#+TITLE: Simscape Model - Micro Station
:DRAWER:
#+LANGUAGE: en
#+EMAIL: dehaeze.thomas@gmail.com
#+AUTHOR: Dehaeze Thomas

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* Build                                                             :noexport:
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* Notes                                                             :noexport:
** Notes
Prefix is =ustation=

From modal analysis: validation of the multi-body model.

*Goals*:
- *Modelling of the micro-station*: Kinematics + Dynamics + Disturbances
- Kinematics of each stage
- Modelling: solid bodies + joints. Show what is used for each stage
- Correlation with the dynamical measurements
- Inclusion of disturbances (correlation with measurements)

*Notes*:
- Do not talk about Nano-Hexapod yet
- Do not talk about external metrology
- Therefore, do not talk about computation of the sample position / error

** DONE [#B] Put colors for each different stage
CLOSED: [2024-10-30 Wed 14:07]

** DONE [#B] Delete Gravity compensation Stage
CLOSED: [2024-10-30 Wed 13:51]

** DONE [#B] Maybe make a simpler Simscape model for this report
CLOSED: [2024-10-30 Wed 13:51]

** DONE [#A] Open the Simscape model and verify it all works
CLOSED: [2024-10-30 Wed 13:33] SCHEDULED: <2024-10-30 Wed>

#+begin_src matlab
initializeSimscapeConfiguration('gravity', false);
initializeLoggingConfiguration('log', 'none');

initializeGround(      'type', 'rigid');
initializeGranite(     'type', 'flexible');
initializeTy(          'type', 'flexible');
initializeRy(          'type', 'flexible');
initializeRz(          'type', 'flexible');
initializeMicroHexapod('type', 'flexible');

initializeReferences();
initializeDisturbances('enable', false);
#+end_src

#+begin_src matlab

% initializeAxisc(       'type', 'flexible');

% initializeMirror(      'type', 'none');
% initializeNanoHexapod( 'type', 'none');
% initializeSample(      'type', 'none');

initializeController(  'type', 'open-loop');

#+end_src

** DONE [#B] Just keep smallest number of variants for each stage
CLOSED: [2024-10-30 Wed 16:15]

- [ ] none
- [ ] rigid
- [ ] flexible
- [X] Init => Removed
- [X] modal analysis => Removed

** DONE [#A] Verify that we get "correct" compliance
CLOSED: [2024-10-30 Wed 21:53] SCHEDULED: <2024-10-30 Wed>

- [ ] Find the compliance measurements
  - The one from modal analysis
  - [[file:~/Cloud/work-projects/ID31-NASS/matlab/nass-measurements/2018-01-12 - Marc]] : Compliance measurement in X,Y,Z with geophone, marble not glued
  - [[file:~/Cloud/work-projects/ID31-NASS/matlab/nass-measurements/2018-10-12 - Marc]]: same but in the hutch with glued marble
  - *08/2020*: [[file:~/Cloud/work-projects/ID31-NASS/matlab/nass-measurements/micro-station-compliance/index.org::+TITLE: Compliance Measurement of the Micro Station]]
- [ ] See if it matches somehow the current model
- [ ] If not, see if model parameters can be tuned to have better match
      For instance from values here: file:/home/thomas/Cloud/meetings/esrf-meetings/2018-04-24-Simscape-Model/2018-04-24-Simscape-Model.pdf

** DONE [#B] Make sure to well model the micro-hexapod
CLOSED: [2024-10-31 Thu 10:36] SCHEDULED: <2024-10-31 Thu>

- [ ] Find total mass (should be 37 kg)
- [ ] Find mass of top platform, bottom platform, struts
- [ ] Estimation of the strut stiffness from the manufacturer stiffness?
  X,Y: 5N/um
  Z: 50N/um
  From the geometry, compute the strut stiffness: should be 10N/um (currently configured at 20N/um)

** TODO [#A] Import all relevant report to this file

Based on:
- [X] [[file:~/Cloud/work-projects/ID31-NASS/matlab/nass-simscape/org/kinematics.org][kinematics]]: compute sample's motion from each stage motion
- [X] [[file:~/Cloud/work-projects/ID31-NASS/matlab/nass-simscape/org/simscape_subsystems.org][simscape_subsystems]]: general presentation of the micro-station. Used model: solid body + joints. Presentation of each stage.
- [X] [[file:~/Cloud/work-projects/ID31-NASS/documents/work-package-1/work-package-1.org::*Specification of requirements][Specification of requirements]]
- [X] [[file:~/Cloud/work-projects/ID31-NASS/matlab/nass-simscape/org/identification.org][identification]]: comparison of measurements and simscape model (not so good?)
- [ ] file:/home/thomas/Cloud/meetings/esrf-meetings/2018-04-24-Simscape-Model/2018-04-24-Simscape-Model.pdf
- [ ] [[file:~/Cloud/work-projects/ID31-NASS/matlab/nass-simscape/org/experiments.org][experiments]]: simulation of experiments with the Simscape model
- [ ] Disturbances: Similar to what was done for the uniaxial model (the same?)
  - [[file:~/Cloud/work-projects/ID31-NASS/matlab/nass-simscape/org/disturbances.org::+TITLE: Identification of the disturbances]]
- [ ] Measurement of disturbances / things that will have to be corrected using the NASS:
  - [ ] [[file:~/Cloud/work-projects/ID31-NASS/matlab/nass-measurements/static-to-dynamic/index.org]]
  - [ ] [[file:~/Cloud/work-projects/ID31-NASS/matlab/nass-measurements/disturbance-control-system/index.org]]
  - [ ] [[file:~/Cloud/work-projects/ID31-NASS/matlab/nass-measurements/disturbance-sr-rz/index.org]]
  - [ ] [[file:~/Cloud/work-projects/ID31-NASS/matlab/nass-measurements/ground-motion/index.org]]
  - [ ] [[file:~/Cloud/work-projects/ID31-NASS/matlab/nass-measurements/static-spindle/index.org]]
  - [ ] Check [[file:~/Cloud/work-projects/ID31-NASS/specifications/id-31-spindle-meas][this directory]] and [[file:~/Cloud/work-projects/ID31-NASS/specifications/stage-by-stage][this directory]]

** TODO [#B] Make good "init" for the Simscape model

In model properties => Callbacks => Init Fct
There are some loading of .mat files.
Is it the best option?

** TODO [#C] Check the effect of each stage on the compliance

- [ ] Put =rigid= mode one by one to see the effect

** TODO [#C] Make a comparison of the measured vibrations of the micro-station with the vibrations of the simscape model of the micro-station
Do we have a correlation? At least in the frequency domain?

Procedure:
- [ ] Take the time domain measurement of the vibrations due to spindle or translation stage
- [ ] Take the estimated PSD of the estimated disturbance force
- [ ] Simulate the Simscape model without the nano-hexapod (same conditions as when measuring the disturbances) and add only the considered disturbance
- [ ] Save the position errors due to the disturbance
- [ ] Compare with the measured vibrations


** TODO [#C] Add two perturbations: =Frz_x= and =Frz_y=
Maybe I can estimate them from the measurements that was made on the spindle?

** TODO [#C] Determine which Degree-Of-Freedom to keep and which to constrain

For instance, for now the granite can not rotate, but in reality, the modes may be linked to the granite's rotation.
By constraining more DoF, the simulation will be faster and the obtain state space will have a lower order.

** TODO [#B] Compute eigenvalues of the model to see if we have similar frequencies than the modal analysis?

* Introduction                                                        :ignore:


# #+name: tab:ustation_section_matlab_code
# #+caption: Report sections and corresponding Matlab files
# #+attr_latex: :environment tabularx :width 0.6\linewidth :align lX
# #+attr_latex: :center t :booktabs t
# | *Sections*       | *Matlab File*   |
# |------------------+-----------------|
# | Section ref:sec: | =ustation_1_.m= |


* Micro-Station Kinematics
:PROPERTIES:
:HEADER-ARGS:matlab+: :tangle matlab/.m
:END:
<<sec:ustation_kinematics>>
** Introduction                                                      :ignore:

[[file:~/Cloud/work-projects/ID31-NASS/matlab/nass-simscape/org/kinematics.org]]

# - Small overview of each stage and associated stiffnesses / inertia
# - schematic that shows to considered DoF
# - import from CAD

** Matlab Init                                              :noexport:ignore:
#+begin_src matlab
%% .m
#+end_src

#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
<<matlab-dir>>
#+end_src

#+begin_src matlab :exports none :results silent :noweb yes
<<matlab-init>>
#+end_src

#+begin_src matlab :tangle no :noweb yes
<<m-init-path>>
#+end_src

#+begin_src matlab :eval no :noweb yes
<<m-init-path-tangle>>
#+end_src

#+begin_src matlab :noweb yes
<<m-init-simscape>>
#+end_src

#+begin_src matlab :noweb yes
<<m-init-other>>
#+end_src

** Motion Stages
*** Translation Stage
*** Tilt Stage
*** Spindle
*** Micro-Hexapod
** Specification for each stage
<<ssec:spec_stage>>

Figure here: file:/home/thomas/Cloud/work-projects/ID31-NASS/documents/work-package-1/figures

For each stage, after a quick presentation, the specifications (stroke, precision, ...) of the stage, the metrology used with that stage and the parasitic motions associated to it are given.

*** Translation along Y
**** Presentation
This is the first stage, it is fixed directly on the granite.
In order to minimize the parasitic motions, two cylinders are used to guide the stage.
The motor used to drive the stage is one linear motor.
The 3D model of this translation stage is shown figure ref:fig:stage_translation.

#+name: fig:stage_translation
#+caption: Translation stage
[[./figures/stages/stage_translation.pdf]]

**** Specifications
#+attr_latex: :align |c|c|c|c|c| :placement [!htpb]
#+CAPTION:   Specifications for the translation stage along Y
#+NAME:      fig:spec-t-y
|--------+------------------------+--------------------------+--------------+--------------|
| Motion | Mode                   | Stroke                   | Repetability | MIM[fn:3]    |
|--------+------------------------+--------------------------+--------------+--------------|
| $T_y$  | P[fn:1]/S[fn:2]        | $\pm5 mm$                | $0.04 \mu m$ | $0.02 \mu m$ |
|        | rotation stage along Y | ($\pm10 mm$ if possible) |              |              |
|--------+------------------------+--------------------------+--------------+--------------|

**** Metrology
A linear digital encoder is used to measure the displacement of the translation stage along the y axis.

**** Parasitic motions
#+attr_latex: :align |c|c|c| :placement [!htpb]
#+CAPTION:   Parasitic motions for the translation stage along Y
#+NAME:      fig:parasitic-motions-t-y
|--------------------+-----------------------+------------------------------|
| Axial Motion ($y$) | Radial Motion ($y-z$) | Rotation motion ($\theta-z$) |
|--------------------+-----------------------+------------------------------|
| $10nm$             | $20nm$                | $< 1.7 \mu rad$              |
|--------------------+-----------------------+------------------------------|

*** Rotation around Y
**** Presentation
This stage is mainly use for the reflectivity experiment. This permit to make the sample rotates around the y axis.
The rotation axis should be parallel to the y-axis and cross the X-ray.

As it is located below the Spindle, it make the axis of rotation of the spindle to also tilt with respect to the X-ray.

4 joints are used in order to pre-stress the system.

The model of the stage is shown figure ref:fig:stage_tilt.

#+name: fig:stage_tilt
#+caption: Tilt Stage
[[./figures/stages/stage_tilt.pdf]]

**** Specifications
#+attr_latex: :align |c|c|c|c|c| :placement [!htpb]
#+CAPTION:   Specifications for the rotation stage along Y
#+NAME:      fig:spec-r-y
|------------+------+----------------------+--------------+-------------|
| Motion     | Mode | Stroke               | Repetability | MIM         |
|------------+------+----------------------+--------------+-------------|
| $\theta_y$ | S    | $\pm51 mrad$         | $5 \mu rad$  | $2 \mu rad$ |
|            |      | Speed: $\pm 3^o/min$ |              |             |
|------------+------+----------------------+--------------+-------------|

**** Metrology
Two linear digital encoders are used (one on each side).

**** Parasitic motions
#+attr_latex: :align |c|c|c| :placement [!htpb]
#+CAPTION:   Parasitic motions for the rotation stage along Y
#+NAME:      fig:parasitic-motions-r-y
|-------------------+--------------------+---------------|
| Axial Error ($y$) | Radial Error ($z$) | Tilt error () |
|-------------------+--------------------+---------------|
| $0.5\mu m$        | $10nm$             | $1.5 \mu rad$ |
|-------------------+--------------------+---------------|

*** Spindle
**** Presentation
The Spindle consist of one stator (attached to the tilt stage) and one rotor.

The rotor is separated from the stator thanks to an air bearing.

The spindle is represented figure ref:fig:stage_spindle.

It has been developed by Lab-Leuven[fn:5].

