nass-simscape/positioning_error/index.org

#+TITLE: Computation of the Positioning Error with respect to the nano-hexapod
:DRAWER:
#+STARTUP: overview

#+LANGUAGE: en
#+EMAIL: dehaeze.thomas@gmail.com
#+AUTHOR: Dehaeze Thomas

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:END:

* Introduction                                                       :ignore:
The global measurement and control schematic is shown in figure [[fig:control-schematic-nass]].

#+name: fig:control-schematic-nass
#+caption: Global Control Schematic for the Station
[[file:figs/control-schematic-nass.png]]

In this document, we develop and verify that the two green blocs are working.

We suppose that we are able to measure perfectly the position of the sample with respect to the granite.
This means that we do not care about the bloc "Compute Sample Position w.r.t. Granite" that makes the transformation from the interferometer measurements to the position of the sample.
We suppose that we can directly measure perfectly the position of the sample with respect to the granite.

Also, all the stages can be perfectly positioned.

In section [[sec:compute_reference]], we verify that the function developed to compute the wanted pose (translation and orientation) of the sample with respect to the granite can be determined from the wanted position of each stage (translation stage, tilt stage, spindle and micro-hexapod). This corresponds to the bloc "Compute Wanted Sample Position w.r.t. Granite" in figure [[fig:control-schematic-nass]].
To do so, we impose a perfect displacement and all the stage, we perfectly measure the position of the sample with respect to the granite, and we verify that this measured position corresponds to the computed wanted pose of the sample.

Then, in section [[sec:compute_pos_error]], we introduce some positioning error in the micro-station's stages.
The positioning error of the sample expressed with respect to the granite frame (the one measured) is expressed in a frame connected to the NASS top platform (corresponding to the green bloc "Compute Sample Position Error w.r.t. NASS" in figure [[fig:control-schematic-nass]]).
Then, we move the NASS such that it compensate for the positioning error that are expressed in the frame of the NASS, and we verify that the positioning error of the sample is well compensated.

* How do we measure the position of the sample with respect to the granite
<<sec:measurement_principle>>
A transform sensor block gives the translation and orientation of the follower frame with respect to the base frame.

The base frame is fixed to the granite and located at the initial sample location that defines the zero position.

The follower frame is attached to the sample (or more precisely to the reflector).

The outputs of the transform sensor are:
- the 3 translations x, y and z in meter
- the *rotation matrix* $\bm{R}$ that permits to rotate the base frame into the follower frame.

We can then determine extract other orientation conventions such that Euler angles or screw axis.

* Verify that the function to compute the reference pose is correct
<<sec:compute_reference>>
** Introduction                                                    :ignore:
The goal here is to perfectly move the station and verify that there is no mismatch between the metrology measurement and the computation of the reference pose.

** 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
  simulinkproject('../');
#+end_src

#+begin_src matlab
  open('positioning_error/matlab/sim_nano_station_metrology.slx')
#+end_src

** Prepare the Simulation
We load the configuration.
#+begin_src matlab
  load('mat/conf_simscape.mat');
#+end_src

We set a small =StopTime=.
#+begin_src matlab
  set_param(conf_simscape, 'StopTime', '0.5');
#+end_src

We initialize all the stages.
#+begin_src matlab
  initializeGround();
  initializeGranite();
  initializeTy();
  initializeRy();
  initializeRz();
  initializeMicroHexapod();
  initializeAxisc();
  initializeMirror();
  initializeNanoHexapod(struct('actuator', 'piezo'));
  initializeSample(struct('mass', 50));
#+end_src

We setup the reference path to be constant.
#+begin_src matlab
  opts = struct( ...
      'Ts',           1e-3, ...       % Sampling Frequency [s]
      'Dy_type',      'constant', ... % Either "constant" / "triangular" / "sinusoidal"
      'Dy_amplitude', 5e-3, ...          % Amplitude of the displacement [m]
      'Dy_period',    1, ...          % Period of the displacement [s]
      'Ry_type',      'constant', ... % Either "constant" / "triangular" / "sinusoidal"
      'Ry_amplitude', -1*pi/180, ...          % Amplitude [rad]
      'Ry_period',    10, ...         % Period of the displacement [s]
      'Rz_type',      'constant', ... % Either "constant" / "rotating"
      'Rz_amplitude', -135*pi/180, ...          % Initial angle [rad]
      'Rz_period',    1, ...          % Period of the rotating [s]
      'Dh_type',      'constant', ... % For now, only constant is implemented
      'Dh_pos',       [0.01; 0.02; -0.03; -3*pi/180; 1*pi/180; 3*pi/180], ...  % Initial position [m,m,m,rad,rad,rad] of the top platform
      'Rm_type',      'constant', ... % For now, only constant is implemented
      'Rm_pos',       [0, pi]', ...   % Initial position of the two masses
      'Dn_type',      'constant', ... % For now, only constant is implemented
      'Dn_pos',       [1e-3; 2e-3; 3e-3; 1*pi/180; 0; 1*pi/180] ...  % Initial position [m,m,m,rad,rad,rad] of the top platform
  );

  initializeReferences(opts);
#+end_src

No position error for now (perfect positioning).
#+begin_src matlab
  Dye = 0; % [m]
  Rye = 0; % [rad]
  Rze = 0; % [rad]
  Dhe = zeros(6,1); % [m,rad]
  Dhle = zeros(6,1); % [m]
  Dne = zeros(6,1); % [m,rad]
#+end_src

