nass-simscape/org/nano_hexapod.org

#+TITLE: Nano-Hexapod
#+SETUPFILE: ./setup/org-setup-file.org

* Introduction                                                        :ignore:

* Nano-Hexapod
** 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
simulinkproject('../');
#+end_src

#+begin_src matlab :eval no
addpath('nano_hexapod');
open('nano_hexapod/nano_hexapod.slx')
#+end_src

#+begin_src matlab :tangle no
addpath('matlab/nano_hexapod');
open('matlab/nano_hexapod/nano_hexapod.slx')
#+end_src

** Nano Hexapod - Configuration
<<sec:nano_hexapod_conf>>

The nano-hexapod can be initialized and configured using the =initializeNanoHexapodFinal= function ([[sec:initializeNanoHexapodFinal][link]]).

#+begin_src matlab
n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
                                       'flex_top_type', '3dof', ...
                                       'motion_sensor_type', 'struts', ...
                                       'actuator_type', '2dof', ...
                                       'MO_B', 150e-3);
#+end_src

We initialize the identification parameters.
#+begin_src matlab
%% Options for Linearized
options = linearizeOptions;
options.SampleTime = 0;

%% Name of the Simulink File
mdl = 'nano_hexapod';

%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/F'],  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
#+end_src

** Effect of encoders on the decentralized plant
<<sec:effect_encoder_location>>

We here wish to compare the plant from actuators to the encoders when the encoders are either fixed on the struts or on the plates.

Identify the plant when the encoders are on the struts:
#+begin_src matlab
n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
                                       'flex_top_type', '3dof', ...
                                       'motion_sensor_type', 'struts', ...
                                       'actuator_type', '2dof');

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

And identify the plant when the encoders are fixed on the plates:
#+begin_src matlab
n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
                                       'flex_top_type', '3dof', ...
                                       'motion_sensor_type', 'plates', ...
                                       'actuator_type', '2dof');

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

The obtained plants are compared in Figure [[fig:nano_hexapod_effect_encoder]].
#+begin_src matlab :exports none
freqs = logspace(0, 3, 1000);

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

ax1 = nexttile([2,1]);
hold on;
set(gca,'ColorOrderIndex',1);
plot(freqs, abs(squeeze(freqresp(Gs('D1', 'F1'), freqs, 'Hz'))), '-', 'DisplayName', 'Struts');
for i = 1:6
    set(gca,'ColorOrderIndex',1);
    plot(freqs, abs(squeeze(freqresp(Gs(['D', num2str(i)], ['F', num2str(i)]), freqs, 'Hz'))), '-', ...
         'HandleVisibility', 'off');
end
set(gca,'ColorOrderIndex',2);
plot(freqs, abs(squeeze(freqresp(Gp('D1', 'F1'), freqs, 'Hz'))), '-', 'DisplayName', 'Plates');
for i = 1:6
    set(gca,'ColorOrderIndex',2);
    plot(freqs, abs(squeeze(freqresp(Gp(['D', num2str(i)], ['F', num2str(i)]), freqs, 'Hz'))), '-', ...
         'HandleVisibility', 'off');
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
legend('location', 'northeast');
ylim([1e-11, 1e-6]);

ax2 = nexttile;
hold on;
for i = 1:6
    set(gca,'ColorOrderIndex',1);
    plot(freqs, 180/pi*angle(squeeze(freqresp(Gs(['D', num2str(i)], ['F', num2str(i)]), freqs, 'Hz'))), '-');
end
for i = 1:6
    set(gca,'ColorOrderIndex',2);
    plot(freqs, 180/pi*angle(squeeze(freqresp(Gp(['D', num2str(i)], ['F', num2str(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/nano_hexapod_effect_encoder.pdf', 'width', 'full', 'height', 'tall');
#+end_src

#+name: fig:nano_hexapod_effect_encoder
#+caption: Comparison of the plants from actuator to associated encoder when the encoders are either fixed to the struts or to the plates
#+RESULTS:
[[file:figs/nano_hexapod_effect_encoder.png]]

#+begin_question
Why do we have zeros at 400Hz and 800Hz when the encoders are fixed on the struts?
#+end_question

** Effect of APA flexibility
<<sec:effect_apa_flexibility>>

First identify the plant for APA represented by 2DoF system:
#+begin_src matlab
n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
                                       'flex_top_type', '3dof', ...
                                       'motion_sensor_type', 'struts', ...
                                       'actuator_type', '2dof', ...
                                       'actuator_Ga', 2);

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

First identify the plant for APA represented by a flexible element:
#+begin_src matlab
n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
                                       'flex_top_type', '3dof', ...
                                       'motion_sensor_type', 'struts', ...
                                       'actuator_type', 'flexible');

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

#+begin_src matlab :exports none
freqs = logspace(0, 3, 1000);

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

ax1 = nexttile([2,1]);
hold on;
set(gca,'ColorOrderIndex',1);
plot(freqs, abs(squeeze(freqresp(Gs('D1', 'F1'), freqs, 'Hz'))), '-', 'DisplayName', '2DoF');
for i = 1:6
    set(gca,'ColorOrderIndex',1);
    plot(freqs, abs(squeeze(freqresp(Gs(['D', num2str(i)], ['F', num2str(i)]), freqs, 'Hz'))), '-', ...
         'HandleVisibility', 'off');
end
set(gca,'ColorOrderIndex',2);
plot(freqs, abs(squeeze(freqresp(Gf('D1', 'F1'), freqs, 'Hz'))), '-', 'DisplayName', 'Flexible');
for i = 1:6
    set(gca,'ColorOrderIndex',2);
    plot(freqs, abs(squeeze(freqresp(Gf(['D', num2str(i)], ['F', num2str(i)]), freqs, 'Hz'))), '-', ...
         'HandleVisibility', 'off');
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
legend('location', 'northeast');
ylim([1e-11, 1e-6]);

ax2 = nexttile;
hold on;
for i = 1:6
    set(gca,'ColorOrderIndex',1);
    plot(freqs, 180/pi*angle(squeeze(freqresp(Gs(['D', num2str(i)], ['F', num2str(i)]), freqs, 'Hz'))), '-');
end
for i = 1:6
    set(gca,'ColorOrderIndex',2);
    plot(freqs, 180/pi*angle(squeeze(freqresp(Gf(['D', num2str(i)], ['F', num2str(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/nano_hexapod_effect_flexible_apa.pdf', 'width', 'full', 'height', 'tall');
#+end_src

#+name: fig:nano_hexapod_effect_flexible_apa
#+caption: Comparison of the plants from actuator to associated strut encoder when the APA are modelled with a 2DoF system of with a flexible one
#+RESULTS:
[[file:figs/nano_hexapod_effect_flexible_apa.png]]

#+begin_important
The first resonance is strange when using the flexible APA model (Figure [[fig:nano_hexapod_effect_flexible_apa]]).
Otherwise, the 2DoF model matches quite well the flexible model considering its simplicity.
#+end_important

** Nano Hexapod - Number of DoF
<<sec:num_states>>

In this section, we wish to see how the configuration of each element changes the number of the states of the obtained system.

The most minimalist model is the following:
#+begin_src matlab
n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '3dof', ...
                                       'flex_top_type', '2dof', ...
                                       'motion_sensor_type', 'struts', ...
                                       'actuator_type', '2dof');
G = linearize(mdl, io, 0.0, options);
#+end_src

#+begin_src matlab :results value replace :exports results
sprintf('There are %i states.', length(G.statename))
#+end_src

#+RESULTS:
: There are 24 states.

These states are summarized on table [[tab:num_states_nano_hexapod]].

#+name: tab:num_states_nano_hexapod
#+caption: Number of states for the minimalist model
#+attr_latex: :environment tabularx :width 0.3\linewidth :align cc
#+attr_latex: :center t :booktabs t :float t
| Element   | States |
|-----------+--------|
| Struts    |    2*6 |
| Top Plate |     12 |
|-----------+--------|
| Total:    |     24 |
#+TBLFM: @>$2=vsum(@I..@II)

If we add axial stiffness on the top joints, we should add 2 states for each struts.
#+begin_src matlab
n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
                                       'flex_top_type', '2dof', ...
                                       'motion_sensor_type', 'struts', ...
                                       'actuator_type', '2dof');
G = linearize(mdl, io, 0.0, options);
#+end_src

#+begin_src matlab :results value replace :exports results
sprintf('There are %i states.', length(G.statename))
#+end_src

#+RESULTS:
: There are 36 states.

If we add torsional stiffness on the bottom joints, we should again add 2 states for each struts.
#+begin_src matlab
n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
                                       'flex_top_type', '3dof', ...
                                       'motion_sensor_type', 'struts', ...
                                       'actuator_type', '2dof');
G = linearize(mdl, io, 0.0, options);
#+end_src

#+begin_src matlab :results value replace :exports results
sprintf('There are %i states.', length(G.statename))
#+end_src

#+RESULTS:
: There are 48 states.

