Review Christophe's comnments

nass-rotating-3dof-model.bib

@@ -141,20 +141,6 @@
 
 
 
-@article{preumont08_trans_zeros_struc_contr_with,
-  author          = {Preumont, Andr{\'e} and De Marneffe, Bruno and Krenk,
-                  Steen},
-  title           = {Transmission Zeros in Structural Control With Collocated
-                  Multi-Input/multi-Output Pairs},
-  journal         = {Journal of guidance, control, and dynamics},
-  volume          = 31,
-  number          = 2,
-  pages           = {428--432},
-  year            = 2008,
-}
-
-
-
 @article{teo15_optim_integ_force_feedb_activ_vibrat_contr,
   author          = {Yik R. Teo and Andrew J. Fleming},
   title           = {Optimal Integral Force Feedback for Active Vibration

nass-rotating-3dof-model.org

@@ -141,7 +141,7 @@ In the last section (Section ref:sec:rotating_nass), a model of the micro-statio
 The goal is to determine if the rotation imposes performance limitation for the NASS.
 
 #+name: fig:rotating_overview
-#+caption: Overview of this report on rotating effects
+#+caption: Overview of this chapter's organization. Sections are indicated by the red circles.
 #+attr_latex: :width \linewidth
 [[file:figs/rotating_overview.png]]
 
@@ -251,7 +251,7 @@ After the dynamics of this system is studied, the objective will be to damp the
 mdl = 'rotating_model';
 #+end_src
 
-** Equations of motion
+** Equations of motion and transfer functions
 To obtain the equations of motion for the system represented in Figure ref:fig:rotating_3dof_model_schematic, the Lagrangian equation eqref:eq:rotating_lagrangian_equations is used.
 $L = T - V$ is the Lagrangian, $T$ the kinetic coenergy, $V$ the potential energy, $D$ the dissipation function, and $Q_i$ the generalized force associated with the generalized variable $\begin{bmatrix}q_1 & q_2\end{bmatrix} = \begin{bmatrix}d_u & d_v\end{bmatrix}$.
 These terms are derived in eqref:eq:rotating_energy_functions_lagrange.
@@ -282,7 +282,6 @@ The uniform rotation of the system induces two /gyroscopic effects/ as shown in
 - /Coriolis forces/: that adds /coupling/ between the two orthogonal directions.
 One can verify that without rotation ($\Omega = 0$) the system becomes equivalent to two /uncoupled/ one degree of freedom mass-spring-damper systems.
 
-** Transfer Functions in the Laplace domain
 To study the dynamics of the system, the two differential equations of motions eqref:eq:rotating_eom_coupled are converted into the Laplace domain and the $2 \times 2$ transfer function matrix $\mathbf{G}_d$ from $\begin{bmatrix}F_u & F_v\end{bmatrix}$ to $\begin{bmatrix}d_u & d_v\end{bmatrix}$ in equation eqref:eq:rotating_Gd_mimo_tf is obtained.
 The four transfer functions in $\mathbf{G}_d$ are shown in equation eqref:eq:rotating_Gd_indiv_el.
 
@@ -593,7 +592,7 @@ The goal is now to damp the two suspension modes of the payload using an active
 As was explained with the uniaxial model, such active damping strategy is key to both reducing the magnification of the response in the vicinity of the resonances cite:collette11_review_activ_vibrat_isolat_strat and to make the plant easier to control for the high authority controller.
 
 Many active damping techniques have been developed over the years such as Positive Position Feedback (PPF) cite:lin06_distur_atten_precis_hexap_point,fanson90_posit_posit_feedb_contr_large_space_struc, Integral Force Feedback (IFF) cite:preumont91_activ and Direct Velocity Feedback (DVF) cite:karnopp74_vibrat_contr_using_semi_activ_force_gener,serrand00_multic_feedb_contr_isolat_base_excit_vibrat,preumont02_force_feedb_versus_accel_feedb.
-In cite:preumont92_activ_dampin_by_local_force, the IFF control scheme has been proposed, where a force sensor, a force actuator and an integral controller are used to increase the damping of a mechanical system.
+In [[cite:&preumont91_activ]], the IFF control scheme has been proposed, where a force sensor, a force actuator and an integral controller are used to increase the damping of a mechanical system.
 When the force sensor is collocated with the actuator, the open-loop transfer function has alternating poles and zeros which facilitates to guarantee the stability of the closed loop system cite:preumont02_force_feedb_versus_accel_feedb.
 It was latter shown that this property holds for multiple collated actuator/sensor pairs cite:preumont08_trans_zeros_struc_contr_with.
 
@@ -1071,16 +1070,12 @@ Two parameters can be tuned for the modified controller in equation eqref:eq:rot
 The optimal values of $\omega_i$ and $g$ are here considered as the values for which the damping of all the closed-loop poles are simultaneously maximized.
 
 In order to visualize how $\omega_i$ does affect the attainable damping, the Root Locus plots for several $\omega_i$ are displayed in Figure ref:fig:rotating_root_locus_iff_modified_effect_wi.
-It is shown that even though small $\omega_i$ seem to allow more damping to be added to the suspension modes (see Root locus in Figure ref:fig:rotating_root_locus_iff_modified_effect_wi_large), the control gain $g$ may be limited to small values due to equation eqref:eq:rotating_gmax_iff_hpf.
+It is shown that even though small $\omega_i$ seem to allow more damping to be added to the suspension modes (see Root locus in Figure ref:fig:rotating_root_locus_iff_modified_effect_wi), the control gain $g$ may be limited to small values due to equation eqref:eq:rotating_gmax_iff_hpf.
 In order to study this trade off, the attainable closed-loop damping ratio $\xi_{\text{cl}}$ is computed as a function of $\omega_i/\omega_0$.
 The gain $g_{\text{opt}}$ at which this maximum damping is obtained is also displayed and compared with the gain $g_{\text{max}}$ at which the system becomes unstable (Figure ref:fig:rotating_iff_hpf_optimal_gain).
 
-# TODO - Maybe comment on these "regions"
-
-Three regions can be observed:
-- $\omega_i/\omega_0 < 0.02$: the added damping is limited by the maximum allowed control gain $g_{\text{max}}$
-- $0.02 < \omega_i/\omega_0 < 0.2$: the attainable damping ratio is maximized and is reached for $g \approx 2$
-- $0.2 < \omega_i/\omega_0$: the added damping decreases as $\omega_i/\omega_0$ increases.
+For small values of $\omega_i$, the added damping is limited by the maximum allowed control gain $g_{\text{max}}$ (red curve and dashed red curve superimposed in Figure ref:fig:rotating_iff_hpf_optimal_gain) at which point the pole corresponding to the controller becomes unstable.
+For larger values of $\omega_i$, the attainable damping ratio decreases as a function of $\omega_i$ as was predicted from the root locus plot of Figure ref:fig:rotating_iff_root_locus_hpf_large.
 
 #+begin_src matlab
 %% High Pass Filter Cut-Off Frequency
@@ -1172,7 +1167,7 @@ exportFig('figs/rotating_iff_hpf_optimal_gain.pdf', 'width', 'half', 'height', 4
 #+end_src
 
 #+name: fig:rotating_iff_modified_effect_wi
-#+caption: Root Locus for several high pass filter cut-off frequency (\subref{fig:rotating_root_locus_iff_modified_effect_wi_large}).
+#+caption: Root Locus for several high pass filter cut-off frequency (\subref{fig:rotating_root_locus_iff_modified_effect_wi}). The achievable damping ratio decreases as $\omega_i$ increases which is confirmed in (\subref{fig:rotating_iff_hpf_optimal_gain})
 #+attr_latex: :options [htbp]
 #+begin_figure
 #+attr_latex: :caption \subcaption{\label{fig:rotating_root_locus_iff_modified_effect_wi}Root Locus}
@@ -1762,7 +1757,7 @@ exportFig('figs/rotating_iff_kp_optimal_gain.pdf', 'width', 'half', 'height', 45
 
 ** Damped plant
 Let's choose a parallel stiffness equal to $k_p = 2 m \Omega^2$ and compute the damped plant.
-The damped and undamped transfer functions from $F_u$ to $d_u$ are compared in Figure ref:fig:rotating_iff_kp_damped_plant.
+The damped and undamped transfer functions from $F_u$ to $d_u$ are compared in Figure ref:fig:rotating_iff_kp_added_hpf_damped_plant.
 Even though the two resonances are well damped, the IFF changes the low frequency behavior of the plant which is usually not wanted.
 This is due to the fact that "pure" integrators are used, and that the low frequency loop gains becomes large below some frequency.
 
