nass-simscape/org/kinematics.org

#+TITLE: Kinematics of the station
#+SETUPFILE: ./setup/org-setup-file.org

* Introduction                                                       :ignore:
In this document, we discuss the way the motion of each stage is defined.

* Micro Hexapod
** How the Symetrie Hexapod is controlled on the micro station
For the Micro-Hexapod, the convention for the angles are defined in =MAN_A_Software API_4.0.150918_EN.pdf= on page 13 (section 2.4 - Rotation Vectors):

#+begin_quote
The *Euler type II convention* is used to express the rotation vector.
This convention is mainly used in the aeronautics field (standard ISO 1151 concerning flight mechanics).

This convention uses the concepts of rotation of vehicles (ship, car and plane).
Generally, we consider that the main movement of the vehicle is following the X-axis and the Z-axis is parallel to the axis of gravity (at the initial  position).
The roll rotation is around the X-axis, the pitch is around the Y-axis and yaw is the rotation around the Z-axis.
*The order of rotation is: Rx, Ry and then Rz.*

In most case, rotations are related to a reference with fixed axis; thus we say the rotations are around fixed axes.
The combination of these three rotations enables to write a rotation matrix.
This writing is unique and equal to:
\[ \bm{R} = \bm{R}_z(\gamma) \cdot \bm{R}_y(\beta) \cdot \bm{R}_x(\alpha) \]

The Euler type II convention corresponding to the *succession of rotations with respect to fixed axes*: first around X0, then Y0 and Z0.
This is equivalent to the succession of rotations with respect to mobile axes: first around Z0, then Y1' and X2'.
#+end_quote

More generally on the Control of the Micro-Hexapod:
#+begin_quote
Note that for all control modes, *the rotation center coincides with Object coordinate system origin*.
Moreover, the movements are controlled with *translation components at first* (Tx, Ty, Tz) *then rotation components* (Rx, Ry, Rz).
#+end_quote

Thus, it does the translations and then the rotation around the new translated frame.

** Control of the Micro-Hexapod using Simscape
*** Introduction                                                   :ignore:
We can think of two main ways to position the Micro-Hexapod using Simscape.

The first one is to use only one Bushing Joint between the base and the mobile platform.
The advantage is that it is very easy to impose the wanted displacement, however, we loose the dynamical properties of the Hexapod.

The second way is to specify the wanted length of the legs of the Hexapod in order to have the wanted position of the mobile platform.
This require a little bit more of mathematical derivations but this is the chosen solution.

*** Using Bushing Joint
In the documentation of the Bushing Joint (=doc "Bushing Joint"=) that is used to position the Hexapods, it is mention that the following frame is positioned with respect to the base frame in a way shown in figure [[fig:bushing_joint_transform]].

#+name: fig:bushing_joint_transform
#+caption: Joint Transformation Sequence for the Bushing Joint
[[file:figs/bushing_joint_transform.png]]

Basically, it performs the translations, and then the rotation along the X, Y and Z axis of the moving frame.
The three rotations that we define thus corresponds to the Euler U-V-W angles.

We should have the *same behavior* for the Micro-Hexapod on Simscape (same inputs at least).
However, the Bushing Joint makes rotations around mobiles axes (X, Y' and then Z'') and not fixed axes (X, Y and Z).

*** Using Inverse Kinematics and Leg Actuators
Here, we can use the Inverse Kinematic of the Hexapod to determine the length of each leg in order to obtain some defined translation and rotation of the mobile platform.

The advantages are:
- we can position the Hexapod as we want by specifying a rotation matrix
- the hexapod keeps its full flexibility as we don't specify any wanted displacements, only leg's rest position

However:
- even though the rest position of each leg (the position where the stiffness force is zero) is set correctly, the hexapod will we deflected due to gravity

Thus, for this simulation, we *remove the gravity*.

**** Theory
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.

**** 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

**** Matlab Implementation
We open the Simulink file.
#+begin_src matlab
  open('nass_model.slx')
#+end_src

We load the configuration and set a small =StopTime=.
#+begin_src matlab
  load('mat/conf_simulink.mat');
  set_param(conf_simulink, 'StopTime', '0.1');
#+end_src

We define the wanted position/orientation of the Hexapod under study.
#+begin_src matlab
  tx = 0.05; % [rad]
  ty = 0.1; % [rad]
  tz = 0.02; % [rad]

  Rx = [1 0        0;
        0 cos(tx) -sin(tx);
        0 sin(tx)  cos(tx)];

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

  Rz = [cos(tz) -sin(tz) 0;
        sin(tz)  cos(tz) 0;
        0        0       1];

  ARB = Rz*Ry*Rx;
  AP = [0.1; 0.005; 0.01]; % [m]
#+end_src

#+begin_src matlab
  initializeSimscapeConfiguration('gravity', false);
  initializeGround('type', 'none');
  initializeGranite('type', 'none');
  initializeTy('type', 'none');
  initializeRy('type', 'none');
  initializeRz('type', 'none');
  initializeMicroHexapod('type', 'rigid', 'AP', AP, 'ARB', ARB);
  initializeAxisc('type', 'none');
  initializeMirror('type', 'none');
  initializeNanoHexapod('type', 'none');
  initializeSample('type', 'none');
  initializeLoggingConfiguration('log', 'all');
#+end_src

We run the simulation.
#+begin_src matlab
  sim('nass_model');
#+end_src

And we verify that we indeed succeed to go to the wanted position.
#+begin_src matlab :results table replace
  [simout.Dhm.x.Data(end) ; simout.Dhm.y.Data(end) ; simout.Dhm.z.Data(end)] - AP
#+end_src

#+RESULTS:
|  8.4655e-16 |
|  1.5586e-15 |
| -2.1337e-16 |

#+begin_src matlab :results table replace
  simout.Dhm.R.Data(:, :, end)-ARB
#+end_src

#+RESULTS:
| -1.1102e-16 |   -1.36e-15 | 4.2744e-15 |
|  1.0651e-15 |  6.6613e-16 | 5.1278e-15 |
| -4.2882e-15 | -4.9336e-15 | 1.1102e-16 |

* TODO Tests on the transformation from reference to wanted position :noexport:
:PROPERTIES:
:header-args:matlab+: :eval no
:END:
** 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