Add analysis on the DVF complex conjugate zero

org/nano_hexapod.org

@@ -44,6 +44,15 @@ addpath('matlab/nano_hexapod');
 open('matlab/nano_hexapod/nano_hexapod.slx')
 #+end_src
 
+#+begin_src matlab
+%% Options for Linearized
+options = linearizeOptions;
+options.SampleTime = 0;
+
+%% Name of the Simulink File
+mdl = 'nano_hexapod';
+#+end_src
+
 ** Nano Hexapod - Configuration
 <<sec:nano_hexapod_conf>>
 *** Introduction                                                    :ignore:
@@ -169,13 +178,6 @@ We here wish to compare the plant from actuators to the encoders when the encode
 
 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
@@ -185,7 +187,7 @@ io(io_i) = linio([mdl, '/D'],  1, 'openoutput'); io_i = io_i + 1; % Relative Mot
 Identify the plant when the encoders are on the struts:
 #+begin_src matlab
 n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
-                                       'flex_top_type', '3dof', ...
+                                       'flex_top_type', '4dof', ...
                                        'motion_sensor_type', 'struts', ...
                                        'actuator_type', '2dof');
 
@@ -197,7 +199,7 @@ Gs.OutputName = {'D1', 'D2', 'D3', 'D4', 'D5', 'D6'};
 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', ...
+                                       'flex_top_type', '4dof', ...
                                        'motion_sensor_type', 'plates', ...
                                        'actuator_type', '2dof');
 
@@ -265,8 +267,9 @@ exportFig('figs/nano_hexapod_effect_encoder.pdf', 'width', 'full', 'height', 'ta
 [[file:figs/nano_hexapod_effect_encoder.png]]
 
 #+begin_important
-The zeros at  400Hz and 800Hz should corresponds to resonances of the system when one of the APA is blocked.
+The zeros at 280Hz should corresponds to resonances of the system when one of the APA is blocked.
 It is linked to the axial stiffness of the flexible joints: increasing the axial stiffness of the joints will increase the frequency of the zeros.
+This will be verified in
 #+end_important
 
 ** Effect of APA flexibility
@@ -453,20 +456,13 @@ Depending on the physical effects we want to model, we therefore know how many s
 
 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 [N]
 io(io_i) = linio([mdl, '/D'], 1, 'openoutput'); io_i = io_i + 1; % Relative displacements [m]
 
 n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
-                                       'flex_top_type', '3dof', ...
+                                       'flex_top_type', '4dof', ...
                                        'motion_sensor_type', 'struts', ...
                                        'actuator_type', '2dof');
 
@@ -560,28 +556,49 @@ exportFig('figs/nano_hexapod_struts_2dof_dvf_plant.pdf', 'width', 'full', 'heigh
 <<sec:iff_plant>>
 
 The transfer function from actuators to force sensors is identified.
+It corresponds to the plant for the Integral Force Feedback (IFF) control strategy.
 #+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', ...
+                                       'flex_top_type', '4dof', ...
                                        '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 DC gain from actuator to the force sensor should be equal to (for the 2dof APA):
+\[ \frac{k_e k_1}{k_1 k_e + k_a k_1 + k_e k_a} \]
+
+Which is equal to:
+#+begin_src matlab :results value replace :exports results :tangle no
+sprintf('DCgain = %.2e [m/N]', n_hexapod.actuator.ke(1)*n_hexapod.actuator.k(1)/(n_hexapod.actuator.k(1)*n_hexapod.actuator.ke(1) + n_hexapod.actuator.ka(1)*n_hexapod.actuator.k(1) + n_hexapod.actuator.ke(1)*n_hexapod.actuator.ka(1)))
+#+end_src
+
+#+RESULTS:
+: DCgain = 6.56e-03 [m/N]
+
+This is indeed close to what we see on DC gain of the $6 \times 6$ IFF plant in Table [[tab:iff_dc_gain]].
+Also, the off diagonal terms are quite small compared to the diagonal terms.
+#+begin_src matlab :exports results :results value table replace :tangle no
+  data2orgtable(dcgain(Giff), {}, {}, ' %.4f ');
+#+end_src
+
+#+name: tab:iff_dc_gain
+#+caption: DC gain of the IFF plant
+#+attr_latex: :environment tabularx :width 0.8\linewidth :align cccccc
+#+attr_latex: :center t :booktabs t :float t
+#+RESULTS:
+|  0.0068 |  0.0002 | -0.0002 |    -0.0 |  0.0002 | -0.0001 |
+|  0.0003 |  0.0068 |  0.0001 | -0.0001 | -0.0001 |  0.0002 |
+|  0.0001 |     0.0 |  0.0068 | -0.0001 | -0.0001 |  0.0001 |
+| -0.0002 | -0.0003 |  0.0001 |  0.0068 | -0.0002 |    -0.0 |
+| -0.0001 | -0.0003 |    -0.0 |  0.0001 |  0.0065 |     0.0 |
+|  0.0001 |  0.0001 | -0.0002 |    -0.0 |  0.0002 |  0.0066 |
 
