_pz_rigid.InputName = {'f1', 'f2', 'f3', 'f4', 'f5', 'f6'};
G_m25_pz_rigid.OutputName = {'l1', 'l2', 'l3', 'l4', 'l5', 'l6'};
#+end_src
#+begin_src matlab :exports none :results none
figure;
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
plot(freqs, abs(squeeze(freqresp(G_m25_pz_rigid(1,1), freqs, 'Hz'))), 'color', colors(1,:), ...
'DisplayName', '$f_{ni}/f_i$ - 25kg')
plot(freqs, abs(squeeze(freqresp(G_m25_pz(1,1), freqs, 'Hz'))), 'color', colors(2,:), ...
'DisplayName', '$f_{ni}/f_i$ - 25kg')
for i = 1:5
for j = i+1:6
plot(freqs, abs(squeeze(freqresp(G_m25_pz_rigid(i,j), freqs, 'Hz'))), 'color', [colors(1,:), 0.2], ...
'HandleVisibility', 'off');
plot(freqs, abs(squeeze(freqresp(G_m25_pz(i,j), freqs, 'Hz'))), 'color', [colors(2,:), 0.2], ...
'HandleVisibility', 'off');
end
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
% ylim([1e-5, 1e1]);
leg = legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 1);
leg.ItemTokenSize(1) = 15;
ax2 = nexttile;
hold on;
for i = 1:6
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_m25_pz_rigid(i,i), freqs, 'Hz')))), 'color', colors(1,:));
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_m25_pz(i,i), freqs, 'Hz')))), 'color', colors(2,:));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-200, 20]);
yticks([-180:45:180]);
linkaxes([ax1,ax2],'x');
xlim([freqs(1), freqs(end)]);
#+end_src
** DONE [#A] Add possibility to configure the nano-hexapod to be fully rigid
CLOSED: [2025-02-12 Wed 14:46]
- Use to compare TF without the NASS
** CANC [#C] What performance metric can we use? :@christophe:
CLOSED: [2024-11-12 Tue 09:22]
- State "CANC" from "QUES" [2024-11-12 Tue 09:22]
This can be nice to have a (scalar) performance metric that can be used for optimization.
In cite:hauge04_sensor_contr_space_based_six, a (scalar) performance metric representing the 6dof transmissibility is used.
** DONE [#C] Identify the sensibility to disturbances without the nano-hexapod and save the results
CLOSED: [2024-11-07 Thu 09:20]
This can then be used to compare with obtained performance with the nano-hexapod.
This should be done in the ustation report (A4).
* Introduction :ignore:
From last sections:
- Uniaxial: No stiff nano-hexapod (should also demonstrate that here)
- Rotating: No soft nano-hexapod, Decentralized IFF can be used robustly by adding parallel stiffness
- Micro-Station multi body model tuned from a modal analysis
- Multi-body model of a nano-hexapod that can be merged with the multi-body model of the micro-station
In this section:
- Take the model of the nano-hexapod described in previous section (stiffness 1um/N)
- Control kinematics: how the external metrology, the nano-hexapod metrology are used to control the sample's position (Section ref:sec:nass_kinematics)
- Apply decentralized IFF (Section ref:sec:nass_active_damping)
- Apply HAC-LAC (Section ref:sec:nass_hac)
- Check robustness to change of payload and to spindle rotation
- Simulation of experiments
- Conclusion of the conceptual phase, validation with simulations
#+name: fig:nass_simscape_model
#+caption: 3D view of the NASS multi-body model
#+attr_latex: :width 0.8\linewidth
[[file:figs/nass_simscape_model.jpg]]
* Control Kinematics
:PROPERTIES:
:HEADER-ARGS:matlab+: :tangle matlab/nass_1_kinematics.m
:END:
<>
** Introduction :ignore:
Figure ref:fig:nass_concept_schematic presents a schematic overview of the NASS.
This section focuses specifically on the components of the "Instrumentation and Real-Time Control" block.
#+name: fig:nass_concept_schematic
#+caption: Schematic of the Nano Active Stabilization System
[[file:figs/nass_concept_schematic.png]]
As established in the previous section on Stewart platforms, the proposed control strategy combines Decentralized Integral Force Feedback with a High Authority Controller performed in the frame of the struts.
For the Nano Active Stabilization System, computing the positioning errors in the frame of the struts involves three key steps.
First, the system computes the desired sample pose relative to a frame representing the point where the X-ray light is focused using micro-station kinematics, as detailed in Section ref:ssec:nass_ustation_kinematics.
Second, it measures the actual sample pose relative to the same fix frame, described in Section ref:ssec:nass_sample_pose_error.
Finally, it determines the sample pose error and maps these errors to the nano-hexapod struts, as explained in Section ref:ssec:nass_error_struts.
The complete control architecture is detailed in Section ref:ssec:nass_control_architecture.
** Matlab Init :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
<>
#+end_src
#+begin_src matlab :exports none :results silent :noweb yes
<>
#+end_src
#+begin_src matlab :tangle no :noweb yes
<>
#+end_src
#+begin_src matlab :eval no :noweb yes
<>
#+end_src
#+begin_src matlab :noweb yes
<>
#+end_src
#+begin_src matlab :noweb yes
<>
#+end_src
** Micro Station Kinematics
<>
The micro-station kinematics enables the computation of the desired sample pose from the reference signals of each micro-station stage.
These reference signals consist of the desired lateral position $r_{D_y}$, tilt angle $r_{R_y}$, and spindle angle $r_{R_z}$.
The micro-hexapod pose is defined by six parameters: three translations ($r_{D_{\mu x}}$, $r_{D_{\mu y}}$, $r_{D_{\mu z}}$) and three rotations ($r_{\theta_{\mu x}}$, $r_{\theta_{\mu y}}$, $r_{\theta_{\mu z}}$).
Using these reference signals, the desired sample position relative to the fixed frame is expressed through the homogeneous transformation matrix $\bm{T}_{\mu\text{-station}}$, as defined in equation eqref:eq:nass_sample_ref.
\begin{equation}\label{eq:nass_sample_ref}
\bm{T}_{\mu\text{-station}} = \bm{T}_{D_y} \cdot \bm{T}_{R_y} \cdot \bm{T}_{R_z} \cdot \bm{T}_{\mu\text{-hexapod}}
\end{equation}
\begin{equation}\label{eq:nass_ustation_matrices}
\begin{align}
\bm{T}_{D_y} &= \begin{bmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & r_{D_y} \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{bmatrix} \quad
\bm{T}_{\mu\text{-hexapod}} =
\left[ \begin{array}{ccc|c}
& & & r_{D_{\mu x}} \\
& \bm{R}_x(r_{\theta_{\mu x}}) \bm{R}_y(r_{\theta_{\mu y}}) \bm{R}_{z}(r_{\theta_{\mu z}}) & & r_{D_{\mu y}} \\
& & & r_{D_{\mu z}} \cr
\hline
0 & 0 & 0 & 1
\end{array} \right] \\
\bm{T}_{R_z} &= \begin{bmatrix}
\cos(r_{R_z}) & -\sin(r_{R_z}) & 0 & 0 \\
\sin(r_{R_z}) & \cos(r_{R_z}) & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{bmatrix} \quad
\bm{T}_{R_y} = \begin{bmatrix}
\cos(r_{R_y}) & 0 & \sin(r_{R_y}) & 0 \\
0 & 1 & 0 & 0 \\
-\sin(r_{R_y}) & 0 & \cos(r_{R_y}) & 0 \\
0 & 0 & 0 & 1
\end{bmatrix}
\end{align}
\end{equation}
** Computation of the sample's pose error
<>
The external metrology system measures the sample position relative to the fixed granite.
Due to the system's symmetry, this metrology provides measurements for five degrees of freedom: three translations ($D_x$, $D_y$, $D_z$) and two rotations ($R_x$, $R_y$).
The sixth degree of freedom ($R_z$) is still required to compute the errors in the frame of the nano-hexapod struts (i.e. to compute the nano-hexapod inverse kinematics).
This $R_z$ rotation is estimated by combining measurements from the spindle encoder and the nano-hexapod's internal metrology, which consists of relative motion sensors in each strut (note that the micro-hexapod is not used for $R_z$ rotation, and is therefore ignore for $R_z$ estimation).
The measured sample pose is represented by the homogeneous transformation matrix $\bm{T}_{\text{sample}}$, as shown in equation eqref:eq:nass_sample_pose.
\begin{equation}\label{eq:nass_sample_pose}
\bm{T}_{\text{sample}} =
\left[ \begin{array}{ccc|c}
& & & D_{x} \\
& \bm{R}_x(R_{x}) \bm{R}_y(R_{y}) \bm{R}_{z}(R_{z}) & & D_{y} \\
& & & D_{z} \cr
\hline
0 & 0 & 0 & 1
\end{array} \right]
\end{equation}
** Position error in the frame of the struts
<>
The homogeneous transformation formalism enables straightforward computation of the sample position error.
This computation involves the previously computed homogeneous $4 \times 4$ matrices: $\bm{T}_{\mu\text{-station}}$ representing the desired pose, and $\bm{T}_{\text{sample}}$ representing the measured pose.
Their combination yields $\bm{T}_{\text{error}}$, which expresses the position error of the sample in the frame of the rotating nano-hexapod, as shown in equation eqref:eq:nass_transformation_error.
\begin{equation}\label{eq:nass_transformation_error}
\bm{T}_{\text{error}} = \bm{T}_{\mu\text{-station}}^{-1} \cdot \bm{T}_{\text{sample}}
\end{equation}
The known structure of the homogeneous transformation matrix facilitates efficient real-time computation of the inverse.
From $\bm{T}_{\text{error}}$, the position and orientation errors $\bm{\epsilon}_{\mathcal{X}} = [\epsilon_{D_x},\ \epsilon_{D_y},\ \epsilon_{D_z},\ \epsilon_{R_x},\ \epsilon_{R_y},\ \epsilon_{R_z}]$ of the sample are extracted using equation eqref:eq:nass_compute_errors:
\begin{equation}\label{eq:nass_compute_errors}
\begin{align}
\epsilon_{D_x} & = \bm{T}_{\text{error}}(1,4) \\
\epsilon_{D_y} & = \bm{T}_{\text{error}}(2,4) \\
\epsilon_{D_z} & = \bm{T}_{\text{error}}(3,4) \\
\epsilon_{R_y} & = \text{atan2}(\bm{T}_{\text{error}}(1,3), \sqrt{\bm{T}_{\text{error}}(1,1)^2 + \bm{T}_{\text{error}}(1,2)^2}) \\
\epsilon_{R_x} & = \text{atan2}(-\bm{T}_{\text{error}}(2,3)/\cos(\epsilon_{R_y}), \bm{T}_{\text{error}}(3,3)/\cos(\epsilon_{R_y})) \\
\epsilon_{R_z} & = \text{atan2}(-\bm{T}_{\text{error}}(1,2)/\cos(\epsilon_{R_y}), \bm{T}_{\text{error}}(1,1)/\cos(\epsilon_{R_y})) \\
\end{align}
\end{equation}
Finally, these errors are mapped to the strut space through the nano-hexapod Jacobian matrix eqref:eq:nass_inverse_kinematics.
\begin{equation}\label{eq:nass_inverse_kinematics}
\bm{\epsilon}_{\mathcal{L}} = \bm{J} \cdot \bm{\epsilon}_{\mathcal{X}}
\end{equation}
** Control Architecture - Summary
<>
The complete control architecture is summarized in Figure ref:fig:nass_control_architecture.
The sample pose is measured using external metrology for five degrees of freedom, while the sixth degree of freedom (Rz) is estimated by combining measurements from the nano-hexapod encoders and spindle encoder.
The sample reference pose is determined by the reference signals of the translation stage, tilt stage, spindle, and micro-hexapod.
Position error computation follows a two-step process: first, homogeneous transformation matrices are used to determine the error in the nano-hexapod frame, then the Jacobian matrix $\bm{J}$ maps these errors to individual strut coordinates.
For control purposes, force sensors mounted on each strut are used in a decentralized way for active damping, as detailed in Section ref:sec:nass_active_damping.
Then, the high authority controller uses the computed errors in the frame of the struts to provides real-time stabilization of the sample position (Section ref:sec:nass_hac).
#+begin_src latex :file nass_control_architecture.pdf
\begin{tikzpicture}
% Blocs
\node[block={2.0cm}{1.0cm}, fill=colorblue!20!white] (metrology) {Metrology};
\node[block={2.0cm}{2.0cm}, below=0.1 of metrology, align=center, fill=colorblue!20!white] (nhexa) {Nano\\Hexapod};
\node[block={3.0cm}{1.5cm}, below=0.1 of nhexa, align=center, fill=colorblue!20!white] (ustation) {Micro\\Station};
\coordinate[] (inputf) at ($(nhexa.south west)!0.5!(nhexa.north west)$);
\coordinate[] (outputfn) at ($(nhexa.south east)!0.3!(nhexa.north east)$);
\coordinate[] (outputde) at ($(nhexa.south east)!0.7!(nhexa.north east)$);
\coordinate[] (outputDy) at ($(ustation.south east)!0.1!(ustation.north east)$);
\coordinate[] (outputRy) at ($(ustation.south east)!0.5!(ustation.north east)$);
\coordinate[] (outputRz) at ($(ustation.south east)!0.9!(ustation.north east)$);
\node[block={1.0cm}{1.0cm}, right=0.5 of outputde, fill=colorred!20!white] (Rz_kinematics) {$\bm{J}_{R_z}^{-1}$};
\node[block={2.0cm}{2.0cm}, right=2.2 of ustation, align=center, fill=colorred!20!white] (ustation_kinematics) {Compute\\Reference\\Position};
\node[block={2.0cm}{2.0cm}, right=0.8 of ustation_kinematics, align=center, fill=colorred!20!white] (compute_error) {Compute\\Error\\Position};
\node[block={2.0cm}{2.0cm}, above=0.8 of compute_error, align=center, fill=colorred!20!white] (compute_pos) {Compute\\Sample\\Position};
\node[block={1.0cm}{1.0cm}, right=0.8 of compute_error, fill=colorred!20!white] (hexa_jacobian) {$\bm{J}$};
\coordinate[] (inputMetrology) at ($(compute_error.north east)!0.3!(compute_error.north west)$);
\coordinate[] (inputRz) at ($(compute_error.north east)!0.7!(compute_error.north west)$);
\node[addb={+}{}{}{}{}, right=0.4 of Rz_kinematics, fill=colorred!20!white] (addRz) {};
\draw[->] (Rz_kinematics.east) -- (addRz.west);
\draw[->] (outputRz-|addRz)node[branch]{} -- (addRz.south);
\draw[->] (outputDy) node[above right]{$r_{D_y}$} -- (outputDy-|ustation_kinematics.west);
\draw[->] (outputRy) node[above right]{$r_{R_y}$} -- (outputRy-|ustation_kinematics.west);
\draw[->] (outputRz) node[above right]{$r_{R_z}$} -- (outputRz-|ustation_kinematics.west);
\draw[->] (metrology.east)node[above right]{$[D_x,\,D_y,\,D_z,\,R_x,\,R_y]$} -- (compute_pos.west|-metrology);
\draw[->] (addRz.east)node[above right]{$R_z$} -- (compute_pos.west|-addRz);
\draw[->] (compute_pos.south)node -- (compute_error.north)node[above right]{$\bm{y}_{\mathcal{X}}$};
\draw[->] (outputde) -- (Rz_kinematics.west) node[above left]{$\bm{\mathcal{L}}$};
\draw[->] (ustation_kinematics.east) -- (compute_error.west) node[above left]{$\bm{r}_{\mathcal{X}}$};
\draw[->] (compute_error.east) -- (hexa_jacobian.west) node[above left]{$\bm{\epsilon\mathcal{X}}$};
\draw[->] (hexa_jacobian.east) -- ++(1.8, 0) node[above left]{$\bm{\epsilon\mathcal{L}}$};
\draw[->] (outputfn) -- ($(outputfn-|hexa_jacobian.east) + (1.0, 0)$)coordinate(fn) node[above left]{$\bm{f}_n$};
\begin{scope}[on background layer]
\node[fit={(metrology.north-|ustation.west) (hexa_jacobian.east|-compute_error.south)}, fill=black!10!white, draw, dashed, inner sep=4pt] (plant) {};
\node[anchor={north east}] at (plant.north east){$\text{Plant}$};
\end{scope}
\node[block, above=0.2 of plant, fill=coloryellow!20!white] (Kiff) {$\bm{K}_{\text{IFF}}$};
\draw[->] ($(fn)-(0.6,0)$)node[branch]{} |- (Kiff.east);
\node[addb={+}{}{}{}{}, left=0.8 of inputf] (addf) {};
\draw[->] (Kiff.west) -| (addf.north);
\begin{scope}[on background layer]
\node[fit={(plant.south-|fn) (addf.west|-Kiff.north)}, fill=black!20!white, draw, dashed, inner sep=4pt] (damped_plant) {};
\node[anchor={north east}] at (damped_plant.north east){$\text{Damped Plant}$};
\end{scope}
\begin{scope}[on background layer]
\node[fit={(metrology.north-|ustation.west) (hexa_jacobian.east|-compute_error.south)}, fill=black!10!white, draw, dashed, inner sep=4pt] (plant) {};
\node[anchor={north east}] at (plant.north east){$\text{Plant}$};
\end{scope}
\node[block, left=0.8 of addf, fill=colorgreen!20!white] (Khac) {$\bm{K}_{\text{HAC}}$};
\draw[->] ($(hexa_jacobian.east)+(1.4,0)$)node[branch]{} |- ($(Khac.west)+(-0.4, -3.4)$) |- (Khac.west);
\draw[->] (Khac.east) -- node[midway, above]{$\bm{f}^{\prime}$} (addf.west);
\draw[->] (addf.east) -- (inputf) node[above left]{$\bm{f}$};
\end{tikzpicture}
#+end_src
#+name: fig:nass_control_architecture
#+caption: The physical systems are shown in blue, the control kinematics in red, the decentralized Integral Force Feedback in yellow and the centralized High Authority Controller in green.
#+attr_latex: :width \linewidth
#+RESULTS:
[[file:figs/nass_control_architecture.png]]
* Decentralized Active Damping
:PROPERTIES:
:HEADER-ARGS:matlab+: :tangle matlab/nass_2_active_damping.m
:END:
<>
** Introduction :ignore:
Building upon the uniaxial model study, this section implements decentralized Integral Force Feedback (IFF) as the first component of the HAC-LAC strategy.
Springs in parallel to the force sensors are used to guarantee the control robustness as was found using the 3DoF rotating model.
The objective here is to design a decentralized IFF controller that provides good damping of the nano-hexapod modes across payload masses ranging from $1$ to $50\,\text{kg}$ and rotational velocity up to $360\,\text{deg/s}$.
Used payloads have a cylindrical shape with 250 mm height and with masses of 1 kg, 25 kg, and 50 kg.
** Matlab Init :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
<>
#+end_src
#+begin_src matlab :exports none :results silent :noweb yes
<>
#+end_src
#+begin_src matlab :tangle no :noweb yes
<>
#+end_src
#+begin_src matlab :eval no :noweb yes
<>
#+end_src
#+begin_src matlab :noweb yes
<>
#+end_src
#+begin_src matlab :noweb yes
<>
#+end_src
** IFF Plant
<>
Transfer functions from actuator forces $f_i$ to force sensor measurements $f_{mi}$ are computed using the multi-body model.
Figure ref:fig:nass_iff_plant_effect_kp examines how parallel stiffness affects the plant dynamics, with identification performed at maximum spindle velocity $\Omega_z = 360\,\text{deg/s}$ and with a payload mass of 25 kg.
Without parallel stiffness (Figure ref:fig:nass_iff_plant_no_kp), the dynamics exhibits non-minimum phase zeros at low frequency, confirming predictions from the three-degree-of-freedom rotating model.
Adding parallel stiffness (Figure ref:fig:nass_iff_plant_kp) transforms these into minimum phase complex conjugate zeros, enabling unconditionally stable decentralized IFF implementation.
Though both cases show significant coupling around resonances, stability is guaranteed by the collocated arrangement of actuators and sensors [[cite:&preumont08_trans_zeros_struc_contr_with]].
#+begin_src matlab
%% Identify the IFF plant dynamics using the Simscape model
% Initialize each Simscape model elements
initializeGround();
initializeGranite();
initializeTy();
initializeRy();
initializeRz();
initializeMicroHexapod();
initializeSimplifiedNanoHexapod();
% Initial Simscape Configuration
initializeSimscapeConfiguration('gravity', false);
initializeDisturbances('enable', false);
initializeLoggingConfiguration('log', 'none');
initializeController('type', 'open-loop');
initializeReferences();
% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Controller'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs [N]
io(io_i) = linio([mdl, '/NASS'], 3, 'openoutput', [], 'fn'); io_i = io_i + 1; % Force Sensors [N]
%% Identify for multi payload masses (no rotation)
initializeReferences(); % No Spindle Rotation
% 1kg Sample
initializeSample('type', 'cylindrical', 'm', 1);
G_iff_m1 = linearize(mdl, io);
G_iff_m1.InputName = {'f1', 'f2', 'f3', 'f4', 'f5', 'f6'};
G_iff_m1.OutputName = {'fn1', 'fn2', 'fn3', 'fn4', 'fn5', 'fn6'};
% 25kg Sample
initializeSample('type', 'cylindrical', 'm', 25);
G_iff_m25 = linearize(mdl, io);
G_iff_m25.InputName = {'f1', 'f2', 'f3', 'f4', 'f5', 'f6'};
G_iff_m25.OutputName = {'fn1', 'fn2', 'fn3', 'fn4', 'fn5', 'fn6'};
% 50kg Sample
initializeSample('type', 'cylindrical', 'm', 50);
G_iff_m50 = linearize(mdl, io);
G_iff_m50.InputName = {'f1', 'f2', 'f3', 'f4', 'f5', 'f6'};
G_iff_m50.OutputName = {'fn1', 'fn2', 'fn3', 'fn4', 'fn5', 'fn6'};
%% Effect of Rotation
initializeReferences(...
'Rz_type', 'rotating', ...
'Rz_period', 1); % 360 deg/s
initializeSample('type', 'cylindrical', 'm', 25);
G_iff_m25_Rz = linearize(mdl, io, 0.1);
G_iff_m25_Rz.InputName = {'f1', 'f2', 'f3', 'f4', 'f5', 'f6'};
G_iff_m25_Rz.OutputName = {'fn1', 'fn2', 'fn3', 'fn4', 'fn5', 'fn6'};
%% Effect of Rotation - No added parallel stiffness
initializeSimplifiedNanoHexapod('actuator_kp', 0);
initializeReferences(...
'Rz_type', 'rotating', ...
'Rz_period', 1); % 360 deg/s
initializeSample('type', 'cylindrical', 'm', 25);
G_iff_m25_Rz_no_kp = linearize(mdl, io, 0.1);
G_iff_m25_Rz_no_kp.InputName = {'f1', 'f2', 'f3', 'f4', 'f5', 'f6'};
G_iff_m25_Rz_no_kp.OutputName = {'fn1', 'fn2', 'fn3', 'fn4', 'fn5', 'fn6'};
#+end_src
#+begin_src matlab :exports none :results none
%% IFF Plant - Without parallel stiffness
f = logspace(-1,3,1000);
figure;
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
for i = 1:5
for j = i+1:6
plot(f, abs(squeeze(freqresp(G_iff_m25_Rz_no_kp(i,j), f, 'Hz'))), 'color', [0, 0, 0, 0.2], ...
'HandleVisibility', 'off');
end
end
plot(f, abs(squeeze(freqresp(G_iff_m25_Rz_no_kp(1,1), f, 'Hz'))), 'color', colors(1,:), ...
'DisplayName', '$f_{ni}/f_i$ - $k_p = 0$')
for i = 2:6
plot(f, abs(squeeze(freqresp(G_iff_m25_Rz_no_kp(i,i), f, 'Hz'))), 'color', colors(1,:), ...
'HandleVisibility', 'off');
end
plot(f, abs(squeeze(freqresp(G_iff_m25_Rz_no_kp(1,2), f, 'Hz'))), 'color', [0, 0, 0, 0.2], ...
'DisplayName', '$f_{ni}/f_j$ - $k_p = 0$')
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [N/N]'); set(gca, 'XTickLabel',[]);
ylim([1e-4, 1e1]);
leg = legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 1);
leg.ItemTokenSize(1) = 15;
ax2 = nexttile;
hold on;
for i = 1:6
plot(f, 180/pi*unwrap(angle(squeeze(freqresp(G_iff_m25_Rz_no_kp(i,i), f, 'Hz')))), 'color', colors(1,:));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-20, 200]);
yticks([0:45:180]);
linkaxes([ax1,ax2],'x');
xlim([f(1), f(end)]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file none
exportFig('figs/nass_iff_plant_no_kp.pdf', 'width', 'half', 'height', 600);
#+end_src
#+begin_src matlab :exports none :results none
%% IFF Plant - With added parallel stiffness
figure;
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
for i = 1:5
for j = i+1:6
plot(f, abs(squeeze(freqresp(G_iff_m25_Rz(i,j), f, 'Hz'))), 'color', [0, 0, 0, 0.2], ...
'HandleVisibility', 'off');
end
end
plot(f, abs(squeeze(freqresp(G_iff_m25_Rz(1,1), f, 'Hz'))), 'color', colors(1,:), ...
