test-bench-nano-hexapod/test-bench-nano-hexapod.org

#+TITLE: Nano-Hexapod - Test Bench
:DRAWER:
#+LANGUAGE: en
#+EMAIL: dehaeze.thomas@gmail.com
#+AUTHOR: Dehaeze Thomas

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:END:

#+begin_export html
  <hr>
  <p>This report is also available as a <a href="./test-bench-nano-hexapod.pdf">pdf</a>.</p>
  <hr>
#+end_export

#+latex: \clearpage

* Introduction                                                        :ignore:
This document is dedicated to the experimental study of the nano-hexapod shown in Figure [[fig:picture_bench_granite_nano_hexapod]].

#+name: fig:picture_bench_granite_nano_hexapod
#+caption: Nano-Hexapod
#+attr_latex: :width \linewidth
[[file:figs/IMG_20210608_152917.jpg]]

#+begin_note
Here are the documentation of the equipment used for this test bench (lots of them are shwon in Figure [[fig:picture_bench_granite_overview]]):
- Voltage Amplifier: PiezoDrive [[file:doc/PD200-V7-R1.pdf][PD200]]
- Amplified Piezoelectric Actuator: Cedrat [[file:doc/APA300ML.pdf][APA300ML]]
- DAC/ADC: Speedgoat [[file:doc/IO131-OEM-Datasheet.pdf][IO313]]
- Encoder: Renishaw [[file:doc/L-9517-9678-05-A_Data_sheet_VIONiC_series_en.pdf][Vionic]] and used [[file:doc/L-9517-9862-01-C_Data_sheet_RKLC_EN.pdf][Ruler]]
- Interferometers: Attocube
#+end_note

#+name: fig:picture_bench_granite_overview
#+caption: Nano-Hexapod and the control electronics
#+attr_latex: :width \linewidth
[[file:figs/IMG_20210608_154722.jpg]]

In Figure [[fig:nano_hexapod_signals]] is shown a block diagram of the experimental setup.
When possible, the notations are consistent with this diagram and summarized in Table [[tab:list_signals]].

#+begin_src latex :file nano_hexapod_signals.pdf
\definecolor{instrumentation}{rgb}{0, 0.447, 0.741}
\definecolor{mechanics}{rgb}{0.8500, 0.325, 0.098}

\begin{tikzpicture}
  % Blocs
  \node[block={4.0cm}{3.0cm}, fill=mechanics!20!white] (nano_hexapod) {Mechanics};
  \coordinate[] (inputF)  at (nano_hexapod.west);
  \coordinate[] (outputL) at ($(nano_hexapod.south east)!0.8!(nano_hexapod.north east)$);
  \coordinate[] (outputF) at ($(nano_hexapod.south east)!0.2!(nano_hexapod.north east)$);

  \node[block, left= 0.8 of inputF,  fill=instrumentation!20!white, align=center] (F_stack) {\tiny Actuator \\ \tiny stacks};
  \node[block, left= 0.8 of F_stack, fill=instrumentation!20!white] (PD200)   {PD200};
  \node[DAC,   left= 0.8 of PD200,   fill=instrumentation!20!white] (F_DAC)   {DAC};
  \node[block, right=0.8 of outputF, fill=instrumentation!20!white, align=center] (Fm_stack){\tiny Sensor \\ \tiny stack};
  \node[ADC,   right=0.8 of Fm_stack,fill=instrumentation!20!white] (Fm_ADC)  {ADC};
  \node[block, right=0.8 of outputL, fill=instrumentation!20!white] (encoder) {\tiny Encoder};

  % Connections and labels
  \draw[->] ($(F_DAC.west)+(-0.8,0)$) node[above right]{$\bm{u}$} node[below right]{$[V]$} -- node[sloped]{$/$} (F_DAC.west);
  \draw[->] (F_DAC.east)   -- node[midway, above]{$\tilde{\bm{u}}$}node[midway, below]{$[V]$} (PD200.west);
  \draw[->] (PD200.east)   -- node[midway, above]{$\bm{u}_a$}node[midway, below]{$[V]$} (F_stack.west);
  \draw[->] (F_stack.east) -- (inputF) node[above left]{$\bm{\tau}$}node[below left]{$[N]$};

  \draw[->] (outputF)       -- (Fm_stack.west) node[above left]{$\bm{\epsilon}$} node[below left]{$[m]$};
  \draw[->] (Fm_stack.east) -- node[midway, above]{$\tilde{\bm{\tau}}_m$}node[midway, below]{$[V]$} (Fm_ADC.west);
  \draw[->] (Fm_ADC.east)   -- node[sloped]{$/$} ++(0.8, 0)coordinate(end) node[above left]{$\bm{\tau}_m$}node[below left]{$[V]$};

  \draw[->] (outputL)      -- (encoder.west) node[above left]{$d\bm{\mathcal{L}}$} node[below left]{$[m]$};
  \draw[->] (encoder.east) -- node[sloped]{$/$} (encoder-|end) node[above left]{$d\bm{\mathcal{L}}_m$}node[below left]{$[m]$};

  % Nano-Hexapod
  \begin{scope}[on background layer]
    \node[fit={(F_stack.west|-nano_hexapod.south) (Fm_stack.east|-nano_hexapod.north)}, fill=black!20!white, draw, inner sep=2pt] (system) {};
    \node[above] at (system.north) {Nano-Hexapod};
  \end{scope}
\end{tikzpicture}
#+end_src

#+name: fig:nano_hexapod_signals
#+caption: Block diagram of the system with named signals
#+attr_latex: :scale 1
[[file:figs/nano_hexapod_signals.png]]

#+name: tab:list_signals
#+caption: List of signals
#+attr_latex: :environment tabularx :width \linewidth :align Xllll
#+attr_latex: :center t :booktabs t :float t
|                                    | *Unit*    | *Matlab*  | *Vector*              | *Elements*           |
|------------------------------------+-----------+-----------+-----------------------+----------------------|
| Control Input (wanted DAC voltage) | =[V]=     | =u=       | $\bm{u}$              | $u_i$                |
| DAC Output Voltage                 | =[V]=     | =u=       | $\tilde{\bm{u}}$      | $\tilde{u}_i$        |
| PD200 Output Voltage               | =[V]=     | =ua=      | $\bm{u}_a$            | $u_{a,i}$            |
| Actuator applied force             | =[N]=     | =tau=     | $\bm{\tau}$           | $\tau_i$             |
|------------------------------------+-----------+-----------+-----------------------+----------------------|
| Strut motion                       | =[m]=     | =dL=      | $d\bm{\mathcal{L}}$   | $d\mathcal{L}_i$     |
| Encoder measured displacement      | =[m]=     | =dLm=     | $d\bm{\mathcal{L}}_m$ | $d\mathcal{L}_{m,i}$ |
|------------------------------------+-----------+-----------+-----------------------+----------------------|
| Force Sensor strain                | =[m]=     | =epsilon= | $\bm{\epsilon}$       | $\epsilon_i$         |
| Force Sensor Generated Voltage     | =[V]=     | =taum=    | $\tilde{\bm{\tau}}_m$ | $\tilde{\tau}_{m,i}$ |
| Measured Generated Voltage         | =[V]=     | =taum=    | $\bm{\tau}_m$         | $\tau_{m,i}$         |
|------------------------------------+-----------+-----------+-----------------------+----------------------|
| Motion of the top platform         | =[m,rad]= | =dX=      | $d\bm{\mathcal{X}}$   | $d\mathcal{X}_i$     |
| Metrology measured displacement    | =[m,rad]= | =dXm=     | $d\bm{\mathcal{X}}_m$ | $d\mathcal{X}_{m,i}$ |

This document is divided in the following sections:
- Section [[sec:encoders_struts]]: the encoders are fixed to the struts
- Section [[sec:encoders_plates]]: the encoders are fixed to the plates

* Encoders fixed to the Struts
<<sec:encoders_struts>>

** Introduction
In this section, the encoders are fixed to the struts.

It is divided in the following sections:
- Section [[sec:enc_struts_plant_id]]: the transfer function matrix from the actuators to the force sensors and to the encoders is experimentally identified.
- Section [[sec:enc_struts_comp_simscape]]: the obtained FRF matrix is compared with the dynamics of the simscape model
- Section [[sec:enc_struts_iff]]: decentralized Integral Force Feedback (IFF) is applied and its performances are evaluated.
- Section [[sec:enc_struts_modal_analysis]]: a modal analysis of the nano-hexapod is performed

** Identification of the dynamics
<<sec:enc_struts_plant_id>>
*** 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
addpath('./matlab/mat/');
addpath('./matlab/src/');
addpath('./matlab/');
#+end_src

#+begin_src matlab :eval no
addpath('./mat/');
addpath('./src/');
#+end_src

*** Load Measurement Data
#+begin_src matlab
%% Load Identification Data
meas_data_lf = {};

for i = 1:6
    meas_data_lf(i) = {load(sprintf('mat/frf_data_exc_strut_%i_noise_lf.mat', i), 't', 'Va', 'Vs', 'de')};
    meas_data_hf(i) = {load(sprintf('mat/frf_data_exc_strut_%i_noise_hf.mat', i), 't', 'Va', 'Vs', 'de')};
end
#+end_src

*** Spectral Analysis - Setup
#+begin_src matlab
%% Setup useful variables
% Sampling Time [s]
Ts = (meas_data_lf{1}.t(end) - (meas_data_lf{1}.t(1)))/(length(meas_data_lf{1}.t)-1);

% Sampling Frequency [Hz]
Fs = 1/Ts;

% Hannning Windows
win = hanning(ceil(1*Fs));

% And we get the frequency vector
[~, f] = tfestimate(meas_data_lf{1}.Va, meas_data_lf{1}.de, win, [], [], 1/Ts);

i_lf = f < 250; % Points for low frequency excitation
i_hf = f > 250; % Points for high frequency excitation
#+end_src

*** DVF Plant
First, let's compute the coherence from the excitation voltage and the displacement as measured by the encoders (Figure [[fig:enc_struts_dvf_coh]]).

#+begin_src matlab
%% Coherence
coh_dvf = zeros(length(f), 6, 6);

for i = 1:6
    coh_dvf_lf = mscohere(meas_data_lf{i}.Va, meas_data_lf{i}.de, win, [], [], 1/Ts);
    coh_dvf_hf = mscohere(meas_data_hf{i}.Va, meas_data_hf{i}.de, win, [], [], 1/Ts);
    coh_dvf(:,:,i) = [coh_dvf_lf(i_lf, :); coh_dvf_hf(i_hf, :)];
end
#+end_src

#+begin_src matlab :exports none
%% Coherence for the transfer function from u to dLm
figure;
hold on;
for i = 1:5
    for j = i+1:6
        plot(f, coh_dvf(:, i, j), 'color', [0, 0, 0, 0.2], ...
             'HandleVisibility', 'off');
    end
end
for i =1:6
    set(gca,'ColorOrderIndex',i)
    plot(f, coh_dvf(:, i, i), ...
         'DisplayName', sprintf('$G_{dvf}(%i,%i)$', i, i));
end
plot(f, coh_dvf(:, 1, 2), 'color', [0, 0, 0, 0.2], ...
     'DisplayName', '$G_{dvf}(i,j)$');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Coherence [-]');
xlim([20, 2e3]); ylim([0, 1]);
legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 3);
#+end_src

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

#+name: fig:enc_struts_dvf_coh
#+caption: Obtained coherence for the DVF plant
#+RESULTS:
[[file:figs/enc_struts_dvf_coh.png]]

Then the 6x6 transfer function matrix is estimated (Figure [[fig:enc_struts_dvf_frf]]).
#+begin_src matlab
%% DVF Plant (transfer function from u to dLm)
G_dvf = zeros(length(f), 6, 6);

for i = 1:6
    G_dvf_lf = tfestimate(meas_data_lf{i}.Va, meas_data_lf{i}.de, win, [], [], 1/Ts);
    G_dvf_hf = tfestimate(meas_data_hf{i}.Va, meas_data_hf{i}.de, win, [], [], 1/Ts);
    G_dvf(:,:,i) = [G_dvf_lf(i_lf, :); G_dvf_hf(i_hf, :)];
end
#+end_src

#+begin_src matlab :exports none
%% Bode plot for the transfer function from u to dLm
figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');

ax1 = nexttile([2,1]);
hold on;
for i = 1:5
    for j = i+1:6
        plot(f, abs(G_dvf(:, i, j)), 'color', [0, 0, 0, 0.2], ...
             'HandleVisibility', 'off');
    end
end
for i =1:6
    set(gca,'ColorOrderIndex',i)
    plot(f, abs(G_dvf(:,i, i)), ...
         'DisplayName', sprintf('$G_{dvf}(%i,%i)$', i, i));
    set(gca,'ColorOrderIndex',i)
    plot(f, abs(G_dvf(:,i, i)), ...
         'HandleVisibility', 'off');
end
plot(f, abs(G_dvf(:, 1, 2)), 'color', [0, 0, 0, 0.2], ...
     'DisplayName', '$G_{dvf}(i,j)$');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude $d_e/V_a$ [m/V]'); set(gca, 'XTickLabel',[]);
ylim([1e-9, 1e-3]);
legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 3);

ax2 = nexttile;
hold on;
for i =1:6
    set(gca,'ColorOrderIndex',i)
    plot(f, 180/pi*angle(G_dvf(:,i, i)));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
yticks(-360:90:360);

linkaxes([ax1,ax2],'x');
xlim([20, 2e3]);
#+end_src

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

#+name: fig:enc_struts_dvf_frf
#+caption: Measured FRF for the DVF plant
#+RESULTS:
[[file:figs/enc_struts_dvf_frf.png]]

*** IFF Plant
First, let's compute the coherence from the excitation voltage and the displacement as measured by the encoders (Figure [[fig:enc_struts_iff_coh]]).

#+begin_src matlab
%% Coherence for the IFF plant
coh_iff = zeros(length(f), 6, 6);

for i = 1:6
    coh_iff_lf = mscohere(meas_data_lf{i}.Va, meas_data_lf{i}.Vs, win, [], [], 1/Ts);
    coh_iff_hf = mscohere(meas_data_hf{i}.Va, meas_data_hf{i}.Vs, win, [], [], 1/Ts);
    coh_iff(:,:,i) = [coh_iff_lf(i_lf, :); coh_iff_hf(i_hf, :)];
end
#+end_src

#+begin_src matlab :exports none
%% Coherence of the IFF Plant (transfer function from u to taum)
figure;
hold on;
for i = 1:5
    for j = i+1:6
        plot(f, coh_iff(:, i, j), 'color', [0, 0, 0, 0.2], ...
             'HandleVisibility', 'off');
    end
end
for i =1:6
    set(gca,'ColorOrderIndex',i)
    plot(f, coh_iff(:,i, i), ...
         'DisplayName', sprintf('$G_{iff}(%i,%i)$', i, i));
end
plot(f, coh_iff(:, 1, 2), 'color', [0, 0, 0, 0.2], ...
     'DisplayName', '$G_{iff}(i,j)$');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Coherence [-]');
xlim([20, 2e3]); ylim([0, 1]);
legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 3);
#+end_src

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

#+name: fig:enc_struts_iff_coh
#+caption: Obtained coherence for the IFF plant
#+RESULTS:
[[file:figs/enc_struts_iff_coh.png]]

Then the 6x6 transfer function matrix is estimated (Figure [[fig:enc_struts_iff_frf]]).
#+begin_src matlab
%% IFF Plant
G_iff = zeros(length(f), 6, 6);

for i = 1:6
    G_iff_lf = tfestimate(meas_data_lf{i}.Va, meas_data_lf{i}.Vs, win, [], [], 1/Ts);
    G_iff_hf = tfestimate(meas_data_hf{i}.Va, meas_data_hf{i}.Vs, win, [], [], 1/Ts);
    G_iff(:,:,i) = [G_iff_lf(i_lf, :); G_iff_hf(i_hf, :)];
end
#+end_src

#+begin_src matlab :exports none
%% Bode plot of the IFF Plant (transfer function from u to taum)
figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');

ax1 = nexttile([2,1]);
hold on;
for i = 1:5
    for j = i+1:6
        plot(f, abs(G_iff(:, i, j)), 'color', [0, 0, 0, 0.2], ...
             'HandleVisibility', 'off');
    end
end
for i =1:6
    set(gca,'ColorOrderIndex',i)
    plot(f, abs(G_iff(:,i , i)), ...
         'DisplayName', sprintf('$G_{iff}(%i,%i)$', i, i));
end
plot(f, abs(G_iff(:, 1, 2)), 'color', [0, 0, 0, 0.2], ...
     'DisplayName', '$G_{iff}(i,j)$');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude $V_s/V_a$ [V/V]'); set(gca, 'XTickLabel',[]);
legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 3);
ylim([1e-3, 1e2]);

ax2 = nexttile;
hold on;
for i =1:6
    set(gca,'ColorOrderIndex',i)
    plot(f, 180/pi*angle(G_iff(:,i, i)));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
yticks(-360:90:360);

linkaxes([ax1,ax2],'x');
xlim([20, 2e3]);
#+end_src

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

#+name: fig:enc_struts_iff_frf
#+caption: Measured FRF for the IFF plant
#+RESULTS:
[[file:figs/enc_struts_iff_frf.png]]

*** Save Identified Plants
#+begin_src matlab :tangle no
save('matlab/mat/identified_plants_enc_struts.mat', 'f', 'Ts', 'G_iff', 'G_dvf')
#+end_src

#+begin_src matlab :exports none :eval no
save('mat/identified_plants_enc_struts.mat', 'f', 'Ts', 'G_iff', 'G_dvf')
#+end_src

** Jacobian                                                        :noexport:
*** Introduction                                                    :ignore:
The Jacobian is used to transform the excitation force in the cartesian frame as well as the displacements.

Consider the plant shown in Figure [[fig:schematic_jacobian_in_out]] with:
- $\tau$ the 6 input voltages (going to the PD200 amplifier and then to the APA)
- $d\mathcal{L}$ the relative motion sensor outputs (encoders)
- $\bm{\tau}_m$ the generated voltage of the force sensor stacks
- $J_a$ and $J_s$ the Jacobians for the actuators and sensors

#+begin_src latex :file schematic_jacobian_in_out.pdf
\begin{tikzpicture}
  % Blocs
  \node[block={2.0cm}{2.0cm}] (P) {Plant};
  \coordinate[] (inputF)  at (P.west);
  \coordinate[] (outputL) at ($(P.south east)!0.8!(P.north east)$);
  \coordinate[] (outputF) at ($(P.south east)!0.2!(P.north east)$);

  \node[block, left= of inputF]   (Ja) {$\bm{J}^{-T}_a$};
  \node[block, right= of outputL] (Js) {$\bm{J}^{-1}_s$};
  \node[block, right= of outputF] (Jf) {$\bm{J}^{-1}_s$};

  % Connections and labels
  \draw[->] ($(Ja.west)+(-1,0)$) -- (Ja.west) node[above left]{$\bm{\mathcal{F}}$};
  \draw[->] (Ja.east) -- (inputF)  node[above left]{$\bm{\tau}$};
  \draw[->] (outputL) -- (Js.west) node[above left]{$d\bm{\mathcal{L}}$};
  \draw[->] (Js.east) -- ++(1, 0) node[above left]{$d\bm{\mathcal{X}}$};
  \draw[->] (outputF) -- (Jf.west) node[above left]{$\bm{\tau}_m$};
  \draw[->] (Jf.east) -- ++(1, 0) node[above left]{$\bm{\mathcal{F}}_m$};
\end{tikzpicture}
#+end_src

#+name: fig:schematic_jacobian_in_out
#+caption: Plant in the cartesian Frame
#+RESULTS:
[[file:figs/schematic_jacobian_in_out.png]]

First, we load the Jacobian matrix (same for the actuators and sensors).
#+begin_src matlab
load('jacobian.mat', 'J');
#+end_src

*** 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
addpath('./matlab/mat/');
addpath('./matlab/src/');
addpath('./matlab/');
#+end_src

#+begin_src matlab :eval no
addpath('./mat/');
addpath('./src/');
#+end_src

#+begin_src matlab
load('identified_plants_enc_struts.mat', 'f', 'Ts', 'G_iff', 'G_dvf')
load('jacobian.mat', 'J');
#+end_src

*** DVF Plant
The transfer function from $\bm{\mathcal{F}}$ to $d\bm{\mathcal{X}}$ is computed and shown in Figure [[fig:enc_struts_dvf_cart_frf]].

#+begin_src matlab
G_dvf_J = permute(pagemtimes(inv(J), pagemtimes(permute(G_dvf, [2 3 1]), inv(J'))), [3 1 2]);
#+end_src

#+begin_src matlab :exports none
labels = {'$D_x/F_{x}$', '$D_y/F_{y}$', '$D_z/F_{z}$', '$R_{x}/M_{x}$', '$R_{y}/M_{y}$', '$R_{R}/M_{z}$'};

figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');

ax1 = nexttile([2,1]);
hold on;
for i = 1:5
    for j = i+1:6
        plot(f, abs(G_dvf_J(:, i, j)), 'color', [0, 0, 0, 0.2], ...
             'HandleVisibility', 'off');
    end
end
for i =1:6
    set(gca,'ColorOrderIndex',i)
    plot(f, abs(G_dvf_J(:,i , i)), ...
         'DisplayName', labels{i});
end
plot(f, abs(G_dvf_J(:, 1, 2)), 'color', [0, 0, 0, 0.2], ...
     'DisplayName', '$D_i/F_j$');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude $d_e/V_a$ [m/V]'); set(gca, 'XTickLabel',[]);
ylim([1e-7, 1e-1]);
legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 3);

ax2 = nexttile;
hold on;
for i =1:6
    set(gca,'ColorOrderIndex',i)
    plot(f, 180/pi*angle(G_dvf_J(:,i , i)));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
yticks(-360:90:360);

linkaxes([ax1,ax2],'x');
xlim([20, 2e3]);
#+end_src

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

#+name: fig:enc_struts_dvf_cart_frf
#+caption: Measured FRF for the DVF plant in the cartesian frame
#+RESULTS:
[[file:figs/enc_struts_dvf_cart_frf.png]]

*** IFF Plant
The transfer function from $\bm{\mathcal{F}}$ to $\bm{\mathcal{F}}_m$ is computed and shown in Figure [[fig:enc_struts_iff_cart_frf]].

