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figs/test_struts_mounting_step_2.svg

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matlab/test_struts_1_flexible_modes.m

@@ -12,7 +12,7 @@ addpath('./src/'); % Path for functions
 colors = colororder;
 
 % Measured results
-% The obtained frequency response functions for the three configurations (X-bending, Y-bending and Z-torsion) are shown in Figure ref:fig:test_struts_spur_res_frf_no_enc when the encoder is not fixed to the strut and in Figure ref:fig:test_struts_spur_res_frf_enc when the encoder is fixed to the strut.
+% The obtained frequency response functions are shown in Figure ref:fig:test_struts_spur_res_frf.
 
 
 %% Load Data (without the encoder)
@@ -41,7 +41,6 @@ hold off;
 set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log');
 xlabel('Frequency [Hz]'); ylabel('Amplitude');
 xlim([50, 8e2]); ylim([5e-7, 3e-4])
-xticks([50, 100, 500]);
 legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 1);
 
 %% Plot the responses (with the encoder)
@@ -60,5 +59,4 @@ hold off;
 set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log');
 xlabel('Frequency [Hz]'); ylabel('Amplitude');
 xlim([50, 8e2]); ylim([5e-7, 3e-4])
-xticks([50, 100, 500]);
 legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 1);

matlab/test_struts_2_dynamical_meas.m

@@ -63,31 +63,11 @@ enc_frf = [frf_sweep(i_lf); frf_noise_hf(i_hf)]; % Combine the FRF
 
 
 
-% System identification is performed without the encoder fixed to the strut (Figure ref:fig:test_struts_bench_leg_front) and with one encoder fixed to the strut (Figure ref:fig:test_struts_bench_leg_coder).
+% Figure ref:fig:test_struts_effect_encoder_int
+% Same goes for the transfer function from excitation voltage $u$ to the axial motion of the strut $d_a$ as measured by the interferometer ().
 
-% #+name: fig:test_struts_bench_leg_with_without_enc
+% The transfer function from the excitation voltage $u$ to the generated voltage $V_s$ by the sensor stack is not influence by the fixation of the encoder (Figure ref:fig:test_struts_effect_encoder_iff).
-% #+caption: Struts fixed to the test bench with clamped flexible joints. The coder can be fixed to the struts (\subref{fig:test_struts_bench_leg_coder}) or removed (\subref{fig:test_struts_bench_leg_front})
+% This means that the IFF control strategy should be as effective whether or not the encoders are fixed to the struts.
-% #+attr_latex: :options [htbp]
-% #+begin_figure
-% #+attr_latex: :caption \subcaption{\label{fig:test_struts_bench_leg_coder}Strut with encoder}
-% #+attr_latex: :options {0.5\textwidth}
-% #+begin_subfigure
-% #+attr_latex: :height 6cm
-% [[file:figs/test_struts_bench_leg_coder.jpg]]
-% #+end_subfigure
-% #+attr_latex: :caption \subcaption{\label{fig:test_struts_bench_leg_front}Strut without encoder}
-% #+attr_latex: :options {0.5\textwidth}
-% #+begin_subfigure
-% #+attr_latex: :height 6cm
-% [[file:figs/test_struts_bench_leg_front.jpg]]
-% #+end_subfigure
-% #+end_figure
-
-% The obtained frequency response functions are compared in Figure ref:fig:test_struts_effect_encoder.
-% It is found that the encoder has very little effect on the transfer function from excitation voltage $u$ to the axial motion of the strut $d_a$ as measured by the interferometer (Figure ref:fig:test_struts_effect_encoder_int).
-% This means that the axial motion of the strut is unaffected by the presence of the encoder.
-% Similarly, it has very little effect on the transfer function from $u$ to the sensor stack voltage $V_s$ (Figure ref:fig:test_struts_effect_encoder_iff).
-% This means that the integral force feedback control strategy should be as effective whether the encoders are fixed to the struts or not.
 
 
 %% Plot the FRF from u to da with and without the encoder
@@ -103,8 +83,7 @@ set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
 ylabel('Amplitude $d_a/u$ [m/V]'); set(gca, 'XTickLabel',[]);
 hold off;
 ylim([1e-7, 1e-3]);
-leg = legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 1);
+legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
-leg.ItemTokenSize(1) = 15;
 
 ax2 = nexttile;
 hold on;
@@ -118,7 +97,6 @@ yticks(-360:90:360); ylim([-180, 180]);
 
 linkaxes([ax1,ax2],'x');
 xlim([10, 2e3]);
-xticks([1e1, 1e2, 1e3]);
 
 %% Compare the IFF plant with and without the encoders
 figure;
@@ -132,8 +110,7 @@ hold off;
 set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
 ylabel('Amplitude $V_s/u$ [V/V]'); set(gca, 'XTickLabel',[]);
 hold off;
-leg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 1);
+legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
-leg.ItemTokenSize(1) = 15;
 ylim([1e-2, 1e2]);
 
 ax2 = nexttile;
@@ -148,18 +125,23 @@ yticks(-360:90:360); ylim([-180, 180]);
 
 linkaxes([ax1,ax2],'x');
 xlim([10, 2e3]);
-xticks([1e1, 1e2, 1e3]);
 
 % Comparison of the encoder and interferometer
 % <<ssec:test_struts_comp_enc_int>>
 
-% The dynamics as measured by the encoder (i.e. $d_e/u$) and by the interferometers (i.e. $d_a/u$) are compared in Figure ref:fig:test_struts_comp_enc_int.
+% The dynamics as measured by the encoder and by the interferometers are compared in Figure ref:fig:test_struts_comp_enc_int.
-% The dynamics from the excitation voltage $u$ to the measured displacement by the encoder $d_e$ presents a behavior that is much more complex than the dynamics to the displacement as measured by the interferometer (comparison made in Figure ref:fig:test_struts_comp_enc_int).
-% Three additional resonance frequencies can be observed at 197Hz, 290Hz and 376Hz.
-% These resonance frequencies are matching the frequencies of the flexible modes that were studied in Section ref:sec:test_struts_flexible_modes.
 
-% The good news is that these resonances are not impacting the axial motion of the strut (which is what is important for the hexapod positioning).
+% The dynamics from the excitation voltage $u$ to the measured displacement by the encoder $d_e$ presents much more complicated behavior than the transfer function to the displacement as measured by the Interferometer (compared in Figure ref:fig:test_struts_comp_enc_int).
-% However, these resonances are making the use of encoder fixed to the strut difficult from a control perspective.
+% It will be further investigated why the two dynamics as so different and what are causing all these resonances.
+
+% As shown in Figure ref:fig:test_struts_comp_enc_int, we can clearly see three spurious resonances at 197Hz, 290Hz and 376Hz.
+% These resonances correspond to parasitic resonances of the strut itself that was estimated using a finite element model of the strut (Figure ref:fig:test_struts_mode_shapes):
+% - Mode in X-bending at 189Hz
+% - Mode in Y-bending at 285Hz
+% - Mode in Z-torsion at 400Hz
+
+% The good news is that these resonances are not seen on the interferometer (they are therefore not impacting the axial motion of the strut).
+% But these resonances are making the use of encoder fixed to the strut difficult.
 
 
 figure;
@@ -167,19 +149,18 @@ tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
 
 ax1 = nexttile([2,1]);
 hold on;
-plot(f, abs(enc_frf),          'DisplayName', '$d_e/u$');
+plot(f, abs(enc_frf),          'DisplayName', 'Encoder');
-plot(f, abs(int_with_enc_frf), 'DisplayName', '$d_a/u$');
+plot(f, abs(int_with_enc_frf), 'DisplayName', 'Interferometer');
-text(85,  4e-4,  {'93Hz'}, 'VerticalAlignment','middle','HorizontalAlignment','right')
+text(93,  4e-4,  {'93Hz'}, 'VerticalAlignment','bottom','HorizontalAlignment','center')
 text(200, 1.3e-4,{'197Hz'},'VerticalAlignment','bottom','HorizontalAlignment','center')
-text(300, 4e-6,  {'290Hz'},'VerticalAlignment','bottom','HorizontalAlignment','left')
+text(300, 4e-6,  {'290Hz'},'VerticalAlignment','bottom','HorizontalAlignment','center')
-text(400, 4e-7,{'376Hz'},'VerticalAlignment','top','HorizontalAlignment','center')
+text(400, 1.4e-6,{'376Hz'},'VerticalAlignment','bottom','HorizontalAlignment','center')
 hold off;
 set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
 ylabel('Amplitude $d/u$ [m/V]'); set(gca, 'XTickLabel',[]);
 hold off;
-leg = legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 1);
+legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
-leg.ItemTokenSize(1) = 15;
+ylim([1e-8, 1e-3]);
-ylim([1e-7, 1e-3]);
 
 ax2 = nexttile;
 hold on;
@@ -193,7 +174,6 @@ yticks(-360:90:360); ylim([-180, 180]);
 
 linkaxes([ax1,ax2],'x');
 xlim([10, 2e3]);
-xticks([1e1, 1e2, 1e3]);
 
 % Comparison of all the Struts
 % <<ssec:test_struts_comp_all_struts>>
@@ -242,9 +222,8 @@ end
 
 
 
-% Then, the dynamics of all the mounted struts (only 5 at the time of the experiment) were all measured using the same test bench.
+% Then, the transfer function from the DAC output voltage $u$ to the measured displacement by the Attocube is computed for all the struts and shown in Figure ref:fig:test_struts_comp_interf_plants.
-% The obtained dynamics from $u$ to $d_a$ are compared in Figure ref:fig:test_struts_comp_interf_plants while is dynamics from $u$ to $V_s$ are compared in Figure ref:fig:test_struts_comp_iff_plants.
+% All the struts are giving very similar FRF.
-% Very good match can be observed between all the struts.
 
 
 %% Plot the FRF from u to de (interferometer)
@@ -261,9 +240,8 @@ hold off;
 set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
 ylabel('Amplitude $d_a/u$ [m/V]'); set(gca, 'XTickLabel',[]);
 hold off;
-leg = legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 1);
+legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 2);
-leg.ItemTokenSize(1) = 15;
+ylim([1e-9, 1e-3]);
-ylim([1e-8, 1e-3]);
 
 ax2 = nexttile;
 hold on;
@@ -278,7 +256,6 @@ yticks(-360:90:360); ylim([-180 180]);
 
 linkaxes([ax1,ax2],'x');
 xlim([10, 2e3]);
-xticks([1e1, 1e2, 1e3]);
 
 %% Plot the FRF from u to Vs
 figure;
@@ -295,8 +272,7 @@ set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
 ylabel('Amplitude $V_s/u$ [V/V]'); set(gca, 'XTickLabel',[]);
 hold off;
 ylim([1e-2, 1e2]);
-leg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 1);
+legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2);
-leg.ItemTokenSize(1) = 15;
 
 ax2 = nexttile;
 hold on;
@@ -311,7 +287,33 @@ yticks(-360:90:360); ylim([-180 180]);
 
 linkaxes([ax1,ax2],'x');
 xlim([10, 2e3]);
-xticks([1e1, 1e2, 1e3]);
+
+
+
+% #+name: fig:test_struts_comp_plants
+% #+caption: Comparison of the measured plants
+% #+attr_latex: :options [htbp]
+% #+begin_figure
+% #+attr_latex: :caption \subcaption{\label{fig:test_struts_comp_interf_plants}$u$ to $d_a$}
+% #+attr_latex: :options {0.49\textwidth}
+% #+begin_subfigure
+% #+attr_latex: :width \linewidth
+% [[file:figs/test_struts_comp_interf_plants.png]]
+% #+end_subfigure
+% #+attr_latex: :caption \subcaption{\label{fig:test_struts_comp_iff_plants}$u$ to $V_s$}
+% #+attr_latex: :options {0.49\textwidth}
+% #+begin_subfigure
+% #+attr_latex: :width \linewidth
+% [[file:figs/test_struts_comp_iff_plants.png]]
+% #+end_subfigure
+% #+end_figure
+
+% There is a very large variability of the dynamics as measured by the encoder as shown in Figure ref:fig:test_struts_comp_enc_plants.
+% Even-though the same peaks are seen for all of the struts (95Hz, 200Hz, 300Hz, 400Hz), the amplitude of the peaks are not the same.
+% Moreover, the location or even the presence of complex conjugate zeros is changing from one strut to the other.
+
+% All of this will be studied in Section ref:sec:test_struts_simscape using the Simscape model.
+
 