'DisplayName', '$f_{ni}/f_i$ - $k_p = 50N/mm$')
for i = 2:6
plot(f, abs(squeeze(freqresp(G_iff_m25_Rz(i,i), f, 'Hz'))), 'color', colors(1,:), ...
'HandleVisibility', 'off');
end
plot(f, abs(squeeze(freqresp(G_iff_m25_Rz(1,2), f, 'Hz'))), 'color', [0, 0, 0, 0.2], ...
'DisplayName', '$f_{ni}/f_j$ - $k_p = 50N/mm$')
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [N/N]'); set(gca, 'XTickLabel',[]);
ylim([1e-4, 1e1]);
leg = legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 1);
leg.ItemTokenSize(1) = 15;
ax2 = nexttile;
hold on;
for i = 1:6
plot(f, 180/pi*angle(squeeze(freqresp(G_iff_m25_Rz(i,i), f, 'Hz'))), 'color', colors(1,:));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-20, 200]);
yticks([0:45:180]);
linkaxes([ax1,ax2],'x');
xlim([f(1), f(end)]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file none
exportFig('figs/nass_iff_plant_kp.pdf', 'width', 'half', 'height', 600);
#+end_src
#+name: fig:nass_iff_plant_effect_kp
#+caption: Effect of stiffness parallel to the force sensor on the IFF plant with $\Omega_z = 360\,\text{deg/s}$ and payload mass of 25kg. The dynamics without parallel stiffness has non-minimum phase zeros at low frequency (\subref{fig:nass_iff_plant_no_kp}). The added parallel stiffness transforms the non-minimum phase zeros to complex conjugate zeros (\subref{fig:nass_iff_plant_kp})
#+attr_latex: :options [htbp]
#+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:nass_iff_plant_no_kp}without parallel stiffness}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
#+attr_latex: :width 0.95\linewidth
[[file:figs/nass_iff_plant_no_kp.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:nass_iff_plant_kp}with parallel stiffness}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
#+attr_latex: :width 0.95\linewidth
[[file:figs/nass_iff_plant_kp.png]]
#+end_subfigure
#+end_figure
The effect of rotation, shown in Figure ref:fig:nass_iff_plant_effect_rotation, is negligible as the actuator stiffness ($k_a = 1\,N/\mu m$) is large compared to the negative stiffness induced by gyroscopic effects (estimated from the 3DoF rotating model).
Figure ref:fig:nass_iff_plant_effect_payload illustrate the effect of payload mass on the plant dynamics.
While the poles and zeros are shifting with payload mass, the alternating pattern of poles and zeros is maintained, ensuring that the phase remains bounded between 0 and 180 degrees, and thus good robustness properties.
#+begin_src matlab :exports none :results none
%% Effect of spindle's rotation on the IFF Plant
figure;
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
for i = 1:5
for j = i+1:6
plot(freqs, abs(squeeze(freqresp(G_iff_m25(i,j), freqs, 'Hz'))), 'color', [colors(1,:), 0.1], ...
'HandleVisibility', 'off');
plot(freqs, abs(squeeze(freqresp(G_iff_m25_Rz(i,j), freqs, 'Hz'))), 'color', [colors(2,:), 0.1], ...
'HandleVisibility', 'off');
end
end
plot(freqs, abs(squeeze(freqresp(G_iff_m25(1,1), freqs, 'Hz'))), 'color', colors(1,:), ...
'DisplayName', '$f_{ni}/f_i$ - $\Omega_z = 0$ deg/s')
plot(freqs, abs(squeeze(freqresp(G_iff_m25_Rz(1,1), freqs, 'Hz'))), 'color', colors(2,:), ...
'DisplayName', '$f_{ni}/f_i$ - $\Omega_z = 360$ deg/s')
for i = 2:6
plot(freqs, abs(squeeze(freqresp(G_iff_m25(i,i), freqs, 'Hz'))), 'color', colors(1,:), ...
'HandleVisibility', 'off');
plot(freqs, abs(squeeze(freqresp(G_iff_m25_Rz(i,i), freqs, 'Hz'))), 'color', colors(2,:), ...
'HandleVisibility', 'off');
end
% plot(freqs, abs(squeeze(freqresp(G_iff_m25_Rz(1,2), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ...
% 'DisplayName', '$f_{ni}/f_j$')
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [N/N]'); set(gca, 'XTickLabel',[]);
ylim([1e-4, 1e2]);
leg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 1);
leg.ItemTokenSize(1) = 15;
ax2 = nexttile;
hold on;
for i = 1:6
plot(freqs, 180/pi*angle(squeeze(freqresp(G_iff_m25(i,i), freqs, 'Hz'))), 'color', colors(1,:));
plot(freqs, 180/pi*angle(squeeze(freqresp(G_iff_m25_Rz(i,i), freqs, 'Hz'))), 'color', colors(2,:));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-20, 200]);
yticks([0:45:180]);
linkaxes([ax1,ax2],'x');
xlim([freqs(1), freqs(end)]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file none
exportFig('figs/nass_iff_plant_effect_rotation.pdf', 'width', 'half', 'height', 600);
#+end_src
#+begin_src matlab :exports none :results none
%% Effect of the payload's mass on the IFF Plant
figure;
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
plot(freqs, abs(squeeze(freqresp(G_iff_m1(1,1), freqs, 'Hz'))), 'color', [colors(1,:), 0.5], ...
'DisplayName', '$f_{ni}/f_i$ - 1kg')
for i = 2:6
plot(freqs, abs(squeeze(freqresp(G_iff_m1(i,i), freqs, 'Hz'))), 'color', [colors(1,:), 0.5], ...
'HandleVisibility', 'off');
end
plot(freqs, abs(squeeze(freqresp(G_iff_m25(1,1), freqs, 'Hz'))), 'color', [colors(2,:), 0.5], ...
'DisplayName', '$f_{ni}/f_i$ - 25kg')
for i = 2:6
plot(freqs, abs(squeeze(freqresp(G_iff_m25(i,i), freqs, 'Hz'))), 'color', [colors(2,:), 0.5], ...
'HandleVisibility', 'off');
end
plot(freqs, abs(squeeze(freqresp(G_iff_m50(1,1), freqs, 'Hz'))), 'color', [colors(3,:), 0.5], ...
'DisplayName', '$f_{ni}/f_i$ - 50kg')
for i = 2:6
plot(freqs, abs(squeeze(freqresp(G_iff_m50(i,i), freqs, 'Hz'))), 'color', [colors(3,:), 0.5], ...
'HandleVisibility', 'off');
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [N/N]'); set(gca, 'XTickLabel',[]);
ylim([1e-4, 1e2]);
leg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 1);
leg.ItemTokenSize(1) = 15;
ax2 = nexttile;
hold on;
for i = 1:6
plot(freqs, 180/pi*angle(squeeze(freqresp(G_iff_m1(i,i), freqs, 'Hz'))), 'color', [colors(1,:), 0.5]);
end
for i = 1:6
plot(freqs, 180/pi*angle(squeeze(freqresp(G_iff_m25(i,i), freqs, 'Hz'))), 'color', [colors(2,:), 0.5]);
end
for i = 1:6
plot(freqs, 180/pi*angle(squeeze(freqresp(G_iff_m50(i,i), freqs, 'Hz'))), 'color', [colors(3,:), 0.5]);
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-20, 200]);
yticks([0:45:180]);
linkaxes([ax1,ax2],'x');
xlim([freqs(1), freqs(end)]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file none
exportFig('figs/nass_iff_plant_effect_payload.pdf', 'width', 'half', 'height', 600);
#+end_src
#+name: fig:nass_iff_plant_effect_rotation_payload
#+caption: Effect of the Spindle's rotational velocity on the IFF plant (\subref{fig:nass_iff_plant_effect_rotation}) and effect of the payload's mass on the IFF plant (\subref{fig:nass_iff_plant_effect_payload})
#+attr_latex: :options [htbp]
#+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:nass_iff_plant_effect_rotation}Effect of Spindle rotation}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
#+attr_latex: :width 0.95\linewidth
[[file:figs/nass_iff_plant_effect_rotation.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:nass_iff_plant_effect_payload}Effect of payload mass}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
#+attr_latex: :width 0.95\linewidth
[[file:figs/nass_iff_plant_effect_payload.png]]
#+end_subfigure
#+end_figure
** Controller Design
<>
Previous analysis using the 3DoF rotating model showed that decentralized Integral Force Feedback (IFF) with pure integrators is unstable due to gyroscopic effects caused by spindle rotation.
This finding is also confirmed with the multi-body model of the NASS: the system is unstable when using pure integrators and without parallel stiffness.
This instability can be mitigated by introducing sufficient stiffness in parallel with the force sensors.
However, as illustrated in Figure ref:fig:nass_iff_plant_kp, adding parallel stiffness increases the low frequency gain.
If using pure integrators, this would results in high loop gain at low frequencies, adversely affecting the damped plant dynamics, which is undesirable.
To resolve this issue, a second-order high-pass filter is introduced to limit the low frequency gain, as shown in Equation eqref:eq:nass_kiff.
\begin{equation}\label{eq:nass_kiff}
\bm{K}_{\text{IFF}}(s) = g \cdot \begin{bmatrix}
K_{\text{IFF}}(s) & & 0 \\
& \ddots & \\
0 & & K_{\text{IFF}}(s)
\end{bmatrix}, \quad K_{\text{IFF}}(s) = \frac{1}{s} \cdot \frac{\frac{s^2}{\omega_z^2}}{\frac{s^2}{\omega_z^2} + 2 \xi_z \frac{s}{\omega_z} + 1}
\end{equation}
The cut-off frequency of the second-order high-pass filter is tuned to be below the frequency of the complex conjugate zero for the highest mass, which is at $5\,\text{Hz}$.
The overall gain is then increased to have large loop gain around resonances to be damped, as illustrated in Figure ref:fig:nass_iff_loop_gain.
#+begin_src matlab
%% Verify that parallel stiffness permits to have a stable plant
Kiff_pure_int = -200/s*eye(6);
isstable(feedback(G_iff_m25_Rz, Kiff_pure_int, 1))
isstable(feedback(G_iff_m25_Rz_no_kp, Kiff_pure_int, 1))
#+end_src
#+begin_src matlab
%% IFF Controller Design
% Second order high pass filter
wz = 2*pi*2;
xiz = 0.7;
Ghpf = (s^2/wz^2)/(s^2/wz^2 + 2*xiz*s/wz + 1);
Kiff = -200 * ... % Gain
1/(0.01*2*pi + s) * ... % LPF: provides integral action
Ghpf * ... % 2nd order HPF (limit low frequency gain)
eye(6); % Diagonal 6x6 controller (i.e. decentralized)
Kiff.InputName = {'fm1', 'fm2', 'fm3', 'fm4', 'fm5', 'fm6'};
Kiff.OutputName = {'f1', 'f2', 'f3', 'f4', 'f5', 'f6'};
#+end_src
#+begin_src matlab :exports none :tangle no
% The designed IFF controller is saved
save('./matlab/mat/nass_K_iff.mat', 'Kiff');
#+end_src
#+begin_src matlab :eval no
% The designed IFF controller is saved
save('./mat/nass_K_iff.mat', 'Kiff');
#+end_src
#+begin_src matlab :exports none :results none
%% Loop gain for the decentralized IFF
figure;
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
plot(freqs, abs(squeeze(freqresp(Kiff(1,1)*G_iff_m1(1,1), freqs, 'Hz'))), 'color', [colors(1,:), 0.5], ...
'DisplayName', '1kg')
for i = 2:6
plot(freqs, abs(squeeze(freqresp(Kiff(1,1)*G_iff_m1(i,i), freqs, 'Hz'))), 'color', [colors(1,:), 0.5], ...
'HandleVisibility', 'off');
end
plot(freqs, abs(squeeze(freqresp(Kiff(1,1)*G_iff_m25(1,1), freqs, 'Hz'))), 'color', [colors(2,:), 0.5], ...
'DisplayName', '25kg')
for i = 2:6
plot(freqs, abs(squeeze(freqresp(Kiff(1,1)*G_iff_m25(i,i), freqs, 'Hz'))), 'color', [colors(2,:), 0.5], ...
'HandleVisibility', 'off');
end
plot(freqs, abs(squeeze(freqresp(Kiff(1,1)*G_iff_m50(1,1), freqs, 'Hz'))), 'color', [colors(3,:), 0.5], ...
'DisplayName', '50kg')
for i = 2:6
plot(freqs, abs(squeeze(freqresp(Kiff(1,1)*G_iff_m50(i,i), freqs, 'Hz'))), 'color', [colors(3,:), 0.5], ...
'HandleVisibility', 'off');
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Loop Gain'); set(gca, 'XTickLabel',[]);
ylim([1e-4, 1e2]);
leg = legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
leg.ItemTokenSize(1) = 15;
ax2 = nexttile;
hold on;
for i = 1:6
plot(freqs, 180/pi*angle(squeeze(freqresp(-Kiff(1,1)*G_iff_m1(i,i), freqs, 'Hz'))), 'color', [colors(1,:), 0.5]);
end
for i = 1:6
plot(freqs, 180/pi*angle(squeeze(freqresp(-Kiff(1,1)*G_iff_m25(i,i), freqs, 'Hz'))), 'color', [colors(2,:), 0.5]);
end
for i = 1:6
plot(freqs, 180/pi*angle(squeeze(freqresp(-Kiff(1,1)*G_iff_m50(i,i), freqs, 'Hz'))), 'color', [colors(3,:), 0.5]);
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-110, 200]);
yticks([-180, -90, 0, 90, 180]);
linkaxes([ax1,ax2],'x');
xlim([freqs(1), freqs(end)]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/nass_iff_loop_gain.pdf', 'width', 'wide', 'height', 'normal');
#+end_src
#+name: fig:nass_iff_loop_gain
#+caption: Loop gain for the decentralized IFF: $K_{\text{IFF}}(s) \cdot \frac{f_{mi}}{f_i}(s)$
#+RESULTS:
[[file:figs/nass_iff_loop_gain.png]]
To verify stability, root loci for the three payload configurations are computed and shown in Figure ref:fig:nass_iff_root_locus.
The results demonstrate that the closed-loop poles remain within the left-half plane, indicating the robust stability properties of the applied decentralized IFF.
#+begin_src matlab :exports none :results none
%% Root Locus for the Decentralized IFF controller - 1kg Payload
figure;
gains = logspace(-2, 1, 200);
figure;
tiledlayout(1, 1, 'TileSpacing', 'compact', 'Padding', 'None');
nexttile();
hold on;
plot(real(pole(G_iff_m1)), imag(pole(G_iff_m1)), 'x', 'color', colors(1,:), ...
'DisplayName', '$g = 0$');
plot(real(tzero(G_iff_m1)), imag(tzero(G_iff_m1)), 'o', 'color', colors(1,:), ...
'HandleVisibility', 'off');
for g = gains
clpoles = pole(feedback(G_iff_m1, g*Kiff, +1));
plot(real(clpoles), imag(clpoles), '.', 'color', colors(1,:), ...
'HandleVisibility', 'off');
end
% Optimal gain
clpoles = pole(feedback(G_iff_m1, Kiff, +1));
plot(real(clpoles), imag(clpoles), 'kx', ...
'DisplayName', '$g_{opt}$');
xline(0);
yline(0);
hold off;
axis equal;
xlim([-900, 100]); ylim([-100, 900]);
xticks([-900:100:0]);
yticks([0:100:900]);
set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]);
xlabel('Real part'); ylabel('Imaginary part');
#+end_src
#+begin_src matlab :tangle no :exports results :results file none
exportFig('figs/nass_iff_root_locus_1kg.pdf', 'width', 'third', 'height', 'normal');
#+end_src
#+begin_src matlab :exports none :results none
%% Root Locus for the Decentralized IFF controller - 25kg Payload
gains = logspace(-2, 1, 200);
figure;
tiledlayout(1, 1, 'TileSpacing', 'compact', 'Padding', 'None');
nexttile();
hold on;
plot(real(pole(G_iff_m25)), imag(pole(G_iff_m25)), 'x', 'color', colors(2,:), ...
'DisplayName', '$g = 0$');
plot(real(tzero(G_iff_m25)), imag(tzero(G_iff_m25)), 'o', 'color', colors(2,:), ...
'HandleVisibility', 'off');
for g = gains
clpoles = pole(feedback(G_iff_m25, g*Kiff, +1));
plot(real(clpoles), imag(clpoles), '.', 'color', colors(2,:), ...
'HandleVisibility', 'off');
end
% Optimal gain
clpoles = pole(feedback(G_iff_m25, Kiff, +1));
plot(real(clpoles), imag(clpoles), 'kx', ...
'DisplayName', '$g_{opt}$');
xline(0);
yline(0);
hold off;
axis equal;
xlim([-900, 100]); ylim([-100, 900]);
xticks([-900:100:0]);
yticks([0:100:900]);
set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]);
xlabel('Real part'); ylabel('Imaginary part');
#+end_src
#+begin_src matlab :tangle no :exports results :results file none
exportFig('figs/nass_iff_root_locus_25kg.pdf', 'width', 'third', 'height', 'normal');
#+end_src
#+begin_src matlab :exports none :results none
%% Root Locus for the Decentralized IFF controller - 50kg Payload
gains = logspace(-2, 1, 200);
figure;
tiledlayout(1, 1, 'TileSpacing', 'compact', 'Padding', 'None');
nexttile();
hold on;
plot(real(pole(G_iff_m50)), imag(pole(G_iff_m50)), 'x', 'color', colors(3,:), ...
'DisplayName', '$g = 0$');
plot(real(tzero(G_iff_m50)), imag(tzero(G_iff_m50)), 'o', 'color', colors(3,:), ...
'HandleVisibility', 'off');
for g = gains
clpoles = pole(feedback(G_iff_m50, g*Kiff, +1));
plot(real(clpoles), imag(clpoles), '.', 'color', colors(3,:), ...
'HandleVisibility', 'off');
end
% Optimal gain
clpoles = pole(feedback(G_iff_m50, Kiff, +1));
plot(real(clpoles), imag(clpoles), 'kx', ...
'DisplayName', '$g_{opt}$');
xline(0);
yline(0);
hold off;
axis equal;
xlim([-900, 100]); ylim([-100, 900]);
xticks([-900:100:0]);
yticks([0:100:900]);
set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]);
xlabel('Real part'); ylabel('Imaginary part');
#+end_src
#+begin_src matlab :tangle no :exports results :results file none
exportFig('figs/nass_iff_root_locus_50kg.pdf', 'width', 'third', 'height', 'normal');
#+end_src
#+name: fig:nass_iff_root_locus
#+caption: Root Loci for Decentralized IFF for three payload masses. Closed-loop poles are shown by the black crosses.
#+attr_latex: :options [htbp]
#+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:nass_iff_root_locus_1kg} $1\,\text{kg}$}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
#+attr_latex: :width 0.9\linewidth
[[file:figs/nass_iff_root_locus_1kg.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:nass_iff_root_locus_25kg} $25\,\text{kg}$}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
#+attr_latex: :width 0.9\linewidth
[[file:figs/nass_iff_root_locus_25kg.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:nass_iff_root_locus_50kg} $50\,\text{kg}$}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
#+attr_latex: :width 0.9\linewidth
[[file:figs/nass_iff_root_locus_50kg.png]]
#+end_subfigure
#+end_figure
* Centralized Active Vibration Control
:PROPERTIES:
:HEADER-ARGS:matlab+: :tangle matlab/nass_3_hac.m
:END:
<>
** Introduction :ignore:
# - [ ] [[file:~/Cloud/work-projects/ID31-NASS/matlab/nass-simscape/org/uncertainty_experiment.org][uncertainty_experiment]]: Effect of experimental conditions on the plant (payload mass, Ry position, Rz position, Rz velocity, etc...)
- [ ] Effect of micro-station compliance
Compare plant with "rigid" u-station and normal u-station
- Effect of IFF
- Effect of payload mass
- Decoupled plant
- Controller design
From control kinematics:
- Talk about issue of not estimating Rz from external metrology? (maybe could be nice to discuss that during the experiments!)
- Show what happens is Rz is not estimated (for instance supposed equaled to zero => increased coupling)
** Matlab Init :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
<>
#+end_src
#+begin_src matlab :exports none :results silent :noweb yes
<>
#+end_src
#+begin_src matlab :tangle no :noweb yes
<>
#+end_src
#+begin_src matlab :eval no :noweb yes
<>
#+end_src
#+begin_src matlab :noweb yes
<>
#+end_src
#+begin_src matlab :noweb yes
<>
#+end_src
** HAC Plant
#+begin_src matlab
%% Identify the IFF plant dynamics using the Simscape model
% Initialize each Simscape model elements
initializeGround();
initializeGranite();
initializeTy();
initializeRy();
initializeRz();
initializeMicroHexapod();
initializeSimplifiedNanoHexapod();
initializeSample('type', 'cylindrical', 'm', 1);
% Initial Simscape Configuration
initializeSimscapeConfiguration('gravity', false);
initializeDisturbances('enable', false);
initializeLoggingConfiguration('log', 'none');
initializeController('type', 'open-loop');
initializeReferences();
% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Controller'], 1, 'input'); io_i = io_i + 1; % Actuator Inputs [N]
io(io_i) = linio([mdl, '/Tracking Error'], 1, 'openoutput', [], 'EdL'); io_i = io_i + 1; % Strut errors [m]
%% Identify HAC Plant without using IFF
initializeSample('type', 'cylindrical', 'm', 1);
G_m1 = linearize(mdl, io);
G_m1.InputName = {'f1', 'f2', 'f3', 'f4', 'f5', 'f6'};
G_m1.OutputName = {'l1', 'l2', 'l3', 'l4', 'l5', 'l6'};
initializeSample('type', 'cylindrical', 'm', 25);
G_m25 = linearize(mdl, io);
G_m25.InputName = {'f1', 'f2', 'f3', 'f4', 'f5', 'f6'};
G_m25.OutputName = {'l1', 'l2', 'l3', 'l4', 'l5', 'l6'};
initializeSample('type', 'cylindrical', 'm', 50);
G_m50 = linearize(mdl, io);
G_m50.InputName = {'f1', 'f2', 'f3', 'f4', 'f5', 'f6'};
G_m50.OutputName = {'l1', 'l2', 'l3', 'l4', 'l5', 'l6'};
%% Effect of Rotation
initializeSample('type', 'cylindrical', 'm', 1);
initializeReferences(...
'Rz_type', 'rotating', ...
'Rz_period', 1); % 360 deg/s
G_m1_Rz = linearize(mdl, io, 0.1);
G_m1_Rz.InputName = {'f1', 'f2', 'f3', 'f4', 'f5', 'f6'};
G_m1_Rz.OutputName = {'l1', 'l2', 'l3', 'l4', 'l5', 'l6'};
#+end_src
- Effect of rotation: ref:fig:nass_undamped_plant_effect_Wz
Add some coupling at low frequency, but still small at the considered velocity.
This is thanks to the relatively stiff nano-hexapod (CF rotating model)
- Effect of payload mass:
Decrease resonance frequencies
Increase coupling: ref:fig:nass_undamped_plant_effect_mass
=> control challenge for high payload masses
- Other effects such as: Ry tilt angle, Rz spindle position, micro-hexapod position are found to have negligible effect on the plant dynamics.
This is thanks to the fact the the plant dynamics is well decoupled from the micro-station dynamics.
#+begin_src matlab :exports none :results none
figure;
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
plot(freqs, abs(squeeze(freqresp(G_m1(1,1), freqs, 'Hz'))), 'color', colors(1,:), ...
'DisplayName', '$\epsilon_{\mathcal{L}i}/f_i$, $\Omega = 0$')
plot(freqs, abs(squeeze(freqresp(G_m1_Rz(1,1), freqs, 'Hz'))), 'color', colors(2,:), ...
'DisplayName', '$\epsilon_{\mathcal{L}i}/f_i$, $\Omega = 360$ deg/s')
plot(freqs, abs(squeeze(freqresp(G_m1(1,2), freqs, 'Hz'))), 'color', [colors(1,:), 0.2], ...
'DisplayName', '$\epsilon_{\mathcal{L}i}/f_j$')
plot(freqs, abs(squeeze(freqresp(G_m1_Rz(1,2), freqs, 'Hz'))), 'color', [colors(2,:), 0.2], ...
'DisplayName', '$\epsilon_{\mathcal{L}i}/f_j$')
for i = 1:5
for j = i+1:6
plot(freqs, abs(squeeze(freqresp(G_m1(i,j), freqs, 'Hz'))), 'color', [colors(1,:), 0.2], ...
'HandleVisibility', 'off');
plot(freqs, abs(squeeze(freqresp(G_m1_Rz(i,j), freqs, 'Hz'))), 'color', [colors(2,:), 0.2], ...
'HandleVisibility', 'off');
end
end
for i = 2:6
plot(freqs, abs(squeeze(freqresp(G_m1(i,i), freqs, 'Hz'))), 'color', colors(1,:), ...
'HandleVisibility', 'off');
plot(freqs, abs(squeeze(freqresp(G_m1_Rz(i,i), freqs, 'Hz'))), 'color', colors(2,:), ...