#+name: fig:stage_spindle
#+caption: Spindle
[[./figures/stages/stage_spindle.pdf]]

**** Specifications
#+attr_latex: :align |c|c|c|c|c| :placement [!htpb]
#+CAPTION:   Specifications for the Spindle
#+NAME:      fig:spec-r-y
|------------+------+----------------------+--------------+-------------|
| Motion     | Mode | Stroke               | Repetability | MIM         |
|------------+------+----------------------+--------------+-------------|
| $\theta_z$ | S    | $\pm51 mrad$         | $5 \mu rad$  | $2 \mu rad$ |
|            |      | Speed: $\pm 3^o/min$ |              |             |
|------------+------+----------------------+--------------+-------------|

**** Metrology
There is an incremental rotation encoder.

**** Parasitic motions
#+attr_latex: :align |c|c| :placement [!htpb]
#+CAPTION:   Parasitic motions for the Spindle
#+NAME:      fig:parasitic-motions-r-z
|------------------------+----------------------|
| Radial Error ($x$-$y$) | Vertical Error ($z$) |
|------------------------+----------------------|
| $0.33\mu m$            | $0.07\mu m$          |
|------------------------+----------------------|

More complete measurements have been conducted at the PEL[fn:7].

*** Long Stroke Hexapod
**** Presentation
The long stroke hexapod consists of 6 legs, each composed of:
- one linear actuators
- one limit switch
- one absolute linear encoder
- one ball joint at each side, one is fixed on the fixed platform, the other on the mobile platform

This long stroke hexapod has been developed by Symetrie[fn:4] and is shown figure ref:fig:stage_hexapod.

#+name: fig:stage_hexapod
#+caption: Hexapod Symetrie
[[./figures/stages/stage_hexapod.pdf]]

**** Specifications
#+attr_latex: :align |c|c|c|c|c| :placement [!htpb]
#+CAPTION:   Specifications for the long stroke hexapod
#+NAME:      fig:spec-hexa
|------------------+--------------+----------------+--------------+---------------|
| Motion           | Stroke       | Repetability   | MIM          | Stiffness     |
|------------------+--------------+----------------+--------------+---------------|
| $T_{\mu_x}$      | $\pm10mm$    | $\pm1\mu m$    | $0.5\mu m$   | $>12N/\mu m$  |
| $T_{\mu_y}$      | $\pm10mm$    | $\pm1\mu m$    | $0.5\mu m$   | $>12N/\mu m$  |
| $T_{\mu_z}$      | $\pm10mm$    | $\pm1\mu m$    | $0.5\mu m$   | $>135N/\mu m$ |
| $\theta_{\mu_x}$ | $\pm3 deg$   | $\pm5 \mu rad$ | $2.5\mu rad$ |               |
| $\theta_{\mu_y}$ | $\pm3 deg$   | $\pm5 \mu rad$ | $2.5\mu rad$ |               |
| $\theta_{\mu_z}$ | $\pm0.5 deg$ | $\pm5 \mu rad$ | $2.5\mu rad$ |               |
|------------------+--------------+----------------+--------------+---------------|

**** Metrology
The metrology consist of one absolute linear encoder in each leg.

**** Parasitic motions
#+attr_latex: :align |c|c|c| :placement [!htpb]
#+CAPTION:   Parasitic motions for the Spindle
#+NAME:      fig:parasitic-motions-r-z
|------------------+--------------+----------------|
| Movement         | Resolution   | Repeatability  |
|------------------+--------------+----------------|
| $T_{\mu_x}$      | $0.5\mu m$   | $\pm 1\mu m$   |
| $T_{\mu_y}$      | $0.5\mu m$   | $\pm 1\mu m$   |
| $T_{\mu_z}$      | $0.1\mu m$   | $\pm 1\mu m$   |
| $\theta_{\mu_x}$ | $2.5\mu rad$ | $\pm 4\mu rad$ |
| $\theta_{\mu_y}$ | $2.5\mu rad$ | $\pm 4\mu rad$ |
|------------------+--------------+----------------|
*** Short Stroke Hexapod
**** Presentation
This last stage is located just below the sample to study.
This is the only stage that is not yet build nor designed.

**** Specifications
#+attr_latex: :align |c|c|c|c|c| :placement [!htpb]
#+CAPTION:   Specifications for the nano-stage
#+NAME:      fig:spec-nano
|------------------+----------------+--------------+-------|
| Motion           | Stroke         | Repetability | MIM   |
|------------------+----------------+--------------+-------|
| $T_{\nu_x}$      | $\pm10\mu m$   | $10nm$       | $3nm$ |
| $T_{\nu_y}$      | $\pm10\mu m$   | $10nm$       | $3nm$ |
| $T_{\nu_z}$      | $\pm10\mu m$   | $10nm$       | $3nm$ |
| $\theta_{\nu_x}$ | $\pm10\mu rad$ | $1.7\mu rad$ |       |
| $\theta_{\nu_y}$ | $\pm10\mu rad$ | $1.7\mu rad$ |       |
| $\theta_{\nu_z}$ | $\pm10\mu rad$ |              |       |
|------------------+----------------+--------------+-------|

**** Dimensions
- Bottom plate :
  - Inner diameter: $>150mm$
  - External diameter: $<305mm$
- Top plate
  - External diameter: $<300mm$
- Height: $90mm$
- Location of the rotation point: Centered, at $175mm$ above the top platform

** Compute sample's motion from stage motion
*** Introduction                                                    :ignore:
#+begin_quote
Rx = [1 0       0;
      0 cos(t) -sin(t);
      0 sin(t)  cos(t)];

Ry = [ cos(t) 0 sin(t);
       0      1 0;
      -sin(t) 0 cos(t)];

Rz = [cos(t) -sin(t) 0;
      sin(t)  cos(t) 0;
      0       0      1];
#+end_quote

Let's define the following frames:
- $\{W\}$ the frame that is *fixed to the granite* and its origin at the theoretical meeting point between the X-ray and the spindle axis.
- $\{S\}$ the frame *attached to the sample* (in reality attached to the top platform of the nano-hexapod) with its origin at 175mm above the top platform of the nano-hexapod.
  Its origin is $O_S$.
- $\{T\}$ the theoretical wanted frame that correspond to the wanted pose of the frame $\{S\}$.
  $\{T\}$ is computed from the wanted position of each stage. It is thus theoretical and does not correspond to a real position.
  The origin of $T$ is $O_T$ and is the wanted position of the sample.

Thus:
- the *measurement* of the position of the sample corresponds to ${}^W O_S = \begin{bmatrix} {}^WP_{x,m} & {}^WP_{y,m} & {}^WP_{z,m} \end{bmatrix}^T$ in translation and to $\theta_m {}^W\bm{s}_m = \theta_m \cdot \begin{bmatrix} {}^Ws_{x,m} & {}^Ws_{y,m} & {}^Ws_{z,m} \end{bmatrix}^T$ in rotations
- the *wanted position* of the sample expressed w.r.t. the granite is ${}^W O_T = \begin{bmatrix} {}^WP_{x,r} & {}^WP_{y,r} & {}^WP_{z,r} \end{bmatrix}^T$ in translation and to $\theta_r {}^W\bm{s}_r = \theta_r \cdot \begin{bmatrix} {}^Ws_{x,r} & {}^Ws_{y,r} & {}^Ws_{z,r} \end{bmatrix}^T$ in rotations

*** Wanted Position of the Sample with respect to the Granite
Let's define the wanted position of each stage.
#+begin_src matlab
  Ty = 0; % [m]
  Ry = 3*pi/180; % [rad]
  Rz = 180*pi/180; % [rad]

  % Hexapod (first consider only translations)
  Thx = 0; % [m]
  Thy = 0; % [m]
  Thz = 0; % [m]
#+end_src

Now, we compute the corresponding wanted translation and rotation of the sample with respect to the granite frame $\{W\}$.
This corresponds to ${}^WO_T$ and $\theta_m {}^Ws_m$.

To do so, we have to define the homogeneous transformation for each stage.
#+begin_src matlab
  % Translation Stage
  Rty = [1 0 0 0;
         0 1 0 Ty;
         0 0 1 0;
         0 0 0 1];

  % Tilt Stage - Pure rotating aligned with Ob
  Rry = [ cos(Ry) 0 sin(Ry) 0;
          0       1 0       0;
         -sin(Ry) 0 cos(Ry) 0;
          0       0 0       1];

  % Spindle - Rotation along the Z axis
  Rrz = [cos(Rz) -sin(Rz) 0 0 ;
         sin(Rz)  cos(Rz) 0 0 ;
         0        0       1 0 ;
         0        0       0 1 ];

  % Micro-Hexapod (only rotations first)
  Rh = [1 0 0 Thx ;
        0 1 0 Thy ;
        0 0 1 Thz ;
        0 0 0 1 ];
#+end_src

We combine the individual homogeneous transformations into one homogeneous transformation for all the station.
#+begin_src matlab
  Ttot = Rty*Rry*Rrz*Rh;
#+end_src

Using this homogeneous transformation, we can compute the wanted position and orientation of the sample with respect to the granite.

Translation.
#+begin_src matlab
  WOr = Ttot*[0;0;0;1];
  WOr = WOr(1:3);
#+end_src

Rotation.
#+begin_src matlab
  thetar = acos((trace(Ttot(1:3, 1:3))-1)/2)
  if thetar == 0
    WSr = [0; 0; 0];
  else
    [V, D] = eig(Ttot(1:3, 1:3));
    WSr = thetar*V(:, abs(diag(D) - 1) < eps(1));
  end
#+end_src

#+begin_src matlab
  WPr = [WOr ; WSr];
#+end_src

*** Measured Position of the Sample with respect to the Granite
The measurement of the position of the sample using the metrology system gives the position and orientation of the sample with respect to the granite.
#+begin_src matlab
  % Measurements: Xm, Ym, Zm, Rx, Ry, Rz
  Dxm = 0; % [m]
  Dym = 0; % [m]
  Dzm = 0; % [m]

  Rxm = 0*pi/180; % [rad]
  Rym = 0*pi/180; % [rad]
  Rzm = 180*pi/180; % [rad]
#+end_src

Let's compute the corresponding orientation using screw axis.
#+begin_src matlab
  Trxm = [1 0         0;
          0 cos(Rxm) -sin(Rxm);
          0 sin(Rxm)  cos(Rxm)];
  Trym = [ cos(Rym) 0 sin(Rym);
           0        1 0;
          -sin(Rym) 0 cos(Rym)];
  Trzm = [cos(Rzm) -sin(Rzm) 0;
          sin(Rzm)  cos(Rzm) 0;
          0         0        1];

  STw = [[ Trym*Trxm*Trzm , [Dxm; Dym; Dzm]]; 0 0 0 1];
#+end_src

We then obtain the orientation measurement in the form of screw coordinate $\theta_m ({}^Ws_{x,m},\ {}^Ws_{y,m},\ {}^Ws_{z,m})^T$ where:
- $\theta_m = \cos^{-1} \frac{\text{Tr}(R) - 1}{2}$
- ${}^W\bm{s}_m$ is the eigen vector of the rotation matrix $R$ corresponding to the eigen value $\lambda = 1$

#+begin_src matlab
  thetam = acos((trace(STw(1:3, 1:3))-1)/2); % [rad]
  if thetam == 0
    WSm = [0; 0; 0];
  else
    [V, D] = eig(STw(1:3, 1:3));
    WSm = thetam*V(:, abs(diag(D) - 1) < eps(1));
  end
#+end_src

#+begin_src matlab
  WPm = [Dxm ; Dym ; Dzm ; WSm];
#+end_src

*** Positioning Error with respect to the Granite
The wanted position expressed with respect to the granite is ${}^WO_T$ and the measured position with respect to the granite is ${}^WO_S$, thus the *position error* expressed in $\{W\}$ is
\[ {}^W E = {}^W O_T - {}^W O_S \]
The same is true for rotations:
\[ \theta_\epsilon {}^W\bm{s}_\epsilon = \theta_r {}^W\bm{s}_r - \theta_m {}^W\bm{s}_m \]

#+begin_src matlab
  WPe = WPr - WPm;
#+end_src

#+begin_quote
Now we want to express this error in a frame attached to the *base of the nano-hexapod* with its origin at the same point where the Jacobian of the nano-hexapod is computed (175mm above the top platform + 90mm of total height of the nano-hexapod).

Or maybe should we want to express this error with respect to the *top platform of the nano-hexapod*?
We are measuring the position of the top-platform, and we don't know exactly the position of the bottom platform.
We could compute the position of the bottom platform in two ways:
- from the encoders of each stage
- from the measurement of the nano-hexapod top platform + the internal metrology in the nano-hexapod (capacitive sensors e.g)

A third option is to say that the maximum stroke of the nano-hexapod is so small that the error should no change to much by the change of base.
#+end_quote

*** Position Error Expressed in the Nano-Hexapod Frame
We now want the position error to be expressed in $\{S\}$ (the frame attach to the sample) for control:
\[ {}^S E = {}^S T_W \cdot {}^W E \]

Thus we need to compute the homogeneous transformation ${}^ST_W$.
Fortunately, this homogeneous transformation can be computed from the measurement of the sample position and orientation with respect to the granite.
#+begin_src matlab
  Trxm = [1 0         0;
          0 cos(Rxm) -sin(Rxm);
          0 sin(Rxm)  cos(Rxm)];
  Trym = [ cos(Rym) 0 sin(Rym);
           0        1 0;
          -sin(Rym) 0 cos(Rym)];
  Trzm = [cos(Rzm) -sin(Rzm) 0;
          sin(Rzm)  cos(Rzm) 0;
          0         0        1];

  STw = [[ Trym*Trxm*Trzm , [Dxm; Dym; Dzm]]; 0 0 0 1];
#+end_src

Translation Error.
#+begin_src matlab
  SEm = STw * [WPe(1:3); 0];
  SEm = SEm(1:3);
#+end_src

Rotation Error.
#+begin_src matlab
  SEr = STw * [WPe(4:6); 0];
  SEr = SEr(1:3);
#+end_src

#+begin_src matlab
  Etot = [SEm ; SEr]
#+end_src
*** Another try
Let's denote:
- $\{W\}$ the initial fixed frame
- $\{R\}$ the reference frame corresponding to the wanted pose of the sample
- $\{M\}$ the frame corresponding to the measured pose of the sample

We have then computed:
- ${}^WT_R$
- ${}^WT_M$

We have:
\begin{align}
  {}^MT_R &= {}^MT_W {}^WT_R \\
          &= {}^WT_M^t {}^WT_R
\end{align}

#+begin_src matlab
  MTr = STw'*Ttot;
#+end_src

Position error:
#+begin_src matlab
  MTr(1:3, 1:4)*[0; 0; 0; 1]
#+end_src

Orientation error:
#+begin_src matlab
  MTr(1:3, 1:3)
#+end_src

*** Verification
How can we verify that the computation is correct?
Options:
- Test with simscape multi-body
  - Impose motion on each stage
  - Measure the position error w.r.t. the NASS
  - Compare with the computation

* Stage Modeling
:PROPERTIES:
:HEADER-ARGS:matlab+: :tangle matlab/.m
:END:
<<sec:ustation_kinematics>>
** Introduction                                                      :ignore:

The goal here is to tune the Simscape model of the station in order to have a good dynamical representation of the real system.