 %% Bode plot of the FRF from u to de
 figure;
@@ -327,8 +329,7 @@ hold off;
 set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
 ylabel('Amplitude $d_e/u$ [m/V]'); set(gca, 'XTickLabel',[]);
 hold off;
-leg = legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 1);
+legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 2);
-leg.ItemTokenSize(1) = 15;
 ylim([1e-8, 1e-3]);
 
 ax2 = nexttile;
@@ -344,7 +345,6 @@ yticks(-360:90:360); ylim([-180, 180]);
 
 linkaxes([ax1,ax2],'x');
 xlim([10, 2e3]);
-xticks([1e1, 1e2, 1e3]);
 
 %% Save the estimated FRF for further analysis
 save('./mat/meas_struts_frf.mat', 'f', 'enc_frf', 'int_frf', 'iff_frf', 'strut_nums');

matlab/test_struts_3_simscape_model.m

@@ -34,7 +34,6 @@ freqs = logspace(1, log10(2000), 1000);
 % Model dynamics
 % <<ssec:test_struts_comp_model>>
 
-
 %% Load measured FRF for comparison
 load('meas_struts_frf.mat', 'f', 'enc_frf', 'int_frf', 'iff_frf', 'strut_nums');
 
@@ -66,16 +65,20 @@ Gs_flex.OutputName = {'Vs', 'de', 'da'};
 
 
 
-% Two models of the APA300ML are used here: a simple two degrees of freedom model and a model using a super element extracted from a finite element model.
+% Two models of the APA300ML are used here for comparison:
+% - a simple two degrees of freedom model
+% - a model using a super element extracted from a finite element model
+
 % These two models of the APA300ML were tuned to best match measured frequency response functions of the APA alone.
 % The flexible joints are here modelled with the 4DoF model (axial stiffness, two bending stiffnesses and one torsion stiffness).
+
 % These two models are compared with the measured frequency responses in Figure ref:fig:test_struts_comp_frf_flexible_model.
 
 % The model dynamics from DAC voltage $u$ to the axial motion of the strut $d_a$ (Figure ref:fig:test_struts_comp_frf_flexible_model_int) and from DAC voltage $u$ to the force sensor voltage $V_s$ (Figure ref:fig:test_struts_comp_frf_flexible_model_iff) are well matching the experimental identification.
 
 % However, the transfer function from $u$ to encoder displacement $d_e$ are not well matching for both models.
 % For the 2DoF model, this is normal as the resonances affecting the dynamics are not modelled at all (the APA300ML is modelled as infinitely rigid in all directions except the translation along it's actuation axis).
-% For the flexible model, it will be shown in the next section that by adding some misalignment between the flexible joints and the APA300ML, this model can better represent the observed dynamics.
+% For the flexible model, it will be shown in the next section that by adding some misalignment betwen the flexible joints and the APA300ML, this model can better represent the observed dynamics.
 
 
 %% Compare the FRF and identified dynamics from u to Vs and da
@@ -117,7 +120,6 @@ yticks(-360:90:360); ylim([-180, 180]);
 
 linkaxes([ax1a,ax2a],'x');
 xlim([10, 2e3]);
-xticks([1e1, 1e2, 1e3]);
 
 %% Compare the FRF and identified dynamics from u to Vs and da
 figure;
@@ -158,7 +160,6 @@ yticks(-360:90:360); ylim([-180, 180]);
 
 linkaxes([ax1a,ax2a],'x');
 xlim([10, 2e3]);
-xticks([1e1, 1e2, 1e3]);
 
 %% Compare the FRF and identified dynamics from u to Vs and da
 figure;
@@ -200,33 +201,33 @@ yticks(-360:90:360); ylim([-180, 180]);
 
 linkaxes([ax1a,ax2a],'x');
 xlim([10, 2e3]);
-xticks([1e1, 1e2, 1e3]);
 
 % Effect of strut misalignment
 % <<ssec:test_struts_effect_misalignment>>
 
 % As was shown in Figure ref:fig:test_struts_comp_enc_plants, the identified dynamics from DAC voltage $u$ to encoder measured displacement $d_e$ are very different from one strut to the other.
+
 % In this section, it is investigated whether poor alignment of the strut (flexible joints with respect to the APA) can explain such dynamics.
 % For instance, consider Figure ref:fig:test_struts_misalign_schematic where there is a misalignment in the $y$ direction between the two flexible joints (well aligned thanks to the mounting procedure in Section ref:sec:test_struts_mounting) and the APA300ML.
-% In such case, the "x-bending" mode at 200Hz (see Figure ref:fig:test_struts_meas_x_bending) can be expected to have more impact on the dynamics from the actuator to the encoder.
+% In such case, the "x-bending" mode at 200Hz (see Figure ref:fig:test_struts_meas_x_bending) can be expected to be more excited, and thus the dynamics from the actuator to the encoder should be affected at frequencies around 200Hz.
 
 % #+name: fig:test_struts_misalign_schematic
 % #+caption: Mis-alignement between the joints and the APA
 % #+attr_latex: :width 0.8\linewidth
 % [[file:figs/test_struts_misalign_schematic.png]]
 
-% To verify this assumption, the dynamics from output DAC voltage $u$ to the measured displacement by the encoder $d_e$ is computed using the flexible APA Simscape model for several misalignment in the $y$ direction.
+% To verify this assumption, the dynamics from output DAC voltage $u$ to the measured displacement by the encoder $d_e$ is computed using the Simscape model with flexible APA for several misalignment in the $y$ direction.
 % Obtained dynamics are shown in Figure ref:fig:test_struts_effect_misalignment_y.
-% The alignment of the APA with the flexible joints as a large influence on the dynamics from actuator voltage to measured displacement by the encoder.
+% The alignment of the APA with the flexible joints as a *huge* influence on the dynamics from actuator voltage to measured displacement by the encoder.
 % The misalignment in the $y$ direction mostly influences:
 % - the presence of the flexible mode at 200Hz (see mode shape in Figure ref:fig:test_struts_mode_shapes_1)
 % - the location of the complex conjugate zero between the first two resonances:
-%   - if $d_{y} < 0$: there is no zero between the two resonances and possibly not even between the second and third ones
+%   - if $d_y < 0$: there is no zero between the two resonances and possibly not even between the second and third ones
-%   - if $d_{y} > 0$: there is a complex conjugate zero between the first two resonances
+%   - if $d_y > 0$: there is a complex conjugate zero between the first two resonances
 % - the location of the high frequency complex conjugate zeros at 500Hz (secondary effect, as the axial stiffness of the joint also has large effect on the position of this zero)
 
 % The same can be done for a misalignment in the $x$ direction.
-% The obtained dynamics (Figure ref:fig:test_struts_effect_misalignment_x) are showing that misalignment in the $x$ direction mostly influences the presence of the flexible mode at 300Hz (see mode shape in Figure ref:fig:test_struts_mode_shapes_2).
+% The obtained dynamics are shown in Figure ref:fig:test_struts_effect_misalignment_x where it is shown that misalignment in the $x$ direction mostly influences the presence of the flexible mode at 300Hz (see mode shape in Figure ref:fig:test_struts_mode_shapes_2).
 
 % Comparing the experimental frequency response functions for all the APA in Figure ref:fig:test_struts_comp_enc_plants with the model dynamics for several $y$ misalignments in Figure ref:fig:test_struts_effect_misalignment_y indicates a clear similarity.
 % This similarity suggests that the identified differences in dynamics are caused by the misalignment.
@@ -300,7 +301,6 @@ yticks(-360:90:360); ylim([-180, 180]);
 
 linkaxes([ax1,ax2],'x');
 xlim([10, 2e3]);
-xticks([1e1, 1e2, 1e3]);
 
 %% Transfer function from Vs to de - effect of x-misalignment
 figure;
@@ -336,18 +336,17 @@ yticks(-360:90:360); ylim([-180, 180]);
 
 linkaxes([ax1,ax2],'x');
 xlim([10, 2e3]);
-xticks([1e1, 1e2, 1e3]);
 
 % Measured strut misalignment
 % <<ssec:test_struts_meas_misalignment>>
 
-% During the initial mounting of the struts as presented in Section ref:sec:test_struts_mounting, the positioning pins that are used to position the APA with respect to the flexible joints in the $y$ directions were not used (not received at the time).
+% During the first mounting of the struts presented in Section ref:sec:test_struts_mounting, the positioning pins used to position the APA with respect to the flexible joints in the $y$ directions were not used (not received at the time).
-% Therefore, large $y$ misalignments is expected.
+% Therefore, large $y$ misalignments may be expected.
 
 % In order to estimate the misalignments between the two flexible joints and the APA:
-% - the struts are fixed horizontally on the mounting bench as shown in Figure ref:fig:test_struts_mounting_step_3 but without the encoder
+% - the struts are fixed horizontally to the mounting bench as shown in Figure ref:fig:test_struts_mounting_step_3 but without the encoder
-% - using a length gauge[fn:2], the height difference from the flexible joints surface and the APA shell surface is measured both for the top and bottom joints and for both sides
+% - using a length gauge[fn:2], the height difference from the flexible joints surface and the APA shell surface is measured both for the top and bottom joints and on both sides
-% - as the thickness of the flexible joint is $21\,mm$ and the thickness of the APA shell is $20\,mm$, $0.5\,mm$ of height different should be measured if the two are perfectly aligned
+% - as the thickness of the flexible joint is $21\,mm$ and the thickness of the APA shell is $20\,mm$, $0.5\,mm$ of height different should be measured is the two are perfectly aligned
 
 % Large variations in the $y$ misalignment are found from one strut to the other (results are summarized in Table ref:tab:test_struts_meas_y_misalignment).
 
@@ -391,11 +390,13 @@ dy_top = (strut_align(:,3) - strut_align(:,4))/2; % [mm]
 % |       4 | -0.01 |  0.54 |
 % |       5 |  0.15 |  0.02 |
 
-% By using the measured $y$ misalignment in the Simscape model with the flexible APA model, the model dynamics from $u$ to $d_e$ is closer to the measured one as shown in Figure ref:fig:test_struts_comp_dy_tuned_model_frf_enc.
+% By using the measured $y$ misalignment in the Simscape model with the flexible APA model, the measured dynamics from $u$ to $d_e$ can be approached as shown in Figure ref:fig:test_struts_comp_dy_tuned_model_frf_enc.
-% Better match in the dynamics can be obtained by fine tuning both the $x$ and $y$ misalignments (yellow curves in Figure ref:fig:test_struts_comp_dy_tuned_model_frf_enc).
+% Even better match in the dynamics can be obtained by fine tuning both the $x$ and $y$ misalignments (yellow curves in Figure ref:fig:test_struts_comp_dy_tuned_model_frf_enc).
+
 
 % This confirms that the misalignment between the APA and the strut axis (determined by the two flexible joints) is critical and is inducing large variations in the dynamics from DAC voltage $u$ to encoder measured displacement $d_e$.
-% If encoders are fixed to the struts, it is important to precisely align the APA and the flexible joints when mounting the struts.
+% If encoders are to be used when fixed on the struts, it is therefore very important to properly align the APA and the flexible joints when mounting the struts.
+
 
 % In the next section, the struts are re-assembled with a "positioning pin" to better align the APA with the flexible joints.
 % With a better alignment, the amplitude of the spurious resonances are expected to decrease as was shown in Figure ref:fig:test_struts_effect_misalignment_y.
@@ -503,7 +504,7 @@ xlim([10, 2e3]); ylim([1e-8, 1e-3]);
 % This should make the APA better aligned with the two flexible joints.
 
