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* 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.
** 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 $\{M\}$).
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 ${}^W\bm{T}_R$.
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 ${}^W\bm{T}_M$.
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
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
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.