'HandleVisibility', 'off');
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
ylim([1e-11, 2e-5]);
leg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2);
leg.ItemTokenSize(1) = 15;
ax2 = nexttile;
hold on;
for i = 1:6
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_m1(i,i), freqs, 'Hz')))), 'color', colors(1,:));
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_m1_Rz(i,i), freqs, 'Hz')))), 'color', colors(2,:));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-200, 20]);
yticks([-180:45:180]);
linkaxes([ax1,ax2],'x');
xlim([freqs(1), freqs(end)]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file none
exportFig('figs/nass_undamped_plant_effect_Wz.pdf', 'width', 'half', 'height', 600);
#+end_src
#+begin_src matlab :exports none :results none
figure;
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
plot(freqs, abs(squeeze(freqresp(G_m1( 1,1), freqs, 'Hz'))), 'color', colors(1,:), ...
'DisplayName', '$\epsilon_{\mathcal{L}i}/f_i$, 1 kg')
plot(freqs, abs(squeeze(freqresp(G_m25(1,1), freqs, 'Hz'))), 'color', colors(2,:), ...
'DisplayName', '$\epsilon_{\mathcal{L}i}/f_i$, 25 kg')
plot(freqs, abs(squeeze(freqresp(G_m50(1,1), freqs, 'Hz'))), 'color', colors(3,:), ...
'DisplayName', '$\epsilon_{\mathcal{L}i}/f_i$, 50 kg')
for i = 1:5
for j = i+1:6
plot(freqs, abs(squeeze(freqresp(G_m1(i,j), freqs, 'Hz'))), 'color', [colors(1,:), 0.2], ...
'HandleVisibility', 'off');
plot(freqs, abs(squeeze(freqresp(G_m25(i,j), freqs, 'Hz'))), 'color', [colors(2,:), 0.2], ...
'HandleVisibility', 'off');
plot(freqs, abs(squeeze(freqresp(G_m50(i,j), freqs, 'Hz'))), 'color', [colors(3,:), 0.2], ...
'HandleVisibility', 'off');
end
end
for i = 2:6
plot(freqs, abs(squeeze(freqresp(G_m1( i,i), freqs, 'Hz'))), 'color', colors(1,:), ...
'HandleVisibility', 'off');
plot(freqs, abs(squeeze(freqresp(G_m25(i,i), freqs, 'Hz'))), 'color', colors(2,:), ...
'HandleVisibility', 'off');
plot(freqs, abs(squeeze(freqresp(G_m50(i,i), freqs, 'Hz'))), 'color', colors(3,:), ...
'HandleVisibility', 'off');
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
ylim([1e-11, 2e-5]);
leg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 1);
leg.ItemTokenSize(1) = 15;
ax2 = nexttile;
hold on;
for i = 1:6
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_m1(i,i), freqs, 'Hz')))), 'color', colors(1,:));
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_m25(i,i), freqs, 'Hz')))), 'color', colors(2,:));
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_m50(i,i), freqs, 'Hz')))), 'color', colors(3,:));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-200, 20]);
yticks([-180:45:180]);
linkaxes([ax1,ax2],'x');
xlim([freqs(1), freqs(end)]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file none
exportFig('figs/nass_undamped_plant_effect_mass.pdf', 'width', 'half', 'height', 600);
#+end_src
#+name: fig:nass_undamped_plant_effect
#+caption: Effect of the Spindle's rotational velocity on the positioning plant (\subref{fig:nass_undamped_plant_effect_Wz}) and effect of the payload's mass on the positioning plant (\subref{fig:nass_undamped_plant_effect_mass})
#+attr_latex: :options [htbp]
#+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:nass_undamped_plant_effect_Wz}Effect of rotational velocity $\Omega_z$}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
#+attr_latex: :width 0.95\linewidth
[[file:figs/nass_undamped_plant_effect_Wz.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:nass_undamped_plant_effect_mass}Effect of payload's mass}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
#+attr_latex: :width 0.95\linewidth
[[file:figs/nass_undamped_plant_effect_mass.png]]
#+end_subfigure
#+end_figure
- Effect of IFF on the plant ref:fig:nass_comp_undamped_damped_plant_m1
Modes are well damped
Small coupling increase at low frequency
- Benefits of using IFF ref:fig:nass_hac_plants
with added damping, the set of plants to be controlled (with payloads from 1kg to 50kg) is more easily controlled.
Between 10 and 50Hz, the plant dynamics does not vary a lot with the frequency, whereas without active damping, it would be impossible to design a robust controller with bandwidth above 10Hz that is robust to the change of payload
#+begin_src matlab
%% Identify HAC Plant without using IFF
initializeReferences(); % No Spindle Rotation
initializeController('type', 'iff'); % Implemented IFF controller
load('nass_K_iff.mat', 'Kiff'); % Load designed IFF controller
% 1kg payload
initializeSample('type', 'cylindrical', 'm', 1);
G_hac_m1 = linearize(mdl, io);
G_hac_m1.InputName = {'f1', 'f2', 'f3', 'f4', 'f5', 'f6'};
G_hac_m1.OutputName = {'l1', 'l2', 'l3', 'l4', 'l5', 'l6'};
% 25kg payload
initializeSample('type', 'cylindrical', 'm', 25);
G_hac_m25 = linearize(mdl, io);
G_hac_m25.InputName = {'f1', 'f2', 'f3', 'f4', 'f5', 'f6'};
G_hac_m25.OutputName = {'l1', 'l2', 'l3', 'l4', 'l5', 'l6'};
% 50kg payload
initializeSample('type', 'cylindrical', 'm', 50);
G_hac_m50 = linearize(mdl, io);
G_hac_m50.InputName = {'f1', 'f2', 'f3', 'f4', 'f5', 'f6'};
G_hac_m50.OutputName = {'l1', 'l2', 'l3', 'l4', 'l5', 'l6'};
% Check stability
if not(isstable(G_hac_m1) && isstable(G_hac_m25) && isstable(G_hac_m50))
warning('One of HAC plant is not stable')
end
#+end_src
#+begin_src matlab :exports none :results none
figure;
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
plot(freqs, abs(squeeze(freqresp(G_m1( 1,1), freqs, 'Hz'))), 'color', colors(1,:), ...
'DisplayName', '$\epsilon_{\mathcal{L}i}/f_i$, OL')
plot(freqs, abs(squeeze(freqresp(G_hac_m1(1,1), freqs, 'Hz'))), 'color', colors(2,:), ...
'DisplayName', '$\epsilon_{\mathcal{L}i}/f_i$, with IFF')
for i = 1:5
for j = i+1:6
plot(freqs, abs(squeeze(freqresp(G_m1(i,j), freqs, 'Hz'))), 'color', [colors(1,:), 0.2], ...
'HandleVisibility', 'off');
plot(freqs, abs(squeeze(freqresp(G_hac_m1(i,j), freqs, 'Hz'))), 'color', [colors(2,:), 0.2], ...
'HandleVisibility', 'off');
end
end
for i = 2:6
plot(freqs, abs(squeeze(freqresp(G_m1( i,i), freqs, 'Hz'))), 'color', colors(1,:), ...
'HandleVisibility', 'off');
plot(freqs, abs(squeeze(freqresp(G_hac_m1(i,i), freqs, 'Hz'))), 'color', colors(2,:), ...
'HandleVisibility', 'off');
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
ylim([1e-10, 2e-5]);
leg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 1);
leg.ItemTokenSize(1) = 15;
ax2 = nexttile;
hold on;
for i = 1:6
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_m1(i,i), freqs, 'Hz')))), 'color', colors(1,:));
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_hac_m1(i,i), freqs, 'Hz')))), 'color', colors(2,:));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-200, 20]);
yticks([-180:45:180]);
linkaxes([ax1,ax2],'x');
xlim([freqs(1), freqs(end)]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file none
exportFig('figs/nass_comp_undamped_damped_plant_m1.pdf', 'width', 'half', 'height', 600);
#+end_src
#+begin_src matlab :exports none :results none
%% Comparison of all the undamped FRF and all the damped FRF
figure;
tiledlayout(3, 1, 'TileSpacing', 'compact', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
plot(freqs, abs(squeeze(freqresp(G_m1( 1,1), freqs, 'Hz'))), 'color', [colors(1,:), 0.5], 'DisplayName', 'Undamped - $\epsilon\mathcal{L}_i/f_i$');
plot(freqs, abs(squeeze(freqresp(G_hac_m1(1,1), freqs, 'Hz'))), 'color', [colors(2,:), 0.5], 'DisplayName', 'Damped - $\epsilon\mathcal{L}_i/f_i^\prime$');
for i = 1:6
plot(freqs, abs(squeeze(freqresp(G_m1( i,i), freqs, 'Hz'))), 'color', [colors(1,:), 0.5], 'HandleVisibility', 'off');
plot(freqs, abs(squeeze(freqresp(G_m25(i,i), freqs, 'Hz'))), 'color', [colors(1,:), 0.5], 'HandleVisibility', 'off');
plot(freqs, abs(squeeze(freqresp(G_m50(i,i), freqs, 'Hz'))), 'color', [colors(1,:), 0.5], 'HandleVisibility', 'off');
end
for i = 1:6
plot(freqs, abs(squeeze(freqresp(G_hac_m1( i,i), freqs, 'Hz'))), 'color', [colors(2,:), 0.5], 'HandleVisibility', 'off');
plot(freqs, abs(squeeze(freqresp(G_hac_m25(i,i), freqs, 'Hz'))), 'color', [colors(2,:), 0.5], 'HandleVisibility', 'off');
plot(freqs, abs(squeeze(freqresp(G_hac_m50(i,i), freqs, 'Hz'))), 'color', [colors(2,:), 0.5], 'HandleVisibility', 'off');
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
leg = legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 1);
leg.ItemTokenSize(1) = 15;
% ylim([1e-8, 1e-4]);
ax2 = nexttile;
hold on;
for i =1:6
plot(freqs, 180/pi*angle(squeeze(freqresp(G_m1( i,i), freqs, 'Hz'))), 'color', [colors(1,:), 0.5]);
plot(freqs, 180/pi*angle(squeeze(freqresp(G_m25(i,i), freqs, 'Hz'))), 'color', [colors(1,:), 0.5]);
plot(freqs, 180/pi*angle(squeeze(freqresp(G_m50(i,i), freqs, 'Hz'))), 'color', [colors(1,:), 0.5]);
end
for i = 1:6
plot(freqs, 180/pi*angle(squeeze(freqresp(G_hac_m1( i,i), freqs, 'Hz'))), 'color', [colors(2,:), 0.5]);
plot(freqs, 180/pi*angle(squeeze(freqresp(G_hac_m25(i,i), freqs, 'Hz'))), 'color', [colors(2,:), 0.5]);
plot(freqs, 180/pi*angle(squeeze(freqresp(G_hac_m50(i,i), freqs, 'Hz'))), 'color', [colors(2,:), 0.5]);
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
ylim([-200, 20]);
yticks([-180:45:180]);
linkaxes([ax1,ax2],'x');
% xlim([1, 5e2]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file none
exportFig('figs/nass_hac_plants.pdf', 'width', 'half', 'height', 600);
#+end_src
#+name: fig:nass_hac_plant
#+caption: Effect of the Spindle's rotational velocity on the positioning plant (\subref{fig:nass_undamped_plant_effect_Wz}) and effect of the payload's mass on the positioning plant (\subref{fig:nass_undamped_plant_effect_mass})
#+attr_latex: :options [htbp]
#+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:nass_comp_undamped_damped_plant_m1}Effect of IFF - $m = 1\,\text{kg}$}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
#+attr_latex: :width 0.95\linewidth
[[file:figs/nass_comp_undamped_damped_plant_m1.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:nass_hac_plants}Effect of IFF on the set of plants to control}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
#+attr_latex: :width 0.95\linewidth
[[file:figs/nass_hac_plants.png]]
#+end_subfigure
#+end_figure
** Effect of micro-station compliance
Micro-Station complex dynamics has almost no effect on the plant dynamics (Figure ref:fig:nass_effect_ustation_compliance):
- adds some alternating poles and zeros above 100Hz, which should not be an issue for control
#+begin_src matlab
%% Identify plant with "rigid" micro-station
initializeGround('type', 'rigid');
initializeGranite('type', 'rigid');
initializeTy('type', 'rigid');
initializeRy('type', 'rigid');
initializeRz('type', 'rigid');
initializeMicroHexapod('type', 'rigid');
initializeSimplifiedNanoHexapod();
initializeSample('type', 'cylindrical', 'm', 25);
initializeReferences();
initializeController('type', 'open-loop'); % Implemented IFF controller
load('nass_K_iff.mat', 'Kiff'); % Load designed IFF controller
% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Controller'], 1, 'input'); io_i = io_i + 1; % Actuator Inputs [N]
io(io_i) = linio([mdl, '/Tracking Error'], 1, 'openoutput', [], 'EdL'); io_i = io_i + 1; % Strut errors [m]
G_m25_rigid = linearize(mdl, io);
G_m25_rigid.InputName = {'f1', 'f2', 'f3', 'f4', 'f5', 'f6'};
G_m25_rigid.OutputName = {'l1', 'l2', 'l3', 'l4', 'l5', 'l6'};
#+end_src
#+begin_src matlab :exports none :results none
%% Effect of the micro-station limited compliance on the plant dynamics
figure;
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
plot(freqs, abs(squeeze(freqresp(G_m25_rigid( 1,1), freqs, 'Hz'))), 'color', colors(1,:), ...
'DisplayName', '$\epsilon_{\mathcal{L}i}/f_i$, OL')
plot(freqs, abs(squeeze(freqresp(G_m25(1,1), freqs, 'Hz'))), 'color', colors(2,:), ...
'DisplayName', '$\epsilon_{\mathcal{L}i}/f_i$, with IFF')
for i = 1:5
for j = i+1:6
plot(freqs, abs(squeeze(freqresp(G_m25_rigid(i,j), freqs, 'Hz'))), 'color', [colors(1,:), 0.2], ...
'HandleVisibility', 'off');
plot(freqs, abs(squeeze(freqresp(G_m25(i,j), freqs, 'Hz'))), 'color', [colors(2,:), 0.2], ...
'HandleVisibility', 'off');
end
end
for i = 2:6
plot(freqs, abs(squeeze(freqresp(G_m25_rigid( i,i), freqs, 'Hz'))), 'color', colors(1,:), ...
'HandleVisibility', 'off');
plot(freqs, abs(squeeze(freqresp(G_m25(i,i), freqs, 'Hz'))), 'color', colors(2,:), ...
'HandleVisibility', 'off');
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
ylim([1e-10, 2e-5]);
leg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 1);
leg.ItemTokenSize(1) = 15;
ax2 = nexttile;
hold on;
for i = 1:6
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_m25_rigid(i,i), freqs, 'Hz')))), 'color', colors(1,:));
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_m25(i,i), freqs, 'Hz')))), 'color', colors(2,:));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-200, 20]);
yticks([-180:45:180]);
linkaxes([ax1,ax2],'x');
xlim([freqs(1), freqs(end)]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/nass_effect_ustation_compliance.pdf', 'width', 'wide', 'height', 600);
#+end_src
#+name: fig:nass_effect_ustation_compliance
#+caption: Effect of the micro-station limited compliance on the plant dynamics
#+RESULTS:
[[file:figs/nass_effect_ustation_compliance.png]]
** Higher or lower nano-hexapod stiffness?
*Goal*: confirm the analysis with simpler models (uniaxial and 3DoF) that a nano-hexapod stiffness of $\approx 1\,N/\mu m$ should give better performances than a very stiff or very soft nano-hexapod.
- *Stiff nano-hexapod*:
uniaxial model: high nano-hexapod stiffness induce coupling between the nano-hexapod and the micro-station dynamics.
considering the complex dynamics of the micro-station as shown by the modal analysis, that would result in a complex system to control
To show that, a nano-hexapod with actuator stiffness equal to 100N/um is initialized, payload of 25kg.
The dynamics from $\bm{f}$ to $\bm{\epsilon}_{\mathcal{L}}$ is identified and compared to the case where the micro-station is infinitely rigid (figure ref:fig:nass_stiff_nano_hexapod_coupling_ustation):
- Coupling induced by the micro-station: much more complex and difficult to model / predict
- Similar to what was predicted using the uniaxial model
- *Soft nano-hexapod*:
Nano-hexapod with stiffness of 0.01N/um is initialized, payload of 25kg.
Dynamics is identified with no spindle rotation, and with spindle rotation of 36deg/s and 360deg/s (Figure ref:fig:nass_soft_nano_hexapod_effect_Wz)
- Rotation as huge effect on the dynamics: unstable for high rotational velocities, added coupling due to gyroscopic effects, and change of resonance frequencies as a function of the rotational velocity
- Simple 3DoF rotating model is helpful to understand the complex effect of the rotation => similar conclusion
- Say that controlling the frame of the struts is not adapted with a soft nano-hexapod, but we should rather control in the frame matching the center of mass of the payload, but we would still obtain large coupling and change of dynamics due to gyroscopic effects.
#+begin_src matlab
%% Identify Dynamics with a Stiff nano-hexapod (100N/um)
% Initialize each Simscape model elements
initializeGround();
initializeGranite();
initializeTy();
initializeRy();
initializeRz();
initializeMicroHexapod();
initializeSimplifiedNanoHexapod('actuator_k', 1e8, 'actuator_kp', 0, 'actuator_c', 1e3);
initializeSample('type', 'cylindrical', 'm', 25);
% Initial Simscape Configuration
initializeSimscapeConfiguration('gravity', false);
initializeDisturbances('enable', false);
initializeLoggingConfiguration('log', 'none');
initializeController('type', 'open-loop');
initializeReferences();
% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Controller'], 1, 'input'); io_i = io_i + 1; % Actuator Inputs [N]
io(io_i) = linio([mdl, '/Tracking Error'], 1, 'openoutput', [], 'EdL'); io_i = io_i + 1; % Strut errors [m]
% Identify Plant
G_m25_pz = linearize(mdl, io);
G_m25_pz.InputName = {'f1', 'f2', 'f3', 'f4', 'f5', 'f6'};
G_m25_pz.OutputName = {'l1', 'l2', 'l3', 'l4', 'l5', 'l6'};
%% Compare with Nano-Hexapod alone (rigid micro-station)
initializeGround('type', 'rigid');
initializeGranite('type', 'rigid');
initializeTy('type', 'rigid');
initializeRy('type', 'rigid');
initializeRz('type', 'rigid');
initializeMicroHexapod('type', 'rigid');
% Identify Plant
G_m25_pz_rigid = linearize(mdl, io);
G_m25_pz_rigid.InputName = {'f1', 'f2', 'f3', 'f4', 'f5', 'f6'};
G_m25_pz_rigid.OutputName = {'l1', 'l2', 'l3', 'l4', 'l5', 'l6'};
#+end_src
#+begin_src matlab :exports none :results none
%% Stiff nano-hexapod - Coupling with the micro-station
figure;
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
plot(freqs, abs(squeeze(freqresp(G_m25_pz_rigid(1,1), freqs, 'Hz'))), 'color', colors(1,:), ...
'DisplayName', '$\epsilon_{\mathcal{L}i}/f_i$ - Rigid')
plot(freqs, abs(squeeze(freqresp(G_m25_pz(1,1), freqs, 'Hz'))), 'color', colors(2,:), ...
'DisplayName', '$\epsilon_{\mathcal{L}i}/f_i$ - $\mu$-station')
plot(freqs, abs(squeeze(freqresp(G_m25_pz_rigid(1,2), freqs, 'Hz'))), 'color', [colors(1,:), 0.1], ...
'DisplayName', '$\epsilon_{\mathcal{L}i}/f_j$ - Rigid')
plot(freqs, abs(squeeze(freqresp(G_m25_pz(1,2), freqs, 'Hz'))), 'color', [colors(2,:), 0.1], ...
'DisplayName', '$\epsilon_{\mathcal{L}i}/f_j$ - $\mu$-station')
for i = 1:5
for j = i+1:6
plot(freqs, abs(squeeze(freqresp(G_m25_pz_rigid(i,j), freqs, 'Hz'))), 'color', [colors(1,:), 0.1], ...
'HandleVisibility', 'off');
plot(freqs, abs(squeeze(freqresp(G_m25_pz(i,j), freqs, 'Hz'))), 'color', [colors(2,:), 0.1], ...
'HandleVisibility', 'off');
end
end
for i = 2:6
plot(freqs, abs(squeeze(freqresp(G_m25_pz_rigid(i,i), freqs, 'Hz'))), 'color', colors(1,:), ...
'HandleVisibility', 'off');
plot(freqs, abs(squeeze(freqresp(G_m25_pz(i,i), freqs, 'Hz'))), 'color', colors(2,:), ...
'HandleVisibility', 'off');
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
ylim([1e-12, 3e-7]);
leg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 1);
leg.ItemTokenSize(1) = 15;
ax2 = nexttile;
hold on;
for i = 1:6
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_m25_pz_rigid(i,i), freqs, 'Hz')))), 'color', colors(1,:));
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G_m25_pz(i,i), freqs, 'Hz')))), 'color', colors(2,:));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-200, 20]);
yticks([-180:45:180]);
linkaxes([ax1,ax2],'x');
xlim([freqs(1), freqs(end)]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file none
exportFig('figs/nass_stiff_nano_hexapod_coupling_ustation.pdf', 'width', 'half', 'height', 600);
#+end_src
#+begin_src matlab
%% Identify Dynamics with a Soft nano-hexapod (0.01N/um)
initializeGround();
initializeGranite();
initializeTy();
initializeRy();
initializeRz();
initializeMicroHexapod();
initializeSimplifiedNanoHexapod('actuator_k', 1e4, 'actuator_kp', 0, 'actuator_c', 1);
% Initialize each Simscape model elements
initializeSample('type', 'cylindrical', 'm', 25); % 25kg payload
initializeController('type', 'open-loop');
% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Controller'], 1, 'input'); io_i = io_i + 1; % Actuator Inputs [N]
io(io_i) = linio([mdl, '/Tracking Error'], 1, 'openoutput', [], 'EdL'); io_i = io_i + 1; % Strut errors [m]
% Identify the dynamics without rotation
initializeReferences();
G_m1_vc = linearize(mdl, io);
G_m1_vc.InputName = {'f1', 'f2', 'f3', 'f4', 'f5', 'f6'};
G_m1_vc.OutputName = {'l1', 'l2', 'l3', 'l4', 'l5', 'l6'};
% Identify the dynamics with 36 deg/s rotation
initializeReferences(...
'Rz_type', 'rotating', ...
'Rz_period', 10); % 36 deg/s
G_m1_vc_Rz_slow = linearize(mdl, io, 0.1);
G_m1_vc_Rz_slow.InputName = {'f1', 'f2', 'f3', 'f4', 'f5', 'f6'};
G_m1_vc_Rz_slow.OutputName = {'l1', 'l2', 'l3', 'l4', 'l5', 'l6'};
% Identify the dynamics with 360 deg/s rotation
initializeReferences(...
'Rz_type', 'rotating', ...
'Rz_period', 1); % 360 deg/s
G_m1_vc_Rz_fast = linearize(mdl, io, 0.1);
G_m1_vc_Rz_fast.InputName = {'f1', 'f2', 'f3', 'f4', 'f5', 'f6'};
G_m1_vc_Rz_fast.OutputName = {'l1', 'l2', 'l3', 'l4', 'l5', 'l6'};
#+end_src
#+begin_src matlab :exports none :results none
%% Soft Nano-Hexapod - effect of rotational velocity on the dynamics
f = logspace(-1,2,200);
figure;
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
plot(f, abs(squeeze(freqresp(G_m1_vc(1,1), f, 'Hz'))), 'color', colors(1,:), ...
'DisplayName', '$f_{ni}/f_i$ - $\Omega_z = 0$')
plot(f, abs(squeeze(freqresp(G_m1_vc_Rz_slow(1,1), f, 'Hz'))), 'color', colors(2,:), ...
'DisplayName', '$f_{ni}/f_i$ - $\Omega_z = 36$ deg/s')
plot(f, abs(squeeze(freqresp(G_m1_vc_Rz_fast(1,1), f, 'Hz'))), 'color', colors(3,:), ...
'DisplayName', '$f_{ni}/f_i$ - $\Omega_z = 360$ deg/s')
for i = 1:5
for j = i+1:6
plot(f, abs(squeeze(freqresp(G_m1_vc(i,j), f, 'Hz'))), 'color', [colors(1,:), 0.2], ...
'HandleVisibility', 'off');
plot(f, abs(squeeze(freqresp(G_m1_vc_Rz_slow(i,j), f, 'Hz'))), 'color', [colors(2,:), 0.2], ...
'HandleVisibility', 'off');
plot(f, abs(squeeze(freqresp(G_m1_vc_Rz_fast(i,j), f, 'Hz'))), 'color', [colors(3,:), 0.2], ...
'HandleVisibility', 'off');
end
end
for i = 2:6
plot(f, abs(squeeze(freqresp(G_m1_vc(i,i), f, 'Hz'))), 'color', colors(1,:), ...
'HandleVisibility', 'off');
end
for i = 2:6
plot(f, abs(squeeze(freqresp(G_m1_vc_Rz_slow(i,i), f, 'Hz'))), 'color', colors(2,:), ...
'HandleVisibility', 'off');
end
for i = 2:6
plot(f, abs(squeeze(freqresp(G_m1_vc_Rz_fast(i,i), f, 'Hz'))), 'color', colors(3,:), ...