 % This alignment is then estimated using a length gauge as in the previous sections.
-% Measured $y$ alignments are summarized in Table ref:tab:test_struts_meas_y_misalignment_with_pin and are found to be bellow $55\mu m$ for all the struts which is much better than before (see Table ref:tab:test_struts_meas_y_misalignment).
+% Measured $y$ alignments are summarized in Table ref:tab:test_struts_meas_y_misalignment_with_pin and are found to be bellow $55\mu m$ for all the struts which is much better than better (see Table ref:tab:test_struts_meas_y_misalignment).
 
 
 %% Measurement of the y misalignment between the APA and the flexible joints after strut better alignment
@@ -547,10 +548,11 @@ dy_top = (strut_align(:,3) - strut_align(:,4))/2; % [mm]
 
 % The dynamics of the re-aligned struts are then measured using the same test bench (Figure ref:fig:test_struts_bench_leg).
 % The comparison of the initial strut dynamics and the dynamics of the re-aligned struts (i.e. with the positioning pin) is made in Figure ref:fig:test_struts_comp_enc_frf_realign.
-% Even though the struts are now much better aligned, not much improvement can be observed.
-% Also, the dynamics of the six aligned struts are quite different from one another.
 
-% Having the encoders fixed to the struts are making the control more challenging.
+% Even though the struts are now much better aligned, not much improvement can be observed.
+% The dynamics of the six aligned struts are quite different from one another.
+
+% Having the encoders fixed to the struts may prove to be difficult to use.
 % Therefore, the encoders may be fixed to the nano-hexapod plates instead.
 
 
@@ -579,6 +581,9 @@ end
 
 %% Bode plot of the FRF from u to de
 figure;
+tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
+
+ax1 = nexttile([2,1]);
 hold on;
 plot(f, abs(enc_frf(:, 1)), 'color', [colors(1,:), 0.5], ...
         'DisplayName', 'Initial alignment');
@@ -594,7 +599,145 @@ for i = 1:length(strut_align_nums)
 end
 hold off;
 set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
-xlabel('Frequency [Hz]'); ylabel('Amplitude $d_e/u$ [m/V]');
+ylabel('Amplitude $d_e/u$ [m/V]'); set(gca, 'XTickLabel',[]);
-legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 1);
+hold off;
-xlim([10, 2e3]);
+legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
 ylim([1e-8, 1e-3]);
+
+ax2 = nexttile;
+hold on;
+for i = 1:length(strut_nums)
+    plot(f, 180/pi*angle(enc_frf(:, i)), 'color', [colors(1,:), 0.5]);
+end
+for i = 1:length(strut_align_nums)
+    plot(f, 180/pi*angle(enc_frf_aligned(:, i)), 'color', [colors(2,:), 0.5]);
+end
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
+hold off;
+yticks(-360:90:360); ylim([-180, 180]);
+
+linkaxes([ax1,ax2],'x');
+xlim([10, 2e3]);
+
+% Effect of the flexible joint
+% <<ssec:test_struts_effect_joint_stiffness>>
+
+% As the struts are composed of one APA and two flexible joints, it is expected that the flexible joint characteristics will change the dynamic behavior of the struts.
+
+% Using the Simscape model, the effect of the flexible joint's characteristics on the dynamics as measured on the test bench are studied.
+% The studied dynamics is between $u$ and the encoder displacement $d_e$.
+
+% Let's initialize an APA which is a little bit misaligned.
+
+% APA Initialization
+n_hexapod.actuator = initializeAPA('type', 'flexible', 'd_align_bot', [0.1e-3; 0.5e-3; 0], 'd_align_top', [0.1e-3; 0.5e-3; 0]);
+
+%% Study the effect of the bending stiffness of the Flexible joints
+% Tested bending stiffnesses [Nm/rad]
+kRs = [3, 4, 5, 6, 7];
+
+% Idenfity the transfer function from actuator to encoder for all bending stiffnesses
+Gs_kRs = {zeros(length(kRs), 1)};
+
+for i = 1:length(kRs)
+    n_hexapod.flex_bot = initializeBotFlexibleJoint(...
+        'type', '4dof', ...
+        'kRx', kRs(i), ...
+        'kRy', kRs(i));
+    n_hexapod.flex_top = initializeTopFlexibleJoint(...
+        'type', '4dof', ...
+        'kRx', kRs(i), ...
+        'kRy', kRs(i));
+
+    G = exp(-s*1e-4)*linearize(mdl, io, 0.0, opts);
+    G.InputName  = {'u'};
+    G.OutputName = {'Vs', 'de', 'da'};
+
+    Gs_kRs(i) = {G};
+end
+
+%% Study the effect of the axial stiffness of the Flexible joints
+% Tested axial stiffnesses [N/m]
+kzs = [5e7 7.5e7 1e8 2.5e8];
+
+% Idenfity the transfer function from actuator to encoder for all bending stiffnesses
+Gs_kzs = {zeros(length(kzs), 1)};
+
+for i = 1:length(kzs)
+    n_hexapod.flex_bot = initializeBotFlexibleJoint(...
+        'type', '4dof', ...
+        'kz', kzs(i));
+    n_hexapod.flex_top = initializeTopFlexibleJoint(...
+        'type', '4dof', ...
+        'kz', kzs(i));
+
+    G = exp(-s*1e-4)*linearize(mdl, io, 0.0, opts);
+    G.InputName  = {'u'};
+    G.OutputName = {'Vs', 'de', 'da'};
+
+    Gs_kzs(i) = {G};
+end
+
+%% Plot the obtained transfer functions for all the bending stiffnesses
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
+
+ax1 = nexttile([2,1]);
+hold on;
+for i = 1:length(kRs)
+    plot(freqs, abs(squeeze(freqresp(Gs_kRs{i}('de', 'u'), freqs, 'Hz'))), ...
+         'DisplayName', sprintf('$k_R = %.0f$ [Nm/rad]', kRs(i)));
+end
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude $d_e/u$ [m/V]'); set(gca, 'XTickLabel',[]);
+hold off;
+ylim([1e-8, 1e-3]);
+legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 1);
+
+ax2 = nexttile;
+hold on;
+for i = 1:length(kRs)
+    plot(freqs, 180/pi*angle(squeeze(freqresp(Gs_kRs{i}('de', 'u'), freqs, 'Hz'))));
+end
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
+hold off;
+yticks(-360:90:360); ylim([-180, 180]);
+
+linkaxes([ax1,ax2],'x');
+xlim([20, 2e3]);
+
+%% Plot the obtained transfer functions for all the axial stiffnesses
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
+
+ax1 = nexttile([2,1]);
+hold on;
+for i = 1:length(kzs)
+    plot(freqs, abs(squeeze(freqresp(Gs_kzs{i}('de', 'u'), freqs, 'Hz'))), ...
+         'DisplayName', sprintf('$k_z = %.1e$ [N/m]', kzs(i)));
+end
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude $d_e/u$ [m/V]'); set(gca, 'XTickLabel',[]);
+hold off;
+ylim([1e-8, 1e-3]);
+legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 1);
+
+ax2 = nexttile;
+hold on;
+for i = 1:length(kzs)
+    plot(freqs, 180/pi*angle(squeeze(freqresp(Gs_kzs{i}('de', 'u'), freqs, 'Hz'))));
+end
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
+hold off;
+yticks(-360:90:360); ylim([-180, 180]);
+
+linkaxes([ax1,ax2],'x');
+xlim([20, 2e3]);

preamble.tex

@@ -1,16 +1,137 @@
-\usepackage[      %
+\usepackage{float}
-    acronym,      % Separate acronyms and glossary
-    toc,          % appear in ToC
-    automake,     % auto-use the makeglossaries command (requires shell-escape)
-    nonumberlist, % don't back reference pages
-    nogroupskip,  % don't group by letter
-    nopostdot     % don't add a dot at the end of each element
-]{glossaries}
 
-\usepackage[stylemods=longextra]{glossaries-extra}
+\usepackage{caption,tabularx,booktabs}
+\usepackage{bm}
 
-\setabbreviationstyle[acronym]{long-short}
+\usepackage{xpatch} % Recommanded for biblatex
-\setglossarystyle{long-name-desc}
+\usepackage[        % use biblatex for bibliography
+    backend=biber,  % use biber backend (bibtex replacement) or bibtex
+    style=ieee,     % bib style
+    hyperref=true,  % activate hyperref support
+    backref=true,   % activate backrefs
+    isbn=false,     % don't show isbn tags
+    url=false,      % don't show url tags
+    doi=false,      % don't show doi tags
+    urldate=long,   % display type for dates
+    maxnames=3,     %
+    minnames=1,     %
+    maxbibnames=5,  %
+    minbibnames=3,  %
+    maxcitenames=2, %
+    mincitenames=1  %
+    ]{biblatex}
 
-\makeindex
+\setlength\bibitemsep{1.1\itemsep}
-\makeglossaries
+
+% \renewcommand*{\bibfont}{\footnotesize}
+
+\usepackage{fontawesome}
+
+\usepackage{caption}
+\usepackage{subcaption}
+
+\captionsetup[figure]{labelfont=bf}
+\captionsetup[subfigure]{labelfont=bf}
+\captionsetup[listing]{labelfont=bf}
+\captionsetup[table]{labelfont=bf}
+
+\usepackage{xcolor}
+
+\definecolor{my-blue}{HTML}{6b7adb}
+\definecolor{my-pale-blue}{HTML}{e6e9f9}
+\definecolor{my-red}{HTML}{db6b6b}
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+\definecolor{my-green}{HTML}{6bdbb6}
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+\definecolor{my-yellow}{HTML}{dbd26b}
+\definecolor{my-pale-yellow}{HTML}{f9f7e6}
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+\definecolor{my-pale-orange}{HTML}{f9f0e6}
+\definecolor{my-grey}{HTML}{a3a3a3}
+\definecolor{my-pale-grey}{HTML}{f0f0f0}
+\definecolor{my-turq}{HTML}{6bc7db}
+\definecolor{my-pale-turq}{HTML}{e6f6f9}
+
+\usepackage{inconsolata}
+
+\usepackage[newfloat=true, chapter]{minted}
+\usemintedstyle{autumn}
+
+\setminted{frame=lines,breaklines=true,tabsize=4,fontsize=\scriptsize,autogobble=true,labelposition=topline,bgcolor=my-pale-grey}
+\setminted[matlab]{label=Matlab}
+\setminted[latex]{label=LaTeX}
+\setminted[bash]{label=Bash}
+\setminted[python]{label=Python}
+\setminted[text]{label=Results}
+\setminted[md]{label=Org Mode}
+
+\setmintedinline{fontsize=\normalsize,bgcolor=my-pale-grey}
+
+\usepackage[most]{tcolorbox}
+
+\tcbuselibrary{minted}
+
+\newtcolorbox{seealso}{   enhanced,breakable,colback=my-pale-grey,colframe=my-grey,fonttitle=\bfseries,title=See Also}
+\newtcolorbox{hint}{      enhanced,breakable,colback=my-pale-grey,colframe=my-grey,fonttitle=\bfseries,title=Hint}
+\newtcolorbox{definition}{enhanced,breakable,colback=my-pale-red, colframe=my-red, fonttitle=\bfseries,title=Definition}
+\newtcolorbox{important}{ enhanced,breakable,colback=my-pale-red, colframe=my-red, fonttitle=\bfseries,title=Important}
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+
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+  breakable,
+  enhanced jigsaw,
+  borderline west={4pt}{0pt}{my-grey}}
+
+\renewenvironment{quote}{\begin{my-quote}}{\end{my-quote}}
+
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+  colback=my-pale-grey,
+  grow to right by=-10mm,
+  grow to left by=-10mm,
+  boxrule=0pt,
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+  breakable,
+  enhanced jigsaw,
+  borderline west={4pt}{0pt}{my-grey}}
+
+\renewenvironment{verse}{\begin{my-verse}}{\end{my-verse}}
+
+\usepackage{environ}% http://ctan.org/pkg/environ
+\NewEnviron{aside}{%
+  \marginpar{\BODY}
+}
+
+\renewenvironment{verbatim}{\VerbatimEnvironment\begin{minted}[]{text}}{\end{minted}}
+
+\usepackage{soul}
+\sethlcolor{my-pale-grey}
+
+\let\OldTexttt\texttt
+\renewcommand{\texttt}[1]{{\ttfamily\hl{\mbox{\,#1\,}}}}
+
+\makeatletter
+\preto\Gin@extensions{png,}
+\DeclareGraphicsRule{.png}{pdf}{.pdf}{\noexpand\Gin@base.pdf}
+\preto\Gin@extensions{gif,}
+\DeclareGraphicsRule{.gif}{png}{.png}{\noexpand\Gin@base.png}
+\makeatother
+
+\usepackage{hyperref}
+\hypersetup{
+  colorlinks = true,
+  allcolors = my-blue
+}
+
+\usepackage{hypcap}