And we run the simulation.
#+begin_src matlab
  sim('sim_nano_station_metrology.slx');
#+end_src

** Verify that the pose of the sample is the same as the computed one
Let's denote:
- $\{W\}$ the initial fixed frame (base in which the interferometric measurement is done)
- $\{R\}$ 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:
- ${}^W\bm{T}_R$ which corresponds to the wanted pose of the sample with respect to the granite
- ${}^W\bm{T}_M$ which corresponds to the measured pose of the sample with respect to the granite

We load the reference and we compute the desired trajectory of the sample in the form of an homogeneous transformation matrix ${}^W\bm{T}_R$.
#+begin_src matlab
  n = length(Dref.Dy.Time);
  WTr = zeros(4, 4, n);
  for i = 1:n
    WTr(:, :, i) = computeReferencePose(Dref.Dy.Data(i), Dref.Ry.Data(i), Dref.Rz.Data(i), Dref.Dh.Data(i,:), Dref.Dn.Data(i,:));
  end
#+end_src

As the displacement is perfect, we also measure in simulation the pose of the sample with respect to the granite.
From that we can compute the homogeneous transformation matrix ${}^W\bm{T}_M$.
#+begin_src matlab
  n = length(Dsm.R.Time);
  WTm = zeros(4, 4, n);
  WTm(1:3, 1:3, :) = Dsm.R.Data;
  WTm(1:3, 4, :) = [Dsm.x.Data' ; Dsm.y.Data' ; Dsm.z.Data'];
  WTm(4, 4, :) = 1;
#+end_src

As the simulation is perfect (no measurement error and no motion error), we should have that
\[ {}^W\bm{T}_R = {}^W\bm{T}_M \]

Or are least:
\[ {}^W\bm{T}_R(1:3, 4) = {}^W\bm{T}_M(1:3, 4) \]
\[ {}^W\bm{R}_R^t \cdot {}^W\bm{R}_M = \bm{I}_3 \]

#+begin_src matlab :results output replace
  WTr(1:3, 4, end)-WTm(1:3, 4, end)
  WTr(1:3, 1:3, end)'*WTm(1:3, 1:3, end)-eye(3)
#+end_src

#+RESULTS:
#+begin_example
WTr(1:3, 4, end)-WTm(1:3, 4, end)
ans =
       1.8027246362351e-14
      1.40408518145563e-14
      6.93889390390723e-17
WTr(1:3, 1:3, end)'*WTm(1:3, 1:3, end)-eye(3)
ans =
      1.59872115546023e-14     -1.56629266848118e-14     -3.71230823859037e-16
      1.56742023874057e-14      1.59872115546023e-14     -2.12330153459561e-15
     -1.14144804719274e-15     -5.51642065360625e-16      9.28146448586631e-14
#+end_example

** Conclusion
#+begin_important
We are able to compute the wanted position and orientation of the sample.
Both the measurement and the theory gives the same result.
#+end_important

* Verify that the function to convert the position error in the frame fixed to the nano-hexapod is working
<<sec:compute_pos_error>>
** Introduction                                                     :ignore:
We now introduce some positioning error in the stage.
This will induce a global positioning error of the sample with respect to the desired pose that we can compute.

We want to verify that we are able to measure this positioning error and convert it in the frame attached to the Nano-hexapod.

** 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
  simulinkproject('../');
#+end_src

#+begin_src matlab
  open('positioning_error/matlab/sim_nano_station_metrology.slx')
#+end_src

** Prepare the Simulation
We load the configuration.
#+begin_src matlab
  load('mat/conf_simscape.mat');
#+end_src

We set a small =StopTime=.
#+begin_src matlab
  set_param(conf_simscape, 'StopTime', '0.5');
#+end_src

We initialize all the stages.
#+begin_src matlab
  initializeGround();
  initializeGranite();
  initializeTy();
  initializeRy();
  initializeRz();
  initializeMicroHexapod();
  initializeAxisc();
  initializeMirror();
  initializeNanoHexapod(struct('actuator', 'piezo'));
  initializeSample(struct('mass', 50));
#+end_src