In order to do so, we reproduce the Modal Analysis done on the station using the Simscape model.

We can then compare the measured Frequency Response Functions with the identified dynamics of the model.

Finally, this should help to tune the parameters of the model such that the dynamics is closer to the measured FRF.

# Validation of the Model
# - Most important metric: support compliance
# - Compare model and measurement

** Matlab Init                                              :noexport:ignore:
#+begin_src matlab
%% .m
#+end_src

#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
<<matlab-dir>>
#+end_src

#+begin_src matlab :exports none :results silent :noweb yes
<<matlab-init>>
#+end_src

#+begin_src matlab :tangle no :noweb yes
<<m-init-path>>
#+end_src

#+begin_src matlab :eval no :noweb yes
<<m-init-path-tangle>>
#+end_src

#+begin_src matlab :noweb yes
<<m-init-simscape>>
#+end_src

#+begin_src matlab :noweb yes
<<m-init-other>>
#+end_src

** Some notes about the Simscape Model

The Simscape Model of the micro-station consists of several solid bodies:
- Bottom Granite
- Top Granite
- Translation Stage
- Tilt Stage
- Spindle
- Hexapod

Each solid body has some characteristics: Center of Mass, mass, moment of inertia, etc...
These parameters are automatically computed from the geometry and from the density of the materials.

Then, the solid bodies are connected with springs and dampers.
Some of the springs and dampers values can be estimated from the joints/stages specifications, however, we here prefer to tune these values based on the measurements.

** Compare with measurements at the CoM of each element
#+begin_src matlab
%% All stages are initialized
initializeGround(      'type', 'rigid');
initializeGranite(     'type', 'flexible');
initializeTy(          'type', 'flexible');
initializeRy(          'type', 'flexible');
initializeRz(          'type', 'flexible');
initializeMicroHexapod('type', 'flexible');

initializeLoggingConfiguration('log', 'none');

initializeReferences();
initializeDisturbances('enable', false);
#+end_src

We now use a new Simscape Model where 6DoF inertial sensors are located at the Center of Mass of each solid body.

We use the =linearize= function in order to estimate the dynamics from forces applied on the Translation stage at the same position used for the real modal analysis to the inertial sensors.
#+begin_src matlab
%% Identification
% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Micro-Station/Translation Stage/Flexible/F_hammer'],          1, 'openinput');  io_i = io_i + 1;
io(io_i) = linio([mdl, '/Micro-Station/Granite/Flexible/accelerometer'],               1, 'openoutput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Micro-Station/Translation Stage/Flexible/accelerometer'],     1, 'openoutput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Micro-Station/Tilt Stage/Flexible/accelerometer'],            1, 'openoutput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Micro-Station/Spindle/Flexible/accelerometer'],               1, 'openoutput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Micro-Station/Micro Hexapod/Flexible/accelerometer'],         1, 'openoutput'); io_i = io_i + 1;

% Run the linearization
G_ms = linearize(mdl, io, 0);

% Input/Output definition
G_ms.InputName  = {'Fx', 'Fy', 'Fz'};
G_ms.OutputName = {'gtop_x', 'gtop_y', 'gtop_z', 'gtop_rx', 'gtop_ry', 'gtop_rz', ...
                   'ty_x', 'ty_y', 'ty_z', 'ty_rx', 'ty_ry', 'ty_rz', ...
                   'ry_x', 'ry_y', 'ry_z', 'ry_rx', 'ry_ry', 'ry_rz', ...
                   'rz_x', 'rz_y', 'rz_z', 'rz_rx', 'rz_ry', 'rz_rz', ...
                   'hexa_x', 'hexa_y', 'hexa_z', 'hexa_rx', 'hexa_ry', 'hexa_rz'};

% Put the output of G in displacements instead of acceleration
G_ms = G_ms/s^2;
#+end_src

#+begin_src matlab
%% Load estimated FRF at CoM
load('/home/thomas/Cloud/work-projects/ID31-NASS/phd-thesis-chapters/A3-micro-station-modal-analysis/matlab/mat/frf_matrix.mat', 'freqs');
load('/home/thomas/Cloud/work-projects/ID31-NASS/phd-thesis-chapters/A3-micro-station-modal-analysis/matlab/mat/frf_com.mat', 'frfs_CoM');
#+end_src

#+begin_src matlab :exports none
%% Compare the two
dirs = {'x', 'y', 'z', 'rx', 'ry', 'rz'};
stages = {'gbot', 'gtop', 'ty', 'ry', 'rz', 'hexa'}

n_stg = 3;
n_dir = 6; % x, y, z, Rx, Ry, Rz
n_exc = 2; % x, y, z

f = logspace(0, 3, 1000);

figure;
hold on;
plot(freqs, abs(squeeze(frfs_CoM(6*(n_stg-1) + n_dir, n_exc, :)))./((2*pi*freqs).^2)');
plot(f, abs(squeeze(freqresp(G_ms([stages{n_stg}, '_', dirs{n_dir}], ['F', dirs{n_exc}]), f, 'Hz'))));
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]');
hold off;
xlim([1, 200]);
#+end_src

#+begin_src matlab :exports none
dirs = {'x', 'y', 'z', 'rx', 'ry', 'rz'};
stages = {'gtop', 'ty', 'ry', 'rz', 'hexa'}

f = logspace(1, 3, 1000);

figure;
for n_stg = 1:2
    for n_dir = 1:3
        subplot(3, 2, (n_dir-1)*2 + n_stg);
        title(['F ', dirs{n_dir}, ' to ', stages{n_stg}, ' ', dirs{n_dir}]);
        hold on;
        plot(freqs, abs(squeeze(frfs_CoM(6*(n_stg) + n_dir, n_dir, :)))./((2*pi*freqs).^2)');
        plot(f, abs(squeeze(freqresp(G_ms([stages{n_stg}, '_', dirs{n_dir}], ['F', dirs{n_dir}]), f, 'Hz'))));
        set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
        ylabel('Amplitude [m/N]');
        if n_dir == 3
            xlabel('Frequency [Hz]');
        end
        hold off;
        xlim([10, 1000]);
        ylim([1e-12, 1e-6]);
    end
end
#+end_src

#+begin_src matlab :exports none
  dirs = {'x', 'y', 'z', 'rx', 'ry', 'rz'};
  stages = {'ry', 'rz', 'hexa'}

  f = logspace(1, 3, 1000);

  figure;
  for n_stg = 1:2
    for n_dir = 1:3
      subplot(3, 2, (n_dir-1)*2 + n_stg);
      title(['F ', dirs{n_dir}, ' to ', stages{n_stg}, ' ', dirs{n_dir}]);
      hold on;
      plot(freqs, abs(squeeze(frfs_CoM(6*(n_stg+2) + n_dir, n_dir, :)))./((2*pi*freqs).^2)');
      plot(f, abs(squeeze(freqresp(G_ms([stages{n_stg}, '_', dirs{n_dir}], ['F', dirs{n_dir}]), f, 'Hz'))));
      set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
      ylabel('Amplitude [m/N]');
      if n_dir == 3
        xlabel('Frequency [Hz]');
      end
      hold off;
      xlim([10, 1000]);
      ylim([1e-12, 1e-6]);
    end
  end
#+end_src

#+begin_src matlab :exports none
  dirs = {'x', 'y', 'z', 'rx', 'ry', 'rz'};
  stages = {'hexa'}

  f = logspace(1, 3, 1000);

  figure;
  for n_stg = 1
    for n_dir = 1:3
      subplot(3, 1, (n_dir-1) + n_stg);
      title(['F ', dirs{n_dir}, ' to ', stages{n_stg}, ' ', dirs{n_dir}]);
      hold on;
      plot(freqs, abs(squeeze(frfs_CoM(6*(n_stg+4) + n_dir, n_dir, :)))./((2*pi*freqs).^2)');
      plot(f, abs(squeeze(freqresp(G_ms([stages{n_stg}, '_', dirs{n_dir}], ['F', dirs{n_dir}]), f, 'Hz'))));
      set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
      ylabel('Amplitude [m/N]');
      if n_dir == 3
        xlabel('Frequency [Hz]');
      end
      hold off;
      xlim([10, 1000]);
      ylim([1e-12, 1e-6]);
    end
  end
#+end_src

** Measured micro-station compliance
*** Introduction                                                    :ignore:

The most important dynamical characteristic of the micro-station that should be well modeled is its compliance.
This is what can impact the nano-hexapod dynamics.

- [ ] Add schematic of the experiment with Micro-Hexapod top platform, location of accelerometers, of impacts, etc...

- 4 3-axis accelerometers
- 10 hammer impacts on the micro-hexapod top plaftorm
- *Was the rotation compensation axis present?* (I don't think so)

*** Position of inertial sensors on top of the micro-hexapod

4 accelerometers are fixed to the micro-hexapod top platform.

# | *Num* | *Position* | *Orientation* | *Sensibility* | *Channels* |
# |-------+------------+---------------+---------------+------------|
# |     1 | [0, +A, 0] | [x, y, z]     | 1V/g          |        1-3 |
# |     2 | [-B, 0, 0] | [x, y, z]     | 1V/g          |        4-6 |
# |     3 | [0, -A, 0] | [x, y, z]     | 0.1V/g        |        7-9 |
# |     4 | [+B, 0, 0] | [x, y, z]     | 1V/g          |      10-12 |

# Instrumented Hammer:
# - Channel 13
# - Sensibility: 230 uV/N

# | Acc Number | Dir | Channel Number |
# |------------+-----+----------------|
# |          1 | x   |              1 |
# |          1 | y   |              2 |
# |          1 | z   |              3 |
# |          2 | x   |              4 |
# |          2 | y   |              5 |
# |          2 | z   |              6 |
# |          3 | x   |              7 |
# |          3 | y   |              8 |
# |          3 | z   |              9 |
# |          4 | x   |             10 |
# |          4 | y   |             11 |
# |          4 | z   |             12 |
# |     Hammer |     |             13 |

To convert the 12 acceleration signals $\mathbf{a}_{\mathcal{L}} = [a_{1x}\ a_{1y}\ a_{1z}\ a_{2x}\ \dots\ a_{4z}]$ to the acceleration expressed in cartesian coordinate $\mathbf{a}_{\mathcal{X}} [a_{dx}\ a_{dy}\ a_{dz}\ a_{rx}\ a_{ry}\ a_{rz}]$, a Jacobian matrix can be written based on the positions and orientations of the accelerometers.

\begin{equation}
\mathbf{a}_{\mathcal{L}} = J_a \cdot \mathbf{a}_{\mathcal{X}}
\end{equation}

\begin{equation}
J_a = \begin{bmatrix}
1 & 0 & 0 & 0 & 0 &-d \\
0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & d & 0 & 0 \\
1 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 &-d \\
0 & 0 & 1 & 0 & d & 0 \\
1 & 0 & 0 & 0 & 0 & d \\
0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 &-d & 0 & 0 \\
1 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & d \\
0 & 0 & 1 & 0 &-d & 0
\end{bmatrix}
\end{equation}

Then, the acceleration in the cartesian frame can be computed
\begin{equation}
\mathbf{a}_{\mathcal{X}} = J_a^\dagger \cdot \mathbf{a}_{\mathcal{L}}
\end{equation}

#+begin_src matlab
%% Jacobian for accelerometers
% L = Ja X
d = 0.14; % [m]
Ja = [1  0  0  0  0 -d;
      0  1  0  0  0  0;
      0  0  1  d  0  0;
      1  0  0  0  0  0;
      0  1  0  0  0 -d;
      0  0  1  0  d  0;
      1  0  0  0  0  d;
      0  1  0  0  0  0;
      0  0  1 -d  0  0;
      1  0  0  0  0  0;
      0  1  0  0  0  d;
      0  0  1  0 -d  0];

Ja_inv = pinv(Ja);
#+end_src

#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
data2orgtable(Ja_inv, {'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'}, {'d1x', 'd1y', 'd1z', 'd2x', 'd2y', 'd2z', 'd3x', 'd3y', 'd3z', 'd4x', 'd4y', 'd4z'}, ' %.5f ');
#+end_src