preamble_extra.tex

@@ -1,134 +0,0 @@
-\usepackage{float}
-\usepackage{enumitem}
-
-\usepackage{caption,tabularx,booktabs}
-\usepackage{bm}
-
-\usepackage{xpatch} % Recommanded for biblatex
-\usepackage[        % use biblatex for bibliography
-    backend=biber,  % use biber backend (bibtex replacement) or bibtex
-    style=ieee,     % bib style
-    hyperref=true,  % activate hyperref support
-    backref=true,   % activate backrefs
-    isbn=false,     % don't show isbn tags
-    url=false,      % don't show url tags
-    doi=false,      % don't show doi tags
-    urldate=long,   % display type for dates
-    maxnames=3,     %
-    minnames=1,     %
-    maxbibnames=5,  %
-    minbibnames=3,  %
-    maxcitenames=2, %
-    mincitenames=1  %
-    ]{biblatex}
-
-\setlength\bibitemsep{1.1\itemsep}
-
-\usepackage{caption}
-\usepackage{subcaption}
-
-\captionsetup[figure]{labelfont=bf}
-\captionsetup[subfigure]{labelfont=bf}
-\captionsetup[listing]{labelfont=bf}
-\captionsetup[table]{labelfont=bf}
-
-\usepackage{xcolor}
-
-\definecolor{my-blue}{HTML}{6b7adb}
-\definecolor{my-pale-blue}{HTML}{e6e9f9}
-\definecolor{my-red}{HTML}{db6b6b}
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-\definecolor{my-pale-grey}{HTML}{f0f0f0}
-\definecolor{my-turq}{HTML}{6bc7db}
-\definecolor{my-pale-turq}{HTML}{e6f6f9}
-
-\usepackage{inconsolata}
-
-\usepackage[newfloat=true, chapter]{minted}
-\usemintedstyle{autumn}
-
-\setminted{frame=lines,breaklines=true,tabsize=4,fontsize=\scriptsize,autogobble=true,labelposition=topline,bgcolor=my-pale-grey}
-\setminted[matlab]{label=Matlab}
-\setminted[latex]{label=LaTeX}
-\setminted[bash]{label=Bash}
-\setminted[python]{label=Python}
-\setminted[text]{label=Results}
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-\preto\Gin@extensions{gif,}
-\DeclareGraphicsRule{.gif}{png}{.png}{\noexpand\Gin@base.png}
-\makeatother
-
-\usepackage{hyperref}
-\hypersetup{
-  colorlinks = true,
-  allcolors = my-blue
-}
-
-\usepackage{hypcap}

test-bench-struts.tex

@@ -1,13 +1,9 @@
-% Created 2024-10-25 Fri 17:22
+% Created 2024-03-27 Wed 22:22
 % Intended LaTeX compiler: pdflatex
 \documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt}
 
 \input{preamble.tex}
-\newacronym{nass}{NASS}{Nano Active Stabilization System}
+\bibliography{test-bench-struts.bib}
-\newacronym{fem}{FEM}{Finite Element Model}
-\newacronym{apa}{APA}{Amplified Piezoelectric Actuator}
-\newacronym{dac}{DAC}{Digital to Analog Converter}
-\input{preamble_extra.tex}
 \author{Dehaeze Thomas}
 \date{\today}
 \title{Test Bench - Nano-Hexapod Struts}
@@ -16,7 +12,7 @@
  pdftitle={Test Bench - Nano-Hexapod Struts},
  pdfkeywords={},
  pdfsubject={},
- pdfcreator={Emacs 29.4 (Org mode 9.6)}, 
+ pdfcreator={Emacs 29.3 (Org mode 9.7)}, 
  pdflang={English}}
 \usepackage{biblatex}
 
@@ -26,8 +22,16 @@
 \tableofcontents
 
 \clearpage
+In this document, a test-bench is used to characterize the struts of the nano-hexapod.
 
-The Nano-Hexapod struts (shown in Figure \ref{fig:test_struts_picture_strut}) are composed of two flexible joints that are fixed at the two ends of the strut, one \acrfull{apa}\footnote{APA300ML from Cedrat Technologies} and one optical encoder\footnote{Vionic from Renishaw}.
+Each strut includes (Figure \ref{fig:test_struts_picture_strut}):
+\begin{itemize}
+\item 2 flexible joints at each ends.
+These flexible joints have been characterized in a separate test bench (see \ldots{}).
+\item 1 Amplified Piezoelectric Actuator (APA300ML) (described in Section \ldots{}).
+Two stacks are used as an actuator and one stack as a (force) sensor.
+\item 1 encoder (Renishaw Vionic) that has been characterized in a separate test bench (see \ldots{}).
+\end{itemize}
 
 \begin{figure}[htbp]
 \centering
@@ -35,57 +39,52 @@ The Nano-Hexapod struts (shown in Figure \ref{fig:test_struts_picture_strut}) ar
 \caption{\label{fig:test_struts_picture_strut}One strut including two flexible joints, an amplified piezoelectric actuator and an encoder}
 \end{figure}
 
-After the strut elements have been individually characterized (see previous sections), the struts are assembled.
+Then the struts are mounted (procedure described in Section \ref{sec:test_struts_mounting}), and are fixed to the same measurement bench.
-The mounting procedure of the struts is explained in Section \ref{sec:test_struts_mounting}.
+The goals are to:
-A mounting bench was used to ensure coaxiality between the two ends of the struts.
+\begin{itemize}
-In this way, no angular stroke is lost when mounted to the nano-hexapod.
+\item Section \ref{sec:test_struts_dynamical_meas}: Identify the dynamics from the generated DAC voltage to:
-
+\begin{itemize}
-The flexible modes of the struts were then experimentally measured and compared with a finite element model (Section \ref{sec:test_struts_flexible_modes}).
+\item the sensors stack generated voltage
-
+\item the measured displacement by the encoder
-Dynamic measurements of the strut are performed with the same test bench used to characterize the APA300ML dynamics (Section \ref{sec:test_struts_dynamical_meas}).
+\item the measured displacement by the interferometer (representing encoders that would be fixed to the nano-hexapod's plates instead of the struts)
-It was found that the dynamics from the \acrshort{dac} voltage to the displacement measured by the encoder is complex due to the flexible modes of the struts (Section \ref{sec:test_struts_flexible_modes}).
+\end{itemize}
-
+\item Section \ref{sec:test_struts_simscape}: Compare the measurements with the Simscape model of the struts and tune the models
-The strut models were then compared with the measured dynamics (Section \ref{sec:test_struts_simscape}).
+\end{itemize}
-The model dynamics from the \acrshort{dac} voltage to the axial motion of the strut (measured by an interferometer) and to the force sensor voltage well match the experimental results.
-However, this is not the case for the dynamics from \acrshort{dac} voltage to the encoder displacement.
-It is found that the complex dynamics is due to a misalignment between the flexible joints and the \acrshort{apa}.
 
+The final goal of the work presented in this document is to have an accurate Simscape model of the struts that can then be included in the Simscape model of the nano-hexapod.
 
+\begin{table}[htbp]
+\caption{\label{tab:test_struts_section_matlab_code}Report sections and corresponding Matlab files}
+\centering
+\begin{tabularx}{0.6\linewidth}{lX}
+\toprule
+\textbf{Sections} & \textbf{Matlab File}\\
+\midrule
+Section \ref{sec:test_struts_flexible_modes} & \texttt{test\_struts\_1\_flexible\_modes.m}\\
+Section \ref{sec:test_struts_dynamical_meas} & \texttt{test\_struts\_2\_dynamical\_meas.m}\\
+Section \ref{sec:test_struts_simscape} & \texttt{test\_struts\_3\_simscape\_model.m}\\
+\bottomrule
+\end{tabularx}
+\end{table}
 \chapter{Mounting Procedure}
 \label{sec:test_struts_mounting}
-A mounting bench was developed to ensure:
-\begin{itemize}
-\item Good coaxial alignment between the interfaces (cylinders) of the flexible joints.
-This is important not to loose to much angular stroke during their mounting into the nano-hexapod
-\item Uniform length across all struts
-\item Precise alignment of the APA with the two flexible joints
-\item Reproducible and consistent assembly between all struts
-\end{itemize}
 \section{Mounting Bench}
 
-A CAD view of the mounting bench is shown in Figure \ref{fig:test_struts_mounting_bench_first_concept}.
+A mounting bench is used to greatly simply the mounting of the struts as well as ensuring the correct strut length and coaxiality of the flexible joint's interfaces.
-It consists of a ``main frame'' (Figure \ref{fig:test_struts_mounting_step_0}) precisely machined to ensure both correct strut length and strut coaxiality.
+This is very important in order to not loose any stroke when the struts will be mounted on the nano-hexapod.
-The coaxiality is ensured by good flatness (specified at \(20\,\mu m\)) between surfaces A and B and between surfaces C and D.
-Such flatness was checked using a Faro arm\footnote{Faro Arm Platinum 4ft, specified accuracy of \(\pm 13\mu m\)} (see Figure \ref{fig:test_struts_check_dimensions_bench}) and was found to comply with the requirements.
-The strut length (defined by the distance between the rotation points of the two flexible joints) was ensured by using precisely machines dowel holes.
 