 The bode plot is shown in Figure [[fig:nano_hexapod_struts_2dof_iff_plant]].
 #+begin_src matlab :exports none
@@ -677,13 +694,6 @@ The "perfect" sensor output $\mathcal{X}$ is used to verify that the sensor Jaco
 
 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
@@ -694,7 +704,7 @@ io(io_i) = linio([mdl, '/X'],  1, 'openoutput'); io_i = io_i + 1; % Relative Mot
 Start when the encoders are fixed on the struts.
 #+begin_src matlab
 n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
-                                       'flex_top_type', '3dof', ...
+                                       'flex_top_type', '4dof', ...
                                        'motion_sensor_type', 'struts', ...
                                        'actuator_type', '2dof', ...
                                        'MO_B', 150e-3);
@@ -767,7 +777,7 @@ exportFig('figs/nano_hexapod_comp_cartesian_plants_struts.pdf', 'width', 'wide',
 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', ...
+                                       'flex_top_type', '4dof', ...
                                        'motion_sensor_type', 'plates', ...
                                        'actuator_type', '2dof', ...
                                        'MO_B', 150e-3);
@@ -900,6 +910,141 @@ exportFig('figs/nano_hexapod_cartesian_plant_encoder_comp.pdf', 'width', 'full',
 #+RESULTS:
 [[file:figs/nano_hexapod_cartesian_plant_encoder_comp.png]]
 
+#+begin_question
+Is the complex conjugate zeros on Figure [[fig:nano_hexapod_cartesian_plant_encoder_comp]] could be due to the axial stiffness of the joints?
+Let's find out by using a model with no axial flexibility on the joints and see if we still have this zero.
+#+end_question
+
+*** Perfect axial stiffness of the joints
+Let's initialize the Nano-Hexapod with perfect joints (no axial stiffness):
+#+begin_src matlab
+n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '3dof', ...
+                                       'flex_top_type', '2dof', ...
+                                       'motion_sensor_type', 'struts', ...
+                                       'actuator_type', '2dof', ...
+                                       'MO_B', 150e-3);
+#+end_src
+
+Identify its dynamics.
+#+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'};
+#+end_src
+
+And compute the transfer function in the cartesian frame.
+#+begin_src matlab
+G = inv(n_hexapod.geometry.Js)*G({'D1', 'D2', 'D3', 'D4', 'D5', 'D6'}, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'})*inv(n_hexapod.geometry.J)';
+#+end_src
+
+The obtained dynamics is shown in Figure [[fig:nano_hexapod_cartesian_plant_perfect_joints]].
+It is shown that there are no zeros at high frequency.
+#+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(G(i,i), freqs, 'Hz'))), '-', ...
+         'DisplayName', labels{i});
+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(G(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_perfect_joints.pdf', 'width', 'full', 'height', 'tall');
+#+end_src
+
+#+name: fig:nano_hexapod_cartesian_plant_perfect_joints
+#+caption: Bode plot of the "cartesian" plant (transfer function from $\mathcal{F}$ to $d\mathcal{X}$) when the encoders are fixed on the struts and with perfect joints (infinite axial stiffness)
+#+RESULTS:
+[[file:figs/nano_hexapod_cartesian_plant_perfect_joints.png]]
+
+#+begin_important
+The zeros appearing between 100Hz and 500Hz are due to the axial stiffness of the flexible joints.
+They appear when the relative motion sensors are fixed to the struts.
+They basically correspond to the resonance frequency where the APA are infinitively stiff.
+#+end_important
+
+*** Frequency of the zeros
+Let's verify this last affirmation by generating a nano-hexapod with very stiff APA.
+#+begin_src matlab
+n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
+                                       'flex_top_type', '4dof', ...
+                                       'motion_sensor_type', 'struts', ...
+                                       'actuator_type', '2dof', ...
+                                       'actuator_k',  1e9*ones(6,1));
+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
+
+#+begin_src matlab
+Gs_p = imag(pole(Gs))/2/pi;
+sort(Gs_p(Gs_p > 0))
+#+end_src
+
+#+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
+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
+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
+
 ** Decentralized Plant - Decoupling at the Center of Stiffness
 <<sec:decoupled_plant_cok>>
 