#+begin_src matlab
G_iff_J = permute(pagemtimes(inv(J), pagemtimes(permute(G_iff, [2 3 1]), inv(J'))), [3 1 2]);
#+end_src

#+begin_src matlab :exports none
labels = {'$F_{m,x}/F_{x}$', '$F_{m,y}/F_{y}$', '$F_{m,z}/F_{z}$', '$M_{m,x}/M_{x}$', '$M_{m,y}/M_{y}$', '$M_{m,z}/M_{z}$'};

figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');

ax1 = nexttile([2,1]);
hold on;
for i = 1:5
    for j = i+1:6
        plot(f, abs(G_iff_J(:, i, j)), 'color', [0, 0, 0, 0.2], ...
             'HandleVisibility', 'off');
    end
end
for i =1:6
    set(gca,'ColorOrderIndex',i)
    plot(f, abs(G_iff_J(:,i, i)), ...
         'DisplayName', labels{i});
end
plot(f, abs(G_iff_J(:, 1, 2)), 'color', [0, 0, 0, 0.2], ...
     'DisplayName', '$D_i/F_j$');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude $d_e/V_a$ [m/V]'); set(gca, 'XTickLabel',[]);
ylim([1e-3, 1e4]);
legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 3);

ax2 = nexttile;
hold on;
for i =1:6
    set(gca,'ColorOrderIndex',i)
    plot(f, 180/pi*angle(G_iff_J(:,i, i)));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
yticks(-360:90:360);

linkaxes([ax1,ax2],'x');
xlim([20, 2e3]);
#+end_src

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

#+name: fig:enc_struts_iff_cart_frf
#+caption: Measured FRF for the IFF plant in the cartesian frame
#+RESULTS:
[[file:figs/enc_struts_iff_cart_frf.png]]

** Comparison with the Simscape Model
<<sec:enc_struts_comp_simscape>>
*** Introduction                                                    :ignore:
In this section, the measured dynamics is compared with the dynamics estimated from the Simscape model.

*** 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
addpath('./matlab/mat/');
addpath('./matlab/src/');
addpath('./matlab/');
#+end_src

#+begin_src matlab :eval no
addpath('./mat/');
addpath('./src/');
#+end_src

#+begin_src matlab :tangle no
%% Add all useful folders to the path
addpath('matlab/nass-simscape/matlab/nano_hexapod/')
addpath('matlab/nass-simscape/STEPS/nano_hexapod/')
addpath('matlab/nass-simscape/STEPS/png/')
addpath('matlab/nass-simscape/src/')
addpath('matlab/nass-simscape/mat/')
#+end_src

#+begin_src matlab :eval no
%% Add all useful folders to the path
addpath('nass-simscape/matlab/nano_hexapod/')
addpath('nass-simscape/STEPS/nano_hexapod/')
addpath('nass-simscape/STEPS/png/')
addpath('nass-simscape/src/')
addpath('nass-simscape/mat/')
#+end_src

#+begin_src matlab
%% Open Simulink Model
mdl = 'nano_hexapod_simscape';

options = linearizeOptions;
options.SampleTime = 0;

Rx = zeros(1, 7);

open(mdl)
#+end_src

*** Load measured FRF

#+begin_src matlab
%% Load data
load('identified_plants_enc_struts.mat', 'f', 'Ts', 'G_iff', 'G_dvf')
#+end_src

*** Dynamics from Actuator to Force Sensors
#+begin_src matlab
%% Initialize Nano-Hexapod
n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
                                       'flex_top_type', '4dof', ...
                                       'motion_sensor_type', 'struts', ...
                                       'actuator_type', '2dof');
#+end_src

#+begin_src matlab
%% Identify the IFF Plant (transfer function from u to taum)
clear io; io_i = 1;
io(io_i) = linio([mdl, '/du'],  1, 'openinput');   io_i = io_i + 1; % Actuator Inputs
io(io_i) = linio([mdl, '/dum'],  1, 'openoutput'); io_i = io_i + 1; % Force Sensors

Giff = exp(-s*Ts)*linearize(mdl, io, 0.0, options);
#+end_src

#+begin_src matlab :exports none
%% Bode plot of the identified IFF Plant (Simscape) and measured FRF data
freqs = 2*logspace(1, 3, 1000);

figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');

ax1 = nexttile([2,1]);
hold on;
plot(f, abs(G_iff(:,1, 1)), 'color', [0,0,0,0.2], ...
     'DisplayName', '$\tau_{m,i}/u_i$ - FRF')
for i = 2:6
    set(gca,'ColorOrderIndex',2)
    plot(f, abs(G_iff(:,i, i)), 'color', [0,0,0,0.2], ...
         'HandleVisibility', 'off');
end
set(gca,'ColorOrderIndex',2);
plot(freqs, abs(squeeze(freqresp(Giff(1,1), freqs, 'Hz'))), '-', ...
     'DisplayName', '$\tau_{m,i}/u_i$ - Model')
for i = 2:6
    set(gca,'ColorOrderIndex',2);
    plot(freqs, abs(squeeze(freqresp(Giff(i,i), freqs, 'Hz'))), '-', ...
         'HandleVisibility', 'off');
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [V/V]'); set(gca, 'XTickLabel',[]);
legend('location', 'southeast');

ax2 = nexttile;
hold on;
for i = 1:6
    plot(f, 180/pi*angle(G_iff(:,i, i)), 'color', [0,0,0,0.2]);
end
for i = 1:6
    set(gca,'ColorOrderIndex',2);
    plot(freqs, 180/pi*angle(squeeze(freqresp(Giff(i,i), freqs, 'Hz'))), '-');
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-180, 180]);
yticks([-180, -90, 0, 90, 180]);

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

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

#+name: fig:enc_struts_iff_comp_simscape
#+caption: Diagonal elements of the IFF Plant
#+RESULTS:
[[file:figs/enc_struts_iff_comp_simscape.png]]

#+begin_src matlab :exports none
%% Bode plot of the identified IFF Plant (Simscape) and measured FRF data (off-diagonal elements)
freqs = 2*logspace(1, 3, 1000);

figure;
hold on;
% Off diagonal terms
plot(f, abs(G_iff(:, 1, 2)), 'color', [0,0,0,0.2], ...
     'DisplayName', '$\tau_{m,i}/u_j$ - FRF')
for i = 1:5
    for j = i+1:6
        plot(f, abs(G_iff(:, i, j)), 'color', [0,0,0,0.2], ...
             'HandleVisibility', 'off');
    end
end
set(gca,'ColorOrderIndex',2);
plot(freqs, abs(squeeze(freqresp(Giff(1, 2), freqs, 'Hz'))), ...
     'DisplayName', '$\tau_{m,i}/u_j$ - Model')
for i = 1:5
    for j = i+1:6
        set(gca,'ColorOrderIndex',2);
        plot(freqs, abs(squeeze(freqresp(Giff(i, j), freqs, 'Hz'))), ...
             'HandleVisibility', 'off');
    end
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Amplitude [V/V]');
xlim([freqs(1), freqs(end)]); ylim([1e-3, 1e2]);
legend('location', 'northeast');
#+end_src

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

#+name: fig:enc_struts_iff_comp_offdiag_simscape
#+caption: Off diagonal elements of the IFF Plant
#+RESULTS:
[[file:figs/enc_struts_iff_comp_offdiag_simscape.png]]

*** Dynamics from Actuator to Encoder
#+begin_src matlab
%% Initialization of the Nano-Hexapod
n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
                                       'flex_top_type', '4dof', ...
                                       'motion_sensor_type', 'struts', ...
                                       'actuator_type', 'flexible');
#+end_src

#+begin_src matlab
%% Identify the DVF Plant (transfer function from u to dLm)
clear io; io_i = 1;
io(io_i) = linio([mdl, '/du'],  1, 'openinput');  io_i = io_i + 1; % Actuator Inputs
io(io_i) = linio([mdl, '/D'],  1, 'openoutput'); io_i = io_i + 1; % Encoders

Gdvf = exp(-s*Ts)*linearize(mdl, io, 0.0, options);
#+end_src

#+begin_src matlab :exports none
%% Diagonal elements of the DVF plant
freqs = 2*logspace(1, 3, 1000);

figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');

ax1 = nexttile([2,1]);
hold on;
plot(f, abs(G_dvf(:,1, 1)), 'color', [0,0,0,0.2], ...
     'DisplayName', '$d\mathcal{L}_{m,i}/u_i$ - FRF')
for i = 2:6
    set(gca,'ColorOrderIndex',2)
    plot(f, abs(G_dvf(:,i, i)), 'color', [0,0,0,0.2], ...
         'HandleVisibility', 'off');
end
set(gca,'ColorOrderIndex',2);
plot(freqs, abs(squeeze(freqresp(Gdvf(1,1), freqs, 'Hz'))), '-', ...
     'DisplayName', '$d\mathcal{L}_{m,i}/u_i$ - Model')
for i = 2:6
    set(gca,'ColorOrderIndex',2);
    plot(freqs, abs(squeeze(freqresp(Gdvf(i,i), freqs, 'Hz'))), '-', ...
         'HandleVisibility', 'off');
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/V]'); set(gca, 'XTickLabel',[]);
ylim([1e-8, 1e-3]);
legend('location', 'northeast');

ax2 = nexttile;
hold on;
for i = 1:6
    plot(f, 180/pi*angle(G_dvf(:,i, i)), 'color', [0,0,0,0.2]);
end
for i = 1:6
    set(gca,'ColorOrderIndex',2);
    plot(freqs, 180/pi*angle(squeeze(freqresp(Gdvf(i,i), freqs, 'Hz'))), '-');
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-180, 180]);
yticks([-180, -90, 0, 90, 180]);

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

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

#+name: fig:enc_struts_dvf_comp_simscape
#+caption: Diagonal elements of the DVF Plant
#+RESULTS:
[[file:figs/enc_struts_dvf_comp_simscape.png]]

#+begin_src matlab :exports none
%% Off-diagonal elements of the DVF plant
freqs = 2*logspace(1, 3, 1000);

figure;
hold on;
% Off diagonal terms
plot(f, abs(G_dvf(:, 1, 2)), 'color', [0,0,0,0.2], ...
     'DisplayName', '$d\mathcal{L}_{m,i}/u_j$ - FRF')
for i = 1:5
    for j = i+1:6
        plot(f, abs(G_dvf(:, i, j)), 'color', [0,0,0,0.2], ...
             'HandleVisibility', 'off');
    end
end
set(gca,'ColorOrderIndex',2);
plot(freqs, abs(squeeze(freqresp(Gdvf(1, 2), freqs, 'Hz'))), ...
     'DisplayName', '$d\mathcal{L}_{m,i}/u_j$ - Model')
for i = 1:5
    for j = i+1:6
        set(gca,'ColorOrderIndex',2);
        plot(freqs, abs(squeeze(freqresp(Gdvf(i, j), freqs, 'Hz'))), ...
             'HandleVisibility', 'off');
    end
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Amplitude [m/V]');
xlim([freqs(1), freqs(end)]); ylim([1e-8, 1e-3]);
legend('location', 'northeast');
#+end_src

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

#+name: fig:enc_struts_dvf_comp_offdiag_simscape
#+caption: Off diagonal elements of the DVF Plant
#+RESULTS:
[[file:figs/enc_struts_dvf_comp_offdiag_simscape.png]]

*** Effect of a change in bending damping of the joints
#+begin_src matlab
%% Tested bending dampings [Nm/(rad/s)]
cRs = [1e-3, 5e-3, 1e-2, 5e-2, 1e-1];
#+end_src

#+begin_src matlab
%% Identify the DVF Plant (transfer function from u to dLm)
clear io; io_i = 1;
io(io_i) = linio([mdl, '/du'],  1, 'openinput');  io_i = io_i + 1; % Actuator Inputs
io(io_i) = linio([mdl, '/D'],  1, 'openoutput'); io_i = io_i + 1; % Encoders
#+end_src

Then the identification is performed for all the values of the bending damping.
#+begin_src matlab
%% Idenfity the transfer function from actuator to encoder for all bending dampins
Gs = {zeros(length(cRs), 1)};

for i = 1:length(cRs)
    n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
                                           'flex_top_type', '4dof', ...
                                           'motion_sensor_type', 'struts', ...
                                           'actuator_type', 'flexible', ...
                                           'flex_bot_cRx', cRs(i), ...
                                           'flex_bot_cRy', cRs(i), ...
                                           'flex_top_cRx', cRs(i), ...
                                           'flex_top_cRy', cRs(i));

    G = exp(-s*Ts)*linearize(mdl, io, 0.0, options);
    G.InputName  = {'Va1', 'Va2', 'Va3', 'Va4', 'Va5', 'Va6'};
    G.OutputName = {'dL1', 'dL2', 'dL3', 'dL4', 'dL5', 'dL6'};

    Gs(i) = {G};
end
#+end_src

#+begin_src matlab :exports none
%% Plot the obtained direct transfer functions for all the bending stiffnesses
freqs = 2*logspace(1, 3, 1000);

figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');

ax1 = nexttile([2,1]);
hold on;
for i = 1:length(cRs)
    plot(freqs, abs(squeeze(freqresp(Gs{i}('dL1', 'Va1'), freqs, 'Hz'))), ...
         'DisplayName', sprintf('$c_R = %.3f\\,[\\frac{Nm}{rad/s}]$', cRs(i)));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude $d_L/V_a$ [m/V]'); set(gca, 'XTickLabel',[]);
hold off;
ylim([1e-8, 1e-3]);
legend('location', 'southwest');

ax2 = nexttile;
hold on;
for i = 1:length(cRs)
    plot(freqs, 180/pi*angle(squeeze(freqresp(Gs{i}('dL1', 'Va1'), freqs, 'Hz'))));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
yticks(-360:90:360); ylim([-180, 180]);

linkaxes([ax1,ax2],'x');
xlim([20, 2e3]);
#+end_src

#+begin_src matlab :exports none
%% Plot the obtained coupling transfer functions for all the bending stiffnesses
freqs = 2*logspace(1, 3, 1000);

figure;
hold on;
for i = 1:length(cRs)
    plot(freqs, abs(squeeze(freqresp(Gs{i}('dL2', 'Va1'), freqs, 'Hz'))), ...
         'DisplayName', sprintf('$c_R = %.3f\\,[\\frac{Nm}{rad/s}]$', cRs(i)));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude $d_L/V_a$ [m/V]'); set(gca, 'XTickLabel',[]);
hold off;
ylim([1e-8, 1e-3]);
legend('location', 'southwest');
xlim([20, 2e3]);
#+end_src

- Could be nice
- Actual damping is very small

*** Effect of a change in damping factor of the APA
#+begin_src matlab
%% Tested bending dampings [Nm/(rad/s)]
xis = [1e-3, 5e-3, 1e-2, 5e-2, 1e-1];
#+end_src

#+begin_src matlab
%% Identify the DVF Plant (transfer function from u to dLm)
clear io; io_i = 1;
io(io_i) = linio([mdl, '/du'],  1, 'openinput');  io_i = io_i + 1; % Actuator Inputs
io(io_i) = linio([mdl, '/D'],  1, 'openoutput'); io_i = io_i + 1; % Encoders
#+end_src

#+begin_src matlab
%% Idenfity the transfer function from actuator to encoder for all bending dampins
Gs = {zeros(length(xis), 1)};

for i = 1:length(xis)
    n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
                                           'flex_top_type', '4dof', ...
                                           'motion_sensor_type', 'struts', ...
                                           'actuator_type', 'flexible', ...
                                           'actuator_xi', xis(i));

    G = exp(-s*Ts)*linearize(mdl, io, 0.0, options);
    G.InputName  = {'Va1', 'Va2', 'Va3', 'Va4', 'Va5', 'Va6'};
    G.OutputName = {'dL1', 'dL2', 'dL3', 'dL4', 'dL5', 'dL6'};

    Gs(i) = {G};
end
#+end_src

#+begin_src matlab :exports none
%% Plot the obtained direct transfer functions for all the bending stiffnesses
freqs = 2*logspace(1, 3, 1000);

figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');

ax1 = nexttile([2,1]);
hold on;
for i = 1:length(xis)
    plot(freqs, abs(squeeze(freqresp(Gs{i}('dL1', 'Va1'), freqs, 'Hz'))), ...
         'DisplayName', sprintf('$\\xi = %.3f$', xis(i)));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude $d_L/V_a$ [m/V]'); set(gca, 'XTickLabel',[]);
hold off;
ylim([1e-8, 1e-3]);
legend('location', 'southwest');

ax2 = nexttile;
hold on;
for i = 1:length(xis)
    plot(freqs, 180/pi*angle(squeeze(freqresp(Gs{i}('dL1', 'Va1'), freqs, 'Hz'))));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
yticks(-360:90:360); ylim([-180, 180]);

linkaxes([ax1,ax2],'x');
xlim([20, 2e3]);
#+end_src

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

#+name: fig:bode_Va_dL_effect_xi_damp
#+caption: Effect of the APA damping factor $\xi$ on the dynamics from $u$ to $d\mathcal{L}$
#+RESULTS:
[[file:figs/bode_Va_dL_effect_xi_damp.png]]

#+begin_src matlab :exports none
%% Plot the obtained coupling transfer functions for all the bending stiffnesses
freqs = 2*logspace(1, 3, 1000);

figure;
hold on;
for i = 1:length(xis)
    plot(freqs, abs(squeeze(freqresp(Gs{i}('dL2', 'Va1'), freqs, 'Hz'))), ...
         'DisplayName', sprintf('$c_R = %.3f\\,[\\frac{Nm}{rad/s}]$', xis(i)));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude $d_L/V_a$ [m/V]'); set(gca, 'XTickLabel',[]);
hold off;
ylim([1e-8, 1e-3]);
legend('location', 'southwest');
xlim([20, 2e3]);
#+end_src

#+begin_important
Damping factor $\xi$ has a large impact on the damping of the "spurious resonances" at 200Hz and 300Hz.
#+end_important

#+begin_question
Why is the damping factor does not change the damping of the first peak?
#+end_question

*** Effect of a change in stiffness damping coef of the APA
#+begin_src matlab
m_coef = 1e1;
#+end_src

#+begin_src matlab
%% Tested bending dampings [Nm/(rad/s)]
k_coefs = [1e-6, 5e-6, 1e-5, 5e-5, 1e-4];
#+end_src

#+begin_src matlab
%% Identify the DVF Plant (transfer function from u to dLm)
clear io; io_i = 1;
io(io_i) = linio([mdl, '/du'],  1, 'openinput');  io_i = io_i + 1; % Actuator Inputs
io(io_i) = linio([mdl, '/D'],  1, 'openoutput'); io_i = io_i + 1; % Encoders
#+end_src

#+begin_src matlab
%% Idenfity the transfer function from actuator to encoder for all bending dampins
Gs = {zeros(length(k_coefs), 1)};
n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
                                       'flex_top_type', '4dof', ...
                                       'motion_sensor_type', 'struts', ...
                                       'actuator_type', 'flexible');

for i = 1:length(k_coefs)
    k_coef = k_coefs(i);

    G = exp(-s*Ts)*linearize(mdl, io, 0.0, options);
    G.InputName  = {'Va1', 'Va2', 'Va3', 'Va4', 'Va5', 'Va6'};
    G.OutputName = {'dL1', 'dL2', 'dL3', 'dL4', 'dL5', 'dL6'};

    Gs(i) = {G};
end
#+end_src

#+begin_src matlab :exports none
%% Plot the obtained direct transfer functions for all the bending stiffnesses
freqs = 2*logspace(1, 3, 1000);

figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');

ax1 = nexttile([2,1]);
hold on;
for i = 1:length(k_coefs)
    plot(freqs, abs(squeeze(freqresp(Gs{i}('dL1', 'Va1'), freqs, 'Hz'))), ...
         'DisplayName', sprintf('kcoef = %.0e', k_coefs(i)));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude $d_L/V_a$ [m/V]'); set(gca, 'XTickLabel',[]);
hold off;
ylim([1e-8, 1e-3]);
legend('location', 'southwest');

ax2 = nexttile;
hold on;
for i = 1:length(k_coefs)
    plot(freqs, 180/pi*angle(squeeze(freqresp(Gs{i}('dL1', 'Va1'), freqs, 'Hz'))));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
yticks(-360:90:360); ylim([-180, 180]);

linkaxes([ax1,ax2],'x');
xlim([20, 2e3]);
#+end_src

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

#+name: fig:bode_Va_dL_effect_k_coef
#+caption: Effect of a change of the damping "stiffness coeficient" on the transfer function from $u$ to $d\mathcal{L}$
#+RESULTS:
[[file:figs/bode_Va_dL_effect_k_coef.png]]

#+begin_src matlab :exports none
%% Plot the obtained coupling transfer functions for all the bending stiffnesses
freqs = 2*logspace(1, 3, 1000);

figure;
hold on;
for i = 1:length(xis)
    plot(freqs, abs(squeeze(freqresp(Gs{i}('dL2', 'Va1'), freqs, 'Hz'))), ...
         'DisplayName', sprintf('$c_R = %.3f\\,[\\frac{Nm}{rad/s}]$', xis(i)));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude $d_L/V_a$ [m/V]'); set(gca, 'XTickLabel',[]);
hold off;
ylim([1e-8, 1e-3]);
legend('location', 'southwest');
xlim([20, 2e3]);
#+end_src

*** Effect of a change in mass damping coef of the APA
#+begin_src matlab
k_coef = 1e-6;
#+end_src

#+begin_src matlab
%% Tested bending dampings [Nm/(rad/s)]
m_coefs = [1e1, 5e1, 1e2, 5e2, 1e3];
#+end_src

#+begin_src matlab
%% Identify the DVF Plant (transfer function from u to dLm)
clear io; io_i = 1;
io(io_i) = linio([mdl, '/du'],  1, 'openinput');  io_i = io_i + 1; % Actuator Inputs
io(io_i) = linio([mdl, '/D'],  1, 'openoutput'); io_i = io_i + 1; % Encoders
#+end_src

#+begin_src matlab
%% Idenfity the transfer function from actuator to encoder for all bending dampins
Gs = {zeros(length(m_coefs), 1)};
n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
                                       'flex_top_type', '4dof', ...
                                       'motion_sensor_type', 'struts', ...
                                       'actuator_type', 'flexible');

for i = 1:length(m_coefs)
    m_coef = m_coefs(i);

    G = exp(-s*Ts)*linearize(mdl, io, 0.0, options);
    G.InputName  = {'Va1', 'Va2', 'Va3', 'Va4', 'Va5', 'Va6'};
    G.OutputName = {'dL1', 'dL2', 'dL3', 'dL4', 'dL5', 'dL6'};

    Gs(i) = {G};
end
#+end_src

#+begin_src matlab :exports none
%% Plot the obtained direct transfer functions for all the bending stiffnesses
freqs = 2*logspace(1, 3, 1000);

figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');

ax1 = nexttile([2,1]);
hold on;
for i = 1:length(m_coefs)
    plot(freqs, abs(squeeze(freqresp(Gs{i}('dL1', 'Va1'), freqs, 'Hz'))), ...
         'DisplayName', sprintf('mcoef = %.0e', m_coefs(i)));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude $d_L/V_a$ [m/V]'); set(gca, 'XTickLabel',[]);
hold off;
ylim([1e-8, 1e-3]);
legend('location', 'southwest');

ax2 = nexttile;
hold on;
for i = 1:length(m_coefs)
    plot(freqs, 180/pi*angle(squeeze(freqresp(Gs{i}('dL1', 'Va1'), freqs, 'Hz'))));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
yticks(-360:90:360); ylim([-180, 180]);

linkaxes([ax1,ax2],'x');
xlim([20, 2e3]);
#+end_src

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

#+name: fig:bode_Va_dL_effect_m_coef
#+caption: Effect of a change of the damping "mass coeficient" on the transfer function from $u$ to $d\mathcal{L}$
#+RESULTS:
[[file:figs/bode_Va_dL_effect_m_coef.png]]

