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@ -2833,7 +2833,7 @@ exportFig('figs/rotating_nano_hexapod_dynamics_pz.pdf', 'width', 'third', 'heigh
#+caption: Effect of rotation on the nano-hexapod dynamics. Dashed lines represent plants without rotation, solid lines represent plants at maximum rotating velocity ($\Omega = 60\,\text{rpm}$), and shaded lines are coupling terms at maximum rotating velocity
#+attr_latex: :options [htbp]
#+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:uniaxial_damped_plant_three_active_damping_techniques_vc}$k_n = 0.01\,N/\mu m$}
#+attr_latex: :caption \subcaption{\label{fig:rotating_nano_hexapod_dynamics_vc}$k_n = 0.01\,N/\mu m$}
#+attr_latex: :options {0.33\textwidth}
#+begin_subfigure
#+attr_latex: :width 0.95\linewidth

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nass-rotating-3dof-model.pdf
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nass-rotating-3dof-model.tex
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@ -1,4 +1,4 @@
% Created 2024-04-30 Tue 15:25
% Created 2025-04-03 Thu 21:55
% Intended LaTeX compiler: pdflatex
\documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt}
@ -26,13 +26,6 @@
\author{Dehaeze Thomas}
\date{\today}
\title{Nano Active Stabilization System - Effect of rotation}
\hypersetup{
pdfauthor={Dehaeze Thomas},
pdftitle={Nano Active Stabilization System - Effect of rotation},
pdfkeywords={},
pdfsubject={},
pdfcreator={Emacs 29.3 (Org mode 9.6)},
pdflang={English}}
\usepackage{biblatex}
\begin{document}
@ -41,7 +34,6 @@
\tableofcontents
\clearpage
An important aspect of the \acrfull{nass} is that the nano-hexapod continuously rotates around a vertical axis, whereas the external metrology is not.
Such rotation induces gyroscopic effects that may impact the system dynamics and obtained performance.
To study these effects, a model of a rotating suspended platform is first presented (Section \ref{sec:rotating_system_description})
@ -70,9 +62,9 @@ The goal is to determine whether the rotation imposes performance limitation on
\includegraphics[scale=1,width=\linewidth]{figs/rotating_overview.png}
\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}
\label{sec:rotating_system_description}
The system used to study gyroscopic effects consists of a 2 degree of freedom translation stage on top of a rotating stage (Figure \ref{fig:rotating_3dof_model_schematic}).
The rotating stage is supposed to be ideal, meaning it induces a perfect rotation \(\theta(t) = \Omega t\) where \(\Omega\) is the rotational speed in \(\si{\radian\per\s}\).
The suspended platform consists of two orthogonal actuators, each represented by three elements in parallel: a spring with a stiffness \(k\) in \(\si{\newton\per\meter}\), a dashpot with a damping coefficient \(c\) in \(\si{\newton\per(\meter\per\second)}\) and an ideal force source \(F_u, F_v\).
@ -86,7 +78,6 @@ After the dynamics of this system is studied, the objective will be to dampen th
\includegraphics[scale=1,scale=0.8]{figs/rotating_3dof_model_schematic.png}
\caption{\label{fig:rotating_3dof_model_schematic}Schematic of the studied system}
\end{figure}
\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 \(\begin{bmatrix}q_1 & q_2\end{bmatrix} = \begin{bmatrix}d_u & d_v\end{bmatrix}\).
@ -146,7 +137,6 @@ The elements of the transfer function matrix \(\mathbf{G}_d\) are described by e
\mathbf{G}_{d}(1,2) &= \frac{\frac{1}{k} \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)}{\left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2}
\end{align}
\end{subequations}
\section{System Poles: Campbell Diagram}
The poles of \(\mathbf{G}_d\) are the complex solutions \(p\) of equation \eqref{eq:rotating_poles} (i.e. the roots of its denominator).
@ -183,7 +173,6 @@ Physically, the negative stiffness term \(-m\Omega^2\) induced by centrifugal fo
\end{subfigure}
\caption{\label{fig:rotating_campbell_diagram}Campbell diagram - Real (\subref{fig:rotating_campbell_diagram_real}) and Imaginary (\subref{fig:rotating_campbell_diagram_imag}) parts of the poles as a function of the rotating velocity \(\Omega\).}
\end{figure}
\section{System Dynamics: Effect of rotation}
The system dynamics from actuator forces \([F_u, F_v]\) to the relative motion \([d_u, d_v]\) is identified for several rotating velocities.