#+RESULTS:
|    |      d1x |  d1y |     d1z |  d2x |      d2y |     d2z |     d3x |  d3y |      d3z |  d4x |     d4y |      d4z |
|----+----------+------+---------+------+----------+---------+---------+------+----------+------+---------+----------|
| Dx |     0.25 |  0.0 |     0.0 | 0.25 |      0.0 |     0.0 |    0.25 |  0.0 |      0.0 | 0.25 |     0.0 |      0.0 |
| Dy |      0.0 | 0.25 |     0.0 |  0.0 |     0.25 |     0.0 |     0.0 | 0.25 |      0.0 |  0.0 |    0.25 |      0.0 |
| Dz |      0.0 |  0.0 |    0.25 |  0.0 |      0.0 |    0.25 |     0.0 |  0.0 |     0.25 |  0.0 |     0.0 |     0.25 |
| Rx |      0.0 |  0.0 | 3.57143 |  0.0 |      0.0 |    -0.0 |     0.0 |  0.0 | -3.57143 |  0.0 |     0.0 |     -0.0 |
| Ry |      0.0 |  0.0 |     0.0 |  0.0 |      0.0 | 3.57143 |     0.0 |  0.0 |      0.0 |  0.0 |     0.0 | -3.57143 |
| Rz | -1.78571 |  0.0 |    -0.0 |  0.0 | -1.78571 |     0.0 | 1.78571 |  0.0 |     -0.0 |  0.0 | 1.78571 |     -0.0 |

*** Hammer blow position/orientation

10 hammer hits are performed as shown in Figure ...

| *Num* | *Direction* | *Position* | Accelerometer position | Jacobian number |
|-------+-------------+------------+------------------------+-----------------|
|     1 | -Y          | [0, +A, 0] |                      1 |              -2 |
|     2 | -Z          | [0, +A, 0] |                      1 |              -3 |
|     3 | X           | [-B, 0, 0] |                      2 |               4 |
|     4 | -Z          | [-B, 0, 0] |                      2 |              -6 |
|     5 | Y           | [0, -A, 0] |                      3 |               8 |
|     6 | -Z          | [0, -A, 0] |                      3 |              -9 |
|     7 | -X          | [+B, 0, 0] |                      4 |             -10 |
|     8 | -Z          | [+B, 0, 0] |                      4 |             -12 |
|     9 | -X          | [0, -A, 0] |                      3 |              -7 |
|    10 | -X          | [0, +A, 0] |                      1 |              -1 |

Similar to what is done for the accelerometers, a Jacobian matrix can be computed to convert the individual hammer forces to force and torques applied at the center of the micro-hexapod top plate.

\begin{equation}
\mathbf{F}_{\mathcal{X}} = J_F^t \cdot \mathbf{F}_{\mathcal{L}}
\end{equation}

\begin{equation}
J_F = \begin{bmatrix}
 0 & -1 &  0 &  0 &  0 &  0\\
 0 &  0 & -1 & -d &  0 &  0\\
 1 &  0 &  0 &  0 &  0 &  0\\
 0 &  0 & -1 &  0 & -d &  0\\
 0 &  1 &  0 &  0 &  0 &  0\\
 0 &  0 & -1 &  d &  0 &  0\\
-1 &  0 &  0 &  0 &  0 &  0\\
 0 &  0 & -1 &  0 &  d &  0\\
-1 &  0 &  0 &  0 &  0 & -d\\
-1 &  0 &  0 &  0 &  0 &  d
\end{bmatrix}
\end{equation}

#+begin_src matlab
%% Jacobian for hammer impacts
% F = Jf' tau
Jf = [0 -1  0  0  0  0;
      0  0 -1 -d  0  0;
      1  0  0  0  0  0;
      0  0 -1  0 -d  0;
      0  1  0  0  0  0;
      0  0 -1  d  0  0;
     -1  0  0  0  0  0;
      0  0 -1  0  d  0;
     -1  0  0  0  0 -d;
     -1  0  0  0  0  d]';
Jf_inv = pinv(Jf);
#+end_src

#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
data2orgtable(Jf, {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'}, {'-F1y', '-F1z', '+F2x', '-F2z', '+F3y', '-F3z', '-F4x', '-F4z', '-F3x', '-F1x'}, ' %.2f ');
#+end_src

#+RESULTS:
|    | -F1y |  -F1z | +F2x |  -F2z | +F3y | -F3z | -F4x | -F4z |  -F3x | -F1x |
|----+------+-------+------+-------+------+------+------+------+-------+------|
| Fx |  0.0 |   0.0 |  1.0 |   0.0 |  0.0 |  0.0 | -1.0 |  0.0 |  -1.0 | -1.0 |
| Fy | -1.0 |   0.0 |  0.0 |   0.0 |  1.0 |  0.0 |  0.0 |  0.0 |   0.0 |  0.0 |
| Fz |  0.0 |  -1.0 |  0.0 |  -1.0 |  0.0 | -1.0 |  0.0 | -1.0 |   0.0 |  0.0 |
| Mx |  0.0 | -0.14 |  0.0 |   0.0 |  0.0 | 0.14 |  0.0 |  0.0 |   0.0 |  0.0 |
| My |  0.0 |   0.0 |  0.0 | -0.14 |  0.0 |  0.0 |  0.0 | 0.14 |   0.0 |  0.0 |
| Mz |  0.0 |   0.0 |  0.0 |   0.0 |  0.0 |  0.0 |  0.0 |  0.0 | -0.14 | 0.14 |

*** Compute FRF

#+begin_src matlab
%% Load raw measurement data
% "Track1" to "Track12" are the 12 accelerometers
% "Track13" is the instrumented hammer force sensor
data = [
    load('ustation_compliance_hammer_1.mat'), ...
    load('ustation_compliance_hammer_2.mat'), ...
    load('ustation_compliance_hammer_3.mat'), ...
    load('ustation_compliance_hammer_4.mat'), ...
    load('ustation_compliance_hammer_5.mat'), ...
    load('ustation_compliance_hammer_6.mat'), ...
    load('ustation_compliance_hammer_7.mat'), ...
    load('ustation_compliance_hammer_8.mat'), ...
    load('ustation_compliance_hammer_9.mat'), ...
    load('ustation_compliance_hammer_10.mat')];

%% Prepare the FRF computation
Ts = 1e-3; % Sampling Time [s]
Nfft = floor(1/Ts); % Number of points for the FFT computation
win = ones(Nfft, 1); % Rectangular window
[~, f] = tfestimate(data(1).Track13, data(1).Track1, win, [], Nfft, 1/Ts);

% Samples taken before and after the hammer "impact"
pre_n = floor(0.1/Ts);
post_n = Nfft - pre_n - 1;

%% Compute the FRFs for the 10 hammer impact locations.
% The FRF from hammer force the 12 accelerometers are computed
% for each of the 10 hammer impacts and averaged
G_raw = zeros(12,10,length(f));

for i = 1:10 % For each impact location
    % Find the 10 impacts
    [~, impacts_i] = find(diff(data(i).Track13 > 50)==1);
    % Only keep the first 10 impacts if there are more than 10 impacts
    if length(impacts_i)>10
        impacts_i(11:end) = [];
    end

    % Average the FRF for the 10 impacts
    for impact_i = impacts_i
        i_start = impact_i - pre_n;
        i_end = impact_i + post_n;
        data_in = data(i).Track13(i_start:i_end); % [N]
        % Remove hammer DC offset
        data_in  = data_in  - mean(data_in(end-pre_n:end));
        % Combine all outputs [m/s^2]
        data_out = [data(i).Track1( i_start:i_end); ...
                    data(i).Track2( i_start:i_end); ...
                    data(i).Track3( i_start:i_end); ...
                    data(i).Track4( i_start:i_end); ...
                    data(i).Track5( i_start:i_end); ...
                    data(i).Track6( i_start:i_end); ...
                    data(i).Track7( i_start:i_end); ...
                    data(i).Track8( i_start:i_end); ...
                    data(i).Track9( i_start:i_end); ...
                    data(i).Track10(i_start:i_end); ...
                    data(i).Track11(i_start:i_end); ...
                    data(i).Track12(i_start:i_end)];

        [frf, ~] = tfestimate(data_in, data_out', win, [], Nfft, 1/Ts);
        G_raw(:,i,:) = squeeze(G_raw(:,i,:)) + 0.1*frf'./(-(2*pi*f').^2);
    end
end

%% Compute transfer function in cartesian frame using Jacobian matrices
% FRF_cartesian = inv(Ja) * FRF * inv(Jf)
FRF_cartesian = pagemtimes(Ja_inv, pagemtimes(G_raw, Jf_inv));
#+end_src

#+begin_src matlab :exports none :results none
%% Measured FRF of the compliance of the micro-station in the Cartesian frame
figure;
hold on;
plot(f, abs(squeeze(FRF_cartesian(1,1,:))), '.', 'color', colors(1,:), 'DisplayName', '$D_x/F_x$')
plot(f, abs(squeeze(FRF_cartesian(2,2,:))), '.', 'color', colors(2,:), 'DisplayName', '$D_y/F_y$')
plot(f, abs(squeeze(FRF_cartesian(3,3,:))), '.', 'color', colors(3,:), 'DisplayName', '$D_z/F_z$')
hold off;
leg = legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 1);
leg.ItemTokenSize(1) = 15;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Magnitude [m/N]');
xlim([20, 500]); ylim([1e-9, 2e-6]);
xticks([20, 50, 100, 200, 500])
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/ustation_frf_compliance_xyz.pdf', 'width', 'wide', 'height', 'normal');
#+end_src

#+name: fig:ustation_frf_compliance_xyz
#+caption: Measured FRF of the compliance of the micro-station in the Cartesian frame
#+RESULTS:
[[file:figs/ustation_frf_compliance_xyz.png]]

#+begin_src matlab
figure;
hold on;
plot(f, abs(squeeze(FRF_cartesian(4,4,:))), '-', 'color', colors(1,:), 'DisplayName', '$D_x/F_x$')
plot(f, abs(squeeze(FRF_cartesian(5,5,:))), '-', 'color', colors(2,:), 'DisplayName', '$D_y/F_y$')
plot(f, abs(squeeze(FRF_cartesian(6,6,:))), '-', 'color', colors(3,:), 'DisplayName', '$D_z/F_z$')
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Magnitude [m/N]');
xlim([30, 300]); ylim([1e-9, 2e-6]);
leg = legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 1);
leg.ItemTokenSize(1) = 15;
#+end_src

** Compare with the Model

#+begin_src matlab
%% Initialize simulation with default parameters (flexible elements)
initializeGround();
initializeGranite();
initializeTy();
initializeRy();
initializeRz();
initializeMicroHexapod();

initializeReferences();
initializeDisturbances();
initializeSimscapeConfiguration();
initializeLoggingConfiguration();
#+end_src

And we identify the dynamics from forces/torques applied on the micro-hexapod top platform to the motion of the micro-hexapod top platform at the same point.

#+begin_src matlab
%% Identification of the compliance
% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Micro-Station/Micro Hexapod/Flexible/Fm'], 1, 'openinput');       io_i = io_i + 1; % Direct Forces/Torques applied on the micro-hexapod top platform
io(io_i) = linio([mdl, '/Micro-Station/Micro Hexapod/Flexible/Dm'], 1, 'output'); io_i = io_i + 1; % Absolute displacement of the top platform

% Run the linearization
Gm = linearize(mdl, io, 0);
Gm.InputName  = {'Fmx', 'Fmy', 'Fmz', 'Mmx', 'Mmy', 'Mmz'};
Gm.OutputName = {'Dx', 'Dy', 'Dz', 'Drx', 'Dry', 'Drz'};
#+end_src

#+begin_src matlab :exports none :results none
%% Extracted FRF of the compliance of the micro-station in the Cartesian frame from the Simscape model
figure;
hold on;
plot(f, abs(squeeze(FRF_cartesian(1,1,:))), '.', 'color', colors(1,:), 'DisplayName', '$D_x/F_x$ - Measured')
plot(f, abs(squeeze(FRF_cartesian(2,2,:))), '.', 'color', colors(2,:), 'DisplayName', '$D_y/F_y$ - Measured')
plot(f, abs(squeeze(FRF_cartesian(3,3,:))), '.', 'color', colors(3,:), 'DisplayName', '$D_z/F_z$ - Measured')
plot(freqs, abs(squeeze(freqresp(Gm(1,1), freqs, 'Hz'))), '--', 'color', colors(1,:), 'DisplayName', '$D_x/F_x$ - Model')
plot(freqs, abs(squeeze(freqresp(Gm(2,2), freqs, 'Hz'))), '--', 'color', colors(2,:), 'DisplayName', '$D_y/F_y$ - Model')
plot(freqs, abs(squeeze(freqresp(Gm(3,3), freqs, 'Hz'))), '--', 'color', colors(3,:), 'DisplayName', '$D_z/F_z$ - Model')
hold off;
leg = legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 2);
leg.ItemTokenSize(1) = 15;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Magnitude [m/N]');
xlim([20, 500]); ylim([2e-10, 1e-6]);
xticks([20, 50, 100, 200, 500])
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/ustation_frf_compliance_xyz_model.pdf', 'width', 'wide', 'height', 'normal');
#+end_src

#+name: fig:ustation_frf_compliance_xyz_model
#+caption: Extracted FRF of the compliance of the micro-station in the Cartesian frame from the Simscape model
#+RESULTS:
[[file:figs/ustation_frf_compliance_xyz_model.png]]