Finally, if we add axial stiffness on the bottom joint, we should add 2 states for each struts.
#+begin_src matlab
n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
                                       'flex_top_type', '4dof', ...
                                       'motion_sensor_type', 'struts', ...
                                       'actuator_type', '2dof');
G = linearize(mdl, io, 0.0, options);
#+end_src

#+begin_src matlab :results value replace :exports results
sprintf('There are %i states.', length(G.statename))
#+end_src

#+RESULTS:
: There are 60 states.

#+begin_important
Obtained number of states is very comprehensible.
Depending on the physical effects we want to model, we therefore know how many states are added when configuring the model.
#+end_important

** Direct Velocity Feedback Plant
<<sec:dvf_plant>>

The transfer function from actuator forces $\tau_i$ to the encoder measurements $\mathcal{L}_i$ is now identified both when the encoders are fixed to the struts.
#+begin_src matlab
%% Options for Linearized
options = linearizeOptions;
options.SampleTime = 0;

%% Name of the Simulink File
mdl = 'nano_hexapod';

%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/F'], 1, 'openinput');  io_i = io_i + 1; % Actuator Inputs
io(io_i) = linio([mdl, '/D'], 1, 'openoutput'); io_i = io_i + 1; % Force Sensors

n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
                                       'flex_top_type', '3dof', ...
                                       'motion_sensor_type', 'struts', ...
                                       'actuator_type', '2dof');

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

The DC gain from actuator to relative motion sensor should be equal to (for the 2dof APA):
\[ \frac{1}{k + k_a + kk_a/k_e} \]

Which is equal to:
#+begin_src matlab :results value replace :exports results :tangle no
sprintf('DCgain = %.2e [m/N]', 1/(n_hexapod.actuator.k(1) + n_hexapod.actuator.ka(1) + n_hexapod.actuator.k(1)*n_hexapod.actuator.ka(1)/n_hexapod.actuator.ke(1)))
#+end_src

#+RESULTS:
: DCgain = 1.87e-08 [m/N]

Let's verify that by looking at the DC gain of the $6 \times 6$ DVF plant in Table [[tab:dvf_dc_gain]].
#+begin_src matlab :results value replace :exports results :tangle no
dcgain(Gdvf)
#+end_src

#+name: tab:dvf_dc_gain
#+caption: DC gain of the DVF plant
#+attr_latex: :environment tabularx :width 0.9\linewidth :align cccccc
#+attr_latex: :center t :booktabs t :float t
#+RESULTS:
|  1.8617e-08 | -1.0408e-10 |  1.3034e-10 | 3.2559e-11 | -1.1188e-10 |  9.0385e-11 |
| -5.1839e-11 |  1.8593e-08 | -4.4868e-11 |  8.016e-11 |  4.3558e-11 | -1.1164e-10 |
|  5.1963e-12 | -6.9001e-12 |  1.8564e-08 | 3.0844e-11 |  4.0097e-11 | -3.4387e-11 |
|  1.9359e-11 |  1.7432e-10 | -5.0928e-11 |  1.855e-08 |  1.6406e-10 |  4.5757e-12 |
| -2.1185e-11 |  2.1724e-10 |  1.5333e-12 | -8.802e-11 |  1.8803e-08 | -4.6946e-11 |
| -1.1728e-11 | -5.7682e-11 |  1.6213e-10 | 2.1934e-12 | -1.6237e-10 |  1.8715e-08 |

And the bode plot of the DVF plant is shown in Figure [[fig:nano_hexapod_struts_2dof_dvf_plant]].
#+begin_src matlab :exports none
freqs = 5*logspace(0, 2, 1000);

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

ax1 = nexttile([2,1]);
hold on;
plot(freqs, abs(squeeze(freqresp(Gdvf(1,1), freqs, 'Hz'))), '-', ...
     'DisplayName', '$\mathcal{L}_{i}/\tau_i$')
for i = 1:6
    set(gca,'ColorOrderIndex',1);
    plot(freqs, abs(squeeze(freqresp(Gdvf(i,i), freqs, 'Hz'))), '-', ...
         'HandleVisibility', 'off');
end
% Off diagonal terms
plot(freqs, abs(squeeze(freqresp(Gdvf(1, 2), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ...
     'DisplayName', '$F_{m,i}/\tau_j \quad i \neq j$');
for i = 1:5
    for j = i+1:6
        plot(freqs, abs(squeeze(freqresp(Gdvf(i, j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ...
             'HandleVisibility', 'off');
    end
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
legend('location', 'southeast');

ax2 = nexttile;
hold on;
for i = 1:6
    set(gca,'ColorOrderIndex',1);
    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/nano_hexapod_struts_2dof_dvf_plant.pdf', 'width', 'full', 'height', 'tall');
#+end_src

#+name: fig:nano_hexapod_struts_2dof_dvf_plant
#+caption: Bode plot of the transfer functions from actuator forces $\tau_i$ to relative motion sensors attached to the struts $\mathcal{L}_i$. Diagonal terms are shown in blue, and off-diagonal terms in black.
#+RESULTS:
[[file:figs/nano_hexapod_struts_2dof_dvf_plant.png]]

** Integral Force Feedback Plant
<<sec:iff_plant>>

The transfer function from actuators to force sensors is identified.
#+begin_src matlab
%% Options for Linearized
options = linearizeOptions;
options.SampleTime = 0;

%% Name of the Simulink File
mdl = 'nano_hexapod';

%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/F'],  1, 'openinput');   io_i = io_i + 1; % Actuator Inputs
io(io_i) = linio([mdl, '/Fm'],  1, 'openoutput'); io_i = io_i + 1; % Force Sensors

n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
                                       'flex_top_type', '3dof', ...
                                       'motion_sensor_type', 'struts', ...
                                       'actuator_type', '2dof');

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

This is corresponding to the dynamics for the Integral Force Feedback (IFF) control strategy.

The bode plot is shown in Figure [[fig:nano_hexapod_struts_2dof_iff_plant]].
#+begin_src matlab :exports none
freqs = 5*logspace(0, 2, 1000);

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

ax1 = nexttile([2,1]);
hold on;
plot(freqs, abs(squeeze(freqresp(Giff(1,1), freqs, 'Hz'))), '-', ...
     'DisplayName', '$F_{m,i}/\tau_i$')
for i = 1:6
    set(gca,'ColorOrderIndex',1);
    plot(freqs, abs(squeeze(freqresp(Giff(i,i), freqs, 'Hz'))), '-', ...
         'HandleVisibility', 'off');
end
% Off diagonal terms
plot(freqs, abs(squeeze(freqresp(Giff(1, 2), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ...
     'DisplayName', '$F_{m,i}/\tau_j \quad i \neq j$');
for i = 1:5
    for j = i+1:6
        plot(freqs, abs(squeeze(freqresp(Giff(i, j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ...
             'HandleVisibility', 'off');
    end
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [N/N]'); set(gca, 'XTickLabel',[]);
legend('location', 'southeast');

ax2 = nexttile;
hold on;
for i = 1:6
    set(gca,'ColorOrderIndex',1);
    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/nano_hexapod_struts_2dof_iff_plant.pdf', 'width', 'full', 'height', 'tall');
#+end_src

#+name: fig:nano_hexapod_struts_2dof_iff_plant
#+caption: Bode plot of the transfer functions from actuator forces $\tau_i$ to force sensors $F_{m,i}$. Diagonal terms are shown in blue, and off-diagonal terms in black.
#+RESULTS:
[[file:figs/nano_hexapod_struts_2dof_iff_plant.png]]

** Decentralized Plant - Cartesian coordinates
<<sec:decoupled_plant>>
*** Introduction                                                    :ignore:
Consider the plant shown in Figure [[fig:nano_hexapod_decentralized_schematic]] with:
- $\tau$ the 6 input forces (APA)
- $d\mathcal{L}$ the relative motion sensor outputs (encoders)
- $\mathcal{X}$ the motion of the top platform measured with "perfect" 6-dof sensor
- $J_a$ and $J_s$ the Jacobians for the actuators and sensors

#+begin_src latex :file nano_hexapod_decentralized_schematic.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[] (outputX) 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) {$\bm{J}^{-1}_s$};

  % 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[->] (outputX.east) -- ++(1, 0) node[above left]{$\bm{\mathcal{X}}$};
\end{tikzpicture}
#+end_src

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

*** Verification of the Sensor Jacobian
The "perfect" sensor output $\mathcal{X}$ is used to verify that the sensor Jacobian is working correctly both when the encoders are fixed to the struts and to the plates.

