#+TITLE: Modal Analysis - Processing of FRF
:DRAWER:
#+STARTUP: overview

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

#+HTML_LINK_HOME: ../index.html
#+HTML_LINK_UP: ./index.html

#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="../css/htmlize.css"/>
#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="../css/readtheorg.css"/>
#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="../css/zenburn.css"/>
#+HTML_HEAD: <script type="text/javascript" src="../js/jquery.min.js"></script>
#+HTML_HEAD: <script type="text/javascript" src="../js/bootstrap.min.js"></script>
#+HTML_HEAD: <script type="text/javascript" src="../js/jquery.stickytableheaders.min.js"></script>
#+HTML_HEAD: <script type="text/javascript" src="../js/readtheorg.js"></script>

#+HTML_MATHJAX: align: center tagside: right font: TeX

#+PROPERTY: header-args:matlab  :session *MATLAB*
#+PROPERTY: header-args:matlab+ :comments org
#+PROPERTY: header-args:matlab+ :results none
#+PROPERTY: header-args:matlab+ :exports both
#+PROPERTY: header-args:matlab+ :eval no-export
#+PROPERTY: header-args:matlab+ :output-dir figs

#+PROPERTY: header-args:shell  :eval no-export

#+PROPERTY: header-args:latex  :headers '("\\usepackage{tikz}" "\\usepackage{import}" "\\import{$HOME/MEGA/These/LaTeX/}{config.tex}")
#+PROPERTY: header-args:latex+ :imagemagick t :fit yes
#+PROPERTY: header-args:latex+ :iminoptions -scale 100% -density 150
#+PROPERTY: header-args:latex+ :imoutoptions -quality 100
#+PROPERTY: header-args:latex+ :results raw replace :buffer no
#+PROPERTY: header-args:latex+ :eval no-export
#+PROPERTY: header-args:latex+ :exports both
#+PROPERTY: header-args:latex+ :mkdirp yes
#+PROPERTY: header-args:latex+ :output-dir figs
:END:

The measurements have been conducted and we have computed the $n \times p \times q$ Frequency Response Functions Matrix with:
- $n$: the number of measurements: $23 \times 3 = 69$ (23 accelerometers measuring 3 directions each)
- $p$: the number of excitation inputs: $3$
- $q$: the number of frequency points $\omega_i$

However, in our model, we only consider 6 solid bodies, namely:
- Bottom Granite
- Top Granite
- Translation Stage
- Tilt Stage
- Spindle
- Hexapod

Thus, we are only interested in $6 \times 6 = 36$ degrees of freedom.

We here process the FRF matrix to go from the 69 measured DOFs to the wanted 36 DOFs.

* 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

* Importation of measured FRF curves
We load the measured FRF and Coherence matrices.
We also load the geometric parameters of the station: solid bodies considered and the position of the accelerometers.

#+begin_src matlab
  load('./mat/frf_coh_matrices.mat', 'FRFs', 'COHs', 'freqs');
  load('mat/geometry.mat', 'solids', 'solid_names', 'acc_pos');
#+end_src

* From accelerometer DOFs to solid body DOFs - Mathematics
Let's consider the schematic shown on figure [[fig:local_to_global_coordinates]] where we are measuring the motion of a (supposed) solid body at 4 distinct points in x-y-z.

The goal here is to link these $4 \times 3 = 12$ measurements to the 6 DOFs of the solid body expressed in the frame $\{O\}$.

#+begin_src latex :file local_to_global_coordinates.pdf :post pdf2svg(file=*this*, ext="png") :exports results
  \newcommand\irregularcircle[2]{% radius, irregularity
    \pgfextra {\pgfmathsetmacro\len{(#1)+rand*(#2)}}
    +(0:\len pt)
    \foreach \a in {10,20,...,350}{
      \pgfextra {\pgfmathsetmacro\len{(#1)+rand*(#2)}}
      -- +(\a:\len pt)
    } -- cycle
  }
  \begin{tikzpicture}
    \draw[rounded corners=1mm, fill=blue!30!white] (0, 0) \irregularcircle{3cm}{1mm};