+A CAD view of the mounting bench is shown in Figure \ref{fig:test_struts_mounting_bench_first_concept}.
+
+Faro arm\footnote{Faro Arm Platinum 4ft, specified accuracy of \(\pm 13\mu m\)}
 
 \begin{figure}[htbp]
-\begin{subfigure}{0.49\textwidth}
+\centering
-\begin{center}
+\includegraphics[scale=1,width=0.6\linewidth]{figs/test_struts_mounting_bench_first_concept.png}
-\includegraphics[scale=1,width=\linewidth]{figs/test_struts_mounting_bench_first_concept.png}
+\caption{\label{fig:test_struts_mounting_bench_first_concept}CAD view of the mounting bench}
-\end{center}
-\subcaption{\label{fig:test_struts_mounting_bench_first_concept}CAD view of the mounting bench}
-\end{subfigure}
-\begin{subfigure}{0.49\textwidth}
-\begin{center}
-\includegraphics[scale=1,width=\linewidth]{figs/test_struts_mounting_overview.jpg}
-\end{center}
-\subcaption{\label{fig:test_struts_mounting_overview}Exploded view}
-\end{subfigure}
-\caption{\label{fig:test_struts_mounting}Strut mounting bench}
 \end{figure}
 
+The main part of the bench is here to ensure both the correct strut length and strut coaxiality as shown in Figure \ref{fig:test_struts_mounting_step_0}.
+
 \begin{figure}[htbp]
 \begin{subfigure}{0.56\textwidth}
 \begin{center}
@@ -99,12 +98,33 @@ The strut length (defined by the distance between the rotation points of the two
 \end{center}
 \subcaption{\label{fig:test_struts_check_dimensions_bench}Dimensional check}
 \end{subfigure}
-\caption{\label{fig:test_struts_mounting_base_part}Main element of the mounting bench for the struts that ensure good coaxiality of the two flexible joints and correct struts length.}
+\caption{\label{fig:test_struts_mounting_base_part}Caption\ldots{}, add foot note with Faro arm}
 \end{figure}
 
-The flexible joints were not directly fixed to the mounting bench but were fixed to a cylindrical ``sleeve'' shown in Figures \ref{fig:test_struts_cylindrical_mounting_part_top} and \ref{fig:test_struts_cylindrical_mounting_part_bot}.
+The tight tolerances of this element has been verified as shown in Figure \ref{fig:test_struts_check_dimensions_bench} and were found to comply with the requirements.
-The goal of these ``sleeves'' is to avoid mechanical stress that could damage the flexible joints during the mounting process.
+
-These ``sleeves'' have one dowel groove (that are fitted to the dowel holes shown in Figure \ref{fig:test_struts_mounting_step_0}) that will determine the length of the mounted strut.
+The flexible joints are rigidly fixed to cylindrical tools shown in Figures \ref{fig:test_struts_cylindrical_mounting_part_top} and \ref{fig:test_struts_cylindrical_mounting_part_bot} which are then mounted on the mounting tool shown in Figure \ref{fig:test_struts_mounting_step_0}.
+This cylindrical tool is here to protect the flexible joints when tightening the screws and therefore applying large torque.
+\section{Mounting Procedure}
+
+\begin{itemize}
+\item[{$\square$}] Better explain the mounting procedure
+\item[{$\square$}] Speak about the ``locating'' pins that are used to aligned the APA with the two flexible joints
+\end{itemize}
+
+The mounting procedure is as follows:
+\begin{enumerate}
+\item Screw flexible joints inside the cylindrical interface element shown in Figure \ref{fig:test_struts_cylindrical_mounting}
+\item Fix the two interface elements. One of the two should be clamped, the other one should have its axial rotation free.
+Visually align the clamped one horizontally. (Figure \ref{fig:test_struts_mounting_step_1})
+\item Put cylindrical washers, APA and interface pieces on top of the flexible joints (Figure \ref{fig:test_struts_mounting_step_2})
+\item Put the 4 screws just in contact such that everything is correctly positioned and such that the ``free'' flexible joint is correctly oriented
+\item Put the 8 lateral screws in contact
+\item Tighten the 4 screws to fix the APA on the two flexible joints (using a torque screwdriver)
+\item Remove the 4 laterals screws
+\item (optional) Put the APA horizontally and fix the encoder and align it to maximize the contrast (Figure \ref{fig:test_struts_mounting_step_3})
+\item Disassemble to have an properly mounted strut (Figure \ref{fig:test_struts_mounting_step_4}) for which the coaxiality between the two flexible joint's interfaces is good
+\end{enumerate}
 
 \begin{figure}[htbp]
 \begin{subfigure}{0.33\textwidth}
@@ -125,23 +145,9 @@ These ``sleeves'' have one dowel groove (that are fitted to the dowel holes show
 \end{center}
 \subcaption{\label{fig:test_struts_mounting_joints}Mounted flexible joints}
 \end{subfigure}
-\caption{\label{fig:test_struts_cylindrical_mounting}Preparation of the flexible joints by fixing them in their cylindrical ``sleeve''}
+\caption{\label{fig:test_struts_cylindrical_mounting}Preparation of the flexible joints by fixing them in their cylindrical interface}
 \end{figure}
 
-\section{Mounting Procedure}
-
-The ``sleeves'' were mounted to the main element as shown in Figure \ref{fig:test_struts_mounting_step_0}.
-The left sleeve has a thigh fit such that its orientation is fixed (it is roughly aligned horizontally), while the right sleeve has a loose fit such that it can rotate (it will get the same orientation as the fixed one when tightening the screws).
-
-The cylindrical washers and the APA300ML are stacked on top of the flexible joints, as shown in Figure \ref{fig:test_struts_mounting_step_2} and screwed together using a torque screwdriver.
-A dowel pin is used to laterally align the APA300ML with the flexible joints (see the dowel slot on the flexible joints in Figure \ref{fig:test_struts_mounting_joints}).
-Two cylindrical washers are used to allow proper mounting even when the two APA interfaces are not parallel.
-
-The encoder and ruler are then fixed to the strut and properly aligned, as shown in Figure \ref{fig:test_struts_mounting_step_3}.
-
-Finally, the strut can be disassembled from the mounting bench (Figure \ref{fig:test_struts_mounting_step_4}).
-Thanks to this mounting procedure, the coaxiality and length between the two flexible joint's interfaces can be obtained within the desired tolerances.
-
 \begin{figure}[htbp]
 \begin{subfigure}{0.5\textwidth}
 \begin{center}
@@ -171,72 +177,67 @@ Thanks to this mounting procedure, the coaxiality and length between the two fle
 \end{subfigure}
 \caption{\label{fig:test_struts_mounting_steps}Steps for mounting the struts.}
 \end{figure}
-
 \chapter{Measurement of flexible modes}
 \label{sec:test_struts_flexible_modes}
 \section{Introduction}
 
-A Finite Element Model\footnote{Using Ansys\textsuperscript{\textregistered}. Flexible Joints and APA Shell are made of a stainless steel allow called \emph{17-4 PH}. Encoder and ruler support material is aluminium.} of the struts is developed and is used to estimate the flexible modes.
+From a Finite Element Model of the struts, it have been found that three main resonances are foreseen to be problematic for the control of the APA300ML (Figure \ref{fig:test_struts_mode_shapes}): an ``X-bending'' mode at 189Hz, a ``Y-bending'' mode at 285Hz and a ``Z-torsion'' mode at 400Hz.
-The inertia of the encoder (estimated at \(15\,g\)) is considered.
-The two cylindrical interfaces were fixed, and the first three flexible modes were computed.
-The mode shapes are displayed in Figure \ref{fig:test_struts_mode_shapes}: an ``X-bending'' mode at 189Hz, a ``Y-bending'' mode at 285Hz and a ``Z-torsion'' mode at 400Hz.
 
 \begin{figure}[htbp]
 \begin{subfigure}{0.33\textwidth}
 \begin{center}
-\includegraphics[scale=1,width=0.85\linewidth]{figs/test_struts_mode_shapes_1.png}
+\includegraphics[scale=1,width=0.9\linewidth]{figs/test_struts_mode_shapes_1.png}
 \end{center}
 \subcaption{\label{fig:test_struts_mode_shapes_1}X-bending mode (189Hz)}
 \end{subfigure}
 \begin{subfigure}{0.33\textwidth}
 \begin{center}
-\includegraphics[scale=1,width=0.85\linewidth]{figs/test_struts_mode_shapes_2.png}
+\includegraphics[scale=1,width=0.9\linewidth]{figs/test_struts_mode_shapes_2.png}
 \end{center}
 \subcaption{\label{fig:test_struts_mode_shapes_2}Y-bending mode (285Hz)}
 \end{subfigure}
 \begin{subfigure}{0.33\textwidth}
 \begin{center}
-\includegraphics[scale=1,width=0.85\linewidth]{figs/test_struts_mode_shapes_3.png}
+\includegraphics[scale=1,width=0.9\linewidth]{figs/test_struts_mode_shapes_3.png}
 \end{center}
 \subcaption{\label{fig:test_struts_mode_shapes_3}Z-torsion mode (400Hz)}
 \end{subfigure}
 \caption{\label{fig:test_struts_mode_shapes}Spurious resonances of the struts estimated from a Finite Element Model}
 \end{figure}
-
 \section{Measurement Setup}
 
-To experimentally measure these mode shapes, a Laser vibrometer\footnote{OFV-3001 controller and OFV512 sensor head from Polytec} was used.
+A Laser vibrometer is measuring the difference of motion between two beam path (red points in Figure \ref{fig:test_struts_meas_modes}).
-It measures the difference of motion between two beam path (red points in Figure \ref{fig:test_struts_meas_modes}).
+The strut is excited with an instrumented hammer and the transfer function from the hammer to the measured rotation is computed.
-The strut is then excited by an instrumented hammer, and the transfer function from the hammer to the measured rotation is computed.
 
-The setup used to measure the ``X-bending'' mode is shown in Figure \ref{fig:test_struts_meas_x_bending}.
+The ``X-bending'' mode is measured as shown in Figure \ref{fig:test_struts_meas_x_bending}.
-The ``Y-bending'' mode is measured as shown in Figure \ref{fig:test_struts_meas_y_bending} and the ``Z-torsion'' measurement setup is shown in Figure \ref{fig:test_struts_meas_z_torsion}.
+The ``Y-bending'' mode is measured as shown in Figure \ref{fig:test_struts_meas_y_bending}.
-These tests were performed with and without the encoder being fixed to the strut.
+Finally, the ``Z-torsion'' is measured as shown in Figure \ref{fig:test_struts_meas_z_torsion}.
+
+This is done with and without the encoder fixed to the strut.
 
 \begin{figure}[htbp]
 \begin{subfigure}{0.33\textwidth}
 \begin{center}
-\includegraphics[scale=1,width=0.85\linewidth]{figs/test_struts_meas_x_bending.jpg}
+\includegraphics[scale=1,width=0.9\linewidth]{figs/test_struts_meas_x_bending.jpg}
 \end{center}
 \subcaption{\label{fig:test_struts_meas_x_bending}X-bending mode}
 \end{subfigure}
 \begin{subfigure}{0.33\textwidth}
 \begin{center}
-\includegraphics[scale=1,width=0.85\linewidth]{figs/test_struts_meas_y_bending.jpg}
+\includegraphics[scale=1,width=0.9\linewidth]{figs/test_struts_meas_y_bending.jpg}
 \end{center}
 \subcaption{\label{fig:test_struts_meas_y_bending}Y-bending mode}
 \end{subfigure}
 \begin{subfigure}{0.33\textwidth}
 \begin{center}
-\includegraphics[scale=1,width=0.85\linewidth]{figs/test_struts_meas_z_torsion.jpg}
+\includegraphics[scale=1,width=0.9\linewidth]{figs/test_struts_meas_z_torsion.jpg}
 \end{center}
 \subcaption{\label{fig:test_struts_meas_z_torsion}Z-torsion mode}
 \end{subfigure}
 \caption{\label{fig:test_struts_meas_modes}Measurement of strut flexible modes}
 \end{figure}
-
 \section{Measured results}
-The obtained frequency response functions for the three configurations (X-bending, Y-bending and Z-torsion) are shown in Figure \ref{fig:test_struts_spur_res_frf_no_enc} when the encoder is not fixed to the strut and in Figure \ref{fig:test_struts_spur_res_frf_enc} when the encoder is fixed to the strut.
+The obtained frequency response functions are shown in Figure \ref{fig:test_struts_spur_res_frf}.
 