'HandleVisibility', 'off');
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
ylim([1e-9, 1e-2]);
leg = legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 1);
leg.ItemTokenSize(1) = 15;
ax2 = nexttile;
hold on;
for i = 1:6
plot(f, 180/pi*angle(squeeze(freqresp(G_m1_vc(i,i), f, 'Hz'))), 'color', colors(1,:));
plot(f, 180/pi*angle(squeeze(freqresp(G_m1_vc_Rz_slow(i,i), f, 'Hz'))), 'color', colors(2,:));
plot(f, 180/pi*angle(squeeze(freqresp(G_m1_vc_Rz_fast(i,i), f, 'Hz'))), 'color', colors(3,:));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-180, 180]);
yticks([-180:90:180]);
linkaxes([ax1,ax2],'x');
xlim([f(1), f(end)]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file none
exportFig('figs/nass_soft_nano_hexapod_effect_Wz.pdf', 'width', 'half', 'height', 600);
#+end_src
#+name: fig:nass_soft_stiff_hexapod
#+caption: Plant dynamics of a stiff ($k_a = 100\,N/\mu m$) nano-hexapod (\subref{fig:nass_stiff_nano_hexapod_coupling_ustation}) and of a soft ($k_a = 0.01\,N/\mu m$) nano-hexapod (\subref{fig:nass_soft_nano_hexapod_effect_Wz})
#+attr_latex: :options [htbp]
#+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:nass_stiff_nano_hexapod_coupling_ustation}Stiff nano-hexapod - Coupling with the micro-station}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
#+attr_latex: :width 0.95\linewidth
[[file:figs/nass_stiff_nano_hexapod_coupling_ustation.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:nass_soft_nano_hexapod_effect_Wz}Soft nano-hexapod - Effect of Spindle rotational velocity}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
#+attr_latex: :width 0.95\linewidth
[[file:figs/nass_soft_nano_hexapod_effect_Wz.png]]
#+end_subfigure
#+end_figure
** Controller design
In this section, a high authority controller is design such that:
- it is robust to the change of payload mass (i.e. is should be stable for all the damped plants of Figure ref:fig:nass_hac_plants)
- it has reasonably high bandwidth to give good performances (here 10Hz)
eqref:eq:nass_robust_hac
\begin{equation}\label{eq:nass_robust_hac}
K_{\text{HAC}}(s) = g_0 \cdot \underbrace{\frac{\omega_c}{s}}_{\text{int}} \cdot \underbrace{\frac{1}{\sqrt{\alpha}}\frac{1 + \frac{s}{\omega_c/\sqrt{\alpha}}}{1 + \frac{s}{\omega_c\sqrt{\alpha}}}}_{\text{lead}} \cdot \underbrace{\frac{1}{1 + \frac{s}{\omega_0}}}_{\text{LPF}}, \quad \left( \omega_c = 2\pi10\,\text{rad/s},\ \alpha = 2,\ \omega_0 = 2\pi80\,\text{rad/s} \right)
\end{equation}
#+begin_src matlab
%% HAC Design
% Wanted crossover
wc = 2*pi*10; % [rad/s]
% Integrator
H_int = wc/s;
% Lead to increase phase margin
a = 2; % Amount of phase lead / width of the phase lead / high frequency gain
H_lead = 1/sqrt(a)*(1 + s/(wc/sqrt(a)))/(1 + s/(wc*sqrt(a)));
% Low Pass filter to increase robustness
H_lpf = 1/(1 + s/2/pi/80);
% Gain to have unitary crossover at wc
H_gain = 1./abs(evalfr(G_hac_m50(1,1), 1j*wc));
% Decentralized HAC
Khac = -H_gain * ... % Gain
H_int * ... % Integrator
H_lead * ... % Low Pass filter
H_lpf * ... % Low Pass filter
eye(6); % 6x6 Diagonal
#+end_src
#+begin_src matlab :exports none :tangle no
% The designed HAC controller is saved
save('./matlab/mat/nass_K_hac.mat', 'Khac');
#+end_src
#+begin_src matlab :eval no
% The designed HAC controller is saved
save('./mat/nass_K_hac.mat', 'Khac');
#+end_src
- "Decentralized" Loop Gain:
Bandwidth around 10Hz
- Characteristic Loci:
Stable for all payloads with acceptable stability margins
#+begin_src matlab :exports none :results none
%% "Diagonal" loop gain for the High Authority Controller
f = logspace(-1, 2, 1000);
figure;
tiledlayout(3, 1, 'TileSpacing', 'compact', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
plot(f, abs(squeeze(freqresp(Khac(i,i)*G_hac_m1( i,i), f, 'Hz'))), ...
'color', [colors(1,:), 0.5], 'DisplayName', '1kg');
plot(f, abs(squeeze(freqresp(Khac(i,i)*G_hac_m25(i,i), f, 'Hz'))), ...
'color', [colors(2,:), 0.5], 'DisplayName', '25kg');
plot(f, abs(squeeze(freqresp(Khac(i,i)*G_hac_m50(i,i), f, 'Hz'))), ...
'color', [colors(3,:), 0.5], 'DisplayName', '50kg');
for i = 2:6
plot(f, abs(squeeze(freqresp(Khac(i,i)*G_hac_m1( i,i), f, 'Hz'))), 'color', [colors(1,:), 0.5], 'HandleVisibility', 'off');
plot(f, abs(squeeze(freqresp(Khac(i,i)*G_hac_m25(i,i), f, 'Hz'))), 'color', [colors(2,:), 0.5], 'HandleVisibility', 'off');
plot(f, abs(squeeze(freqresp(Khac(i,i)*G_hac_m50(i,i), f, 'Hz'))), 'color', [colors(3,:), 0.5], 'HandleVisibility', 'off');
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Loop Gain'); set(gca, 'XTickLabel',[]);
ylim([1e-2, 1e2]);
leg = legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
leg.ItemTokenSize(1) = 15;
ax2 = nexttile;
hold on;
for i = 1:6
plot(f, 180/pi*angle(squeeze(freqresp(-Khac(i,i)*G_hac_m1( i,i), f, 'Hz'))), 'color', [colors(1,:), 0.5]);
plot(f, 180/pi*angle(squeeze(freqresp(-Khac(i,i)*G_hac_m25(i,i), f, 'Hz'))), 'color', [colors(2,:), 0.5]);
plot(f, 180/pi*angle(squeeze(freqresp(-Khac(i,i)*G_hac_m50(i,i), f, 'Hz'))), 'color', [colors(3,:), 0.5]);
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
yticks(-360:90:360);
ylim([-180, 180])
linkaxes([ax1,ax2],'x');
xlim([0.1, 100]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file none
exportFig('figs/nass_hac_loop_gain.pdf', 'width', 'half', 'height', 600);
#+end_src
#+begin_src matlab :exports none :results none
%% Characteristic Loci for the High Authority Controller
Ldet_m1 = zeros(6, length(freqs));
Lmimo_m1 = squeeze(freqresp(-G_hac_m1*Khac, freqs, 'Hz'));
for i_f = 2:length(freqs)
Ldet_m1(:, i_f) = eig(squeeze(Lmimo_m1(:,:,i_f)));
end
Ldet_m25 = zeros(6, length(freqs));
Lmimo_m25 = squeeze(freqresp(-G_hac_m25*Khac, freqs, 'Hz'));
for i_f = 2:length(freqs)
Ldet_m25(:, i_f) = eig(squeeze(Lmimo_m25(:,:,i_f)));
end
Ldet_m50 = zeros(6, length(freqs));
Lmimo_m50 = squeeze(freqresp(-G_hac_m50*Khac, freqs, 'Hz'));
for i_f = 2:length(freqs)
Ldet_m50(:, i_f) = eig(squeeze(Lmimo_m50(:,:,i_f)));
end
figure;
hold on;
plot(real(squeeze(Ldet_m1(1,:))), imag(squeeze(Ldet_m1(1,:))), ...
'.', 'color', colors(1, :), ...
'DisplayName', '1kg');
plot(real(squeeze(Ldet_m1(1,:))),-imag(squeeze(Ldet_m1(1,:))), ...
'.', 'color', colors(1, :), ...
'HandleVisibility', 'off');
plot(real(squeeze(Ldet_m25(1,:))), imag(squeeze(Ldet_m25(1,:))), ...
'.', 'color', colors(2, :), ...
'DisplayName', '25kg');
plot(real(squeeze(Ldet_m25(1,:))),-imag(squeeze(Ldet_m25(1,:))), ...
'.', 'color', colors(2, :), ...
'HandleVisibility', 'off');
plot(real(squeeze(Ldet_m50(1,:))), imag(squeeze(Ldet_m50(1,:))), ...
'.', 'color', colors(3, :), ...
'DisplayName', '50kg');
plot(real(squeeze(Ldet_m50(1,:))),-imag(squeeze(Ldet_m50(1,:))), ...
'.', 'color', colors(3, :), ...
'HandleVisibility', 'off');
for i = 2:6
plot(real(squeeze(Ldet_m1(i,:))), imag(squeeze(Ldet_m1(i,:))), ...
'.', 'color', colors(1, :), ...
'HandleVisibility', 'off');
plot(real(squeeze(Ldet_m1(i,:))), -imag(squeeze(Ldet_m1(i,:))), ...
'.', 'color', colors(1, :), ...
'HandleVisibility', 'off');
plot(real(squeeze(Ldet_m25(i,:))), imag(squeeze(Ldet_m25(i,:))), ...
'.', 'color', colors(2, :), ...
'HandleVisibility', 'off');
plot(real(squeeze(Ldet_m25(i,:))), -imag(squeeze(Ldet_m25(i,:))), ...
'.', 'color', colors(2, :), ...
'HandleVisibility', 'off');
plot(real(squeeze(Ldet_m50(i,:))), imag(squeeze(Ldet_m50(i,:))), ...
'.', 'color', colors(3, :), ...
'HandleVisibility', 'off');
plot(real(squeeze(Ldet_m50(i,:))), -imag(squeeze(Ldet_m50(i,:))), ...
'.', 'color', colors(3, :), ...
'HandleVisibility', 'off');
end
plot(-1, 0, 'kx', 'HandleVisibility', 'off');
hold off;
set(gca, 'XScale', 'lin'); set(gca, 'YScale', 'lin');
xlabel('Real Part'); ylabel('Imaginary Part');
axis square
xlim([-1.8, 0.2]); ylim([-1, 1]);
leg = legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
leg.ItemTokenSize(1) = 15;
#+end_src
#+begin_src matlab :tangle no :exports results :results file none
exportFig('figs/nass_hac_loci.pdf', 'width', 'half', 'height', 600);
#+end_src
#+name: fig:nass_hac_controller
#+caption: High Authority Controller - "Diagonal Loop Gain" (\subref{fig:nass_hac_loop_gain}) and Characteristic Loci (\subref{fig:nass_hac_loci})
#+attr_latex: :options [htbp]
#+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:nass_hac_loop_gain}Loop Gain}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
#+attr_latex: :width 0.95\linewidth
[[file:figs/nass_hac_loop_gain.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:nass_hac_loci}Characteristic Loci}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
#+attr_latex: :width 0.95\linewidth
[[file:figs/nass_hac_loci.png]]
#+end_subfigure
#+end_figure
** TODO Sensitivity to disturbances :noexport:
- Compute transfer functions from spindle vertical error to sample vertical error with HAC-IFF
Compare without the NASS, and with just IFF
- Same for horizontal
#+begin_src matlab
% Initialize each Simscape model elements
initializeGround();
initializeGranite();
initializeTy();
initializeRy();
initializeRz();
initializeMicroHexapod();
initializeSimplifiedNanoHexapod();
initializeSample('type', 'cylindrical', 'm', 1);
% Initial Simscape Configuration
initializeSimscapeConfiguration('gravity', false);
initializeDisturbances('enable', false);
initializeLoggingConfiguration('log', 'none');
initializeController('type', 'open-loop');
initializeReferences();
% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Disturbances'], 1, 'openinput', [], 'Frz_y'); io_i = io_i + 1; % Spindle Lateral Vibration [N]
io(io_i) = linio([mdl, '/Disturbances'], 1, 'openinput', [], 'Frz_z'); io_i = io_i + 1; % Spindle Vertical Vibration [N]
io(io_i) = linio([mdl, '/Disturbances'], 1, 'openinput', [], 'Fdy_z'); io_i = io_i + 1; % Vertical Ground Motion [m]
io(io_i) = linio([mdl, '/Disturbances'], 1, 'openinput', [], 'Dwy'); io_i = io_i + 1; % Vertical Ground Motion [m]
io(io_i) = linio([mdl, '/Disturbances'], 1, 'openinput', [], 'Dwz'); io_i = io_i + 1; % Vertical Ground Motion [m]
io(io_i) = linio([mdl, '/NASS'], 2, 'output', [], 'y'); io_i = io_i + 1; % Lateral Displacement [m]
io(io_i) = linio([mdl, '/NASS'], 2, 'output', [], 'z'); io_i = io_i + 1; % Vertical Displacement [m]
Gd_ol = linearize(mdl, io);
Gd_ol.InputName = {'Frz_y', 'Frz_z', 'Fdy_z', 'Dwy', 'Dwz'};
Gd_ol.OutputName = {'Dy', 'Dz'};
initializeController('type', 'iff'); % Implemented IFF controller
load('nass_K_iff.mat', 'Kiff'); % Load designed IFF controller
Gd_iff = linearize(mdl, io);
Gd_iff.InputName = {'Frz_y', 'Frz_z', 'Fdy_z', 'Dwy', 'Dwz'};
Gd_iff.OutputName = {'Dy', 'Dz'};
initializeController('type', 'hac-iff'); % Implemented IFF controller
load('nass_K_hac.mat', 'Khac'); % Load designed HAC controller
Gd_hac_iff = linearize(mdl, io);
Gd_hac_iff.InputName = {'Frz_y', 'Frz_z', 'Fdy_z', 'Dwy', 'Dwz'};
Gd_hac_iff.OutputName = {'Dy', 'Dz'};
#+end_src
#+begin_src matlab
dist = load('ustation_disturbance_psd.mat');
#+end_src
Spindle, lateral:
#+begin_src matlab
figure;
hold on;
plot(freqs, abs(squeeze(freqresp(Gd_ol( 'Dy', 'Frz_y'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gd_iff('Dy', 'Frz_y'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gd_hac_iff('Dy', 'Frz_y'), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude $D_z/F_{R_z,z}$ [m/N]'); xlabel('Frequency [Hz]');
xticks([1e0, 1e1, 1e2]);
xlim([1, 500]);
#+end_src
Spindle, vertical:
#+begin_src matlab
freqs = logspace(-1,3,1000);
figure;
hold on;
plot(freqs, abs(squeeze(freqresp(Gd_ol( 'Dz', 'Frz_z'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gd_iff('Dz', 'Frz_z'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gd_hac_iff('Dz', 'Frz_z'), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude $D_z/F_{R_z,z}$ [m/N]'); xlabel('Frequency [Hz]');
#+end_src
Ground motion, vertical:
#+begin_src matlab
freqs = logspace(-1,3,1000);
figure;
hold on;
plot(freqs, abs(squeeze(freqresp(Gd_ol( 'Dz', 'Dwz'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gd_iff('Dz', 'Dwz'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gd_hac_iff('Dz', 'Dwz'), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude $D_z/F_{R_z,z}$ [m/N]'); xlabel('Frequency [Hz]');
xticks([1e0, 1e1, 1e2]);
% xlim([1, 500]);
#+end_src
Ground motion, lateral:
#+begin_src matlab
figure;
hold on;
plot(freqs, abs(squeeze(freqresp(Gd_ol( 'Dy', 'Dwy'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gd_iff('Dy', 'Dwy'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gd_hac_iff('Dy', 'Dwy'), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude $D_y/F_{R_z,z}$ [m/N]'); xlabel('Frequency [Hz]');
xticks([1e0, 1e1, 1e2]);
xlim([1, 500]);
#+end_src
Noise Budget:
#+begin_src matlab
figure;
hold on;
plot(dist.gm_dist.f, sqrt(flip(-cumtrapz(flip(dist.gm_dist.f), flip(dist.gm_dist.pxx_y.*abs(squeeze(freqresp(Gd_ol( 'Dy', 'Dwy'), dist.gm_dist.f, 'Hz'))).^2)))));
plot(dist.gm_dist.f, sqrt(flip(-cumtrapz(flip(dist.gm_dist.f), flip(dist.gm_dist.pxx_y.*abs(squeeze(freqresp(Gd_iff( 'Dy', 'Dwy'), dist.gm_dist.f, 'Hz'))).^2)))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('ASD [m/sqrt(Hz)]'); xlabel('Frequency [Hz]');
xticks([1e0, 1e1, 1e2]);
xlim([1, 500]);
#+end_src
** Tomography experiment
- Validation of concept with tomography scans at the highest rotational velocity of $\Omega_z = 360\,\text{deg/s}$
- Compare obtained results with the smallest beam size that is expected with future beamline upgrade: 200nm (horizontal size) x 100nm (vertical size)
- Take into account the two main sources of disturbances: ground motion, spindle vibrations
Other noise sources are not taken into account here as they will be optimized latter (detail design phase): measurement noise, electrical noise for DAC and voltage amplifiers, ...
The open-loop errors and the closed-loop errors for the tomography scan with the light sample $1\,kg$ are shown in Figure ref:fig:nass_tomo_1kg_60rpm.
#+begin_src matlab
% Sample is not centered with the rotation axis
% This is done by offsetfing the micro-hexapod by 0.9um
P_micro_hexapod = [0.9e-6; 0; 0]; % [m]
open(mdl);
set_param(mdl, 'StopTime', '2');
initializeGround();
initializeGranite();
initializeTy();
initializeRy();
initializeRz();
initializeMicroHexapod('AP', P_micro_hexapod);
initializeSample('type', 'cylindrical', 'm', 1);
initializeSimscapeConfiguration('gravity', false);
initializeLoggingConfiguration('log', 'all', 'Ts', 1e-3);
initializeDisturbances(...
'Dw_x', true, ... % Ground Motion - X direction
'Dw_y', true, ... % Ground Motion - Y direction
'Dw_z', true, ... % Ground Motion - Z direction
'Fdy_x', false, ... % Translation Stage - X direction
'Fdy_z', false, ... % Translation Stage - Z direction
'Frz_x', true, ... % Spindle - X direction
'Frz_y', true, ... % Spindle - Y direction
'Frz_z', true); % Spindle - Z direction
initializeReferences(...
'Rz_type', 'rotating', ...
'Rz_period', 1, ...
'Dh_pos', [P_micro_hexapod; 0; 0; 0]);
% Open-Loop Simulation without Nano-Hexapod - 1kg payload
initializeSimplifiedNanoHexapod('type', 'none');
initializeController('type', 'open-loop');
sim(mdl);
exp_tomo_ol_m1 = simout;
% Closed-Loop Simulation with NASS
initializeSimplifiedNanoHexapod();
initializeController('type', 'hac-iff');
load('nass_K_iff.mat', 'Kiff');
load('nass_K_hac.mat', 'Khac');
% 1kg payload
initializeSample('type', 'cylindrical', 'm', 1);
sim(mdl);
exp_tomo_cl_m1 = simout;
% 25kg payload
initializeSample('type', 'cylindrical', 'm', 25);
sim(mdl);
exp_tomo_cl_m25 = simout;
% 50kg payload
initializeSample('type', 'cylindrical', 'm', 50);
sim(mdl);
exp_tomo_cl_m50 = simout;
% Slower tomography for high payload mass
% initializeReferences(...
% 'Rz_type', 'rotating', ...
% 'Rz_period', 10, ... % 36deg/s
% 'Dh_pos', [P_micro_hexapod; 0; 0; 0]);
% initializeSample('type', 'cylindrical', 'm', 50);
% set_param(mdl, 'StopTime', '5');
% sim(mdl);
% exp_tomo_cl_m50_slow = simout;
#+end_src
#+begin_src matlab :exports none :results none
%% Simulation of tomography experiment - 1kg payload - 360deg/s - XY errors
figure;
hold on;
plot(1e6*exp_tomo_ol_m1.y.x.Data, 1e6*exp_tomo_ol_m1.y.y.Data, 'DisplayName', 'OL')
plot(1e6*exp_tomo_cl_m1.y.x.Data(1e3:end), 1e6*exp_tomo_cl_m1.y.y.Data(1e3:end), 'color', colors(2,:), 'DisplayName', 'CL')
hold off;
xlabel('$D_x$ [$\mu$m]'); ylabel('$D_y$ [$\mu$m]');
axis equal
xlim([-2, 2]); ylim([-2, 2]);
xticks([-2:1:2]);
yticks([-2:1:2]);
leg = legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
leg.ItemTokenSize(1) = 15;
#+end_src
#+begin_src matlab :tangle no :exports results :results file none
exportFig('figs/nass_tomo_1kg_60rpm_xy.pdf', 'width', 'half', 'height', 'normal');
#+end_src
#+begin_src matlab :exports none :results none
%% Simulation of tomography experiment - no payload, 30rpm - YZ errors
figure;
tiledlayout(2, 1, 'TileSpacing', 'compact', 'Padding', 'None');
ax1 = nexttile();
hold on;
plot(1e6*exp_tomo_ol_m1.y.y.Data, 1e6*exp_tomo_ol_m1.y.z.Data, 'DisplayName', 'OL')
plot(1e6*exp_tomo_cl_m1.y.y.Data(1e3:end), 1e6*exp_tomo_cl_m1.y.z.Data(1e3:end), 'color', colors(2,:), 'DisplayName', 'CL')
hold off;
xlabel('$D_y$ [$\mu$m]'); ylabel('$D_z$ [$\mu$m]');
axis equal
xlim([-2, 2]); ylim([-0.4, 0.4]);
xticks([-2:1:2]);
yticks([-2:0.2:2]);
leg = legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
leg.ItemTokenSize(1) = 15;
ax2 = nexttile();
hold on;
plot(1e9*exp_tomo_cl_m1.y.y.Data(1e3:end), 1e9*exp_tomo_cl_m1.y.z.Data(1e3:end), 'color', colors(2,:), 'DisplayName', 'CL')
theta = linspace(0, 2*pi, 500); % Angle to plot the circle [rad]
plot(100*cos(theta), 50*sin(theta), 'k--', 'DisplayName', 'Beam size')
hold off;
xlabel('$D_y$ [nm]'); ylabel('$D_z$ [nm]');
axis equal
xlim([-500, 500]); ylim([-100, 100]);
xticks([-500:100:500]);
yticks([-100:50:100]);
leg = legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
leg.ItemTokenSize(1) = 15;
#+end_src
#+begin_src matlab :tangle no :exports results :results file none
exportFig('figs/nass_tomo_1kg_60rpm_yz.pdf', 'width', 'half', 'height', 'normal');
#+end_src
#+name: fig:nass_tomo_1kg_60rpm
#+caption: Position error of the sample in the XY (\subref{fig:nass_tomo_1kg_60rpm_xy}) and YZ (\subref{fig:nass_tomo_1kg_60rpm_yz}) planes during a simulation of a tomography experiment at $360\,\text{deg/s}$. 1kg payload is placed on top of the nano-hexapod.
#+attr_latex: :options [htbp]
#+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:nass_tomo_1kg_60rpm_xy}XY plane}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
#+attr_latex: :scale 0.9
[[file:figs/nass_tomo_1kg_60rpm_xy.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:nass_tomo_1kg_60rpm_yz}YZ plane}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
#+attr_latex: :scale 0.9
[[file:figs/nass_tomo_1kg_60rpm_yz.png]]
#+end_subfigure
#+end_figure
- Effect of payload mass (Figure ref:fig:nass_tomography_hac_iff):
Worse performance for high masses, as expected from the control analysis, but still acceptable considering that the rotational velocity of 360deg/s is only used for light payloads.