    \node[] (origin) at (4, -1) {$\bullet$};
    \begin{scope}[shift={(origin)}]
      \def\axissize{0.8cm}
      \draw[->] (0, 0) -- ++(\axissize, 0) node[above left]{$x$};
      \draw[->] (0, 0) -- ++(0, \axissize) node[below right]{$y$};
      \draw[fill, color=black] (0, 0) circle (0.05*\axissize);
      \node[draw, circle, inner sep=0pt, minimum size=0.4*\axissize, label=left:$z$] (yaxis) at (0, 0){};
      \node[below right] at (0, 0){$\{O\}$};
    \end{scope}

    \coordinate[] (p1) at (-1.5, -1.5);
    \coordinate[] (p2) at (-1.5,  1.5);
    \coordinate[] (p3) at ( 1.5,  1.5);
    \coordinate[] (p4) at ( 1.5, -1.5);

    \draw[->] (p1)node[]{$\bullet$}node[above]{$p_1$} -- ++(1, 0.5)node[right]{$v_1$};
    \draw[->] (p2)node[]{$\bullet$}node[above]{$p_2$} -- ++(-0.5, 1)node[right]{$v_2$};
    \draw[->] (p3)node[]{$\bullet$}node[above]{$p_3$} -- ++(1, 0.5)node[right]{$v_3$};
    \draw[->] (p4)node[]{$\bullet$}node[above]{$p_4$} -- ++(0.5, 1)node[right]{$v_4$};
  \end{tikzpicture}
#+end_src

#+name: fig:local_to_global_coordinates
#+caption: Schematic of the measured motions of a solid body
#+RESULTS:
[[file:figs/local_to_global_coordinates.png]]

From the figure [[fig:local_to_global_coordinates]], we can write:
\begin{align*}
  \vec{v}_1 &= \vec{v} + \Omega \vec{p}_1\\
  \vec{v}_2 &= \vec{v} + \Omega \vec{p}_2\\
  \vec{v}_3 &= \vec{v} + \Omega \vec{p}_3\\
  \vec{v}_4 &= \vec{v} + \Omega \vec{p}_4
\end{align*}

With
\begin{equation}
  \Omega = \begin{bmatrix}
  0 & -\Omega_z & \Omega_y \\
  \Omega_z & 0 & -\Omega_x \\
  -\Omega_y & \Omega_x & 0
\end{bmatrix}
\end{equation}
$\vec{v}$ and $\Omega$ represent to velocity and rotation of the solid expressed in the frame $\{O\}$.

We can rearrange the equations in a matrix form:

\begin{equation}
\left[\begin{array}{ccc|ccc}
  1 & 0 & 0 & 0 & p_{1z} & -p_{1y} \\
  0 & 1 & 0 & -p_{1z} & 0 & p_{1x} \\
  0 & 0 & 1 & p_{1y} & -p_{1x} & 0 \\ \hline
    & \vdots &   &    & \vdots &   \\ \hline
  1 & 0 & 0 & 0 & p_{4z} & -p_{4y} \\
  0 & 1 & 0 & -p_{4z} & 0 & p_{4x} \\
  0 & 0 & 1 & p_{4y} & -p_{4x} & 0
\end{array}\right] \begin{bmatrix}
  v_x \\ v_y \\ v_z \\ \hline \Omega_x \\ \Omega_y \\ \Omega_z
\end{bmatrix} = \begin{bmatrix}
  v_{1x} \\ v_{1y} \\ v_{1z} \\\hline \vdots \\\hline v_{4x} \\ v_{4y} \\ v_{4z}
\end{bmatrix}
\end{equation}

and then we obtain the velocity and rotation of the solid in the wanted frame $\{O\}$:
\begin{equation}
\begin{bmatrix}
  v_x \\ v_y \\ v_z \\ \hline \Omega_x \\ \Omega_y \\ \Omega_z
\end{bmatrix} =
\left[\begin{array}{ccc|ccc}
  1 & 0 & 0 & 0 & p_{1z} & -p_{1y} \\
  0 & 1 & 0 & -p_{1z} & 0 & p_{1x} \\
  0 & 0 & 1 & p_{1y} & -p_{1x} & 0 \\ \hline
    & \vdots &   &    & \vdots &   \\ \hline
  1 & 0 & 0 & 0 & p_{4z} & -p_{4y} \\
  0 & 1 & 0 & -p_{4z} & 0 & p_{4x} \\
  0 & 0 & 1 & p_{4y} & -p_{4x} & 0
\end{array}\right]^{-1} \begin{bmatrix}
  v_{1x} \\ v_{1y} \\ v_{1z} \\\hline \vdots \\\hline v_{4x} \\ v_{4y} \\ v_{4z}
\end{bmatrix}
\end{equation}

This inversion is equivalent to resolving a mean square problem.