Let's then identify the plant for both configuration, and compare the transfer functions from $\mathcal{F}$ to $d\mathcal{X}$ and to $\mathcal{X}$.
#+begin_src matlab
%% Options for Linearized
options = linearizeOptions;
options.SampleTime = 0;

%% Name of the Simulink File
mdl = 'nano_hexapod';

%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/F'],  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
io(io_i) = linio([mdl, '/X'],  1, 'openoutput'); io_i = io_i + 1; % Relative Motion Outputs
#+end_src

Start when the encoders are fixed on the struts.
#+begin_src matlab
n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
                                       'flex_top_type', '3dof', ...
                                       'motion_sensor_type', 'struts', ...
                                       'actuator_type', '2dof', ...
                                       'MO_B', 150e-3);
Gs = linearize(mdl, io, 0.0, options);
Gs.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
Gs.OutputName = {'D1', 'D2', 'D3', 'D4', 'D5', 'D6', ...
                 'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'};

% Cartesian plant using the Jacobians
Gsc = inv(n_hexapod.geometry.Js)*Gs({'D1', 'D2', 'D3', 'D4', 'D5', 'D6'}, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'})*inv(n_hexapod.geometry.J)';
% Cartesian plant using the perfect sensor
Gsp = -Gs({'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'}, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'})*inv(n_hexapod.geometry.J)';
#+end_src

The diagonal elements of the plant are shown in Figure [[fig:nano_hexapod_comp_cartesian_plants_struts]].
#+begin_src matlab :exports none
freqs = 5*logspace(0, 2, 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;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');

ax1 = nexttile([2,1]);
hold on;
for i = 1:6
    plot(freqs, abs(squeeze(freqresp(Gsc(i,i), freqs, 'Hz'))), '-', ...
         'DisplayName', labels{i});
end
set(gca,'ColorOrderIndex',1);
for i = 1:6
    plot(freqs, abs(squeeze(freqresp(Gsp(i,i), freqs, 'Hz'))), '--', ...
         'HandleVisibility', 'off');
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
legend('location', 'southwest');
ylim([1e-11, 1e-4]);

ax2 = nexttile;
hold on;
for i = 1:6
    plot(freqs, 180/pi*angle(squeeze(freqresp(Gsc(i,i), freqs, 'Hz'))), '-');
end
set(gca,'ColorOrderIndex',1);
for i = 1:6
    plot(freqs, 180/pi*angle(squeeze(freqresp(Gsp(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/nano_hexapod_comp_cartesian_plants_struts.pdf', 'width', 'wide', 'height', 'tall');
#+end_src

#+name: fig:nano_hexapod_comp_cartesian_plants_struts
#+caption: Bode plot of the diagonal elements of the decentralized (cartesian) plant when using the sensor Jacobian (solid) and when using "perfect" 6dof sensor (dashed). The encoders are fixed on the struts.
#+RESULTS:
[[file:figs/nano_hexapod_comp_cartesian_plants_struts.png]]

The same if performed when the encoders are fixed to the plates.
#+begin_src matlab
n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
                                       'flex_top_type', '3dof', ...
                                       'motion_sensor_type', 'plates', ...
                                       'actuator_type', '2dof', ...
                                       'MO_B', 150e-3);
Gp = linearize(mdl, io, 0.0, options);
Gp.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
Gp.OutputName = {'D1', 'D2', 'D3', 'D4', 'D5', 'D6', ...
                 'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'};

% Cartesian plant using the Jacobians
Gpc = inv(n_hexapod.geometry.Js)*Gp({'D1', 'D2', 'D3', 'D4', 'D5', 'D6'}, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'})*inv(n_hexapod.geometry.J)';
% Cartesian plant using the perfect sensor
Gpp = -Gp({'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'}, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'})*inv(n_hexapod.geometry.J)';
#+end_src

The obtained bode plots are shown in Figure [[fig:nano_hexapod_comp_cartesian_plants_plates]].
#+begin_src matlab :exports none
freqs = 5*logspace(0, 2, 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;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');

ax1 = nexttile([2,1]);
hold on;
for i = 1:6
    plot(freqs, abs(squeeze(freqresp(Gpc(i,i), freqs, 'Hz'))), '-', ...
         'DisplayName', labels{i});
end
set(gca,'ColorOrderIndex',1);
for i = 1:6
    plot(freqs, abs(squeeze(freqresp(Gpp(i,i), freqs, 'Hz'))), '--', ...
         'HandleVisibility', 'off');
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
legend('location', 'southwest');
ylim([1e-11, 1e-4]);

ax2 = nexttile;
hold on;
for i = 1:6
    plot(freqs, 180/pi*angle(squeeze(freqresp(Gpc(i,i), freqs, 'Hz'))), '-');
end
set(gca,'ColorOrderIndex',1);
for i = 1:6
    plot(freqs, 180/pi*angle(squeeze(freqresp(Gpp(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/nano_hexapod_comp_cartesian_plants_plates.pdf', 'width', 'wide', 'height', 'tall');
#+end_src

#+name: fig:nano_hexapod_comp_cartesian_plants_plates
#+caption: Bode plot of the diagonal elements of the decentralized (cartesian) plant when using the sensor Jacobian (solid) and when using "perfect" 6dof sensor (dashed). The encoders are fixed on the plates.
#+RESULTS:
[[file:figs/nano_hexapod_comp_cartesian_plants_plates.png]]

#+begin_important
The Jacobian for the encoders is working properly both when the encoders are fixed to the plates or to the struts.

However, then the encoders are fixed to the struts, there is a mismatch between the estimated motion and the measured motion above 100Hz due to a complex conjugate zero.
#+end_important

*** Comparison of the decentralized plants

The decentralized plants are now compared whether the encoders are fixed on the struts or on the plates in Figure [[fig:nano_hexapod_cartesian_plant_encoder_comp]].
#+begin_src matlab :exports none
freqs = 5*logspace(0, 2, 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;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');

ax1 = nexttile([2,1]);
hold on;
for i = 1:6
    plot(freqs, abs(squeeze(freqresp(Gsc(i,i), freqs, 'Hz'))), '-', ...
         'DisplayName', labels{i});
end
set(gca,'ColorOrderIndex',1);
for i = 1:6
    plot(freqs, abs(squeeze(freqresp(Gpc(i,i), freqs, 'Hz'))), '--', ...
         'HandleVisibility', 'off');
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
legend('location', 'southwest');
ylim([1e-11, 1e-4]);

ax2 = nexttile;
hold on;
for i = 1:6
    plot(freqs, 180/pi*angle(squeeze(freqresp(Gsc(i,i), freqs, 'Hz'))), '-');
end
set(gca,'ColorOrderIndex',1);
for i = 1:6
    plot(freqs, 180/pi*angle(squeeze(freqresp(Gpc(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/nano_hexapod_cartesian_plant_encoder_comp.pdf', 'width', 'full', 'height', 'tall');
#+end_src

#+name: fig:nano_hexapod_cartesian_plant_encoder_comp
#+caption: Bode plot of the "cartesian" plant (transfer function from $\mathcal{F}$ to $d\mathcal{X}$) when the encoders are fixed on the struts (solid) and on the plates (dashed)
#+RESULTS:
[[file:figs/nano_hexapod_cartesian_plant_encoder_comp.png]]

** Decentralized Plant - Decoupling at the Center of Stiffness
<<sec:decoupled_plant_cok>>

*** Center of Stiffness
<<sec:center_of_stiffness>>

Let's define some parameters:
#+begin_src matlab
si = n_hexapod.geometry.si; % Orientation of struts
bi = n_hexapod.geometry.Fb; % Location of bi w.r.t. {F}
ki = ones(1,6); % Normalized strut stiffness
#+end_src

In order to find is the Center of Stiffness (CoK) exists, we have to verify is the following is diagonal:
#+begin_src matlab :results value replace :exports both
ki.*si*si'
#+end_src

#+RESULTS:
|     1.8977 |  2.4659e-17 |  5.1838e-19 |
| 2.4659e-17 |      1.8977 | -2.3143e-05 |
| 5.1838e-19 | -2.3143e-05 |      2.2046 |

And we can find the location of the CoK with respect to {F}:
#+begin_src matlab
OkX = (ki.*cross(bi, si)*si')/(ki.*si*si');
Ok = [OkX(3,2);OkX(1,3);OkX(2,1)]
#+end_src

#+begin_src matlab :results value replace :exports results :tangle no
ans = Ok
#+end_src

#+RESULTS:
| -1.7444e-18 |
|  2.1511e-06 |
|    0.052707 |

The center of the cube is therefore 52.7mm above the bottom platform {F} frame.