@@ -1894,7 +1889,7 @@ xlim([freqs(1), freqs(end)]);
 exportFig('figs/rotating_iff_kp_added_hpf_damped_plant.pdf', 'width', 700, 'height', 600);
 #+end_src
 
-#+name: fig:rotating_iff_optimal_kp
+#+name: fig:rotating_iff_optimal_hpf
 #+caption:Effect of the high pass filter cut-off frequency on the obtained damping
 #+attr_latex: :options [htbp]
 #+begin_figure
@@ -2191,7 +2186,7 @@ exportFig('figs/rotating_rdc_damped_plant.pdf', 'width', 'half', 'height', 500);
 #+end_src
 
 #+name: fig:rotating_rdc_result
-#+caption: Relative Damping Control. Root Locus (\subref{fig:rotating_rdc_root_locus}) and obtained damped plant (\subref{rotating_rdc_damped_plant})
+#+caption: Relative Damping Control. Root Locus (\subref{fig:rotating_rdc_root_locus}) and obtained damped plant (\subref{fig:rotating_rdc_damped_plant})
 #+attr_latex: :options [htbp]
 #+begin_figure
 #+attr_latex: :caption \subcaption{\label{fig:rotating_rdc_root_locus}Root Locus for Relative Damping Control}
@@ -2753,7 +2748,7 @@ linkaxes([ax1,ax2],'x');
 #+end_src
 
 #+begin_src matlab :tangle no :exports results :results file none
-exportFig('figs/rotating_nano_hexapod_dynamics_vc.pdf', 'width', 'third', 'height', 600);
+exportFig('figs/rotating_nano_hexapod_dynamics_vc.pdf', 'width', 'third', 'height', 500);
 #+end_src
 
 #+begin_src matlab :results none
@@ -2791,7 +2786,7 @@ linkaxes([ax1,ax2],'x');
 #+end_src
 
 #+begin_src matlab :tangle no :exports results :results file none
-exportFig('figs/rotating_nano_hexapod_dynamics_md.pdf', 'width', 'third', 'height', 600);
+exportFig('figs/rotating_nano_hexapod_dynamics_md.pdf', 'width', 'third', 'height', 500);
 #+end_src
 
 #+begin_src matlab :results none
@@ -2829,7 +2824,7 @@ linkaxes([ax1,ax2],'x');
 #+end_src
 
 #+begin_src matlab :tangle no :exports results :results file none
-exportFig('figs/rotating_nano_hexapod_dynamics_pz.pdf', 'width', 'third', 'height', 600);
+exportFig('figs/rotating_nano_hexapod_dynamics_pz.pdf', 'width', 'third', 'height', 500);
 #+end_src
 
 #+name: fig:rotating_nano_hexapod_dynamics
@@ -3462,7 +3457,7 @@ xlim([freqs_vc(1), freqs_vc(end)]);
 #+end_src
 
 #+begin_src matlab :tangle no :exports results :results file replace
-exportFig('figs/rotating_nass_damped_plant_comp_vc.pdf', 'width', 'third', 'height', 600);
+exportFig('figs/rotating_nass_damped_plant_comp_vc.pdf', 'width', 'third', 'height', 500);
 #+end_src
 
 #+begin_src matlab :exports none :results none
@@ -3504,7 +3499,7 @@ xlim([freqs_md(1), freqs_md(end)]);
 #+end_src
 
 #+begin_src matlab :tangle no :exports results :results file replace
-exportFig('figs/rotating_nass_damped_plant_comp_md.pdf', 'width', 'third', 'height', 600);
+exportFig('figs/rotating_nass_damped_plant_comp_md.pdf', 'width', 'third', 'height', 500);
 #+end_src
 
 #+begin_src matlab :exports none :results none
@@ -3556,7 +3551,7 @@ xlim([freqs_pz(1), freqs_pz(end)]);
 #+end_src
 
 #+begin_src matlab :tangle no :exports results :results file replace
-exportFig('figs/rotating_nass_damped_plant_comp_pz.pdf', 'width', 'third', 'height', 600);
+exportFig('figs/rotating_nass_damped_plant_comp_pz.pdf', 'width', 'third', 'height', 500);
 #+end_src
 
 #+name: fig:rotating_nass_damped_plant_comp
@@ -3629,9 +3624,9 @@ load('nass_controllers.mat');
 In order to have a more realistic dynamics model of the NASS, the 2-DoF nano-hexapod (modelled as shown in Figure ref:fig:rotating_3dof_model_schematic) is now located on top of a model of the micro-station including (see Figure ref:fig:rotating_nass_model for a 3D view):
 - the floor whose motion is imposed
 - a 2-DoF granite ($k_{g,x} = k_{g,y} = \SI{950}{\N\per\mu\m}$, $m_g = \SI{2500}{\kg}$)
-- a 2-DoF $T_y$ stage ($k_{t,x} = k_{t,y} = \SI{520}{\N\per\mu\m}$, $m_g = \SI{600}{\kg}$)
+- a 2-DoF $T_y$ stage ($k_{t,x} = k_{t,y} = \SI{520}{\N\per\mu\m}$, $m_t = \SI{600}{\kg}$)
 - a spindle (vertical rotation) stage whose rotation is imposed ($m_s = \SI{600}{\kg}$)
-- a 2-DoF micro-hexapod ($k_{h,x} = k_{h,y} = \SI{61}{\N\per\mu\m}$, $m_g = \SI{15}{\kg}$)
+- a 2-DoF micro-hexapod ($k_{h,x} = k_{h,y} = \SI{61}{\N\per\mu\m}$, $m_h = \SI{15}{\kg}$)
 
 A payload is rigidly fixed to the nano-hexapod and the $x,y$ motion of the payload is measured with respect to the granite.
 
@@ -3643,9 +3638,9 @@ A payload is rigidly fixed to the nano-hexapod and the $x,y$ motion of the paylo
 ** System dynamics
 
 The dynamics of the un-damped and damped plants are identified using the optimal parameters found in Section ref:sec:rotating_nano_hexapod.
-The obtained dynamics are compared in Figure ref:fig:rotating_nass_plant_comp_stiffness where the direct terms are shown by the solid curves while the coupling terms are shown by the shaded ones.
+The obtained dynamics are compared in Figure ref:fig:rotating_nass_plant_comp_stiffness in which the direct terms are shown by the solid curves while the coupling terms are shown by the shaded ones.
 It can be observed that:
-- Coupling (quantified by the ratio between the off-diagonal and direct terms) is higher for the soft nano-hexapod
+- The coupling (quantified by the ratio between the off-diagonal and direct terms) is higher for the soft nano-hexapod
 - Damping added by the three proposed techniques is quite high and the obtained plant is rather easy to control
 - There is some coupling between nano-hexapod and micro-station dynamics for the stiff nano-hexapod (mode at 200Hz)
 - The two proposed IFF modification yields similar results
@@ -3827,7 +3822,7 @@ plot(freqs_vc, abs(squeeze(freqresp(G_vc_fast_rdc('Dy', 'Fu'), freqs_vc, 'Hz')))
 hold off;
 set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
 ylabel('Magnitude [m/N]'); set(gca, 'XTickLabel',[]);
-ylim([1e-12, 1e-2])
+ylim([1e-8, 1e-2])
 
 ax2 = nexttile;
 hold on;
@@ -3840,7 +3835,7 @@ set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
 xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
 hold off;
 yticks(-360:90:360);
-ylim([-270, 90]);
+ylim([ -200, 20]);
 
 linkaxes([ax,ax2],'x');
 xlim([freqs_vc(1), freqs_vc(end)]);
@@ -3848,7 +3843,7 @@ xticks([1e-1, 1e0, 1e1]);
 #+end_src
 
 #+begin_src matlab :tangle no :exports results :results file none
-exportFig('figs/rotating_nass_plant_comp_stiffness_vc.pdf', 'width', 'third', 'height', 'tall');
+exportFig('figs/rotating_nass_plant_comp_stiffness_vc.pdf', 'width', 'third', 'height', 600);
 #+end_src
 
 #+begin_src matlab :exports none :results none
@@ -3877,7 +3872,7 @@ plot(freqs_md, abs(squeeze(freqresp(G_md_fast_rdc('Dy', 'Fu'), freqs_md, 'Hz')))
 hold off;
 set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
 ylabel('Magnitude [m/N]'); set(gca, 'XTickLabel',[]);
-ylim([1e-12, 1e-2])
+ylim([1e-10, 1e-4])
 
 ax2 = nexttile;
 hold on;
@@ -3890,7 +3885,7 @@ set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
 xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
 hold off;
 yticks(-360:90:360);
-ylim([-270, 90]);
+ylim([ -200, 20]);
 
 linkaxes([ax1,ax2],'x');
 xlim([freqs_md(1), freqs_md(end)]);
@@ -3898,7 +3893,7 @@ xticks([1e0, 1e1, 1e2]);
 #+end_src
 
 #+begin_src matlab :tangle no :exports results :results file none
-exportFig('figs/rotating_nass_plant_comp_stiffness_md.pdf', 'width', 'third', 'height', 'tall');
+exportFig('figs/rotating_nass_plant_comp_stiffness_md.pdf', 'width', 'third', 'height', 600);
 #+end_src
 
