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 \documentclass[10pt]{iopart}
 
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 \usetikzlibrary{shapes.misc,arrows,arrows.meta}
 \usepackage[square,comma,sort&compress,numbers]{natbib}
 \renewcommand{\bibsection}{\section*{References}}
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 \begin{document}
 
 \title{Active damping of rotating platforms using integral force feedback}
@@ -42,30 +44,34 @@ The results reveal that, despite their different implementations, both modified
 \maketitle
 % Two Columns output
 \ioptwocol
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 \section{Introduction}
 \label{sec:introduction}
 There is an increasing need to reduce the undesirable vibration of sensitive equipment in many different fields such as in aerospace industry \cite{hanieh03_activ_stewar,hauge04_sensor_contr_space_based_six,souleille18_concep_activ_mount_space_applic}, semi conductor industry \cite{yoshioka01_activ_microv_isolat_system_hi}, microscopy \cite{lan08_activ_vibrat_isolat_long_range,fleming15_low_order_dampin_track_contr}, gravitational wave detectors \cite{matichard15_seism_isolat_advan_ligo} and particles accelerators \cite{collette10_activ_quadr_stabil_futur_linear_partic_collid,dehaeze18_sampl_stabil_for_tomog_exper}.
 
 A common method to reduce vibration is to mount the sensitive equipment on a suspended platform which attenuates the vibrations above the frequency of the suspension modes \cite{reilly06_critic,poel10_explor_activ_hard_mount_vibrat}. \par
+
 In order to further decrease the residual vibrations, active damping can be used for reducing the magnification of the response in the vicinity of the resonances \cite{collette11_review_activ_vibrat_isolat_strat}.
 
 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}. \par
+
 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 directly augment 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 facilitate 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}. \par
+
 The main advantages of IFF over other active damping techniques are the guaranteed stability even in presence of flexible dynamics, good performances and robustness properties \cite{preumont02_force_feedb_versus_accel_feedb}. \par
+
 Several improvements of the classical IFF have been proposed, such as adding a feed-through term to increase the achievable damping \cite{teo15_optim_integ_force_feedb_activ_vibrat_contr} or adding an high pass filter to recover the loss of compliance at low frequency \cite{chesne16_enhan_dampin_flexib_struc_using_force_feedb}.
 Recently, an \(\mathcal{H}_\infty\) optimization criterion has been used to derive optimal gains for the IFF controller \cite{zhao19_optim_integ_force_feedb_contr}. \par
+
 However, when the platform is rotating, gyroscopic effects alter the system dynamics and IFF cannot be applied as is.
 The purpose of this paper is to study how the IFF strategy can be adapted to deal with these gyroscopic effects. \par
+
 The paper is structured as follows.
 Section \ref{sec:dynamics} presents a simple model of a rotating suspended platform that will be used throughout this study.
 Section \ref{sec:iff} explains how the unconditional stability of IFF is lost due to gyroscopic effects induced by the rotation.
 Section \ref{sec:iff_hpf} suggests a simple modification of the control law such that damping can be added to the suspension modes in a robust way.
 Section \ref{sec:iff_kp} proposes to add springs in parallel with the force sensors to regain the unconditional stability of IFF.
 Section \ref{sec:comparison} compares both proposed modifications to the classical IFF in terms of damping authority and closed-loop system behavior.
-
 \section{Dynamics of Rotating Platforms}
 \label{sec:dynamics}
 In order to study how the rotation affects the use of IFF, a model of a suspended platform on top of a rotating stage is used.
@@ -86,6 +92,7 @@ Two reference frames are used: an inertial frame \((\vec{i}_x, \vec{i}_y, \vec{i
 The position of the payload is represented by \((d_u, d_v, 0)\) expressed in the rotating frame.
 
 \par
+
 To obtain the equations of motion for the system represented in Figure \ref{fig:system}, the Lagrangian equations are used:
 \begin{equation}
 \label{eq:lagrangian_equations}
@@ -120,6 +127,7 @@ The uniform rotation of the system induces two gyroscopic effects as shown in Eq
 One can verify that without rotation (\(\Omega = 0\)) the system becomes equivalent to two uncoupled one degree of freedom mass-spring-damper systems.
 
 \par
+
 To study the dynamics of the system, the differential equations of motions \eqref{eq: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 Eq. \eqref{eq:Gd_mimo_tf} is obtained.
 Its elements are shown in Eq. \eqref{eq:Gd_indiv_el}.
 
@@ -158,6 +166,7 @@ For all further numerical analysis in this study, we consider \(\omega_0 = \SI{1
 Even though no system with such parameters will be encountered in practice, conclusions can be drawn relative to these parameters such that they can be generalized to any other set of parameters.
 
 \par
+
 The poles of \(\mathbf{G}_d\) are the complex solutions \(p\) of Eq. \eqref{eq:poles}.
 
 \begin{equation}
@@ -220,7 +229,6 @@ For \(\Omega > \omega_0\), the low frequency pair of complex conjugate poles \(p
 \caption{\label{fig:plant_compare_rotating_speed}Bode plots for \(\mathbf{G}_d\) for several rotational speed \(\Omega\)}
 \centering
 \end{figure}
-
 \section{Decentralized Integral Force Feedback}
 \label{sec:iff}
 In order to apply IFF to the rotating system, force sensors are added in series with the two actuators (Figure \ref{fig:system_iff}).
@@ -240,6 +248,7 @@ The control diagram is schematically shown in Figure \ref{fig:control_diagram_if
 \end{figure}
 