#+begin_src matlab :exports none
%% Plot the obtained coupling transfer functions for all the bending stiffnesses
freqs = 2*logspace(1, 3, 1000);

figure;
hold on;
for i = 1:length(xis)
    plot(freqs, abs(squeeze(freqresp(Gs{i}('dL2', 'Va1'), freqs, 'Hz'))), ...
         'DisplayName', sprintf('$c_R = %.3f\\,[\\frac{Nm}{rad/s}]$', xis(i)));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude $d_L/V_a$ [m/V]'); set(gca, 'XTickLabel',[]);
hold off;
ylim([1e-8, 1e-3]);
legend('location', 'southwest');
xlim([20, 2e3]);
#+end_src

*** TODO Using Flexible model
#+begin_src matlab
d_aligns = [[-0.05,  -0.3,  0];
            [ 0,      0.5,  0];
            [-0.1,   -0.3,  0];
            [ 0,      0.3,  0];
            [-0.05,   0.05, 0];
            [0,       0,    0]]*1e-3;
#+end_src

#+begin_src matlab
d_aligns = zeros(6,3);
% d_aligns(1,:) = [-0.05,  -0.3,  0]*1e-3;
d_aligns(2,:) = [ 0,      0.3,  0]*1e-3;
#+end_src

#+begin_src matlab
%% Initialize Nano-Hexapod
n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
                                       'flex_top_type', '4dof', ...
                                       'motion_sensor_type', 'struts', ...
                                       'actuator_type', 'flexible', ...
                                       'actuator_d_align', d_aligns);
#+end_src

#+begin_question
Why do we have smaller resonances when using flexible APA?
On the test bench we have the same resonance as the 2DoF model.
Could it be due to the compliance in other dof of the flexible model?
#+end_question

#+begin_src matlab
%% Identify the DVF Plant (transfer function from u to dLm)
clear io; io_i = 1;
io(io_i) = linio([mdl, '/du'],  1, 'openinput');  io_i = io_i + 1; % Actuator Inputs
io(io_i) = linio([mdl, '/D'],  1, 'openoutput'); io_i = io_i + 1; % Encoders

Gdvf = exp(-s*Ts)*linearize(mdl, io, 0.0, options);
#+end_src

#+begin_src matlab :exports none
%% Comparison of the plants (encoder output) when tuning the misalignment
freqs = 2*logspace(0, 3, 1000);

figure;
tiledlayout(2, 3, 'TileSpacing', 'Compact', 'Padding', 'None');
ax1 = nexttile();
hold on;
plot(f, abs(G_dvf(:, 1, 1)));
plot(freqs, abs(squeeze(freqresp(Gdvf(1,1), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
set(gca, 'XTickLabel',[]); ylabel('Amplitude [m/V]');

ax2 = nexttile();
hold on;
plot(f, abs(G_dvf(:, 2, 2)));
plot(freqs, abs(squeeze(freqresp(Gdvf(2,2), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]);

ax3 = nexttile();
hold on;
plot(f, abs(G_dvf(:, 3, 3)), 'DisplayName', 'Meas.');
plot(freqs, abs(squeeze(freqresp(Gdvf(3,3), freqs, 'Hz'))), ...
     'DisplayName', 'Model');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]);
legend('location', 'southwest', 'FontSize', 8);

ax4 = nexttile();
hold on;
plot(f, abs(G_dvf(:, 4, 4)));
plot(freqs, abs(squeeze(freqresp(Gdvf(4,4), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Amplitude [m/V]');

ax5 = nexttile();
hold on;
plot(f, abs(G_dvf(:, 5, 5)));
plot(freqs, abs(squeeze(freqresp(Gdvf(5,5), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]);

ax6 = nexttile();
hold on;
plot(f, abs(G_dvf(:, 6, 6)));
plot(freqs, abs(squeeze(freqresp(Gdvf(6,6), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]);

linkaxes([ax1,ax2,ax3,ax4,ax5,ax6],'xy');
% xlim([20, 2e3]); ylim([1e-8, 1e-3]);
xlim([50, 5e2]); ylim([1e-6, 1e-3]);
#+end_src

#+begin_src matlab :exports none
%% Diagonal elements of the DVF plant
freqs = 6*logspace(1, 2, 2000);

i_strut = 1;

figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');

ax1 = nexttile([2,1]);
hold on;
plot(f, abs(G_dvf(:,i_strut, 2)), 'color', [0,0,0,0.2], ...
     'DisplayName', '$d\mathcal{L}_{m,i}/u_i$ - FRF')
plot(freqs, abs(squeeze(freqresp(Gdvf(2,2), freqs, 'Hz'))), '-', ...
     'DisplayName', '$d\mathcal{L}_{m,i}/u_i$ - Model')
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/V]'); set(gca, 'XTickLabel',[]);
ylim([1e-8, 1e-3]);
legend('location', 'northeast');

ax2 = nexttile;
hold on;
plot(f, 180/pi*angle(G_dvf(:,2, 2)), 'color', [0,0,0,0.2]);
plot(freqs, 180/pi*angle(squeeze(freqresp(Gdvf(2,2), freqs, 'Hz'))), '-');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-180, 180]);
yticks([-180, -90, 0, 90, 180]);

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

#+begin_src matlab :exports none
%% Diagonal elements of the DVF plant
freqs = 6*logspace(1, 2, 2000);

figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');

ax1 = nexttile([2,1]);
hold on;
plot(f, abs(G_dvf(:,1, 1)), 'color', [0,0,0,0.2], ...
     'DisplayName', '$d\mathcal{L}_{m,i}/u_i$ - FRF')
for i = 2:6
    set(gca,'ColorOrderIndex',2)
    plot(f, abs(G_dvf(:,i, i)), 'color', [0,0,0,0.2], ...
         'HandleVisibility', 'off');
end
set(gca,'ColorOrderIndex',2);
plot(freqs, abs(squeeze(freqresp(Gdvf(1,1), freqs, 'Hz'))), '-', ...
     'DisplayName', '$d\mathcal{L}_{m,i}/u_i$ - Model')
for i = 2:6
    set(gca,'ColorOrderIndex',2);
    plot(freqs, abs(squeeze(freqresp(Gdvf(i,i), freqs, 'Hz'))), '-', ...
         'HandleVisibility', 'off');
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/V]'); set(gca, 'XTickLabel',[]);
ylim([1e-8, 1e-3]);
legend('location', 'northeast');

ax2 = nexttile;
hold on;
for i = 1:6
    plot(f, 180/pi*angle(G_dvf(:,i, i)), 'color', [0,0,0,0.2]);
end
for i = 1:6
    set(gca,'ColorOrderIndex',2);
    plot(freqs, 180/pi*angle(squeeze(freqresp(Gdvf(i,i), freqs, 'Hz'))), '-');
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-180, 180]);
yticks([-180, -90, 0, 90, 180]);

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

#+begin_src matlab
%% Identify the IFF Plant (transfer function from u to taum)
clear io; io_i = 1;
io(io_i) = linio([mdl, '/du'],  1, 'openinput');   io_i = io_i + 1; % Actuator Inputs
io(io_i) = linio([mdl, '/dum'],  1, 'openoutput'); io_i = io_i + 1; % Force Sensors

Giff = exp(-s*Ts)*linearize(mdl, io, 0.0, options);
#+end_src

#+begin_src matlab :exports none
%% Bode plot of the identified IFF Plant (Simscape) and measured FRF data
freqs = 2*logspace(1, 3, 1000);

figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');

ax1 = nexttile([2,1]);
hold on;
plot(f, abs(G_iff(:,1, 1)), 'color', [0,0,0,0.2], ...
     'DisplayName', '$\tau_{m,i}/u_i$ - FRF')
for i = 2:6
    set(gca,'ColorOrderIndex',2)
    plot(f, abs(G_iff(:,i, i)), 'color', [0,0,0,0.2], ...
         'HandleVisibility', 'off');
end
set(gca,'ColorOrderIndex',2);
plot(freqs, abs(squeeze(freqresp(Giff(1,1), freqs, 'Hz'))), '-', ...
     'DisplayName', '$\tau_{m,i}/u_i$ - Model')
for i = 2:6
    set(gca,'ColorOrderIndex',2);
    plot(freqs, abs(squeeze(freqresp(Giff(i,i), freqs, 'Hz'))), '-', ...
         'HandleVisibility', 'off');
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [V/V]'); set(gca, 'XTickLabel',[]);
legend('location', 'southeast');

ax2 = nexttile;
hold on;
for i = 1:6
    plot(f, 180/pi*angle(G_iff(:,i, i)), 'color', [0,0,0,0.2]);
end
for i = 1:6
    set(gca,'ColorOrderIndex',2);
    plot(freqs, 180/pi*angle(squeeze(freqresp(Giff(i,i), freqs, 'Hz'))), '-');
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-180, 180]);
yticks([-180, -90, 0, 90, 180]);

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

#+begin_src matlab
#+end_src

*** Flexible model + encoders fixed to the plates
#+begin_src matlab
%% Identify the IFF Plant (transfer function from u to taum)
clear io; io_i = 1;
io(io_i) = linio([mdl, '/du'],  1, 'openinput');   io_i = io_i + 1; % Actuator Inputs
io(io_i) = linio([mdl, '/D'],  1, 'openoutput'); io_i = io_i + 1; % Force Sensors
#+end_src

#+begin_src matlab
d_aligns = [[-0.05,  -0.3,  0];
            [ 0,      0.5,  0];
            [-0.1,   -0.3,  0];
            [ 0,      0.3,  0];
            [-0.05,   0.05, 0];
            [0,       0,    0]]*1e-3;
#+end_src

#+begin_src matlab
%% Initialize Nano-Hexapod
n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
                                       'flex_top_type', '4dof', ...
                                       'motion_sensor_type', 'struts', ...
                                       'actuator_type', 'flexible', ...
                                       'actuator_d_align', d_aligns);
#+end_src

#+begin_src matlab
Gdvf_struts = exp(-s*Ts)*linearize(mdl, io, 0.0, options);
#+end_src

#+begin_src matlab
%% Initialize Nano-Hexapod
n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
                                       'flex_top_type', '4dof', ...
                                       'motion_sensor_type', 'plates', ...
                                       'actuator_type', 'flexible', ...
                                       'actuator_d_align', d_aligns);
#+end_src

#+begin_src matlab
Gdvf_plates = exp(-s*Ts)*linearize(mdl, io, 0.0, options);
#+end_src

#+begin_src matlab :exports none
%% Plot the obtained direct transfer functions for all the bending stiffnesses
freqs = 2*logspace(1, 3, 1000);

figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');

ax1 = nexttile([2,1]);
hold on;
plot(freqs, abs(squeeze(freqresp(Gdvf_struts(1, 1), freqs, 'Hz'))), ...
     'DisplayName', 'Struts');
plot(freqs, abs(squeeze(freqresp(Gdvf_plates(1, 1), freqs, 'Hz'))), ...
     'DisplayName', 'Plates');
for i = 2:6
    set(gca,'ColorOrderIndex',1);
    plot(freqs, abs(squeeze(freqresp(Gdvf_struts(i, i), freqs, 'Hz'))), ...
         'HandleVisibility', 'off');
    set(gca,'ColorOrderIndex',2);
    plot(freqs, abs(squeeze(freqresp(Gdvf_plates(i, i), freqs, 'Hz'))), ...
         'HandleVisibility', 'off');
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude $d_L/V_a$ [m/V]'); set(gca, 'XTickLabel',[]);
hold off;
ylim([1e-8, 1e-3]);
legend('location', 'southwest');

ax2 = nexttile;
hold on;
for i = 1:6
    set(gca,'ColorOrderIndex',1);
    plot(freqs, 180/pi*angle(squeeze(freqresp(Gdvf_struts(i, i), freqs, 'Hz'))));
    set(gca,'ColorOrderIndex',2);
    plot(freqs, 180/pi*angle(squeeze(freqresp(Gdvf_plates(i, i), freqs, 'Hz'))));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
yticks(-360:90:360); ylim([-180, 180]);

linkaxes([ax1,ax2],'x');
xlim([20, 2e3]);
#+end_src

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

#+name: fig:dvf_plant_comp_struts_plates
#+caption: Comparison of the dynamics from $V_a$ to $d_L$ when the encoders are fixed to the struts (blue) and to the plates (red). APA are modeled as a flexible element.
#+RESULTS:
[[file:figs/dvf_plant_comp_struts_plates.png]]

** Integral Force Feedback
<<sec:enc_struts_iff>>
*** 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
addpath('./matlab/mat/');
addpath('./matlab/src/');
addpath('./matlab/');
#+end_src

#+begin_src matlab :eval no
addpath('./mat/');
addpath('./src/');
#+end_src

#+begin_src matlab
load('identified_plants_enc_struts.mat', 'f', 'Ts', 'G_iff', 'G_dvf')
#+end_src

#+begin_src matlab :tangle no
%% Add all useful folders to the path
addpath('matlab/nass-simscape/matlab/nano_hexapod/')
addpath('matlab/nass-simscape/STEPS/nano_hexapod/')
addpath('matlab/nass-simscape/STEPS/png/')
addpath('matlab/nass-simscape/src/')
addpath('matlab/nass-simscape/mat/')
#+end_src

#+begin_src matlab :eval no
%% Add all useful folders to the path
addpath('nass-simscape/matlab/nano_hexapod/')
addpath('nass-simscape/STEPS/nano_hexapod/')
addpath('nass-simscape/STEPS/png/')
addpath('nass-simscape/src/')
addpath('nass-simscape/mat/')
#+end_src

#+begin_src matlab
%% Open Simulink Model
mdl = 'nano_hexapod_simscape';

options = linearizeOptions;
options.SampleTime = 0;

Rx = zeros(1, 7);

open(mdl)
#+end_src

*** Identification of the IFF Plant
#+begin_src matlab
%% Initialize Nano-Hexapod
n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
                                       'flex_top_type', '4dof', ...
                                       'motion_sensor_type', 'struts', ...
                                       'actuator_type', '2dof');
#+end_src

#+begin_src matlab
%% Identify the IFF Plant (transfer function from u to taum)
clear io; io_i = 1;
io(io_i) = linio([mdl, '/du'],  1, 'openinput');   io_i = io_i + 1; % Actuator Inputs
io(io_i) = linio([mdl, '/dum'],  1, 'openoutput'); io_i = io_i + 1; % Force Sensors

Giff = exp(-s*Ts)*linearize(mdl, io, 0.0, options);
#+end_src

*** Root Locus and Decentralized Loop gain
#+begin_src matlab
%% IFF Controller
Kiff_g1 = -(1/(s + 2*pi*40))*... % Low pass filter (provides integral action above 40Hz)
           (s/(s + 2*pi*30))*... % High pass filter to limit low frequency gain
           (1/(1 + s/2/pi/500))*... % Low pass filter to be more robust to high frequency resonances
           eye(6); % Diagonal 6x6 controller
#+end_src

#+begin_src matlab :exports none
%% Root Locus for IFF
gains = logspace(1, 4, 100);

figure;

hold on;
% Pure Integrator
set(gca,'ColorOrderIndex',1);
plot(real(pole(Giff)),  imag(pole(Giff)), 'x', 'DisplayName', '$g = 0$');
set(gca,'ColorOrderIndex',1);
plot(real(tzero(Giff)),  imag(tzero(Giff)), 'o', 'HandleVisibility', 'off');

for g = gains
    clpoles = pole(feedback(Giff, g*Kiff_g1*eye(6)));
    set(gca,'ColorOrderIndex',1);
    plot(real(clpoles), imag(clpoles), '.', 'HandleVisibility', 'off');
end

g = 4e2;
clpoles = pole(feedback(Giff, g*Kiff_g1*eye(6)));
set(gca,'ColorOrderIndex',2);
plot(real(clpoles), imag(clpoles), 'x', 'DisplayName', sprintf('$g=%.0f$', g));
hold off;
axis square;
xlim([-1250, 0]); ylim([0, 1250]);
xlabel('Real Part'); ylabel('Imaginary Part');
legend('location', 'northwest');
#+end_src

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

#+name: fig:enc_struts_iff_root_locus
#+caption: Root Locus for the IFF control strategy
#+RESULTS:
[[file:figs/enc_struts_iff_root_locus.png]]

Then the "optimal" IFF controller is:
#+begin_src matlab
%% IFF controller with Optimal gain
Kiff = g*Kiff_g1;
#+end_src

#+begin_src matlab :tangle no
save('matlab/mat/Kiff.mat', 'Kiff')
#+end_src

#+begin_src matlab :exports none :eval no
save('mat/Kiff.mat', 'Kiff')
#+end_src

#+begin_src matlab :exports none
%% Bode plot of the "decentralized loop gain"
freqs = 2*logspace(1, 3, 1000);

figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');

ax1 = nexttile([2,1]);
hold on;
plot(f, abs(squeeze(freqresp(Kiff(1,1), f, 'Hz')).*G_iff(:, 1, 1)), 'color', [0,0,0,0.2], ...
     'DisplayName', '$\tau_{m,i}/u_i \cdot K_{iff}$ - FRF')
for i = 2:6
    set(gca,'ColorOrderIndex',2)
    plot(f, abs(squeeze(freqresp(Kiff(1,1), f, 'Hz')).*G_iff(:, i, i)), 'color', [0,0,0,0.2], ...
         'HandleVisibility', 'off');
end
set(gca,'ColorOrderIndex',2);
plot(freqs, abs(squeeze(freqresp(Kiff(1,1)*Giff(1,1), freqs, 'Hz'))), '-', ...
     'DisplayName', '$\tau_{m,i}/u_i \cdot K_{iff}$ - Model')
for i = 2:6
    set(gca,'ColorOrderIndex',2);
    plot(freqs, abs(squeeze(freqresp(Kiff(1,1)*Giff(i,i), freqs, 'Hz'))), '-', ...
         'HandleVisibility', 'off');
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [V/V]'); set(gca, 'XTickLabel',[]);
legend('location', 'northeast');

ax2 = nexttile;
hold on;
for i = 1:6
    plot(f, 180/pi*angle(squeeze(freqresp(Kiff(1,1), f, 'Hz')).*G_iff(:, i, i)), 'color', [0,0,0,0.2]);
end
for i = 1:6
    set(gca,'ColorOrderIndex',2);
    plot(freqs, 180/pi*angle(squeeze(freqresp(Kiff(1,1)*Giff(i,i), freqs, 'Hz'))), '-');
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-180, 180]);
yticks([-180, -90, 0, 90, 180]);

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

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

#+name: fig:enc_struts_iff_opt_loop_gain
#+caption: Bode plot of the "decentralized loop gain" $G_\text{iff}(i,i) \times K_\text{iff}(i,i)$
#+RESULTS:
[[file:figs/enc_struts_iff_opt_loop_gain.png]]

*** Multiple Gains - Simulation
#+begin_src matlab
%% Tested IFF gains
iff_gains = [4, 10, 20, 40, 100, 200, 400];
#+end_src

#+begin_src matlab
%% Initialize the Simscape model in closed loop
n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
                                       'flex_top_type', '4dof', ...
                                       'motion_sensor_type', 'struts', ...
                                       'actuator_type', '2dof', ...
                                       'controller_type', 'iff');
#+end_src

#+begin_src matlab
%% Identify the (damped) transfer function from u to dLm for different values of the IFF gain
Gd_iff = {zeros(1, length(iff_gains))};

clear io; io_i = 1;
io(io_i) = linio([mdl, '/du'],  1, 'openinput');  io_i = io_i + 1; % Actuator Inputs
io(io_i) = linio([mdl, '/D'],  1, 'openoutput'); io_i = io_i + 1; % Strut Displacement (encoder)

for i = 1:length(iff_gains)
    Kiff = iff_gains(i)*Kiff_g1*eye(6); % IFF Controller
    Gd_iff(i) = {exp(-s*Ts)*linearize(mdl, io, 0.0, options)};

    isstable(Gd_iff{i})
end
#+end_src

#+begin_src matlab :exports none
%% Bode plot of the transfer function from u to dLm for tested values of the IFF gain
freqs = 2*logspace(1, 3, 1000);

figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');

ax1 = nexttile([2,1]);
hold on;
for i = 1:length(iff_gains)
    plot(freqs, abs(squeeze(freqresp(Gd_iff{i}(1,1), freqs, 'Hz'))), '-', ...
         'DisplayName', sprintf('$g = %.0f$', iff_gains(i)));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/V]'); set(gca, 'XTickLabel',[]);
legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 2);

ax2 = nexttile;
hold on;
for i = 1:length(iff_gains)
    plot(freqs, 180/pi*angle(squeeze(freqresp(Gd_iff{i}(1,1), freqs, 'Hz'))), '-');
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-180, 180]);
yticks([-180, -90, 0, 90, 180]);

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

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

#+name: fig:enc_struts_iff_gains_effect_dvf_plant
#+caption: Effect of the IFF gain $g$ on the transfer function from $\bm{\tau}$ to $d\bm{\mathcal{L}}_m$
#+RESULTS:
[[file:figs/enc_struts_iff_gains_effect_dvf_plant.png]]

*** Experimental Results - Gains
**** Introduction                                                  :ignore:
Let's look at the damping introduced by IFF as a function of the IFF gain and compare that with the results obtained using the Simscape model.

**** Load Data
#+begin_src matlab
%% Load Identification Data
meas_iff_gains = {};

for i = 1:length(iff_gains)
    meas_iff_gains(i) = {load(sprintf('mat/iff_strut_1_noise_g_%i.mat', iff_gains(i)), 't', 'Vexc', 'Vs', 'de', 'u')};
end
#+end_src

**** Spectral Analysis - Setup
#+begin_src matlab
%% Setup useful variables
% Sampling Time [s]
Ts = (meas_iff_gains{1}.t(end) - (meas_iff_gains{1}.t(1)))/(length(meas_iff_gains{1}.t)-1);

% Sampling Frequency [Hz]
Fs = 1/Ts;

% Hannning Windows
win = hanning(ceil(1*Fs));

% And we get the frequency vector
[~, f] = tfestimate(meas_iff_gains{1}.Vexc, meas_iff_gains{1}.de, win, [], [], 1/Ts);
#+end_src

**** DVF Plant
#+begin_src matlab
%% DVF Plant (transfer function from u to dLm)
G_iff_gains = {};

for i = 1:length(iff_gains)
    G_iff_gains{i} = tfestimate(meas_iff_gains{i}.Vexc, meas_iff_gains{i}.de(:,1), win, [], [], 1/Ts);
end
#+end_src

#+begin_src matlab :exports none
%% Bode plot of the transfer function from u to dLm for tested values of the IFF gain
freqs = 2*logspace(1, 3, 1000);

figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');

ax1 = nexttile([2,1]);
hold on;
for i = 1:length(iff_gains)
    plot(f, abs(G_iff_gains{i}), '-', ...
         'DisplayName', sprintf('$g_{iff} = %.0f$', iff_gains(i)));
end
set(gca,'ColorOrderIndex',1)
for i = 1:length(iff_gains)
    plot(freqs, abs(squeeze(freqresp(Gd_iff{i}(1,1), freqs, 'Hz'))), '--', ...
         'HandleVisibility', 'off');
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/V]'); set(gca, 'XTickLabel',[]);
legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 2);

ax2 = nexttile;
hold on;
for i =1:length(iff_gains)
    plot(f, 180/pi*angle(G_iff_gains{i}), '-');
end
set(gca,'ColorOrderIndex',1)
for i = 1:length(iff_gains)
    plot(freqs, 180/pi*angle(squeeze(freqresp(Gd_iff{i}(1,1), freqs, 'Hz'))), '--');
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-180, 180]);
yticks([-180, -90, 0, 90, 180]);

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

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

#+name: fig:comp_iff_gains_dvf_plant
#+caption: Transfer function from $u$ to $d\mathcal{L}_m$ for multiple values of the IFF gain
#+RESULTS:
[[file:figs/comp_iff_gains_dvf_plant.png]]

#+begin_src matlab :exports none
xlim([20, 200]);
#+end_src

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

#+name: fig:comp_iff_gains_dvf_plant_zoom
#+caption: Transfer function from $u$ to $d\mathcal{L}_m$ for multiple values of the IFF gain (Zoom)
#+RESULTS:
[[file:figs/comp_iff_gains_dvf_plant_zoom.png]]

#+begin_important
The IFF control strategy is very effective for the damping of the suspension modes.
It however does not damp the modes at 200Hz, 300Hz and 400Hz (flexible modes of the APA).
This is very logical.