#+begin_src matlab :exports none :results none
%% Simulation of tomography experiment - no payload, 30rpm - YZ errors
figure;
tiledlayout(1, 1, 'TileSpacing', 'compact', 'Padding', 'None');
ax1 = nexttile();
hold on;
plot(1e9*exp_tomo_cl_m1.y.y.Data(1e3:end), 1e9*exp_tomo_cl_m1.y.z.Data(1e3:end), 'color', colors(1,:), 'DisplayName', '$m = 1$ kg')
theta = linspace(0, 2*pi, 500); % Angle to plot the circle [rad]
plot(100*cos(theta), 50*sin(theta), 'k--', 'DisplayName', 'Beam size')
hold off;
xlabel('$D_y$ [$\mu$m]'); ylabel('$D_z$ [$\mu$m]');
axis equal
xlim([-200, 200]); ylim([-100, 100]);
xticks([-200:50:200]); yticks([-100:50:100]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file none
exportFig('figs/nass_tomography_hac_iff_m1.pdf', 'width', 'third', 'height', 'normal');
#+end_src
#+begin_src matlab :exports none :results none
%% Simulation of tomography experiment - no payload, 30rpm - YZ errors
figure;
tiledlayout(1, 1, 'TileSpacing', 'compact', 'Padding', 'None');
ax1 = nexttile();
hold on;
plot(1e9*exp_tomo_cl_m25.y.y.Data(1e3:end), 1e9*exp_tomo_cl_m25.y.z.Data(1e3:end), 'color', colors(2,:), 'DisplayName', '$m = 25$ kg')
theta = linspace(0, 2*pi, 500); % Angle to plot the circle [rad]
plot(100*cos(theta), 50*sin(theta), 'k--', 'DisplayName', 'Beam size')
hold off;
xlabel('$D_y$ [$\mu$m]'); ylabel('$D_z$ [$\mu$m]');
axis equal
xlim([-200, 200]); ylim([-100, 100]);
xticks([-200:50:200]); yticks([-100:50:100]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file none
exportFig('figs/nass_tomography_hac_iff_m25.pdf', 'width', 'third', 'height', 'normal');
#+end_src
#+begin_src matlab :exports none :results none
%% Simulation of tomography experiment - no payload, 30rpm - YZ errors
figure;
tiledlayout(1, 1, 'TileSpacing', 'compact', 'Padding', 'None');
ax1 = nexttile();
hold on;
plot(1e9*exp_tomo_cl_m50.y.y.Data(1e3:end), 1e9*exp_tomo_cl_m50.y.z.Data(1e3:end), 'color', colors(3,:), 'DisplayName', '$m = 50$ kg')
theta = linspace(0, 2*pi, 500); % Angle to plot the circle [rad]
plot(100*cos(theta), 50*sin(theta), 'k--', 'DisplayName', 'Beam size')
hold off;
xlabel('$D_y$ [$\mu$m]'); ylabel('$D_z$ [$\mu$m]');
axis equal
xlim([-200, 200]); ylim([-100, 100]);
xticks([-200:50:200]); yticks([-100:50:100]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file none
exportFig('figs/nass_tomography_hac_iff_m50.pdf', 'width', 'third', 'height', 'normal');
#+end_src
#+name: fig:nass_tomography_hac_iff
#+caption: Simulation of tomography experiments - 360deg/s. Beam size shown by dashed black
#+attr_latex: :options [htbp]
#+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:nass_tomography_hac_iff_m1} $m = 1\,kg$}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
#+attr_latex: :scale 1
[[file:figs/nass_tomography_hac_iff_m1.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:nass_tomography_hac_iff_m25} $m = 25\,kg$}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
#+attr_latex: :scale 1
[[file:figs/nass_tomography_hac_iff_m25.png]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:nass_tomography_hac_iff_m50} $m = 50\,kg$}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
#+attr_latex: :scale 1
[[file:figs/nass_tomography_hac_iff_m50.png]]
#+end_subfigure
#+end_figure
** Conclusion
:PROPERTIES:
:UNNUMBERED: t
:END:
* Conclusion
:PROPERTIES:
:UNNUMBERED: t
:END:
<>
* Bibliography :ignore:
#+latex: \printbibliography[heading=bibintoc,title={Bibliography}]
* Helping Functions :noexport:
** Initialize Path
#+NAME: m-init-path
#+BEGIN_SRC matlab
addpath('./matlab/'); % Path for scripts
%% Path for functions, data and scripts
addpath('./matlab/mat/'); % Path for Computed FRF
addpath('./matlab/src/'); % Path for functions
addpath('./matlab/STEPS/'); % Path for STEPS
addpath('./matlab/subsystems/'); % Path for Subsystems Simulink files
%% Data directory
data_dir = './matlab/mat/'
#+END_SRC
#+NAME: m-init-path-tangle
#+BEGIN_SRC matlab
%% Path for functions, data and scripts
addpath('./mat/'); % Path for Data
addpath('./src/'); % Path for functions
addpath('./STEPS/'); % Path for STEPS
addpath('./subsystems/'); % Path for Subsystems Simulink files
%% Data directory
data_dir = './mat/';
#+END_SRC
** Initialize Simscape Model
#+NAME: m-init-simscape
#+begin_src matlab
% Simulink Model name
mdl = 'nass_model';
#+end_src
** Initialize other elements
#+NAME: m-init-other
#+BEGIN_SRC matlab
%% Colors for the figures
colors = colororder;
%% Frequency Vector [Hz]
freqs = logspace(0, 3, 1000);
#+END_SRC
* Matlab Functions :noexport:
** =initializeSimscapeConfiguration=: Simscape Configuration
:PROPERTIES:
:header-args:matlab+: :tangle matlab/src/initializeSimscapeConfiguration.m
:header-args:matlab+: :comments none :mkdirp yes :eval no
:END:
*** Function description
#+begin_src matlab
function [] = initializeSimscapeConfiguration(args)
#+end_src
*** Optional Parameters
#+begin_src matlab
arguments
args.gravity logical {mustBeNumericOrLogical} = true
end
#+end_src
*** Structure initialization
#+begin_src matlab
conf_simscape = struct();
#+end_src
*** Add Type
#+begin_src matlab
if args.gravity
conf_simscape.type = 1;
else
conf_simscape.type = 2;
end
#+end_src
*** Save the Structure
#+begin_src matlab
if exist('./mat', 'dir')
if exist('./mat/nass_model_conf_simscape.mat', 'file')
save('mat/nass_model_conf_simscape.mat', 'conf_simscape', '-append');
else
save('mat/nass_model_conf_simscape.mat', 'conf_simscape');
end
elseif exist('./matlab', 'dir')
if exist('./matlab/mat/nass_model_conf_simscape.mat', 'file')
save('matlab/mat/nass_model_conf_simscape.mat', 'conf_simscape', '-append');
else
save('matlab/mat/nass_model_conf_simscape.mat', 'conf_simscape');
end
end
#+end_src
** =initializeLoggingConfiguration=: Logging Configuration
:PROPERTIES:
:header-args:matlab+: :tangle matlab/src/initializeLoggingConfiguration.m
:header-args:matlab+: :comments none :mkdirp yes :eval no
:END:
*** Function description
#+begin_src matlab
function [] = initializeLoggingConfiguration(args)
#+end_src
*** Optional Parameters
#+begin_src matlab
arguments
args.log char {mustBeMember(args.log,{'none', 'all', 'forces'})} = 'none'
args.Ts (1,1) double {mustBeNumeric, mustBePositive} = 1e-3
end
#+end_src
*** Structure initialization
#+begin_src matlab
conf_log = struct();
#+end_src
*** Add Type
#+begin_src matlab
switch args.log
case 'none'
conf_log.type = 0;
case 'all'
conf_log.type = 1;
case 'forces'
conf_log.type = 2;
end
#+end_src
*** Sampling Time
#+begin_src matlab
conf_log.Ts = args.Ts;
#+end_src
*** Save the Structure
#+begin_src matlab
if exist('./mat', 'dir')
if exist('./mat/nass_model_conf_log.mat', 'file')
save('mat/nass_model_conf_log.mat', 'conf_log', '-append');
else
save('mat/nass_model_conf_log.mat', 'conf_log');
end
elseif exist('./matlab', 'dir')
if exist('./matlab/mat/nass_model_conf_log.mat', 'file')
save('matlab/mat/nass_model_conf_log.mat', 'conf_log', '-append');
else
save('matlab/mat/nass_model_conf_log.mat', 'conf_log');
end
end
#+end_src
** =initializeReferences=: Generate Reference Signals
:PROPERTIES:
:header-args:matlab+: :tangle matlab/src/initializeReferences.m
:header-args:matlab+: :comments none :mkdirp yes :eval no
:END:
*** Function Declaration and Documentation
#+begin_src matlab
function [ref] = initializeReferences(args)
#+end_src
*** Optional Parameters
#+begin_src matlab
arguments
% Sampling Frequency [s]
args.Ts (1,1) double {mustBeNumeric, mustBePositive} = 1e-3
% Maximum simulation time [s]
args.Tmax (1,1) double {mustBeNumeric, mustBePositive} = 100
% Either "constant" / "triangular" / "sinusoidal"
args.Dy_type char {mustBeMember(args.Dy_type,{'constant', 'triangular', 'sinusoidal'})} = 'constant'
% Amplitude of the displacement [m]
args.Dy_amplitude (1,1) double {mustBeNumeric} = 0
% Period of the displacement [s]
args.Dy_period (1,1) double {mustBeNumeric, mustBePositive} = 1
% Either "constant" / "triangular" / "sinusoidal"
args.Ry_type char {mustBeMember(args.Ry_type,{'constant', 'triangular', 'sinusoidal'})} = 'constant'
% Amplitude [rad]
args.Ry_amplitude (1,1) double {mustBeNumeric} = 0
% Period of the displacement [s]
args.Ry_period (1,1) double {mustBeNumeric, mustBePositive} = 1
% Either "constant" / "rotating"
args.Rz_type char {mustBeMember(args.Rz_type,{'constant', 'rotating', 'rotating-not-filtered'})} = 'constant'
% Initial angle [rad]
args.Rz_amplitude (1,1) double {mustBeNumeric} = 0
% Period of the rotating [s]
args.Rz_period (1,1) double {mustBeNumeric, mustBePositive} = 1
% For now, only constant is implemented
args.Dh_type char {mustBeMember(args.Dh_type,{'constant'})} = 'constant'
% Initial position [m,m,m,rad,rad,rad] of the top platform (Pitch-Roll-Yaw Euler angles)
args.Dh_pos (6,1) double {mustBeNumeric} = zeros(6, 1), ...
% For now, only constant is implemented
args.Rm_type char {mustBeMember(args.Rm_type,{'constant'})} = 'constant'
% Initial position of the two masses
args.Rm_pos (2,1) double {mustBeNumeric} = [0; pi]
% For now, only constant is implemented
args.Dn_type char {mustBeMember(args.Dn_type,{'constant'})} = 'constant'
% Initial position [m,m,m,rad,rad,rad] of the top platform
args.Dn_pos (6,1) double {mustBeNumeric} = zeros(6,1)
end
#+end_src
*** Initialize Parameters
#+begin_src matlab
%% Set Sampling Time
Ts = args.Ts;
Tmax = args.Tmax;
%% Low Pass Filter to filter out the references
s = zpk('s');
w0 = 2*pi*10;
xi = 1;
H_lpf = 1/(1 + 2*xi/w0*s + s^2/w0^2);
#+end_src
*** Translation Stage
#+begin_src matlab
%% Translation stage - Dy
t = 0:Ts:Tmax; % Time Vector [s]
Dy = zeros(length(t), 1);
Dyd = zeros(length(t), 1);
Dydd = zeros(length(t), 1);
switch args.Dy_type
case 'constant'
Dy(:) = args.Dy_amplitude;
Dyd(:) = 0;
Dydd(:) = 0;
case 'triangular'
% This is done to unsure that we start with no displacement
Dy_raw = args.Dy_amplitude*sawtooth(2*pi*t/args.Dy_period,1/2);
i0 = find(t>=args.Dy_period/4,1);
Dy(1:end-i0+1) = Dy_raw(i0:end);
Dy(end-i0+2:end) = Dy_raw(end); % we fix the last value
% The signal is filtered out
Dy = lsim(H_lpf, Dy, t);
Dyd = lsim(H_lpf*s, Dy, t);
Dydd = lsim(H_lpf*s^2, Dy, t);
case 'sinusoidal'
Dy(:) = args.Dy_amplitude*sin(2*pi/args.Dy_period*t);
Dyd = args.Dy_amplitude*2*pi/args.Dy_period*cos(2*pi/args.Dy_period*t);
Dydd = -args.Dy_amplitude*(2*pi/args.Dy_period)^2*sin(2*pi/args.Dy_period*t);
otherwise
warning('Dy_type is not set correctly');
end
Dy = struct('time', t, 'signals', struct('values', Dy), 'deriv', Dyd, 'dderiv', Dydd);
#+end_src
*** Tilt Stage
#+begin_src matlab
%% Tilt Stage - Ry
t = 0:Ts:Tmax; % Time Vector [s]
Ry = zeros(length(t), 1);
Ryd = zeros(length(t), 1);
Rydd = zeros(length(t), 1);
switch args.Ry_type
case 'constant'
Ry(:) = args.Ry_amplitude;
Ryd(:) = 0;
Rydd(:) = 0;
case 'triangular'
Ry_raw = args.Ry_amplitude*sawtooth(2*pi*t/args.Ry_period,1/2);
i0 = find(t>=args.Ry_period/4,1);
Ry(1:end-i0+1) = Ry_raw(i0:end);
Ry(end-i0+2:end) = Ry_raw(end); % we fix the last value
% The signal is filtered out
Ry = lsim(H_lpf, Ry, t);
Ryd = lsim(H_lpf*s, Ry, t);
Rydd = lsim(H_lpf*s^2, Ry, t);
case 'sinusoidal'
Ry(:) = args.Ry_amplitude*sin(2*pi/args.Ry_period*t);
Ryd = args.Ry_amplitude*2*pi/args.Ry_period*cos(2*pi/args.Ry_period*t);
Rydd = -args.Ry_amplitude*(2*pi/args.Ry_period)^2*sin(2*pi/args.Ry_period*t);
otherwise
warning('Ry_type is not set correctly');
end
Ry = struct('time', t, 'signals', struct('values', Ry), 'deriv', Ryd, 'dderiv', Rydd);
#+end_src
*** Spindle
#+begin_src matlab
%% Spindle - Rz
t = 0:Ts:Tmax; % Time Vector [s]
Rz = zeros(length(t), 1);
Rzd = zeros(length(t), 1);
Rzdd = zeros(length(t), 1);
switch args.Rz_type
case 'constant'
Rz(:) = args.Rz_amplitude;
Rzd(:) = 0;
Rzdd(:) = 0;
case 'rotating-not-filtered'
Rz(:) = 2*pi/args.Rz_period*t;
% The signal is filtered out
Rz(:) = 2*pi/args.Rz_period*t;
Rzd(:) = 2*pi/args.Rz_period;
Rzdd(:) = 0;
% We add the angle offset
Rz = Rz + args.Rz_amplitude;
case 'rotating'
Rz(:) = 2*pi/args.Rz_period*t;
% The signal is filtered out
Rz = lsim(H_lpf, Rz, t);
Rzd = lsim(H_lpf*s, Rz, t);
Rzdd = lsim(H_lpf*s^2, Rz, t);
% We add the angle offset
Rz = Rz + args.Rz_amplitude;
otherwise
warning('Rz_type is not set correctly');
end
Rz = struct('time', t, 'signals', struct('values', Rz), 'deriv', Rzd, 'dderiv', Rzdd);
#+end_src
*** Micro Hexapod
#+begin_src matlab
%% Micro-Hexapod
t = [0, Ts];
Dh = zeros(length(t), 6);
Dhl = zeros(length(t), 6);
switch args.Dh_type
case 'constant'
Dh = [args.Dh_pos, args.Dh_pos];
load('nass_model_stages.mat', 'micro_hexapod');
AP = [args.Dh_pos(1) ; args.Dh_pos(2) ; args.Dh_pos(3)];
tx = args.Dh_pos(4);
ty = args.Dh_pos(5);
tz = args.Dh_pos(6);
ARB = [cos(tz) -sin(tz) 0;
sin(tz) cos(tz) 0;
0 0 1]*...
[ cos(ty) 0 sin(ty);
0 1 0;
-sin(ty) 0 cos(ty)]*...
[1 0 0;
0 cos(tx) -sin(tx);
0 sin(tx) cos(tx)];
[~, Dhl] = inverseKinematics(micro_hexapod, 'AP', AP, 'ARB', ARB);
Dhl = [Dhl, Dhl];
otherwise
warning('Dh_type is not set correctly');
end
Dh = struct('time', t, 'signals', struct('values', Dh));
Dhl = struct('time', t, 'signals', struct('values', Dhl));
#+end_src
*** Save the Structure
#+begin_src matlab
if exist('./mat', 'dir')
if exist('./mat/nass_model_references.mat', 'file')
save('mat/nass_model_references.mat', 'Dy', 'Ry', 'Rz', 'Dh', 'Dhl', 'args', 'Ts', '-append');
else
save('mat/nass_model_references.mat', 'Dy', 'Ry', 'Rz', 'Dh', 'Dhl', 'args', 'Ts');
end
elseif exist('./matlab', 'dir')
if exist('./matlab/mat/nass_model_references.mat', 'file')
save('matlab/mat/nass_model_references.mat', 'Dy', 'Ry', 'Rz', 'Dh', 'Dhl', 'args', 'Ts', '-append');
else
save('matlab/mat/nass_model_references.mat', 'Dy', 'Ry', 'Rz', 'Dh', 'Dhl', 'args', 'Ts');
end
end
#+end_src
** =initializeDisturbances=: Initialize Disturbances
:PROPERTIES:
:header-args:matlab+: :tangle matlab/src/initializeDisturbances.m
:header-args:matlab+: :comments none :mkdirp yes
:header-args:matlab+: :eval no :results none
:END:
*** Function Declaration and Documentation
#+begin_src matlab
function [] = initializeDisturbances(args)
% initializeDisturbances - Initialize the disturbances
%
% Syntax: [] = initializeDisturbances(args)
%
% Inputs:
% - args -
#+end_src
*** Optional Parameters
#+begin_src matlab
arguments
% Global parameter to enable or disable the disturbances
args.enable logical {mustBeNumericOrLogical} = true
% Ground Motion - X direction
args.Dw_x logical {mustBeNumericOrLogical} = true
% Ground Motion - Y direction
args.Dw_y logical {mustBeNumericOrLogical} = true
% Ground Motion - Z direction
args.Dw_z logical {mustBeNumericOrLogical} = true
% Translation Stage - X direction
args.Fdy_x logical {mustBeNumericOrLogical} = true
% Translation Stage - Z direction
args.Fdy_z logical {mustBeNumericOrLogical} = true
% Spindle - X direction
args.Frz_x logical {mustBeNumericOrLogical} = true
% Spindle - Y direction
args.Frz_y logical {mustBeNumericOrLogical} = true
% Spindle - Z direction
args.Frz_z logical {mustBeNumericOrLogical} = true
end
#+end_src
#+begin_src matlab
% Initialization of random numbers
rng("shuffle");
#+end_src
*** Ground Motion
#+begin_src matlab
%% Ground Motion
if args.enable
% Load the PSD of disturbance
load('ustation_disturbance_psd.mat', 'gm_dist')
% Frequency Data
Dw.f = gm_dist.f;
Dw.psd_x = gm_dist.pxx_x;
Dw.psd_y = gm_dist.pxx_y;
Dw.psd_z = gm_dist.pxx_z;
% Time data
Fs = 2*Dw.f(end); % Sampling Frequency of data is twice the maximum frequency of the PSD vector [Hz]
N = 2*length(Dw.f); % Number of Samples match the one of the wanted PSD
T0 = N/Fs; % Signal Duration [s]
Dw.t = linspace(0, T0, N+1)'; % Time Vector [s]
% ASD representation of the ground motion
C = zeros(N/2,1);
for i = 1:N/2
C(i) = sqrt(Dw.psd_x(i)/T0);
end
if args.Dw_x
theta = 2*pi*rand(N/2,1); % Generate random phase [rad]
Cx = [0 ; C.*complex(cos(theta),sin(theta))];
Cx = [Cx; flipud(conj(Cx(2:end)))];;
Dw.x = N/sqrt(2)*ifft(Cx); % Ground Motion - x direction [m]
else
Dw.x = zeros(length(Dw.t), 1);
end
if args.Dw_y
theta = 2*pi*rand(N/2,1); % Generate random phase [rad]
Cx = [0 ; C.*complex(cos(theta),sin(theta))];
Cx = [Cx; flipud(conj(Cx(2:end)))];;
Dw.y = N/sqrt(2)*ifft(Cx); % Ground Motion - y direction [m]
else
Dw.y = zeros(length(Dw.t), 1);
end
if args.Dw_y
theta = 2*pi*rand(N/2,1); % Generate random phase [rad]
Cx = [0 ; C.*complex(cos(theta),sin(theta))];
Cx = [Cx; flipud(conj(Cx(2:end)))];;
Dw.z = N/sqrt(2)*ifft(Cx); % Ground Motion - z direction [m]
else
Dw.z = zeros(length(Dw.t), 1);
end
else
Dw.t = [0,1]; % Time Vector [s]
Dw.x = [0,0]; % Ground Motion - X [m]
Dw.y = [0,0]; % Ground Motion - Y [m]
Dw.z = [0,0]; % Ground Motion - Z [m]
end
#+end_src
*** Translation stage
#+begin_src matlab
%% Translation stage
if args.enable
% Load the PSD of disturbance
load('ustation_disturbance_psd.mat', 'dy_dist')
% Frequency Data
Dy.f = dy_dist.f;
Dy.psd_x = dy_dist.pxx_fx;
Dy.psd_z = dy_dist.pxx_fz;
% Time data
Fs = 2*Dy.f(end); % Sampling Frequency of data is twice the maximum frequency of the PSD vector [Hz]
N = 2*length(Dy.f); % Number of Samples match the one of the wanted PSD
T0 = N/Fs; % Signal Duration [s]
Dy.t = linspace(0, T0, N+1)'; % Time Vector [s]
% ASD representation of the disturbance voice
C = zeros(N/2,1);
for i = 1:N/2
C(i) = sqrt(Dy.psd_x(i)/T0);
end
if args.Fdy_x
theta = 2*pi*rand(N/2,1); % Generate random phase [rad]
Cx = [0 ; C.*complex(cos(theta),sin(theta))];
Cx = [Cx; flipud(conj(Cx(2:end)))];;
Dy.x = N/sqrt(2)*ifft(Cx); % Translation stage disturbances - X direction [N]
else
Dy.x = zeros(length(Dy.t), 1);
end
if args.Fdy_z
theta = 2*pi*rand(N/2,1); % Generate random phase [rad]
Cx = [0 ; C.*complex(cos(theta),sin(theta))];
Cx = [Cx; flipud(conj(Cx(2:end)))];;
Dy.z = N/sqrt(2)*ifft(Cx); % Translation stage disturbances - Z direction [N]
else
Dy.z = zeros(length(Dy.t), 1);
end
else
Dy.t = [0,1]; % Time Vector [s]
Dy.x = [0,0]; % Translation Stage disturbances - X [N]
Dy.z = [0,0]; % Translation Stage disturbances - Z [N]
end
#+end_src
*** Spindle
#+begin_src matlab
%% Spindle
if args.enable
% Load the PSD of disturbance
load('ustation_disturbance_psd.mat', 'rz_dist')
% Frequency Data
Rz.f = rz_dist.f;
Rz.psd_x = rz_dist.pxx_fx;
Rz.psd_y = rz_dist.pxx_fy;
Rz.psd_z = rz_dist.pxx_fz;
% Time data
Fs = 2*Rz.f(end); % Sampling Frequency of data is twice the maximum frequency of the PSD vector [Hz]
N = 2*length(Rz.f); % Number of Samples match the one of the wanted PSD
T0 = N/Fs; % Signal Duration [s]
Rz.t = linspace(0, T0, N+1)'; % Time Vector [s]
% ASD representation of the disturbance voice
C = zeros(N/2,1);
for i = 1:N/2
C(i) = sqrt(Rz.psd_x(i)/T0);
end
if args.Frz_x
theta = 2*pi*rand(N/2,1); % Generate random phase [rad]
Cx = [0 ; C.*complex(cos(theta),sin(theta))];
Cx = [Cx; flipud(conj(Cx(2:end)))];;
Rz.x = N/sqrt(2)*ifft(Cx); % spindle disturbances - X direction [N]
else
Rz.x = zeros(length(Rz.t), 1);
end
if args.Frz_y
theta = 2*pi*rand(N/2,1); % Generate random phase [rad]
Cx = [0 ; C.*complex(cos(theta),sin(theta))];
Cx = [Cx; flipud(conj(Cx(2:end)))];;
Rz.y = N/sqrt(2)*ifft(Cx); % spindle disturbances - Y direction [N]
else
Rz.y = zeros(length(Rz.t), 1);
end
if args.Frz_z
theta = 2*pi*rand(N/2,1); % Generate random phase [rad]
Cx = [0 ; C.*complex(cos(theta),sin(theta))];
Cx = [Cx; flipud(conj(Cx(2:end)))];;
Rz.z = N/sqrt(2)*ifft(Cx); % spindle disturbances - Z direction [N]
else
Rz.z = zeros(length(Rz.t), 1);
end
else
Rz.t = [0,1]; % Time Vector [s]
Rz.x = [0,0]; % Spindle disturbances - X [N]
Rz.y = [0,0]; % Spindle disturbances - X [N]
Rz.z = [0,0]; % Spindle disturbances - Z [N]
end
#+end_src
*** Direct Forces
#+begin_src matlab
u = zeros(100, 6);
Fd = u;
#+end_src
*** Set initial value to zero
#+begin_src matlab
Dw.x = Dw.x - Dw.x(1);
Dw.y = Dw.y - Dw.y(1);
Dw.z = Dw.z - Dw.z(1);
Dy.x = Dy.x - Dy.x(1);
Dy.z = Dy.z - Dy.z(1);
Rz.x = Rz.x - Rz.x(1);
Rz.y = Rz.y - Rz.y(1);
Rz.z = Rz.z - Rz.z(1);
#+end_src
*** Save the Structure
#+begin_src matlab
if exist('./mat', 'dir')
save('mat/nass_model_disturbances.mat', 'Dw', 'Dy', 'Rz', 'Fd', 'args');
elseif exist('./matlab', 'dir')
save('matlab/mat/nass_model_disturbances.mat', 'Dw', 'Dy', 'Rz', 'Fd', 'args');
end
#+end_src
** =initializeController=: Initialize Controller
#+begin_src matlab :tangle matlab/src/initializeController.m :comments none :mkdirp yes :eval no
function [] = initializeController(args)
arguments
args.type char {mustBeMember(args.type,{'open-loop', 'iff', 'dvf', 'hac-dvf', 'ref-track-L', 'ref-track-iff-L', 'cascade-hac-lac', 'hac-iff', 'stabilizing'})} = 'open-loop'
end
controller = struct();
switch args.type
case 'open-loop'
controller.type = 1;
controller.name = 'Open-Loop';
case 'dvf'
controller.type = 2;
controller.name = 'Decentralized Direct Velocity Feedback';
case 'iff'
controller.type = 3;
controller.name = 'Decentralized Integral Force Feedback';
case 'hac-dvf'
controller.type = 4;
controller.name = 'HAC-DVF';
case 'ref-track-L'
controller.type = 5;
controller.name = 'Reference Tracking in the frame of the legs';
case 'ref-track-iff-L'
controller.type = 6;
controller.name = 'Reference Tracking in the frame of the legs + IFF';
case 'cascade-hac-lac'
controller.type = 7;
controller.name = 'Cascade Control + HAC-LAC';
case 'hac-iff'
controller.type = 8;
controller.name = 'HAC-IFF';
case 'stabilizing'
controller.type = 9;
controller.name = 'Stabilizing Controller';
end
if exist('./mat', 'dir')
save('mat/nass_model_controller.mat', 'controller');
elseif exist('./matlab', 'dir')
save('matlab/mat/nass_model_controller.mat', 'controller');
end
end
#+end_src
** =describeMicroStationSetup=
:PROPERTIES:
:header-args:matlab+: :tangle matlab/src/describeMicroStationSetup.m
:header-args:matlab+: :comments none :mkdirp yes :eval no
:END:
*** Function description
#+begin_src matlab
function [] = describeMicroStationSetup()
% describeMicroStationSetup -
%
% Syntax: [] = describeMicroStationSetup()
%
% Inputs:
% - -
%
% Outputs:
% - -
#+end_src
*** Simscape Configuration
#+begin_src matlab
load('./mat/nass_model_conf_simscape.mat', 'conf_simscape');
#+end_src
#+begin_src matlab
fprintf('Simscape Configuration:\n');
if conf_simscape.type == 1
fprintf('- Gravity is included\n');
else
fprintf('- Gravity is not included\n');
end
fprintf('\n');
#+end_src
*** Disturbances
#+begin_src matlab
load('./mat/nass_model_disturbances.mat', 'args');
#+end_src
#+begin_src matlab
fprintf('Disturbances:\n');
if ~args.enable
fprintf('- No disturbance is included\n');
else
if args.Dwx && args.Dwy && args.Dwz
fprintf('- Ground motion\n');
end
if args.Fdy_x && args.Fdy_z
fprintf('- Vibrations of the Translation Stage\n');
end
if args.Frz_z
fprintf('- Vibrations of the Spindle\n');
end
end
fprintf('\n');
#+end_src
*** References
#+begin_src matlab
load('./mat/nass_model_references.mat', 'args');
#+end_src
#+begin_src matlab
fprintf('Reference Tracking:\n');
fprintf('- Translation Stage:\n');
switch args.Dy_type
case 'constant'
fprintf(' - Constant Position\n');
fprintf(' - Dy = %.0f [mm]\n', args.Dy_amplitude*1e3);
case 'triangular'
fprintf(' - Triangular Path\n');
fprintf(' - Amplitude = %.0f [mm]\n', args.Dy_amplitude*1e3);
fprintf(' - Period = %.0f [s]\n', args.Dy_period);
case 'sinusoidal'
fprintf(' - Sinusoidal Path\n');
fprintf(' - Amplitude = %.0f [mm]\n', args.Dy_amplitude*1e3);
fprintf(' - Period = %.0f [s]\n', args.Dy_period);
end
fprintf('- Tilt Stage:\n');
switch args.Ry_type
case 'constant'
fprintf(' - Constant Position\n');
fprintf(' - Ry = %.0f [mm]\n', args.Ry_amplitude*1e3);
case 'triangular'
fprintf(' - Triangular Path\n');
fprintf(' - Amplitude = %.0f [mm]\n', args.Ry_amplitude*1e3);
fprintf(' - Period = %.0f [s]\n', args.Ry_period);
case 'sinusoidal'
fprintf(' - Sinusoidal Path\n');
fprintf(' - Amplitude = %.0f [mm]\n', args.Ry_amplitude*1e3);
fprintf(' - Period = %.0f [s]\n', args.Ry_period);
end
fprintf('- Spindle:\n');
switch args.Rz_type
case 'constant'
fprintf(' - Constant Position\n');
fprintf(' - Rz = %.0f [deg]\n', 180/pi*args.Rz_amplitude);
case { 'rotating', 'rotating-not-filtered' }
fprintf(' - Rotating\n');
fprintf(' - Speed = %.0f [rpm]\n', 60/args.Rz_period);
end
fprintf('- Micro Hexapod:\n');
switch args.Dh_type
case 'constant'
fprintf(' - Constant Position\n');
fprintf(' - Dh = %.0f, %.0f, %.0f [mm]\n', args.Dh_pos(1), args.Dh_pos(2), args.Dh_pos(3));
fprintf(' - Rh = %.0f, %.0f, %.0f [deg]\n', args.Dh_pos(4), args.Dh_pos(5), args.Dh_pos(6));
end
fprintf('\n');
#+end_src
*** Micro-Station
#+begin_src matlab
load('./mat/nass_model_stages.mat', 'ground', 'granite', 'ty', 'ry', 'rz', 'micro_hexapod', 'axisc');
#+end_src
#+begin_src matlab
fprintf('Micro Station:\n');
if granite.type == 1 && ...