* What reference frame to choose?
The question we wish here to answer is how to choose the reference frame $\{O\}$ in which the DOFs of the solid bodies are defined.

The goal is to compare the motion of each solid body to see which relative DOFs between solid bodies can be neglected, that is to say, which joint between solid bodies can be regarded as perfect (and this in all the frequency range of interest).
Ideally, we would like to have the same number of degrees of freedom than the number of identified modes.

The possibles choices are:
- *One frame for each solid body* which is located at its center of mass
- *One common frame*, for instance located at the point of interest ($270mm$ above the Hexapod)
- *Base located at the joint position*: this is where we want to see the motion and estimate stiffness

#+name: tab:frame_comparison
#+caption: Advantages and disadvantages for the choice of reference frame
| Chosen Frame             | Advantages                                          | Disadvantages                                        |
|--------------------------+-----------------------------------------------------+------------------------------------------------------|
| Frames at CoM            | Physically, it makes more sense                     | How to compare the motion of the solid bodies?       |
| Common Frame             | We can compare the motion of each solid body        | Small $\theta_{x, y}$ may result in large $T_{x, y}$ |
| Frames at joint position | Directly gives which joint direction can be blocked | How to choose the joint position?                    |

As the easiest choice is to choose a common frame, we start with that solution.

* From accelerometer DOFs to solid body DOFs - Matlab Implementation
First, we initialize a new FRF matrix =FRFs_O= which is an $n \times p \times q$ with:
- $n$ is the number of DOFs of the considered 6 solid-bodies: $6 \times 6 = 36$
- $p$ is the number of excitation inputs: $3$
- $q$ is the number of frequency points $\omega_i$

#+begin_src matlab
  FRFs_O = zeros(length(solid_names)*6, 3, 801);
#+end_src

Then, as we know the positions of the accelerometers on each solid body, and we have the response of those accelerometers, we can use the equations derived in the previous section to determine the response of each solid body expressed in the frame $\{O\}$.
#+begin_src matlab
  for solid_i = 1:length(solid_names)
    solids_i = solids.(solid_names{solid_i});

    A = zeros(3*length(solids_i), 6);
    for i = 1:length(solids_i)
      acc_i = solids_i(i);

      A(3*(i-1)+1:3*i, 1:3) = eye(3);
      A(3*(i-1)+1:3*i, 4:6) = [ 0                  acc_pos(acc_i, 3) -acc_pos(acc_i, 2) ;
                               -acc_pos(acc_i, 3)  0                  acc_pos(acc_i, 1) ;
                                acc_pos(acc_i, 2) -acc_pos(acc_i, 1)  0];
    end

    for exc_dir = 1:3
      FRFs_O((solid_i-1)*6+1:solid_i*6, exc_dir, :) = A\squeeze(FRFs((solids_i(1)-1)*3+1:solids_i(end)*3, exc_dir, :));
    end
  end
#+end_src

* Analysis of some FRF in the global coordinates
First, we can compare the motions of the 6 solid bodies in one direction (figure [[fig:frf_all_bodies_one_direction]])

We can also compare all the DOFs of one solid body (figure [[fig:frf_one_body_all_directions]]).