 \begin{figure}[htbp]
 \begin{subfigure}{0.49\textwidth}
@@ -253,18 +254,19 @@ The obtained frequency response functions for the three configurations (X-bendin
 \end{subfigure}
 \caption{\label{fig:test_struts_spur_res_frf}Measured frequency response functions without the encoder \ref{fig:test_struts_spur_res_frf} and with the encoder \ref{fig:test_struts_spur_res_frf_enc}}
 \end{figure}
-
+Table \ref{tab:test_struts_spur_mode_freqs} summarizes the measured resonance frequencies as well as the computed ones using the Finite Element Model.
-Table \ref{tab:test_struts_spur_mode_freqs} summarizes the measured resonance frequencies and the computed ones using the \acrfull{fem}.
+It is shown that:
-The resonance frequencies of the 3 modes are only slightly decreased when the encoder is fixed to the strut.
+\begin{itemize}
-In addition, the computed resonance frequencies from the \acrshort{fem} are very close to the measured frequencies when the encoder is fixed to the strut.
+\item the resonance frequencies of the 3 modes are only slightly increasing when the encoder is removed
-This validates the quality of the \acrshort{fem}.
+\item the computed resonance frequencies from the FEM are very close to the measured one when the encoder is fixed to the strut
+\end{itemize}
 
 \begin{table}[htbp]
-\caption{\label{tab:test_struts_spur_mode_freqs}Measured frequency of the flexible modes of the strut}
+\caption{\label{tab:test_struts_spur_mode_freqs}Measured frequency of the strut spurious modes}
 \centering
-\begin{tabularx}{0.9\linewidth}{Xccc}
+\begin{tabularx}{0.7\linewidth}{Xccc}
 \toprule
-\textbf{Mode} & \textbf{FEM with Encoder} & \textbf{Exp. with Encoder} & \textbf{Exp. without Encoder}\\
+\textbf{Mode} & \textbf{Struts (FEM)} & \textbf{Struts (exp)} & \textbf{Plates (exp)}\\
 \midrule
 X-Bending & 189Hz & 198Hz & 226Hz\\
 Y-Bending & 285Hz & 293Hz & 337Hz\\
@@ -272,38 +274,33 @@ Z-Torsion & 400Hz & 381Hz & 398Hz\\
 \bottomrule
 \end{tabularx}
 \end{table}
-
 \chapter{Dynamical measurements}
 \label{sec:test_struts_dynamical_meas}
-In order to measure the dynamics of the strut, the test bench used to measure the APA300ML dynamics is being used again.
+The bench is shown in Figure \ref{fig:test_struts_bench_leg}.
-
-The strut mounted on the bench is shown in Figure \ref{fig:test_struts_bench_leg_overview}
-A schematic of the bench and the associated signals are shown in Figure \ref{fig:test_struts_bench_schematic}.
-A fiber interferometer\footnote{Two fiber intereferometers were used: an IDS3010 from Attocube and a quDIS from QuTools} is used to measure the motion of the granite (i.e. the axial motion of the strut).
 
 \begin{figure}[htbp]
 \begin{subfigure}{0.32\textwidth}
 \begin{center}
-\includegraphics[scale=1,height=210px]{figs/test_struts_bench_leg_overview.jpg}
+\includegraphics[scale=1,height=214px]{figs/test_struts_bench_leg_overview.jpg}
 \end{center}
 \subcaption{\label{fig:test_struts_bench_leg_overview}Overview Picture}
 \end{subfigure}
 \begin{subfigure}{0.68\textwidth}
 \begin{center}
-\includegraphics[scale=1,height=210px]{figs/test_struts_bench_schematic.png}
+\includegraphics[scale=1,scale=1]{figs/test_struts_bench_schematic.png}
 \end{center}
 \subcaption{\label{fig:test_struts_bench_schematic}Schematic}
 \end{subfigure}
-\caption{\label{fig:test_struts_bench_leg}Experimental setup used to measure the dynamics of the struts.}
+\caption{\label{fig:test_struts_bench_leg}Experimental setup used to measured the dynamics of the struts.}
 \end{figure}
 
-First, the effect of the encoder on the measured dynamics is investigated in Section \ref{ssec:test_struts_effect_encoder}.
+First, the effect of the encoder on the measured dynamics is studied in Section \ref{ssec:test_struts_effect_encoder}.
-The dynamics observed by the encoder and interferometers are compared in Section \ref{ssec:test_struts_comp_enc_int}.
+Then, the dynamics seen by the encoder and by the interferometers are compared in Section \ref{ssec:test_struts_comp_enc_int}.
-Finally, all measured struts are compared in terms of dynamics in Section \ref{ssec:test_struts_comp_all_struts}.
+Finally, all the measured struts are compared in terms of dynamics in Section \ref{ssec:test_struts_comp_all_struts}.
 \section{Effect of the Encoder on the measured dynamics}
 \label{ssec:test_struts_effect_encoder}
 
-System identification was performed without the encoder being fixed to the strut (Figure \ref{fig:test_struts_bench_leg_front}) and with one encoder being fixed to the strut (Figure \ref{fig:test_struts_bench_leg_coder}).
+Measurements are performed either when no encoder is fixed to the strut (Figure \ref{fig:test_struts_bench_leg_front}) or when one encoder is fixed to the strut (Figure \ref{fig:test_struts_bench_leg_coder}).
 
 \begin{figure}[htbp]
 \begin{subfigure}{0.5\textwidth}
@@ -321,112 +318,131 @@ System identification was performed without the encoder being fixed to the strut
 \caption{\label{fig:test_struts_bench_leg_with_without_enc}Struts fixed to the test bench with clamped flexible joints. The coder can be fixed to the struts (\subref{fig:test_struts_bench_leg_coder}) or removed (\subref{fig:test_struts_bench_leg_front})}
 \end{figure}
 
-The obtained frequency response functions are compared in Figure \ref{fig:test_struts_effect_encoder}.
+Figure \ref{fig:test_struts_effect_encoder_int}
-It was found that the encoder had very little effect on the transfer function from excitation voltage \(u\) to the axial motion of the strut \(d_a\) as measured by the interferometer (Figure \ref{fig:test_struts_effect_encoder_int}).
+Same goes for the transfer function from excitation voltage \(u\) to the axial motion of the strut \(d_a\) as measured by the interferometer ().
-This means that the axial motion of the strut is unaffected by the presence of the encoder.
+
-Similarly, it has little effect on the transfer function from \(u\) to the sensor stack voltage \(V_s\) (Figure \ref{fig:test_struts_effect_encoder_iff}).
+The transfer function from the excitation voltage \(u\) to the generated voltage \(V_s\) by the sensor stack is not influence by the fixation of the encoder (Figure \ref{fig:test_struts_effect_encoder_iff}).
-This means that the encoder should have little effect on the effectiveness of the integral force feedback control strategy.
+This means that the IFF control strategy should be as effective whether or not the encoders are fixed to the struts.
 
 \begin{figure}[htbp]
-\begin{subfigure}{0.33\textwidth}
+\begin{subfigure}{0.49\textwidth}
 \begin{center}
 \includegraphics[scale=1,width=0.95\linewidth]{figs/test_struts_effect_encoder_int.png}
 \end{center}
 \subcaption{\label{fig:test_struts_effect_encoder_int}$u$ to $d_a$}
 \end{subfigure}
-\begin{subfigure}{0.33\textwidth}
+\begin{subfigure}{0.49\textwidth}
 \begin{center}
 \includegraphics[scale=1,width=0.95\linewidth]{figs/test_struts_effect_encoder_iff.png}
 \end{center}
 \subcaption{\label{fig:test_struts_effect_encoder_iff}$u$ to $V_s$}
 \end{subfigure}
-\begin{subfigure}{0.33\textwidth}
+\caption{\label{fig:test_struts_effect_encoder}Effect of having the encoder fixed to the struts on the measured dynamics from \(u\) to \(d_a\) (\subref{fig:test_struts_effect_encoder_int}) and from \(u\) to \(V_s\) (\subref{fig:test_struts_effect_encoder_iff})}
-\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/test_struts_comp_enc_int.png}
-\end{center}
-\subcaption{\label{fig:test_struts_comp_enc_int}$u$ to $d_e$, $d_a$}
-\end{subfigure}
-\caption{\label{fig:test_struts_effect_encoder}Effect of having the encoder fixed to the struts on the measured dynamics from \(u\) to \(d_a\) (\subref{fig:test_struts_effect_encoder_int}) and from \(u\) to \(V_s\) (\subref{fig:test_struts_effect_encoder_iff}). Comparison of the observed dynamics by the encoder and interferometers (\subref{fig:test_struts_comp_enc_int})}
 \end{figure}
-
 \section{Comparison of the encoder and interferometer}
 \label{ssec:test_struts_comp_enc_int}
 
-The dynamics measured by the encoder (i.e. \(d_e/u\)) and interferometers (i.e. \(d_a/u\)) are compared in Figure \ref{fig:test_struts_comp_enc_int}.
+The dynamics as measured by the encoder and by the interferometers are compared in Figure \ref{fig:test_struts_comp_enc_int}.
-The dynamics from the excitation voltage \(u\) to the displacement measured by the encoder \(d_e\) presents a behavior that is much more complex than the dynamics of the displacement measured by the interferometer (comparison made in Figure \ref{fig:test_struts_comp_enc_int}).
-Three additional resonance frequencies can be observed at 197Hz, 290Hz and 376Hz.
-These resonance frequencies match the frequencies of the flexible modes studied in Section \ref{sec:test_struts_flexible_modes}.
 
-The good news is that these resonances are not impacting the axial motion of the strut (which is what is important for the hexapod positioning).
+The dynamics from the excitation voltage \(u\) to the measured displacement by the encoder \(d_e\) presents much more complicated behavior than the transfer function to the displacement as measured by the Interferometer (compared in Figure \ref{fig:test_struts_comp_enc_int}).
-However, these resonances make the use of an encoder fixed to the strut difficult from a control perspective.
+It will be further investigated why the two dynamics as so different and what are causing all these resonances.
 
+As shown in Figure \ref{fig:test_struts_comp_enc_int}, we can clearly see three spurious resonances at 197Hz, 290Hz and 376Hz.
+These resonances correspond to parasitic resonances of the strut itself that was estimated using a finite element model of the strut (Figure \ref{fig:test_struts_mode_shapes}):
+\begin{itemize}
+\item Mode in X-bending at 189Hz
+\item Mode in Y-bending at 285Hz
+\item Mode in Z-torsion at 400Hz
+\end{itemize}
+
+The good news is that these resonances are not seen on the interferometer (they are therefore not impacting the axial motion of the strut).
+But these resonances are making the use of encoder fixed to the strut difficult.
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1]{figs/test_struts_comp_enc_int.png}
+\caption{\label{fig:test_struts_comp_enc_int}Comparison of the transfer functions from excitation voltage \(u\) to either the encoder \(d_e\) or the interferometer \(d_a\)}
+\end{figure}
 \section{Comparison of all the Struts}
 \label{ssec:test_struts_comp_all_struts}
 
-The dynamics of all the mounted struts (only 5 at the time of the experiment) were then measured on the same test bench.
+Then, the transfer function from the DAC output voltage \(u\) to the measured displacement by the Attocube is computed for all the struts and shown in Figure \ref{fig:test_struts_comp_interf_plants}.
-The obtained dynamics from \(u\) to \(d_a\) are compared in Figure \ref{fig:test_struts_comp_interf_plants} while is dynamics from \(u\) to \(V_s\) are compared in Figure \ref{fig:test_struts_comp_iff_plants}.
+All the struts are giving very similar FRF.
-A very good match can be observed between the struts.
 
 \begin{figure}[htbp]
-\begin{subfigure}{0.33\textwidth}
+\begin{subfigure}{0.49\textwidth}
 \begin{center}
 \includegraphics[scale=1,width=\linewidth]{figs/test_struts_comp_interf_plants.png}
 \end{center}
 \subcaption{\label{fig:test_struts_comp_interf_plants}$u$ to $d_a$}
 \end{subfigure}
-\begin{subfigure}{0.33\textwidth}
+\begin{subfigure}{0.49\textwidth}
 \begin{center}
 \includegraphics[scale=1,width=\linewidth]{figs/test_struts_comp_iff_plants.png}
 \end{center}
 \subcaption{\label{fig:test_struts_comp_iff_plants}$u$ to $V_s$}
 \end{subfigure}
-\begin{subfigure}{0.33\textwidth}
-\begin{center}
-\includegraphics[scale=1,width=\linewidth]{figs/test_struts_comp_enc_plants.png}
-\end{center}
-\subcaption{\label{fig:test_struts_comp_enc_plants}$u$ to $d_e$}
-\end{subfigure}
 \caption{\label{fig:test_struts_comp_plants}Comparison of the measured plants}
 \end{figure}
 
-The same comparison is made for the transfer function from \(u\) to \(d_e\) (encoder output) in Figure \ref{fig:test_struts_comp_enc_plants}.
+There is a very large variability of the dynamics as measured by the encoder as shown in Figure \ref{fig:test_struts_comp_enc_plants}.
-In this study, large dynamics differences were observed between the 5 struts.
+Even-though the same peaks are seen for all of the struts (95Hz, 200Hz, 300Hz, 400Hz), the amplitude of the peaks are not the same.
-Although the same resonance frequencies were seen for all of the struts (95Hz, 200Hz, 300Hz and 400Hz), the amplitude of the peaks were not the same.
+Moreover, the location or even the presence of complex conjugate zeros is changing from one strut to the other.
-In addition, the location or even presence of complex conjugate zeros changes from one strut to another.
 