 \par
+
 The forces \(\begin{bmatrix}f_u & f_v\end{bmatrix}\) measured by the two force sensors represented in Figure \ref{fig:system_iff} are described by Eq. \eqref{eq:measured_force}.
 \begin{equation}
 \label{eq:measured_force}
@@ -298,6 +307,7 @@ This can be explained as follows: a constant force \(F_u\) induces a small displ
 \end{figure}
 
 \par
+
 \label{sec:iff_pure_int}
 The two IFF controllers \(K_{F}\) consist of a pure integrator as shown in Eq. \eqref{eq:Kf_pure_int} where \(g\) is a scalar representing the gain of the controller.
 \begin{equation}
@@ -326,7 +336,6 @@ The control system is thus canceling the spring forces which makes the suspended
 
 In order to apply decentralized IFF to rotating platforms, two solutions are proposed to deal with this instability problem.
 The first one consists of slightly modifying the control law (Section \ref{sec:iff_hpf}) while the second one consists of adding springs in parallel with the force sensors (Section \ref{sec:iff_kp}).
-
 \section{Integral Force Feedback with High Pass Filter}
 \label{sec:iff_hpf}
 As was explained in the previous section, the instability comes in part from the high gain at low frequency caused by the pure integrators.
@@ -344,6 +353,7 @@ This modification of the IFF controller is typically done to avoid saturation as
 This is however not the case in this study as it will become clear in the next section.
 
 \par
+
 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:loop_gain_modified_iff}.
 The effect of the added HPF limits the low frequency gain as expected.
 
@@ -370,6 +380,7 @@ It is interesting to note that \(g_{\text{max}}\) also corresponds to the contro
 \end{figure}
 
 \par
+
 Two parameters can be tuned for the modified controller in Eq. \eqref{eq:IFF_LHF}: the gain \(g\) and the pole's location \(\omega_i\).
 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.
 
@@ -397,7 +408,6 @@ Three regions can be observed:
 \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}
-
 \section{Integral Force Feedback with Parallel Springs}
 \label{sec:iff_kp}
 In this section additional springs in parallel with the force sensors are added to counteract the negative stiffness induced by the gyroscopic effects.
@@ -413,6 +423,7 @@ The parallel stiffness \(k_p\) then corresponds to the mechanical amplification
 \end{figure}
 
 \par
+
 The forces measured by the two force sensors represented in Figure \ref{fig:system_parallel_springs} are described by Eq. \eqref{eq:measured_force_kp}.
 
 \begin{equation}
@@ -476,6 +487,7 @@ It is shown that if the added stiffness is higher than the maximum negative stif
 \end{figure}
 
 \par
+
 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 foreseen to have a large impact on the attainable damping.
 
@@ -497,7 +509,6 @@ This is confirmed by the Figure \ref{fig:opt_damp_alpha} where the attainable cl
 \includegraphics[scale=1,scale=0.95]{figs/fig16.pdf}
 \caption{\label{fig:opt_damp_alpha}Optimal damping ratio \(\xi_\text{opt}\) and the corresponding optimal gain \(g_\text{opt}\) as a function of \(\alpha\)}
 \end{figure}
-
 \section{Comparison and Discussion}
 \label{sec:comparison}
 In order to overcome the instability issue of the classical IFF strategy when applied to rotating platforms, two modifications of IFF have been proposed in Sections \ref{sec:iff_hpf} and \ref{sec:iff_kp}.
@@ -507,6 +518,7 @@ For the following comparisons, the cut-off frequency for the HPF is set to \(\om
 These values are chosen based on the discussion about optimal parameters.
 
 \par
+
 Figure \ref{fig:comp_root_locus} shows the Root Locus plots for the two proposed IFF modifications.
 While the two pairs of complex conjugate open-loop poles are identical for both techniques, the transmission zeros are not.
 This means that the closed-loop behavior of both systems will differ when large control gains are used.
@@ -523,6 +535,7 @@ It is interesting to note that the maximum added damping is very similar for bot
 \end{figure}
 
 \par
+
 The two proposed techniques are now compared in terms of closed-loop transmissibility and compliance.
 
 The transmissibility is here defined as the transfer function from a displacement of the rotating stage along \(\vec{i}_x\) to the displacement of the payload along the same direction.
@@ -557,7 +570,6 @@ It is also confirmed that these two techniques can significantly damp the suspen
 On can see in Figure \ref{fig:comp_transmissibility} that the problem of the degradation of the transmissibility at high frequency when using passive damping techniques is overcome by the use of IFF.
 
 The addition of the HPF or the use of the parallel stiffness also permits to limit the degradation of the compliance as compared with classical IFF (Figure \ref{fig:comp_compliance}).
-
 \section{Conclusion}
 \label{sec:conclusion}
 
@@ -575,15 +587,12 @@ It was shown that if the stiffness \(k_p\) of the additional springs is larger t
 While having very different implementations, both proposed modifications are very similar when it comes to the attainable damping and the obtained closed loop system behavior.
 
 Future work will focus on the experimental validation of the proposed IFF modifications.
-
 \section*{Data Availability}
 Matlab \cite{matlab20} was used for this study.
 The source code is available under a MIT License and archived in Zenodo \cite{dehaeze20_activ_dampin_rotat_posit_platf}.
-
 \section*{Acknowledgments}
 This research benefited from a FRIA grant (grant number: FC 31597) from the French Community of Belgium.
 This paper is based on a paper previously presented at the ISMA conference \cite{dehaeze20_activ_dampin_rotat_platf_integ_force_feedb}.
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