Let's initialize the hexapod with frame {A} and {B} at the CoK:
#+begin_src matlab
n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
                                       'flex_top_type', '3dof', ...
                                       'motion_sensor_type', 'struts', ...
                                       'actuator_type', '2dof', ...
                                       'MO_B', Ok(3)-95e-3);
#+end_src

And the (normalized) stiffness matrix is computed as follows:
#+begin_src matlab :results value replace
n_hexapod.geometry.J'*diag(ki)*n_hexapod.geometry.J
#+end_src

#+name: tab:stiffness_matrix_cok
#+caption: Normalized Stiffness Matrix - Center of Stiffness
#+attr_latex: :environment tabularx :width 0.9\linewidth :align cccccc
#+attr_latex: :center t :booktabs t :float t
#+RESULTS:
|      1.8977 |           0 |           0 |           0 | -2.0817e-17 | -1.5311e-06 |
|           0 |      1.8977 | -2.3143e-05 |   4.175e-06 |           0 |           0 |
|           0 | -2.3143e-05 |      2.2046 |  4.7422e-06 |           0 |           0 |
|           0 |   4.175e-06 |  4.7422e-06 |    0.012594 |  2.1684e-19 | -8.6736e-19 |
| -1.8521e-17 |           0 |           0 |           0 |    0.012594 | -9.3183e-08 |
| -1.5311e-06 | -6.9389e-18 |  2.7756e-17 | -8.6736e-19 | -9.3183e-08 |    0.043362 |

And we indeed obtain a diagonal stiffness matrix.

*** Obtained plant

#+begin_src matlab
%% Options for Linearized
options = linearizeOptions;
options.SampleTime = 0;

%% Name of the Simulink File
mdl = 'nano_hexapod';

%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/F'],  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
io(io_i) = linio([mdl, '/X'],  1, 'openoutput'); io_i = io_i + 1; % Relative Motion Outputs
#+end_src

#+begin_src matlab
G = linearize(mdl, io, 0.0, options);
G.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
G.OutputName = {'D1', 'D2', 'D3', 'D4', 'D5', 'D6', ...
                'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'};

% % Cartesian plant using the Jacobians
% Gsc = inv(n_hexapod.geometry.Js)*Gs({'D1', 'D2', 'D3', 'D4', 'D5', 'D6'}, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'})*inv(n_hexapod.geometry.J)';
% % Cartesian plant using the perfect sensor
% Gsp = -Gs({'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'}, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'})*inv(n_hexapod.geometry.J)';
#+end_src

#+begin_src matlab
Gc = inv(n_hexapod.geometry.J)*G({'D1', 'D2', 'D3', 'D4', 'D5', 'D6'}, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'})*inv(n_hexapod.geometry.J');
#+end_src

#+begin_src matlab :results value replace :exports results :tangle no
dcgain(Gc)
#+end_src

#+RESULTS:
|  9.8602e-09 |  7.8692e-11 |  1.4426e-11 |  -2.663e-10 |     2.8e-10 | -4.7559e-11 |
|   8.457e-11 |  9.8788e-09 | -2.4002e-11 | -2.9502e-11 | -1.3262e-10 | -8.7346e-11 |
| -4.3244e-11 |  2.4075e-13 |  8.4775e-09 |  1.1442e-11 | -2.5809e-10 |  2.8796e-11 |
| -1.8326e-09 |  -9.318e-10 |  6.8188e-10 |  1.4697e-06 |  5.5936e-09 |  8.7632e-10 |
|  4.6906e-10 |  1.5911e-09 |  1.6989e-10 |  -5.223e-09 |  1.4729e-06 | -2.6059e-10 |
| -6.5754e-11 | -3.0408e-12 |   5.394e-11 | -1.0917e-10 |  6.9479e-10 |  4.2979e-07 |

As the rotations and translations have very different gains, we normalize each motion to one.
#+begin_src matlab
Gc = diag(1./diag(dcgain(Gc)))*Gc;
#+end_src

The diagonal and off-diagonal elements are shown in Figure [[fig:nano_hexapod_diagonal_plant_cok]], and we can see good decoupling at low frequency.
#+begin_src matlab :exports none
freqs = logspace(0, 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');
xlim([freqs(1), freqs(end)]); ylim([1e-8, 1e3])
legend('location', 'southwest');
#+end_src

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

#+name: fig:nano_hexapod_diagonal_plant_cok
#+caption: Diagonal and off-diagonal elements of the (normalized) decentralized plant with the Jacobians estimated at the "center of stiffness"
#+RESULTS:
[[file:figs/nano_hexapod_diagonal_plant_cok.png]]

#+begin_important
The Jacobian matrices can be used to decoupled the plant at low frequency.
#+end_important

** Stiffness matrix
*** Introduction                                                    :ignore:

The stiffness matrix of the nano-hexapod describes its induced static displacement/rotation when a force/torque is applied on its top platform.
The location of the applied force/torque and the expressed displacement/rotation can be defined as wanted.
Such location (or frame) is then used for the computation of the Jacobian which in turns is used to compute the stiffness matrix.

*** Compute the theoretical stiffness of the nano-hexapod

Neglecting stiffness of the joints, we have:
\[ K = J^t \mathcal{K} J \]
where $\mathcal{K}$ is a diagonal 6x6 matrix with axial stiffness of the struts on the diagonal.

Let's note the axial stiffness of the APA:
\[ k_{\text{APA}} = k + \frac{k_e k_a}{k_e + k_a} \]

Them axial stiffness of the struts $k_s$:
\[ k_s = \frac{k_z k_{\text{APA}}}{k_z + 2 k_{\text{APA}}} \]
with $k_z$ the axial stiffness of the flexible joints.

#+begin_src matlab
n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
                                       'flex_top_type', '4dof', ...
                                       'motion_sensor_type', 'struts', ...
                                       'actuator_type', '2dof');
#+end_src

#+begin_src matlab
k  = n_hexapod.actuator.k(1);
ke = n_hexapod.actuator.ke(1);
ka = n_hexapod.actuator.ka(1);
kz = n_hexapod.flex_top.kz(1);
#+end_src

#+begin_src matlab
kAPA = k + ke*ka/(ke + ka);
#+end_src

#+begin_src matlab :results value replace :exports results :tangle no
sprintf('kAPA = %.3e [N/m]', kAPA)
#+end_src

#+RESULTS:
: kAPA = 1.799e+06 [N/m]

#+begin_src matlab
ks = kz*kAPA/(kz + 2*kAPA);
#+end_src

#+begin_src matlab :results value replace :exports results :tangle no
sprintf('ks = %.3e [N/m]', ks)
#+end_src

#+RESULTS:
: ks = 1.737e+06 [N/m]

We can see that the axial stiffness of the flexible joint as little impact on the total axial stiffness of the struts.

Let's now compute the stiffness matrix corresponding to an hexapod with perfect joints and the above computed axial stiffness:
#+begin_src matlab
Ks = n_hexapod.geometry.J'*(ks*eye(6))*n_hexapod.geometry.J;
#+end_src

And the compliance matrix can be computed as the inverse of the stiffness matrix:
#+begin_src matlab
C = inv(Ks);
#+end_src

#+begin_src matlab :results value replace :exports results :tangle no
ans = C
#+end_src

#+name: tab:compliance_matrix_perfect_joints
#+caption: Compliance Matrix - Perfect Joints
#+attr_latex: :environment tabularx :width 0.9\linewidth :align cccccc
#+attr_latex: :center t :booktabs t :float t
#+RESULTS:
|  1.9938e-06 | -2.3138e-22 |  3.3403e-23 |  1.0202e-21 |  8.7906e-06 |  2.9603e-11 |
| -3.1875e-23 |  1.9938e-06 |  2.2094e-11 | -8.7909e-06 | -1.6576e-22 | -3.5622e-28 |
|  6.6811e-23 |  2.2094e-11 |  2.6115e-07 | -9.8337e-11 |  3.4744e-22 |  7.4663e-28 |
|  1.4054e-22 | -8.7909e-06 | -9.8337e-11 |  4.5715e-05 |  7.3086e-22 |  1.5706e-27 |
|  8.7906e-06 | -1.0202e-21 |  1.7371e-22 |   4.498e-21 |  4.5714e-05 |  9.8237e-11 |
|  2.9603e-11 | -1.9261e-22 | -1.7611e-27 |  8.4925e-22 |  9.8237e-11 |  1.3277e-05 |

*** Comparison with Simscape Model
#+begin_src matlab
%% Options for Linearized
options = linearizeOptions;
options.SampleTime = 0;

%% Name of the Simulink File
mdl = 'nano_hexapod';

%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Fe'],  1, 'openinput');  io_i = io_i + 1;
io(io_i) = linio([mdl, '/X'],   1, 'openoutput'); io_i = io_i + 1;

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

#+begin_src matlab :results value replace :exports results :tangle no
dcgain(G)
#+end_src