#+begin_src matlab :exports none :results none
%% Extracted FRF of the compliance of the micro-station in the Cartesian frame from the Simscape model
figure;
hold on;
plot(f, abs(squeeze(FRF_cartesian(4,4,:))), '.', 'color', colors(1,:), 'DisplayName', '$R_x/M_x$ - Measured')
plot(f, abs(squeeze(FRF_cartesian(5,5,:))), '.', 'color', colors(2,:), 'DisplayName', '$R_y/M_y$ - Measured')
plot(f, abs(squeeze(FRF_cartesian(6,6,:))), '.', 'color', colors(3,:), 'DisplayName', '$R_z/M_z$ - Measured')
plot(freqs, abs(squeeze(freqresp(Gm(4,4), freqs, 'Hz'))), '--', 'color', colors(1,:), 'DisplayName', '$R_x/M_x$ - Model')
plot(freqs, abs(squeeze(freqresp(Gm(5,5), freqs, 'Hz'))), '--', 'color', colors(2,:), 'DisplayName', '$R_y/M_y$ - Model')
plot(freqs, abs(squeeze(freqresp(Gm(6,6), freqs, 'Hz'))), '--', 'color', colors(3,:), 'DisplayName', '$R_z/M_z$ - Model')
hold off;
leg = legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 2);
leg.ItemTokenSize(1) = 15;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Magnitude [m/N]');
xlim([20, 500]); ylim([2e-7, 1e-4]);
xticks([20, 50, 100, 200, 500])
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/ustation_frf_compliance_Rxyz_model.pdf', 'width', 'wide', 'height', 'normal');
#+end_src

#+name: fig:ustation_frf_compliance_Rxyz_model
#+caption: Extracted FRF of the compliance of the micro-station in the Cartesian frame from the Simscape model
#+RESULTS:
[[file:figs/ustation_frf_compliance_Rxyz_model.png]]

** Get resonance frequencies
#+begin_src matlab
%% Initialize simulation with default parameters (flexible elements)
initializeGround();
initializeGranite();
initializeTy();
initializeRy();
initializeRz();
initializeMicroHexapod();

initializeReferences();
initializeDisturbances();
initializeSimscapeConfiguration();
initializeLoggingConfiguration();
#+end_src

And we identify the dynamics from forces/torques applied on the micro-hexapod top platform to the motion of the micro-hexapod top platform at the same point.

#+begin_src matlab
%% Identification of the compliance
% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Micro-Station/Micro Hexapod/Flexible/Fm'], 1, 'openinput');       io_i = io_i + 1; % Direct Forces/Torques applied on the micro-hexapod top platform
io(io_i) = linio([mdl, '/Micro-Station/Micro Hexapod/Flexible/Dm'], 1, 'output'); io_i = io_i + 1; % Absolute displacement of the top platform

% Run the linearization
Gm = linearize(mdl, io, 0);
Gm.InputName  = {'Fmx', 'Fmy', 'Fmz', 'Mmx', 'Mmy', 'Mmz'};
Gm.OutputName = {'Dx', 'Dy', 'Dz', 'Drx', 'Dry', 'Drz'};
#+end_src

#+begin_src matlab
modes_freq = imag(eig(Gm))/2/pi;
modes_freq = sort(modes_freq(modes_freq>0));
#+end_src

#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
data2orgtable([modes_freq(1:16), [11.9, 18.6, 37.8, 39.1, 56.3, 69.8, 72.5, 84.8, 91.3, 105.5, 106.6, 112.7, 124.2, 145.3, 150.5, 165.4]'], {'1', '2', '3', '4', '5', '6', '7', '8', '9', '10', '11', '12', '13', '14', '15', '16'}, {'Mode', 'Simscape', 'Modal analysis'}, ' %.1f ');
#+end_src

#+RESULTS:
| Mode | Simscape | Modal analysis |
|------+----------+----------------|
|    1 |     11.7 |           11.9 |
|    2 |     21.3 |           18.6 |
|    3 |     26.1 |           37.8 |
|    4 |     57.5 |           39.1 |
|    5 |     60.6 |           56.3 |
|    6 |     73.0 |           69.8 |
|    7 |     97.9 |           72.5 |
|    8 |    120.2 |           84.8 |
|    9 |    126.2 |           91.3 |
|   10 |    142.4 |          105.5 |
|   11 |    155.9 |          106.6 |
|   12 |    178.5 |          112.7 |
|   13 |    179.3 |          124.2 |
|   14 |    182.6 |          145.3 |
|   15 |    223.6 |          150.5 |
|   16 |    226.4 |          165.4 |

** Conclusion
For such a complex system, we believe that the Simscape Model represents the dynamics of the system with enough fidelity.

* Measurement of Positioning Errors
:PROPERTIES:
:HEADER-ARGS:matlab+: :tangle matlab/ustation_2_kinematics.m
:END:
<<sec:ustation_kinematics>>
** Introduction                                                      :ignore:

[[file:~/Cloud/work-projects/ID31-NASS/matlab/nass-simscape/org/kinematics.org]]

** Matlab Init                                              :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
<<matlab-dir>>
#+end_src

#+begin_src matlab :exports none :results silent :noweb yes
<<matlab-init>>
#+end_src

#+begin_src matlab :tangle no :noweb yes
<<m-init-path>>
#+end_src

#+begin_src matlab :eval no :noweb yes
<<m-init-path-tangle>>
#+end_src

#+begin_src matlab :noweb yes
<<m-init-other>>
#+end_src



* Simulation of Scientific Experiments
:PROPERTIES:
:HEADER-ARGS:matlab+: :tangle matlab/.m
:END:
<<sec:ustation_kinematics>>
** Introduction                                                      :ignore:

** Matlab Init                                              :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
<<matlab-dir>>
#+end_src

#+begin_src matlab :exports none :results silent :noweb yes
<<matlab-init>>
#+end_src

#+begin_src matlab :tangle no :noweb yes
<<m-init-path>>
#+end_src

#+begin_src matlab :eval no :noweb yes
<<m-init-path-tangle>>
#+end_src

#+begin_src matlab :noweb yes
<<m-init-other>>
#+end_src


* Estimation of disturbances
This was already done in uni-axial model.

* Conclusion
<<sec:uniaxial_conclusion>>


* 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
#+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
#+END_SRC

** Initialize Simscape Model
#+NAME: m-init-simscape
#+begin_src matlab
% Simulink Model name
mdl = 'ustation_simscape';

load('conf_simulink.mat');
#+end_src

** Initialize other elements
#+NAME: m-init-other
#+BEGIN_SRC matlab
%% Colors for the figures
colors = colororder;

%% Frequency Vector
freqs = logspace(log10(10), log10(2e3), 1000);
#+END_SRC
* Matlab Functions                                                  :noexport:
** Simscape Configuration
:PROPERTIES:
:header-args:matlab+: :tangle matlab/src/initializeSimscapeConfiguration.m
:header-args:matlab+: :comments none :mkdirp yes :eval no
:END:

*** Function description
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  function [] = initializeSimscapeConfiguration(args)
#+end_src

*** Optional Parameters
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  arguments
    args.gravity logical {mustBeNumericOrLogical} = true
  end
#+end_src

*** Structure initialization
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  conf_simscape = struct();
#+end_src

*** Add Type
:PROPERTIES:
:UNNUMBERED: t
:END:
#+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/conf_simscape.mat', 'file')
        save('mat/conf_simscape.mat', 'conf_simscape', '-append');
    else
        save('mat/conf_simscape.mat', 'conf_simscape');
    end
elseif exist('./matlab', 'dir')
    if exist('./matlab/mat/conf_simscape.mat', 'file')
        save('matlab/mat/conf_simscape.mat', 'conf_simscape', '-append');
    else
        save('matlab/mat/conf_simscape.mat', 'conf_simscape');
    end
end
#+end_src

** Logging Configuration
:PROPERTIES:
:header-args:matlab+: :tangle matlab/src/initializeLoggingConfiguration.m
:header-args:matlab+: :comments none :mkdirp yes :eval no
:END:

*** Function description
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  function [] = initializeLoggingConfiguration(args)
#+end_src

*** Optional Parameters
:PROPERTIES:
:UNNUMBERED: t
:END:
#+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
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  conf_log = struct();
#+end_src

*** Add Type
:PROPERTIES:
:UNNUMBERED: t
:END:
#+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
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  conf_log.Ts = args.Ts;
#+end_src

*** Save the Structure
#+begin_src matlab
if exist('./mat', 'dir')
    if exist('./mat/conf_log.mat', 'file')
        save('mat/conf_log.mat', 'conf_log', '-append');
    else
        save('mat/conf_log.mat', 'conf_log');
    end
elseif exist('./matlab', 'dir')
    if exist('./matlab/mat/conf_log.mat', 'file')
        save('matlab/mat/conf_log.mat', 'conf_log', '-append');
    else
        save('matlab/mat/conf_log.mat', 'conf_log');
    end
end
#+end_src

** Ground
:PROPERTIES:
:header-args:matlab+: :tangle matlab/src/initializeGround.m
:header-args:matlab+: :comments none :mkdirp yes :eval no
:END:

*** Function description
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  function [ground] = initializeGround(args)
#+end_src

*** Optional Parameters
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  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
#+end_src

*** Structure initialization
:PROPERTIES:
:UNNUMBERED: t
:END:
First, we initialize the =granite= structure.
#+begin_src matlab
  ground = struct();
#+end_src

*** Add Type
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  switch args.type
    case 'none'
      ground.type = 0;
    case 'rigid'
      ground.type = 1;
  end
#+end_src

*** Ground Solid properties
:PROPERTIES:
:UNNUMBERED: t
:END:
We set the shape and density of the ground solid element.
#+begin_src matlab
  ground.shape   = [2, 2, 0.5]; % [m]
  ground.density = 2800;        % [kg/m3]
#+end_src

*** Rotation Point
:PROPERTIES:
:UNNUMBERED: t
:END:

#+begin_src matlab
  ground.rot_point = args.rot_point;
#+end_src

*** Save the Structure
#+begin_src matlab
if exist('./mat', 'dir')
    if exist('./mat/nass_stages.mat', 'file')
        save('mat/nass_stages.mat', 'ground', '-append');
    else
        save('mat/nass_stages.mat', 'ground');
    end
elseif exist('./matlab', 'dir')
    if exist('./matlab/mat/nass_stages.mat', 'file')
        save('matlab/mat/nass_stages.mat', 'ground', '-append');
    else
        save('matlab/mat/nass_stages.mat', 'ground');
    end
end
#+end_src

** Granite
:PROPERTIES:
:header-args:matlab+: :tangle matlab/src/initializeGranite.m
:header-args:matlab+: :comments none :mkdirp yes :eval no
:END:

*** Function description
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  function [granite] = initializeGranite(args)
#+end_src

*** Optional Parameters
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  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} = 0.8 % Height of the measurment point [m]
  end
#+end_src

*** Structure initialization
:PROPERTIES:
:UNNUMBERED: t
:END:
First, we initialize the =granite= structure.
#+begin_src matlab
  granite = struct();
#+end_src

*** Add Granite Type
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  switch args.type
    case 'none'
      granite.type = 0;
    case 'rigid'
      granite.type = 1;
    case 'flexible'
      granite.type = 2;
  end
#+end_src

*** Material and Geometry
:PROPERTIES:
:UNNUMBERED: t
:END:

Properties of the Material and link to the geometry of the granite.
#+begin_src matlab
  granite.density = args.density; % [kg/m3]
  granite.STEP    = 'granite.STEP';
#+end_src

Z-offset for the initial position of the sample with respect to the granite top surface.
#+begin_src matlab
  granite.sample_pos = args.sample_pos; % [m]
#+end_src

*** Stiffness and Damping properties
:PROPERTIES:
:UNNUMBERED: t
:END:

#+begin_src matlab
  granite.K = args.K; % [N/m]
  granite.C = args.C; % [N/(m/s)]
#+end_src

*** Save the Structure
#+begin_src matlab
if exist('./mat', 'dir')
    if exist('./mat/nass_stages.mat', 'file')
        save('mat/nass_stages.mat', 'granite', '-append');
    else
        save('mat/nass_stages.mat', 'granite');
    end
elseif exist('./matlab', 'dir')
    if exist('./matlab/mat/nass_stages.mat', 'file')
        save('matlab/mat/nass_stages.mat', 'granite', '-append');
    else
        save('matlab/mat/nass_stages.mat', 'granite');
    end
end
#+end_src

** Translation Stage
:PROPERTIES:
:header-args:matlab+: :tangle matlab/src/initializeTy.m
:header-args:matlab+: :comments none :mkdirp yes :eval no
:END:

*** Function description
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  function [ty] = initializeTy(args)
#+end_src

*** Optional Parameters
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  arguments
    args.type      char   {mustBeMember(args.type,{'none', 'rigid', 'flexible'})} = 'flexible'
  end
#+end_src

*** Structure initialization
:PROPERTIES:
:UNNUMBERED: t
:END:
First, we initialize the =ty= structure.
#+begin_src matlab
  ty = struct();
#+end_src

*** Add Translation Stage Type
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  switch args.type
    case 'none'
      ty.type = 0;
    case 'rigid'
      ty.type = 1;
    case 'flexible'
      ty.type = 2;
  end
#+end_src

*** Material and Geometry
:PROPERTIES:
:UNNUMBERED: t
:END:
Define the density of the materials as well as the geometry (STEP files).
#+begin_src matlab
  % 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';
#+end_src