ty.type == 1 && ...
ry.type == 1 && ...
rz.type == 1 && ...
micro_hexapod.type == 1;
fprintf('- All stages are rigid\n');
elseif granite.type == 2 && ...
ty.type == 2 && ...
ry.type == 2 && ...
rz.type == 2 && ...
micro_hexapod.type == 2;
fprintf('- All stages are flexible\n');
else
if granite.type == 1 || granite.type == 4
fprintf('- Granite is rigid\n');
else
fprintf('- Granite is flexible\n');
end
if ty.type == 1 || ty.type == 4
fprintf('- Translation Stage is rigid\n');
else
fprintf('- Translation Stage is flexible\n');
end
if ry.type == 1 || ry.type == 4
fprintf('- Tilt Stage is rigid\n');
else
fprintf('- Tilt Stage is flexible\n');
end
if rz.type == 1 || rz.type == 4
fprintf('- Spindle is rigid\n');
else
fprintf('- Spindle is flexible\n');
end
if micro_hexapod.type == 1 || micro_hexapod.type == 4
fprintf('- Micro Hexapod is rigid\n');
else
fprintf('- Micro Hexapod is flexible\n');
end
end
fprintf('\n');
#+end_src
** =computeReferencePose=
:PROPERTIES:
:header-args:matlab+: :tangle matlab/src/computeReferencePose.m
:header-args:matlab+: :comments none :mkdirp yes :eval no
:END:
#+begin_src matlab
function [WTr] = computeReferencePose(Dy, Ry, Rz, Dh, Dn)
% computeReferencePose - Compute the homogeneous transformation matrix corresponding to the wanted pose of the sample
%
% Syntax: [WTr] = computeReferencePose(Dy, Ry, Rz, Dh, Dn)
%
% Inputs:
% - Dy - Reference of the Translation Stage [m]
% - Ry - Reference of the Tilt Stage [rad]
% - Rz - Reference of the Spindle [rad]
% - Dh - Reference of the Micro Hexapod (Pitch, Roll, Yaw angles) [m, m, m, rad, rad, rad]
% - Dn - Reference of the Nano Hexapod [m, m, m, rad, rad, rad]
%
% Outputs:
% - WTr -
%% Translation Stage
Rty = [1 0 0 0;
0 1 0 Dy;
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
Rhx = [1 0 0;
0 cos(Dh(4)) -sin(Dh(4));
0 sin(Dh(4)) cos(Dh(4))];
Rhy = [ cos(Dh(5)) 0 sin(Dh(5));
0 1 0;
-sin(Dh(5)) 0 cos(Dh(5))];
Rhz = [cos(Dh(6)) -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(1:3, 1:3) = Rhz*Rhy*Rhx;
%% Nano-Hexapod
Rnx = [1 0 0;
0 cos(Dn(4)) -sin(Dn(4));
0 sin(Dn(4)) cos(Dn(4))];
Rny = [ cos(Dn(5)) 0 sin(Dn(5));
0 1 0;
-sin(Dn(5)) 0 cos(Dn(5))];
Rnz = [cos(Dn(6)) -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(1:3, 1:3) = Rnz*Rny*Rnx;
%% Total Homogeneous transformation
WTr = Rty*Rry*Rrz*Rh*Rn;
end
#+end_src
** =circlefit=
:PROPERTIES:
:header-args:matlab+: :tangle matlab/src/circlefit.m
:header-args:matlab+: :comments none :mkdirp yes :eval no
:END:
#+begin_src matlab
function [xc,yc,R,a] = circlefit(x,y)
%
% [xc yx R] = circfit(x,y)
%
% fits a circle in x,y plane in a more accurate
% (less prone to ill condition )
% procedure than circfit2 but using more memory
% x,y are column vector where (x(i),y(i)) is a measured point
%
% result is center point (yc,xc) and radius R
% an optional output is the vector of coeficient a
% describing the circle's equation
%
% x^2+y^2+a(1)*x+a(2)*y+a(3)=0
%
% By: Izhak bucher 25/oct /1991,
x=x(:); y=y(:);
a=[x y ones(size(x))]\[-(x.^2+y.^2)];
xc = -.5*a(1);
yc = -.5*a(2);
R = sqrt((a(1)^2+a(2)^2)/4-a(3));
#+end_src
** Initialize Micro-Station Stages
*** =initializeGround=: Ground
#+begin_src matlab :tangle matlab/src/initializeGround.m :comments none :mkdirp yes :eval no
function [ground] = initializeGround(args)
arguments
args.type char {mustBeMember(args.type,{'none', 'rigid'})} = 'rigid'
args.rot_point (3,1) double {mustBeNumeric} = zeros(3,1) % Rotation point for the ground motion [m]
end
ground = struct();
switch args.type
case 'none'
ground.type = 0;
case 'rigid'
ground.type = 1;
end
ground.shape = [2, 2, 0.5]; % [m]
ground.density = 2800; % [kg/m3]
ground.rot_point = args.rot_point;
if exist('./mat', 'dir')
if exist('./mat/nass_model_stages.mat', 'file')
save('mat/nass_model_stages.mat', 'ground', '-append');
else
save('mat/nass_model_stages.mat', 'ground');
end
elseif exist('./matlab', 'dir')
if exist('./matlab/mat/nass_model_stages.mat', 'file')
save('matlab/mat/nass_model_stages.mat', 'ground', '-append');
else
save('matlab/mat/nass_model_stages.mat', 'ground');
end
end
end
#+end_src
*** =initializeGranite=: Granite
#+begin_src matlab :tangle matlab/src/initializeGranite.m :comments none :mkdirp yes :eval no
function [granite] = initializeGranite(args)
arguments
args.type char {mustBeMember(args.type,{'rigid', 'flexible', 'none'})} = 'flexible'
args.density (1,1) double {mustBeNumeric, mustBeNonnegative} = 2800 % Density [kg/m3]
args.K (6,1) double {mustBeNumeric, mustBeNonnegative} = [5e9; 5e9; 5e9; 2.5e7; 2.5e7; 1e7] % [N/m]
args.C (6,1) double {mustBeNumeric, mustBeNonnegative} = [4.0e5; 1.1e5; 9.0e5; 2e4; 2e4; 1e4] % [N/(m/s)]
args.x0 (1,1) double {mustBeNumeric} = 0 % Rest position of the Joint in the X direction [m]
args.y0 (1,1) double {mustBeNumeric} = 0 % Rest position of the Joint in the Y direction [m]
args.z0 (1,1) double {mustBeNumeric} = 0 % Rest position of the Joint in the Z direction [m]
args.sample_pos (1,1) double {mustBeNumeric} = 0.775 % Height of the measurment point [m]
end
granite = struct();
switch args.type
case 'none'
granite.type = 0;
case 'rigid'
granite.type = 1;
case 'flexible'
granite.type = 2;
end
granite.density = args.density; % [kg/m3]
granite.STEP = 'granite.STEP';
% Z-offset for the initial position of the sample with respect to the granite top surface.
granite.sample_pos = args.sample_pos; % [m]
granite.K = args.K; % [N/m]
granite.C = args.C; % [N/(m/s)]
if exist('./mat', 'dir')
if exist('./mat/nass_model_stages.mat', 'file')
save('mat/nass_model_stages.mat', 'granite', '-append');
else
save('mat/nass_model_stages.mat', 'granite');
end
elseif exist('./matlab', 'dir')
if exist('./matlab/mat/nass_model_stages.mat', 'file')
save('matlab/mat/nass_model_stages.mat', 'granite', '-append');
else
save('matlab/mat/nass_model_stages.mat', 'granite');
end
end
end
#+end_src
*** =initializeTy=: Translation Stage
#+begin_src matlab :tangle matlab/src/initializeTy.m :comments none :mkdirp yes :eval no
function [ty] = initializeTy(args)
arguments
args.type char {mustBeMember(args.type,{'none', 'rigid', 'flexible'})} = 'flexible'
end
ty = struct();
switch args.type
case 'none'
ty.type = 0;
case 'rigid'
ty.type = 1;
case 'flexible'
ty.type = 2;
end
% Ty Granite frame
ty.granite_frame.density = 7800; % [kg/m3] => 43kg
ty.granite_frame.STEP = 'Ty_Granite_Frame.STEP';
% Guide Translation Ty
ty.guide.density = 7800; % [kg/m3] => 76kg
ty.guide.STEP = 'Ty_Guide.STEP';
% Ty - Guide_Translation12
ty.guide12.density = 7800; % [kg/m3]
ty.guide12.STEP = 'Ty_Guide_12.STEP';
% Ty - Guide_Translation11
ty.guide11.density = 7800; % [kg/m3]
ty.guide11.STEP = 'Ty_Guide_11.STEP';
% Ty - Guide_Translation22
ty.guide22.density = 7800; % [kg/m3]
ty.guide22.STEP = 'Ty_Guide_22.STEP';
% Ty - Guide_Translation21
ty.guide21.density = 7800; % [kg/m3]
ty.guide21.STEP = 'Ty_Guide_21.STEP';
% Ty - Plateau translation
ty.frame.density = 7800; % [kg/m3]
ty.frame.STEP = 'Ty_Stage.STEP';
% Ty Stator Part
ty.stator.density = 5400; % [kg/m3]
ty.stator.STEP = 'Ty_Motor_Stator.STEP';
% Ty Rotor Part
ty.rotor.density = 5400; % [kg/m3]
ty.rotor.STEP = 'Ty_Motor_Rotor.STEP';
ty.K = [2e8; 1e8; 2e8; 6e7; 9e7; 6e7]; % [N/m, N*m/rad]
ty.C = [8e4; 5e4; 8e4; 2e4; 3e4; 1e4]; % [N/(m/s), N*m/(rad/s)]
if exist('./mat', 'dir')
if exist('./mat/nass_model_stages.mat', 'file')
save('mat/nass_model_stages.mat', 'ty', '-append');
else
save('mat/nass_model_stages.mat', 'ty');
end
elseif exist('./matlab', 'dir')
if exist('./matlab/mat/nass_model_stages.mat', 'file')
save('matlab/mat/nass_model_stages.mat', 'ty', '-append');
else
save('matlab/mat/nass_model_stages.mat', 'ty');
end
end
end
#+end_src
*** =initializeRy=: Tilt Stage
#+begin_src matlab :tangle matlab/src/initializeRy.m :comments none :mkdirp yes :eval no
function [ry] = initializeRy(args)
arguments
args.type char {mustBeMember(args.type,{'none', 'rigid', 'flexible'})} = 'flexible'
args.Ry_init (1,1) double {mustBeNumeric} = 0
end
ry = struct();
switch args.type
case 'none'
ry.type = 0;
case 'rigid'
ry.type = 1;
case 'flexible'
ry.type = 2;
end
% Ry - Guide for the tilt stage
ry.guide.density = 7800; % [kg/m3]
ry.guide.STEP = 'Tilt_Guide.STEP';
% Ry - Rotor of the motor
ry.rotor.density = 2400; % [kg/m3]
ry.rotor.STEP = 'Tilt_Motor_Axis.STEP';
% Ry - Motor
ry.motor.density = 3200; % [kg/m3]
ry.motor.STEP = 'Tilt_Motor.STEP';
% Ry - Plateau Tilt
ry.stage.density = 7800; % [kg/m3]
ry.stage.STEP = 'Tilt_Stage.STEP';
% Z-Offset so that the center of rotation matches the sample center;
ry.z_offset = 0.58178; % [m]
ry.Ry_init = args.Ry_init; % [rad]
ry.K = [3.8e8; 4e8; 3.8e8; 1.2e8; 6e4; 1.2e8];
ry.C = [1e5; 1e5; 1e5; 3e4; 1e3; 3e4];
if exist('./mat', 'dir')
if exist('./mat/nass_model_stages.mat', 'file')
save('mat/nass_model_stages.mat', 'ry', '-append');
else
save('mat/nass_model_stages.mat', 'ry');
end
elseif exist('./matlab', 'dir')
if exist('./matlab/mat/nass_model_stages.mat', 'file')
save('matlab/mat/nass_model_stages.mat', 'ry', '-append');
else
save('matlab/mat/nass_model_stages.mat', 'ry');
end
end
end
#+end_src
*** =initializeRz=: Spindle
#+begin_src matlab :tangle matlab/src/initializeRz.m :comments none :mkdirp yes :eval no
function [rz] = initializeRz(args)
arguments
args.type char {mustBeMember(args.type,{'none', 'rigid', 'flexible'})} = 'flexible'
end
rz = struct();
switch args.type
case 'none'
rz.type = 0;
case 'rigid'
rz.type = 1;
case 'flexible'
rz.type = 2;
end
% Spindle - Slip Ring
rz.slipring.density = 7800; % [kg/m3]
rz.slipring.STEP = 'Spindle_Slip_Ring.STEP';
% Spindle - Rotor
rz.rotor.density = 7800; % [kg/m3]
rz.rotor.STEP = 'Spindle_Rotor.STEP';
% Spindle - Stator
rz.stator.density = 7800; % [kg/m3]
rz.stator.STEP = 'Spindle_Stator.STEP';
rz.K = [7e8; 7e8; 2e9; 1e7; 1e7; 1e7];
rz.C = [4e4; 4e4; 7e4; 1e4; 1e4; 1e4];
if exist('./mat', 'dir')
if exist('./mat/nass_model_stages.mat', 'file')
save('mat/nass_model_stages.mat', 'rz', '-append');
else
save('mat/nass_model_stages.mat', 'rz');
end
elseif exist('./matlab', 'dir')
if exist('./matlab/mat/nass_model_stages.mat', 'file')
save('matlab/mat/nass_model_stages.mat', 'rz', '-append');
else
save('matlab/mat/nass_model_stages.mat', 'rz');
end
end
end
#+end_src
*** =initializeMicroHexapod=: Micro Hexapod
#+begin_src matlab :tangle matlab/src/initializeMicroHexapod.m :comments none :mkdirp yes :eval no
function [micro_hexapod] = initializeMicroHexapod(args)
arguments
args.type char {mustBeMember(args.type,{'none', 'rigid', 'flexible'})} = 'flexible'
% initializeFramesPositions
args.H (1,1) double {mustBeNumeric, mustBePositive} = 350e-3
args.MO_B (1,1) double {mustBeNumeric} = 270e-3
% generateGeneralConfiguration
args.FH (1,1) double {mustBeNumeric, mustBePositive} = 50e-3
args.FR (1,1) double {mustBeNumeric, mustBePositive} = 175.5e-3
args.FTh (6,1) double {mustBeNumeric} = [-10, 10, 120-10, 120+10, 240-10, 240+10]*(pi/180)
args.MH (1,1) double {mustBeNumeric, mustBePositive} = 45e-3
args.MR (1,1) double {mustBeNumeric, mustBePositive} = 118e-3
args.MTh (6,1) double {mustBeNumeric} = [-60+10, 60-10, 60+10, 180-10, 180+10, -60-10]*(pi/180)
% initializeStrutDynamics
args.Ki (1,1) double {mustBeNumeric, mustBeNonnegative} = 2e7
args.Ci (1,1) double {mustBeNumeric, mustBeNonnegative} = 1.4e3
% initializeCylindricalPlatforms
args.Fpm (1,1) double {mustBeNumeric, mustBePositive} = 10
args.Fph (1,1) double {mustBeNumeric, mustBePositive} = 26e-3
args.Fpr (1,1) double {mustBeNumeric, mustBePositive} = 207.5e-3
args.Mpm (1,1) double {mustBeNumeric, mustBePositive} = 10
args.Mph (1,1) double {mustBeNumeric, mustBePositive} = 26e-3
args.Mpr (1,1) double {mustBeNumeric, mustBePositive} = 150e-3
% initializeCylindricalStruts
args.Fsm (1,1) double {mustBeNumeric, mustBePositive} = 1
args.Fsh (1,1) double {mustBeNumeric, mustBePositive} = 100e-3
args.Fsr (1,1) double {mustBeNumeric, mustBePositive} = 25e-3
args.Msm (1,1) double {mustBeNumeric, mustBePositive} = 1
args.Msh (1,1) double {mustBeNumeric, mustBePositive} = 100e-3
args.Msr (1,1) double {mustBeNumeric, mustBePositive} = 25e-3
% inverseKinematics
args.AP (3,1) double {mustBeNumeric} = zeros(3,1)
args.ARB (3,3) double {mustBeNumeric} = eye(3)
end
stewart = initializeStewartPlatform();
stewart = initializeFramesPositions(stewart, ...
'H', args.H, ...
'MO_B', args.MO_B);
stewart = generateGeneralConfiguration(stewart, ...
'FH', args.FH, ...
'FR', args.FR, ...
'FTh', args.FTh, ...
'MH', args.MH, ...
'MR', args.MR, ...
'MTh', args.MTh);
stewart = computeJointsPose(stewart);
stewart = initializeStrutDynamics(stewart, ...
'k', args.Ki, ...
'c', args.Ci);
stewart = initializeJointDynamics(stewart, ...
'type_F', '2dof', ...
'type_M', '3dof');
stewart = initializeCylindricalPlatforms(stewart, ...
'Fpm', args.Fpm, ...
'Fph', args.Fph, ...
'Fpr', args.Fpr, ...
'Mpm', args.Mpm, ...
'Mph', args.Mph, ...
'Mpr', args.Mpr);
stewart = initializeCylindricalStruts(stewart, ...
'Fsm', args.Fsm, ...
'Fsh', args.Fsh, ...
'Fsr', args.Fsr, ...
'Msm', args.Msm, ...
'Msh', args.Msh, ...
'Msr', args.Msr);
stewart = computeJacobian(stewart);
stewart = initializeStewartPose(stewart, ...
'AP', args.AP, ...