#+name: tab:compliance_matrix_simscape_joints
#+caption: Compliance Matrix - Estimated from Simscape
#+attr_latex: :environment tabularx :width 0.9\linewidth :align cccccc
#+attr_latex: :center t :booktabs t :float t
#+RESULTS:
|  1.9863e-06 |  2.4683e-08 |  2.2859e-09 | -6.4768e-08 |  8.7307e-06 |  3.2754e-10 |
|  -3.693e-09 |  1.9663e-06 | -7.4503e-09 |  -8.615e-06 | -6.7926e-08 | -2.0193e-08 |
| -3.5525e-09 | -8.8671e-10 |  2.6042e-07 |  4.7504e-09 | -9.9677e-09 |  1.7242e-10 |
|  2.1133e-08 | -8.6554e-06 |  3.2841e-08 |  4.4849e-05 |  3.0873e-07 |  8.1525e-08 |
|  8.7375e-06 |  9.9165e-08 |   8.135e-09 | -2.5208e-07 |  4.5331e-05 |  3.0602e-09 |
|  6.1611e-09 |  6.1733e-09 |  2.6778e-09 | -3.3188e-08 |  3.3887e-08 |  1.3212e-05 |

* To-order                                                          :noexport:

** Verify why unstable strut
#+begin_src matlab :results value replace
n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '3dof', ...
                                       'flex_top_type', '2dof', ...
                                       'motion_sensor_type', 'plates', ...
                                       'actuator_type', 'flexible');
isstable(linearize(mdl, io, 0.0, options))
#+end_src

#+RESULTS:
: 1

#+begin_src matlab :results value replace
n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '3dof', ...
                                       'flex_top_type', '2dof', ...
                                       'motion_sensor_type', 'struts', ...
                                       'actuator_type', 'flexible');
isstable(linearize(mdl, io, 0.0, options))
#+end_src

#+RESULTS:
: 0

#+begin_src matlab :results value replace
n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
                                       'flex_top_type', '3dof', ...
                                       'motion_sensor_type', 'plates', ...
                                       'actuator_type', 'flexible');
isstable(linearize(mdl, io, 0.0, options))
#+end_src

#+RESULTS:
: 0

#+begin_src matlab :results value replace
n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
                                       'flex_top_type', '3dof', ...
                                       'motion_sensor_type', 'struts', ...
                                       'actuator_type', 'flexible');
isstable(linearize(mdl, io, 0.0, options))
#+end_src

#+RESULTS:
: 0


** Verify why unstable strut
#+begin_src matlab :results value replace
n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '3dof', ...
                                       'flex_top_type', '2dof', ...
                                       'motion_sensor_type', 'plates', ...
                                       'actuator_type', '2dof');
isstable(linearize(mdl, io, 0.0, options))
#+end_src

#+RESULTS:
: 1

#+begin_src matlab :results value replace
n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '3dof', ...
                                       'flex_top_type', '2dof', ...
                                       'motion_sensor_type', 'struts', ...
                                       'actuator_type', '2dof');
isstable(linearize(mdl, io, 0.0, options))
#+end_src

#+RESULTS:
: 1

#+begin_src matlab :results value replace
n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
                                       'flex_top_type', '3dof', ...
                                       'motion_sensor_type', 'plates', ...
                                       'actuator_type', '2dof');
isstable(linearize(mdl, io, 0.0, options))
#+end_src

#+RESULTS:
: 1

#+begin_src matlab :results value replace
n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
                                       'flex_top_type', '3dof', ...
                                       'motion_sensor_type', 'struts', ...
                                       'actuator_type', '2dof');
isstable(linearize(mdl, io, 0.0, options))
#+end_src

#+RESULTS:
: 1


** APA300ML structure

#+begin_src matlab
open('matlab/nano_hexapod/apa300ml.slx')
#+end_src

#+begin_src matlab
%% Options for Linearized
options = linearizeOptions;
options.SampleTime = 0;

%% Name of the Simulink File
mdl = 'apa300ml_structure';

%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/F'],  1, 'openinput');  io_i = io_i + 1; % Actuator Inputs
io(io_i) = linio([mdl, '/Fa'],  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
io(io_i) = linio([mdl, '/x'],  1, 'openoutput'); io_i = io_i + 1; % Relative Motion Outputs

%% Run the linearization
G = linearize(mdl, io, 0.0, options);
G.InputName  = {'F', 'Fa'};
G.OutputName = {'D', 'x'};
#+end_src

#+begin_src matlab
dcgain(G('D', 'Fa'))/dcgain(G('x', 'Fa'))
#+end_src

#+begin_src matlab :exports none
freqs = logspace(1, 4, 1000);

figure;
tiledlayout(1, 2, 'TileSpacing', 'None', 'Padding', 'None');

ax1 = nexttile;
hold on;
plot(freqs, abs(squeeze(freqresp(G('D', 'Fa'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(G('x', 'Fa'), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);

ax2 = nexttile;
hold on;
plot(freqs, 180/pi*angle(squeeze(freqresp(G('D', 'Fa'), freqs, 'Hz'))));
plot(freqs, 180/pi*angle(squeeze(freqresp(G('x', 'Fa'), 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');
#+end_src

#+begin_src matlab :exports none
freqs = logspace(1, 4, 1000);

figure;

ax1 = subplot(2, 1, 1);
hold on;
plot(freqs, abs(squeeze(freqresp(G('D', 'Fa'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(G('x', 'Fa'), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);

ax2 = subplot(2, 1, 2);
hold on;
plot(freqs, 180/pi*angle(squeeze(freqresp(G('D', 'Fa'), freqs, 'Hz'))));
plot(freqs, 180/pi*angle(squeeze(freqresp(G('x', 'Fa'), 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');
#+end_src

** Full APA300ML
*** Matlab Init                                            :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
<<matlab-dir>>
#+end_src

#+begin_src matlab :exports none :results silent :noweb yes
<<matlab-init>>
#+end_src

#+begin_src matlab :tangle no
simulinkproject('../');
#+end_src

*** APA
#+begin_src matlab
K = readmatrix('APA300ML_full_mat_K.CSV');
M = readmatrix('APA300ML_full_mat_M.CSV');
[int_xyz, int_i, n_xyz, n_i, nodes] = extractNodes('APA300ML_full_out_nodes_3D.txt');
#+end_src

#+begin_src matlab
open('matlab/nano_hexapod/apa300ml.slx')
#+end_src

*** APA Stiffness
#+begin_src matlab
%% Options for Linearized
options = linearizeOptions;
options.SampleTime = 0;

%% Name of the Simulink File
mdl = 'apa300ml';

%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Fext'],  1, 'openinput');  io_i = io_i + 1;
io(io_i) = linio([mdl, '/L'],  1, 'openoutput'); io_i = io_i + 1;

%% Run the linearization
G = linearize(mdl, io, 0.0, options);
G.InputName  = {'Fe'};
G.OutputName = {'L'};
#+end_src

#+begin_src matlab :results value replace
1./dcgain(G)
#+end_src

#+RESULTS:
: 1796000.0

Same as documentation

*** Amplification Factor
#+begin_src matlab
%% Options for Linearized
options = linearizeOptions;
options.SampleTime = 0;

%% Name of the Simulink File
mdl = 'apa300ml';

%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/F'],  1, 'openinput');  io_i = io_i + 1;
io(io_i) = linio([mdl, '/L'],  1, 'openoutput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Ls'],  1, 'openoutput'); io_i = io_i + 1;
% io(io_i) = linio([mdl, '/Sensor Stack'],  1, 'openoutput'); io_i = io_i + 1;
% io(io_i) = linio([mdl, '/Actuator Stack'],  1, 'openoutput'); io_i = io_i + 1;

%% Run the linearization
G = linearize(mdl, io, 0.0, options);
G.InputName  = {'F'};
G.OutputName = {'L', 'Ls'};
#+end_src

#+begin_src matlab :results value replace
abs(dcgain(G('L', 'F'))./dcgain(G('Ls', 'F')))
#+end_src

#+RESULTS:
: 5.1155

Same as estimated.