@@ -971,13 +1116,6 @@ And we indeed obtain a diagonal stiffness matrix.
 *** Obtained plant
 Let's identify the transfer function from $\tau$ to $d\mathcal{L}$ and from $\tau$ 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
@@ -1068,14 +1206,14 @@ The location of the applied force/torque and the expressed displacement/rotation
 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:
+Neglecting the torsional/bending 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.
+where $\mathcal{K}$ is a diagonal 6x6 matrix with the 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 $k_s$:
+The axial stiffness of the struts $k_s$ is then:
 \[ k_s = \frac{k_z k_{\text{APA}}}{k_z + 2 k_{\text{APA}}} \]
 with $k_z$ the axial stiffness of the flexible joints.
 
@@ -1087,32 +1225,18 @@ n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
                                        'actuator_type', '2dof');
 #+end_src
 
-The axial stiffness of the joints and stiffnesses of the 2-DoF actuators are defined below.
-#+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); % Joint's axial stiffness [m/N]
-#+end_src
-
 The total axial stiffness of the APA is:
-#+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)
+kAPA = n_hexapod.actuator.k(1) + n_hexapod.actuator.ke(1)*n_hexapod.actuator.ka(1)/(n_hexapod.actuator.ke(1) + n_hexapod.actuator.ka(1));
+sprintf('kAPA = %.3e [N/m]', kAPA);
 #+end_src
 
 #+RESULTS:
 : kAPA = 1.799e+06 [N/m]
 
 And the total axial stiffness of the struts is:
-#+begin_src matlab
-ks = kz*kAPA/(kz + 2*kAPA);
-#+end_src
-
 #+begin_src matlab :results value replace :exports results :tangle no
+ks = n_hexapod.flex_top.kz(1)*kAPA/(n_hexapod.flex_top.kz(1) + 2*kAPA);
 sprintf('ks = %.3e [N/m]', ks)
 #+end_src
 
@@ -1120,7 +1244,7 @@ sprintf('ks = %.3e [N/m]', ks)
 : ks = 1.737e+06 [N/m]
 
 #+begin_important
-We can see that the axial stiffness of the flexible joint as little impact on the total axial stiffness of the struts.
+We can see that the axial stiffness of the flexible joint as little impact on the total axial stiffness of the struts as it is much stiffer than the APA.
 #+end_important
 
 Let's now compute the stiffness matrix corresponding to an hexapod with perfect joints and the above computed axial strut stiffness:
@@ -1153,13 +1277,6 @@ ans = C
 *** Comparison with Simscape Model
 Let's now identify the compliance matrix using Simscape.
 #+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; % External forces [N, Nm/rad]
@@ -1189,10 +1306,27 @@ dcgain(G)
 |  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 |
 
-#+begin_important
+Let's compare the two obtained compliance matrices.
-The bending and torsional stiffness of the flexible joints induces a lot of coupling between forces/torques applied to the to platform to its displacement/rotation.
+The results are shown in Table [[tab:compliance_matrices_compare]].
-It can be seen by comparison the compliance matrices in Tables [[tab:compliance_matrix_perfect_joints]] and [[tab:compliance_matrix_simscape_joints]].
+The diagonal terms are matching quite good, as well as the 1/4 and 2/5 of diagonal terms that are the most important ones.
-#+end_important
+
+All the other terms are zero in the ideal case (Table [[tab:compliance_matrix_perfect_joints]]).
+There are not zero when considering the bending/torsional stiffness of the joints (Table [[tab:compliance_matrix_simscape_joints]]), however they are still small as compared to the diagonal terms.
+#+begin_src matlab :results value replace :exports both :tangle no
+C./dcgain(G)
+#+end_src
+
+#+name: tab:compliance_matrices_compare
+#+caption: Compliance Matrices - Comparison
+#+attr_latex: :environment tabularx :width 0.85\linewidth :align cccccc
+#+attr_latex: :center t :booktabs t :float t
+#+RESULTS:
+|      1.0037 | -9.3741e-15 |  1.4613e-14 | -1.5751e-14 |      1.0069 |   0.090379 |
+|  8.6313e-15 |       1.014 |  -0.0029656 |      1.0204 |  2.4404e-15 | 1.7641e-20 |
+| -1.8807e-14 |   -0.024917 |      1.0028 |   -0.020701 | -3.4857e-14 | 4.3304e-18 |
+|  6.6502e-15 |      1.0157 |  -0.0029943 |      1.0193 |  2.3673e-15 | 1.9265e-20 |
+|      1.0061 | -1.0288e-14 |  2.1354e-14 | -1.7844e-14 |      1.0085 |   0.032102 |
+|   0.0048048 | -3.1201e-14 | -6.5769e-19 | -2.5589e-14 |    0.002899 |      1.005 |
 