#+begin_src matlab :exports none
  exc_names = {'$F_x$', '$F_y$', '$F_z$'};
  DOFs = {'$T_x$', '$T_y$', '$T_z$', '$\theta_x$', '$\theta_y$', '$\theta_z$'}
  solids_i = 1:6;
  dir_i = 1;
  exc_dir = 1;

  figure;

  ax1 = subaxis(2, 1, 1);
  hold on;
  for solid_i = solids_i
    plot(freqs, abs(squeeze(FRFs_O((solid_i-1)*6+dir_i, exc_dir, :))), 'DisplayName', solid_names{solid_i});
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  set(gca, 'XTickLabel',[]);
  ylabel('Amplitude');
  legend('Location', 'northwest');
  title(sprintf('FRF between %s and %s', exc_names{exc_dir}, DOFs{dir_i}));

  ax2 = subaxis(2, 1, 2);
  hold on;
  for solid_i = solids_i
    plot(freqs, mod(180+180/pi*phase(squeeze(FRFs_O((solid_i-1)*6+dir_i, exc_dir, :))), 360)-180);
  end
  hold off;
  ylim([-180, 180]); yticks(-180:90:180);
  xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
  set(gca, 'xscale', 'log');

  linkaxes([ax1,ax2],'x');
  xlim([1, 200]);
#+end_src

#+HEADER: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/frf_all_bodies_one_direction.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
  <<plt-matlab>>
#+end_src

#+NAME: fig:frf_all_bodies_one_direction
#+CAPTION: FRFs of all the 6 solid bodies in one direction
[[file:figs/frf_all_bodies_one_direction.png]]

#+begin_src matlab :exports none
  DOFs = {'$T_x$', '$T_y$', '$T_z$', '$\theta_x$', '$\theta_y$', '$\theta_z$'}
  solid_i = 3;
  dirs_i = 1:6;
  exc_dir = 1;

  figure;

  ax1 = subplot(2, 1, 1);
  hold on;
  for dir_i = dirs_i
    plot(freqs, abs(squeeze(FRFs_O((solid_i-1)*6+dir_i, exc_dir, :))), 'DisplayName', DOFs{dir_i});
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  set(gca, 'XTickLabel',[]);
  ylabel('Amplitude');
  legend('Location', 'northwest');
  title(sprintf('Motion of %s due to %s', solid_names{solid_i}, exc_names{exc_dir}));

  ax2 = subplot(2, 1, 2);
  hold on;
  for dir_i = dirs_i
    plot(freqs, mod(180+180/pi*phase(squeeze(FRFs_O((solid_i-1)*6+dir_i, exc_dir, :))), 360)-180);
  end
  hold off;
  ylim([-180, 180]); yticks(-180:90:180);
  xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
  set(gca, 'xscale', 'log');

  linkaxes([ax1,ax2],'x');
  xlim([1, 200]);
#+end_src

#+HEADER: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/frf_one_body_all_directions.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
  <<plt-matlab>>
#+end_src

#+NAME: fig:frf_one_body_all_directions
#+CAPTION: FRFs of one solid body in all its DOFs
[[file:figs/frf_one_body_all_directions.png]]

* TODO How to compare the relative motion of solid bodies
We have some of elements of the full FRF matrix:
\[ \frac{D_{1x}}{F_x},\ \frac{D_{1y}}{F_x},\ \frac{D_{1z}}{F_x},\ \frac{D_{2x}}{F_x},\ \dots \]

\[ \frac{D_{1x}}{D_{2x}} = \frac{\frac{D_{1x}}{F_x}}{\frac{D_{2x}}{F_x}} \]
Then, if $\left| \frac{D_{1x}}{D_{2x}} \right| \approx 1$ in all the frequency band of interest, we can block the $x$ motion between the solids 1 and 2.

\[ \frac{D_{2x} - D_{1x}}{D_{1x} + D_{2x}} = \frac{\frac{D_{2x}}{F_x} - \frac{D_{1x}}{F_x}}{\frac{D_{1x}}{F_x} + \frac{D_{2x}}{F_x}} \]

Then if $\left| \frac{D_{2x} - D_{1x}}{D_{1x} + D_{2x}} \right| \ll 1$ in all the frequency band of interest, we can block the $x$ motion between the solids 1 and 2.