*** Plant
Identification
#+begin_src matlab
%% Options for Linearized
options = linearizeOptions;
options.SampleTime = 0;

%% Name of the Simulink File
mdl = 'apa300ml';

%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/F'],  1, 'openinput');  io_i = io_i + 1; % Actuator Inputs
io(io_i) = linio([mdl, '/L'],  1, 'openoutput'); io_i = io_i + 1; % Relative Motion Outputs
io(io_i) = linio([mdl, '/Sensor Stack'],  1, 'openoutput'); io_i = io_i + 1; % Force Sensor

%% Run the linearization
G = linearize(mdl, io, 0.0, options);
G.InputName  = {'F'};
G.OutputName = {'L', 'dL'};
#+end_src

IFF Plant
#+begin_src matlab :exports none
freqs = logspace(1, 4, 1000);

figure;

ax1 = subplot(2, 1, 1);
hold on;
plot(freqs, abs(squeeze(freqresp(G('dL', 'F'), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [N/N]'); set(gca, 'XTickLabel',[]);

ax2 = subplot(2, 1, 2);
hold on;
plot(freqs, 180/pi*angle(squeeze(freqresp(G('dL', 'F'), 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');
#+end_src

DVF Plant
#+begin_src matlab :exports none
freqs = logspace(1, 4, 1000);

figure;

ax1 = subplot(2, 1, 1);
hold on;
plot(freqs, abs(squeeze(freqresp(G('L', 'F'), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);

ax2 = subplot(2, 1, 2);
hold on;
plot(freqs, 180/pi*angle(squeeze(freqresp(G('L', 'F'), 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');
#+end_src

*** Reduction from Full APA to 2DoF system

#+begin_src matlab
m = 1; % [kg]
#+end_src

Identification of the full APA model:
#+begin_src matlab
%% Options for Linearized
options = linearizeOptions;
options.SampleTime = 0;

%% Name of the Simulink File
mdl = 'apa300ml';

%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Fa'], 1, 'openinput');  io_i = io_i + 1; % Actuator Force
io(io_i) = linio([mdl, '/Fe'], 1, 'openinput');  io_i = io_i + 1; % External Force
io(io_i) = linio([mdl, '/Fm'], 1, 'openoutput'); io_i = io_i + 1; % Force Sensor
io(io_i) = linio([mdl, '/dL'], 1, 'openoutput'); io_i = io_i + 1; % Relative Motion Sensor

%% Run the linearization
G = linearize(mdl, io, 0.0, options);
G.InputName  = {'Fa', 'Fe'};
G.OutputName = {'Fm', 'dL'};
#+end_src

#+begin_src matlab :results output replace :exports results :tangle no
size(G)
#+end_src

#+RESULTS:
: size(G)
: State-space model with 2 outputs, 2 inputs, and 50 states.

#+begin_src matlab
k  = 0.35e6;
c  = 3e1;
ka = 43e6;
ke = 1.5e6;

Leq = 0.056; % [m]
#+end_src

Identification of the 2-DoF APA model:
#+begin_src matlab
open('matlab/nano_hexapod/apa300ml_2dof.slx')

%% Options for Linearized
options = linearizeOptions;
options.SampleTime = 0;

%% Name of the Simulink File
mdl = 'apa300ml_2dof';

%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Fa'], 1, 'openinput');  io_i = io_i + 1; % Actuator Force
io(io_i) = linio([mdl, '/Fe'], 1, 'openinput');  io_i = io_i + 1; % External Force
io(io_i) = linio([mdl, '/Fm'], 1, 'openoutput'); io_i = io_i + 1; % Force Sensor
io(io_i) = linio([mdl, '/dL'], 1, 'openoutput'); io_i = io_i + 1; % Relative Motion Sensor

%% Run the linearization
Gb = linearize(mdl, io, 0.0, options);
Gb.InputName  = {'Fa', 'Fe'};
Gb.OutputName = {'Fm', 'dL'};
#+end_src

#+begin_src matlab :results output replace :exports results :tangle no
size(Gb)
#+end_src

#+RESULTS:
: size(Gb)
: State-space model with 2 outputs, 2 inputs, and 4 states.

From external force to displacement
#+begin_src matlab :exports none
freqs = logspace(1, 4, 1000);

figure;

ax1 = subplot(2, 1, 1);
hold on;
plot(freqs, abs(squeeze(freqresp(G( 'dL', 'Fe'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gb('dL', 'Fe'), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [N/N]'); set(gca, 'XTickLabel',[]);

ax2 = subplot(2, 1, 2);
hold on;
plot(freqs, 180/pi*angle(squeeze(freqresp(G( 'dL', 'Fe'), freqs, 'Hz'))));
plot(freqs, 180/pi*angle(squeeze(freqresp(Gb('dL', 'Fe'), 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');
#+end_src

From external force to force sensor
#+begin_src matlab :exports none
freqs = logspace(1, 4, 1000);

figure;

ax1 = subplot(2, 1, 1);
hold on;
plot(freqs, abs(squeeze(freqresp(G( 'Fm', 'Fe'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gb('Fm', 'Fe'), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [N/N]'); set(gca, 'XTickLabel',[]);

ax2 = subplot(2, 1, 2);
hold on;
plot(freqs, 180/pi*angle(squeeze(freqresp(G( 'Fm', 'Fe'), freqs, 'Hz'))));
plot(freqs, 180/pi*angle(squeeze(freqresp(Gb('Fm', 'Fe'), 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');
#+end_src

From Actuator to displacement
#+begin_src matlab :exports none
freqs = logspace(1, 4, 1000);

figure;

ax1 = subplot(2, 1, 1);
hold on;
plot(freqs, abs(squeeze(freqresp(G( 'dL', 'Fa'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gb('dL', 'Fa'), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [N/N]'); set(gca, 'XTickLabel',[]);

ax2 = subplot(2, 1, 2);
hold on;
plot(freqs, 180/pi*angle(squeeze(freqresp(G( 'dL', 'Fa'), freqs, 'Hz'))));
plot(freqs, 180/pi*angle(squeeze(freqresp(Gb('dL', 'Fa'), 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');
#+end_src

From Actuator to Force Sensor
#+begin_src matlab :exports none
freqs = logspace(1, 4, 1000);

figure;

ax1 = subplot(2, 1, 1);
hold on;
plot(freqs, abs(squeeze(freqresp(G( 'Fm', 'Fa'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gb('Fm', 'Fa'), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [N/N]'); set(gca, 'XTickLabel',[]);

ax2 = subplot(2, 1, 2);
hold on;
plot(freqs, 180/pi*angle(squeeze(freqresp(G( 'Fm', 'Fa'), freqs, 'Hz'))));
plot(freqs, 180/pi*angle(squeeze(freqresp(Gb('Fm', 'Fa'), 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');
#+end_src

#+begin_src matlab
tiledlayout(2, 2, 'TileSpacing', 'None', 'Padding', 'None');

ax1 = nexttile;
hold on;
plot(freqs, abs(squeeze(freqresp(G( 'dL', 'Fa'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gb('dL', 'Fa'), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
set(gca, 'XTickLabel',[]); ylabel('To vertical motion [m]');
title('From Actuator $F_a$ [N]');

ax2 = nexttile;
hold on;
plot(freqs, abs(squeeze(freqresp(G( 'dL', 'Fe'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gb('dL', 'Fe'), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]);
title('From External $F_e$ [N]');

ax3 = nexttile;
hold on;
plot(freqs, abs(squeeze(freqresp(G( 'Fm', 'Fa'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gb('Fm', 'Fa'), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('To Force Sensor [N]');

ax4 = nexttile;
hold on;
plot(freqs, abs(squeeze(freqresp(G( 'Fm', 'Fe'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gb('Fm', 'Fe'), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]);

linkaxes([ax1,ax2,ax3,ax4],'x');
linkaxes([ax1,ax2],'y');
linkaxes([ax3,ax4],'y');
#+end_src

*** Reduction from Full APA to 2DoF system

#+begin_src matlab
Ms = [0.1, 1, 5, 10]; % [kg]
#+end_src

Identification of the full APA model:
#+begin_src matlab
%% Options for Linearized
options = linearizeOptions;
options.SampleTime = 0;

%% Name of the Simulink File
mdl = 'apa300ml';

%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Fa'], 1, 'openinput');  io_i = io_i + 1; % Actuator Force
io(io_i) = linio([mdl, '/Fe'], 1, 'openinput');  io_i = io_i + 1; % External Force
io(io_i) = linio([mdl, '/Fm'], 1, 'openoutput'); io_i = io_i + 1; % Force Sensor
io(io_i) = linio([mdl, '/dL'], 1, 'openoutput'); io_i = io_i + 1; % Relative Motion Sensor

%% Run the linearization
G = linearize(mdl, io, 0.0, options);
G.InputName  = {'Fa', 'Fe'};
G.OutputName = {'Fm', 'dL'};
#+end_src

#+begin_src matlab :results output replace :exports results :tangle no
size(G)
#+end_src

#+RESULTS:
: size(G)
: State-space model with 2 outputs, 2 inputs, and 50 states.

#+begin_src matlab
k  = 0.35e6;
c  = 3e1;
ka = 43e6;
ke = 1.5e6;

Leq = 0.056; % [m]
#+end_src

Identification of the 2-DoF APA model:
#+begin_src matlab
open('matlab/nano_hexapod/apa300ml_2dof.slx')

%% Options for Linearized
options = linearizeOptions;
options.SampleTime = 0;

%% Name of the Simulink File
mdl = 'apa300ml_2dof';

%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Fa'], 1, 'openinput');  io_i = io_i + 1; % Actuator Force
io(io_i) = linio([mdl, '/Fe'], 1, 'openinput');  io_i = io_i + 1; % External Force
io(io_i) = linio([mdl, '/Fm'], 1, 'openoutput'); io_i = io_i + 1; % Force Sensor
io(io_i) = linio([mdl, '/dL'], 1, 'openoutput'); io_i = io_i + 1; % Relative Motion Sensor

%% Run the linearization
Gb = linearize(mdl, io, 0.0, options);
Gb.InputName  = {'Fa', 'Fe'};
Gb.OutputName = {'Fm', 'dL'};
#+end_src

#+begin_src matlab :results output replace :exports results :tangle no
size(Gb)
#+end_src

#+RESULTS:
: size(Gb)
: State-space model with 2 outputs, 2 inputs, and 4 states.