Also, the experimental results and the models obtained from the Simscape model are in agreement.
#+end_important

**** Experimental Results - Comparison of the un-damped and fully damped system
#+begin_src matlab :exports none
%% Bode plot for the transfer function from u to dLm
freqs = 2*logspace(1, 3, 1000);

figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');

ax1 = nexttile([2,1]);
hold on;
% Un Damped measurement
set(gca,'ColorOrderIndex',1)
plot(f, abs(G_dvf(:, 1, 1)), ...
     'DisplayName', 'Un-Damped')
for i = 2:6
    set(gca,'ColorOrderIndex',1)
    plot(f, abs(G_dvf(:,i , i)), ...
         'HandleVisibility', 'off');
end

% IFF Damped measurement
set(gca,'ColorOrderIndex',2)
plot(f, abs(G_iff_opt{1}(:,1)), ...
     'DisplayName', 'Optimal gain')
for i = 2:6
    set(gca,'ColorOrderIndex',2)
    plot(f, abs(G_iff_opt{i}(:,i)), ...
         'HandleVisibility', 'off');
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude $d_e/V_{exc}$ [m/V]'); set(gca, 'XTickLabel',[]);
ylim([1e-9, 1e-3]);
legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 3);

ax2 = nexttile;
hold on;
for i =1:6
    set(gca,'ColorOrderIndex',1)
    plot(f, 180/pi*angle(G_dvf(i,i, i)));
    set(gca,'ColorOrderIndex',2)
    plot(f, 180/pi*angle(G_iff_opt{i}(:,i)));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
yticks(-360:90:360);

linkaxes([ax1,ax2],'x');
xlim([20, 2e3]);
#+end_src

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

#+name: fig:comp_undamped_opt_iff_gain_diagonal
#+caption: Comparison of the diagonal elements of the tranfer function from $\bm{u}$ to $d\bm{\mathcal{L}}_m$ without active damping and with optimal IFF gain
#+RESULTS:
[[file:figs/comp_undamped_opt_iff_gain_diagonal.png]]

#+begin_question
A series of modes at around 205Hz are also damped.

Are these damped modes at 205Hz additional "suspension" modes or flexible modes of the struts?
#+end_question

*** Experimental Results - Damped Plant with Optimal gain
**** Introduction                                                  :ignore:
Let's now look at the $6 \times 6$ damped plant with the optimal gain $g = 400$.

**** Load Data
#+begin_src matlab
%% Load Identification Data
meas_iff_struts = {};

for i = 1:6
    meas_iff_struts(i) = {load(sprintf('mat/iff_strut_%i_noise_g_400.mat', i), 't', 'Vexc', 'Vs', 'de', 'u')};
end
#+end_src

**** Spectral Analysis - Setup
#+begin_src matlab
%% Setup useful variables
% Sampling Time [s]
Ts = (meas_iff_struts{1}.t(end) - (meas_iff_struts{1}.t(1)))/(length(meas_iff_struts{1}.t)-1);

% Sampling Frequency [Hz]
Fs = 1/Ts;

% Hannning Windows
win = hanning(ceil(1*Fs));

% And we get the frequency vector
[~, f] = tfestimate(meas_iff_struts{1}.Vexc, meas_iff_struts{1}.de, win, [], [], 1/Ts);
#+end_src

**** DVF Plant
#+begin_src matlab
%% DVF Plant (transfer function from u to dLm)
G_iff_opt = {};

for i = 1:6
    G_iff_opt{i} = tfestimate(meas_iff_struts{i}.Vexc, meas_iff_struts{i}.de, win, [], [], 1/Ts);
end
#+end_src

#+begin_src matlab :exports none
%% Bode plot for the transfer function from u to dLm
freqs = 2*logspace(1, 3, 1000);

figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');

ax1 = nexttile([2,1]);
hold on;
% Diagonal Elements FRF
plot(f, abs(G_iff_opt{1}(:,1)), 'color', [0,0,0,0.2], ...
     'DisplayName', '$d\mathcal{L}_{m,i}/u_i$ - FRF')
for i = 2:6
    plot(f, abs(G_iff_opt{i}(:,i)), 'color', [0,0,0,0.2], ...
         'HandleVisibility', 'off');
end

% Diagonal Elements Model
set(gca,'ColorOrderIndex',2)
plot(freqs, abs(squeeze(freqresp(Gd_iff{end}(1,1), freqs, 'Hz'))), '-', ...
     'DisplayName', '$d\mathcal{L}_{m,i}/u_i$ - Model')
for i = 2:6
    set(gca,'ColorOrderIndex',2)
    plot(freqs, abs(squeeze(freqresp(Gd_iff{end}(i,i), freqs, 'Hz'))), '-', ...
         'HandleVisibility', 'off');
end

hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude $d_e/V_{exc}$ [m/V]'); set(gca, 'XTickLabel',[]);
ylim([1e-9, 1e-3]);
legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 3);

ax2 = nexttile;
hold on;
for i =1:6
    plot(f, 180/pi*angle(G_iff_opt{i}(:,i)), 'color', [0,0,0,0.2]);
    set(gca,'ColorOrderIndex',2)
    plot(freqs, 180/pi*angle(squeeze(freqresp(Gd_iff{end}(i,i), freqs, 'Hz'))));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
yticks(-360:90:360);

linkaxes([ax1,ax2],'x');
xlim([20, 2e3]);
#+end_src

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

#+name: fig:damped_iff_plant_comp_diagonal
#+caption: Comparison of the diagonal elements of the transfer functions from $\bm{u}$ to $d\bm{\mathcal{L}}_m$ with active damping (IFF) applied with an optimal gain $g = 400$
#+RESULTS:
[[file:figs/damped_iff_plant_comp_diagonal.png]]

#+begin_src matlab :exports none
%% Bode plot for the transfer function from u to dLm
freqs = 2*logspace(1, 3, 1000);

figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');

ax1 = nexttile([2,1]);
hold on;
% Off diagonal FRF
plot(f, abs(G_iff_opt{1}(:,2)), 'color', [0,0,0,0.2], ...
     'DisplayName', '$d\mathcal{L}_{m,i}/u_j$ - FRF')
for i = 1:5
    for j = i+1:6
        plot(f, abs(G_iff_opt{i}(:,j)), 'color', [0, 0, 0, 0.2], ...
             'HandleVisibility', 'off');
    end
end

% Off diagonal Model
set(gca,'ColorOrderIndex',2)
plot(freqs, abs(squeeze(freqresp(Gd_iff{end}(1,2), freqs, 'Hz'))), '-', ...
     'DisplayName', '$d\mathcal{L}_{m,i}/u_j$ - Model')
for i = 1:5
    for j = i+1:6
        set(gca,'ColorOrderIndex',2)
        plot(freqs, abs(squeeze(freqresp(Gd_iff{end}(i,j), freqs, 'Hz'))), ...
             'HandleVisibility', 'off');
    end
end

hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude $d_e/V_{exc}$ [m/V]'); set(gca, 'XTickLabel',[]);
ylim([1e-9, 1e-3]);
legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 3);

ax2 = nexttile;
hold on;
% Off diagonal FRF
for i = 1:5
    for j = i+1:6
        plot(f, 180/pi*angle(G_iff_opt{i}(:,j)), 'color', [0, 0, 0, 0.2]);
    end
end

% Off diagonal Model
for i = 1:5
    for j = i+1:6
        set(gca,'ColorOrderIndex',2)
        plot(freqs, 180/pi*angle(squeeze(freqresp(Gd_iff{end}(i,j), freqs, 'Hz'))));
    end
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
yticks(-360:90:360);

linkaxes([ax1,ax2],'x');
xlim([20, 2e3]);
#+end_src

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

#+name: fig:damped_iff_plant_comp_off_diagonal
#+caption: Comparison of the off-diagonal elements of the transfer functions from $\bm{u}$ to $d\bm{\mathcal{L}}_m$ with active damping (IFF) applied with an optimal gain $g = 400$
#+RESULTS:
[[file:figs/damped_iff_plant_comp_off_diagonal.png]]

#+begin_important
With the IFF control strategy applied and the optimal gain used, the suspension modes are very well damped.
Remains the undamped flexible modes of the APA (200Hz, 300Hz, 400Hz), and the modes of the plates (700Hz).

The Simscape model and the experimental results are in very good agreement.
#+end_important

** Modal Analysis
<<sec:enc_struts_modal_analysis>>
*** Introduction                                                    :ignore:
Several 3-axis accelerometers are fixed on the top platform of the nano-hexapod as shown in Figure [[fig:compliance_vertical_comp_iff]].

#+name: fig:accelerometers_nano_hexapod
#+caption: Location of the accelerometers on top of the nano-hexapod
#+attr_latex: :width \linewidth
[[file:figs/accelerometers_nano_hexapod.jpg]]

The top platform is then excited using an instrumented hammer as shown in Figure [[fig:hammer_excitation_compliance_meas]].

#+name: fig:hammer_excitation_compliance_meas
#+caption: Example of an excitation using an instrumented hammer
#+attr_latex: :width \linewidth
[[file:figs/hammer_excitation_compliance_meas.jpg]]

*** 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
addpath('./matlab/mat/');
addpath('./matlab/src/');
addpath('./matlab/');
#+end_src

#+begin_src matlab :eval no
addpath('./mat/');
addpath('./src/');
#+end_src

#+begin_src matlab :tangle no
%% Add all useful folders to the path
addpath('matlab/nass-simscape/matlab/nano_hexapod/')
addpath('matlab/nass-simscape/STEPS/nano_hexapod/')
addpath('matlab/nass-simscape/STEPS/png/')
addpath('matlab/nass-simscape/src/')
addpath('matlab/nass-simscape/mat/')
#+end_src

#+begin_src matlab :eval no
%% Add all useful folders to the path
addpath('nass-simscape/matlab/nano_hexapod/')
addpath('nass-simscape/STEPS/nano_hexapod/')
addpath('nass-simscape/STEPS/png/')
addpath('nass-simscape/src/')
addpath('nass-simscape/mat/')
#+end_src

#+begin_src matlab
%% Open Simulink Model
mdl = 'nano_hexapod_simscape';

options = linearizeOptions;
options.SampleTime = 0;

Rx = zeros(1, 7);

open(mdl)
#+end_src

*** Effectiveness of the IFF Strategy - Compliance
In this section, we wish to estimated the effectiveness of the IFF strategy concerning the compliance.

The top plate is excited vertically using the instrumented hammer two times:
1. no control loop is used
2. decentralized IFF is used

The data is loaded.
#+begin_src matlab
frf_ol  = load('Measurement_Z_axis.mat'); % Open-Loop
frf_iff = load('Measurement_Z_axis_damped.mat'); % IFF
#+end_src

The mean vertical motion of the top platform is computed by averaging all 5 accelerometers.
#+begin_src matlab
%% Multiply by 10 (gain in m/s^2/V) and divide by 5 (number of accelerometers)
d_frf_ol = 10/5*(frf_ol.FFT1_H1_4_1_RMS_Y_Mod + frf_ol.FFT1_H1_7_1_RMS_Y_Mod + frf_ol.FFT1_H1_10_1_RMS_Y_Mod + frf_ol.FFT1_H1_13_1_RMS_Y_Mod + frf_ol.FFT1_H1_16_1_RMS_Y_Mod)./(2*pi*frf_ol.FFT1_H1_16_1_RMS_X_Val).^2;
d_frf_iff = 10/5*(frf_iff.FFT1_H1_4_1_RMS_Y_Mod + frf_iff.FFT1_H1_7_1_RMS_Y_Mod + frf_iff.FFT1_H1_10_1_RMS_Y_Mod + frf_iff.FFT1_H1_13_1_RMS_Y_Mod + frf_iff.FFT1_H1_16_1_RMS_Y_Mod)./(2*pi*frf_iff.FFT1_H1_16_1_RMS_X_Val).^2;
#+end_src

The vertical compliance (magnitude of the transfer function from a vertical force applied on the top plate to the vertical motion of the top plate) is shown in Figure [[fig:compliance_vertical_comp_iff]].

#+begin_src matlab :exports none
figure;
hold on;
plot(frf_ol.FFT1_H1_16_1_RMS_X_Val,  d_frf_ol, 'DisplayName', 'OL');
plot(frf_iff.FFT1_H1_16_1_RMS_X_Val, d_frf_iff, 'DisplayName', 'IFF');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Vertical Compliance [$m/N$]');
xlim([20, 2e3]); ylim([2e-9, 2e-5]);
legend('location', 'northeast');
#+end_src

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

#+name: fig:compliance_vertical_comp_iff
#+caption: Measured vertical compliance with and without IFF
#+RESULTS:
[[file:figs/compliance_vertical_comp_iff.png]]

#+begin_important
From Figure [[fig:compliance_vertical_comp_iff]], it is clear that the IFF control strategy is very effective in damping the suspensions modes of the nano-hexapode.
It also has the effect of degrading (slightly) the vertical compliance at low frequency.

It also seems some damping can be added to the modes at around 205Hz which are flexible modes of the struts.
#+end_important

*** Comparison with the Simscape Model
Let's now compare the measured vertical compliance with the vertical compliance as estimated from the Simscape model.

The transfer function from a vertical external force to the absolute motion of the top platform is identified (with and without IFF) using the Simscape model.
#+begin_src matlab :exports none
%% Identify the IFF Plant (transfer function from u to taum)
clear io; io_i = 1;
io(io_i) = linio([mdl, '/duz_ext'],     1, 'openinput');  io_i = io_i + 1; % External - Vertical force
io(io_i) = linio([mdl, '/Z_top_plat'], 1, 'openoutput'); io_i = io_i + 1; % Absolute vertical motion of top platform

%% Initialize Nano-Hexapod in Open Loop
n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
                                       'flex_top_type', '4dof', ...
                                       'motion_sensor_type', 'struts', ...
                                       'actuator_type', '2dof');

G_compl_z_ol = linearize(mdl, io, 0.0, options);

%% Initialize Nano-Hexapod with IFF
Kiff = 400*(1/(s + 2*pi*40))*... % Low pass filter (provides integral action above 40Hz)
           (s/(s + 2*pi*30))*... % High pass filter to limit low frequency gain
           (1/(1 + s/2/pi/500))*... % Low pass filter to be more robust to high frequency resonances
           eye(6); % Diagonal 6x6 controller

n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
                                       'flex_top_type', '4dof', ...
                                       'motion_sensor_type', 'struts', ...
                                       'actuator_type', '2dof', ...
                                       'controller_type', 'iff');

G_compl_z_iff = linearize(mdl, io, 0.0, options);
#+end_src

The comparison is done in Figure [[fig:compliance_vertical_comp_model_iff]].
Again, the model is quite accurate!
#+begin_src matlab :exports none
%% Comparison of the measured compliance and the one obtained from the model
freqs = 2*logspace(1,3,1000);

figure;
hold on;
plot(frf_ol.FFT1_H1_16_1_RMS_X_Val,  d_frf_ol,  '-', 'DisplayName', 'OL - Meas.');
plot(frf_iff.FFT1_H1_16_1_RMS_X_Val, d_frf_iff, '-', 'DisplayName', 'IFF - Meas.');
set(gca,'ColorOrderIndex',1)
plot(freqs, abs(squeeze(freqresp(G_compl_z_ol,  freqs, 'Hz'))), '--', 'DisplayName', 'OL - Model')
plot(freqs, abs(squeeze(freqresp(G_compl_z_iff, freqs, 'Hz'))), '--', 'DisplayName', 'IFF - Model')
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Vertical Compliance [$m/N$]');
xlim([20, 2e3]); ylim([2e-9, 2e-5]);
legend('location', 'northeast', 'FontSize', 8);
#+end_src

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

#+name: fig:compliance_vertical_comp_model_iff
#+caption: Measured vertical compliance with and without IFF
#+RESULTS:
[[file:figs/compliance_vertical_comp_model_iff.png]]

*** Obtained Mode Shapes
Then, several excitation are performed using the instrumented Hammer and the mode shapes are extracted.

We can observe the mode shapes of the first 6 modes that are the suspension modes (the plate is behaving as a solid body) in Figure [[fig:mode_shapes_annotated]].

#+name: fig:mode_shapes_annotated
#+caption: Measured mode shapes for the first six modes
#+attr_latex: :width \linewidth
[[file:figs/mode_shapes_annotated.gif]]

Then, there is a mode at 692Hz which corresponds to a flexible mode of the top plate (Figure [[fig:mode_shapes_flexible_annotated]]).

#+name: fig:mode_shapes_flexible_annotated
#+caption: First flexible mode at 692Hz
#+attr_latex: :width 0.3\linewidth
[[file:figs/ModeShapeFlex1_crop.gif]]

The obtained modes are summarized in Table [[tab:description_modes]].

#+name: tab:description_modes
#+caption: Description of the identified modes
#+attr_latex: :environment tabularx :width 0.7\linewidth :align ccX
#+attr_latex: :center t :booktabs t :float t
| Mode | Freq. [Hz] | Description                                  |
|------+------------+----------------------------------------------|
|    1 |        105 | Suspension Mode: Y-translation               |
|    2 |        107 | Suspension Mode: X-translation               |
|    3 |        131 | Suspension Mode: Z-translation               |
|    4 |        161 | Suspension Mode: Y-tilt                      |
|    5 |        162 | Suspension Mode: X-tilt                      |
|    6 |        180 | Suspension Mode: Z-rotation                  |
|    7 |        692 | (flexible) Membrane mode of the top platform |

** Accelerometers fixed on the top platform
*** Introduction                                                    :ignore:

#+name: fig:acc_top_plat_top_view
#+caption: Accelerometers fixed on the top platform
#+attr_latex: :width \linewidth
[[file:figs/acc_top_plat_top_view.jpg]]

*** 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
addpath('./matlab/mat/');
addpath('./matlab/src/');
addpath('./matlab/');
#+end_src

#+begin_src matlab :eval no
addpath('./mat/');
addpath('./src/');
#+end_src

#+begin_src matlab :tangle no
%% Add all useful folders to the path
addpath('matlab/nass-simscape/matlab/nano_hexapod/')
addpath('matlab/nass-simscape/STEPS/nano_hexapod/')
addpath('matlab/nass-simscape/STEPS/png/')
addpath('matlab/nass-simscape/src/')
addpath('matlab/nass-simscape/mat/')
#+end_src

#+begin_src matlab :eval no
%% Add all useful folders to the path
addpath('nass-simscape/matlab/nano_hexapod/')
addpath('nass-simscape/STEPS/nano_hexapod/')
addpath('nass-simscape/STEPS/png/')
addpath('nass-simscape/src/')
addpath('nass-simscape/mat/')
#+end_src

#+begin_src matlab
%% Open Simulink Model
mdl = 'nano_hexapod_simscape';

options = linearizeOptions;
options.SampleTime = 0;

Rx = zeros(1, 7);

open(mdl)
#+end_src

*** Experimental Identification
#+begin_src matlab
%% Load Identification Data
meas_acc = {};

for i = 1:6
    meas_acc(i) = {load(sprintf('mat/meas_acc_top_plat_strut_%i.mat', i), 't', 'Va', 'de', 'Am')};
end
#+end_src

#+begin_src matlab
%% Setup useful variables
% Sampling Time [s]
Ts = (meas_acc{1}.t(end) - (meas_acc{1}.t(1)))/(length(meas_acc{1}.t)-1);

% Sampling Frequency [Hz]
Fs = 1/Ts;

% Hannning Windows
win = hanning(ceil(1*Fs));

% And we get the frequency vector
[~, f] = tfestimate(meas_acc{1}.Va, meas_acc{1}.de, win, [], [], 1/Ts);
#+end_src

The sensibility of the accelerometers are $0.1 V/g \approx 0.01 V/(m/s^2)$.
#+begin_src matlab
%% Compute the 6x6 transfer function matrix
G_acc = zeros(length(f), 6, 6);

for i = 1:6
    G_acc(:,:,i) = tfestimate(meas_acc{i}.Va, 1/0.01*meas_acc{i}.Am, win, [], [], 1/Ts);
end
#+end_src

*** Location and orientation of accelerometers
#+begin_src matlab
Opm = [ 0.047, -0.112, 10e-3;
        0.047, -0.112, 10e-3;
       -0.113,  0.011, 10e-3;
       -0.113,  0.011, 10e-3;
        0.040,  0.113, 10e-3;
        0.040,  0.113, 10e-3]';

Osm = [-1,  0, 0;
        0,  0, 1;
        0, -1, 0;
        0,  0, 1;
       -1,  0, 0;
        0,  0, 1]';

#+end_src

*** COM
#+begin_src matlab
Hbm = -15e-3;

M = getTransformationMatrixAcc(Opm-[0;0;Hbm], Osm);
J = getJacobianNanoHexapod(Hbm);
#+end_src

#+begin_src matlab
G_acc_CoM = zeros(size(G_acc));

for i = 1:length(f)
    G_acc_CoM(i, :, :) = inv(M)*squeeze(G_acc(i, :, :))*inv(J');
end
#+end_src