* Relative Motion in the global coordinates
Below we plot the normalized relative motion between each stage:
\[ 0 < \frac{\left| D_{ix} - D_{jx} \right|}{|D_{ix}| + |D_{jx}|} < 1 \]

#+begin_src matlab
  DOFs = {'$T_x$', '$T_y$', '$T_z$', '$\theta_x$', '$\theta_y$', '$\theta_z$'}

  dirs_i = 1:6;
  exc_dir = 1;

  figure;

  for i = 2:6
    subaxis(3, 2, i);
    hold on;
    for dir_i = dirs_i
      H = (squeeze(FRFs_O((i-1)*6+dir_i, exc_dir, :))-squeeze(FRFs_O((i-2)*6+dir_i, exc_dir, :)))./(abs(squeeze(FRFs_O((i-1)*6+dir_i, exc_dir, :)))+abs(squeeze(FRFs_O((i-2)*6+dir_i, exc_dir, :))));
      plot(freqs, abs(H));
    end
    hold off;
    set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
    xlim([1, 200]); ylim([0, 1]);
    % xlabel('Frequency [Hz]'); ylabel('Relative Motion');
    title(sprintf('Normalized motion %s - %s', solid_names{i-1}, solid_names{i}));
    if i > 4
      xlabel('Frequency [Hz]');
    else
      set(gca, 'XTickLabel',[]);
    end
  end

  for i = 1:length(dirs_i)
    legend_names{i} = DOFs{dirs_i(i)};
  end
  lgd = legend(legend_names);

  hL = subplot(3, 2, 1);
  poshL = get(hL,'position');

  set(lgd,'position', poshL);
  axis(hL, 'off');
#+end_src

#+HEADER: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/relative_motion_comparison.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
  <<plt-matlab>>
#+end_src

#+NAME: fig:relative_motion_comparison
#+CAPTION: Relative motion between each stage
[[file:figs/relative_motion_comparison.png]]

* TODO Compare original FRF measurements to transformed FRF in the global frame
We wish here to compare the FRF in order to verify if there is any mistake.

#+begin_src matlab
  dir_names = {'X', 'Y', 'Z', '$\theta_X$', '$\theta_Y$', '$\theta_Z$'};

  solid_i = 6;
  acc_dir_O = 1;
  acc_dir = 1;
  exc_dir = 1;

  figure;

  ax1 = subplot(2, 1, 1);
  hold on;
  for i = solids.(solid_names{solid_i})
    plot(freqs, abs(squeeze(FRFs(acc_dir+3*(i-1), exc_dir, :))));
  end
  plot(freqs, abs(squeeze(FRFs_O((solid_i-1)*6+acc_dir_O, exc_dir, :))), '-k');
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  set(gca, 'XTickLabel',[]);
  ylabel('Amplitude');
  title(sprintf('%s motion measured by the Acc. vs %s motion computed in the common frame - %s', dir_names{acc_dir}, dir_names{acc_dir_O}, solid_names{solid_i}));

  ax2 = subplot(2, 1, 2);
  hold on;
  for i = solids.(solid_names{solid_i})
    plot(freqs, mod(180+180/pi*phase(squeeze(FRFs(acc_dir+3*(i-1), exc_dir, :))), 360)-180);
  end
  plot(freqs, mod(180+180/pi*phase(squeeze(FRFs_O((solid_i-1)*6+acc_dir_O, exc_dir, :))), 360)-180, '-k');
  hold off;
  ylim([-180, 180]); yticks(-180:90:180);
  xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
  set(gca, 'xscale', 'log');

  linkaxes([ax1,ax2],'x');
  xlim([1, 200]);
#+end_src

* Verify that we find the original FRF from the FRF in the global coordinates
We have computed the Frequency Response Functions Matrix =FRFs_O= representing the response of the 6 solid bodies in their 6 DOFs.

From the response of one body in its 6 DOFs, we should be able to compute the FRF of each of its accelerometer fixed to it during the measurement.