Looking at the transfer function matrix \(\mathbf{G}_d\) in equation \eqref{eq:rotating_Gd_w0_xi_k}, one can see that the two diagonal (direct) terms are equal and that the two off-diagonal (coupling) terms are opposite.
@ -206,9 +195,9 @@ For \(\Omega > \omega_0\), the low-frequency pair of complex conjugate poles \(p
\end{subfigure}
\caption{\label{fig:rotating_bode_plot}Bode plot of the direct (\subref{fig:rotating_bode_plot_direct}) and coupling (\subref{fig:rotating_bode_plot_direct}) terms for several rotating velocities}
\end{figure}
\chapter{Integral Force Feedback}
\label{sec:rotating_iff_pure_int}
The goal is now to damp the two suspension modes of the payload using an active damping strategy while the rotating stage performs a constant rotation.
As was explained with the uniaxial model, such an 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.
@ -224,7 +213,6 @@ Recently, an \(\mathcal{H}_\infty\) optimization criterion has been used to deri
However, none of these studies have been applied to rotating systems.
In this section, the \acrshort{iff} strategy is applied on the rotating suspended platform, and it is shown that gyroscopic effects alter the system dynamics and that IFF cannot be applied as is.
\section{System and Equations of motion}
To apply Integral Force Feedback, two force sensors are added in series with the actuators (Figure \ref{fig:rotating_3dof_model_schematic_iff}).
Two identical controllers \(K_F\) described by \eqref{eq:rotating_iff_controller} are then used to feedback each of the sensed force to its associated actuator.
@ -296,7 +284,6 @@ This small displacement then increases the centrifugal force \(m\Omega^2d_u = \f
0 & \frac{\Omega^2}{{\omega_0}^2 - \Omega^2}
\end{bmatrix}
\end{equation}
\section{Effect of rotation speed on IFF plant dynamics}
The transfer functions from actuator forces \([F_u,\ F_v]\) to the measured force sensors \([f_u,\ f_v]\) are identified for several rotating velocities and are shown in Figure \ref{fig:rotating_iff_bode_plot_effect_rot}.
As expected from the derived equations of motion:
@ -322,7 +309,6 @@ A pair of (minimum phase) complex conjugate zeros appears between the two comple
\end{subfigure}
\caption{\label{fig:rotating_iff_bode_plot_effect_rot}Effect of the rotation velocity on the bode plot of the direct terms (\subref{fig:rotating_iff_bode_plot_effect_rot_direct}) and on the IFF root locus (\subref{fig:rotating_root_locus_iff_pure_int})}
\end{figure}
\section{Decentralized Integral Force Feedback}
The control diagram for decentralized \acrshort{iff} is shown in Figure \ref{fig:rotating_iff_diagram}.
The decentralized \acrshort{iff} controller \(\bm{K}_F\) corresponds to a diagonal controller with integrators \eqref{eq:rotating_Kf_pure_int}.
@ -341,9 +327,9 @@ Whereas collocated IFF is usually associated with unconditional stability \cite{
This can be seen in the Root Locus plot (Figure \ref{fig:rotating_root_locus_iff_pure_int}) where poles corresponding to the controller are bound to the right half plane implying closed-loop system instability.
Physically, this can be explained as follows: at low frequencies, the loop gain is huge due to the pure integrator in \(K_{F}\) and the finite gain of the plant (Figure \ref{fig:rotating_iff_bode_plot_effect_rot}).
The control system is thus cancels the spring forces, which makes the suspended platform not capable to hold the payload against centrifugal forces, hence the instability.
\chapter{Integral Force Feedback with a High-Pass Filter}
\label{sec:rotating_iff_pseudo_int}
As explained in the previous section, the instability of the IFF controller applied to the rotating system is due to the high gain of the integrator at low-frequency.
To limit the low-frequency controller gain, a \acrfull{hpf} can be added to the controller, as shown in equation \eqref{eq:rotating_iff_lhf}.
This is equivalent to slightly shifting the controller pole to the left along the real axis.
@ -353,7 +339,6 @@ This is however not the reason why this high-pass filter is added here.
\begin{equation}\label{eq:rotating_iff_lhf}
\boxed{K_{F}(s) = g \cdot \frac{1}{s} \cdot \underbrace{\frac{s/\omega_i}{1 + s/\omega_i}}_{\text{HPF}} = g \cdot \frac{1}{s + \omega_i}}
\end{equation}
\section{Modified Integral Force Feedback Controller}
The Integral Force Feedback Controller is modified such that instead of using pure integrators, pseudo integrators (i.e. low pass filters) are used \eqref{eq:rotating_iff_lhf} where \(\omega_i\) characterize the frequency down to which the signal is integrated.