#+begin_src matlab :exports none
labels = {'$D_x/F_{x}$', '$D_y/F_{y}$', '$D_z/F_{z}$', '$R_{x}/M_{x}$', '$R_{y}/M_{y}$', '$R_{R}/M_{z}$'};

figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');

ax1 = nexttile([2,1]);
hold on;
for i = 1:2
    for j = i+1:3
        plot(f, abs(G_acc_CoM(:, i, j)./(-(2*pi*f).^2)), 'color', [0, 0, 0, 0.2], ...
             'HandleVisibility', 'off');
    end
end
for i =1:3
    set(gca,'ColorOrderIndex',i)
    plot(f, abs(G_acc_CoM(:,i , i)./(-(2*pi*f).^2)), ...
         'DisplayName', labels{i});
end
plot(f, abs(G_acc_CoM(:, 1, 2)./(-(2*pi*f).^2)), 'color', [0, 0, 0, 0.2], ...
     'DisplayName', '$D_i/F_j$');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude $A_m/V_a$ [m/V]'); set(gca, 'XTickLabel',[]);
ylim([1e-9, 1e-5]);
legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 3);

ax2 = nexttile;
hold on;
for i =1:3
    set(gca,'ColorOrderIndex',i)
    plot(f, 180/pi*angle(G_acc_CoM(:,i , i)./(-(2*pi*f).^2)));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
yticks(-360:90:360);

linkaxes([ax1,ax2],'x');
xlim([20, 2e3]);
#+end_src

#+begin_src matlab :exports none
labels = {'$D_x/F_{x}$', '$D_y/F_{y}$', '$D_z/F_{z}$', '$R_{x}/M_{x}$', '$R_{y}/M_{y}$', '$R_{R}/M_{z}$'};

figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');

ax1 = nexttile([2,1]);
hold on;
for i =1:6
    set(gca,'ColorOrderIndex',i)
    plot(f, abs(G_acc_CoM(:,i , i)./(-(2*pi*f).^2)), ...
         'DisplayName', labels{i});
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude $A_m/V_a$ [m/V]'); set(gca, 'XTickLabel',[]);
ylim([1e-9, 1e-3]);
legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 3);

ax2 = nexttile;
hold on;
for i =1:6
    set(gca,'ColorOrderIndex',i)
    plot(f, 180/pi*angle(G_acc_CoM(:,i , i)./(-(2*pi*f).^2)));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
yticks(-360:90:360);

linkaxes([ax1,ax2],'x');
xlim([50, 5e2]);
#+end_src

*** COK
#+begin_src matlab
Hbm = -42.3e-3;

M = getTransformationMatrixAcc(Opm-[0;0;Hbm], Osm);
J = getJacobianNanoHexapod(Hbm);
#+end_src

#+begin_src matlab
G_acc_CoK = zeros(size(G_acc));

for i = 1:length(f)
    G_acc_CoK(i, :, :) = inv(M)*squeeze(G_acc(i, :, :))*inv(J');
end
#+end_src

#+begin_src matlab :exports none
labels = {'$D_x/F_{x}$', '$D_y/F_{y}$', '$D_z/F_{z}$', '$R_{x}/M_{x}$', '$R_{y}/M_{y}$', '$R_{R}/M_{z}$'};

figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');

ax1 = nexttile([2,1]);
hold on;
for i = 1:2
    for j = i+1:3
        plot(f, abs(G_acc_CoK(:, i, j)./(-(2*pi*f).^2)), 'color', [0, 0, 0, 0.2], ...
             'HandleVisibility', 'off');
    end
end
for i =1:3
    set(gca,'ColorOrderIndex',i)
    plot(f, abs(G_acc_CoK(:,i , i)./(-(2*pi*f).^2)), ...
         'DisplayName', labels{i});
end
plot(f, abs(G_acc_CoK(:, 1, 2)./(-(2*pi*f).^2)), 'color', [0, 0, 0, 0.2], ...
     'DisplayName', '$D_i/F_j$');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude $A_m/V_a$ [m/V]'); set(gca, 'XTickLabel',[]);
ylim([1e-9, 1e-5]);
legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 3);

ax2 = nexttile;
hold on;
for i =1:3
    set(gca,'ColorOrderIndex',i)
    plot(f, 180/pi*angle(G_acc_CoK(:,i , i)./(-(2*pi*f).^2)));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
yticks(-360:90:360);

linkaxes([ax1,ax2],'x');
xlim([20, 2e3]);
#+end_src

#+begin_src matlab :exports none
labels = {'$D_x/\mathcal{F}_x$', '$D_y/\mathcal{F}_y$', '$D_z/\mathcal{F}_z$', ...
          '$R_x/\mathcal{M}_x$', '$R_y/\mathcal{M}_y$', '$R_z/\mathcal{M}_z$'};

figure;
hold on;
for i =1:6
    set(gca,'ColorOrderIndex',i)
    plot(f, abs(G_acc_CoK(:,i,i)./(-(2*pi*f).^2)), ...
         'DisplayName', labels{i});
end
plot(f, abs(G_acc_CoK(:,1,2)./(-(2*pi*f).^2)), 'color', [0, 0, 0, 0.2], ...
    'DisplayName', 'Off-Diagonal');
for i = 1:5
    for j = i+1:6
        plot(f, abs(G_acc_CoK(:,i,j)./(-(2*pi*f).^2)), 'color', [0, 0, 0, 0.2], ...
             'HandleVisibility', 'off');
    end
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Amplitude $X_m/V_a$ [m/V]');
xlim([50, 5e2]); ylim([1e-7, 1e-1]);
legend('location', 'southwest');
#+end_src

*** Comp with the Simscape Model
#+begin_src matlab
n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
                                       'flex_top_type', '4dof', ...
                                       'motion_sensor_type', 'struts', ...
                                       'actuator_type', 'flexible', ...
                                       'MO_B', -42.3e-3);
#+end_src

#+begin_src matlab
%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/du'],  1, 'openinput');  io_i = io_i + 1; % Actuator Inputs
io(io_i) = linio([mdl, '/D'],  1, 'openoutput'); io_i = io_i + 1; % Relative Motion Outputs

G = linearize(mdl, io, 0.0, options);
G.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
G.OutputName = {'D1', 'D2', 'D3', 'D4', 'D5', 'D6'};
#+end_src

Then use the Jacobian matrices to obtain the "cartesian" centralized plant.
#+begin_src matlab
Gc = inv(n_hexapod.geometry.J)*...
     G*...
     inv(n_hexapod.geometry.J');
#+end_src

#+begin_src matlab :exports none
freqs = 2*logspace(1, 3, 1000);

labels = {'$D_x/\mathcal{F}_x$', '$D_y/\mathcal{F}_y$', '$D_z/\mathcal{F}_z$', ...
          '$R_x/\mathcal{M}_x$', '$R_y/\mathcal{M}_y$', '$R_z/\mathcal{M}_z$'};

figure;
hold on;
for i = 1:6
    plot(freqs, abs(squeeze(freqresp(Gc(i,i), freqs, 'Hz'))), '-', ...
         'DisplayName', labels{i});
end
plot(freqs, abs(squeeze(freqresp(Gc(1, 2), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ...
    'DisplayName', 'Off-Diagonal');
for i = 1:5
    for j = i+1:6
        plot(freqs, abs(squeeze(freqresp(Gc(i, j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ...
             'HandleVisibility', 'off');
    end
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Amplitude [m/N,rad/N/m]');
xlim([50, 5e2]); ylim([1e-7, 1e-1]);
legend('location', 'southwest');
#+end_src

* Encoders fixed to the plates
<<sec:encoders_plates>>

** Introduction                                                      :ignore:
In this section, the encoders are fixed to the plates rather than to the struts as shown in Figure [[fig:enc_fixed_to_struts]].

#+name: fig:enc_fixed_to_struts
#+caption: Nano-Hexapod with encoders fixed to the struts
#+attr_latex: :width \linewidth
[[file:figs/IMG_20210625_083801.jpg]]

It is structured as follow:
- Section [[sec:enc_plates_plant_id]]: The dynamics of the nano-hexapod is identified
- Section [[sec:enc_plates_comp_simscape]]: The identified dynamics is compared with the Simscape model
- Section [[sec:enc_plates_iff]]: The Integral Force Feedback (IFF) control strategy is applied and the dynamics of the damped nano-hexapod is identified and compare with the Simscape model
- Section [[sec:hac_iff_struts]]: The High Authority Control (HAC) in the frame of the struts is developed
- Section [[sec:hac_iff_struts_ref_track]]: Some reference tracking tests are performed in order to experimentally validate the HAC-LAC control strategy.

** Identification of the dynamics
<<sec:enc_plates_plant_id>>
*** 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
addpath('./matlab/mat/');
addpath('./matlab/src/');
addpath('./matlab/');
#+end_src

#+begin_src matlab :eval no
addpath('./mat/');
addpath('./src/');
#+end_src

*** Load Measurement Data
#+begin_src matlab
%% Load Identification Data
meas_data_lf = {};

for i = 1:6
    meas_data_lf(i) = {load(sprintf('mat/frf_exc_strut_%i_enc_plates_noise.mat', i), 't', 'Va', 'Vs', 'de')};
end
#+end_src

*** Spectral Analysis - Setup
#+begin_src matlab
%% Setup useful variables
% Sampling Time [s]
Ts = (meas_data_lf{1}.t(end) - (meas_data_lf{1}.t(1)))/(length(meas_data_lf{1}.t)-1);

% Sampling Frequency [Hz]
Fs = 1/Ts;

% Hannning Windows
win = hanning(ceil(1*Fs));

% And we get the frequency vector
[~, f] = tfestimate(meas_data_lf{1}.Va, meas_data_lf{1}.de, win, [], [], 1/Ts);
#+end_src

*** DVF Plant
First, let's compute the coherence from the excitation voltage and the displacement as measured by the encoders (Figure [[fig:enc_plates_dvf_coh]]).

#+begin_src matlab
%% Coherence
coh_dvf = zeros(length(f), 6, 6);

for i = 1:6
    coh_dvf(:, :, i) = mscohere(meas_data_lf{i}.Va, meas_data_lf{i}.de, win, [], [], 1/Ts);
end
#+end_src

#+begin_src matlab :exports none
%% Coherence for the transfer function from u to dLm
figure;
hold on;
for i = 1:5
    for j = i+1:6
        plot(f, coh_dvf(:, i, j), 'color', [0, 0, 0, 0.2], ...
             'HandleVisibility', 'off');
    end
end
for i =1:6
    set(gca,'ColorOrderIndex',i)
    plot(f, coh_dvf(:, i, i), ...
         'DisplayName', sprintf('$G_{dvf}(%i,%i)$', i, i));
end
plot(f, coh_dvf(:, 1, 2), 'color', [0, 0, 0, 0.2], ...
     'DisplayName', '$G_{dvf}(i,j)$');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Coherence [-]');
xlim([20, 2e3]); ylim([0, 1]);
legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 3);
#+end_src

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

#+name: fig:enc_plates_dvf_coh
#+caption: Obtained coherence for the DVF plant
#+RESULTS:
[[file:figs/enc_plates_dvf_coh.png]]

Then the 6x6 transfer function matrix is estimated (Figure [[fig:enc_plates_dvf_frf]]).
#+begin_src matlab
%% DVF Plant (transfer function from u to dLm)
G_dvf = zeros(length(f), 6, 6);

for i = 1:6
    G_dvf(:,:,i) = tfestimate(meas_data_lf{i}.Va, meas_data_lf{i}.de, win, [], [], 1/Ts);
end
#+end_src

#+begin_src matlab :exports none
%% Bode plot for the transfer function from u to dLm
figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');

ax1 = nexttile([2,1]);
hold on;
for i = 1:5
    for j = i+1:6
        plot(f, abs(G_dvf(:, i, j)), 'color', [0, 0, 0, 0.2], ...
             'HandleVisibility', 'off');
    end
end
for i =1:6
    set(gca,'ColorOrderIndex',i)
    plot(f, abs(G_dvf(:,i, i)), ...
         'DisplayName', sprintf('$G_{dvf}(%i,%i)$', i, i));
    set(gca,'ColorOrderIndex',i)
    plot(f, abs(G_dvf(:,i, i)), ...
         'HandleVisibility', 'off');
end
plot(f, abs(G_dvf(:, 1, 2)), 'color', [0, 0, 0, 0.2], ...
     'DisplayName', '$G_{dvf}(i,j)$');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude $d_e/V_a$ [m/V]'); set(gca, 'XTickLabel',[]);
ylim([1e-9, 1e-3]);
legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 3);

ax2 = nexttile;
hold on;
for i =1:6
    set(gca,'ColorOrderIndex',i)
    plot(f, 180/pi*angle(G_dvf(:,i, i)));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
yticks(-360:90:360);

linkaxes([ax1,ax2],'x');
xlim([20, 2e3]);
#+end_src

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

#+name: fig:enc_plates_dvf_frf
#+caption: Measured FRF for the DVF plant
#+RESULTS:
[[file:figs/enc_plates_dvf_frf.png]]

*** IFF Plant
First, let's compute the coherence from the excitation voltage and the displacement as measured by the encoders (Figure [[fig:enc_plates_iff_coh]]).

#+begin_src matlab
%% Coherence for the IFF plant
coh_iff = zeros(length(f), 6, 6);

for i = 1:6
    coh_iff(:,:,i) = mscohere(meas_data_lf{i}.Va, meas_data_lf{i}.Vs, win, [], [], 1/Ts);
end
#+end_src

#+begin_src matlab :exports none
%% Coherence of the IFF Plant (transfer function from u to taum)
figure;
hold on;
for i = 1:5
    for j = i+1:6
        plot(f, coh_iff(:, i, j), 'color', [0, 0, 0, 0.2], ...
             'HandleVisibility', 'off');
    end
end
for i =1:6
    set(gca,'ColorOrderIndex',i)
    plot(f, coh_iff(:,i, i), ...
         'DisplayName', sprintf('$G_{iff}(%i,%i)$', i, i));
end
plot(f, coh_iff(:, 1, 2), 'color', [0, 0, 0, 0.2], ...
     'DisplayName', '$G_{iff}(i,j)$');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Coherence [-]');
xlim([20, 2e3]); ylim([0, 1]);
legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 3);
#+end_src

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

#+name: fig:enc_plates_iff_coh
#+caption: Obtained coherence for the IFF plant
#+RESULTS:
[[file:figs/enc_plates_iff_coh.png]]

Then the 6x6 transfer function matrix is estimated (Figure [[fig:enc_plates_iff_frf]]).
#+begin_src matlab
%% IFF Plant
G_iff = zeros(length(f), 6, 6);

for i = 1:6
    G_iff(:,:,i) = tfestimate(meas_data_lf{i}.Va, meas_data_lf{i}.Vs, win, [], [], 1/Ts);
end
#+end_src

#+begin_src matlab :exports none
%% Bode plot of the IFF Plant (transfer function from u to taum)
figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');

ax1 = nexttile([2,1]);
hold on;
for i = 1:5
    for j = i+1:6
        plot(f, abs(G_iff(:, i, j)), 'color', [0, 0, 0, 0.2], ...
             'HandleVisibility', 'off');
    end
end
for i =1:6
    set(gca,'ColorOrderIndex',i)
    plot(f, abs(G_iff(:,i , i)), ...
         'DisplayName', sprintf('$G_{iff}(%i,%i)$', i, i));
end
plot(f, abs(G_iff(:, 1, 2)), 'color', [0, 0, 0, 0.2], ...
     'DisplayName', '$G_{iff}(i,j)$');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude $V_s/V_a$ [V/V]'); set(gca, 'XTickLabel',[]);
legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 3);
ylim([1e-3, 1e2]);

ax2 = nexttile;
hold on;
for i =1:6
    set(gca,'ColorOrderIndex',i)
    plot(f, 180/pi*angle(G_iff(:,i, i)));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
yticks(-360:90:360);

linkaxes([ax1,ax2],'x');
xlim([20, 2e3]);
#+end_src

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

#+name: fig:enc_plates_iff_frf
#+caption: Measured FRF for the IFF plant
#+RESULTS:
[[file:figs/enc_plates_iff_frf.png]]

*** Save Identified Plants
#+begin_src matlab :tangle no
save('matlab/mat/identified_plants_enc_plates.mat', 'f', 'Ts', 'G_iff', 'G_dvf')
#+end_src

#+begin_src matlab :exports none :eval no
save('mat/identified_plants_enc_plates.mat', 'f', 'Ts', 'G_iff', 'G_dvf')
#+end_src

** Comparison with the Simscape Model
<<sec:enc_plates_comp_simscape>>
*** Introduction                                                    :ignore:
In this section, the measured dynamics is compared with the dynamics estimated from the Simscape model.

*** 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
addpath('./matlab/mat/');
addpath('./matlab/src/');
addpath('./matlab/');
#+end_src

#+begin_src matlab :eval no
addpath('./mat/');
addpath('./src/');
#+end_src

#+begin_src matlab :tangle no
%% Add all useful folders to the path
addpath('matlab/nass-simscape/matlab/nano_hexapod/')
addpath('matlab/nass-simscape/STEPS/nano_hexapod/')
addpath('matlab/nass-simscape/STEPS/png/')
addpath('matlab/nass-simscape/src/')
addpath('matlab/nass-simscape/mat/')
#+end_src

#+begin_src matlab :eval no
%% Add all useful folders to the path
addpath('nass-simscape/matlab/nano_hexapod/')
addpath('nass-simscape/STEPS/nano_hexapod/')
addpath('nass-simscape/STEPS/png/')
addpath('nass-simscape/src/')
addpath('nass-simscape/mat/')
#+end_src

#+begin_src matlab
%% Open Simulink Model
mdl = 'nano_hexapod_simscape';

options = linearizeOptions;
options.SampleTime = 0;

Rx = zeros(1, 7);

open(mdl)
#+end_src

*** Load measured FRF
#+begin_src matlab
%% Load data
load('identified_plants_enc_plates.mat', 'f', 'Ts', 'G_iff', 'G_dvf')
#+end_src

*** Dynamics from Actuator to Force Sensors
#+begin_src matlab
%% Initialize Nano-Hexapod
n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
                                       'flex_top_type', '4dof', ...
                                       'motion_sensor_type', 'plates', ...
                                       'actuator_type', 'flexible');
#+end_src

#+begin_src matlab
%% Identify the IFF Plant (transfer function from u to taum)
clear io; io_i = 1;
io(io_i) = linio([mdl, '/du'],  1, 'openinput');   io_i = io_i + 1; % Actuator Inputs
io(io_i) = linio([mdl, '/dum'],  1, 'openoutput'); io_i = io_i + 1; % Force Sensors

Giff = exp(-s*Ts)*linearize(mdl, io, 0.0, options);
#+end_src

#+begin_src matlab :exports none
%% Comparison of the plants (encoder output) when tuning the misalignment
freqs = 2*logspace(1, 3, 1000);

i_input = 1;

figure;
tiledlayout(2, 3, 'TileSpacing', 'Compact', 'Padding', 'None');

ax1 = nexttile();
hold on;
plot(f, abs(G_iff(:, 1, i_input)));
plot(freqs, abs(squeeze(freqresp(Giff(1, i_input), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
set(gca, 'XTickLabel',[]); ylabel('Amplitude [m/V]');
title(sprintf('$d\\tau_{m1}/u_{%i}$', i_input));

ax2 = nexttile();
hold on;
plot(f, abs(G_iff(:, 2, i_input)));
plot(freqs, abs(squeeze(freqresp(Giff(2, i_input), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]);
title(sprintf('$d\\tau_{m2}/u_{%i}$', i_input));

ax3 = nexttile();
hold on;
plot(f, abs(G_iff(:, 3, i_input)), ...
     'DisplayName', 'Meas.');
plot(freqs, abs(squeeze(freqresp(Giff(3, i_input), freqs, 'Hz'))), ...
     'DisplayName', 'Model');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]);
legend('location', 'southeast', 'FontSize', 8);
title(sprintf('$d\\tau_{m3}/u_{%i}$', i_input));

ax4 = nexttile();
hold on;
plot(f, abs(G_iff(:, 4, i_input)));
plot(freqs, abs(squeeze(freqresp(Giff(4, i_input), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Amplitude [m/V]');
title(sprintf('$d\\tau_{m4}/u_{%i}$', i_input));

ax5 = nexttile();
hold on;
plot(f, abs(G_iff(:, 5, i_input)));
plot(freqs, abs(squeeze(freqresp(Giff(5, i_input), freqs, 'Hz'))));
hold off;
xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]);
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
title(sprintf('$d\\tau_{m5}/u_{%i}$', i_input));

ax6 = nexttile();
hold on;
plot(f, abs(G_iff(:, 6, i_input)));
plot(freqs, abs(squeeze(freqresp(Giff(6, i_input), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]);
title(sprintf('$d\\tau_{m6}/u_{%i}$', i_input));

linkaxes([ax1,ax2,ax3,ax4,ax5,ax6],'xy');
xlim([20, 2e3]); ylim([1e-2, 1e2]);
#+end_src

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

#+name: fig:enc_plates_iff_comp_simscape_all
#+caption: IFF Plant for the first actuator input and all the force senosrs
#+RESULTS:
[[file:figs/enc_plates_iff_comp_simscape_all.png]]

#+begin_src matlab :exports none
%% Bode plot of the identified IFF Plant (Simscape) and measured FRF data
freqs = 2*logspace(1, 3, 1000);

figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');

ax1 = nexttile([2,1]);
hold on;
plot(f, abs(G_iff(:,1, 1)), 'color', [0,0,0,0.2], ...
     'DisplayName', '$\tau_{m,i}/u_i$ - FRF')
for i = 2:6
    set(gca,'ColorOrderIndex',2)
    plot(f, abs(G_iff(:,i, i)), 'color', [0,0,0,0.2], ...
         'HandleVisibility', 'off');
end
set(gca,'ColorOrderIndex',2);
plot(freqs, abs(squeeze(freqresp(Giff(1,1), freqs, 'Hz'))), '-', ...
     'DisplayName', '$\tau_{m,i}/u_i$ - Model')
for i = 2:6
    set(gca,'ColorOrderIndex',2);
    plot(freqs, abs(squeeze(freqresp(Giff(i,i), freqs, 'Hz'))), '-', ...
         'HandleVisibility', 'off');
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [V/V]'); set(gca, 'XTickLabel',[]);
legend('location', 'southeast');

ax2 = nexttile;
hold on;
for i = 1:6
    plot(f, 180/pi*angle(G_iff(:,i, i)), 'color', [0,0,0,0.2]);
end
for i = 1:6
    set(gca,'ColorOrderIndex',2);
    plot(freqs, 180/pi*angle(squeeze(freqresp(Giff(i,i), freqs, 'Hz'))), '-');
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-180, 180]);
yticks([-180, -90, 0, 90, 180]);

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

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

#+name: fig:enc_plates_iff_comp_simscape
#+caption: Diagonal elements of the IFF Plant
#+RESULTS:
[[file:figs/enc_plates_iff_comp_simscape.png]]

#+begin_src matlab :exports none
%% Bode plot of the identified IFF Plant (Simscape) and measured FRF data (off-diagonal elements)
freqs = 2*logspace(1, 3, 1000);

figure;
hold on;
% Off diagonal terms
plot(f, abs(G_iff(:, 1, 2)), 'color', [0,0,0,0.2], ...
     'DisplayName', '$\tau_{m,i}/u_j$ - FRF')
for i = 1:5
    for j = i+1:6
        plot(f, abs(G_iff(:, i, j)), 'color', [0,0,0,0.2], ...
             'HandleVisibility', 'off');
    end
end
set(gca,'ColorOrderIndex',2);
plot(freqs, abs(squeeze(freqresp(Giff(1, 2), freqs, 'Hz'))), ...
     'DisplayName', '$\tau_{m,i}/u_j$ - Model')
for i = 1:5
    for j = i+1:6
        set(gca,'ColorOrderIndex',2);
        plot(freqs, abs(squeeze(freqresp(Giff(i, j), freqs, 'Hz'))), ...
             'HandleVisibility', 'off');
    end
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Amplitude [V/V]');
xlim([freqs(1), freqs(end)]); ylim([1e-3, 1e2]);
legend('location', 'northeast');
#+end_src