We can then compare the result with the original measurements.
This will help us to determine if:
- the previous inversion used is correct
- the solid body assumption is correct in the frequency band of interest

#+begin_src matlab
  FRF_recovered = zeros(size(FRFs));

  % For each excitation direction
  for exc_dir = 1:3
    % For each solid
    for solid_i = 1:length(solid_names)
      v0 = squeeze(FRFs_O((solid_i-1)*6+1:(solid_i-1)*6+3, exc_dir, :));
      W0 = squeeze(FRFs_O((solid_i-1)*6+4:(solid_i-1)*6+6, exc_dir, :));

      % For each accelerometer attached to the current solid
      for acc_i = solids.(solid_names{solid_i})
        % We get the position of the accelerometer expressed in frame O
        pos = acc_pos(acc_i, :)';
        posX = [0 pos(3) -pos(2); -pos(3) 0 pos(1) ; pos(2) -pos(1) 0];
               [0 acc_pos(i, 3) -acc_pos(i, 2) ; -acc_pos(i, 3) 0 acc_pos(i, 1) ; acc_pos(i, 2) -acc_pos(i, 1) 0]

        FRF_recovered(3*(acc_i-1)+1:3*(acc_i-1)+3, exc_dir, :) = v0 + posX*W0;
      end
    end
  end
#+end_src

We then compare the original FRF measured for each accelerometer with the recovered FRF from the global FRF matrix in the common frame.

The FRF for the 4 accelerometers on the Hexapod are compared on figure [[fig:recovered_frf_comparison_hexa]].
All the FRF are matching very well in all the frequency range displayed.

The FRF for accelerometers located on the translation stage are compared on figure [[fig:recovered_frf_comparison_ty]].
The FRF are matching well until 100Hz.

#+begin_src matlab :exports none
  exc_names = {'$F_x$', '$F_y$', '$F_z$'};
  DOFs = {'$T_x$', '$T_y$', '$T_z$', '$\theta_x$', '$\theta_y$', '$\theta_z$'}

  solid_i = 6;
  exc_dir = 1;

  accs_i = solids.(solid_names{solid_i});

  figure;

  for i = 1:length(accs_i)
    acc_i = accs_i(i);

    subaxis(2, 2, i);

    hold on;
    for dir_i = 1:3
      plot(freqs, abs(squeeze(FRFs(3*(acc_i-1)+dir_i, exc_dir, :))), '-', 'DisplayName', DOFs{dir_i});
    end
    set(gca,'ColorOrderIndex',1)
    for dir_i = 1:3
      plot(freqs, abs(squeeze(FRF_recovered(3*(acc_i-1)+dir_i, exc_dir, :))), '--', 'HandleVisibility', 'off');
    end
    hold off;

    set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
    if i > 2
      xlabel('Frequency [Hz]');
    else
      set(gca, 'XTickLabel',[]);
    end

    if rem(i, 2) == 1
      ylabel('Amplitude');
    end

    xlim([1, 200]);
    title(sprintf('Accelerometer %i', accs_i(i)));
    legend('location', 'northwest');
  end
#+end_src

#+HEADER: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/recovered_frf_comparison_hexa.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
  <<plt-matlab>>
#+end_src

#+NAME: fig:recovered_frf_comparison_hexa
#+CAPTION: Comparison of the original FRF with the recovered ones - Hexapod
[[file:figs/recovered_frf_comparison_hexa.png]]

#+begin_src matlab :exports none
  exc_names = {'$F_x$', '$F_y$', '$F_z$'};
  DOFs = {'$T_x$', '$T_y$', '$T_z$', '$\theta_x$', '$\theta_y$', '$\theta_z$'}

  solid_i = 3;
  exc_dir = 1;

  accs_i = solids.(solid_names{solid_i});

  figure;

  for i = 1:length(accs_i)
    acc_i = accs_i(i);

    subaxis(2, 2, i);

    hold on;
    for dir_i = 1:3
      plot(freqs, abs(squeeze(FRFs(3*(acc_i-1)+dir_i, exc_dir, :))), '-', 'DisplayName', DOFs{dir_i});
    end
    set(gca,'ColorOrderIndex',1)
    for dir_i = 1:3
      plot(freqs, abs(squeeze(FRF_recovered(3*(acc_i-1)+dir_i, exc_dir, :))), '--', 'HandleVisibility', 'off');
    end
    hold off;

    set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
    if i > 2
      xlabel('Frequency [Hz]');
    else
      set(gca, 'XTickLabel',[]);
    end

    if rem(i, 2) == 1
      ylabel('Amplitude');
    end

    xlim([1, 200]);
    title(sprintf('Accelerometer %i', accs_i(i)));
    legend('location', 'northwest');
  end
#+end_src

#+HEADER: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/recovered_frf_comparison_ty.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
  <<plt-matlab>>
#+end_src

#+NAME: fig:recovered_frf_comparison_ty
#+CAPTION: Comparison of the original FRF with the recovered ones - Ty
[[file:figs/recovered_frf_comparison_ty.png]]


#+begin_important
  The reduction of the number of degrees of freedom from 69 (23 accelerometers with each 3DOF) to 36 (6 solid bodies with 6 DOF) seems to work well.