-\section*{Conclusion}
+All of this will be studied in Section \ref{sec:test_struts_simscape} using the Simscape model.
-All the struts exhibit very consistent behavior from the excitation voltage \(u\) to the force sensor generated voltage \(V_s\) and to the interferometer measured displacement \(d_a\).
-However, the dynamics from \(u\) to the encoder measurement \(d_e\) is much more complex and vary from one strut to the another.
-The reason for this variability will be studied in the next section thanks to the strut model.
-
-\chapter{Strut Model}
-\label{sec:test_struts_simscape}
-The Simscape model of the strut was included in the Simscape model of the test bench (see Figure \ref{fig:test_struts_simscape_model}).
-The obtained model was first used to compare the measured FRF with the existing model (Section \ref{ssec:test_struts_comp_model}).
-
-Using a flexible APA model (extracted from a \acrshort{fem}), the effect of a misalignment of the APA with respect to flexible joints is studied (Section \ref{ssec:test_struts_effect_misalignment}).
-It was found that misalignment has a large impact on the dynamics from \(u\) to \(d_e\).
-This misalignment is estimated and measured in Section \ref{ssec:test_struts_meas_misalignment}.
-The struts were then disassembled and reassemble a second time to optimize alignment (Section \ref{sec:test_struts_meas_all_aligned_struts}).
 
 \begin{figure}[htbp]
 \centering
-\includegraphics[scale=1,width=0.65\linewidth]{figs/test_struts_simscape_model.png}
+\includegraphics[scale=1]{figs/test_struts_comp_enc_plants.png}
+\caption{\label{fig:test_struts_comp_enc_plants}Estimated frequency response functions from \(u\) to the encoder \(d_e\) for all the mounted struts}
+\end{figure}
+\begin{important}
+All the struts are giving very consistent behavior from the excitation voltage \(u\) to the force sensor generated voltage \(V_s\) and to the interferometer measured displacement \(d_a\).
+However, the dynamics from \(u\) to the encoder measurement \(d_e\) is much more complex and variable from one strut to the other most likely due to poor alignment of the APA with respect to the flexible joints.
+\end{important}
+\chapter{Strut Model}
+\label{sec:test_struts_simscape}
+However, now the full strut is put instead of only the APA (see Figure \ref{fig:test_struts_simscape_model}).
+
+\begin{figure}[htbp]
+\centering
+\includegraphics[scale=1,width=0.5\linewidth]{figs/test_struts_simscape_model.png}
 \caption{\label{fig:test_struts_simscape_model}Screenshot of the Simscape model of the strut fixed to the bench}
 \end{figure}
+
+This Simscape model is used to:
+\begin{itemize}
+\item compare the measured FRF with the modelled FRF
+\item help the correct understanding/interpretation of the results
+\item tune the model of the struts (APA, flexible joints, encoder)
+\end{itemize}
+
+This study is structured as follow:
+\begin{itemize}
+\item Section \ref{ssec:test_struts_comp_model}: the measured FRF are compared with the Simscape model.
+\item Section \ref{ssec:test_struts_effect_misalignment}: the flexible APA model is used, and the effect of a misalignment of the APA and flexible joints is studied.
+It is found that the misalignment has a large impact on the dynamics from \(u\) to \(d_e\).
+\item Section \ref{ssec:test_struts_effect_joint_stiffness}: the effect of the flexible joint's stiffness on the dynamics is studied.
+It is found that the axial stiffness of the joints has a large impact on the location of the zeros on the transfer function from \(V_s\) to \(d_e\).
+\end{itemize}
 \section{Model dynamics}
 \label{ssec:test_struts_comp_model}
+Two models of the APA300ML are used here for comparison:
+\begin{itemize}
+\item a simple two degrees of freedom model
+\item a model using a super element extracted from a finite element model
+\end{itemize}
+
+These two models of the APA300ML were tuned to best match measured frequency response functions of the APA alone.
+The flexible joints are here modelled with the 4DoF model (axial stiffness, two bending stiffnesses and one torsion stiffness).
 
-Two models of the APA300ML are used here: a simple two-degrees-of-freedom model and a model using a super-element extracted from a \acrlong{fem}.
-These two models of the APA300ML were tuned to best match the measured frequency response functions of the APA alone.
-The flexible joints were modelled with the 4DoF model (axial stiffness, two bending stiffnesses and one torsion stiffness).
 These two models are compared with the measured frequency responses in Figure \ref{fig:test_struts_comp_frf_flexible_model}.
 
 The model dynamics from DAC voltage \(u\) to the axial motion of the strut \(d_a\) (Figure \ref{fig:test_struts_comp_frf_flexible_model_int}) and from DAC voltage \(u\) to the force sensor voltage \(V_s\) (Figure \ref{fig:test_struts_comp_frf_flexible_model_iff}) are well matching the experimental identification.
 
 However, the transfer function from \(u\) to encoder displacement \(d_e\) are not well matching for both models.
-For the 2DoF model, this is normal because the resonances affecting the dynamics are not modelled at all (the APA300ML is modeled as infinitely rigid in all directions except the translation along it's actuation axis).
+For the 2DoF model, this is normal as the resonances affecting the dynamics are not modelled at all (the APA300ML is modelled as infinitely rigid in all directions except the translation along it's actuation axis).
-For the flexible model, it will be shown in the next section that by adding some misalignment between the flexible joints and the APA300ML, this model can better represent the observed dynamics.
+For the flexible model, it will be shown in the next section that by adding some misalignment betwen the flexible joints and the APA300ML, this model can better represent the observed dynamics.
 
 \begin{figure}[htbp]
 \begin{subfigure}{0.33\textwidth}
@@ -447,16 +463,16 @@ For the flexible model, it will be shown in the next section that by adding some
 \end{center}
 \subcaption{\label{fig:test_struts_comp_frf_flexible_model_iff}$u$ to $V_s$}
 \end{subfigure}
-\caption{\label{fig:test_struts_comp_frf_flexible_model}Comparison of the measured frequency response functions, the Simscape model using the 2 DoF APA model, and using the ``flexible'' APA300ML model (Super-Element extracted from a Finite Element Model).}
+\caption{\label{fig:test_struts_comp_frf_flexible_model}Comparison of the measured dynamics and of the Simscape dynamics using the ``flexible'' APA300ML model (Super-Element extracted from a Finite Element Model).}
 \end{figure}
-
 \section{Effect of strut misalignment}
 \label{ssec:test_struts_effect_misalignment}
 
-As shown in Figure \ref{fig:test_struts_comp_enc_plants}, the identified dynamics from DAC voltage \(u\) to encoder measured displacement \(d_e\) are very different from one strut to the other.
+As was shown in Figure \ref{fig:test_struts_comp_enc_plants}, the identified dynamics from DAC voltage \(u\) to encoder measured displacement \(d_e\) are very different from one strut to the other.
+
 In this section, it is investigated whether poor alignment of the strut (flexible joints with respect to the APA) can explain such dynamics.
 For instance, consider Figure \ref{fig:test_struts_misalign_schematic} where there is a misalignment in the \(y\) direction between the two flexible joints (well aligned thanks to the mounting procedure in Section \ref{sec:test_struts_mounting}) and the APA300ML.
-In this case, the ``x-bending'' mode at 200Hz (see Figure \ref{fig:test_struts_meas_x_bending}) can be expected to have greater impact on the dynamics from the actuator to the encoder.
+In such case, the ``x-bending'' mode at 200Hz (see Figure \ref{fig:test_struts_meas_x_bending}) can be expected to be more excited, and thus the dynamics from the actuator to the encoder should be affected at frequencies around 200Hz.
 
 \begin{figure}[htbp]
 \centering
@@ -464,25 +480,25 @@ In this case, the ``x-bending'' mode at 200Hz (see Figure \ref{fig:test_struts_m
 \caption{\label{fig:test_struts_misalign_schematic}Mis-alignement between the joints and the APA}
 \end{figure}
 
-To verify this assumption, the dynamics from the output DAC voltage \(u\) to the measured displacement by the encoder \(d_e\) is computed using the flexible APA Simscape model for several misalignments in the \(y\) direction.
+To verify this assumption, the dynamics from output DAC voltage \(u\) to the measured displacement by the encoder \(d_e\) is computed using the Simscape model with flexible APA for several misalignment in the \(y\) direction.
-The obtained dynamics are shown in Figure \ref{fig:test_struts_effect_misalignment_y}.
+Obtained dynamics are shown in Figure \ref{fig:test_struts_effect_misalignment_y}.
-The alignment of the APA with the flexible joints has a large influence on the dynamics from actuator voltage to the measured displacement by the encoder.
+The alignment of the APA with the flexible joints as a \textbf{huge} influence on the dynamics from actuator voltage to measured displacement by the encoder.
 The misalignment in the \(y\) direction mostly influences:
 \begin{itemize}
 \item the presence of the flexible mode at 200Hz (see mode shape in Figure \ref{fig:test_struts_mode_shapes_1})
 \item the location of the complex conjugate zero between the first two resonances:
 \begin{itemize}
-\item if \(d_{y} < 0\): there is no zero between the two resonances and possibly not even between the second and third resonances
+\item if \(d_y < 0\): there is no zero between the two resonances and possibly not even between the second and third ones
-\item if \(d_{y} > 0\): there is a complex conjugate zero between the first two resonances
+\item if \(d_y > 0\): there is a complex conjugate zero between the first two resonances
 \end{itemize}
 \item the location of the high frequency complex conjugate zeros at 500Hz (secondary effect, as the axial stiffness of the joint also has large effect on the position of this zero)
 \end{itemize}
 
-The same can be done for misalignments in the \(x\) direction.
+The same can be done for a misalignment in the \(x\) direction.
-The obtained dynamics (Figure \ref{fig:test_struts_effect_misalignment_x}) are showing that misalignment in the \(x\) direction mostly influences the presence of the flexible mode at 300Hz (see mode shape in Figure \ref{fig:test_struts_mode_shapes_2}).
+The obtained dynamics are shown in Figure \ref{fig:test_struts_effect_misalignment_x} where it is shown that misalignment in the \(x\) direction mostly influences the presence of the flexible mode at 300Hz (see mode shape in Figure \ref{fig:test_struts_mode_shapes_2}).
 
-A comparison of the experimental frequency response functions in Figure \ref{fig:test_struts_comp_enc_plants} with the model dynamics for several \(y\) misalignments in Figure \ref{fig:test_struts_effect_misalignment_y} indicates a clear similarity.
+Comparing the experimental frequency response functions for all the APA in Figure \ref{fig:test_struts_comp_enc_plants} with the model dynamics for several \(y\) misalignments in Figure \ref{fig:test_struts_effect_misalignment_y} indicates a clear similarity.
-This similarity suggests that the identified differences in dynamics are caused by misalignment.
+This similarity suggests that the identified differences in dynamics are caused by the misalignment.
 
 \begin{figure}[htbp]
 \begin{subfigure}{0.49\textwidth}
@@ -499,24 +515,23 @@ This similarity suggests that the identified differences in dynamics are caused
 \end{subfigure}
 \caption{\label{fig:test_struts_effect_misalignment}Effect of a misalignment between the flexible joints and the APA300ML in the \(y\) direction (\subref{fig:test_struts_effect_misalignment_y}) and in the \(x\) direction (\subref{fig:test_struts_effect_misalignment_x})}
 \end{figure}
-
 \section{Measured strut misalignment}
 \label{ssec:test_struts_meas_misalignment}
 
-During the initial mounting of the struts, as presented in Section \ref{sec:test_struts_mounting}, the positioning pins that were used to position the APA with respect to the flexible joints in the \(y\) directions were not used (not received at the time).
+During the first mounting of the struts presented in Section \ref{sec:test_struts_mounting}, the positioning pins used to position the APA with respect to the flexible joints in the \(y\) directions were not used (not received at the time).
-Therefore, large \(y\) misalignments are expected.
+Therefore, large \(y\) misalignments may be expected.
 