From external force to displacement
#+begin_src matlab :exports none
freqs = logspace(1, 4, 1000);

figure;

ax1 = subplot(2, 1, 1);
hold on;
plot(freqs, abs(squeeze(freqresp(G( 'dL', 'Fe'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gb('dL', 'Fe'), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [N/N]'); set(gca, 'XTickLabel',[]);

ax2 = subplot(2, 1, 2);
hold on;
plot(freqs, 180/pi*angle(squeeze(freqresp(G( 'dL', 'Fe'), freqs, 'Hz'))));
plot(freqs, 180/pi*angle(squeeze(freqresp(Gb('dL', 'Fe'), 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');
#+end_src

From external force to force sensor
#+begin_src matlab :exports none
freqs = logspace(1, 4, 1000);

figure;

ax1 = subplot(2, 1, 1);
hold on;
plot(freqs, abs(squeeze(freqresp(G( 'Fm', 'Fe'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gb('Fm', 'Fe'), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [N/N]'); set(gca, 'XTickLabel',[]);

ax2 = subplot(2, 1, 2);
hold on;
plot(freqs, 180/pi*angle(squeeze(freqresp(G( 'Fm', 'Fe'), freqs, 'Hz'))));
plot(freqs, 180/pi*angle(squeeze(freqresp(Gb('Fm', 'Fe'), 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');
#+end_src

From Actuator to displacement
#+begin_src matlab :exports none
freqs = logspace(1, 4, 1000);

figure;

ax1 = subplot(2, 1, 1);
hold on;
plot(freqs, abs(squeeze(freqresp(G( 'dL', 'Fa'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gb('dL', 'Fa'), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [N/N]'); set(gca, 'XTickLabel',[]);

ax2 = subplot(2, 1, 2);
hold on;
plot(freqs, 180/pi*angle(squeeze(freqresp(G( 'dL', 'Fa'), freqs, 'Hz'))));
plot(freqs, 180/pi*angle(squeeze(freqresp(Gb('dL', 'Fa'), 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');
#+end_src

From Actuator to Force Sensor
#+begin_src matlab :exports none
freqs = logspace(1, 4, 1000);

figure;

ax1 = subplot(2, 1, 1);
hold on;
plot(freqs, abs(squeeze(freqresp(G( 'Fm', 'Fa'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gb('Fm', 'Fa'), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [N/N]'); set(gca, 'XTickLabel',[]);

ax2 = subplot(2, 1, 2);
hold on;
plot(freqs, 180/pi*angle(squeeze(freqresp(G( 'Fm', 'Fa'), freqs, 'Hz'))));
plot(freqs, 180/pi*angle(squeeze(freqresp(Gb('Fm', 'Fa'), 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');
#+end_src

#+begin_src matlab
tiledlayout(2, 2, 'TileSpacing', 'None', 'Padding', 'None');

ax1 = nexttile;
hold on;
plot(freqs, abs(squeeze(freqresp(G( 'dL', 'Fa'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gb('dL', 'Fa'), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
set(gca, 'XTickLabel',[]); ylabel('To vertical motion [m]');
title('From Actuator $F_a$ [N]');

ax2 = nexttile;
hold on;
plot(freqs, abs(squeeze(freqresp(G( 'dL', 'Fe'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gb('dL', 'Fe'), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]);
title('From External $F_e$ [N]');

ax3 = nexttile;
hold on;
plot(freqs, abs(squeeze(freqresp(G( 'Fm', 'Fa'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gb('Fm', 'Fa'), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('To Force Sensor [N]');

ax4 = nexttile;
hold on;
plot(freqs, abs(squeeze(freqresp(G( 'Fm', 'Fe'), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Gb('Fm', 'Fe'), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]);

linkaxes([ax1,ax2,ax3,ax4],'x');
linkaxes([ax1,ax2],'y');
linkaxes([ax3,ax4],'y');
#+end_src

* Function - Initialize Nano Hexapod
:PROPERTIES:
:header-args:matlab+: :tangle ../src/initializeNanoHexapodFinal.m
:header-args:matlab+: :comments none :mkdirp yes :eval no
:END:
<<sec:initializeNanoHexapodFinal>>

** Function description
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
function [nano_hexapod] = initializeNanoHexapodFinal(args)
#+end_src

** Optional Parameters
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
arguments
    %% Bottom Flexible Joints
    args.flex_bot_type      char {mustBeMember(args.flex_bot_type,{'2dof', '3dof', '4dof', 'flexible'})} = '4dof'
    args.flex_bot_kRx (6,1) double {mustBeNumeric} = ones(6,1)*5 % X bending stiffness [Nm/rad]
    args.flex_bot_kRy (6,1) double {mustBeNumeric} = ones(6,1)*5 % Y bending stiffness [Nm/rad]
    args.flex_bot_kRz (6,1) double {mustBeNumeric} = ones(6,1)*260 % Torsionnal stiffness [Nm/rad]
    args.flex_bot_kz  (6,1) double {mustBeNumeric} = ones(6,1)*1e8 % Axial Stiffness [N/m]
    args.flex_bot_cRx (6,1) double {mustBeNumeric} = ones(6,1)*0.1 % X bending Damping [Nm/(rad/s)]
    args.flex_bot_cRy (6,1) double {mustBeNumeric} = ones(6,1)*0.1 % Y bending Damping [Nm/(rad/s)]
    args.flex_bot_cRz (6,1) double {mustBeNumeric} = ones(6,1)*0.1 % Torsionnal Damping [Nm/(rad/s)]
    args.flex_bot_cz  (6,1) double {mustBeNumeric} = ones(6,1)*1e2 % Axial Damping [N/(m/s)]
    %% Top Flexible Joints
    args.flex_top_type      char {mustBeMember(args.flex_top_type,{'2dof', '3dof', '4dof', 'flexible'})} = '4dof'
    args.flex_top_kRx (6,1) double {mustBeNumeric} = ones(6,1)*5 % X bending stiffness [Nm/rad]
    args.flex_top_kRy (6,1) double {mustBeNumeric} = ones(6,1)*5 % Y bending stiffness [Nm/rad]
    args.flex_top_kRz (6,1) double {mustBeNumeric} = ones(6,1)*260 % Torsionnal stiffness [Nm/rad]
    args.flex_top_kz  (6,1) double {mustBeNumeric} = ones(6,1)*1e8 % Axial Stiffness [N/m]
    args.flex_top_cRx (6,1) double {mustBeNumeric} = ones(6,1)*0.1 % X bending Damping [Nm/(rad/s)]
    args.flex_top_cRy (6,1) double {mustBeNumeric} = ones(6,1)*0.1 % Y bending Damping [Nm/(rad/s)]
    args.flex_top_cRz (6,1) double {mustBeNumeric} = ones(6,1)*0.1 % Torsionnal Damping [Nm/(rad/s)]
    args.flex_top_cz  (6,1) double {mustBeNumeric} = ones(6,1)*1e2 % Axial Damping [N/(m/s)]
    %% Jacobian - Location of frame {A} and {B}
    args.MO_B (1,1) double {mustBeNumeric} = 150e-3 % Height of {B} w.r.t. {M} [m]
    %% Relative Motion Sensor
    args.motion_sensor_type char {mustBeMember(args.motion_sensor_type,{'struts', 'plates'})} = 'struts'
    %% Actuators
    args.actuator_type char {mustBeMember(args.actuator_type,{'2dof', 'flexible frame', 'flexible'})} = 'flexible'
    args.actuator_Ga  (6,1) double {mustBeNumeric} = ones(6,1)*1 % Actuator gain [N/V]
    args.actuator_Gs  (6,1) double {mustBeNumeric} = ones(6,1)*1 % Sensor gain [V/m]
                                                                 % For 2DoF
    args.actuator_k   (6,1) double {mustBeNumeric} = ones(6,1)*0.35e6 % [N/m]
    args.actuator_ke  (6,1) double {mustBeNumeric} = ones(6,1)*1.5e6 % [N/m]
    args.actuator_ka  (6,1) double {mustBeNumeric} = ones(6,1)*43e6 % [N/m]
    args.actuator_c   (6,1) double {mustBeNumeric} = ones(6,1)*3e1 % [N/(m/s)]
    args.actuator_ce  (6,1) double {mustBeNumeric} = ones(6,1)*1e1 % [N/(m/s)]
    args.actuator_ca  (6,1) double {mustBeNumeric} = ones(6,1)*1e1 % [N/(m/s)]
    args.actuator_Leq (6,1) double {mustBeNumeric} = ones(6,1)*0.056 % [m]
                                                                     % For Flexible Frame
    args.actuator_ks (6,1) double {mustBeNumeric} = ones(6,1)*235e6 % Stiffness of one stack [N/m]
    args.actuator_cs (6,1) double {mustBeNumeric} = ones(6,1)*1e1 % Stiffness of one stack [N/m]
                                                                  % For Flexible
    args.actuator_xi  (1,1) double {mustBeNumeric} = 0.01 % Damping Ratio
end
#+end_src