 * Active Damping using Integral Force Feedback
 <<sec:integral_force_feedback>>
@@ -1228,6 +1362,15 @@ addpath('matlab/nano_hexapod');
 open('matlab/nano_hexapod/nano_hexapod.slx')
 #+end_src
 
+#+begin_src matlab
+%% Options for Linearized
+options = linearizeOptions;
+options.SampleTime = 0;
+
+%% Name of the Simulink File
+mdl = 'nano_hexapod';
+#+end_src
+
 ** Plant Identification
 <<sec:iff_identification>>
 
@@ -1241,13 +1384,6 @@ n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
 
 The transfer function from actuator inputs to force sensors outputs 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
@@ -1422,13 +1558,6 @@ exportFig('figs/nano_hexapod_iff_loop_gain.pdf', 'width', 'wide', 'height', 'tal
 
 Let's now see how the IFF control strategy effectively damps the plant and how it affects the transfer functions from the actuator forces to the relative motion sensors (encoders).
 #+begin_src matlab :exports none
-%% 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
@@ -1538,13 +1667,6 @@ The Integral Force Feedback Strategy is very effective to damp the 6 suspension
 The IFF strategy has the well known drawback of degrading the compliance (transfer function from external forces/torques applied to the top platform to the motion of the top platform), especially at low frequency where the control gain is large.
 Let's quantify that for the nano-hexapod.
 #+begin_src matlab :exports none
-%% 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; % Actuator Inputs
@@ -1650,6 +1772,15 @@ addpath('matlab/nano_hexapod');
 open('matlab/nano_hexapod/nano_hexapod.slx')
 #+end_src
 
+#+begin_src matlab
+%% Options for Linearized
+options = linearizeOptions;
+options.SampleTime = 0;
+
+%% Name of the Simulink File
+mdl = 'nano_hexapod';
+#+end_src
+
 ** Plant Identification
 <<sec:dvf_identification_struts>>
 
@@ -1663,13 +1794,6 @@ n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
 
 The transfer function from actuator inputs to force sensors outputs 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
@@ -1843,13 +1967,6 @@ exportFig('figs/nano_hexapod_dvf_loop_gain_struts.pdf', 'width', 'wide', 'height
 
 Let's now see how the DVF control strategy effectively damps the plant and how it affects the transfer functions from the actuator forces to the relative motion sensors (encoders).
 #+begin_src matlab :exports none
-%% 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
@@ -1959,13 +2076,6 @@ The Direct Velocity Feedback Strategy is very effective to damp the 6 suspension
 The DVF strategy has the well known drawback of degrading the compliance (transfer function from external forces/torques applied to the top platform to the motion of the top platform), especially at low frequency where the control gain is large.
 Let's quantify that for the nano-hexapod.
 #+begin_src matlab :exports none
-%% 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; % Actuator Inputs
@@ -2065,6 +2175,15 @@ addpath('matlab/nano_hexapod');
 open('matlab/nano_hexapod/nano_hexapod.slx')
 #+end_src
 
+#+begin_src matlab
+%% Options for Linearized
+options = linearizeOptions;
+options.SampleTime = 0;
+
+%% Name of the Simulink File
+mdl = 'nano_hexapod';
+#+end_src
+
 ** Plant Identification
 <<sec:dvf_identification_plates>>
 
@@ -2078,13 +2197,6 @@ n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
 
 The transfer function from actuator inputs to force sensors outputs 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
@@ -2258,13 +2370,6 @@ exportFig('figs/nano_hexapod_dvf_loop_gain_plates.pdf', 'width', 'wide', 'height
 
 Let's now see how the DVF control strategy effectively damps the plant and how it affects the transfer functions from the actuator forces to the relative motion sensors (encoders).
 #+begin_src matlab :exports none
-%% 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
@@ -2374,13 +2479,6 @@ The Direct Velocity Feedback Strategy is very effective in damping the 6 suspens
 The DVF strategy has the well known drawback of degrading the compliance (transfer function from external forces/torques applied to the top platform to the motion of the top platform), especially at low frequency where the control gain is large.
 Let's quantify that for the nano-hexapod.
 #+begin_src matlab :exports none
-%% 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; % Actuator Inputs
@@ -3325,7 +3423,7 @@ arguments
     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
+    % 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]
@@ -3333,7 +3431,7 @@ arguments
     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
+    % 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]