-To estimate the misalignments between the two flexible joints and the APA:
+In order to estimate the misalignments between the two flexible joints and the APA:
 \begin{itemize}
-\item the struts were fixed horizontally on the mounting bench, as shown in Figure \ref{fig:test_struts_mounting_step_3} but without the encoder
+\item the struts are fixed horizontally to the mounting bench as shown in Figure \ref{fig:test_struts_mounting_step_3} but without the encoder
-\item using a length gauge\footnote{Heidenhain MT25, specified accuracy of \(\pm 0.5\,\mu m\)}, the height difference between the flexible joints surface and the APA shell surface was measured for both the top and bottom joints and for both sides
+\item using a length gauge\footnote{Heidenhain MT25, specified accuracy of \(\pm 0.5\,\mu m\)}, the height difference from the flexible joints surface and the APA shell surface is measured both for the top and bottom joints and on both sides
-\item as the thickness of the flexible joint is \(21\,mm\) and the thickness of the APA shell is \(20\,mm\), \(0.5\,mm\) of height difference should be measured if the two are perfectly aligned
+\item as the thickness of the flexible joint is \(21\,mm\) and the thickness of the APA shell is \(20\,mm\), \(0.5\,mm\) of height different should be measured is the two are perfectly aligned
 \end{itemize}
 
 Large variations in the \(y\) misalignment are found from one strut to the other (results are summarized in Table \ref{tab:test_struts_meas_y_misalignment}).
 
-To check the validity of the measurement, it can be verified that the sum of the measured thickness difference on each side is \(1\,mm\) (equal to the thickness difference between the flexible joint and the APA).
+To check the validity of the measurement, it can be verified that sum of the measured thickness difference on each side is \(1\,mm\) (equal to the thickness difference between the flexible joint and the APA).
-Thickness differences for all the struts were found to be between \(0.94\,mm\) and \(1.00\,mm\) which indicate low errors compared to the misalignments found in Table \ref{tab:test_struts_meas_y_misalignment}.
+This thickness differences for all the struts were found to be between \(0.94\,mm\) and \(1.00\,mm\) which indicate low errors as compared to the misalignments found in Table \ref{tab:test_struts_meas_y_misalignment}.
 
 \begin{table}[htbp]
 \caption{\label{tab:test_struts_meas_y_misalignment}Measured \(y\) misalignment at the top and bottom of the APA. Measurements are in \(mm\)}
@@ -534,29 +549,30 @@ Thickness differences for all the struts were found to be between \(0.94\,mm\) a
 \end{tabularx}
 \end{table}
 
-By using the measured \(y\) misalignment in the Simscape model with the flexible APA model, the model dynamics from \(u\) to \(d_e\) is closer to the measured dynamics, as shown in Figure \ref{fig:test_struts_comp_dy_tuned_model_frf_enc}.
+By using the measured \(y\) misalignment in the Simscape model with the flexible APA model, the measured dynamics from \(u\) to \(d_e\) can be approached as shown in Figure \ref{fig:test_struts_comp_dy_tuned_model_frf_enc}.
-A better match in the dynamics can be obtained by fine-tuning both the \(x\) and \(y\) misalignments (yellow curves in Figure \ref{fig:test_struts_comp_dy_tuned_model_frf_enc}).
+Even better match in the dynamics can be obtained by fine tuning both the \(x\) and \(y\) misalignments (yellow curves in Figure \ref{fig:test_struts_comp_dy_tuned_model_frf_enc}).
+
+
+This confirms that the misalignment between the APA and the strut axis (determined by the two flexible joints) is critical and is inducing large variations in the dynamics from DAC voltage \(u\) to encoder measured displacement \(d_e\).
+If encoders are to be used when fixed on the struts, it is therefore very important to properly align the APA and the flexible joints when mounting the struts.
 
-This confirms that misalignment between the APA and the strut axis (determined by the two flexible joints) is critical and inducing large variations in the dynamics from DAC voltage \(u\) to encoder measured displacement \(d_e\).
-If encoders are fixed to the struts, the APA and flexible joints must be precisely aligned when mounting the struts.
 
 In the next section, the struts are re-assembled with a ``positioning pin'' to better align the APA with the flexible joints.
-With a better alignment, the amplitude of the spurious resonances is expected to decrease, as shown in Figure \ref{fig:test_struts_effect_misalignment_y}.
+With a better alignment, the amplitude of the spurious resonances are expected to decrease as was shown in Figure \ref{fig:test_struts_effect_misalignment_y}.
 
 \begin{figure}[htbp]
 \centering
 \includegraphics[scale=1]{figs/test_struts_comp_dy_tuned_model_frf_enc.png}
-\caption{\label{fig:test_struts_comp_dy_tuned_model_frf_enc}Comparison of the frequency response functions from DAC voltage \(u\) to measured displacement \(d_e\) by the encoders for the three struts. In blue, the measured dynamics is represted, in red the dynamics extracted from the model with the \(y\) misalignment estimated from measurements, and in yellow, the dynamics extracted from the model when both the \(x\) and \(y\) misalignments are tuned}
+\caption{\label{fig:test_struts_comp_dy_tuned_model_frf_enc}Comparison of the frequency response functions from DAC voltage \(u\) to measured displacement \(d_e\) by the encoders for three struts. In blue the measured dynamics, in red the dynamics extracted from the model with the \(y\) misalignment estimated from measurements, in yellow the dynamics extracted from the model when both the \(x\) and \(y\) misalignments are tuned}
 \end{figure}
-
 \section{Proper struts alignment}
 \label{sec:test_struts_meas_all_aligned_struts}
 
-After receiving the positioning pins, the struts were mounted again with the positioning pins.
+After the positioning pins had been received, the struts were mounted again with the positioning pins.
-This should improve the alignment of the APA with the two flexible joints.
+This should make the APA better aligned with the two flexible joints.
 
-The alignment is then estimated using a length gauge, as described in the previous sections.
+This alignment is then estimated using a length gauge as in the previous sections.
-Measured \(y\) alignments are summarized in Table \ref{tab:test_struts_meas_y_misalignment_with_pin} and are found to be bellow \(55\mu m\) for all the struts, which is much better than before (see Table \ref{tab:test_struts_meas_y_misalignment}).
+Measured \(y\) alignments are summarized in Table \ref{tab:test_struts_meas_y_misalignment_with_pin} and are found to be bellow \(55\mu m\) for all the struts which is much better than better (see Table \ref{tab:test_struts_meas_y_misalignment}).
 
 \begin{table}[htbp]
 \caption{\label{tab:test_struts_meas_y_misalignment_with_pin}Measured \(y\) misalignment at the top and bottom of the APA after realigning the struts using a positioning pin. Measurements are in \(mm\).}
@@ -575,31 +591,54 @@ Measured \(y\) alignments are summarized in Table \ref{tab:test_struts_meas_y_mi
 \end{tabularx}
 \end{table}
 
-The dynamics of the re-aligned struts were then measured on the same test bench (Figure \ref{fig:test_struts_bench_leg}).
+The dynamics of the re-aligned struts are then measured using the same test bench (Figure \ref{fig:test_struts_bench_leg}).
-A comparison of the initial strut dynamics and the dynamics of the re-aligned struts (i.e. with the positioning pin) is presented in Figure \ref{fig:test_struts_comp_enc_frf_realign}.
+The comparison of the initial strut dynamics and the dynamics of the re-aligned struts (i.e. with the positioning pin) is made in Figure \ref{fig:test_struts_comp_enc_frf_realign}.
-Even though the struts are now much better aligned, not much improvement can be observed.
-The dynamics of the six aligned struts were also quite different from one another.
 
-The fact that the encoders are fixed to the struts makes the control more challenging.
+Even though the struts are now much better aligned, not much improvement can be observed.
-Therefore, fixing the encoders to the nano-hexapod plates instead may be an interesting option.
+The dynamics of the six aligned struts are quite different from one another.
+
+Having the encoders fixed to the struts may prove to be difficult to use.
+Therefore, the encoders may be fixed to the nano-hexapod plates instead.
 
 \begin{figure}[htbp]
 \centering
 \includegraphics[scale=1]{figs/test_struts_comp_enc_frf_realign.png}
-\caption{\label{fig:test_struts_comp_enc_frf_realign}Comparison of the dynamics from \(u\) to \(d_e\) before and after proper alignment using the dowel pins}
+\label{fig:test_struts_comp_enc_frf_realign}
+\end{figure}
+\section{Effect of the flexible joint}
+\label{ssec:test_struts_effect_joint_stiffness}
+
+As the struts are composed of one APA and two flexible joints, it is expected that the flexible joint characteristics will change the dynamic behavior of the struts.
+
+Using the Simscape model, the effect of the flexible joint's characteristics on the dynamics as measured on the test bench are studied.
+The studied dynamics is between \(u\) and the encoder displacement \(d_e\).
+
+Let's initialize an APA which is a little bit misaligned.
+\begin{figure}[htbp]
+\begin{subfigure}{0.49\textwidth}
+\begin{center}
+\includegraphics[scale=1,width=0.95\linewidth]{figs/test_struts_effect_flex_bending_stiffness_enc.png}
+\end{center}
+\subcaption{\label{fig:test_struts_effect_flex_bending_stiffness_enc}Effect of bending stiffness}
+\end{subfigure}
+\begin{subfigure}{0.49\textwidth}
+\begin{center}
+\includegraphics[scale=1,width=0.95\linewidth]{figs/test_struts_effect_flex_axial_stiffness_enc.png}
+\end{center}
+\subcaption{\label{fig:test_struts_effect_flex_axial_stiffness_enc}Effect of axial stiffness}
+\end{subfigure}
+\caption{\label{fig:test_struts_effect_flex_stiffness_enc}Effect of the flexible joints' bending (\subref{fig:test_struts_effect_flex_bending_stiffness_enc}) and axial (\subref{fig:test_struts_effect_flex_axial_stiffness_enc}) stiffnesses on the strut dynamics from \(u\) to \(d_e\)}
 \end{figure}
 
+The bending stiffness of the joints has little impact on the transfer function from \(u\) to \(d_e\).
+
+The axial stiffness of the flexible joint has a large impact on the frequency of the complex conjugate zero.
+Using the measured FRF on the test-bench, if is therefore possible to estimate the axial stiffness of the flexible joints from the location of the zero.
+
+This method gives nice match between the measured FRF and the one extracted from the simscape model, however it could give not so accurate values of the joint's axial stiffness as other factors are also influencing the location of the zero.
+
+Using this method, an axial stiffness of \(70 N/\mu m\) is found to give good results (and is reasonable based on the finite element models).
 \chapter{Conclusion}
 \label{sec:test_struts_conclusion}
-
+\printbibliography[heading=bibintoc,title={Bibliography}]
-The Hano-Hexapod struts are a key component of the developed \acrfull{nass}.
-A mounting bench was used to obtain struts with good interface coaxiality, equal lengths, and ideally the same dynamics.
-Using a test bench, it was found that while all the mounted struts had extremely similar dynamics when considering the axial motion and the integrated force sensor, the dynamics as seen by the encoder is much more complex and varied from one strut to the other.
-
-Thanks to a \acrshort{fem} and experimental measurements, the modes inducing this complex dynamics was identified.
-The variability in the dynamics was attributed to the poor alignment of the \acrshort{apa} with respect to the flexible joints.
-Even with better alignment using dowel pins, the observed dynamics by the encoder remained problematic.
-Therefore, the encoders will be fixed directly to the nano-hexapod plates rather than being fixed to the struts.
-
-\printglossaries
 \end{document}