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

#+name: fig:enc_plates_iff_comp_offdiag_simscape
#+caption: Off diagonal elements of the IFF Plant
#+RESULTS:
[[file:figs/enc_plates_iff_comp_offdiag_simscape.png]]

*** Dynamics from Actuator to Encoder
#+begin_src matlab
%% Initialization of the Nano-Hexapod
n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
                                       'flex_top_type', '4dof', ...
                                       'motion_sensor_type', 'plates', ...
                                       'actuator_type', 'flexible');
#+end_src

#+begin_src matlab
%% Identify the DVF Plant (transfer function from u to dLm)
clear io; io_i = 1;
io(io_i) = linio([mdl, '/du'],  1, 'openinput');  io_i = io_i + 1; % Actuator Inputs
io(io_i) = linio([mdl, '/dL'], 1, 'openoutput'); io_i = io_i + 1; % Encoders

Gdvf = exp(-s*Ts)*linearize(mdl, io, 0.0, options);
#+end_src

#+begin_src matlab :exports none
%% Comparison of the plants (encoder output) when tuning the misalignment
freqs = 2*logspace(1, 3, 1000);

i_input = 3;

figure;
tiledlayout(2, 3, 'TileSpacing', 'Compact', 'Padding', 'None');

ax1 = nexttile();
hold on;
plot(f, abs(G_dvf(:, 1, i_input)));
plot(freqs, abs(squeeze(freqresp(Gdvf(1, i_input), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
set(gca, 'XTickLabel',[]); ylabel('Amplitude [m/V]');
title(sprintf('$d\\mathcal{L}_{m1}/u_{%i}$', i_input));

ax2 = nexttile();
hold on;
plot(f, abs(G_dvf(:, 2, i_input)));
plot(freqs, abs(squeeze(freqresp(Gdvf(2, i_input), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]);
title(sprintf('$d\\mathcal{L}_{m2}/u_{%i}$', i_input));

ax3 = nexttile();
hold on;
plot(f, abs(G_dvf(:, 3, i_input)), ...
     'DisplayName', 'Meas.');
plot(freqs, abs(squeeze(freqresp(Gdvf(3, i_input), freqs, 'Hz'))), ...
     'DisplayName', 'Model');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]);
legend('location', 'southeast', 'FontSize', 8);
title(sprintf('$d\\mathcal{L}_{m3}/u_{%i}$', i_input));

ax4 = nexttile();
hold on;
plot(f, abs(G_dvf(:, 4, i_input)));
plot(freqs, abs(squeeze(freqresp(Gdvf(4, i_input), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Amplitude [m/V]');
title(sprintf('$d\\mathcal{L}_{m4}/u_{%i}$', i_input));

ax5 = nexttile();
hold on;
plot(f, abs(G_dvf(:, 5, i_input)));
plot(freqs, abs(squeeze(freqresp(Gdvf(5, i_input), freqs, 'Hz'))));
hold off;
xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]);
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
title(sprintf('$d\\mathcal{L}_{m5}/u_{%i}$', i_input));

ax6 = nexttile();
hold on;
plot(f, abs(G_dvf(:, 6, i_input)));
plot(freqs, abs(squeeze(freqresp(Gdvf(6, i_input), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]);
title(sprintf('$d\\mathcal{L}_{m6}/u_{%i}$', i_input));

linkaxes([ax1,ax2,ax3,ax4,ax5,ax6],'xy');
xlim([40, 4e2]); ylim([1e-8, 1e-2]);
#+end_src

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

#+name: fig:enc_plates_dvf_comp_simscape_all
#+caption: DVF Plant for the first actuator input and all the encoders
#+RESULTS:
[[file:figs/enc_plates_dvf_comp_simscape_all.png]]

#+begin_src matlab :exports none
%% Diagonal elements of the DVF plant
freqs = 2*logspace(1, 3, 1000);

figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');

ax1 = nexttile([2,1]);
hold on;
plot(f, abs(G_dvf(:,1, 1)), 'color', [0,0,0,0.2], ...
     'DisplayName', '$d\mathcal{L}_{m,i}/u_i$ - FRF')
for i = 2:6
    set(gca,'ColorOrderIndex',2)
    plot(f, abs(G_dvf(:,i, i)), 'color', [0,0,0,0.2], ...
         'HandleVisibility', 'off');
end
set(gca,'ColorOrderIndex',2);
plot(freqs, abs(squeeze(freqresp(Gdvf(1,1), freqs, 'Hz'))), '-', ...
     'DisplayName', '$d\mathcal{L}_{m,i}/u_i$ - Model')
for i = 2:6
    set(gca,'ColorOrderIndex',2);
    plot(freqs, abs(squeeze(freqresp(Gdvf(i,i), freqs, 'Hz'))), '-', ...
         'HandleVisibility', 'off');
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/V]'); set(gca, 'XTickLabel',[]);
ylim([1e-8, 1e-3]);
legend('location', 'northeast');

ax2 = nexttile;
hold on;
for i = 1:6
    plot(f, 180/pi*angle(G_dvf(:,i, i)), 'color', [0,0,0,0.2]);
end
for i = 1:6
    set(gca,'ColorOrderIndex',2);
    plot(freqs, 180/pi*angle(squeeze(freqresp(Gdvf(i,i), freqs, 'Hz'))), '-');
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-180, 180]);
yticks([-180, -90, 0, 90, 180]);

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

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

#+name: fig:enc_plates_dvf_comp_simscape
#+caption: Diagonal elements of the DVF Plant
#+RESULTS:
[[file:figs/enc_plates_dvf_comp_simscape.png]]

#+begin_src matlab :exports none
%% Off-diagonal elements of the DVF plant
freqs = 2*logspace(1, 3, 1000);

figure;
hold on;
% Off diagonal terms
plot(f, abs(G_dvf(:, 1, 2)), 'color', [0,0,0,0.2], ...
     'DisplayName', '$d\mathcal{L}_{m,i}/u_j$ - FRF')
for i = 1:5
    for j = i+1:6
        plot(f, abs(G_dvf(:, i, j)), 'color', [0,0,0,0.2], ...
             'HandleVisibility', 'off');
    end
end
set(gca,'ColorOrderIndex',2);
plot(freqs, abs(squeeze(freqresp(Gdvf(1, 2), freqs, 'Hz'))), ...
     'DisplayName', '$d\mathcal{L}_{m,i}/u_j$ - Model')
for i = 1:5
    for j = i+1:6
        set(gca,'ColorOrderIndex',2);
        plot(freqs, abs(squeeze(freqresp(Gdvf(i, j), freqs, 'Hz'))), ...
             'HandleVisibility', 'off');
    end
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Amplitude [m/V]');
xlim([freqs(1), freqs(end)]); ylim([1e-8, 1e-3]);
legend('location', 'northeast');
#+end_src

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

#+name: fig:enc_plates_dvf_comp_offdiag_simscape
#+caption: Off diagonal elements of the DVF Plant
#+RESULTS:
[[file:figs/enc_plates_dvf_comp_offdiag_simscape.png]]

** Integral Force Feedback
<<sec:enc_plates_iff>>
*** Introduction                                                    :ignore:


#+begin_src latex :file control_architecture_iff.pdf
  \begin{tikzpicture}
    % Blocs
    \node[block={3.0cm}{3.0cm}] (P) {Plant};
    \coordinate[] (inputF) at ($(P.south west)!0.5!(P.north west)$);
    \coordinate[] (outputF) at ($(P.south east)!0.8!(P.north east)$);
    \coordinate[] (outputX) at ($(P.south east)!0.5!(P.north east)$);
    \coordinate[] (outputL) at ($(P.south east)!0.2!(P.north east)$);

    \node[block, above=0.4 of P] (Kiff) {$\bm{K}_\text{IFF}$};
    \node[addb={+}{}{-}{}{}, left= of inputF] (addF) {};

    % Connections and labels
    \draw[->] (outputF) -- ++(1, 0) node[below left]{$\bm{\tau}_m$};
    \draw[->] (outputL) -- ++(1, 0) node[above left]{$d\bm{\mathcal{L}}$};
    \draw[->] (outputX) -- ++(1, 0) node[above left]{$\bm{\mathcal{X}}$};

    \draw[->] ($(outputF) + (0.6, 0)$)node[branch]{} |- (Kiff.east);
    \draw[->] (Kiff.west) -| (addF.north);
    \draw[->] (addF.east) -- (inputF) node[above left]{$\bm{u}$};
    \draw[<-] (addF.west) -- ++(-1, 0) node[above right]{$\bm{u}^\prime$};
  \end{tikzpicture}
#+end_src

#+name: fig:control_architecture_iff
#+caption: Integral Force Feedback Strategy
#+RESULTS:
[[file:figs/control_architecture_iff.png]]

*** 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
addpath('./matlab/mat/');
addpath('./matlab/src/');
addpath('./matlab/');
#+end_src

#+begin_src matlab :eval no
addpath('./mat/');
addpath('./src/');
#+end_src

#+begin_src matlab
load('identified_plants_enc_plates.mat', 'f', 'Ts', 'G_iff', 'G_dvf')
#+end_src

#+begin_src matlab :tangle no
%% Add all useful folders to the path
addpath('matlab/nass-simscape/matlab/nano_hexapod/')
addpath('matlab/nass-simscape/STEPS/nano_hexapod/')
addpath('matlab/nass-simscape/STEPS/png/')
addpath('matlab/nass-simscape/src/')
addpath('matlab/nass-simscape/mat/')
#+end_src

#+begin_src matlab :eval no
%% Add all useful folders to the path
addpath('nass-simscape/matlab/nano_hexapod/')
addpath('nass-simscape/STEPS/nano_hexapod/')
addpath('nass-simscape/STEPS/png/')
addpath('nass-simscape/src/')
addpath('nass-simscape/mat/')
#+end_src

#+begin_src matlab
%% Open Simulink Model
mdl = 'nano_hexapod_simscape';

options = linearizeOptions;
options.SampleTime = 0;

open(mdl)

Rx = zeros(1, 7);

colors = colororder;
#+end_src

*** Identification of the IFF Plant
#+begin_src matlab
%% Initialize Nano-Hexapod
n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
                                       'flex_top_type', '4dof', ...
                                       'motion_sensor_type', 'plates', ...
                                       'actuator_type', '2dof');
#+end_src

#+begin_src matlab
%% Identify the IFF Plant (transfer function from u to taum)
clear io; io_i = 1;
io(io_i) = linio([mdl, '/du'],  1, 'openinput');   io_i = io_i + 1; % Actuator Inputs
io(io_i) = linio([mdl, '/Fm'],  1, 'openoutput'); io_i = io_i + 1; % Force Sensors

Giff = exp(-s*Ts)*linearize(mdl, io, 0.0, options);
#+end_src

*** Effect of IFF on the plant - Simscape Model
#+begin_src matlab
load('Kiff.mat', 'Kiff')
#+end_src

#+begin_src matlab
%% Identify the (damped) transfer function from u to dLm for different values of the IFF gain
clear io; io_i = 1;
io(io_i) = linio([mdl, '/du'],  1, 'openinput');  io_i = io_i + 1; % Actuator Inputs
io(io_i) = linio([mdl, '/dL'], 1, 'openoutput'); io_i = io_i + 1; % Plate Displacement (encoder)
#+end_src

#+begin_src matlab
%% Initialize the Simscape model in closed loop
n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
                                       'flex_top_type', '4dof', ...
                                       'motion_sensor_type', 'plates', ...
                                       'actuator_type', 'flexible');
#+end_src

#+begin_src matlab
Gd_ol = exp(-s*Ts)*linearize(mdl, io, 0.0, options);
#+end_src

#+begin_src matlab
%% Initialize the Simscape model in closed loop
n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
                                       'flex_top_type', '4dof', ...
                                       'motion_sensor_type', 'plates', ...
                                       'actuator_type', 'flexible', ...
                                       'controller_type', 'iff');
#+end_src

#+begin_src matlab
Gd_iff = exp(-s*Ts)*linearize(mdl, io, 0.0, options);
#+end_src

#+begin_src matlab :results value replace :exports results :tangle no
isstable(Gd_iff)
#+end_src

#+RESULTS:
: 1

#+begin_src matlab :exports none
%% Bode plot of the transfer function from u to dLm for tested values of the IFF gain
freqs = 2*logspace(1, 3, 1000);

figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');

ax1 = nexttile([2,1]);
hold on;
plot(freqs, abs(squeeze(freqresp(Gd_ol(1,1),  freqs, 'Hz'))), '-', ...
     'DisplayName', 'OL - Diag');
plot(freqs, abs(squeeze(freqresp(Gd_iff(1,1), freqs, 'Hz'))), '-', ...
     'DisplayName', 'IFF - Diag');
for i = 2:6
    set(gca,'ColorOrderIndex',1);
    plot(freqs, abs(squeeze(freqresp(Gd_ol(1,1), freqs, 'Hz'))), '-', ...
         'HandleVisibility', 'off');
end
for i = 2:6
    set(gca,'ColorOrderIndex',2);
    plot(freqs, abs(squeeze(freqresp(Gd_iff(i,i), freqs, 'Hz'))), '-', ...
         'HandleVisibility', 'off');
end

plot(freqs, abs(squeeze(freqresp(Gd_ol(1,2), freqs, 'Hz'))), 'color', [colors(1,:), 0.2], ...
     'DisplayName', 'OL - Off-diag')
for i = 1:5
    for j = i+1:6
        plot(freqs, abs(squeeze(freqresp(Gd_ol(i,j), freqs, 'Hz'))), 'color', [colors(1,:), 0.2], ...
             'HandleVisibility', 'off');
    end
end
plot(freqs, abs(squeeze(freqresp(Gd_iff(1,2), freqs, 'Hz'))), 'color', [colors(2,:), 0.2], ...
     'DisplayName', 'IFF - Off-diag')
for i = 1:5
    for j = i+1:6
        plot(freqs, abs(squeeze(freqresp(Gd_iff(i,j), freqs, 'Hz'))), 'color', [colors(2,:), 0.2], ...
             'HandleVisibility', 'off');
    end
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/V]'); set(gca, 'XTickLabel',[]);
legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 2);

ax2 = nexttile;
hold on;
for i = 1:6
    set(gca,'ColorOrderIndex',1);
    plot(freqs, 180/pi*angle(squeeze(freqresp(Gd_ol(1,1), freqs, 'Hz'))), '-', ...
         'HandleVisibility', 'off');
    set(gca,'ColorOrderIndex',2);
    plot(freqs, 180/pi*angle(squeeze(freqresp(Gd_iff(i,i), freqs, 'Hz'))), '-', ...
         'HandleVisibility', 'off');
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-180, 180]);
yticks([-180, -90, 0, 90, 180]);

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

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

#+name: fig:enc_plates_iff_gains_effect_dvf_plant
#+caption: Effect of the IFF control strategy on the transfer function from $\bm{\tau}$ to $d\bm{\mathcal{L}}_m$
#+RESULTS:
[[file:figs/enc_plates_iff_gains_effect_dvf_plant.png]]

*** Experimental Results - Damped Plant with Optimal gain
**** Introduction                                                  :ignore:
Let's now look at the $6 \times 6$ damped plant with the optimal gain $g = 400$.

**** Load Data
#+begin_src matlab
%% Load Identification Data
meas_iff_plates = {};

for i = 1:6
    meas_iff_plates(i) = {load(sprintf('mat/frf_exc_iff_strut_%i_enc_plates_noise.mat', i), 't', 'Va', 'Vs', 'de', 'u')};
end
#+end_src

**** Spectral Analysis - Setup
#+begin_src matlab
%% Setup useful variables
% Sampling Time [s]
Ts = (meas_iff_plates{1}.t(end) - (meas_iff_plates{1}.t(1)))/(length(meas_iff_plates{1}.t)-1);

% Sampling Frequency [Hz]
Fs = 1/Ts;

% Hannning Windows
win = hanning(ceil(1*Fs));

% And we get the frequency vector
[~, f] = tfestimate(meas_iff_plates{1}.Va, meas_iff_plates{1}.de, win, [], [], 1/Ts);
#+end_src

**** Simscape Model
#+begin_src matlab
load('Kiff.mat', 'Kiff')
#+end_src

#+begin_src matlab
%% Initialize the Simscape model in closed loop
n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
                                       'flex_top_type', '4dof', ...
                                       'motion_sensor_type', 'plates', ...
                                       'actuator_type', 'flexible', ...
                                       'controller_type', 'iff');
#+end_src

#+begin_src matlab
%% Identify the (damped) transfer function from u to dLm for different values of the IFF gain
clear io; io_i = 1;
io(io_i) = linio([mdl, '/du'], 1, 'openinput');  io_i = io_i + 1; % Actuator Inputs
io(io_i) = linio([mdl, '/dL'], 1, 'openoutput'); io_i = io_i + 1; % Plate Displacement (encoder)
#+end_src

#+begin_src matlab
Gd_iff_opt = exp(-s*Ts)*linearize(mdl, io, 0.0, options);
#+end_src

**** DVF Plant
#+begin_src matlab
%% IFF Plant
G_enc_iff_opt = zeros(length(f), 6, 6);

for i = 1:6
    G_enc_iff_opt(:,:,i) = tfestimate(meas_iff_plates{i}.Va, meas_iff_plates{i}.de, win, [], [], 1/Ts);
end
#+end_src

#+begin_src matlab :exports none
%% Comparison of the plants (encoder output) when tuning the misalignment
freqs = 2*logspace(1, 3, 1000);

i_input = 1;

figure;
tiledlayout(2, 3, 'TileSpacing', 'Compact', 'Padding', 'None');

ax1 = nexttile();
hold on;
plot(f, abs(G_enc_iff_opt(:, 1, i_input)));
plot(freqs, abs(squeeze(freqresp(Gd_iff_opt(1, i_input), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
set(gca, 'XTickLabel',[]); ylabel('Amplitude [m/V]');
title(sprintf('$d\\tau_{m1}/u_{%i}$', i_input));

ax2 = nexttile();
hold on;
plot(f, abs(G_enc_iff_opt(:, 2, i_input)));
plot(freqs, abs(squeeze(freqresp(Gd_iff_opt(2, i_input), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]);
title(sprintf('$d\\tau_{m2}/u_{%i}$', i_input));

ax3 = nexttile();
hold on;
plot(f, abs(G_enc_iff_opt(:, 3, i_input)), ...
     'DisplayName', 'Meas.');
plot(freqs, abs(squeeze(freqresp(Gd_iff_opt(3, i_input), freqs, 'Hz'))), ...
     'DisplayName', 'Model');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]);
legend('location', 'southeast', 'FontSize', 8);
title(sprintf('$d\\tau_{m3}/u_{%i}$', i_input));

ax4 = nexttile();
hold on;
plot(f, abs(G_enc_iff_opt(:, 4, i_input)));
plot(freqs, abs(squeeze(freqresp(Gd_iff_opt(4, i_input), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Amplitude [m/V]');
title(sprintf('$d\\tau_{m4}/u_{%i}$', i_input));

ax5 = nexttile();
hold on;
plot(f, abs(G_enc_iff_opt(:, 5, i_input)));
plot(freqs, abs(squeeze(freqresp(Gd_iff_opt(5, i_input), freqs, 'Hz'))));
hold off;
xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]);
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
title(sprintf('$d\\tau_{m5}/u_{%i}$', i_input));

ax6 = nexttile();
hold on;
plot(f, abs(G_enc_iff_opt(:, 6, i_input)));
plot(freqs, abs(squeeze(freqresp(Gd_iff_opt(6, i_input), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]);
title(sprintf('$d\\tau_{m6}/u_{%i}$', i_input));

linkaxes([ax1,ax2,ax3,ax4,ax5,ax6],'xy');
xlim([20, 2e3]); ylim([1e-8, 1e-4]);
#+end_src

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

#+name: fig:enc_plates_opt_iff_comp_simscape_all
#+caption: FRF from one actuator to all the encoders when the plant is damped using IFF
#+RESULTS:
[[file:figs/enc_plates_opt_iff_comp_simscape_all.png]]

#+begin_src matlab :exports none
%% Bode plot for the transfer function from u to dLm
freqs = 2*logspace(1, 3, 1000);

figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');

ax1 = nexttile([2,1]);
hold on;
% Diagonal Elements FRF
plot(f, abs(G_enc_iff_opt(:,1,1)), 'color', [colors(1,:), 0.2], ...
     'DisplayName', '$d\mathcal{L}_{m,i}/u_i$ - FRF')
for i = 2:6
    plot(f, abs(G_enc_iff_opt(:,i,i)), 'color', [colors(1,:), 0.2], ...
         'HandleVisibility', 'off');
end

% Diagonal Elements Model
set(gca,'ColorOrderIndex',2)
plot(freqs, abs(squeeze(freqresp(Gd_iff_opt(1,1), freqs, 'Hz'))), '-', ...
     'DisplayName', '$d\mathcal{L}_{m,i}/u_i$ - Model')
for i = 2:6
    set(gca,'ColorOrderIndex',2)
    plot(freqs, abs(squeeze(freqresp(Gd_iff_opt(i,i), freqs, 'Hz'))), '-', ...
         'HandleVisibility', 'off');
end

hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude $d_e/V_{exc}$ [m/V]'); set(gca, 'XTickLabel',[]);
ylim([1e-8, 1e-4]);
legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 3);

ax2 = nexttile;
hold on;
for i =1:6
    plot(f, 180/pi*angle(G_enc_iff_opt(:,i,i)), 'color', [colors(1,:), 0.2]);
    set(gca,'ColorOrderIndex',2)
    plot(freqs, 180/pi*angle(squeeze(freqresp(Gd_iff_opt(i,i), freqs, 'Hz'))));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
yticks(-360:90:360);
ylim([-180, 180]);

linkaxes([ax1,ax2],'x');
xlim([20, 2e3]);
#+end_src

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

#+name: fig:damped_iff_plates_plant_comp_diagonal
#+caption: Comparison of the diagonal elements of the transfer functions from $\bm{u}$ to $d\bm{\mathcal{L}}_m$ with active damping (IFF) applied with an optimal gain $g = 400$
#+RESULTS:
[[file:figs/damped_iff_plates_plant_comp_diagonal.png]]

#+begin_src matlab :exports none
%% Bode plot for the transfer function from u to dLm
freqs = 2*logspace(1, 3, 1000);

figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');

ax1 = nexttile([2,1]);
hold on;
% Off diagonal FRF
plot(f, abs(G_enc_iff_opt(:,1,2)), 'color', [colors(1,:), 0.2], ...
     'DisplayName', '$d\mathcal{L}_{m,i}/u_j$ - FRF')
for i = 1:5
    for j = i+1:6
        plot(f, abs(G_enc_iff_opt(:,i,j)), 'color', [colors(1,:), 0.2], ...
             'HandleVisibility', 'off');
    end
end