'ARB', args.ARB);
stewart = initializeInertialSensor(stewart, 'type', 'none');
switch args.type
case 'none'
stewart.type = 0;
case 'rigid'
stewart.type = 1;
case 'flexible'
stewart.type = 2;
end
micro_hexapod = stewart;
if exist('./mat', 'dir')
if exist('./mat/nass_model_stages.mat', 'file')
save('mat/nass_model_stages.mat', 'micro_hexapod', '-append');
else
save('mat/nass_model_stages.mat', 'micro_hexapod');
end
elseif exist('./matlab', 'dir')
if exist('./matlab/mat/nass_model_stages.mat', 'file')
save('matlab/mat/nass_model_stages.mat', 'micro_hexapod', '-append');
else
save('matlab/mat/nass_model_stages.mat', 'micro_hexapod');
end
end
end
#+end_src
*** =initializeSimplifiedNanoHexapod=: Nano Hexapod
#+begin_src matlab :tangle matlab/src/initializeSimplifiedNanoHexapod.m :comments none :mkdirp yes :eval no
function [nano_hexapod] = initializeSimplifiedNanoHexapod(args)
arguments
args.type char {mustBeMember(args.type,{'none', 'stewart'})} = 'stewart'
%% initializeFramesPositions
args.H (1,1) double {mustBeNumeric, mustBePositive} = 95e-3 % Height of the nano-hexapod [m]
args.MO_B (1,1) double {mustBeNumeric} = 150e-3 % Height of {B} w.r.t. {M} [m]
%% generateGeneralConfiguration
args.FH (1,1) double {mustBeNumeric, mustBePositive} = 15e-3 % Height of fixed joints [m]
args.FR (1,1) double {mustBeNumeric, mustBePositive} = 120e-3 % Radius of fixed joints [m]
args.FTh (6,1) double {mustBeNumeric} = [220, 320, 340, 80, 100, 200]*(pi/180) % Angles of fixed joints [rad]
args.MH (1,1) double {mustBeNumeric, mustBePositive} = 15e-3 % Height of mobile joints [m]
args.MR (1,1) double {mustBeNumeric, mustBePositive} = 110e-3 % Radius of mobile joints [m]
args.MTh (6,1) double {mustBeNumeric} = [255, 285, 15, 45, 135, 165]*(pi/180) % Angles of fixed joints [rad]
%% Actuators
args.actuator_type char {mustBeMember(args.actuator_type,{'1dof', '2dof', 'flexible'})} = '1dof'
args.actuator_k (1,1) double {mustBeNumeric, mustBePositive} = 1e6
args.actuator_kp (1,1) double {mustBeNumeric, mustBeNonnegative} = 5e4
args.actuator_ke (1,1) double {mustBeNumeric, mustBePositive} = 4952605
args.actuator_ka (1,1) double {mustBeNumeric, mustBePositive} = 2476302
args.actuator_c (1,1) double {mustBeNumeric, mustBePositive} = 50
args.actuator_cp (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
args.actuator_ce (1,1) double {mustBeNumeric, mustBePositive} = 100
args.actuator_ca (1,1) double {mustBeNumeric, mustBePositive} = 50
%% initializeCylindricalPlatforms
args.Fpm (1,1) double {mustBeNumeric, mustBePositive} = 5 % Mass of the fixed plate [kg]
args.Fph (1,1) double {mustBeNumeric, mustBePositive} = 10e-3 % Thickness of the fixed plate [m]
args.Fpr (1,1) double {mustBeNumeric, mustBePositive} = 150e-3 % Radius of the fixed plate [m]
args.Mpm (1,1) double {mustBeNumeric, mustBePositive} = 5 % Mass of the mobile plate [kg]
args.Mph (1,1) double {mustBeNumeric, mustBePositive} = 10e-3 % Thickness of the mobile plate [m]
args.Mpr (1,1) double {mustBeNumeric, mustBePositive} = 150e-3 % Radius of the mobile plate [m]
%% initializeCylindricalStruts
args.Fsm (1,1) double {mustBeNumeric, mustBePositive} = 1e-3 % Mass of the fixed part of the strut [kg]
args.Fsh (1,1) double {mustBeNumeric, mustBePositive} = 60e-3 % Length of the fixed part of the struts [m]
args.Fsr (1,1) double {mustBeNumeric, mustBePositive} = 5e-3 % Radius of the fixed part of the struts [m]
args.Msm (1,1) double {mustBeNumeric, mustBePositive} = 1e-3 % Mass of the mobile part of the strut [kg]
args.Msh (1,1) double {mustBeNumeric, mustBePositive} = 60e-3 % Length of the mobile part of the struts [m]
args.Msr (1,1) double {mustBeNumeric, mustBePositive} = 5e-3 % Radius of the fixed part of the struts [m]
%% Bottom and Top Flexible Joints
args.flex_type_F char {mustBeMember(args.flex_type_F,{'2dof', '3dof', '4dof', '6dof', 'flexible'})} = '2dof'
args.flex_type_M char {mustBeMember(args.flex_type_M,{'2dof', '3dof', '4dof', '6dof', 'flexible'})} = '3dof'
args.Kf_M (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
args.Cf_M (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
args.Kt_M (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
args.Ct_M (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
args.Kf_F (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
args.Cf_F (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
args.Kt_F (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
args.Ct_F (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
args.Ka_F (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
args.Ca_F (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
args.Kr_F (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
args.Cr_F (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
args.Ka_M (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
args.Ca_M (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
args.Kr_M (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
args.Cr_M (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
%% inverseKinematics
args.AP (3,1) double {mustBeNumeric} = zeros(3,1)
args.ARB (3,3) double {mustBeNumeric} = eye(3)
end
stewart = initializeStewartPlatform();
switch args.type
case 'none'
stewart.type = 0;
case 'stewart'
stewart.type = 1;
end
stewart = initializeFramesPositions(stewart, ...
'H', args.H, ...
'MO_B', args.MO_B);
stewart = generateGeneralConfiguration(stewart, ...
'FH', args.FH, ...
'FR', args.FR, ...
'FTh', args.FTh, ...
'MH', args.MH, ...
'MR', args.MR, ...
'MTh', args.MTh);
stewart = computeJointsPose(stewart);
stewart = initializeStrutDynamics(stewart, ...
'type', args.actuator_type, ...
'k', args.actuator_k, ...
'kp', args.actuator_kp, ...
'ke', args.actuator_ke, ...
'ka', args.actuator_ka, ...
'c', args.actuator_c, ...
'cp', args.actuator_cp, ...
'ce', args.actuator_ce, ...
'ca', args.actuator_ca);
stewart = initializeJointDynamics(stewart, ...
'type_F', args.flex_type_F, ...
'type_M', args.flex_type_M, ...
'Kf_M', args.Kf_M, ...
'Cf_M', args.Cf_M, ...
'Kt_M', args.Kt_M, ...
'Ct_M', args.Ct_M, ...
'Kf_F', args.Kf_F, ...
'Cf_F', args.Cf_F, ...
'Kt_F', args.Kt_F, ...
'Ct_F', args.Ct_F, ...
'Ka_F', args.Ka_F, ...
'Ca_F', args.Ca_F, ...
'Kr_F', args.Kr_F, ...
'Cr_F', args.Cr_F, ...
'Ka_M', args.Ka_M, ...
'Ca_M', args.Ca_M, ...
'Kr_M', args.Kr_M, ...
'Cr_M', args.Cr_M);
stewart = initializeCylindricalPlatforms(stewart, ...
'Fpm', args.Fpm, ...
'Fph', args.Fph, ...
'Fpr', args.Fpr, ...
'Mpm', args.Mpm, ...
'Mph', args.Mph, ...
'Mpr', args.Mpr);
stewart = initializeCylindricalStruts(stewart, ...
'Fsm', args.Fsm, ...
'Fsh', args.Fsh, ...
'Fsr', args.Fsr, ...
'Msm', args.Msm, ...
'Msh', args.Msh, ...
'Msr', args.Msr);
stewart = computeJacobian(stewart);
stewart = initializeStewartPose(stewart, ...
'AP', args.AP, ...
'ARB', args.ARB);
nano_hexapod = stewart;
if exist('./mat', 'dir')
if exist('./mat/nass_model_stages.mat', 'file')
save('mat/nass_model_stages.mat', 'nano_hexapod', '-append');
else
save('mat/nass_model_stages.mat', 'nano_hexapod');
end
elseif exist('./matlab', 'dir')
if exist('./matlab/mat/nass_model_stages.mat', 'file')
save('matlab/mat/nass_model_stages.mat', 'nano_hexapod', '-append');
else
save('matlab/mat/nass_model_stages.mat', 'nano_hexapod');
end
end
end
#+end_src
*** =initializeSample=: Sample
#+begin_src matlab :tangle matlab/src/initializeSample.m :comments none :mkdirp yes :eval no
function [sample] = initializeSample(args)
arguments
args.type char {mustBeMember(args.type,{'none', 'cylindrical'})} = 'none'
args.H (1,1) double {mustBeNumeric, mustBePositive} = 250e-3 % Height [m]
args.R (1,1) double {mustBeNumeric, mustBePositive} = 110e-3 % Radius [m]
args.m (1,1) double {mustBeNumeric, mustBePositive} = 1 % Mass [kg]
end
sample = struct();
switch args.type
case 'none'
sample.type = 0;
sample.m = 0;
case 'cylindrical'
sample.type = 1;
sample.H = args.H;
sample.R = args.R;
sample.m = args.m;
end
if exist('./mat', 'dir')
if exist('./mat/nass_model_stages.mat', 'file')
save('mat/nass_model_stages.mat', 'sample', '-append');
else
save('mat/nass_model_stages.mat', 'sample');
end
elseif exist('./matlab', 'dir')
if exist('./matlab/mat/nass_model_stages.mat', 'file')
save('matlab/mat/nass_model_stages.mat', 'sample', '-append');
else
save('matlab/mat/nass_model_stages.mat', 'sample');
end
end
end
#+end_src
** Initialize Nano-Hexapod
*** =initializeStewartPlatform=: Initialize the Stewart Platform structure
#+begin_src matlab :tangle matlab/src/initializeStewartPlatform.m :comments none :mkdirp yes :eval no
function [stewart] = initializeStewartPlatform()
% initializeStewartPlatform - Initialize the stewart structure
%
% Syntax: [stewart] = initializeStewartPlatform(args)
%
% Outputs:
% - stewart - A structure with the following sub-structures:
% - platform_F -
% - platform_M -
% - joints_F -
% - joints_M -
% - struts_F -
% - struts_M -
% - actuators -
% - geometry -
% - properties -
stewart = struct();
stewart.platform_F = struct();
stewart.platform_M = struct();
stewart.joints_F = struct();
stewart.joints_M = struct();
stewart.struts_F = struct();
stewart.struts_M = struct();
stewart.actuators = struct();
stewart.sensors = struct();
stewart.sensors.inertial = struct();
stewart.sensors.force = struct();
stewart.sensors.relative = struct();
stewart.geometry = struct();
stewart.kinematics = struct();
end
#+end_src
*** =initializeFramesPositions=: Initialize the positions of frames {A}, {B}, {F} and {M}
#+begin_src matlab :tangle matlab/src/initializeFramesPositions.m :comments none :mkdirp yes :eval no
function [stewart] = initializeFramesPositions(stewart, args)
% initializeFramesPositions - Initialize the positions of frames {A}, {B}, {F} and {M}
%
% Syntax: [stewart] = initializeFramesPositions(stewart, args)
%
% Inputs:
% - args - Can have the following fields:
% - H [1x1] - Total Height of the Stewart Platform (height from {F} to {M}) [m]
% - MO_B [1x1] - Height of the frame {B} with respect to {M} [m]
%
% Outputs:
% - stewart - A structure with the following fields:
% - geometry.H [1x1] - Total Height of the Stewart Platform [m]
% - geometry.FO_M [3x1] - Position of {M} with respect to {F} [m]
% - platform_M.MO_B [3x1] - Position of {B} with respect to {M} [m]
% - platform_F.FO_A [3x1] - Position of {A} with respect to {F} [m]
arguments
stewart
args.H (1,1) double {mustBeNumeric, mustBePositive} = 90e-3
args.MO_B (1,1) double {mustBeNumeric} = 50e-3
end
H = args.H; % Total Height of the Stewart Platform [m]
FO_M = [0; 0; H]; % Position of {M} with respect to {F} [m]
MO_B = [0; 0; args.MO_B]; % Position of {B} with respect to {M} [m]
FO_A = MO_B + FO_M; % Position of {A} with respect to {F} [m]
stewart.geometry.H = H;
stewart.geometry.FO_M = FO_M;
stewart.platform_M.MO_B = MO_B;
stewart.platform_F.FO_A = FO_A;
end
#+end_src
*** =generateGeneralConfiguration=: Generate a Very General Configuration
#+begin_src matlab :tangle matlab/src/generateGeneralConfiguration.m :comments none :mkdirp yes :eval no
function [stewart] = generateGeneralConfiguration(stewart, args)
% generateGeneralConfiguration - Generate a Very General Configuration
%
% Syntax: [stewart] = generateGeneralConfiguration(stewart, args)
%
% Inputs:
% - args - Can have the following fields:
% - FH [1x1] - Height of the position of the fixed joints with respect to the frame {F} [m]
% - FR [1x1] - Radius of the position of the fixed joints in the X-Y [m]
% - FTh [6x1] - Angles of the fixed joints in the X-Y plane with respect to the X axis [rad]
% - MH [1x1] - Height of the position of the mobile joints with respect to the frame {M} [m]
% - FR [1x1] - Radius of the position of the mobile joints in the X-Y [m]
% - MTh [6x1] - Angles of the mobile joints in the X-Y plane with respect to the X axis [rad]
%
% Outputs:
% - stewart - updated Stewart structure with the added fields:
% - platform_F.Fa [3x6] - Its i'th column is the position vector of joint ai with respect to {F}
% - platform_M.Mb [3x6] - Its i'th column is the position vector of joint bi with respect to {M}
arguments
stewart
args.FH (1,1) double {mustBeNumeric, mustBePositive} = 15e-3
args.FR (1,1) double {mustBeNumeric, mustBePositive} = 115e-3;
args.FTh (6,1) double {mustBeNumeric} = [-10, 10, 120-10, 120+10, 240-10, 240+10]*(pi/180);
args.MH (1,1) double {mustBeNumeric, mustBePositive} = 15e-3
args.MR (1,1) double {mustBeNumeric, mustBePositive} = 90e-3;
args.MTh (6,1) double {mustBeNumeric} = [-60+10, 60-10, 60+10, 180-10, 180+10, -60-10]*(pi/180);
end
Fa = zeros(3,6);
Mb = zeros(3,6);
for i = 1:6
Fa(:,i) = [args.FR*cos(args.FTh(i)); args.FR*sin(args.FTh(i)); args.FH];
Mb(:,i) = [args.MR*cos(args.MTh(i)); args.MR*sin(args.MTh(i)); -args.MH];
end
stewart.platform_F.Fa = Fa;
stewart.platform_M.Mb = Mb;
end
#+end_src
*** =computeJointsPose=: Compute the Pose of the Joints
#+begin_src matlab :tangle matlab/src/computeJointsPose.m :comments none :mkdirp yes :eval no
function [stewart] = computeJointsPose(stewart)
% computeJointsPose -
%
% Syntax: [stewart] = computeJointsPose(stewart)
%
% Inputs:
% - stewart - A structure with the following fields
% - platform_F.Fa [3x6] - Its i'th column is the position vector of joint ai with respect to {F}
% - platform_M.Mb [3x6] - Its i'th column is the position vector of joint bi with respect to {M}
% - platform_F.FO_A [3x1] - Position of {A} with respect to {F}
% - platform_M.MO_B [3x1] - Position of {B} with respect to {M}
% - geometry.FO_M [3x1] - Position of {M} with respect to {F}
%
% Outputs:
% - stewart - A structure with the following added fields
% - geometry.Aa [3x6] - The i'th column is the position of ai with respect to {A}
% - geometry.Ab [3x6] - The i'th column is the position of bi with respect to {A}
% - geometry.Ba [3x6] - The i'th column is the position of ai with respect to {B}
% - geometry.Bb [3x6] - The i'th column is the position of bi with respect to {B}
% - geometry.l [6x1] - The i'th element is the initial length of strut i
% - geometry.As [3x6] - The i'th column is the unit vector of strut i expressed in {A}
% - geometry.Bs [3x6] - The i'th column is the unit vector of strut i expressed in {B}
% - struts_F.l [6x1] - Length of the Fixed part of the i'th strut
% - struts_M.l [6x1] - Length of the Mobile part of the i'th strut
% - platform_F.FRa [3x3x6] - The i'th 3x3 array is the rotation matrix to orientate the bottom of the i'th strut from {F}
% - platform_M.MRb [3x3x6] - The i'th 3x3 array is the rotation matrix to orientate the top of the i'th strut from {M}
assert(isfield(stewart.platform_F, 'Fa'), 'stewart.platform_F should have attribute Fa')
Fa = stewart.platform_F.Fa;
assert(isfield(stewart.platform_M, 'Mb'), 'stewart.platform_M should have attribute Mb')
Mb = stewart.platform_M.Mb;
assert(isfield(stewart.platform_F, 'FO_A'), 'stewart.platform_F should have attribute FO_A')
FO_A = stewart.platform_F.FO_A;
assert(isfield(stewart.platform_M, 'MO_B'), 'stewart.platform_M should have attribute MO_B')
MO_B = stewart.platform_M.MO_B;
assert(isfield(stewart.geometry, 'FO_M'), 'stewart.geometry should have attribute FO_M')
FO_M = stewart.geometry.FO_M;
Aa = Fa - repmat(FO_A, [1, 6]);
Bb = Mb - repmat(MO_B, [1, 6]);
Ab = Bb - repmat(-MO_B-FO_M+FO_A, [1, 6]);
Ba = Aa - repmat( MO_B+FO_M-FO_A, [1, 6]);
As = (Ab - Aa)./vecnorm(Ab - Aa); % As_i is the i'th vector of As
l = vecnorm(Ab - Aa)';
Bs = (Bb - Ba)./vecnorm(Bb - Ba);
FRa = zeros(3,3,6);
MRb = zeros(3,3,6);
for i = 1:6
FRa(:,:,i) = [cross([0;1;0], As(:,i)) , cross(As(:,i), cross([0;1;0], As(:,i))) , As(:,i)];
FRa(:,:,i) = FRa(:,:,i)./vecnorm(FRa(:,:,i));
MRb(:,:,i) = [cross([0;1;0], Bs(:,i)) , cross(Bs(:,i), cross([0;1;0], Bs(:,i))) , Bs(:,i)];
MRb(:,:,i) = MRb(:,:,i)./vecnorm(MRb(:,:,i));
end
stewart.geometry.Aa = Aa;
stewart.geometry.Ab = Ab;
stewart.geometry.Ba = Ba;
stewart.geometry.Bb = Bb;
stewart.geometry.As = As;
stewart.geometry.Bs = Bs;
stewart.geometry.l = l;
stewart.struts_F.l = l/2;
stewart.struts_M.l = l/2;
stewart.platform_F.FRa = FRa;
stewart.platform_M.MRb = MRb;
end
#+end_src
*** =initializeCylindricalPlatforms=: Initialize the geometry of the Fixed and Mobile Platforms
#+begin_src matlab :tangle matlab/src/initializeCylindricalPlatforms.m :comments none :mkdirp yes :eval no
function [stewart] = initializeCylindricalPlatforms(stewart, args)
% initializeCylindricalPlatforms - Initialize the geometry of the Fixed and Mobile Platforms
%
% Syntax: [stewart] = initializeCylindricalPlatforms(args)
%
% Inputs:
% - args - Structure with the following fields:
% - Fpm [1x1] - Fixed Platform Mass [kg]
% - Fph [1x1] - Fixed Platform Height [m]
% - Fpr [1x1] - Fixed Platform Radius [m]
% - Mpm [1x1] - Mobile Platform Mass [kg]
% - Mph [1x1] - Mobile Platform Height [m]
% - Mpr [1x1] - Mobile Platform Radius [m]
%
% Outputs:
% - stewart - updated Stewart structure with the added fields:
% - platform_F [struct] - structure with the following fields:
% - type = 1
% - M [1x1] - Fixed Platform Mass [kg]
% - I [3x3] - Fixed Platform Inertia matrix [kg*m^2]
% - H [1x1] - Fixed Platform Height [m]
% - R [1x1] - Fixed Platform Radius [m]
% - platform_M [struct] - structure with the following fields:
% - M [1x1] - Mobile Platform Mass [kg]
% - I [3x3] - Mobile Platform Inertia matrix [kg*m^2]
% - H [1x1] - Mobile Platform Height [m]
% - R [1x1] - Mobile Platform Radius [m]
arguments
stewart
args.Fpm (1,1) double {mustBeNumeric, mustBePositive} = 1
args.Fph (1,1) double {mustBeNumeric, mustBePositive} = 10e-3
args.Fpr (1,1) double {mustBeNumeric, mustBePositive} = 125e-3
args.Mpm (1,1) double {mustBeNumeric, mustBePositive} = 1
args.Mph (1,1) double {mustBeNumeric, mustBePositive} = 10e-3
args.Mpr (1,1) double {mustBeNumeric, mustBePositive} = 100e-3
end
I_F = diag([1/12*args.Fpm * (3*args.Fpr^2 + args.Fph^2), ...
1/12*args.Fpm * (3*args.Fpr^2 + args.Fph^2), ...
1/2 *args.Fpm * args.Fpr^2]);
I_M = diag([1/12*args.Mpm * (3*args.Mpr^2 + args.Mph^2), ...
1/12*args.Mpm * (3*args.Mpr^2 + args.Mph^2), ...
1/2 *args.Mpm * args.Mpr^2]);
stewart.platform_F.type = 1;
stewart.platform_F.I = I_F;
stewart.platform_F.M = args.Fpm;
stewart.platform_F.R = args.Fpr;
stewart.platform_F.H = args.Fph;
stewart.platform_M.type = 1;
stewart.platform_M.I = I_M;
stewart.platform_M.M = args.Mpm;
stewart.platform_M.R = args.Mpr;
stewart.platform_M.H = args.Mph;
end
#+end_src
*** =initializeCylindricalStruts=: Define the inertia of cylindrical struts
#+begin_src matlab :tangle matlab/src/initializeCylindricalStruts.m :comments none :mkdirp yes :eval no
function [stewart] = initializeCylindricalStruts(stewart, args)
% initializeCylindricalStruts - Define the mass and moment of inertia of cylindrical struts
%
% Syntax: [stewart] = initializeCylindricalStruts(args)
%
% Inputs:
% - args - Structure with the following fields:
% - Fsm [1x1] - Mass of the Fixed part of the struts [kg]
% - Fsh [1x1] - Height of cylinder for the Fixed part of the struts [m]
% - Fsr [1x1] - Radius of cylinder for the Fixed part of the struts [m]
% - Msm [1x1] - Mass of the Mobile part of the struts [kg]
% - Msh [1x1] - Height of cylinder for the Mobile part of the struts [m]
% - Msr [1x1] - Radius of cylinder for the Mobile part of the struts [m]
%
% Outputs:
% - stewart - updated Stewart structure with the added fields:
% - struts_F [struct] - structure with the following fields:
% - M [6x1] - Mass of the Fixed part of the struts [kg]
% - I [3x3x6] - Moment of Inertia for the Fixed part of the struts [kg*m^2]
% - H [6x1] - Height of cylinder for the Fixed part of the struts [m]
% - R [6x1] - Radius of cylinder for the Fixed part of the struts [m]
% - struts_M [struct] - structure with the following fields:
% - M [6x1] - Mass of the Mobile part of the struts [kg]
% - I [3x3x6] - Moment of Inertia for the Mobile part of the struts [kg*m^2]
% - H [6x1] - Height of cylinder for the Mobile part of the struts [m]
% - R [6x1] - Radius of cylinder for the Mobile part of the struts [m]
arguments
stewart
args.Fsm (1,1) double {mustBeNumeric, mustBePositive} = 0.1
args.Fsh (1,1) double {mustBeNumeric, mustBePositive} = 50e-3
args.Fsr (1,1) double {mustBeNumeric, mustBePositive} = 5e-3
args.Msm (1,1) double {mustBeNumeric, mustBePositive} = 0.1
args.Msh (1,1) double {mustBeNumeric, mustBePositive} = 50e-3
args.Msr (1,1) double {mustBeNumeric, mustBePositive} = 5e-3
end
stewart.struts_M.type = 1;
%% Compute the properties of the cylindrical struts
Fsm = args.Fsm;
Fsh = args.Fsh;
Fsr = args.Fsr;
Msm = args.Msm;
Msh = args.Msh;
Msr = args.Msr;
I_F = [1/12 * Fsm * (3*Fsr^2 + Fsh^2), ...
1/12 * Fsm * (3*Fsr^2 + Fsh^2), ...
1/2 * Fsm * Fsr^2];
I_M = [1/12 * Msm * (3*Msr^2 + Msh^2), ...
1/12 * Msm * (3*Msr^2 + Msh^2), ...