 #+begin_src matlab :exports none :results none
@@ -3927,8 +3922,8 @@ plot(freqs_pz, abs(squeeze(freqresp(G_pz_fast_rdc('Dy', 'Fu'), freqs_pz, 'Hz')))
 hold off;
 set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
 ylabel('Magnitude [m/N]'); set(gca, 'XTickLabel',[]);
-ylim([1e-12, 1e-2])
-ldg = legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
+ylim([1e-12, 1e-6])
+ldg = legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 1);
 ldg.ItemTokenSize = [20, 1];
 
 ax2 = nexttile;
@@ -3942,7 +3937,7 @@ set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
 xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
 hold off;
 yticks(-360:90:360);
-ylim([-270, 90]);
+ylim([ -200, 20]);
 
 linkaxes([ax1,ax2],'x');
 xlim([freqs_pz(1), freqs_pz(end)]);
@@ -3950,7 +3945,7 @@ xticks([1e0, 1e1, 1e2]);
 #+end_src
 
 #+begin_src matlab :tangle no :exports results :results file none
-exportFig('figs/rotating_nass_plant_comp_stiffness_pz.pdf', 'width', 'third', 'height', 'tall');
+exportFig('figs/rotating_nass_plant_comp_stiffness_pz.pdf', 'width', 'third', 'height', 600);
 #+end_src
 
 #+name: fig:rotating_nass_plant_comp_stiffness
@@ -3977,119 +3972,17 @@ exportFig('figs/rotating_nass_plant_comp_stiffness_pz.pdf', 'width', 'third', 'h
 #+end_subfigure
 #+end_figure
 
-To confirm that the coupling is smaller when the stiffness of the nano-hexapod is increase, the /coupling ratio/ for the three nano-hexapod stiffnesses are shown in Figure ref:fig:rotating_nass_plant_coupling_comp.
-
-#+begin_src matlab :exports none :results none
-%% Coupling ratio for the proposed active damping techniques evaluated for the three nano-hexapod stiffnesses
-freqs_vc = logspace(-1, 2, 1000);
-figure;
-hold on;
-plot(freqs_vc, abs(squeeze(freqresp(G_vc_fast('Dy', 'Fu')/G_vc_fast('Dx', 'Fu'), freqs_vc, 'Hz'))), 'color', zeros(1,3), ...
-    'DisplayName', 'OL');
-plot(freqs_vc, abs(squeeze(freqresp(G_vc_fast_iff_hpf('Dy', 'Fu')/G_vc_fast_iff_hpf('Dx', 'Fu'), freqs_vc, 'Hz'))), 'color', colors(1,:), ...
-    'DisplayName', 'IFF + $k_p$');
-plot(freqs_vc, abs(squeeze(freqresp(G_vc_fast_iff_kp('Dy', 'Fu')/G_vc_fast_iff_kp('Dx', 'Fu'), freqs_vc, 'Hz'))), 'color', colors(2,:), ...
-    'DisplayName', 'IFF + HPF');
-plot(freqs_vc, abs(squeeze(freqresp(G_vc_fast_rdc('Dy', 'Fu')/G_vc_fast_rdc('Dx', 'Fu'), freqs_vc, 'Hz'))), 'color', colors(3,:), ...
-    'DisplayName', 'RDC');
-hold off;
-set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
-xlabel('Frequency [Hz]'); ylabel('Coupling Ratio');
-ylim([1e-4, 1e2]);
-xticks([1e-1, 1e0, 1e1]);
-#+end_src
-
-#+begin_src matlab :tangle no :exports results :results file none
-exportFig('figs/rotating_nass_plant_coupling_comp_vc.pdf', 'width', 'third', 'height', 'normal');
-#+end_src
-
-#+begin_src matlab :exports none :results none
-freqs_md = logspace(0, 3, 1000);
-figure;
-hold on;
-plot(freqs_md, abs(squeeze(freqresp(G_md_fast('Dy', 'Fu')/G_md_fast('Dx', 'Fu'), freqs_md, 'Hz'))), 'color', zeros(1,3), ...
-    'DisplayName', 'OL');
-plot(freqs_md, abs(squeeze(freqresp(G_md_fast_iff_hpf('Dy', 'Fu')/G_md_fast_iff_hpf('Dx', 'Fu'), freqs_md, 'Hz'))), 'color', colors(1,:), ...
-    'DisplayName', 'IFF + $k_p$');
-plot(freqs_md, abs(squeeze(freqresp(G_md_fast_iff_kp('Dy', 'Fu')/G_md_fast_iff_kp('Dx', 'Fu'), freqs_md, 'Hz'))), 'color', colors(2,:), ...
-    'DisplayName', 'IFF + HPF');
-plot(freqs_md, abs(squeeze(freqresp(G_md_fast_rdc('Dy', 'Fu')/G_md_fast_rdc('Dx', 'Fu'), freqs_md, 'Hz'))), 'color', colors(3,:), ...
-    'DisplayName', 'RDC');
-hold off;
-set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
-xlabel('Frequency [Hz]'); ylabel('Coupling Ratio');
-ylim([1e-4, 1e2]);
-xticks([1e0, 1e1, 1e2]);
-#+end_src
-
-#+begin_src matlab :tangle no :exports results :results file none
-exportFig('figs/rotating_nass_plant_coupling_comp_md.pdf', 'width', 'third', 'height', 'normal');
-#+end_src
-
-#+begin_src matlab :exports none :results none
-freqs_pz = logspace(0, 3, 1000);
-figure;
-hold on;
-plot(freqs_pz, abs(squeeze(freqresp(G_pz_fast('Dy', 'Fu')/G_pz_fast('Dx', 'Fu'), freqs_pz, 'Hz'))), 'color', zeros(1,3), ...
-    'DisplayName', 'OL');
-plot(freqs_pz, abs(squeeze(freqresp(G_pz_fast_iff_hpf('Dy', 'Fu')/G_pz_fast_iff_hpf('Dx', 'Fu'), freqs_pz, 'Hz'))), 'color', colors(1,:), ...
-    'DisplayName', 'IFF + $k_p$');
-plot(freqs_pz, abs(squeeze(freqresp(G_pz_fast_iff_kp('Dy', 'Fu')/G_pz_fast_iff_kp('Dx', 'Fu'), freqs_pz, 'Hz'))), 'color', colors(2,:), ...
-    'DisplayName', 'IFF + HPF');
-plot(freqs_pz, abs(squeeze(freqresp(G_pz_fast_rdc('Dy', 'Fu')/G_pz_fast_rdc('Dx', 'Fu'), freqs_pz, 'Hz'))), 'color', colors(3,:), ...
-    'DisplayName', 'RDC');
-hold off;
-set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
-xlabel('Frequency [Hz]'); ylabel('Coupling Ratio');
-ldg = legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 1);
-ldg.ItemTokenSize = [20, 1];
-ylim([1e-4, 1e2]);
-xticks([1e0, 1e1, 1e2]);
-#+end_src
-
-#+begin_src matlab :tangle no :exports results :results file none
-exportFig('figs/rotating_nass_plant_coupling_comp_pz.pdf', 'width', 'third', 'height', 'normal');
-#+end_src
-
-#+name: fig:rotating_nass_plant_coupling_comp
-#+caption: Coupling ratio for the proposed active damping techniques evaluated for the three nano-hexapod stiffnesses
-#+attr_latex: :options [htbp]
-#+begin_figure
-#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_plant_coupling_comp_vc}$k_n = 0.01\,N/\mu m$}
-#+attr_latex: :options {0.33\textwidth}
-#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
-[[file:figs/rotating_nass_plant_coupling_comp_vc.png]]
-#+end_subfigure
-#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_plant_coupling_comp_md}$k_n = 1\,N/\mu m$}
-#+attr_latex: :options {0.33\textwidth}
-#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
-[[file:figs/rotating_nass_plant_coupling_comp_md.png]]
-#+end_subfigure
-#+attr_latex: :caption \subcaption{\label{fig:rotating_nass_plant_coupling_comp_pz}$k_n = 100\,N/\mu m$}
-#+attr_latex: :options {0.33\textwidth}
-#+begin_subfigure
-#+attr_latex: :width 0.95\linewidth
-[[file:figs/rotating_nass_plant_coupling_comp_pz.png]]
-#+end_subfigure
-#+end_figure
-
 ** Effect of disturbances
 
-The effect of three disturbances are considered:
-- Floor motion (Figure ref:fig:rotating_nass_effect_floor_motion)
-- Micro-Station vibrations (Figure ref:fig:rotating_nass_effect_stage_vibration)
-- Direct force applied on the payload (Figure ref:fig:rotating_nass_effect_direct_forces)
+The effect of three disturbances are considered (as for the uniaxial model), floor motion $[x_{f,x},\ x_{f,y}]$ (Figure ref:fig:rotating_nass_effect_floor_motion), micro-Station vibrations $[f_{t,x},\ f_{t,y}]$ (Figure ref:fig:rotating_nass_effect_stage_vibration) and direct forces applied on the sample $[f_{s,x},\ f_{s,y}]$ (Figure ref:fig:rotating_nass_effect_direct_forces).
+Note that only the transfer function from the disturbances in the $x$ direction to the relative position $d_x$ between the sample and the granite in the $x$ direction are displayed as the transfer functions in the $y$ direction are the same due to the system symmetry.
 