We setup the reference path to be constant.
#+begin_src matlab
  opts = struct(   ...
      'Ts',           1e-3, ...       % Sampling Frequency [s]
      'Dy_type',      'constant', ... % Either "constant" / "triangular" / "sinusoidal"
      'Dy_amplitude', 0, ...          % Amplitude of the displacement [m]
      'Ry_type',      'constant', ... % Either "constant" / "triangular" / "sinusoidal"
      'Ry_amplitude', 0, ...          % Amplitude [rad]
      'Rz_type',      'constant', ... % Either "constant" / "rotating"
      'Rz_amplitude', 0*pi/180, ...          % Initial angle [rad]
      'Dh_type',      'constant', ... % For now, only constant is implemented
      'Dh_pos',       [0; 0; 0; 0; 0; 0],  ...  % Initial position [m,m,m,rad,rad,rad] of the top platform
      'Rm_type',      'constant', ... % For now, only constant is implemented
      'Rm_pos',       [0, pi]', ...   % Initial position of the two masses
      'Dn_type',      'constant', ... % For now, only constant is implemented
      'Dn_pos',       [0; 0; 0; 0; 0; 0]  ...  % Initial position [m,m,m,rad,rad,rad] of the top platform
  );

  initializeReferences(opts);
#+end_src

Now we introduce some positioning error.
#+begin_src matlab
  Dye = 1e-6; % [m]
  Rye = 2e-4; % [rad]
  Rze = 1e-5; % [rad]
  Dhe = zeros(6,1);
  % Dne = [1e-3 ; 0 ; 2e-3 ; 0 ; 3e-3 ; 1e-3]; % [m,rad]
  Dhle = [1e-6 ; 2e-6 ; 3e-6 ; -2e-6 ; 1e-6 ; 2e-6]; % [m]
  Dne = zeros(6,1);
#+end_src

And we run the simulation.
#+begin_src matlab
  sim('sim_nano_station_metrology.slx');
#+end_src

** Compute the wanted pose of the sample in the NASS Base from the metrology and the reference
Now that we have introduced some positioning error, the computed wanted pose and the measured pose will not be the same.

We would like to compute ${}^M\bm{T}_R$ which corresponds to the wanted pose of the sample expressed in a frame attached to the top platform of the nano-hexapod (frame $\{M\}$).

We have:
\begin{align}
  {}^M\bm{T}_R &= {}^M\bm{T}_W \cdot {}^W\bm{T}_R \\
          &= {}^W{\bm{T}_M}^{-1} \cdot {}^W\bm{T}_R
\end{align}

The top platform of the nano-hexapod is considered to be rigidly connected to the sample, thus, ${}^M\bm{T}_R$ corresponds to the pose error of the sample with respect to the nano-hexapod platform.

We load the reference and we compute the desired trajectory of the sample in the form of an homogeneous transformation matrix ${}^W\bm{T}_R$.
#+begin_src matlab
  n = length(Dref.Dy.Time);
  WTr = zeros(4, 4, n);
  for i = 1:n
    WTr(:, :, i) = computeReferencePose(Dref.Dy.Data(i), Dref.Ry.Data(i), Dref.Rz.Data(i), Dref.Dh.Data(i,:), Dref.Dn.Data(i,:));
  end
#+end_src

We also measure in simulation the pose of the sample with respect to the granite.
From that we can compute the homogeneous transformation matrix ${}^W\bm{T}_M$.
#+begin_src matlab
  n = length(Dsm.R.Time);
  WTm = zeros(4, 4, n);
  WTm(1:3, 1:3, :) = Dsm.R.Data;
  WTm(1:3, 4, :) = [Dsm.x.Data' ; Dsm.y.Data' ; Dsm.z.Data'];
  WTm(4, 4, :) = 1;
#+end_src

The *inverse of the transformation matrix* can be obtain by (it is less computation intensive than doing a full inverse)
\begin{equation}
  {}^B\bm{T}_A = {}^A\bm{T}_B^{-1} =
  \left[ \begin{array}{ccc|c}
      &   &   & \\
      & {}^A\bm{R}_B^T  &   & -{}^A \bm{R}_B^T {}^A\bm{P}_{O_B} \\
      &   &   & \\
    \hline
    0 & 0 & 0 & 1 \\
  \end{array} \right]
\end{equation}

Finally, we compute ${}^M\bm{T}_R$.
#+begin_src matlab
  MTr = zeros(4, 4, n);
  for i = 1:n
    MTr(:, :, i) = [WTm(1:3,1:3,i)', -WTm(1:3,1:3,i)'*WTm(1:3,4,i) ; 0 0 0 1]*WTr(:,:,i);
  end
#+end_src