% Off diagonal Model
set(gca,'ColorOrderIndex',2)
plot(freqs, abs(squeeze(freqresp(Gd_iff_opt(1,2), freqs, 'Hz'))), '-', ...
     'DisplayName', '$d\mathcal{L}_{m,i}/u_j$ - Model')
for i = 1:5
    for j = i+1:6
        set(gca,'ColorOrderIndex',2)
        plot(freqs, abs(squeeze(freqresp(Gd_iff_opt(i,j), freqs, 'Hz'))), ...
             'HandleVisibility', 'off');
    end
end

hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude $d_e/V_{exc}$ [m/V]'); set(gca, 'XTickLabel',[]);
ylim([1e-8, 1e-4]);
legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 3);

ax2 = nexttile;
hold on;
% Off diagonal FRF
for i = 1:5
    for j = i+1:6
        plot(f, 180/pi*angle(G_enc_iff_opt(:,i,j)), 'color', [colors(1,:), 0.2]);
    end
end

% Off diagonal Model
for i = 1:5
    for j = i+1:6
        set(gca,'ColorOrderIndex',2)
        plot(freqs, 180/pi*angle(squeeze(freqresp(Gd_iff_opt(i,j), freqs, 'Hz'))));
    end
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
yticks(-360:90:360);
ylim([-180, 180]);

linkaxes([ax1,ax2],'x');
xlim([20, 2e3]);
#+end_src

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

#+name: fig:damped_iff_plates_plant_comp_off_diagonal
#+caption: Comparison of the off-diagonal elements of the transfer functions from $\bm{u}$ to $d\bm{\mathcal{L}}_m$ with active damping (IFF) applied with an optimal gain $g = 400$
#+RESULTS:
[[file:figs/damped_iff_plates_plant_comp_off_diagonal.png]]

*** Effect of IFF on the plant - FRF
#+begin_src matlab :tangle no
load('identified_plants_enc_plates.mat', 'f', 'G_dvf');
#+end_src

#+begin_src matlab :exports none
%% Bode plot of the transfer function from u to dLm for tested values of the IFF gain
freqs = 2*logspace(1, 3, 1000);

figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');

ax1 = nexttile([2,1]);
hold on;
plot(f, abs(G_dvf(:,1,1)), '-', ...
     'DisplayName', 'OL - Diag');
plot(f, abs(G_enc_iff_opt(:,1,1)), '-', ...
     'DisplayName', 'IFF - Diag');
for i = 2:6
    set(gca,'ColorOrderIndex',1);
    plot(f, abs(G_dvf(:,1,1)), '-', ...
         'HandleVisibility', 'off');
end
for i = 2:6
    set(gca,'ColorOrderIndex',2);
    plot(f, abs(G_enc_iff_opt(:,i,i)), '-', ...
         'HandleVisibility', 'off');
end

plot(f, abs(G_dvf(:,1,2)), 'color', [colors(1,:), 0.2], ...
     'DisplayName', 'OL - Off-diag')
for i = 1:5
    for j = i+1:6
        plot(f, abs(G_dvf(:,i,j)), 'color', [colors(1,:), 0.2], ...
             'HandleVisibility', 'off');
    end
end
plot(f, abs(G_enc_iff_opt(:,1,2)), 'color', [colors(2,:), 0.2], ...
     'DisplayName', 'IFF - Off-diag')
for i = 1:5
    for j = i+1:6
        plot(f, abs(G_enc_iff_opt(:,i,j)), 'color', [colors(2,:), 0.2], ...
             'HandleVisibility', 'off');
    end
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/V]'); set(gca, 'XTickLabel',[]);
legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2);

ax2 = nexttile;
hold on;
for i = 1:6
    set(gca,'ColorOrderIndex',1);
    plot(f, 180/pi*angle(G_dvf(:,1,1)), '-')
    set(gca,'ColorOrderIndex',2);
    plot(f, 180/pi*angle(G_enc_iff_opt(:,i,i)), '-')
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-180, 180]);
yticks([-180, -90, 0, 90, 180]);

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

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

#+name: fig:enc_plant_plates_effect_iff
#+caption: Effect of the IFF control strategy on the transfer function from $\bm{\tau}$ to $d\bm{\mathcal{L}}_m$
#+RESULTS:
[[file:figs/enc_plant_plates_effect_iff.png]]

*** Save Damped Plant
#+begin_src matlab :tangle no
save('matlab/mat/damped_plant_enc_plates.mat', 'f', 'Ts', 'G_enc_iff_opt')
#+end_src

#+begin_src matlab :exports none :eval no
save('mat/damped_plant_enc_plates.mat', 'f', 'Ts', 'G_enc_iff_opt')
#+end_src

** HAC Control - Frame of the struts
<<sec:hac_iff_struts>>
*** Introduction                                                    :ignore:

In a first approximation, the Jacobian matrix can be used instead of using the inverse kinematic equations.

#+begin_src latex :file control_architecture_hac_iff_L.pdf
\begin{tikzpicture}
  % Blocs
  \node[block={3.0cm}{3.0cm}] (P) {Plant};
  \coordinate[] (inputF) at ($(P.south west)!0.5!(P.north west)$);
  \coordinate[] (outputF) at ($(P.south east)!0.8!(P.north east)$);
  \coordinate[] (outputX) at ($(P.south east)!0.5!(P.north east)$);
  \coordinate[] (outputL) at ($(P.south east)!0.2!(P.north east)$);

  \node[block, above=0.4 of P] (Kiff) {$\bm{K}_\text{IFF}$};
  \node[addb, left= of inputF] (addF) {};
  \node[block, left= of addF] (K) {$\bm{K}_\mathcal{L}$};
  \node[addb, left= of K] (subr) {};
  \node[block, align=center, left= of subr] (J) {Inverse\\Kinematics};

  % Connections and labels
  \draw[->] (outputF) -- ++(1, 0) node[below left]{$\bm{\tau}_m$};
  \draw[->] ($(outputF) + (0.6, 0)$)node[branch]{} |- (Kiff.east);
  \draw[->] (Kiff.west) -| (addF.north);
  \draw[->] (addF.east) -- (inputF) node[above left]{$\bm{u}$};

  \draw[->] (outputL) -- ++(1, 0) node[above left]{$d\bm{\mathcal{L}}$};
  \draw[->] ($(outputL) + (0.6, 0)$)node[branch]{} -- ++(0, -1) -| (subr.south);
  \draw[->] (subr.east) -- (K.west) node[above left]{$\bm{\epsilon}_{d\mathcal{L}}$};
  \draw[->] (K.east) -- (addF.west) node[above left]{$\bm{u}^\prime$};

  \draw[->] (outputX) -- ++(1, 0) node[above left]{$\bm{\mathcal{X}}$};

  \draw[->] (J.east) -- (subr.west) node[above left]{$\bm{r}_{d\mathcal{L}}$};
  \draw[<-] (J.west)node[above left]{$\bm{r}_{\mathcal{X}_n}$} -- ++(-1, 0);
\end{tikzpicture}
#+end_src

#+name: fig:control_architecture_hac_iff_L
#+caption: HAC-LAC: IFF + Control in the frame of the legs
#+RESULTS:
[[file:figs/control_architecture_hac_iff_L.png]]

*** 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
addpath('./matlab/mat/');
addpath('./matlab/src/');
addpath('./matlab/');
#+end_src

#+begin_src matlab :eval no
addpath('./mat/');
addpath('./src/');
#+end_src

#+begin_src matlab
load('damped_plant_enc_plates.mat', 'f', 'Ts', 'G_enc_iff_opt')
#+end_src

#+begin_src matlab :tangle no
%% Add all useful folders to the path
addpath('matlab/nass-simscape/matlab/nano_hexapod/')
addpath('matlab/nass-simscape/STEPS/nano_hexapod/')
addpath('matlab/nass-simscape/STEPS/png/')
addpath('matlab/nass-simscape/src/')
addpath('matlab/nass-simscape/mat/')
#+end_src

#+begin_src matlab :eval no
%% Add all useful folders to the path
addpath('nass-simscape/matlab/nano_hexapod/')
addpath('nass-simscape/STEPS/nano_hexapod/')
addpath('nass-simscape/STEPS/png/')
addpath('nass-simscape/src/')
addpath('nass-simscape/mat/')
#+end_src

#+begin_src matlab
%% Open Simulink Model
mdl = 'nano_hexapod_simscape';

options = linearizeOptions;
options.SampleTime = 0;

open(mdl)

Rx = zeros(1, 7);

colors = colororder;
#+end_src

*** Simscape Model
Let's start with the Simscape model and the damped plant.

Apply HAC control and verify the system is stable.

Then, try the control strategy on the real plant.
#+begin_src matlab
load('Kiff.mat', 'Kiff')
#+end_src

#+begin_src matlab
%% Initialize the Simscape model in closed loop
n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
                                       'flex_top_type', '4dof', ...
                                       'motion_sensor_type', 'plates', ...
                                       'actuator_type', 'flexible', ...
                                       'controller_type', 'iff');
#+end_src

#+begin_src matlab
%% Identify the (damped) transfer function from u to dLm for different values of the IFF gain
clear io; io_i = 1;
io(io_i) = linio([mdl, '/du'],  1, 'openinput');  io_i = io_i + 1; % Actuator Inputs
io(io_i) = linio([mdl, '/dL'], 1, 'openoutput'); io_i = io_i + 1; % Plate Displacement (encoder)
#+end_src

#+begin_src matlab
Gd_iff_opt = exp(-s*Ts)*linearize(mdl, io, 0.0, options);
#+end_src

#+begin_src matlab :results value replace :exports both :tangle no
isstable(Gd_iff_opt)
#+end_src

#+RESULTS:
: 1

#+begin_src matlab :exports none
%% Bode plot for the transfer function from u to dLm
freqs = 2*logspace(1, 3, 1000);

figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');

ax1 = nexttile([2,1]);
hold on;
% Diagonal Elements Model
for i = 1:6
    plot(freqs, abs(squeeze(freqresp(Gd_iff_opt(i,i), freqs, 'Hz'))), 'k-');
end

hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude $d\mathcal{L}_m/u$ [m/V]'); set(gca, 'XTickLabel',[]);
ylim([1e-8, 1e-4]);

ax2 = nexttile;
hold on;
for i =1:6
    plot(freqs, 180/pi*angle(squeeze(freqresp(Gd_iff_opt(i,i), freqs, 'Hz'))), 'k-');
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
yticks(-360:90:360);
ylim([-180, 180]);

linkaxes([ax1,ax2],'x');
xlim([20, 2e3]);
#+end_src

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

#+name: fig:hac_iff_struts_enc_plates_plant_bode
#+caption: Transfer function from $u$ to $d\mathcal{L}_m$ with IFF (diagonal elements)
#+RESULTS:
[[file:figs/hac_iff_struts_enc_plates_plant_bode.png]]

*** HAC Controller
Let's try to have 100Hz bandwidth:
#+begin_src matlab
%% Lead
a  = 2;  % Amount of phase lead / width of the phase lead / high frequency gain
wc = 2*pi*100; % Frequency with the maximum phase lead [rad/s]

H_lead = (1 + s/(wc/sqrt(a)))/(1 + s/(wc*sqrt(a)));

%% Low Pass filter
H_lpf = 1/(1 + s/2/pi/200);

%% Notch
gm = 0.02;
xi = 0.3;
wn = 2*pi*700;

H_notch = (s^2 + 2*gm*xi*wn*s + wn^2)/(s^2 + 2*xi*wn*s + wn^2);
#+end_src

#+begin_src matlab
Khac_iff_struts = -(1/(2.87e-5)) * ... % Gain
                  H_lead * ...       % Lead
                  H_notch * ...      % Notch
                  (2*pi*100/s) * ... % Integrator
                  eye(6);            % 6x6 Diagonal
#+end_src

#+begin_src matlab :tangle no
save('matlab/mat/Khac_iff_struts.mat', 'Khac_iff_struts')
#+end_src

#+begin_src matlab :exports none :eval no
save('mat/Khac_iff_struts.mat', 'Khac_iff_struts')
#+end_src

#+begin_src matlab
Lhac_iff_struts = Khac_iff_struts*Gd_iff_opt;
#+end_src

#+begin_src matlab :exports none
%% Bode plot for the transfer function from u to dLm
freqs = 2*logspace(0, 3, 1000);

figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');

ax1 = nexttile([2,1]);
hold on;
% Diagonal Elements Model
for i = 1:6
    plot(freqs, abs(squeeze(freqresp(Lhac_iff_struts(i,i), freqs, 'Hz'))), 'k-');
end
for i = 1:5
    for j = i+1:6
        plot(freqs, abs(squeeze(freqresp(Lhac_iff_struts(i,j), freqs, 'Hz'))), 'color', [colors(2,:), 0.2]);
    end
end

hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude $d_e/V_{exc}$ [m/V]'); set(gca, 'XTickLabel',[]);
ylim([1e-3, 1e2]);

ax2 = nexttile;
hold on;
for i =1:6
    plot(freqs, 180/pi*angle(squeeze(freqresp(Lhac_iff_struts(i,i), freqs, 'Hz'))), 'k-');
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
yticks(-360:90:360);
ylim([-180, 180]);

linkaxes([ax1,ax2],'x');
xlim([2, 2e3]);
#+end_src

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

#+name: fig:loop_gain_hac_iff_struts
#+caption: Diagonal and off-diagonal elements of the Loop gain for "HAC-IFF-Struts"
#+RESULTS:
[[file:figs/loop_gain_hac_iff_struts.png]]


*** Experimental Loop Gain
- [ ] Find a more clever way to do the multiplication

#+begin_src matlab
L_frf = zeros(size(G_enc_iff_opt));

for i = 1:size(G_enc_iff_opt, 1)
    L_frf(i, :, :) = squeeze(G_enc_iff_opt(i,:,:))*freqresp(Khac_iff_struts, f(i), 'Hz');
end
#+end_src

#+begin_src matlab :exports none
%% Bode plot for the transfer function from u to dLm
freqs = 2*logspace(1, 3, 1000);

figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');

ax1 = nexttile([2,1]);
hold on;
% Diagonal Elements FRF
plot(f, abs(L_frf(:,1,1)), 'color', colors(1,:), ...
     'DisplayName', 'Diagonal');
for i = 2:6
    plot(f, abs(L_frf(:,i,i)), 'color', colors(1,:), ...
         'HandleVisibility', 'off');
end
plot(f, abs(L_frf(:,1,2)), 'color', [colors(2,:), 0.2], ...
     'DisplayName', 'Off-Diag');
for i = 1:5
    for j = i+1:6
        plot(f, abs(L_frf(:,i,j)), 'color', [colors(2,:), 0.2], ...
             'HandleVisibility', 'off');
    end
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Loop Gain [-]'); set(gca, 'XTickLabel',[]);
ylim([1e-3, 1e2]);
legend('location', 'northeast');

ax2 = nexttile;
hold on;
for i =1:6
    plot(f, 180/pi*angle(L_frf(:,i,i)), 'color', colors(1,:));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
yticks(-360:90:360);
ylim([-180, 180]);

linkaxes([ax1,ax2],'x');
xlim([1, 2e3]);
#+end_src

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

#+name: fig:hac_iff_plates_exp_loop_gain_diag
#+caption: Diagonal and Off-diagonal elements of the Loop gain (experimental data)
#+RESULTS:
[[file:figs/hac_iff_plates_exp_loop_gain_diag.png]]

*** Verification of the Stability using the Simscape model
#+begin_src matlab
%% Initialize the Simscape model in closed loop
n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
                                       'flex_top_type', '4dof', ...
                                       'motion_sensor_type', 'plates', ...
                                       'actuator_type', 'flexible', ...
                                       'controller_type', 'hac-iff-struts');
#+end_src

#+begin_src matlab
Gd_iff_hac_opt = linearize(mdl, io, 0.0, options);
#+end_src

#+begin_src matlab :results value replace :exports both
isstable(Gd_iff_hac_opt)
#+end_src

#+RESULTS:
: 1

** Reference Tracking
<<sec:hac_iff_struts_ref_track>>
*** 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
addpath('./matlab/mat/');
addpath('./matlab/src/');
addpath('./matlab/');
#+end_src

#+begin_src matlab :eval no
addpath('./mat/');
addpath('./src/');
#+end_src

#+begin_src matlab :tangle no
%% Add all useful folders to the path
addpath('matlab/nass-simscape/matlab/nano_hexapod/')
addpath('matlab/nass-simscape/STEPS/nano_hexapod/')
addpath('matlab/nass-simscape/STEPS/png/')
addpath('matlab/nass-simscape/src/')
addpath('matlab/nass-simscape/mat/')
#+end_src

#+begin_src matlab :eval no
%% Add all useful folders to the path
addpath('nass-simscape/matlab/nano_hexapod/')
addpath('nass-simscape/STEPS/nano_hexapod/')
addpath('nass-simscape/STEPS/png/')
addpath('nass-simscape/src/')
addpath('nass-simscape/mat/')
#+end_src

#+begin_src matlab
%% Open Simulink Model
mdl = 'nano_hexapod_simscape';

options = linearizeOptions;
options.SampleTime = 0;

open(mdl)

colors = colororder;
#+end_src

*** Load
#+begin_src matlab
load('Khac_iff_struts.mat', 'Khac_iff_struts')
#+end_src

*** Y-Z Scans
**** Generate the Scan
#+begin_src matlab
Rx_yz = generateYZScanTrajectory(...
    'y_tot', 4e-6, ...
    'z_tot', 8e-6, ...
    'n', 5, ...
    'Ts', 1e-3, ...
    'ti', 2, ...
    'tw', 0.5, ...
    'ty', 2, ...
    'tz', 1);
#+end_src

#+begin_src matlab
figure;
hold on;
plot(Rx_yz(:,1), Rx_yz(:,3), ...
     'DisplayName', 'Y motion')
plot(Rx_yz(:,1), Rx_yz(:,4), ...
     'DisplayName', 'Z motion')
hold off;
xlabel('Time [s]');
ylabel('Displacement [m]');
legend('location', 'northeast');
#+end_src

#+begin_src matlab
figure;
plot(Rx_yz(:,3), Rx_yz(:,4));
xlabel('y [m]'); ylabel('z [m]');
#+end_src

**** Reference Signal for the Strut lengths
Let's use the Jacobian to estimate the wanted strut length as a function of time.
#+begin_src matlab
dL_ref = [n_hexapod.geometry.J*Rx_yz(:, 2:7)']';
#+end_src

#+begin_src matlab
figure;
hold on;
for i=1:6
    plot(Rx_yz(:,1), dL_ref(:, i))
end
xlabel('Time [s]'); ylabel('Displacement [m]');
#+end_src

**** Time domain simulation with 2DoF model
#+begin_src matlab
%% Initialize the Simscape model in closed loop
n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '2dof', ...
                                       'flex_top_type', '3dof', ...
                                       'motion_sensor_type', 'plates', ...
                                       'actuator_type', '2dof', ...
                                       'controller_type', 'hac-iff-struts');
#+end_src

#+begin_src matlab
set_param(mdl,'StopTime', num2str(Rx_yz(end,1)))
set_param(mdl,'SimulationCommand','start')
#+end_src

#+begin_src matlab
out.X.Data = out.X.Data - out.X.Data(1,:);
#+end_src

#+begin_src matlab :exports none
figure;
hold on;
set(gca,'ColorOrderIndex',2)
plot(1e6*out.X.Data(:,2), 1e6*out.X.Data(:,3), '-', ...
     'DisplayName', 'Meas. Motion')
plot(1e6*Rx_yz(:,3), 1e6*Rx_yz(:,4), 'k--', ...
     'DisplayName', 'Reference Path')
hold off;
xlabel('X displacement [$\mu m$]'); ylabel('Y displacement [$\mu m$]');
legend('location', 'southwest');
#+end_src

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

#+name: fig:ref_track_hac_iff_struts_yz_plane
#+caption: Simulated Y-Z motion
#+RESULTS:
[[file:figs/ref_track_hac_iff_struts_yz_plane.png]]

#+begin_src matlab :exports none
figure;
hold on;
set(gca,'ColorOrderIndex',2)
plot(out.X.Time, out.X.Data(:,2), '-', ...
     'DisplayName', 'Meas. - Y')
plot(Rx_yz(:,1), Rx_yz(:,3), 'k--', ...
     'DisplayName', 'Ref - Y')
plot(out.X.Time, out.X.Data(:,3), '-', ...
     'DisplayName', 'Meas - Z')
plot(Rx_yz(:,1), Rx_yz(:,4), 'k-.', ...
     'DisplayName', 'Ref - Z')
hold off;
xlabel('Time [s]'); ylabel('Y Displacement [$\mu m$]');
legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 2);
#+end_src

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

#+name: fig:ref_track_hac_iff_struts_yz_time
#+caption: Y and Z motion as a function of time as well as the reference signals
#+RESULTS:
[[file:figs/ref_track_hac_iff_struts_yz_time.png]]

#+begin_src matlab :exports none
figure;
tiledlayout(1, 2, 'TileSpacing', 'None', 'Padding', 'None');

ax1 = nexttile;
hold on;
plot(out.X.Time, 1e9*(out.X.Data(:,1) - Rx_yz(:,2)), 'DisplayName', '$\epsilon_x$')
plot(out.X.Time, 1e9*(out.X.Data(:,2) - Rx_yz(:,3)), 'DisplayName', '$\epsilon_y$')
plot(out.X.Time, 1e9*(out.X.Data(:,3) - Rx_yz(:,4)), 'DisplayName', '$\epsilon_z$')
hold off;
xlabel('Time [s]'); ylabel('Position Errors [nm]');
legend('location', 'northeast');

ax2 = nexttile;
hold on;
plot(out.X.Time, 1e6*(out.X.Data(:,4) - Rx_yz(:,5)), 'DisplayName', '$\epsilon_{R_x}$')
plot(out.X.Time, 1e6*(out.X.Data(:,5) - Rx_yz(:,6)), 'DisplayName', '$\epsilon_{R_y}$')
plot(out.X.Time, 1e6*(out.X.Data(:,6) - Rx_yz(:,7)), 'DisplayName', '$\epsilon_{R_z}$')
hold off;
xlabel('Time [s]'); ylabel('Orientation Errors [$\mu rad$]');
legend('location', 'northeast');
#+end_src

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

#+name: fig:ref_track_hac_iff_struts_pos_error
#+caption: Positioning errors as a function of time
#+RESULTS:
[[file:figs/ref_track_hac_iff_struts_pos_error.png]]

*** "NASS" reference path
**** Generate Path
#+begin_src matlab
ref_path = [ ...
    0,  0, 0;
    0,  0, 1; % N
    0,  4, 1;
    3,  0, 1;
    3,  4, 1;
    3,  4, 0;
    4,  0, 0;
    4,  0, 1; % A
    4,  3, 1;
    5,  4, 1;
    6,  4, 1;
    7,  3, 1;
    7,  2, 1;
    4,  2, 1;
    4,  3, 1;
    5,  4, 1;
    6,  4, 1;
    7,  3, 1;
    7,  0, 1;
    7,  0, 0;
    8,  0, 0;
    8,  0, 1; % S
    11, 0, 1;
    11, 2, 1;
    8,  2, 1;
    8,  4, 1;
    11, 4, 1;
    11, 4, 0;
    12, 0, 0;
    12, 0, 1; % S
    15, 0, 1;
    15, 2, 1;
    12, 2, 1;
    12, 4, 1;
    15, 4, 1;
    15, 4, 0;
           ];

% Center the trajectory arround zero
ref_path = ref_path - (max(ref_path) - min(ref_path))/2;