-#+begin_important
 Conclusions are similar than with the uniaxial (non-rotating) model:
 - Regarding the effect of floor motion and forces applied on the payload:
-  - The stiffer, the better (magnitudes are lower for the right curves, Figures ref:fig:rotating_nass_effect_floor_motion and ref:fig:rotating_nass_effect_direct_forces)
-  - Integral Force Feedback degrades the performance at low frequency compared to relative damping control
+  - The stiffer, the better. This can be seen in Figures ref:fig:rotating_nass_effect_floor_motion and ref:fig:rotating_nass_effect_direct_forces where the magnitudes for the stiff-hexapod are lower than for the soft one
+  - acrshort:iff degrades the performance at low frequency compared to acrshort:rdc
 - Regarding the effect of micro-station vibrations:
-  - Having a soft nano-hexapod allows to filter these vibrations between the suspensions modes of the nano-hexapod and some flexible modes of the micro-station. Using relative damping control reduce this filtering (Figure ref:fig:rotating_nass_effect_stage_vibration, left).
-#+end_important
+  - Having a soft nano-hexapod allows to filter these vibrations between the suspensions modes of the nano-hexapod and some flexible modes of the micro-station. Using relative damping control reduces this filtering (Figure ref:fig:rotating_nass_effect_stage_vibration_vc).
 
 #+begin_src matlab :exports none :results none
 %% Effect of Floor motion on the position error - Comparison of active damping techniques for the three nano-hexapod stiffnesses
@@ -4107,14 +4000,14 @@ plot(freqs, abs(squeeze(freqresp(G_vc_fast_rdc('Dx', 'Dfx'), freqs, 'Hz'))), 'co
     'DisplayName', 'RDC');
 hold off;
 set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
-xlabel('Frequency [Hz]'); ylabel('Magnitude $d_x/D_{f,x}$ [m/N]');
+xlabel('Frequency [Hz]'); ylabel('Magnitude $d_x/x_{f,x}$ [m/N]');
 xticks([1e-1, 1e0, 1e1, 1e2, 1e3]);
 xtickangle(0)
 ylim([1e-4, 1e2]);
 #+end_src
 
 #+begin_src matlab :tangle no :exports results :results file none
-exportFig('figs/rotating_nass_effect_floor_motion_vc.pdf', 'width', 'third', 'height', 'normal');
+exportFig('figs/rotating_nass_effect_floor_motion_vc.pdf', 'width', 'third', 'height', 450);
 #+end_src
 
 #+begin_src matlab :exports none :results none
@@ -4130,14 +4023,14 @@ plot(freqs, abs(squeeze(freqresp(G_md_fast_rdc('Dx', 'Dfx'), freqs, 'Hz'))), 'co
     'DisplayName', 'RDC');
 hold off;
 set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
-xlabel('Frequency [Hz]'); ylabel('Magnitude $d_x/D_{f,x}$ [m/N]');
+xlabel('Frequency [Hz]'); ylabel('Magnitude $d_x/x_{f,x}$ [m/N]');
 xticks([1e-1, 1e0, 1e1, 1e2, 1e3]);
 xtickangle(0)
 ylim([1e-4, 1e2]);
 #+end_src
 
 #+begin_src matlab :tangle no :exports results :results file none
-exportFig('figs/rotating_nass_effect_floor_motion_md.pdf', 'width', 'third', 'height', 'normal');
+exportFig('figs/rotating_nass_effect_floor_motion_md.pdf', 'width', 'third', 'height', 450);
 #+end_src
 
 #+begin_src matlab :exports none :results none
@@ -4153,7 +4046,7 @@ plot(freqs, abs(squeeze(freqresp(G_pz_fast_rdc('Dx', 'Dfx'), freqs, 'Hz'))), 'co
     'DisplayName', 'RDC');
 hold off;
 set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
-xlabel('Frequency [Hz]'); ylabel('Magnitude $d_x/D_{f,x}$ [m/N]');
+xlabel('Frequency [Hz]'); ylabel('Magnitude $d_x/x_{f,x}$ [m/N]');
 xticks([1e-1, 1e0, 1e1, 1e2, 1e3]);
 xtickangle(0)
 ldg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 1);
@@ -4162,11 +4055,11 @@ ylim([1e-4, 1e2]);
 #+end_src
 
 #+begin_src matlab :tangle no :exports results :results file none
-exportFig('figs/rotating_nass_effect_floor_motion_pz.pdf', 'width', 'third', 'height', 'normal');
+exportFig('figs/rotating_nass_effect_floor_motion_pz.pdf', 'width', 'third', 'height', 450);
 #+end_src
 
 #+name: fig:rotating_nass_effect_floor_motion
-#+caption: Effect of Floor motion on the position error - Comparison of active damping techniques for the three nano-hexapod stiffnesses
+#+caption: Effect of floor motion $x_{f,x}$ on the position error $d_x$ - Comparison of active damping techniques for the three nano-hexapod stiffnesses. IFF is shown to increase the sensitivity to floor motion at low frequency.
 #+attr_latex: :options [htbp]
 #+begin_figure
 #+attr_latex: :caption \subcaption{\label{fig:rotating_nass_effect_floor_motion_vc}$k_n = 0.01\,N/\mu m$}
@@ -4210,7 +4103,7 @@ ylim([1e-12, 2e-7]);
 #+end_src
 
 #+begin_src matlab :tangle no :exports results :results file none
-exportFig('figs/rotating_nass_effect_stage_vibration_vc.pdf', 'width', 'third', 'height', 'normal');
+exportFig('figs/rotating_nass_effect_stage_vibration_vc.pdf', 'width', 'third', 'height', 450);
 #+end_src
 
 #+begin_src matlab :exports none :results none
@@ -4233,7 +4126,7 @@ ylim([1e-12, 2e-7]);
 #+end_src
 
 #+begin_src matlab :tangle no :exports results :results file none
-exportFig('figs/rotating_nass_effect_stage_vibration_md.pdf', 'width', 'third', 'height', 'normal');
+exportFig('figs/rotating_nass_effect_stage_vibration_md.pdf', 'width', 'third', 'height', 450);
 #+end_src
 
 #+begin_src matlab :exports none :results none
@@ -4258,11 +4151,11 @@ ylim([1e-12, 2e-7]);
 #+end_src
 
 #+begin_src matlab :tangle no :exports results :results file none
-exportFig('figs/rotating_nass_effect_stage_vibration_pz.pdf', 'width', 'third', 'height', 'normal');
+exportFig('figs/rotating_nass_effect_stage_vibration_pz.pdf', 'width', 'third', 'height', 450);
 #+end_src
 
 #+name: fig:rotating_nass_effect_stage_vibration
-#+caption: Effect of micro-station vibrations on the position error - Comparison of active damping techniques for the three nano-hexapod stiffnesses
+#+caption: Effect of micro-station vibrations $f_{t,x}$ on the position error $d_x$ - Comparison of active damping techniques for the three nano-hexapod stiffnesses. Relative Damping Control increases the sensitivity to micro-station vibrations between the soft nano-hexapod suspension modes and the micro-station modes (\subref{fig:rotating_nass_effect_stage_vibration_vc})
 #+attr_latex: :options [htbp]
 #+begin_figure
 #+attr_latex: :caption \subcaption{\label{fig:rotating_nass_effect_stage_vibration_vc}$k_n = 0.01\,N/\mu m$}
@@ -4303,11 +4196,11 @@ set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
 xlabel('Frequency [Hz]'); ylabel('Magnitude $d_x/f_{s,x}$ [m/N]');
 xticks([1e-1, 1e0, 1e1, 1e2, 1e3]);
 xtickangle(0)
-ylim([1e-9, 1e-2])
+ylim([1e-8, 1e-2])
 #+end_src
 
 #+begin_src matlab :tangle no :exports results :results file none
-exportFig('figs/rotating_nass_effect_direct_forces_vc.pdf', 'width', 'third', 'height', 'normal');
+exportFig('figs/rotating_nass_effect_direct_forces_vc.pdf', 'width', 'third', 'height', 450);
 #+end_src
 
 #+begin_src matlab :exports none :results none
@@ -4326,11 +4219,11 @@ set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
 xlabel('Frequency [Hz]'); ylabel('Magnitude $d_x/f_{s,x}$ [m/N]');
 xticks([1e-1, 1e0, 1e1, 1e2, 1e3]);
 xtickangle(0)
-ylim([1e-9, 1e-2])
+ylim([1e-8, 1e-2])
 #+end_src
 
 #+begin_src matlab :tangle no :exports results :results file none
-exportFig('figs/rotating_nass_effect_direct_forces_md.pdf', 'width', 'third', 'height', 'normal');
+exportFig('figs/rotating_nass_effect_direct_forces_md.pdf', 'width', 'third', 'height', 450);
 #+end_src
 
 #+begin_src matlab :exports none :results none
@@ -4353,15 +4246,15 @@ ldg = legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 1);
 ldg.ItemTokenSize = [20, 1];
 
 linkaxes([ax1,ax2,ax3], 'y')
-ylim([1e-9, 1e-2])
+ylim([1e-8, 1e-2])
 #+end_src
 
 #+begin_src matlab :tangle no :exports results :results file none
-exportFig('figs/rotating_nass_effect_direct_forces_pz.pdf', 'width', 'third', 'height', 'normal');
+exportFig('figs/rotating_nass_effect_direct_forces_pz.pdf', 'width', 'third', 'height', 450);
 #+end_src
 