** Nano Hexapod Object
:PROPERTIES:
:UNNUMBERED: t
:END:

#+begin_src matlab
nano_hexapod = struct();
#+end_src

** Flexible Joints - Bot
:PROPERTIES:
:UNNUMBERED: t
:END:

#+begin_src matlab
nano_hexapod.flex_bot = struct();

switch args.flex_bot_type
  case '2dof'
    nano_hexapod.flex_bot.type = 1;
  case '3dof'
    nano_hexapod.flex_bot.type = 2;
  case '4dof'
    nano_hexapod.flex_bot.type = 3;
  case 'flexible'
    nano_hexapod.flex_bot.type = 4;
end

nano_hexapod.flex_bot.kRx = args.flex_bot_kRx; % X bending stiffness [Nm/rad]
nano_hexapod.flex_bot.kRy = args.flex_bot_kRy; % Y bending stiffness [Nm/rad]
nano_hexapod.flex_bot.kRz = args.flex_bot_kRz; % Torsionnal stiffness [Nm/rad]
nano_hexapod.flex_bot.kz  = args.flex_bot_kz;  % Axial stiffness [N/m]

nano_hexapod.flex_bot.cRx = args.flex_bot_cRx; % [Nm/(rad/s)]
nano_hexapod.flex_bot.cRy = args.flex_bot_cRy; % [Nm/(rad/s)]
nano_hexapod.flex_bot.cRz = args.flex_bot_cRz; % [Nm/(rad/s)]
nano_hexapod.flex_bot.cz  = args.flex_bot_cz;  %[N/(m/s)]
#+end_src

** Flexible Joints - Top
:PROPERTIES:
:UNNUMBERED: t
:END:

#+begin_src matlab
nano_hexapod.flex_top = struct();

switch args.flex_top_type
  case '2dof'
    nano_hexapod.flex_top.type = 1;
  case '3dof'
    nano_hexapod.flex_top.type = 2;
  case '4dof'
    nano_hexapod.flex_top.type = 3;
  case 'flexible'
    nano_hexapod.flex_top.type = 4;
end

nano_hexapod.flex_top.kRx = args.flex_top_kRx; % X bending stiffness [Nm/rad]
nano_hexapod.flex_top.kRy = args.flex_top_kRy; % Y bending stiffness [Nm/rad]
nano_hexapod.flex_top.kRz = args.flex_top_kRz; % Torsionnal stiffness [Nm/rad]
nano_hexapod.flex_top.kz  = args.flex_top_kz;  % Axial stiffness [N/m]

nano_hexapod.flex_top.cRx = args.flex_top_cRx; % [Nm/(rad/s)]
nano_hexapod.flex_top.cRy = args.flex_top_cRy; % [Nm/(rad/s)]
nano_hexapod.flex_top.cRz = args.flex_top_cRz; % [Nm/(rad/s)]
nano_hexapod.flex_top.cz  = args.flex_top_cz;  %[N/(m/s)]
#+end_src

** Relative Motion Sensor
:PROPERTIES:
:UNNUMBERED: t
:END:

#+begin_src matlab
nano_hexapod.motion_sensor = struct();

switch args.motion_sensor_type
  case 'struts'
    nano_hexapod.motion_sensor.type = 1;
  case 'plates'
    nano_hexapod.motion_sensor.type = 2;
end
#+end_src

** Amplified Piezoelectric Actuator
:PROPERTIES:
:UNNUMBERED: t
:END:

#+begin_src matlab
nano_hexapod.actuator = struct();

switch args.actuator_type
  case '2dof'
    nano_hexapod.actuator.type = 1;
  case 'flexible frame'
    nano_hexapod.actuator.type = 2;
  case 'flexible'
    nano_hexapod.actuator.type = 3;
end
#+end_src

#+begin_src matlab
nano_hexapod.actuator.Ga = args.actuator_Ga; % Actuator gain [N/V]
nano_hexapod.actuator.Gs = args.actuator_Gs; % Sensor gain [V/m]
#+end_src

2dof
#+begin_src matlab
nano_hexapod.actuator.k  = args.actuator_k;  % [N/m]
nano_hexapod.actuator.ke = args.actuator_ke; % [N/m]
nano_hexapod.actuator.ka = args.actuator_ka; % [N/m]

nano_hexapod.actuator.c  = args.actuator_c;  % [N/(m/s)]
nano_hexapod.actuator.ce = args.actuator_ce; % [N/(m/s)]
nano_hexapod.actuator.ca = args.actuator_ca; % [N/(m/s)]

nano_hexapod.actuator.Leq = args.actuator_Leq; % [m]
#+end_src

Flexible frame and fully flexible
#+begin_src matlab
switch args.actuator_type
  case 'flexible frame'
    nano_hexapod.actuator.K = readmatrix('APA300ML_b_mat_K.CSV'); % Stiffness Matrix
    nano_hexapod.actuator.M = readmatrix('APA300ML_b_mat_M.CSV'); % Mass Matrix
    nano_hexapod.actuator.P = extractNodes('APA300ML_b_out_nodes_3D.txt'); % Node coordinates [m]
  case 'flexible'
    nano_hexapod.actuator.K = readmatrix('full_APA300ML_K.CSV'); % Stiffness Matrix
    nano_hexapod.actuator.M = readmatrix('full_APA300ML_M.CSV'); % Mass Matrix
    nano_hexapod.actuator.P = extractNodes('full_APA300ML_out_nodes_3D.txt'); % Node coordiantes [m]
end

nano_hexapod.actuator.xi = args.actuator_xi; % Damping ratio

nano_hexapod.actuator.ks = args.actuator_ks; % Stiffness of one stack [N/m]
nano_hexapod.actuator.cs = args.actuator_cs; % Damping of one stack [N/m]
#+end_src

** Geometry
:PROPERTIES:
:UNNUMBERED: t
:END:

#+begin_src matlab
nano_hexapod.geometry = struct();
#+end_src

Center of joints $a_i$ with respect to {F}:
#+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]
#+end_src

Center of joints $b_i$ with respect to {M}:
#+begin_src matlab
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]
#+end_src

Now compute the positions $b_i$ with respect to {F}:
#+begin_src matlab
Fb = Mb + [0; 0; 95e-3]; % Bi w.r.t. {F} [m]
#+end_src

The unit vector representing the orientation of the struts can then be computed:
#+begin_src matlab
si = Fb - Fa;
si = si./vecnorm(si); % Normalize
#+end_src

Location of encoder measurement points when fixed on the plates:
#+begin_src matlab
Fc = [[-29.362, -105.765, 52.605]
      [ 29.362, -105.765, 52.605]
      [ 106.276, 27.454,  52.605]
      [ 76.914,  78.31,   52.605]
      [-76.914,  78.31,   52.605]
      [-106.276, 27.454,  52.605]]'*1e-3; % Meas pos w.r.t. {F}
Mc = Fc - [0; 0; 95e-3]; % Meas pos w.r.t. {M}
#+end_src

#+begin_src matlab
nano_hexapod.geometry.Fa = Fa;
nano_hexapod.geometry.Fb = Fb;
nano_hexapod.geometry.Fc = Fc;
nano_hexapod.geometry.Mb = Mb;
nano_hexapod.geometry.Mc = Mc;
nano_hexapod.geometry.si = si;
nano_hexapod.geometry.MO_B = args.MO_B;
#+end_src

** Jacobian for Actuators
:PROPERTIES:
:UNNUMBERED: t
:END:

#+begin_src matlab
Bb = Mb - [0; 0; args.MO_B];

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

** Jacobian for Sensors
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
switch args.motion_sensor_type
  case 'struts'
    nano_hexapod.geometry.Js = nano_hexapod.geometry.J;
  case 'plates'
    Bc = Mc - [0; 0; args.MO_B];
    nano_hexapod.geometry.Js = [nano_hexapod.geometry.si', cross(Bc, nano_hexapod.geometry.si)'];
end
#+end_src

** Save the Structure
:PROPERTIES:
:UNNUMBERED: t
:END:

#+begin_src matlab
if nargout == 0
  save('./mat/stages.mat', 'nano_hexapod', '-append');
end
#+end_src