Verify that the pose error corresponds to the positioning error of the stages.
#+begin_src matlab :exports none
  Edx = MTr(1, 4, end);
  Edy = MTr(2, 4, end);
  Edz = MTr(3, 4, end);
  % The angles obtained are u-v-w Euler angles (rotations in the moving frame)
  Ery = atan2( MTr(1, 3, end),          sqrt(MTr(1, 1, end)^2 + MTr(1, 2, end)^2));
  Erx = atan2(-MTr(2, 3, end)/cos(Ery), MTr(3, 3, end)/cos(Ery));
  Erz = atan2(-MTr(1, 2, end)/cos(Ery), MTr(1, 1, end)/cos(Ery));
#+end_src

#+begin_src matlab
  MTr(1:3, 1:3, end)
  Rx = [1 0       0;
        0 cos(Erx) -sin(Erx);
        0 sin(Erx)  cos(Erx)];

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

  Rz = [cos(Erz) -sin(Erz) 0;
        sin(Erz)  cos(Erz) 0;
        0       0      1];
#+end_src

#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
  data2orgtable([Edx, Edy, Edz, Erx, Ery, Erz], {'Error'}, {'Edx [m]', 'Edy [m]', 'Edz [m]', 'Erx [rad]', 'Ery [rad]', 'Erz [rad]'}, ' %.1e ');
#+end_src

#+RESULTS:
|       | Edx [m] |  Edy [m] |  Edz [m] | Erx [rad] | Ery [rad] | Erz [rad] |
|-------+---------+----------+----------+-----------+-----------+-----------|
| Error | 2.8e-06 | -2.0e-06 | -1.3e-06 |  -5.1e-06 |  -1.8e-04 |   4.2e-07 |

** Verify that be imposing the error motion on the nano-hexapod, we indeed have zero error at the end
We now keep the wanted pose but we impose a displacement of the nano hexapod corresponding to the measured position error.
#+begin_src matlab
  opts.Dn_pos = [Edx, Edy, Edz, Erx, Ery, Erz]';
  initializeReferences(opts);
#+end_src

And we run the simulation.
#+begin_src matlab
  sim('sim_nano_station_metrology.slx');
#+end_src

We keep the old computed computed reference pose ${}^W\bm{T}_r$ even though we have change the nano hexapod reference, but this is not a real wanted reference but rather a adaptation to reject the positioning errors.

As the displacement is perfect, we also measure in simulation the pose of the sample with respect to the granite.
From that we can compute the homogeneous transformation matrix ${}^W\bm{T}_M$.
#+begin_src matlab
  n = length(Dsm.R.Time);
  WTm = zeros(4, 4, n);
  WTm(1:3, 1:3, :) = Dsm.R.Data;
  WTm(1:3, 4, :) = [Dsm.x.Data' ; Dsm.y.Data' ; Dsm.z.Data'];
  WTm(4, 4, :) = 1;
#+end_src

Finally, we compute ${}^M\bm{T}_R$.
#+begin_src matlab
  MTr = zeros(4, 4, n);
  for i = 1:n
    MTr(:, :, i) = [WTm(1:3,1:3,i)', -WTm(1:3,1:3,i)'*WTm(1:3,4,i) ; 0 0 0 1]*WTr(:,:,i);
  end
#+end_src

Verify that the pose error is small.
#+begin_src matlab :exports none
  Edx = MTr(1, 4, end);
  Edy = MTr(2, 4, end);
  Edz = MTr(3, 4, end);
  Ery = atan2(MTr(1, 3, end), sqrt(MTr(1, 1, end)^2 + MTr(1, 2, end)^2));
  Erx = atan2(-MTr(2, 3, end)/cos(Ery), MTr(3, 3, end)/cos(Ery));
  Erz = atan2(-MTr(1, 2, end)/cos(Ery), MTr(1, 1, end)/cos(Ery));
#+end_src

#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
  data2orgtable([Edx, Edy, Edz, Erx, Ery, Erz], {'Error'}, {'Edx [m]', 'Edy [m]', 'Edz [m]', 'Erx [rad]', 'Ery [rad]', 'Erz [rad]'}, ' %.1e ');
#+end_src

#+RESULTS:
|       | Edx [m] |  Edy [m] | Edz [m] | Erx [rad] | Ery [rad] | Erz [rad] |
|-------+---------+----------+---------+-----------+-----------+-----------|
| Error | 1.0e-16 | -1.1e-18 | 3.3e-20 |   1.5e-16 |   5.8e-17 |  -6.1e-16 |

** Conclusion
#+begin_important
  Indeed, we are able to convert the position error in the frame of the NASS and then compensate these errors with the NASS.
#+end_important

* Verify that we are able to compensate the errors using the nano-hexapod
* Tests on the transformation from reference to wanted position    :noexport:
** 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

** 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
  simulinkproject('../');
#+end_src

#+begin_src matlab
  open('positioning_error/matlab/sim_nano_station_metrology.slx')
#+end_src

** 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 => seems better
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 \\
          &= {}^W{T_M}^{-1} {}^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