*** Stiffness and Damping properties
:PROPERTIES:
:UNNUMBERED: t
:END:

#+begin_src matlab
  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)]
#+end_src

*** Save the Structure
#+begin_src matlab
if exist('./mat', 'dir')
    if exist('./mat/nass_stages.mat', 'file')
        save('mat/nass_stages.mat', 'ty', '-append');
    else
        save('mat/nass_stages.mat', 'ty');
    end
elseif exist('./matlab', 'dir')
    if exist('./matlab/mat/nass_stages.mat', 'file')
        save('matlab/mat/nass_stages.mat', 'ty', '-append');
    else
        save('matlab/mat/nass_stages.mat', 'ty');
    end
end
#+end_src

** Tilt Stage
:PROPERTIES:
:header-args:matlab+: :tangle matlab/src/initializeRy.m
:header-args:matlab+: :comments none :mkdirp yes :eval no
:END:

*** Function description
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  function [ry] = initializeRy(args)
#+end_src

*** Optional Parameters
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  arguments
    args.type          char    {mustBeMember(args.type,{'none', 'rigid', 'flexible'})} = 'flexible'
    args.Ry_init (1,1) double  {mustBeNumeric} = 0
  end
#+end_src

*** Structure initialization
:PROPERTIES:
:UNNUMBERED: t
:END:
First, we initialize the =ry= structure.
#+begin_src matlab
  ry = struct();
#+end_src


*** Add Tilt Type
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  switch args.type
    case 'none'
      ry.type = 0;
    case 'rigid'
      ry.type = 1;
    case 'flexible'
      ry.type = 2;
  end
#+end_src

*** Material and Geometry
:PROPERTIES:
:UNNUMBERED: t
:END:
Properties of the Material and link to the geometry of the Tilt stage.
#+begin_src matlab
  % 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';
#+end_src

Z-Offset so that the center of rotation matches the sample center;
#+begin_src matlab
  ry.z_offset = 0.58178; % [m]
#+end_src

#+begin_src matlab
  ry.Ry_init = args.Ry_init; % [rad]
#+end_src

*** Stiffness and Damping properties
:PROPERTIES:
:UNNUMBERED: t
:END:

#+begin_src matlab
  ry.K = [3.8e8; 4e8; 3.8e8; 1.2e8; 6e4; 1.2e8];
  ry.C = [1e5;   1e5; 1e5;   3e4;   1e3; 3e4];
#+end_src

*** Save the Structure
#+begin_src matlab
if exist('./mat', 'dir')
    if exist('./mat/nass_stages.mat', 'file')
        save('mat/nass_stages.mat', 'ry', '-append');
    else
        save('mat/nass_stages.mat', 'ry');
    end
elseif exist('./matlab', 'dir')
    if exist('./matlab/mat/nass_stages.mat', 'file')
        save('matlab/mat/nass_stages.mat', 'ry', '-append');
    else
        save('matlab/mat/nass_stages.mat', 'ry');
    end
end
#+end_src

** Spindle
:PROPERTIES:
:header-args:matlab+: :tangle matlab/src/initializeRz.m
:header-args:matlab+: :comments none :mkdirp yes :eval no
:END:

*** Function description
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  function [rz] = initializeRz(args)
#+end_src

*** Optional Parameters
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  arguments
    args.type    char    {mustBeMember(args.type,{'none', 'rigid', 'flexible'})} = 'flexible'
  end
#+end_src

*** Structure initialization
:PROPERTIES:
:UNNUMBERED: t
:END:
First, we initialize the =rz= structure.
#+begin_src matlab
  rz = struct();
#+end_src

*** Add Spindle Type
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  switch args.type
    case 'none'
      rz.type = 0;
    case 'rigid'
      rz.type = 1;
    case 'flexible'
      rz.type = 2;
  end
#+end_src

*** Material and Geometry
:PROPERTIES:
:UNNUMBERED: t
:END:

Properties of the Material and link to the geometry of the spindle.
#+begin_src matlab
  % 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';
#+end_src

*** Stiffness and Damping properties
:PROPERTIES:
:UNNUMBERED: t
:END:

#+begin_src matlab
  rz.K = [7e8; 7e8; 2e9; 1e7; 1e7; 1e7];
  rz.C = [4e4; 4e4; 7e4; 1e4; 1e4; 1e4];
#+end_src

*** Save the Structure
#+begin_src matlab
if exist('./mat', 'dir')
    if exist('./mat/nass_stages.mat', 'file')
        save('mat/nass_stages.mat', 'rz', '-append');
    else
        save('mat/nass_stages.mat', 'rz');
    end
elseif exist('./matlab', 'dir')
    if exist('./matlab/mat/nass_stages.mat', 'file')
        save('matlab/mat/nass_stages.mat', 'rz', '-append');
    else
        save('matlab/mat/nass_stages.mat', 'rz');
    end
end
#+end_src

** Micro Hexapod
:PROPERTIES:
:header-args:matlab+: :tangle matlab/src/initializeMicroHexapod.m
:header-args:matlab+: :comments none :mkdirp yes :eval no
:END:

*** Function description
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  function [micro_hexapod] = initializeMicroHexapod(args)
#+end_src

*** Optional Parameters
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  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 (6,1) double {mustBeNumeric, mustBeNonnegative} = 2e7*ones(6,1)
      args.Ci (6,1) double {mustBeNumeric, mustBeNonnegative} = 1.4e3*ones(6,1)
      % 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
#+end_src

*** Function content
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  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);
#+end_src

#+begin_src matlab
  stewart = initializeStrutDynamics(stewart, ...
                                    'K', args.Ki, ...
                                    'C', args.Ci);

  stewart = initializeJointDynamics(stewart, ...
                                    'type_F', 'universal_p', ...
                                    'type_M', 'spherical_p');
#+end_src

#+begin_src matlab
  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);
#+end_src

#+begin_src matlab
  stewart = initializeInertialSensor(stewart, 'type', 'none');
#+end_src

*** Add Type
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  switch args.type
    case 'none'
      stewart.type = 0;
    case 'rigid'
      stewart.type = 1;
    case 'flexible'
      stewart.type = 2;
  end
#+end_src

*** Save the Structure
#+begin_src matlab
micro_hexapod = stewart;
if exist('./mat', 'dir')
    if exist('./mat/nass_stages.mat', 'file')
        save('mat/nass_stages.mat', 'micro_hexapod', '-append');
    else
        save('mat/nass_stages.mat', 'micro_hexapod');
    end
elseif exist('./matlab', 'dir')
    if exist('./matlab/mat/nass_stages.mat', 'file')
        save('matlab/mat/nass_stages.mat', 'micro_hexapod', '-append');
    else
        save('matlab/mat/nass_stages.mat', 'micro_hexapod');
    end
end
#+end_src

** 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
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  function [ref] = initializeReferences(args)
#+end_src

*** Optional Parameters
:PROPERTIES:
:UNNUMBERED: t
:END:
#+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
:PROPERTIES:
:UNNUMBERED: t
:END:
#+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
:PROPERTIES:
:UNNUMBERED: t
:END:
#+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
:PROPERTIES:
:UNNUMBERED: t
:END:
#+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
:PROPERTIES:
:UNNUMBERED: t
:END:
#+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
:PROPERTIES:
:UNNUMBERED: t
:END:
#+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_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_references.mat', 'file')
        save('mat/nass_references.mat', 'Dy', 'Ry', 'Rz', 'Dh', 'Dhl', 'args', 'Ts', '-append');
    else
        save('mat/nass_references.mat', 'Dy', 'Ry', 'Rz', 'Dh', 'Dhl', 'args', 'Ts');
    end
elseif exist('./matlab', 'dir')
    if exist('./matlab/mat/nass_references.mat', 'file')
        save('matlab/mat/nass_references.mat', 'Dy', 'Ry', 'Rz', 'Dh', 'Dhl', 'args', 'Ts', '-append');
    else
        save('matlab/mat/nass_references.mat', 'Dy', 'Ry', 'Rz', 'Dh', 'Dhl', 'args', 'Ts');
    end
end
#+end_src


** 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
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  function [] = initializeDisturbances(args)
  % initializeDisturbances - Initialize the disturbances
  %
  % Syntax: [] = initializeDisturbances(args)
  %
  % Inputs:
  %    - args -

#+end_src

*** Optional Parameters
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  arguments
      % Global parameter to enable or disable the disturbances
      args.enable logical {mustBeNumericOrLogical} = true
      % Ground Motion - X direction
      args.Dwx logical {mustBeNumericOrLogical} = true
      % Ground Motion - Y direction
      args.Dwy logical {mustBeNumericOrLogical} = true
      % Ground Motion - Z direction
      args.Dwz logical {mustBeNumericOrLogical} = true
      % Translation Stage - X direction
      args.Fty_x logical {mustBeNumericOrLogical} = true
      % Translation Stage - Z direction
      args.Fty_z logical {mustBeNumericOrLogical} = true
      % Spindle - Z direction
      args.Frz_z logical {mustBeNumericOrLogical} = true
  end
#+end_src


*** Load Data
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  load('dist_psd.mat', 'dist_f');
#+end_src

We remove the first frequency point that usually is very large.
#+begin_src matlab :exports none
  dist_f.f = dist_f.f(2:end);
  dist_f.psd_gm = dist_f.psd_gm(2:end);
  dist_f.psd_ty = dist_f.psd_ty(2:end);
  dist_f.psd_rz = dist_f.psd_rz(2:end);
#+end_src

*** Parameters
:PROPERTIES:
:UNNUMBERED: t
:END:
We define some parameters that will be used in the algorithm.
#+begin_src matlab
  Fs = 2*dist_f.f(end);      % Sampling Frequency of data is twice the maximum frequency of the PSD vector [Hz]
  N  = 2*length(dist_f.f);   % Number of Samples match the one of the wanted PSD
  T0 = N/Fs;                 % Signal Duration [s]
  df = 1/T0;                 % Frequency resolution of the DFT [Hz]
                             % Also equal to (dist_f.f(2)-dist_f.f(1))
  t = linspace(0, T0, N+1)'; % Time Vector [s]
  Ts = 1/Fs;                 % Sampling Time [s]
#+end_src

*** Ground Motion
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  phi = dist_f.psd_gm;
  C = zeros(N/2,1);
  for i = 1:N/2
    C(i) = sqrt(phi(i)*df);
  end
#+end_src

#+begin_src matlab
  if args.Dwx && args.enable
    rng(111);
    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)))];;
    Dwx = N/sqrt(2)*ifft(Cx); % Ground Motion - x direction [m]
  else
    Dwx = zeros(length(t), 1);
  end
#+end_src

#+begin_src matlab
  if args.Dwy && args.enable
    rng(112);
    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)))];;
    Dwy = N/sqrt(2)*ifft(Cx); % Ground Motion - y direction [m]
  else
    Dwy = zeros(length(t), 1);
  end
#+end_src

#+begin_src matlab
  if args.Dwy && args.enable
    rng(113);
    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)))];;
    Dwz = N/sqrt(2)*ifft(Cx); % Ground Motion - z direction [m]
  else
    Dwz = zeros(length(t), 1);
  end
#+end_src

*** Translation Stage - X direction
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  if args.Fty_x && args.enable
    phi = dist_f.psd_ty; % TODO - we take here the vertical direction which is wrong but approximate
    C = zeros(N/2,1);
    for i = 1:N/2
      C(i) = sqrt(phi(i)*df);
    end
    rng(121);
    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)))];;
    u = N/sqrt(2)*ifft(Cx); % Disturbance Force Ty x [N]
    Fty_x = u;
  else
    Fty_x = zeros(length(t), 1);
  end
#+end_src

*** Translation Stage - Z direction
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  if args.Fty_z && args.enable
    phi = dist_f.psd_ty;
    C = zeros(N/2,1);
    for i = 1:N/2
      C(i) = sqrt(phi(i)*df);
    end
    rng(122);
    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)))];;
    u = N/sqrt(2)*ifft(Cx); % Disturbance Force Ty z [N]
    Fty_z = u;
  else
    Fty_z = zeros(length(t), 1);
  end
#+end_src

*** Spindle - Z direction
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  if args.Frz_z && args.enable
    phi = dist_f.psd_rz;
    C = zeros(N/2,1);
    for i = 1:N/2
      C(i) = sqrt(phi(i)*df);
    end
    rng(131);
    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)))];;
    u = N/sqrt(2)*ifft(Cx); % Disturbance Force Rz z [N]
    Frz_z = u;
  else
    Frz_z = zeros(length(t), 1);
  end
#+end_src

*** Direct Forces
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  u = zeros(length(t), 6);
  Fd = u;
#+end_src

*** Set initial value to zero
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  Dwx    = Dwx   - Dwx(1);
  Dwy    = Dwy   - Dwy(1);
  Dwz    = Dwz   - Dwz(1);
  Fty_x  = Fty_x - Fty_x(1);
  Fty_z  = Fty_z - Fty_z(1);
  Frz_z  = Frz_z - Frz_z(1);
#+end_src