The loop gains (\(K_F(s)\) times the direct dynamics \(f_u/F_u\)) with and without the added HPF are shown in Figure \ref{fig:rotating_iff_modified_loop_gain}.
@ -382,7 +367,6 @@ It is interesting to note that \(g_{\text{max}}\) also corresponds to the contro
\end{subfigure}
\caption{\label{fig:rotating_iff_modified_loop_gain_root_locus}Comparison of the IFF with pure integrator and modified IFF with added high-pass filter (\(\Omega = 0.1\omega_0\)). The loop gain is shown in (\subref{fig:rotating_iff_modified_loop_gain}) with \(\omega_i = 0.1 \omega_0\) and \(g = 2\). The root locus is shown in (\subref{fig:rotating_iff_root_locus_hpf_large})}
\end{figure}
\section{Optimal IFF with HPF parameters \(\omega_i\) and \(g\)}
Two parameters can be tuned for the modified controller in equation \eqref{eq:rotating_iff_lhf}: the gain \(g\) and the pole's location \(\omega_i\).
The optimal values of \(\omega_i\) and \(g\) are considered here as the values for which the damping of all the closed-loop poles is simultaneously maximized.
@ -410,7 +394,6 @@ For larger values of \(\omega_i\), the attainable damping ratio decreases as a f
\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}). The achievable damping ratio decreases as \(\omega_i\) increases, as confirmed in (\subref{fig:rotating_iff_hpf_optimal_gain})}
\end{figure}
\section{Obtained Damped Plant}
To study how the parameter \(\omega_i\) affects the damped plant, the obtained damped plants for several \(\omega_i\) are compared in Figure \ref{fig:rotating_iff_hpf_damped_plant_effect_wi_plant}.
It can be seen that the low-frequency coupling increases as \(\omega_i\) increases.
@ -432,9 +415,9 @@ The same trade-off can be seen between achievable damping and loss of compliance
\end{subfigure}
\caption{\label{fig:rotating_iff_hpf_damped_plant_effect_wi}Effect of \(\omega_i\) on the damped plant coupling}
\end{figure}
\chapter{IFF with a stiffness in parallel with the force sensor}
\label{sec:rotating_iff_parallel_stiffness}
In this section it is proposed to add springs in parallel with the force sensors to counteract the negative stiffness induced by the gyroscopic effects.
Such springs are schematically shown in Figure \ref{fig:rotating_3dof_model_schematic_iff_parallel_springs} where \(k_a\) is the stiffness of the actuator and \(k_p\) the added stiffness in parallel with the actuator and force sensor.
@ -443,7 +426,6 @@ Such springs are schematically shown in Figure \ref{fig:rotating_3dof_model_sche
\includegraphics[scale=1,scale=0.8]{figs/rotating_3dof_model_schematic_iff_parallel_springs.png}
\caption{\label{fig:rotating_3dof_model_schematic_iff_parallel_springs}Studied system with additional springs in parallel with the actuators and force sensors (shown in red)}
\end{figure}
\section{Equations}
The forces measured by the two force sensors represented in Figure \ref{fig:rotating_3dof_model_schematic_iff_parallel_springs} are described by \eqref{eq:rotating_measured_force_kp}.
@ -482,7 +464,6 @@ Thus, if the added \emph{parallel stiffness} \(k_p\) is higher than the \emph{ne
\begin{equation}\label{eq:rotating_kp_cond_cc_zeros}
\boxed{\alpha > \frac{\Omega^2}{{\omega_0}^2} \quad \Leftrightarrow \quad k_p > m \Omega^2}
\end{equation}
\section{Effect of parallel stiffness on the IFF plant}
The IFF plant (transfer function from \([F_u, F_v]\) to \([f_u, f_v]\)) is identified without parallel stiffness \(k_p = 0\), with a small parallel stiffness \(k_p < m \Omega^2\) and with a large parallel stiffness \(k_p > m \Omega^2\).
Bode plots of the obtained dynamics are shown in Figure \ref{fig:rotating_iff_effect_kp}.
@ -507,7 +488,6 @@ It is shown that if the added stiffness is higher than the maximum negative stif
\end{subfigure}
\caption{\label{fig:rotating_iff_plant_effect_kp}Effect of parallel stiffness on the IFF plant}
\end{figure}
\section{Effect of \(k_p\) on the attainable damping}
Even though the parallel stiffness \(k_p\) has no impact on the open-loop poles (as the overall stiffness \(k\) is kept constant), it has a large impact on the transmission zeros.
Moreover, as the attainable damping is generally proportional to the distance between poles and zeros \cite{preumont18_vibrat_contr_activ_struc_fourt_edition}, the parallel stiffness \(k_p\) is expected to have some impact on the attainable damping.