 #+name: fig:rotating_nass_effect_direct_forces
-#+caption: Effect of sample forces on the position error - Comparison of active damping techniques for the three nano-hexapod stiffnesses
+#+caption: Effect of sample forces $f_{s,x}$ on the position error $d_x$ - Comparison of active damping techniques for the three nano-hexapod stiffnesses. Integral Force Feedback degrades this compliance at low frequency.
 #+attr_latex: :options [htbp]
 #+begin_figure
 #+attr_latex: :caption \subcaption{\label{fig:rotating_nass_effect_direct_forces_vc}$k_n = 0.01\,N/\mu m$}
@@ -4389,14 +4282,9 @@ exportFig('figs/rotating_nass_effect_direct_forces_pz.pdf', 'width', 'third', 'h
 :UNNUMBERED: t
 :END:
 
-# - problem with voice coil actuator
-# - Two solutions: add parallel stiffness, or change controller
-# - Conclusion: minimum stiffness is required
-# - APA is a nice architecture for parallel stiffness + integrated force sensor (have to speak about IFF before that)
-
-In this study, the gyroscopic effects induced by the spindle's rotation have been studied using a spindle model (Section ref:sec:rotating_system_description).
-Decentralized IFF with pure integrators was shown to be unstable when applied to rotating platforms (Section ref:sec:rotating_iff_pure_int).
-Two modifications of the classical IFF control have been proposed to overcome this issue.
+In this study, the gyroscopic effects induced by the spindle's rotation have been studied using a simplified model (Section ref:sec:rotating_system_description).
+Decentralized acrlong:iff with pure integrators was shown to be unstable when applied to rotating platforms (Section ref:sec:rotating_iff_pure_int).
+Two modifications of the classical acrshort:iff control have been proposed to overcome this issue.
 
 The first modification concerns the controller and consists of adding a high pass filter to the pure integrators.
 This is equivalent to moving the controller pole to the left along the real axis.
@@ -4404,14 +4292,19 @@ This allows the closed loop system to be stable up to some value of the controll
 
 The second proposed modification concerns the mechanical system.
 Additional springs are added in parallel with the actuators and force sensors.
-It was shown that if the stiffness $k_p$ of the additional springs is larger than the negative stiffness $m \Omega^2$ induced by centrifugal forces, the classical decentralized IFF regains its unconditional stability property (Section ref:sec:rotating_iff_parallel_stiffness).
+It was shown that if the stiffness $k_p$ of the additional springs is larger than the negative stiffness $m \Omega^2$ induced by centrifugal forces, the classical decentralized acrshort:iff regains its unconditional stability property (Section ref:sec:rotating_iff_parallel_stiffness).
 
-These two modifications were compared with relative damping control in Section ref:sec:rotating_comp_act_damp.
+These two modifications were compared with acrlong:rdc in Section ref:sec:rotating_comp_act_damp.
 While having very different implementations, both proposed modifications were found to be very similar when it comes to the attainable damping and the obtained closed loop system behavior.
 
-Then, this study has been applied to a rotating system that corresponds to the nano-hexapod parameters (Section ref:sec:rotating_nano_hexapod).
-To be closer to the real system dynamics, the limited compliance of the micro-station has been taken into account.
-Results show that the two proposed IFF modifications can be applied for the NASS even in the presence of spindle rotation.
+Then, this study has been applied to a rotating platform that corresponds to the nano-hexapod parameters (Section ref:sec:rotating_nano_hexapod).
+As for the uniaxial model, three nano-hexapod stiffness are considered.
+The dynamics of the soft nano-hexapod ($k_n = 0.01\,N/\mu m$) was shown to be more depend on the rotation velocity (higher coupling and change of dynamics due to gyroscopic effects).
+Also, the attainable damping ratio of the soft nano-hexapod when using acrshort:iff is limited by gyroscopic effects.
+
+To be closer to the acrlong:nass dynamics, the limited compliance of the micro-station has been taken into account (Section ref:sec:rotating_nass).
+Results are similar to that of the uniaxial model except that come complexity is added for the soft nano-hexapod due to the spindle's rotation.
+For the moderately stiff nano-hexapod ($k_n = 1\,N/\mu m$), the gyroscopic effects are only slightly affecting the system dynamics, and therefore could represent a good alternative to the soft nano-hexapod that was showing better results with the uniaxial model.
 
 * Bibliography                                                        :ignore:
 #+latex: \printbibliography[heading=bibintoc,title={Bibliography}]

nass-rotating-3dof-model.tex

@@ -1,4 +1,4 @@
-% Created 2024-04-16 Tue 22:57
+% Created 2024-04-29 Mon 21:05
 % Intended LaTeX compiler: pdflatex
 \documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt}
 
@@ -14,6 +14,7 @@
 \newacronym{frf}{FRF}{Frequency Response Function}
 \newacronym{iff}{IFF}{Integral Force Feedback}
 \newacronym{rdc}{RDC}{Relative Damping Control}
+\newacronym{drga}{DRGA}{Dynamical Relative Gain Array}
 \newglossaryentry{psdx}{name=\ensuremath{\Phi_{x}},description={{Power spectral density of signal $x$}}}
 \newglossaryentry{asdx}{name=\ensuremath{\Gamma_{x}},description={{Amplitude spectral density of signal $x$}}}
 \newglossaryentry{cpsx}{name=\ensuremath{\Phi_{x}},description={{Cumulative Power Spectrum of signal $x$}}}
@@ -65,7 +66,7 @@ The goal is to determine if the rotation imposes performance limitation for the
 \begin{figure}[htbp]
 \centering
 \includegraphics[scale=1,width=\linewidth]{figs/rotating_overview.png}
-\caption{\label{fig:rotating_overview}Overview of this report on rotating effects}
+\caption{\label{fig:rotating_overview}Overview of this chapter's organization. Sections are indicated by the red circles.}
 \end{figure}
 
 \chapter{System Description and Analysis}
@@ -84,7 +85,7 @@ After the dynamics of this system is studied, the objective will be to damp the
 \caption{\label{fig:rotating_3dof_model_schematic}Schematic of the studied system}
 \end{figure}
 
-\section{Equations of motion}
+\section{Equations of motion and transfer functions}
 To obtain the equations of motion for the system represented in Figure \ref{fig:rotating_3dof_model_schematic}, the Lagrangian equation \eqref{eq:rotating_lagrangian_equations} is used.
 \(L = T - V\) is the Lagrangian, \(T\) the kinetic coenergy, \(V\) the potential energy, \(D\) the dissipation function, and \(Q_i\) the generalized force associated with the generalized variable \(\begin{bmatrix}q_1 & q_2\end{bmatrix} = \begin{bmatrix}d_u & d_v\end{bmatrix}\).
 These terms are derived in \eqref{eq:rotating_energy_functions_lagrange}.
@@ -117,7 +118,6 @@ The uniform rotation of the system induces two \emph{gyroscopic effects} as show
 \end{itemize}
 One can verify that without rotation (\(\Omega = 0\)) the system becomes equivalent to two \emph{uncoupled} one degree of freedom mass-spring-damper systems.
 
-\section{Transfer Functions in the Laplace domain}
 To study the dynamics of the system, the two differential equations of motions \eqref{eq:rotating_eom_coupled} are converted into the Laplace domain and the \(2 \times 2\) transfer function matrix \(\mathbf{G}_d\) from \(\begin{bmatrix}F_u & F_v\end{bmatrix}\) to \(\begin{bmatrix}d_u & d_v\end{bmatrix}\) in equation \eqref{eq:rotating_Gd_mimo_tf} is obtained.
 The four transfer functions in \(\mathbf{G}_d\) are shown in equation \eqref{eq:rotating_Gd_indiv_el}.
 
@@ -211,7 +211,7 @@ The goal is now to damp the two suspension modes of the payload using an active
 As was explained with the uniaxial model, such active damping strategy is key to both reducing the magnification of the response in the vicinity of the resonances \cite{collette11_review_activ_vibrat_isolat_strat} and to make the plant easier to control for the high authority controller.
 
 Many active damping techniques have been developed over the years such as Positive Position Feedback (PPF) \cite{lin06_distur_atten_precis_hexap_point,fanson90_posit_posit_feedb_contr_large_space_struc}, Integral Force Feedback (IFF) \cite{preumont91_activ} and Direct Velocity Feedback (DVF) \cite{karnopp74_vibrat_contr_using_semi_activ_force_gener,serrand00_multic_feedb_contr_isolat_base_excit_vibrat,preumont02_force_feedb_versus_accel_feedb}.
-In \cite{preumont92_activ_dampin_by_local_force}, the IFF control scheme has been proposed, where a force sensor, a force actuator and an integral controller are used to increase the damping of a mechanical system.
+In \cite{preumont91_activ}, the IFF control scheme has been proposed, where a force sensor, a force actuator and an integral controller are used to increase the damping of a mechanical system.
 When the force sensor is collocated with the actuator, the open-loop transfer function has alternating poles and zeros which facilitates to guarantee the stability of the closed loop system \cite{preumont02_force_feedb_versus_accel_feedb}.
 It was latter shown that this property holds for multiple collated actuator/sensor pairs \cite{preumont08_trans_zeros_struc_contr_with}.
 