% Define the X-Y-Z cuboid dimensions containing the trajectory
X_max = 10e-6;
Y_max =  4e-6;
Z_max =  2e-6;

ref_path = ([X_max, Y_max, Z_max]./max(ref_path)).*ref_path; % [m]
#+end_src

#+begin_src matlab
Rx_nass = generateXYZTrajectory('points', ref_path);
#+end_src

#+begin_src matlab
figure;
plot(1e6*Rx_nass(Rx_nass(:,4)>0, 2), 1e6*Rx_nass(Rx_nass(:,4)>0, 3), 'k.')
xlabel('X [$\mu m$]');
ylabel('Y [$\mu m$]');
axis equal;
xlim(1e6*[min(Rx_nass(:,2)), max(Rx_nass(:,2))]);
ylim(1e6*[min(Rx_nass(:,3)), max(Rx_nass(:,3))]);
#+end_src

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

#+name: fig:ref_track_test_nass
#+caption: Reference path corresponding to the "NASS" acronym
#+RESULTS:
[[file:figs/ref_track_test_nass.png]]

#+begin_src matlab
figure;
plot3(Rx_nass(:,2), Rx_nass(:,3), Rx_nass(:,4), 'k-');
xlabel('x');
ylabel('y');
zlabel('z');
#+end_src

#+begin_src matlab
figure;
hold on;
plot(Rx_nass(:,1), Rx_nass(:,2));
plot(Rx_nass(:,1), Rx_nass(:,3));
plot(Rx_nass(:,1), Rx_nass(:,4));
hold off;
#+end_src

#+begin_src matlab
figure;
scatter(Rx_nass(:,2), Rx_nass(:,3), 1, Rx_nass(:,4), 'filled')
colormap winter
#+end_src
**** Simscape Simulations
#+begin_src matlab
%% Initialize the Simscape model in closed loop
n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '2dof', ...
                                       'flex_top_type', '3dof', ...
                                       'motion_sensor_type', 'plates', ...
                                       'actuator_type', '2dof', ...
                                       'controller_type', 'hac-iff-struts');
#+end_src

#+begin_src matlab
set_param(mdl,'StopTime', num2str(Rx_nass(end,1)))
set_param(mdl,'SimulationCommand','start')
#+end_src

#+begin_src matlab
out.X.Data = out.X.Data - out.X.Data(1,:);
#+end_src

#+begin_src matlab :exports none
figure;
hold on;
set(gca,'ColorOrderIndex',2)
plot(1e6*out.X.Data(out.X.Data(:,3)>0, 1), 1e6*out.X.Data(out.X.Data(:,3)>0, 2), '-', ...
     'DisplayName', 'Meas. Motion')
plot(1e6*Rx_nass(Rx_nass(:,4)>0, 2), 1e6*Rx_nass(Rx_nass(:,4)>0, 3), 'k--', ...
     'DisplayName', 'Reference Path')
hold off;
xlabel('X displacement [$\mu m$]'); ylabel('Y displacement [$\mu m$]');
legend('location', 'southwest');
#+end_src

*** Save Reference paths
#+begin_src matlab :tangle no
save('matlab/mat/reference_path.mat', 'Rx_yz', 'Rx_nass')
#+end_src

#+begin_src matlab :exports none :eval no
save('mat/reference_path.mat', 'Rx_yz', 'Rx_nass')
#+end_src

*** Experimental Results

** Feedforward (Open-Loop) Control
*** Introduction

#+begin_src latex :file control_architecture_iff_feedforward.pdf
\begin{tikzpicture}
  % Blocs
  \node[block={3.0cm}{3.0cm}] (P) {Plant};
  \coordinate[] (inputF) at ($(P.south west)!0.5!(P.north west)$);
  \coordinate[] (outputF) at ($(P.south east)!0.8!(P.north east)$);
  \coordinate[] (outputX) at ($(P.south east)!0.5!(P.north east)$);
  \coordinate[] (outputL) at ($(P.south east)!0.2!(P.north east)$);

  \node[block, above=0.4 of P] (Kiff) {$\bm{K}_\text{IFF}$};
  \node[addb, left= of inputF] (addF) {};
  \node[block, left= of addF] (Kff) {$\bm{K}_{\mathcal{L},\text{ff}}$};
  \node[block, align=center, left= of Kff] (J) {Inverse\\Kinematics};

  % Connections and labels
  \draw[->] (outputF) -- ++(1, 0) node[below left]{$\bm{\tau}_m$};
  \draw[->] ($(outputF) + (0.6, 0)$)node[branch]{} |- (Kiff.east);
  \draw[->] (Kiff.west) -| (addF.north);
  \draw[->] (addF.east) -- (inputF) node[above left]{$\bm{u}$};

  \draw[->] (outputL) -- ++(1, 0) node[above left]{$d\bm{\mathcal{L}}$};
  \draw[->] (outputX) -- ++(1, 0) node[above left]{$\bm{\mathcal{X}}$};

  \draw[->] (Kff.east) -- (addF.west) node[above left]{$\bm{u}_{\text{ff}}$};
  \draw[->] (J.east) --   (Kff.west)  node[above left]{$\bm{r}_{d\mathcal{L}}$};
  \draw[<-] (J.west)node[above left]{$\bm{r}_{\mathcal{X}}$} -- ++(-1, 0);
\end{tikzpicture}
#+end_src

#+name: fig:control_architecture_iff_feedforward
#+caption: Feedforward control in the frame of the legs
#+RESULTS:
[[file:figs/control_architecture_iff_feedforward.png]]

*** 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
addpath('./matlab/mat/');
addpath('./matlab/src/');
addpath('./matlab/');
#+end_src

#+begin_src matlab :eval no
addpath('./mat/');
addpath('./src/');
#+end_src

#+begin_src matlab
load('damped_plant_enc_plates.mat', 'f', 'Ts', 'G_enc_iff_opt')
#+end_src

#+begin_src matlab :tangle no
%% Add all useful folders to the path
addpath('matlab/nass-simscape/matlab/nano_hexapod/')
addpath('matlab/nass-simscape/STEPS/nano_hexapod/')
addpath('matlab/nass-simscape/STEPS/png/')
addpath('matlab/nass-simscape/src/')
addpath('matlab/nass-simscape/mat/')
#+end_src

#+begin_src matlab :eval no
%% Add all useful folders to the path
addpath('nass-simscape/matlab/nano_hexapod/')
addpath('nass-simscape/STEPS/nano_hexapod/')
addpath('nass-simscape/STEPS/png/')
addpath('nass-simscape/src/')
addpath('nass-simscape/mat/')
#+end_src

#+begin_src matlab
%% Open Simulink Model
mdl = 'nano_hexapod_simscape';

options = linearizeOptions;
options.SampleTime = 0;

open(mdl)

Rx = zeros(1, 7);

colors = colororder;
#+end_src

*** Simple Feedforward Controller
Let's estimate the mean DC gain for the damped plant (diagonal elements:)
#+begin_src matlab :results value replace :exports results :tangle no
mean(diag(abs(squeeze(mean(G_enc_iff_opt(f>2 & f<4,:,:))))))
#+end_src

#+RESULTS:
: 1.773e-05

The feedforward controller is then taken as the inverse of this gain (the minus sign is there manually added as it is "removed" by the =abs= function):
#+begin_src matlab
Kff_iff_L = -1/mean(diag(abs(squeeze(mean(G_enc_iff_opt(f>2 & f<4,:,:))))));
#+end_src

The open-loop gain (feedforward controller times the damped plant) is shown in Figure [[fig:open_loop_gain_feedforward_iff_struts]].

#+begin_src matlab :exports none
%% Bode plot of the transfer function from u to dLm for tested values of the IFF gain
figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');

ax1 = nexttile([2,1]);
hold on;
for i = 1:6
    set(gca,'ColorOrderIndex',1);
    plot(f, abs(Kff_iff_L*G_enc_iff_opt(:,i,i)), 'k-');
end

hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [-]'); set(gca, 'XTickLabel',[]);
ylim([1e-2, 1e1]);

ax2 = nexttile;
hold on;
for i = 1:6
    set(gca,'ColorOrderIndex',1);
    plot(f, 180/pi*angle(Kff_iff_L*G_enc_iff_opt(:,i,i)), 'k-')
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-180, 180]);
yticks([-180, -90, 0, 90, 180]);

linkaxes([ax1,ax2],'x');
xlim([1, 2e3]);
#+end_src

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

#+name: fig:open_loop_gain_feedforward_iff_struts
#+caption: Diagonal elements of the "open loop gain"
#+RESULTS:
[[file:figs/open_loop_gain_feedforward_iff_struts.png]]

And save the feedforward controller for further use:
#+begin_src matlab
Kff_iff_L = zpk(Kff_iff_L)*eye(6);
#+end_src

#+begin_src matlab :tangle no
save('matlab/mat/feedforward_iff.mat', 'Kff_iff_L')
#+end_src

#+begin_src matlab :exports none :eval no
save('mat/feedforward_iff.mat', 'Kff_iff_L')
#+end_src

*** Test with Simscape Model
#+begin_src matlab
load('reference_path.mat', 'Rx_yz');
#+end_src

** Feedback/Feedforward control in the frame of the struts
*** Introduction                                                    :ignore:

#+begin_src latex :file control_architecture_hac_iff_L_feedforward.pdf
\begin{tikzpicture}
  % Blocs
  \node[block={3.0cm}{3.0cm}] (P) {Plant};
  \coordinate[] (inputF) at ($(P.south west)!0.5!(P.north west)$);
  \coordinate[] (outputF) at ($(P.south east)!0.8!(P.north east)$);
  \coordinate[] (outputX) at ($(P.south east)!0.5!(P.north east)$);
  \coordinate[] (outputL) at ($(P.south east)!0.2!(P.north east)$);

  \node[block, above=0.4 of P] (Kiff) {$\bm{K}_\text{IFF}$};
  \node[addb, left= of inputF] (addF) {};
  \node[block, left= of addF] (K) {$\bm{K}_\mathcal{L}$};
  \node[block, above= of K] (Kff) {$\bm{K}_{\mathcal{L},\text{ff}}$};
  \node[addb, left= of K] (subr) {};
  \node[block, align=center, left= of subr] (J) {Inverse\\Kinematics};

  % Connections and labels
  \draw[->] (outputF) -- ++(1, 0) node[below left]{$\bm{\tau}_m$};
  \draw[->] ($(outputF) + (0.6, 0)$)node[branch]{} |- (Kiff.east);
  \draw[->] (Kiff.west) -| (addF.north);
  \draw[->] (addF.east) -- (inputF) node[above left]{$\bm{u}$};

  \draw[->] (outputL) -- ++(1, 0) node[above left]{$d\bm{\mathcal{L}}$};
  \draw[->] ($(outputL) + (0.6, 0)$)node[branch]{} -- ++(0, -1) -| (subr.south);
  \draw[->] (subr.east) -- (K.west) node[above left]{$\bm{\epsilon}_{d\mathcal{L}}$};
  \draw[->] (K.east) -- (addF.west) node[above left]{$\bm{u}^\prime$};

  \draw[->] (outputX) -- ++(1, 0) node[above left]{$\bm{\mathcal{X}}_n$};

  \draw[->] (J.east) -- (subr.west);
  \draw[->] ($(J.east) + (0.4, 0)$)node[branch]{} node[below]{$\bm{r}_{d\mathcal{L}}$} |- (Kff.west);
  \draw[->] (Kff.east) -- ++(0.5, 0) -- (addF.north west);

  \draw[<-] (J.west)node[above left]{$\bm{r}_{\mathcal{X}_n}$} -- ++(-1, 0);
\end{tikzpicture}
#+end_src

#+name: fig:control_architecture_hac_iff_L_feedforward
#+caption: Feedback/Feedforward control in the frame of the legs
#+RESULTS:
[[file:figs/control_architecture_hac_iff_L_feedforward.png]]


* Functions
** =generateXYZTrajectory=
:PROPERTIES:
:header-args:matlab+: :tangle matlab/src/generateXYZTrajectory.m
:header-args:matlab+: :comments none :mkdirp yes :eval no
:END:
<<sec:generateXYZTrajectory>>

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

*** Optional Parameters
:PROPERTIES:
:UNNUMBERED: t
:END:

#+begin_src matlab
arguments
    args.points double {mustBeNumeric} = zeros(2, 3) % [m]

    args.ti    (1,1) double {mustBeNumeric, mustBePositive} = 1 % Time to go to first point and after last point [s]
    args.tw    (1,1) double {mustBeNumeric, mustBePositive} = 0.5 % Time wait between each point [s]
    args.tm    (1,1) double {mustBeNumeric, mustBePositive} = 1 % Motion time between points [s]

    args.Ts    (1,1) double {mustBeNumeric, mustBePositive} = 1e-3 % Sampling Time [s]
end
#+end_src

*** Initialize Time Vectors
:PROPERTIES:
:UNNUMBERED: t
:END:

#+begin_src matlab
time_i = 0:args.Ts:args.ti;
time_w = 0:args.Ts:args.tw;
time_m = 0:args.Ts:args.tm;
#+end_src

*** XYZ Trajectory
:PROPERTIES:
:UNNUMBERED: t
:END:

#+begin_src matlab
% Go to initial position
xyz = (args.points(1,:))'*(time_i/args.ti);

% Wait
xyz = [xyz, xyz(:,end).*ones(size(time_w))];

% Scans
for i = 2:size(args.points, 1)
    % Go to next point
    xyz = [xyz, xyz(:,end) + (args.points(i,:)' - xyz(:,end))*(time_m/args.tm)];
    % Wait a litle bit
    xyz = [xyz, xyz(:,end).*ones(size(time_w))];
end

% End motion
xyz = [xyz, xyz(:,end) - xyz(:,end)*(time_i/args.ti)];
#+end_src

*** Reference Signal
:PROPERTIES:
:UNNUMBERED: t
:END:

#+begin_src matlab
t = 0:args.Ts:args.Ts*(length(xyz) - 1);
#+end_src

#+begin_src matlab
ref = zeros(length(xyz), 7);

ref(:, 1) = t;
ref(:, 2:4) = xyz';
#+end_src

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

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

*** Optional Parameters
:PROPERTIES:
:UNNUMBERED: t
:END:

#+begin_src matlab
arguments
    args.y_tot (1,1) double {mustBeNumeric} = 10e-6 % [m]
    args.z_tot (1,1) double {mustBeNumeric} = 10e-6 % [m]

    args.n     (1,1) double {mustBeInteger, mustBePositive} = 10 % [-]

    args.Ts    (1,1) double {mustBeNumeric, mustBePositive} = 1e-4 % [s]

    args.ti    (1,1) double {mustBeNumeric, mustBePositive} = 1 % [s]
    args.tw    (1,1) double {mustBeNumeric, mustBePositive} = 1 % [s]
    args.ty    (1,1) double {mustBeNumeric, mustBePositive} = 1 % [s]
    args.tz    (1,1) double {mustBeNumeric, mustBePositive} = 1 % [s]
end
#+end_src

*** Initialize Time Vectors
:PROPERTIES:
:UNNUMBERED: t
:END:

#+begin_src matlab
time_i = 0:args.Ts:args.ti;
time_w = 0:args.Ts:args.tw;
time_y = 0:args.Ts:args.ty;
time_z = 0:args.Ts:args.tz;
#+end_src

*** Y and Z vectors
:PROPERTIES:
:UNNUMBERED: t
:END:

#+begin_src matlab
% Go to initial position
y = (time_i/args.ti)*(args.y_tot/2);

% Wait
y = [y, y(end)*ones(size(time_w))];

% Scans
for i = 1:args.n
    if mod(i,2) == 0
        y = [y, -(args.y_tot/2) + (time_y/args.ty)*args.y_tot];
    else
        y = [y,  (args.y_tot/2) - (time_y/args.ty)*args.y_tot];
    end

    if i < args.n
        y = [y, y(end)*ones(size(time_z))];
    end
end

% Wait a litle bit
y = [y, y(end)*ones(size(time_w))];

% End motion
y = [y, y(end) - y(end)*time_i/args.ti];
#+end_src

#+begin_src matlab
% Go to initial position
z = (time_i/args.ti)*(args.z_tot/2);

% Wait
z = [z, z(end)*ones(size(time_w))];

% Scans
for i = 1:args.n
    z = [z, z(end)*ones(size(time_y))];

    if i < args.n
        z = [z, z(end) - (time_z/args.tz)*args.z_tot/(args.n-1)];
    end
end

% Wait a litle bit
z = [z, z(end)*ones(size(time_w))];

% End motion
z = [z, z(end) - z(end)*time_i/args.ti];
#+end_src

*** Reference Signal
:PROPERTIES:
:UNNUMBERED: t
:END:

#+begin_src matlab
t = 0:args.Ts:args.Ts*(length(y) - 1);
#+end_src

#+begin_src matlab
ref = zeros(length(y), 7);

ref(:, 1) = t;
ref(:, 3) = y;
ref(:, 4) = z;
#+end_src

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

*** Function description
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
function [M] = getTransformationMatrixAcc(Opm, Osm)
% getTransformationMatrixAcc -
%
% Syntax: [M] = getTransformationMatrixAcc(Opm, Osm)
%
% Inputs:
%    - Opm - Nx3 (N = number of accelerometer measurements) X,Y,Z position of accelerometers
%    - Opm - Nx3 (N = number of accelerometer measurements) Unit vectors representing the accelerometer orientation
%
% Outputs:
%    - M - Transformation Matrix
#+end_src

*** Transformation matrix from motion of the solid body to accelerometer measurements
:PROPERTIES:
:UNNUMBERED: t
:END:

Let's try to estimate the x-y-z acceleration of any point of the solid body from the acceleration/angular acceleration of the solid body expressed in $\{O\}$.
For any point $p_i$ of the solid body (corresponding to an accelerometer), we can write:
\begin{equation}
\begin{bmatrix}
a_{i,x} \\ a_{i,y} \\ a_{i,z}
\end{bmatrix} = \begin{bmatrix}
\dot{v}_x \\ \dot{v}_y \\ \dot{v}_z
\end{bmatrix} + p_i \times \begin{bmatrix}
\dot{\omega}_x \\ \dot{\omega}_y \\ \dot{\omega}_z
\end{bmatrix}
\end{equation}

We can write the cross product as a matrix product using the skew-symmetric transformation:
\begin{equation}
\begin{bmatrix}
a_{i,x} \\ a_{i,y} \\ a_{i,z}
\end{bmatrix} = \begin{bmatrix}
\dot{v}_x \\ \dot{v}_y \\ \dot{v}_z
\end{bmatrix} + \underbrace{\begin{bmatrix}
  0       &  p_{i,z} & -p_{i,y} \\
 -p_{i,z} &  0       &  p_{i,x} \\
  p_{i,y} & -p_{i,x} &  0
\end{bmatrix}}_{P_{i,[\times]}} \cdot \begin{bmatrix}
\dot{\omega}_x \\ \dot{\omega}_y \\ \dot{\omega}_z
\end{bmatrix}
\end{equation}

If we now want to know the (scalar) acceleration $a_i$ of the point $p_i$ in the direction of the accelerometer direction $\hat{s}_i$, we can just project the 3d acceleration on $\hat{s}_i$:
\begin{equation}
a_i = \hat{s}_i^T \cdot \begin{bmatrix}
a_{i,x} \\ a_{i,y} \\ a_{i,z}
\end{bmatrix} = \hat{s}_i^T \cdot \begin{bmatrix}
\dot{v}_x \\ \dot{v}_y \\ \dot{v}_z
\end{bmatrix} + \left( \hat{s}_i^T \cdot P_{i,[\times]} \right) \cdot \begin{bmatrix}
\dot{\omega}_x \\ \dot{\omega}_y \\ \dot{\omega}_z
\end{bmatrix}
\end{equation}

Which is equivalent as a simple vector multiplication:
\begin{equation}
a_i = \begin{bmatrix}
\hat{s}_i^T & \hat{s}_i^T \cdot P_{i,[\times]}
\end{bmatrix}
\begin{bmatrix}
\dot{v}_x \\ \dot{v}_y \\ \dot{v}_z \\ \dot{\omega}_x \\ \dot{\omega}_y \\ \dot{\omega}_z
\end{bmatrix} = \begin{bmatrix}
\hat{s}_i^T & \hat{s}_i^T \cdot P_{i,[\times]}
\end{bmatrix} {}^O\vec{x}
\end{equation}

And finally we can combine the 6 (line) vectors for the 6 accelerometers to write that in a matrix form.
We obtain Eq. eqref:eq:M_matrix.
#+begin_important
The transformation from solid body acceleration ${}^O\vec{x}$ from sensor measured acceleration $\vec{a}$ is:
\begin{equation} \label{eq:M_matrix}
\vec{a} = \underbrace{\begin{bmatrix}
\hat{s}_1^T & \hat{s}_1^T \cdot P_{1,[\times]} \\
\vdots & \vdots \\
\hat{s}_6^T & \hat{s}_6^T \cdot P_{6,[\times]}
\end{bmatrix}}_{M} {}^O\vec{x}
\end{equation}

with $\hat{s}_i$ the unit vector representing the measured direction of the i'th accelerometer expressed in frame $\{O\}$ and $P_{i,[\times]}$ the skew-symmetric matrix representing the cross product of the position of the i'th accelerometer expressed in frame $\{O\}$.
#+end_important

Let's define such matrix using matlab:
#+begin_src matlab
M = zeros(length(Opm), 6);

for i = 1:length(Opm)
    Ri = [0,         Opm(3,i), -Opm(2,i);
         -Opm(3,i),  0,         Opm(1,i);
          Opm(2,i), -Opm(1,i),  0];
    M(i, 1:3) = Osm(:,i)';
    M(i, 4:6) = Osm(:,i)'*Ri;
end
#+end_src

#+begin_src matlab
end
#+end_src


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

*** Function description
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
function [J] = getJacobianNanoHexapod(Hbm)
% getJacobianNanoHexapod -
%
% Syntax: [J] = getJacobianNanoHexapod(Hbm)
%
% Inputs:
%    - Hbm - Height of {B} w.r.t. {M} [m]
%
% Outputs:
%    - J - Jacobian Matrix
#+end_src

*** Transformation matrix from motion of the solid body to accelerometer measurements
:PROPERTIES:
:UNNUMBERED: t
:END:

#+begin_src matlab
Fa = [[-86.05,  -74.78, 22.49],
      [ 86.05,  -74.78, 22.49],
      [ 107.79, -37.13, 22.49],
      [ 21.74,  111.91, 22.49],
      [-21.74,  111.91, 22.49],
      [-107.79, -37.13, 22.49]]'*1e-3; % Ai w.r.t. {F} [m]

Mb = [[-28.47, -106.25, -22.50],
      [ 28.47, -106.25, -22.50],
      [ 106.25, 28.47,  -22.50],
      [ 77.78,  77.78,  -22.50],
      [-77.78,  77.78,  -22.50],
      [-106.25, 28.47,  -22.50]]'*1e-3; % Bi w.r.t. {M} [m]

H = 95e-3; % Stewart platform height [m]
Fb = Mb + [0; 0; H]; % Bi w.r.t. {F} [m]

si = Fb - Fa;
si = si./vecnorm(si); % Normalize

Bb = Mb - [0; 0; Hbm];

J = [si', cross(Bb, si)'];
#+end_src