@ -531,7 +511,6 @@ This is confirmed by the Figure \ref{fig:rotating_iff_kp_optimal_gain} where the
\end{subfigure}
\caption{\label{fig:rotating_iff_optimal_kp}Effect of parallel stiffness on the IFF plant}
\end{figure}
\section{Damped plant}
The parallel stiffness are chosen to be \(k_p = 2 m \Omega^2\) and the damped plant is computed.
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}.
@ -567,9 +546,9 @@ The added high-pass filter gives almost the same damping properties to the suspe
\end{subfigure}
\caption{\label{fig:rotating_iff_optimal_hpf}Effect of high-pass filter cut-off frequency on the obtained damping}
\end{figure}
\chapter{Relative Damping Control}
\label{sec:rotating_relative_damp_control}
To apply a ``Relative Damping Control'' strategy, relative motion sensors are added in parallel with the actuators as shown in Figure \ref{fig:rotating_3dof_model_schematic_rdc}.
Two controllers \(K_d\) are used to feed back the relative motion to the actuator.
These controllers are in principle pure derivators (\(K_d = s\)), but to be implemented in practice they are usually replaced by a high-pass filter \eqref{eq:rotating_rdc_controller}.
@ -583,7 +562,6 @@ K_d(s) = g \cdot \frac{s}{s + \omega_d}
\includegraphics[scale=1,scale=0.8]{figs/rotating_3dof_model_schematic_rdc.png}
\caption{\label{fig:rotating_3dof_model_schematic_rdc}System with relative motion sensor and decentralized ``relative damping control'' applied.}
\end{figure}
\section{Equations of motion}
Let's note \(\bm{G}_d\) the transfer function between actuator forces and measured relative motion in parallel with the actuators \eqref{eq:rotating_rdc_plant_matrix}.
The elements of \(\bm{G}_d\) were derived in Section \ref{sec:rotating_system_description} are shown in \eqref{eq:rotating_rdc_plant_elements}.
@ -605,7 +583,6 @@ Therefore, for \(\Omega < \sqrt{k/m}\) (i.e. stable system), the transfer functi
\begin{equation}\label{eq:rotating_rdc_zeros_poles}
z = \pm j \sqrt{\omega_0^2 - \omega^2}, \quad p_1 = \pm j (\omega_0 - \omega), \quad p_2 = \pm j (\omega_0 + \omega)
\end{equation}
\section{Decentralized Relative Damping Control}
The transfer functions from \([F_u,\ F_v]\) to \([d_u,\ d_v]\) were identified for several rotating velocities in Section \ref{sec:rotating_system_description} and are shown in Figure \ref{fig:rotating_bode_plot} (page \pageref{fig:rotating_bode_plot}).
@ -632,13 +609,12 @@ It does not increase the low-frequency coupling as compared to the Integral Forc
\end{subfigure}
\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}
\label{sec:rotating_comp_act_damp}
These two proposed IFF modifications and relative damping control are compared in terms of added damping and closed-loop behavior.
For the following comparisons, the cut-off frequency for the added HPF is set to \(\omega_i = 0.1 \omega_0\) and the stiffness of the parallel springs is set to \(k_p = 5 m \Omega^2\) (corresponding to \(\alpha = 0.05\)).
These values are chosen one the basis of previous discussions about optimal parameters.
\section{Root Locus}
Figure \ref{fig:rotating_comp_techniques_root_locus} shows the Root Locus plots for the two proposed IFF modifications and the relative damping control.
While the two pairs of complex conjugate open-loop poles are identical for both IFF modifications, the transmission zeros are not.
@ -663,12 +639,10 @@ It is interesting to note that the maximum added damping is very similar for bot
\end{subfigure}
\caption{\label{fig:rotating_comp_techniques}Comparison of active damping techniques for rotating platform}
\end{figure}
\section{Obtained Damped Plant}
The actively damped plants are computed for the three techniques and compared in Figure \ref{fig:rotating_comp_techniques_dampled_plants}.
It is shown that while the diagonal (direct) terms of the damped plants are similar for the three active damping techniques, the off-diagonal (coupling) terms are not.
The \acrshort{iff} strategy is adding some coupling at low-frequency, which may negatively impact the positioning performance.
\section{Transmissibility And Compliance}
The proposed active damping techniques are now compared in terms of closed-loop transmissibility and compliance.
The transmissibility is defined as the transfer function from the displacement of the rotating stage along \(\vec{i}_x\) to the displacement of the payload along the same direction.