  This confirms the fact that the stages are indeed behaving as a solid body in the frequency band of interest.
  This valid the fact that a multi-body model can be used to represent the dynamics of the micro-station.
#+end_important

* Importation of measured FRF curves                        :noexport:ignore:
There are 24 measurements files corresponding to 24 series of impacts:
- 3 directions, 8 sets of 3 accelerometers
For each measurement file, the FRF and coherence between the impact and the 9 accelerations measured.

In reality: 4 sets of 10 things

#+begin_src matlab
  a = load('mat/meas_frf_coh_1.mat');
#+end_src

#+begin_src matlab
  figure;

  ax1 = subplot(2, 1, 1);
  hold on;
  plot(a.FFT1_AvXSpc_2_1_RMS_X_Val, a.FFT1_AvXSpc_2_1_RMS_Y_Mod)
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  set(gca, 'XTickLabel',[]);
  ylabel('Amplitude');
  title(sprintf('From %s, to %s', FFT1_AvXSpc_2_1_RfName, FFT1_AvXSpc_2_1_RpName))

  ax2 = subplot(2, 1, 2);
  hold on;
  plot(a.FFT1_AvXSpc_2_1_RMS_X_Val, a.FFT1_AvXSpc_2_1_RMS_Y_Phas)
  hold off;
  ylim([-180, 180]); yticks(-180:90:180);
  xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
  set(gca, 'xscale', 'log');

  linkaxes([ax1,ax2],'x');
  xlim([1, 200]);
#+end_src

* Analysis of some FRFs                                     :noexport:ignore:
#+begin_src matlab
  acc_i = 3;
  acc_dir = 1;
  exc_dir = 1;

  figure;

  ax1 = subplot(2, 1, 1);
  hold on;
  plot(freqs, abs(squeeze(FRFs(acc_dir+3*(acc_i-1), exc_dir, :))));
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  set(gca, 'XTickLabel',[]);
  ylabel('Amplitude');

  ax2 = subplot(2, 1, 2);
  hold on;
  plot(freqs, mod(180+180/pi*phase(squeeze(FRFs(acc_dir+3*(acc_i-1), exc_dir, :))), 360)-180);
  hold off;
  ylim([-180, 180]); yticks(-180:90:180);
  xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
  set(gca, 'xscale', 'log');

  linkaxes([ax1,ax2],'x');
  xlim([1, 200]);
#+end_src


#+begin_src matlab
  figure;
  hold on;
  for i = 1:3*n_acc
    plot(freqs, squeeze(COHs(i, 1, :)), 'color', [0, 0, 0, 0.2]);
  end
  hold off;
  xlabel('Frequency [Hz]');
  ylabel('Coherence [\%]');
#+end_src


Composite Response Function.

We here sum the norm instead of the complex numbers.

#+begin_src matlab
  HHx = squeeze(sum(abs(FRFs(:, 1, :))));
  HHy = squeeze(sum(abs(FRFs(:, 2, :))));
  HHz = squeeze(sum(abs(FRFs(:, 3, :))));
  HH  = squeeze(sum([HHx, HHy, HHz], 2));
#+end_src

#+begin_src matlab
  exc_dir = 3;

  figure;
  hold on;
  for i = 1:3*n_acc
    plot(freqs, abs(squeeze(FRFs(i, exc_dir, :))), 'color', [0, 0, 0, 0.2]);
  end
  plot(freqs, abs(HHx));
  plot(freqs, abs(HHy));
  plot(freqs, abs(HHz));
  plot(freqs, abs(HH), 'k');
  hold off;
  set(gca, 'XScale', 'lin'); set(gca, 'YScale', 'lin');
  xlabel('Frequency [Hz]'); ylabel('Amplitude');
  xlim([1, 200]);
#+end_src