*** Save the Structure
#+begin_src matlab
if exist('./mat', 'dir')
    if exist('./mat/nass_disturbances.mat', 'file')
        save('mat/nass_disturbances.mat', 'Dwx', 'Dwy', 'Dwz', 'Fty_x', 'Fty_z', 'Frz_z', 'Fd', 'Ts', 't', 'args', '-append');
    else
        save('mat/nass_disturbances.mat', 'Dwx', 'Dwy', 'Dwz', 'Fty_x', 'Fty_z', 'Frz_z', 'Fd', 'Ts', 't', 'args');
    end
elseif exist('./matlab', 'dir')
    if exist('./matlab/mat/nass_disturbances.mat', 'file')
        save('matlab/mat/nass_disturbances.mat', 'Dwx', 'Dwy', 'Dwz', 'Fty_x', 'Fty_z', 'Frz_z', 'Fd', 'Ts', 't', 'args', '-append');
    else
        save('matlab/mat/nass_disturbances.mat', 'Dwx', 'Dwy', 'Dwz', 'Fty_x', 'Fty_z', 'Frz_z', 'Fd', 'Ts', 't', 'args');
    end
end
#+end_src

** =initializeStewartPlatform=: Initialize the Stewart Platform structure
:PROPERTIES:
:header-args:matlab+: :tangle matlab/src/initializeStewartPlatform.m
:header-args:matlab+: :comments none :mkdirp yes :eval no
:END:

*** Documentation
:PROPERTIES:
:UNNUMBERED: t
:END:

#+name: fig:stewart-frames-position
#+caption: Definition of the position of the frames
[[file:figs/stewart-frames-position.png]]

*** Function description
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  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 -
#+end_src

*** Initialize the Stewart structure
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  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_src

** =initializeFramesPositions=: Initialize the positions of frames {A}, {B}, {F} and {M}
:PROPERTIES:
:header-args:matlab+: :tangle matlab/src/initializeFramesPositions.m
:header-args:matlab+: :comments none :mkdirp yes :eval no
:END:

*** Documentation
:PROPERTIES:
:UNNUMBERED: t
:END:

#+name: fig:stewart-frames-position
#+caption: Definition of the position of the frames
[[file:figs/stewart-frames-position.png]]

*** Function description
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  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]
#+end_src

*** Optional Parameters
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  arguments
      stewart
      args.H    (1,1) double {mustBeNumeric, mustBePositive} = 90e-3
      args.MO_B (1,1) double {mustBeNumeric} = 50e-3
  end
#+end_src

*** Compute the position of each frame
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  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]
#+end_src

*** Populate the =stewart= structure
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  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_src

** =generateGeneralConfiguration=: Generate a Very General Configuration
:PROPERTIES:
:header-args:matlab+: :tangle matlab/src/generateGeneralConfiguration.m
:header-args:matlab+: :comments none :mkdirp yes :eval no
:END:

*** Documentation
:PROPERTIES:
:UNNUMBERED: t
:END:
Joints are positions on a circle centered with the Z axis of {F} and {M} and at a chosen distance from {F} and {M}.
The radius of the circles can be chosen as well as the angles where the joints are located (see Figure [[fig:joint_position_general]]).

#+begin_src latex :file stewart_bottom_plate.pdf :tangle no
  \begin{tikzpicture}
    % Internal and external limit
    \draw[fill=white!80!black] (0, 0) circle [radius=3];
    % Circle where the joints are located
    \draw[dashed] (0, 0) circle [radius=2.5];

    % Bullets for the positions of the joints
    \node[] (J1) at ( 80:2.5){$\bullet$};
    \node[] (J2) at (100:2.5){$\bullet$};
    \node[] (J3) at (200:2.5){$\bullet$};
    \node[] (J4) at (220:2.5){$\bullet$};
    \node[] (J5) at (320:2.5){$\bullet$};
    \node[] (J6) at (340:2.5){$\bullet$};

    % Name of the points
    \node[above right] at (J1) {$a_{1}$};
    \node[above left]  at (J2) {$a_{2}$};
    \node[above left]  at (J3) {$a_{3}$};
    \node[right     ]  at (J4) {$a_{4}$};
    \node[left      ]  at (J5) {$a_{5}$};
    \node[above right] at (J6) {$a_{6}$};

    % First 2 angles
    \draw[dashed, ->] (0:1)   arc [start angle=0, end angle=80, radius=1]    node[below right]{$\theta_{1}$};
    \draw[dashed, ->] (0:1.5) arc [start angle=0, end angle=100, radius=1.5] node[left       ]{$\theta_{2}$};

    % Division of 360 degrees by 3
    \draw[dashed] (0, 0) -- ( 80:3.2);
    \draw[dashed] (0, 0) -- (100:3.2);
    \draw[dashed] (0, 0) -- (200:3.2);
    \draw[dashed] (0, 0) -- (220:3.2);
    \draw[dashed] (0, 0) -- (320:3.2);
    \draw[dashed] (0, 0) -- (340:3.2);

    % Radius for the position of the joints
    \draw[<->] (0, 0) --node[near end, above]{$R$} (180:2.5);

    \draw[->] (0, 0) -- ++(3.4, 0) node[above]{$x$};
    \draw[->] (0, 0) -- ++(0, 3.4) node[left]{$y$};
  \end{tikzpicture}
#+end_src

#+name: fig:joint_position_general
#+caption: Position of the joints
#+RESULTS:
[[file:figs/stewart_bottom_plate.png]]

*** Function description
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  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}
#+end_src

*** Optional Parameters
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  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
#+end_src

*** Compute the pose
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  Fa = zeros(3,6);
  Mb = zeros(3,6);
#+end_src

#+begin_src matlab
  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
#+end_src

*** Populate the =stewart= structure
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  stewart.platform_F.Fa = Fa;
  stewart.platform_M.Mb = Mb;
#+end_src

** =computeJointsPose=: Compute the Pose of the Joints
:PROPERTIES:
:header-args:matlab+: :tangle matlab/src/computeJointsPose.m
:header-args:matlab+: :comments none :mkdirp yes :eval no
:END:

*** Documentation
:PROPERTIES:
:UNNUMBERED: t
:END:

#+name: fig:stewart-struts
#+caption: Position and orientation of the struts
[[file:figs/stewart-struts.png]]

*** Function description
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  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}
#+end_src

*** Check the =stewart= structure elements
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  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;
#+end_src

*** Compute the position of the Joints
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  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]);
#+end_src

*** Compute the strut length and orientation
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  As = (Ab - Aa)./vecnorm(Ab - Aa); % As_i is the i'th vector of As

  l = vecnorm(Ab - Aa)';
#+end_src

#+begin_src matlab
  Bs = (Bb - Ba)./vecnorm(Bb - Ba);
#+end_src

*** Compute the orientation of the Joints
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  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
#+end_src

*** Populate the =stewart= structure
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  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_src

** =initializeCylindricalPlatforms=: Initialize the geometry of the Fixed and Mobile Platforms
:PROPERTIES:
:header-args:matlab+: :tangle matlab/src/initializeCylindricalPlatforms.m
:header-args:matlab+: :comments none :mkdirp yes :eval no
:END:

*** Function description
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  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]
#+end_src

*** Optional Parameters
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  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
#+end_src

*** Compute the Inertia matrices of platforms
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  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]);
#+end_src

#+begin_src matlab
  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]);
#+end_src

*** Populate the =stewart= structure
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  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;
#+end_src

#+begin_src matlab
  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_src

** =initializeCylindricalStruts=: Define the inertia of cylindrical struts
:PROPERTIES:
:header-args:matlab+: :tangle matlab/src/initializeCylindricalStruts.m
:header-args:matlab+: :comments none :mkdirp yes :eval no
:END:

*** Function description
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  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]
#+end_src

*** Optional Parameters
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  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
#+end_src

*** Compute the properties of the cylindrical struts
:PROPERTIES:
:UNNUMBERED: t
:END:

#+begin_src matlab
  Fsm = ones(6,1).*args.Fsm;
  Fsh = ones(6,1).*args.Fsh;
  Fsr = ones(6,1).*args.Fsr;

  Msm = ones(6,1).*args.Msm;
  Msh = ones(6,1).*args.Msh;
  Msr = ones(6,1).*args.Msr;
#+end_src

#+begin_src matlab
  I_F = zeros(3, 3, 6); % Inertia of the "fixed" part of the strut
  I_M = zeros(3, 3, 6); % Inertia of the "mobile" part of the strut

  for i = 1:6
    I_F(:,:,i) = diag([1/12 * Fsm(i) * (3*Fsr(i)^2 + Fsh(i)^2), ...
                       1/12 * Fsm(i) * (3*Fsr(i)^2 + Fsh(i)^2), ...
                       1/2  * Fsm(i) * Fsr(i)^2]);

    I_M(:,:,i) = diag([1/12 * Msm(i) * (3*Msr(i)^2 + Msh(i)^2), ...
                       1/12 * Msm(i) * (3*Msr(i)^2 + Msh(i)^2), ...
                       1/2  * Msm(i) * Msr(i)^2]);
  end
#+end_src

*** Populate the =stewart= structure
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  stewart.struts_M.type = 1;

  stewart.struts_M.I = I_M;
  stewart.struts_M.M = Msm;
  stewart.struts_M.R = Msr;
  stewart.struts_M.H = Msh;
#+end_src

#+begin_src matlab
  stewart.struts_F.type = 1;

  stewart.struts_F.I = I_F;
  stewart.struts_F.M = Fsm;
  stewart.struts_F.R = Fsr;
  stewart.struts_F.H = Fsh;
#+end_src

** =initializeStrutDynamics=: Add Stiffness and Damping properties of each strut
:PROPERTIES:
:header-args:matlab+: :tangle matlab/src/initializeStrutDynamics.m
:header-args:matlab+: :comments none :mkdirp yes :eval no
:END:

*** Documentation
:PROPERTIES:
:UNNUMBERED: t
:END:

#+name: fig:piezoelectric_stack
#+attr_html: :width 500px
#+caption: Example of a piezoelectric stach actuator (PI)
[[file:figs/piezoelectric_stack.jpg]]

A simplistic model of such amplified actuator is shown in Figure [[fig:actuator_model_simple]] where:
- $K$ represent the vertical stiffness of the actuator
- $C$ represent the vertical damping of the actuator
- $F$ represents the force applied by the actuator
- $F_{m}$ represents the total measured force
- $v_{m}$ represents the absolute velocity of the top part of the actuator
- $d_{m}$ represents the total relative displacement of the actuator

#+begin_src latex :file actuator_model_simple.pdf :tangle no
  \begin{tikzpicture}
    \draw (-1, 0) -- (1, 0);

    % Spring, Damper, and Actuator
    \draw[spring]   (-1, 0) -- (-1, 1.5) node[midway, left=0.1]{$K$};
    \draw[damper]   ( 0, 0) -- ( 0, 1.5) node[midway, left=0.2]{$C$};
    \draw[actuator] ( 1, 0) -- ( 1, 1.5) node[midway, left=0.1](F){$F$};

    \node[forcesensor={2}{0.2}] (fsens) at (0, 1.5){};

    \node[left] at (fsens.west) {$F_{m}$};

    \draw[dashed] (1, 0) -- ++(0.4, 0);
    \draw[dashed] (1, 1.7) -- ++(0.4, 0);

    \draw[->] (0, 1.7)node[]{$\bullet$} -- ++(0, 0.5) node[right]{$v_{m}$};

    \draw[<->] (1.4, 0) -- ++(0, 1.7) node[midway, right]{$d_{m}$};
  \end{tikzpicture}
#+end_src

#+name: fig:actuator_model_simple
#+caption: Simple model of an Actuator
#+RESULTS:
[[file:figs/actuator_model_simple.png]]

*** Function description
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  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)]
#+end_src

*** Optional Parameters
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  arguments
      stewart
      args.type char {mustBeMember(args.type,{'classical', 'amplified'})} = 'classical'
      args.K (6,1) double {mustBeNumeric, mustBeNonnegative} = 20e6*ones(6,1)
      args.C (6,1) double {mustBeNumeric, mustBeNonnegative} = 2e1*ones(6,1)
      args.k1 (6,1) double {mustBeNumeric} = 1e6*ones(6,1)
      args.ke (6,1) double {mustBeNumeric} = 5e6*ones(6,1)
      args.ka (6,1) double {mustBeNumeric} = 60e6*ones(6,1)
      args.c1 (6,1) double {mustBeNumeric} = 10*ones(6,1)
      args.F_gain (6,1) double {mustBeNumeric} = 1*ones(6,1)
      args.me (6,1) double {mustBeNumeric} = 0.01*ones(6,1)
      args.ma (6,1) double {mustBeNumeric} = 0.01*ones(6,1)
  end
#+end_src

*** Add Stiffness and Damping properties of each strut
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  if strcmp(args.type, 'classical')
      stewart.actuators.type = 1;
  elseif strcmp(args.type, 'amplified')
      stewart.actuators.type = 2;
  end

  stewart.actuators.K = args.K;
  stewart.actuators.C = args.C;

  stewart.actuators.k1 = args.k1;
  stewart.actuators.c1 = args.c1;

  stewart.actuators.ka = args.ka;
  stewart.actuators.ke = args.ke;

  stewart.actuators.F_gain = args.F_gain;

  stewart.actuators.ma = args.ma;
  stewart.actuators.me = args.me;
#+end_src

** =initializeJointDynamics=: Add Stiffness and Damping properties for spherical joints
:PROPERTIES:
:header-args:matlab+: :tangle matlab/src/initializeJointDynamics.m
:header-args:matlab+: :comments none :mkdirp yes :eval no
:END:

*** Function description
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  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)]
#+end_src