1/2 * Msm * Msr^2];
stewart.struts_M.I = I_M;
stewart.struts_F.I = I_F;
stewart.struts_M.M = args.Msm;
stewart.struts_M.R = args.Msr;
stewart.struts_M.H = args.Msh;
stewart.struts_F.type = 1;
stewart.struts_F.M = args.Fsm;
stewart.struts_F.R = args.Fsr;
stewart.struts_F.H = args.Fsh;
end
#+end_src
*** =initializeStrutDynamics=: Add Stiffness and Damping properties of each strut
#+begin_src matlab :tangle matlab/src/initializeStrutDynamics.m :comments none :mkdirp yes :eval no
function [stewart] = initializeStrutDynamics(stewart, args)
% initializeStrutDynamics - Add Stiffness and Damping properties of each strut
%
% Syntax: [stewart] = initializeStrutDynamics(args)
%
% Inputs:
% - args - Structure with the following fields:
% - K [6x1] - Stiffness of each strut [N/m]
% - C [6x1] - Damping of each strut [N/(m/s)]
%
% Outputs:
% - stewart - updated Stewart structure with the added fields:
% - actuators.type = 1
% - actuators.K [6x1] - Stiffness of each strut [N/m]
% - actuators.C [6x1] - Damping of each strut [N/(m/s)]
arguments
stewart
args.type char {mustBeMember(args.type,{'1dof', '2dof', 'flexible'})} = '1dof'
args.k (1,1) double {mustBeNumeric, mustBeNonnegative} = 20e6
args.kp (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
args.ke (1,1) double {mustBeNumeric, mustBeNonnegative} = 5e6
args.ka (1,1) double {mustBeNumeric, mustBeNonnegative} = 60e6
args.c (1,1) double {mustBeNumeric, mustBeNonnegative} = 2e1
args.cp (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
args.ce (1,1) double {mustBeNumeric, mustBeNonnegative} = 1e6
args.ca (1,1) double {mustBeNumeric, mustBeNonnegative} = 10
args.F_gain (1,1) double {mustBeNumeric} = 1
args.me (1,1) double {mustBeNumeric} = 0.01
args.ma (1,1) double {mustBeNumeric} = 0.01
end
if strcmp(args.type, '1dof')
stewart.actuators.type = 1;
elseif strcmp(args.type, '2dof')
stewart.actuators.type = 2;
elseif strcmp(args.type, 'flexible')
stewart.actuators.type = 3;
end
stewart.actuators.k = args.k;
stewart.actuators.c = args.c;
% Parallel stiffness
stewart.actuators.kp = args.kp;
stewart.actuators.cp = args.cp;
stewart.actuators.ka = args.ka;
stewart.actuators.ca = args.ca;
stewart.actuators.ke = args.ke;
stewart.actuators.ce = args.ce;
stewart.actuators.F_gain = args.F_gain;
stewart.actuators.ma = args.ma;
stewart.actuators.me = args.me;
end
#+end_src
*** =initializeJointDynamics=: Add Stiffness and Damping properties for spherical joints
#+begin_src matlab :tangle matlab/src/initializeJointDynamics.m :comments none :mkdirp yes :eval no
function [stewart] = initializeJointDynamics(stewart, args)
% initializeJointDynamics - Add Stiffness and Damping properties for the spherical joints
%
% Syntax: [stewart] = initializeJointDynamics(args)
%
% Inputs:
% - args - Structure with the following fields:
% - type_F - 'universal', 'spherical', 'universal_p', 'spherical_p'
% - type_M - 'universal', 'spherical', 'universal_p', 'spherical_p'
% - Kf_M [6x1] - Bending (Rx, Ry) Stiffness for each top joints [(N.m)/rad]
% - Kt_M [6x1] - Torsion (Rz) Stiffness for each top joints [(N.m)/rad]
% - Cf_M [6x1] - Bending (Rx, Ry) Damping of each top joint [(N.m)/(rad/s)]
% - Ct_M [6x1] - Torsion (Rz) Damping of each top joint [(N.m)/(rad/s)]
% - Kf_F [6x1] - Bending (Rx, Ry) Stiffness for each bottom joints [(N.m)/rad]
% - Kt_F [6x1] - Torsion (Rz) Stiffness for each bottom joints [(N.m)/rad]
% - Cf_F [6x1] - Bending (Rx, Ry) Damping of each bottom joint [(N.m)/(rad/s)]
% - Cf_F [6x1] - Torsion (Rz) Damping of each bottom joint [(N.m)/(rad/s)]
%
% Outputs:
% - stewart - updated Stewart structure with the added fields:
% - stewart.joints_F and stewart.joints_M:
% - type - 1 (universal), 2 (spherical), 3 (universal perfect), 4 (spherical perfect)
% - Kx, Ky, Kz [6x1] - Translation (Tx, Ty, Tz) Stiffness [N/m]
% - Kf [6x1] - Flexion (Rx, Ry) Stiffness [(N.m)/rad]
% - Kt [6x1] - Torsion (Rz) Stiffness [(N.m)/rad]
% - Cx, Cy, Cz [6x1] - Translation (Rx, Ry) Damping [N/(m/s)]
% - Cf [6x1] - Flexion (Rx, Ry) Damping [(N.m)/(rad/s)]
% - Cb [6x1] - Torsion (Rz) Damping [(N.m)/(rad/s)]
arguments
stewart
args.type_F char {mustBeMember(args.type_F,{'2dof', '3dof', '4dof', '6dof', 'flexible'})} = '2dof'
args.type_M char {mustBeMember(args.type_M,{'2dof', '3dof', '4dof', '6dof', 'flexible'})} = '3dof'
args.Kf_M (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
args.Cf_M (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
args.Kt_M (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
args.Ct_M (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
args.Kf_F (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
args.Cf_F (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
args.Kt_F (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
args.Ct_F (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
args.Ka_F (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
args.Ca_F (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
args.Kr_F (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
args.Cr_F (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
args.Ka_M (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
args.Ca_M (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
args.Kr_M (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
args.Cr_M (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
args.K_M double {mustBeNumeric} = zeros(6,6)
args.M_M double {mustBeNumeric} = zeros(6,6)
args.n_xyz_M double {mustBeNumeric} = zeros(2,3)
args.xi_M double {mustBeNumeric} = 0.1
args.step_file_M char {} = ''
args.K_F double {mustBeNumeric} = zeros(6,6)
args.M_F double {mustBeNumeric} = zeros(6,6)
args.n_xyz_F double {mustBeNumeric} = zeros(2,3)
args.xi_F double {mustBeNumeric} = 0.1
args.step_file_F char {} = ''
end
switch args.type_F
case '2dof'
stewart.joints_F.type = 1;
case '3dof'
stewart.joints_F.type = 2;
case '4dof'
stewart.joints_F.type = 3;
case '6dof'
stewart.joints_F.type = 4;
case 'flexible'
stewart.joints_F.type = 5;
otherwise
error("joints_F are not correctly defined")
end
switch args.type_M
case '2dof'
stewart.joints_M.type = 1;
case '3dof'
stewart.joints_M.type = 2;
case '4dof'
stewart.joints_M.type = 3;
case '6dof'
stewart.joints_M.type = 4;
case 'flexible'
stewart.joints_M.type = 5;
otherwise
error("joints_M are not correctly defined")
end
stewart.joints_M.Ka = args.Ka_M;
stewart.joints_M.Kr = args.Kr_M;
stewart.joints_F.Ka = args.Ka_F;
stewart.joints_F.Kr = args.Kr_F;
stewart.joints_M.Ca = args.Ca_M;
stewart.joints_M.Cr = args.Cr_M;
stewart.joints_F.Ca = args.Ca_F;
stewart.joints_F.Cr = args.Cr_F;
stewart.joints_M.Kf = args.Kf_M;
stewart.joints_M.Kt = args.Kt_M;
stewart.joints_F.Kf = args.Kf_F;
stewart.joints_F.Kt = args.Kt_F;
stewart.joints_M.Cf = args.Cf_M;
stewart.joints_M.Ct = args.Ct_M;
stewart.joints_F.Cf = args.Cf_F;
stewart.joints_F.Ct = args.Ct_F;
stewart.joints_F.M = args.M_F;
stewart.joints_F.K = args.K_F;
stewart.joints_F.n_xyz = args.n_xyz_F;
stewart.joints_F.xi = args.xi_F;
stewart.joints_F.xi = args.xi_F;
stewart.joints_F.step_file = args.step_file_F;
stewart.joints_M.M = args.M_M;
stewart.joints_M.K = args.K_M;
stewart.joints_M.n_xyz = args.n_xyz_M;
stewart.joints_M.xi = args.xi_M;
stewart.joints_M.step_file = args.step_file_M;
end
#+end_src
*** =initializeStewartPose=: Determine the initial stroke in each leg to have the wanted pose
#+begin_src matlab :tangle matlab/src/initializeStewartPose.m :comments none :mkdirp yes :eval no
function [stewart] = initializeStewartPose(stewart, args)
% initializeStewartPose - Determine the initial stroke in each leg to have the wanted pose
% It uses the inverse kinematic
%
% Syntax: [stewart] = initializeStewartPose(stewart, args)
%
% Inputs:
% - stewart - A structure with the following fields
% - Aa [3x6] - The positions ai expressed in {A}
% - Bb [3x6] - The positions bi expressed in {B}
% - args - Can have the following fields:
% - AP [3x1] - The wanted position of {B} with respect to {A}
% - ARB [3x3] - The rotation matrix that gives the wanted orientation of {B} with respect to {A}
%
% Outputs:
% - stewart - updated Stewart structure with the added fields:
% - actuators.Leq [6x1] - The 6 needed displacement of the struts from the initial position in [m] to have the wanted pose of {B} w.r.t. {A}
arguments
stewart
args.AP (3,1) double {mustBeNumeric} = zeros(3,1)
args.ARB (3,3) double {mustBeNumeric} = eye(3)
end
[Li, dLi] = inverseKinematics(stewart, 'AP', args.AP, 'ARB', args.ARB);
stewart.actuators.Leq = dLi;
end
#+end_src
*** =computeJacobian=: Compute the Jacobian Matrix
#+begin_src matlab :tangle matlab/src/computeJacobian.m :comments none :mkdirp yes :eval no
function [stewart] = computeJacobian(stewart)
% computeJacobian -
%
% Syntax: [stewart] = computeJacobian(stewart)
%
% Inputs:
% - stewart - With at least the following fields:
% - geometry.As [3x6] - The 6 unit vectors for each strut expressed in {A}
% - geometry.Ab [3x6] - The 6 position of the joints bi expressed in {A}
% - actuators.K [6x1] - Total stiffness of the actuators
%
% Outputs:
% - stewart - With the 3 added field:
% - geometry.J [6x6] - The Jacobian Matrix
% - geometry.K [6x6] - The Stiffness Matrix
% - geometry.C [6x6] - The Compliance Matrix
assert(isfield(stewart.geometry, 'As'), 'stewart.geometry should have attribute As')
As = stewart.geometry.As;
assert(isfield(stewart.geometry, 'Ab'), 'stewart.geometry should have attribute Ab')
Ab = stewart.geometry.Ab;
assert(isfield(stewart.actuators, 'k'), 'stewart.actuators should have attribute k')
Ki = stewart.actuators.k;
J = [As' , cross(Ab, As)'];
K = J'*diag(Ki)*J;
C = inv(K);
stewart.geometry.J = J;
stewart.geometry.K = K;
stewart.geometry.C = C;
end
#+end_src
*** =inverseKinematics=: Compute Inverse Kinematics
#+begin_src matlab :tangle matlab/src/inverseKinematics.m :comments none :mkdirp yes :eval no
function [Li, dLi] = inverseKinematics(stewart, args)
% inverseKinematics - Compute the needed length of each strut to have the wanted position and orientation of {B} with respect to {A}
%
% Syntax: [stewart] = inverseKinematics(stewart)
%
% Inputs:
% - stewart - A structure with the following fields
% - geometry.Aa [3x6] - The positions ai expressed in {A}
% - geometry.Bb [3x6] - The positions bi expressed in {B}
% - geometry.l [6x1] - Length of each strut
% - args - Can have the following fields:
% - AP [3x1] - The wanted position of {B} with respect to {A}
% - ARB [3x3] - The rotation matrix that gives the wanted orientation of {B} with respect to {A}
%
% Outputs:
% - Li [6x1] - The 6 needed length of the struts in [m] to have the wanted pose of {B} w.r.t. {A}
% - dLi [6x1] - The 6 needed displacement of the struts from the initial position in [m] to have the wanted pose of {B} w.r.t. {A}
arguments
stewart
args.AP (3,1) double {mustBeNumeric} = zeros(3,1)
args.ARB (3,3) double {mustBeNumeric} = eye(3)
end
assert(isfield(stewart.geometry, 'Aa'), 'stewart.geometry should have attribute Aa')
Aa = stewart.geometry.Aa;
assert(isfield(stewart.geometry, 'Bb'), 'stewart.geometry should have attribute Bb')
Bb = stewart.geometry.Bb;
assert(isfield(stewart.geometry, 'l'), 'stewart.geometry should have attribute l')
l = stewart.geometry.l;
Li = sqrt(args.AP'*args.AP + diag(Bb'*Bb) + diag(Aa'*Aa) - (2*args.AP'*Aa)' + (2*args.AP'*(args.ARB*Bb))' - diag(2*(args.ARB*Bb)'*Aa));
dLi = Li-l;
end
#+end_src
*** =displayArchitecture=: 3D plot of the Stewart platform architecture
:PROPERTIES:
:header-args:matlab+: :tangle matlab/src/displayArchitecture.m
:header-args:matlab+: :comments none :mkdirp yes :eval no
:END:
<>
This Matlab function is accessible [[file:../src/displayArchitecture.m][here]].
**** Function description
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
function [] = displayArchitecture(stewart, args)
% displayArchitecture - 3D plot of the Stewart platform architecture
%
% Syntax: [] = displayArchitecture(args)
%
% Inputs:
% - stewart
% - args - Structure with the following fields:
% - AP [3x1] - The wanted position of {B} with respect to {A}
% - ARB [3x3] - The rotation matrix that gives the wanted orientation of {B} with respect to {A}
% - ARB [3x3] - The rotation matrix that gives the wanted orientation of {B} with respect to {A}
% - F_color [color] - Color used for the Fixed elements
% - M_color [color] - Color used for the Mobile elements
% - L_color [color] - Color used for the Legs elements
% - frames [true/false] - Display the Frames
% - legs [true/false] - Display the Legs
% - joints [true/false] - Display the Joints
% - labels [true/false] - Display the Labels
% - platforms [true/false] - Display the Platforms
% - views ['all', 'xy', 'yz', 'xz', 'default'] -
%
% Outputs:
#+end_src
**** Optional Parameters
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
arguments
stewart
args.AP (3,1) double {mustBeNumeric} = zeros(3,1)
args.ARB (3,3) double {mustBeNumeric} = eye(3)
args.F_color = [0 0.4470 0.7410]
args.M_color = [0.8500 0.3250 0.0980]
args.L_color = [0 0 0]
args.frames logical {mustBeNumericOrLogical} = true
args.legs logical {mustBeNumericOrLogical} = true
args.joints logical {mustBeNumericOrLogical} = true
args.labels logical {mustBeNumericOrLogical} = true
args.platforms logical {mustBeNumericOrLogical} = true
args.views char {mustBeMember(args.views,{'all', 'xy', 'xz', 'yz', 'default'})} = 'default'
end
#+end_src
**** Check the =stewart= structure elements
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
assert(isfield(stewart.platform_F, 'FO_A'), 'stewart.platform_F should have attribute FO_A')
FO_A = stewart.platform_F.FO_A;
assert(isfield(stewart.platform_M, 'MO_B'), 'stewart.platform_M should have attribute MO_B')
MO_B = stewart.platform_M.MO_B;
assert(isfield(stewart.geometry, 'H'), 'stewart.geometry should have attribute H')
H = stewart.geometry.H;
assert(isfield(stewart.platform_F, 'Fa'), 'stewart.platform_F should have attribute Fa')
Fa = stewart.platform_F.Fa;
assert(isfield(stewart.platform_M, 'Mb'), 'stewart.platform_M should have attribute Mb')
Mb = stewart.platform_M.Mb;
#+end_src
**** Figure Creation, Frames and Homogeneous transformations
:PROPERTIES:
:UNNUMBERED: t
:END:
The reference frame of the 3d plot corresponds to the frame $\{F\}$.
#+begin_src matlab
if ~strcmp(args.views, 'all')
figure;
else
f = figure('visible', 'off');
end
hold on;
#+end_src
We first compute homogeneous matrices that will be useful to position elements on the figure where the reference frame is $\{F\}$.
#+begin_src matlab
FTa = [eye(3), FO_A; ...
zeros(1,3), 1];
ATb = [args.ARB, args.AP; ...
zeros(1,3), 1];
BTm = [eye(3), -MO_B; ...
zeros(1,3), 1];
FTm = FTa*ATb*BTm;
#+end_src
Let's define a parameter that define the length of the unit vectors used to display the frames.
#+begin_src matlab
d_unit_vector = H/4;
#+end_src
Let's define a parameter used to position the labels with respect to the center of the element.
#+begin_src matlab
d_label = H/20;
#+end_src
**** Fixed Base elements
:PROPERTIES:
:UNNUMBERED: t
:END:
Let's first plot the frame $\{F\}$.
#+begin_src matlab
Ff = [0, 0, 0];
if args.frames
quiver3(Ff(1)*ones(1,3), Ff(2)*ones(1,3), Ff(3)*ones(1,3), ...
[d_unit_vector 0 0], [0 d_unit_vector 0], [0 0 d_unit_vector], '-', 'Color', args.F_color)
if args.labels
text(Ff(1) + d_label, ...
Ff(2) + d_label, ...
Ff(3) + d_label, '$\{F\}$', 'Color', args.F_color);
end
end
#+end_src
Now plot the frame $\{A\}$ fixed to the Base.
#+begin_src matlab
if args.frames
quiver3(FO_A(1)*ones(1,3), FO_A(2)*ones(1,3), FO_A(3)*ones(1,3), ...
[d_unit_vector 0 0], [0 d_unit_vector 0], [0 0 d_unit_vector], '-', 'Color', args.F_color)
if args.labels
text(FO_A(1) + d_label, ...
FO_A(2) + d_label, ...
FO_A(3) + d_label, '$\{A\}$', 'Color', args.F_color);
end
end
#+end_src
Let's then plot the circle corresponding to the shape of the Fixed base.
#+begin_src matlab
if args.platforms && stewart.platform_F.type == 1
theta = [0:0.01:2*pi+0.01]; % Angles [rad]
v = null([0; 0; 1]'); % Two vectors that are perpendicular to the circle normal
center = [0; 0; 0]; % Center of the circle
radius = stewart.platform_F.R; % Radius of the circle [m]
points = center*ones(1, length(theta)) + radius*(v(:,1)*cos(theta) + v(:,2)*sin(theta));
plot3(points(1,:), ...
points(2,:), ...
points(3,:), '-', 'Color', args.F_color);
end
#+end_src
Let's now plot the position and labels of the Fixed Joints
#+begin_src matlab
if args.joints
scatter3(Fa(1,:), ...
Fa(2,:), ...
Fa(3,:), 'MarkerEdgeColor', args.F_color);
if args.labels
for i = 1:size(Fa,2)
text(Fa(1,i) + d_label, ...
Fa(2,i), ...
Fa(3,i), sprintf('$a_{%i}$', i), 'Color', args.F_color);
end
end
end
#+end_src
**** Mobile Platform elements
:PROPERTIES:
:UNNUMBERED: t
:END:
Plot the frame $\{M\}$.
#+begin_src matlab
Fm = FTm*[0; 0; 0; 1]; % Get the position of frame {M} w.r.t. {F}
if args.frames
FM_uv = FTm*[d_unit_vector*eye(3); zeros(1,3)]; % Rotated Unit vectors
quiver3(Fm(1)*ones(1,3), Fm(2)*ones(1,3), Fm(3)*ones(1,3), ...
FM_uv(1,1:3), FM_uv(2,1:3), FM_uv(3,1:3), '-', 'Color', args.M_color)
if args.labels
text(Fm(1) + d_label, ...
Fm(2) + d_label, ...
Fm(3) + d_label, '$\{M\}$', 'Color', args.M_color);
end
end
#+end_src
Plot the frame $\{B\}$.
#+begin_src matlab
FB = FO_A + args.AP;
if args.frames
FB_uv = FTm*[d_unit_vector*eye(3); zeros(1,3)]; % Rotated Unit vectors
quiver3(FB(1)*ones(1,3), FB(2)*ones(1,3), FB(3)*ones(1,3), ...
FB_uv(1,1:3), FB_uv(2,1:3), FB_uv(3,1:3), '-', 'Color', args.M_color)
if args.labels
text(FB(1) - d_label, ...
FB(2) + d_label, ...
FB(3) + d_label, '$\{B\}$', 'Color', args.M_color);
end
end
#+end_src
Let's then plot the circle corresponding to the shape of the Mobile platform.
#+begin_src matlab
if args.platforms && stewart.platform_M.type == 1
theta = [0:0.01:2*pi+0.01]; % Angles [rad]
v = null((FTm(1:3,1:3)*[0;0;1])'); % Two vectors that are perpendicular to the circle normal
center = Fm(1:3); % Center of the circle
radius = stewart.platform_M.R; % Radius of the circle [m]
points = center*ones(1, length(theta)) + radius*(v(:,1)*cos(theta) + v(:,2)*sin(theta));
plot3(points(1,:), ...
points(2,:), ...
points(3,:), '-', 'Color', args.M_color);
end
#+end_src
Plot the position and labels of the rotation joints fixed to the mobile platform.
#+begin_src matlab
if args.joints
Fb = FTm*[Mb;ones(1,6)];
scatter3(Fb(1,:), ...
Fb(2,:), ...
Fb(3,:), 'MarkerEdgeColor', args.M_color);
if args.labels
for i = 1:size(Fb,2)
text(Fb(1,i) + d_label, ...
Fb(2,i), ...
Fb(3,i), sprintf('$b_{%i}$', i), 'Color', args.M_color);
end
end
end
#+end_src
**** Legs
:PROPERTIES:
:UNNUMBERED: t
:END:
Plot the legs connecting the joints of the fixed base to the joints of the mobile platform.
#+begin_src matlab
if args.legs
for i = 1:6
plot3([Fa(1,i), Fb(1,i)], ...
[Fa(2,i), Fb(2,i)], ...
[Fa(3,i), Fb(3,i)], '-', 'Color', args.L_color);
if args.labels
text((Fa(1,i)+Fb(1,i))/2 + d_label, ...
(Fa(2,i)+Fb(2,i))/2, ...
(Fa(3,i)+Fb(3,i))/2, sprintf('$%i$', i), 'Color', args.L_color);
end
end
end
#+end_src
**** Figure parameters
#+begin_src matlab
switch args.views
case 'default'
view([1 -0.6 0.4]);
case 'xy'
view([0 0 1]);
case 'xz'
view([0 -1 0]);
case 'yz'
view([1 0 0]);
end
axis equal;
axis off;
#+end_src
**** Subplots
#+begin_src matlab
if strcmp(args.views, 'all')
hAx = findobj('type', 'axes');
figure;
s1 = subplot(2,2,1);
copyobj(get(hAx(1), 'Children'), s1);
view([0 0 1]);
axis equal;
axis off;
title('Top')
s2 = subplot(2,2,2);
copyobj(get(hAx(1), 'Children'), s2);
view([1 -0.6 0.4]);
axis equal;
axis off;
s3 = subplot(2,2,3);
copyobj(get(hAx(1), 'Children'), s3);
view([1 0 0]);
axis equal;
axis off;
title('Front')
s4 = subplot(2,2,4);
copyobj(get(hAx(1), 'Children'), s4);
view([0 -1 0]);
axis equal;
axis off;
title('Side')
close(f);
end
#+end_src
*** =describeStewartPlatform=: Display some text describing the current defined Stewart Platform
:PROPERTIES:
:header-args:matlab+: :tangle matlab/src/describeStewartPlatform.m
:header-args:matlab+: :comments none :mkdirp yes :eval no
:END:
<>
This Matlab function is accessible [[file:../src/describeStewartPlatform.m][here]].
**** Function description
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
function [] = describeStewartPlatform(stewart)
% describeStewartPlatform - Display some text describing the current defined Stewart Platform
%
% Syntax: [] = describeStewartPlatform(args)
%
% Inputs:
% - stewart
%
% Outputs:
#+end_src
**** Optional Parameters
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
arguments
stewart
end
#+end_src
**** Geometry
#+begin_src matlab
fprintf('GEOMETRY:\n')
fprintf('- The height between the fixed based and the top platform is %.3g [mm].\n', 1e3*stewart.geometry.H)
if stewart.platform_M.MO_B(3) > 0
fprintf('- Frame {A} is located %.3g [mm] above the top platform.\n', 1e3*stewart.platform_M.MO_B(3))
else
fprintf('- Frame {A} is located %.3g [mm] below the top platform.\n', - 1e3*stewart.platform_M.MO_B(3))
end
fprintf('- The initial length of the struts are:\n')
fprintf('\t %.3g, %.3g, %.3g, %.3g, %.3g, %.3g [mm]\n', 1e3*stewart.geometry.l)
fprintf('\n')
#+end_src
**** Actuators
#+begin_src matlab
fprintf('ACTUATORS:\n')
if stewart.actuators.type == 1
fprintf('- The actuators are modelled as 1DoF.\n')
fprintf('- The Stiffness and Damping of each actuators is:\n')
fprintf('\t k = %.0e [N/m] \t c = %.0e [N/(m/s)]\n', stewart.actuators.k(1), stewart.actuators.c(1))
if stewart.actuators.kp > 0
fprintf('\t Added parallel stiffness: kp = %.0e [N/m] \t c = %.0e [N/(m/s)]\n', stewart.actuators.kp(1))
end
elseif stewart.actuators.type == 2
fprintf('- The actuators are modelled as 2DoF (APA).\n')
fprintf('- The vertical stiffness and damping contribution of the piezoelectric stack is:\n')
fprintf('\t ka = %.0e [N/m] \t ca = %.0e [N/(m/s)]\n', stewart.actuators.ka(1), stewart.actuators.ca(1))
fprintf('- Vertical stiffness when the piezoelectric stack is removed is:\n')
fprintf('\t kr = %.0e [N/m] \t cr = %.0e [N/(m/s)]\n', stewart.actuators.kr(1), stewart.actuators.cr(1))
elseif stewart.actuators.type == 3
fprintf('- The actuators are modelled with a flexible element (FEM).\n')
end
fprintf('\n')
#+end_src
**** Joints
#+begin_src matlab
fprintf('JOINTS:\n')
#+end_src
Type of the joints on the fixed base.
#+begin_src matlab
switch stewart.joints_F.type
case 1
fprintf('- The joints on the fixed based are universal joints (2DoF)\n')
case 2
fprintf('- The joints on the fixed based are spherical joints (3DoF)\n')
end
#+end_src
Type of the joints on the mobile platform.
#+begin_src matlab
switch stewart.joints_M.type
case 1
fprintf('- The joints on the mobile based are universal joints (2DoF)\n')
case 2
fprintf('- The joints on the mobile based are spherical joints (3DoF)\n')
end
#+end_src
Position of the fixed joints
#+begin_src matlab
fprintf('- The position of the joints on the fixed based with respect to {F} are (in [mm]):\n')
fprintf('\t % .3g \t % .3g \t % .3g\n', 1e3*stewart.platform_F.Fa)
#+end_src
Position of the mobile joints
#+begin_src matlab
fprintf('- The position of the joints on the mobile based with respect to {M} are (in [mm]):\n')
fprintf('\t % .3g \t % .3g \t % .3g\n', 1e3*stewart.platform_M.Mb)
fprintf('\n')
#+end_src
**** Kinematics
#+begin_src matlab
fprintf('KINEMATICS:\n')
if isfield(stewart.kinematics, 'K')
fprintf('- The Stiffness matrix K is (in [N/m]):\n')
fprintf('\t % .0e \t % .0e \t % .0e \t % .0e \t % .0e \t % .0e\n', stewart.kinematics.K)
end
if isfield(stewart.kinematics, 'C')
fprintf('- The Damping matrix C is (in [m/N]):\n')
fprintf('\t % .0e \t % .0e \t % .0e \t % .0e \t % .0e \t % .0e\n', stewart.kinematics.C)
end
#+end_src