@@ -386,16 +386,12 @@ Two parameters can be tuned for the modified controller in equation \eqref{eq:ro
 The optimal values of \(\omega_i\) and \(g\) are here considered as the values for which the damping of all the closed-loop poles are simultaneously maximized.
 
 In order to visualize how \(\omega_i\) does affect the attainable damping, the Root Locus plots for several \(\omega_i\) are displayed in Figure \ref{fig:rotating_root_locus_iff_modified_effect_wi}.
-It is shown that even though small \(\omega_i\) seem to allow more damping to be added to the suspension modes (see Root locus in Figure \ref{fig:rotating_root_locus_iff_modified_effect_wi_large}), the control gain \(g\) may be limited to small values due to equation \eqref{eq:rotating_gmax_iff_hpf}.
+It is shown that even though small \(\omega_i\) seem to allow more damping to be added to the suspension modes (see Root locus in Figure \ref{fig:rotating_root_locus_iff_modified_effect_wi}), the control gain \(g\) may be limited to small values due to equation \eqref{eq:rotating_gmax_iff_hpf}.
 In order to study this trade off, the attainable closed-loop damping ratio \(\xi_{\text{cl}}\) is computed as a function of \(\omega_i/\omega_0\).
 The gain \(g_{\text{opt}}\) at which this maximum damping is obtained is also displayed and compared with the gain \(g_{\text{max}}\) at which the system becomes unstable (Figure \ref{fig:rotating_iff_hpf_optimal_gain}).
 
-Three regions can be observed:
-\begin{itemize}
-\item \(\omega_i/\omega_0 < 0.02\): the added damping is limited by the maximum allowed control gain \(g_{\text{max}}\)
-\item \(0.02 < \omega_i/\omega_0 < 0.2\): the attainable damping ratio is maximized and is reached for \(g \approx 2\)
-\item \(0.2 < \omega_i/\omega_0\): the added damping decreases as \(\omega_i/\omega_0\) increases.
-\end{itemize}
+For small values of \(\omega_i\), the added damping is limited by the maximum allowed control gain \(g_{\text{max}}\) (red curve and dashed red curve superimposed in Figure \ref{fig:rotating_iff_hpf_optimal_gain}) at which point the pole corresponding to the controller becomes unstable.
+For larger values of \(\omega_i\), the attainable damping ratio decreases as a function of \(\omega_i\) as was predicted from the root locus plot of Figure \ref{fig:rotating_iff_root_locus_hpf_large}.
 
 \begin{figure}[htbp]
 \begin{subfigure}{0.49\textwidth}
@@ -410,7 +406,7 @@ Three regions can be observed:
 \end{center}
 \subcaption{\label{fig:rotating_iff_hpf_optimal_gain}Attainable damping ratio $\xi_\text{cl}$ as a function of $\omega_i/\omega_0$. Corresponding control gain $g_\text{opt}$ and $g_\text{max}$ are also shown}
 \end{subfigure}
-\caption{\label{fig:rotating_iff_modified_effect_wi}Root Locus for several high pass filter cut-off frequency (\subref{fig:rotating_root_locus_iff_modified_effect_wi_large}).}
+\caption{\label{fig:rotating_iff_modified_effect_wi}Root Locus for several high pass filter cut-off frequency (\subref{fig:rotating_root_locus_iff_modified_effect_wi}). The achievable damping ratio decreases as \(\omega_i\) increases which is confirmed in (\subref{fig:rotating_iff_hpf_optimal_gain})}
 \end{figure}
 
 \section{Obtained Damped Plant}
@@ -503,7 +499,7 @@ It is shown that if the added stiffness is higher than the maximum negative stif
 \end{subfigure}
 \begin{subfigure}{0.44\linewidth}
 \begin{center}
-\includegraphics[scale=1,scale=0.9]{figs/rotating_iff_kp_root_locus.png}
+\includegraphics[scale=1,width=0.95\linewidth]{figs/rotating_iff_kp_root_locus.png}
 \end{center}
 \subcaption{\label{fig:rotating_iff_kp_root_locus}Root Locus for IFF without parallel spring, with small parallel spring and with large parallel spring}
 \end{subfigure}
@@ -536,7 +532,7 @@ This is confirmed by the Figure \ref{fig:rotating_iff_kp_optimal_gain} where the
 
 \section{Damped plant}
 Let's choose a parallel stiffness equal to \(k_p = 2 m \Omega^2\) and compute the damped plant.
-The damped and undamped transfer functions from \(F_u\) to \(d_u\) are compared in Figure \ref{fig:rotating_iff_kp_damped_plant}.
+The damped and undamped transfer functions from \(F_u\) to \(d_u\) are compared in Figure \ref{fig:rotating_iff_kp_added_hpf_damped_plant}.
 Even though the two resonances are well damped, the IFF changes the low frequency behavior of the plant which is usually not wanted.
 This is due to the fact that ``pure'' integrators are used, and that the low frequency loop gains becomes large below some frequency.
 
@@ -567,7 +563,7 @@ The added high pass filter gives almost the same damping properties to the suspe
 \end{center}
 \subcaption{\label{fig:rotating_iff_kp_added_hpf_damped_plant}Damped plant with the parallel stiffness, effect of the added HPF}
 \end{subfigure}
-\caption{\label{fig:rotating_iff_optimal_kp}Effect of the high pass filter cut-off frequency on the obtained damping}
+\caption{\label{fig:rotating_iff_optimal_hpf}Effect of the high pass filter cut-off frequency on the obtained damping}
 \end{figure}
 
 \chapter{Relative Damping Control}
@@ -632,7 +628,7 @@ It does not increase the low frequency coupling as compared to Integral Force Fe
 \end{center}
 \subcaption{\label{fig:rotating_rdc_damped_plant}Damped plant using Relative Damping Control}
 \end{subfigure}
-\caption{\label{fig:rotating_rdc_result}Relative Damping Control. Root Locus (\subref{fig:rotating_rdc_root_locus}) and obtained damped plant (\subref{rotating_rdc_damped_plant})}
+\caption{\label{fig:rotating_rdc_result}Relative Damping Control. Root Locus (\subref{fig:rotating_rdc_root_locus}) and obtained damped plant (\subref{fig:rotating_rdc_damped_plant})}
 \end{figure}
 
 \chapter{Comparison of Active Damping Techniques}
@@ -710,7 +706,8 @@ The parallel stiffness corresponding to the centrifugal forces is \(m \Omega^2 \
 
 The transfer functions from nano-hexapod actuator force \(F_u\) to the displacement of the nano-hexapod in the same direction \(d_u\) as well as in the orthogonal direction \(d_v\) (coupling) are shown in Figure \ref{fig:rotating_nano_hexapod_dynamics} for all three considered nano-hexapod stiffnesses.
 The soft nano-hexapod is the most affected by the rotation.
-This can be seen by the large shift of the resonance frequencies, and by the induced coupling (the ratio between the direct term and the coupling term) which is larger than for the stiffer nano-hexapods.
+This can be seen by the large shift of the resonance frequencies, and by the induced coupling which is larger than for the stiffer nano-hexapods.
+The coupling (or interaction) in a MIMO \(2 \times 2\) system can be visually estimated as the ratio between the diagonal term and the off-diagonal terms (see corresponding Appendix).
 
 \begin{figure}[htbp]
 \begin{subfigure}{0.33\textwidth}
@@ -735,7 +732,7 @@ This can be seen by the large shift of the resonance frequencies, and by the ind
 \end{figure}
 
 \section{Optimal IFF with High Pass Filter}
-In this section, Integral Force Feedback with an added High Pass Filter is applied to the three nano-hexapods.
+Integral Force Feedback with an added High Pass Filter is applied to the three nano-hexapods.
 First, the parameters (\(\omega_i\) and \(g\)) of the IFF controller that yield best simultaneous damping are determined from Figure \ref{fig:rotating_iff_hpf_nass_optimal_gain}.
 The IFF parameters are chosen as follow:
 \begin{itemize}
@@ -788,8 +785,11 @@ In order to keep the overall stiffness constant, the actuator stiffness \(k_a\)
 A high pass filter is also added to limit the low frequency gain with a cut-off frequency \(\omega_i\) equal to one tenth of the system resonance (\(\omega_i = \omega_0/10\)).
 