*** Optional Parameters
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  arguments
      stewart
      args.type_F     char   {mustBeMember(args.type_F,{'universal', 'spherical', 'universal_p', 'spherical_p', 'universal_3dof', 'spherical_3dof', 'flexible'})} = 'universal'
      args.type_M     char   {mustBeMember(args.type_M,{'universal', 'spherical', 'universal_p', 'spherical_p', 'universal_3dof', 'spherical_3dof', 'flexible'})} = 'spherical'
      args.Kf_M (6,1) double {mustBeNumeric, mustBeNonnegative} = 33*ones(6,1)
      args.Cf_M (6,1) double {mustBeNumeric, mustBeNonnegative} = 1e-4*ones(6,1)
      args.Kt_M (6,1) double {mustBeNumeric, mustBeNonnegative} = 236*ones(6,1)
      args.Ct_M (6,1) double {mustBeNumeric, mustBeNonnegative} = 1e-3*ones(6,1)
      args.Kf_F (6,1) double {mustBeNumeric, mustBeNonnegative} = 33*ones(6,1)
      args.Cf_F (6,1) double {mustBeNumeric, mustBeNonnegative} = 1e-4*ones(6,1)
      args.Kt_F (6,1) double {mustBeNumeric, mustBeNonnegative} = 236*ones(6,1)
      args.Ct_F (6,1) double {mustBeNumeric, mustBeNonnegative} = 1e-3*ones(6,1)
      args.Ka_F (6,1) double {mustBeNumeric, mustBeNonnegative} = 1.2e8*ones(6,1)
      args.Ca_F (6,1) double {mustBeNumeric, mustBeNonnegative} = 1e1*ones(6,1)
      args.Kr_F (6,1) double {mustBeNumeric, mustBeNonnegative} = 1.1e7*ones(6,1)
      args.Cr_F (6,1) double {mustBeNumeric, mustBeNonnegative} = 1e1*ones(6,1)
      args.Ka_M (6,1) double {mustBeNumeric, mustBeNonnegative} = 1.2e8*ones(6,1)
      args.Ca_M (6,1) double {mustBeNumeric, mustBeNonnegative} = 1e1*ones(6,1)
      args.Kr_M (6,1) double {mustBeNumeric, mustBeNonnegative} = 1.1e7*ones(6,1)
      args.Cr_M (6,1) double {mustBeNumeric, mustBeNonnegative} = 1e1*ones(6,1)
      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
#+end_src

*** Add Actuator Type
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  switch args.type_F
    case 'universal'
      stewart.joints_F.type = 1;
    case 'spherical'
      stewart.joints_F.type = 2;
    case 'universal_p'
      stewart.joints_F.type = 3;
    case 'spherical_p'
      stewart.joints_F.type = 4;
    case 'flexible'
      stewart.joints_F.type = 5;
    case 'universal_3dof'
      stewart.joints_F.type = 6;
    case 'spherical_3dof'
      stewart.joints_F.type = 7;
  end

  switch args.type_M
    case 'universal'
      stewart.joints_M.type = 1;
    case 'spherical'
      stewart.joints_M.type = 2;
    case 'universal_p'
      stewart.joints_M.type = 3;
    case 'spherical_p'
      stewart.joints_M.type = 4;
    case 'flexible'
      stewart.joints_M.type = 5;
    case 'universal_3dof'
      stewart.joints_M.type = 6;
    case 'spherical_3dof'
      stewart.joints_M.type = 7;
  end
#+end_src

*** Add Stiffness and Damping in Translation of each strut
:PROPERTIES:
:UNNUMBERED: t
:END:
Axial and Radial (shear) Stiffness
#+begin_src matlab
  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;
#+end_src

Translation Damping
#+begin_src matlab
  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;
#+end_src

*** Add Stiffness and Damping in Rotation of each strut
:PROPERTIES:
:UNNUMBERED: t
:END:
Rotational Stiffness
#+begin_src matlab
  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;
#+end_src

Rotational Damping
#+begin_src matlab
  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;
#+end_src

*** Stiffness and Mass matrices for flexible joint
:PROPERTIES:
:UNNUMBERED: t
:END:

#+begin_src matlab
  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_src

** =initializeStewartPose=: Determine the initial stroke in each leg to have the wanted pose
:PROPERTIES:
:header-args:matlab+: :tangle matlab/src/initializeStewartPose.m
:header-args:matlab+: :comments none :mkdirp yes :eval no
:END:

*** Function description
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  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}
#+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)
  end
#+end_src

*** Use the Inverse Kinematic function
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  [Li, dLi] = inverseKinematics(stewart, 'AP', args.AP, 'ARB', args.ARB);
#+end_src

*** Populate the =stewart= structure
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  stewart.actuators.Leq = dLi;
#+end_src

** =computeJacobian=: Compute the Jacobian Matrix
:PROPERTIES:
:header-args:matlab+: :tangle matlab/src/computeJacobian.m
:header-args:matlab+: :comments none :mkdirp yes :eval no
:END:

*** Function description
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  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:
  %        - kinematics.J [6x6] - The Jacobian Matrix
  %        - kinematics.K [6x6] - The Stiffness Matrix
  %        - kinematics.C [6x6] - The Compliance Matrix
#+end_src

*** Check the =stewart= structure elements
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  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;
#+end_src


*** Compute Jacobian Matrix
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  J = [As' , cross(Ab, As)'];
#+end_src

*** Compute Stiffness Matrix
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  K = J'*diag(Ki)*J;
#+end_src

*** Compute Compliance Matrix
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  C = inv(K);
#+end_src

*** Populate the =stewart= structure
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  stewart.kinematics.J = J;
  stewart.kinematics.K = K;
  stewart.kinematics.C = C;
#+end_src

** =initializeInertialSensor=: Initialize the inertial sensor in each strut
:PROPERTIES:
:header-args:matlab+: :tangle matlab/src/initializeInertialSensor.m
:header-args:matlab+: :comments none :mkdirp yes :eval no
:END:

*** Geophone - Working Principle
:PROPERTIES:
:UNNUMBERED: t
:END:
From the schematic of the Z-axis geophone shown in Figure [[fig:z_axis_geophone]], we can write the transfer function from the support velocity $\dot{w}$ to the relative velocity of the inertial mass $\dot{d}$:
\[ \frac{\dot{d}}{\dot{w}} = \frac{-\frac{s^2}{{\omega_0}^2}}{\frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1} \]
with:
- $\omega_0 = \sqrt{\frac{k}{m}}$
- $\xi = \frac{1}{2} \sqrt{\frac{m}{k}}$

#+name: fig:z_axis_geophone
#+caption: Schematic of a Z-Axis geophone
[[file:figs/inertial_sensor.png]]

We see that at frequencies above $\omega_0$:
\[ \frac{\dot{d}}{\dot{w}} \approx -1 \]

And thus, the measurement of the relative velocity of the mass with respect to its support gives the absolute velocity of the support.

We generally want to have the smallest resonant frequency $\omega_0$ to measure low frequency absolute velocity, however there is a trade-off between $\omega_0$ and the mass of the inertial mass.

*** Accelerometer - Working Principle
:PROPERTIES:
:UNNUMBERED: t
:END:
From the schematic of the Z-axis accelerometer shown in Figure [[fig:z_axis_accelerometer]], we can write the transfer function from the support acceleration $\ddot{w}$ to the relative position of the inertial mass $d$:
\[ \frac{d}{\ddot{w}} = \frac{-\frac{1}{{\omega_0}^2}}{\frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1} \]
with:
- $\omega_0 = \sqrt{\frac{k}{m}}$
- $\xi = \frac{1}{2} \sqrt{\frac{m}{k}}$

#+name: fig:z_axis_accelerometer
#+caption: Schematic of a Z-Axis geophone
[[file:figs/inertial_sensor.png]]

We see that at frequencies below $\omega_0$:
\[ \frac{d}{\ddot{w}} \approx -\frac{1}{{\omega_0}^2} \]

And thus, the measurement of the relative displacement of the mass with respect to its support gives the absolute acceleration of the support.

Note that there is trade-off between:
- the highest measurable acceleration $\omega_0$
- the sensitivity of the accelerometer which is equal to $-\frac{1}{{\omega_0}^2}$

*** Function description
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  function [stewart] = initializeInertialSensor(stewart, args)
  % initializeInertialSensor - Initialize the inertial sensor in each strut
  %
  % Syntax: [stewart] = initializeInertialSensor(args)
  %
  % Inputs:
  %    - args - Structure with the following fields:
  %        - type       - 'geophone', 'accelerometer', 'none'
  %        - mass [1x1] - Weight of the inertial mass [kg]
  %        - freq [1x1] - Cutoff frequency [Hz]
  %
  % Outputs:
  %    - stewart - updated Stewart structure with the added fields:
  %      - stewart.sensors.inertial
  %        - type    - 1 (geophone), 2 (accelerometer), 3 (none)
  %        - K [1x1] - Stiffness [N/m]
  %        - C [1x1] - Damping [N/(m/s)]
  %        - M [1x1] - Inertial Mass [kg]
  %        - G [1x1] - Gain
#+end_src

*** Optional Parameters
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  arguments
      stewart
      args.type       char   {mustBeMember(args.type,{'geophone', 'accelerometer', 'none'})} = 'none'
      args.mass (1,1) double {mustBeNumeric, mustBeNonnegative} = 1e-2
      args.freq (1,1) double {mustBeNumeric, mustBeNonnegative} = 1e3
  end
#+end_src

*** Compute the properties of the sensor
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  sensor = struct();

  switch args.type
    case 'geophone'
      sensor.type = 1;

      sensor.M = args.mass;
      sensor.K = sensor.M * (2*pi*args.freq)^2;
      sensor.C = 2*sqrt(sensor.M * sensor.K);
    case 'accelerometer'
      sensor.type = 2;

      sensor.M = args.mass;
      sensor.K = sensor.M * (2*pi*args.freq)^2;
      sensor.C = 2*sqrt(sensor.M * sensor.K);
      sensor.G = -sensor.K/sensor.M;
    case 'none'
      sensor.type = 3;
  end
#+end_src

*** Populate the =stewart= structure
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  stewart.sensors.inertial = sensor;
#+end_src



** =inverseKinematics=: Compute Inverse Kinematics
:PROPERTIES:
:header-args:matlab+: :tangle matlab/src/inverseKinematics.m
:header-args:matlab+: :comments none :mkdirp yes :eval no
:END:

*** Theory
:PROPERTIES:
:UNNUMBERED: t
:END:
For inverse kinematic analysis, it is assumed that the position ${}^A\bm{P}$ and orientation of the moving platform ${}^A\bm{R}_B$ are given and the problem is to obtain the joint variables, namely, $\bm{L} = [l_1, l_2, \dots, l_6]^T$.

From the geometry of the manipulator, the loop closure for each limb, $i = 1, 2, \dots, 6$ can be written as
\begin{align*}
  l_i {}^A\hat{\bm{s}}_i &= {}^A\bm{A} + {}^A\bm{b}_i - {}^A\bm{a}_i \\
                         &= {}^A\bm{A} + {}^A\bm{R}_b {}^B\bm{b}_i - {}^A\bm{a}_i
\end{align*}

To obtain the length of each actuator and eliminate $\hat{\bm{s}}_i$, it is sufficient to dot multiply each side by itself:
\begin{equation}
  l_i^2 \left[ {}^A\hat{\bm{s}}_i^T {}^A\hat{\bm{s}}_i \right] = \left[ {}^A\bm{P} + {}^A\bm{R}_B {}^B\bm{b}_i - {}^A\bm{a}_i \right]^T \left[ {}^A\bm{P} + {}^A\bm{R}_B {}^B\bm{b}_i - {}^A\bm{a}_i \right]
\end{equation}

Hence, for $i = 1, 2, \dots, 6$, each limb length can be uniquely determined by:
\begin{equation}
  l_i = \sqrt{{}^A\bm{P}^T {}^A\bm{P} + {}^B\bm{b}_i^T {}^B\bm{b}_i + {}^A\bm{a}_i^T {}^A\bm{a}_i - 2 {}^A\bm{P}^T {}^A\bm{a}_i + 2 {}^A\bm{P}^T \left[{}^A\bm{R}_B {}^B\bm{b}_i\right] - 2 \left[{}^A\bm{R}_B {}^B\bm{b}_i\right]^T {}^A\bm{a}_i}
\end{equation}

If the position and orientation of the moving platform lie in the feasible workspace of the manipulator, one unique solution to the limb length is determined by the above equation.
Otherwise, when the limbs' lengths derived yield complex numbers, then the position or orientation of the moving platform is not reachable.

*** Function description
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  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}
#+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)
  end
#+end_src

*** Check the =stewart= structure elements
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  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;
#+end_src


*** Compute
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  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));
#+end_src

#+begin_src matlab
  dLi = Li-l;
#+end_src

** =describeMicroStationSetup=
:PROPERTIES:
:header-args:matlab+: :tangle matlab/src/describeMicroStationSetup.m
:header-args:matlab+: :comments none :mkdirp yes :eval no
:END:

*** Function description
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  function [] = describeMicroStationSetup()
  % describeMicroStationSetup -
  %
  % Syntax: [] = describeMicroStationSetup()
  %
  % Inputs:
  %    -  -
  %
  % Outputs:
  %    -  -
#+end_src

*** Simscape Configuration
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  load('./mat/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
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  load('./mat/nass_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.Fty_x && args.Fty_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
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  load('./mat/nass_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

*** Controller
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  load('./mat/controller.mat', 'controller');
#+end_src

#+begin_src matlab
  fprintf('Controller:\n');
  fprintf('- %s\n', controller.name);
  fprintf('\n');
#+end_src

*** Micro-Station
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  load('./mat/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