@ -696,7 +670,6 @@ This is very well known characteristics of these common active damping technique
\end{subfigure}
\caption{\label{fig:rotating_comp_techniques_trans_compliance}Comparison of the obtained transmissibility (\subref{fig:rotating_comp_techniques_transmissibility}) and compliance (\subref{fig:rotating_comp_techniques_compliance}) for the three tested active damping techniques}
\end{figure}
\chapter{Rotating Nano-Hexapod}
\label{sec:rotating_nano_hexapod}
The previous analysis is now applied to a model representing a rotating nano-hexapod.
@ -716,7 +689,7 @@ The coupling (or interaction) in a MIMO \(2 \times 2\) system can be visually es
\begin{center}
\includegraphics[scale=1,width=0.95\linewidth]{figs/rotating_nano_hexapod_dynamics_vc.png}
\end{center}
\subcaption{\label{fig:uniaxial_damped_plant_three_active_damping_techniques_vc}$k_n = 0.01\,N/\mu m$}
\subcaption{\label{fig:rotating_nano_hexapod_dynamics_vc}$k_n = 0.01\,N/\mu m$}
\end{subfigure}
\begin{subfigure}{0.33\textwidth}
\begin{center}
@ -732,7 +705,6 @@ The coupling (or interaction) in a MIMO \(2 \times 2\) system can be visually es
\end{subfigure}
\caption{\label{fig:rotating_nano_hexapod_dynamics}Effect of rotation on the nano-hexapod dynamics. Dashed lines represent plants without rotation, solid lines represent plants at maximum rotating velocity (\(\Omega = 60\,\text{rpm}\)), and shaded lines are coupling terms at maximum rotating velocity}
\end{figure}
\section{Optimal IFF with a High-Pass Filter}
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 the best simultaneous damping are determined from Figure \ref{fig:rotating_iff_hpf_nass_optimal_gain}.
@ -780,7 +752,6 @@ The obtained IFF parameters and the achievable damping are visually shown by lar
\bottomrule
\end{tabularx}
\end{table}
\section{Optimal IFF with Parallel Stiffness}
For each considered nano-hexapod stiffness, the parallel stiffness \(k_p\) is varied from \(k_{p,\text{min}} = m\Omega^2\) (the minimum stiffness that yields unconditional stability) to \(k_{p,\text{max}} = k_n\) (the total nano-hexapod stiffness).
To keep the overall stiffness constant, the actuator stiffness \(k_a\) is decreased when \(k_p\) is increased (\(k_a = k_n - k_p\), with \(k_n\) the total nano-hexapod stiffness).
@ -816,7 +787,6 @@ The corresponding optimal controller gains and achievable damping are summarized
\end{tabularx}
\end{center}
\end{minipage}
\section{Optimal Relative Motion Control}
For each considered nano-hexapod stiffness, relative damping control is applied and the achievable damping ratio as a function of the controller gain is computed (Figure \ref{fig:rotating_rdc_optimal_gain}).
The gain is chosen such that 99\% of modal damping is obtained (obtained gains are summarized in Table \ref{tab:rotating_rdc_opt_params_nass}).
@ -842,7 +812,6 @@ The gain is chosen such that 99\% of modal damping is obtained (obtained gains a
\end{tabularx}
\end{center}
\end{minipage}
\section{Comparison of the obtained damped plants}
Now that the 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}.
@ -874,7 +843,6 @@ Similar to what was concluded in the previous analysis:
\end{subfigure}
\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 solid lines, and the 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}
Until now, the model used to study gyroscopic effects consisted of an infinitely stiff rotating stage with a X-Y suspended stage on top.
@ -897,7 +865,6 @@ A payload is rigidly fixed to the nano-hexapod and the \(x,y\) motion of the pay
\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 using the optimal parameters found in Section \ref{sec:rotating_nano_hexapod}.
@ -931,7 +898,6 @@ It can be observed that:
\end{subfigure}
\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}
\section{Effect of disturbances}
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}).
@ -1016,7 +982,6 @@ Conclusions are similar than those of the uniaxial (non-rotating) model:
\end{subfigure}
\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 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}).
@ -1041,8 +1006,6 @@ In addition, the attainable damping ratio of the soft nano-hexapod when using \a
To be closer to the \acrlong{nass} dynamics, the limited compliance of the micro-station has been considered (Section \ref{sec:rotating_nass}).
Results are similar to those 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 only slightly affect the system dynamics, and therefore could represent a good alternative to the soft nano-hexapod that showed better results with the uniaxial model.
\printbibliography[heading=bibintoc,title={Bibliography}]
\printglossaries
\end{document}