 The achievable maximum simultaneous damping of all the modes is computed as a function of the parallel stiffnesses (Figure \ref{fig:rotating_iff_kp_nass_optimal_gain}).
-It is shown that the soft nano-hexapod cannot yield good damping.
-For the two stiff options, the achievable damping starts to significantly decrease when the parallel stiffness is one tenth of the total stiffness \(k_p = k_n/10\).
+It is shown that the soft nano-hexapod cannot yield good damping as the parallel stiffness cannot be made large enough compared to the negative stiffness induced by the rotation.
+For the two stiff options, the achievable damping decreases when the parallel stiffness is chosen too high as explained in Section \ref{sec:rotating_iff_parallel_stiffness}.
+Such behavior can be explain by the fact that the achievable damping can be approximated by the distance between the open-loop pole and the open-loop zero \cite[chapt 7.2]{preumont18_vibrat_contr_activ_struc_fourt_edition}.
+This distance is larger for stiff nano-hexapod as the open-loop pole will be at higher frequencies while the open-loop zero, which depends on the value of the parallel stiffness, can only be made large for stiff nano-hexapods.
+
 Let's choose \(k_p = 1\,N/mm\), \(k_p = 0.01\,N/\mu m\) and \(k_p = 1\,N/\mu m\) for the three considered nano-hexapods.
 The corresponding optimal controller gains and achievable damping are summarized in Table \ref{tab:rotating_iff_kp_opt_iff_kp_params_nass}.
 
@@ -842,16 +842,14 @@ The gain is chosen is chosen such that 99\% of modal damping is obtained (obtain
 \end{minipage}
 
 \section{Comparison of the obtained damped plants}
-Let's now compare the obtained damped plants for the three active damping techniques applied on the three nano-hexapod stiffnesses (Figure \ref{fig:rotating_nass_damped_plant_comp}).
+Now that optimal parameters for the three considered active damping techniques have been determined, the obtained damped plants are computed and compared in Figure \ref{fig:rotating_nass_damped_plant_comp}.
 
-\begin{important}
 Similarly to what was concluded in previous analysis:
 \begin{itemize}
-\item IFF adds coupling below the resonance frequency as compared to the open-loop and RDC cases
-\item All three methods are yielding good damping, except for IFF applied on the soft nano-hexapod where things are more complicated
+\item \acrshort{iff} adds coupling below the resonance frequency as compared to the open-loop and \acrshort{rdc} cases
+\item All three methods are yielding good damping, except for \acrshort{iff} applied on the soft nano-hexapod
 \item Coupling is smaller for stiff nano-hexapods
 \end{itemize}
-\end{important}
 
 \begin{figure}[htbp]
 \begin{subfigure}{0.33\textwidth}
@@ -872,53 +870,43 @@ Similarly to what was concluded in previous analysis:
 \end{center}
 \subcaption{\label{fig:rotating_nass_damped_plant_comp_pz}$k_n = 100\,N/\mu m$}
 \end{subfigure}
-\caption{\label{fig:rotating_nass_damped_plant_comp}Comparison of the damped plants (direct and coupling terms) for the three proposed active damping techniques (IFF with HPF, IFF with \(k_p\) and RDC) applied on the three nano-hexapod stiffnesses. \(\Omega = 60\,\text{rmp}\) and \(m_n + m_s = \SI{16}{\kg}\).}
+\caption{\label{fig:rotating_nass_damped_plant_comp}Comparison of the damped plants for the three proposed active damping techniques (IFF with HPF in blue, IFF with \(k_p\) in red and RDC in yellow). The direct terms are shown by the solid lines and coupling terms are shown by the shaded lines. Three nano-hexapod stiffnesses are considered. For this analysis the rotating velocity is \(\Omega = 60\,\text{rpm}\) and the suspended mass is \(m_n + m_s = \SI{16}{\kg}\).}
 \end{figure}
 
 \chapter{Nano-Active-Stabilization-System with rotation}
 \label{sec:rotating_nass}
-Up until now, the model used consisted of an infinitely stiff vertical rotating stage with a X-Y suspended stage.
-
+Up until now, the model used to study gyroscopic effects consisted of an infinitely stiff rotating stage with a X-Y suspended stage on top.
 While quite simplistic, this allowed to study the effects of rotation and the associated limitations when active damping is to be applied.
-
 In this section, the limited compliance of the micro-station is taken into account as well as the rotation of the spindle.
-\section{NASS model}
-
-In order to be a bit closer to the NASS application, the 2DoF nano-hexapod (modelled as shown in Figure \ref{fig:rotating_3dof_model_schematic}) is now located on top of a model of the micro-station including (see Figure \ref{fig:rotating_nass_model} for a 3D view):
+\section{Nano Active Stabilization System model}
+In order to have a more realistic dynamics model of the NASS, the 2-DoF nano-hexapod (modelled as shown in Figure \ref{fig:rotating_3dof_model_schematic}) is now located on top of a model of the micro-station including (see Figure \ref{fig:rotating_nass_model} for a 3D view):
 \begin{itemize}
 \item the floor whose motion is imposed
-\item a 2DoF granite (\(k_{g,x} = k_{g,y} = \SI{950}{\N\per\mu\m}\), \(m_g = \SI{2500}{\kg}\))
-\item a 2DoF \(T_y\) stage (\(k_{t,x} = k_{t,y} = \SI{520}{\N\per\mu\m}\), \(m_g = \SI{600}{\kg}\))
+\item a 2-DoF granite (\(k_{g,x} = k_{g,y} = \SI{950}{\N\per\mu\m}\), \(m_g = \SI{2500}{\kg}\))
+\item a 2-DoF \(T_y\) stage (\(k_{t,x} = k_{t,y} = \SI{520}{\N\per\mu\m}\), \(m_t = \SI{600}{\kg}\))
 \item a spindle (vertical rotation) stage whose rotation is imposed (\(m_s = \SI{600}{\kg}\))
-\item a 2DoF micro-hexapod (\(k_{h,x} = k_{h,y} = \SI{61}{\N\per\mu\m}\), \(m_g = \SI{15}{\kg}\))
+\item a 2-DoF micro-hexapod (\(k_{h,x} = k_{h,y} = \SI{61}{\N\per\mu\m}\), \(m_h = \SI{15}{\kg}\))
 \end{itemize}
 
 A payload is rigidly fixed to the nano-hexapod and the \(x,y\) motion of the payload is measured with respect to the granite.
 
 \begin{figure}[htbp]
 \centering
-\includegraphics[scale=1]{figs/rotating_nass_model.png}
+\includegraphics[scale=1,scale=0.7]{figs/rotating_nass_model.png}
 \caption{\label{fig:rotating_nass_model}3D view of the Nano-Active-Stabilization-System model.}
 \end{figure}
 
 \section{System dynamics}
 
-The dynamics of the undamped and damped plants are identified.
-The active damping parameters used are the optimal ones previously identified (i.e. for the rotating nano-hexapod fixed on a rigid platform).
-
-The undamped and damped plants are shown in Figure \ref{fig:rotating_nass_plant_comp_stiffness}.
-Three nano-hexapod velocities are shown (from left to right): \(k_n = \SI{0.01}{\N\per\mu\m}\), \(k_n = \SI{1}{\N\per\mu\m}\) and \(k_n = \SI{100}{\N\per\mu\m}\).
-The direct terms are shown by the solid curves while the coupling terms are shown by the shaded ones.
-
-\begin{important}
-It can be observed on Figure \ref{fig:rotating_nass_plant_comp_stiffness} that:
+The dynamics of the un-damped and damped plants are identified using the optimal parameters found in Section \ref{sec:rotating_nano_hexapod}.
+The obtained dynamics are compared in Figure \ref{fig:rotating_nass_plant_comp_stiffness} in which the direct terms are shown by the solid curves while the coupling terms are shown by the shaded ones.
+It can be observed that:
 \begin{itemize}
-\item Coupling (ratio between the off-diagonal and direct terms) is larger for the soft nano-hexapod
+\item The coupling (quantified by the ratio between the off-diagonal and direct terms) is higher for the soft nano-hexapod
 \item Damping added by the three proposed techniques is quite high and the obtained plant is rather easy to control
 \item There is some coupling between nano-hexapod and micro-station dynamics for the stiff nano-hexapod (mode at 200Hz)
 \item The two proposed IFF modification yields similar results
 \end{itemize}
-\end{important}
 
 \begin{figure}[htbp]
 \begin{subfigure}{0.33\textwidth}
@@ -942,53 +930,23 @@ It can be observed on Figure \ref{fig:rotating_nass_plant_comp_stiffness} that:
 \caption{\label{fig:rotating_nass_plant_comp_stiffness}Bode plot of the transfer function from nano-hexapod actuator to measured motion by the external metrology}
 \end{figure}
 
-To confirm that the coupling is smaller when the stiffness of the nano-hexapod is increase, the \emph{coupling ratio} for the three nano-hexapod stiffnesses are shown in Figure \ref{fig:rotating_nass_plant_coupling_comp}.
-
-\begin{figure}[htbp]
-\begin{subfigure}{0.33\textwidth}
-\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/rotating_nass_plant_coupling_comp_vc.png}
-\end{center}
-\subcaption{\label{fig:rotating_nass_plant_coupling_comp_vc}$k_n = 0.01\,N/\mu m$}
-\end{subfigure}
-\begin{subfigure}{0.33\textwidth}
-\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/rotating_nass_plant_coupling_comp_md.png}
-\end{center}
-\subcaption{\label{fig:rotating_nass_plant_coupling_comp_md}$k_n = 1\,N/\mu m$}
-\end{subfigure}
-\begin{subfigure}{0.33\textwidth}
-\begin{center}
-\includegraphics[scale=1,width=0.95\linewidth]{figs/rotating_nass_plant_coupling_comp_pz.png}
-\end{center}
-\subcaption{\label{fig:rotating_nass_plant_coupling_comp_pz}$k_n = 100\,N/\mu m$}
-\end{subfigure}
-\caption{\label{fig:rotating_nass_plant_coupling_comp}Coupling ratio for the proposed active damping techniques evaluated for the three nano-hexapod stiffnesses}
-\end{figure}
-
 \section{Effect of disturbances}
 
-The effect of three disturbances are considered:
-\begin{itemize}
-\item Floor motion (Figure \ref{fig:rotating_nass_effect_floor_motion})
-\item Micro-Station vibrations (Figure \ref{fig:rotating_nass_effect_stage_vibration})
-\item Direct force applied on the payload (Figure \ref{fig:rotating_nass_effect_direct_forces})
-\end{itemize}
+The effect of three disturbances are considered (as for the uniaxial model), floor motion \([x_{f,x},\ x_{f,y}]\) (Figure \ref{fig:rotating_nass_effect_floor_motion}), micro-Station vibrations \([f_{t,x},\ f_{t,y}]\) (Figure \ref{fig:rotating_nass_effect_stage_vibration}) and direct forces applied on the sample \([f_{s,x},\ f_{s,y}]\) (Figure \ref{fig:rotating_nass_effect_direct_forces}).
+Note that only the transfer function from the disturbances in the \(x\) direction to the relative position \(d_x\) between the sample and the granite in the \(x\) direction are displayed as the transfer functions in the \(y\) direction are the same due to the system symmetry.
 
-\begin{important}
 Conclusions are similar than with the uniaxial (non-rotating) model:
 \begin{itemize}
 \item Regarding the effect of floor motion and forces applied on the payload:
 \begin{itemize}
-\item The stiffer, the better (magnitudes are lower for the right curves, Figures \ref{fig:rotating_nass_effect_floor_motion} and \ref{fig:rotating_nass_effect_direct_forces})
-\item Integral Force Feedback degrades the performance at low frequency compared to relative damping control
+\item The stiffer, the better. This can be seen in Figures \ref{fig:rotating_nass_effect_floor_motion} and \ref{fig:rotating_nass_effect_direct_forces} where the magnitudes for the stiff-hexapod are lower than for the soft one
+\item \acrshort{iff} degrades the performance at low frequency compared to \acrshort{rdc}
 \end{itemize}
 \item Regarding the effect of micro-station vibrations:
 \begin{itemize}
-\item Having a soft nano-hexapod allows to filter these vibrations between the suspensions modes of the nano-hexapod and some flexible modes of the micro-station. Using relative damping control reduce this filtering (Figure \ref{fig:rotating_nass_effect_stage_vibration}, left).
+\item Having a soft nano-hexapod allows to filter these vibrations between the suspensions modes of the nano-hexapod and some flexible modes of the micro-station. Using relative damping control reduces this filtering (Figure \ref{fig:rotating_nass_effect_stage_vibration_vc}).
 \end{itemize}
 \end{itemize}
-\end{important}
 
 \begin{figure}[htbp]
 \begin{subfigure}{0.33\textwidth}
@@ -1009,7 +967,7 @@ Conclusions are similar than with the uniaxial (non-rotating) model:
 \end{center}
 \subcaption{\label{fig:rotating_nass_effect_floor_motion_pz}$k_n = 100\,N/\mu m$}
 \end{subfigure}
-\caption{\label{fig:rotating_nass_effect_floor_motion}Effect of Floor motion on the position error - Comparison of active damping techniques for the three nano-hexapod stiffnesses}
+\caption{\label{fig:rotating_nass_effect_floor_motion}Effect of floor motion \(x_{f,x}\) on the position error \(d_x\) - Comparison of active damping techniques for the three nano-hexapod stiffnesses. IFF is shown to increase the sensitivity to floor motion at low frequency.}
 \end{figure}
 
 \begin{figure}[htbp]
@@ -1031,7 +989,7 @@ Conclusions are similar than with the uniaxial (non-rotating) model:
 \end{center}
 \subcaption{\label{fig:rotating_nass_effect_stage_vibration_pz}$k_n = 100\,N/\mu m$}
 \end{subfigure}
-\caption{\label{fig:rotating_nass_effect_stage_vibration}Effect of micro-station vibrations on the position error - Comparison of active damping techniques for the three nano-hexapod stiffnesses}
+\caption{\label{fig:rotating_nass_effect_stage_vibration}Effect of micro-station vibrations \(f_{t,x}\) on the position error \(d_x\) - Comparison of active damping techniques for the three nano-hexapod stiffnesses. Relative Damping Control increases the sensitivity to micro-station vibrations between the soft nano-hexapod suspension modes and the micro-station modes (\subref{fig:rotating_nass_effect_stage_vibration_vc})}
 \end{figure}
 
 
@@ -1054,13 +1012,13 @@ Conclusions are similar than with the uniaxial (non-rotating) model:
 \end{center}
 \subcaption{\label{fig:rotating_nass_effect_direct_forces_pz}$k_n = 100\,N/\mu m$}
 \end{subfigure}
-\caption{\label{fig:rotating_nass_effect_direct_forces}Effect of sample forces on the position error - Comparison of active damping techniques for the three nano-hexapod stiffnesses}
+\caption{\label{fig:rotating_nass_effect_direct_forces}Effect of sample forces \(f_{s,x}\) on the position error \(d_x\) - Comparison of active damping techniques for the three nano-hexapod stiffnesses. Integral Force Feedback degrades this compliance at low frequency.}
 \end{figure}
 
 \chapter*{Conclusion}
-In this study, the gyroscopic effects induced by the spindle's rotation have been studied using a spindle model (Section \ref{sec:rotating_system_description}).
-Decentralized IFF with pure integrators was shown to be unstable when applied to rotating platforms (Section \ref{sec:rotating_iff_pure_int}).
-Two modifications of the classical IFF control have been proposed to overcome this issue.
+In this study, the gyroscopic effects induced by the spindle's rotation have been studied using a simplified model (Section \ref{sec:rotating_system_description}).
+Decentralized \acrlong{iff} with pure integrators was shown to be unstable when applied to rotating platforms (Section \ref{sec:rotating_iff_pure_int}).
+Two modifications of the classical \acrshort{iff} control have been proposed to overcome this issue.
 
 The first modification concerns the controller and consists of adding a high pass filter to the pure integrators.
 This is equivalent to moving the controller pole to the left along the real axis.
@@ -1068,14 +1026,19 @@ This allows the closed loop system to be stable up to some value of the controll
 
 The second proposed modification concerns the mechanical system.
 Additional springs are added in parallel with the actuators and force sensors.
-It was shown that if the stiffness \(k_p\) of the additional springs is larger than the negative stiffness \(m \Omega^2\) induced by centrifugal forces, the classical decentralized IFF regains its unconditional stability property (Section \ref{sec:rotating_iff_parallel_stiffness}).
+It was shown that if the stiffness \(k_p\) of the additional springs is larger than the negative stiffness \(m \Omega^2\) induced by centrifugal forces, the classical decentralized \acrshort{iff} regains its unconditional stability property (Section \ref{sec:rotating_iff_parallel_stiffness}).
 
-These two modifications were compared with relative damping control in Section \ref{sec:rotating_comp_act_damp}.
+These two modifications were compared with \acrlong{rdc} in Section \ref{sec:rotating_comp_act_damp}.
 While having very different implementations, both proposed modifications were found to be very similar when it comes to the attainable damping and the obtained closed loop system behavior.
 
-Then, this study has been applied to a rotating system that corresponds to the nano-hexapod parameters (Section \ref{sec:rotating_nano_hexapod}).
-To be closer to the real system dynamics, the limited compliance of the micro-station has been taken into account.
-Results show that the two proposed IFF modifications can be applied for the NASS even in the presence of spindle rotation.
+Then, this study has been applied to a rotating platform that corresponds to the nano-hexapod parameters (Section \ref{sec:rotating_nano_hexapod}).
+As for the uniaxial model, three nano-hexapod stiffness are considered.
+The dynamics of the soft nano-hexapod (\(k_n = 0.01\,N/\mu m\)) was shown to be more depend on the rotation velocity (higher coupling and change of dynamics due to gyroscopic effects).
+Also, the attainable damping ratio of the soft nano-hexapod when using \acrshort{iff} is limited by gyroscopic effects.
+
+To be closer to the \acrlong{nass} dynamics, the limited compliance of the micro-station has been taken into account (Section \ref{sec:rotating_nass}).
+Results are similar to that of the uniaxial model except that come complexity is added for the soft nano-hexapod due to the spindle's rotation.
+For the moderately stiff nano-hexapod (\(k_n = 1\,N/\mu m\)), the gyroscopic effects are only slightly affecting the system dynamics, and therefore could represent a good alternative to the soft nano-hexapod that was showing better results with the uniaxial model.
 
 \printbibliography[heading=bibintoc,title={Bibliography}]