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phd-thesis.tex
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% Created 2025-04-21 Mon 23:35
% Created 2025-04-22 Tue 16:24
% Intended LaTeX compiler: pdflatex
\documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt}
@@ -41,7 +41,7 @@
\addbibresource{ref.bib}
\addbibresource{phd-thesis.bib}
\author{Dehaeze Thomas}
\date{2025-04-21}
\date{2025-04-22}
\title{Nano Active Stabilization of samples for tomography experiments: A mechatronic design approach}
\subtitle{PhD Thesis}
\hypersetup{
@@ -669,7 +669,7 @@ During conceptual design, it was found that the guaranteed stability property of
To address this instability issue, two modifications to the classical IFF control scheme were proposed and analyzed.
The first involves a minor adjustment to the control law itself, while the second incorporates physical springs in parallel with the force sensors.
Stability conditions and optimal parameter tuning guidelines were derived for both modified schemes.
This is further discussed in Section~\ref{sec:rotating} and was the subject of publications ~\cite{dehaeze20_activ_dampin_rotat_platf_integ_force_feedb,dehaeze21_activ_dampin_rotat_platf_using}.
This is further discussed in Section~\ref{sec:rotating} and was the subject of a publication~\cite{dehaeze21_activ_dampin_rotat_platf_using}.
\paragraph{Design of complementary filters using \(\mathcal{H}_\infty\) Synthesis}
For implementing sensor fusion, where signals from multiple sensors are combined, complementary filters are often employed.
@@ -4666,7 +4666,7 @@ Through coordinate transformation using the Jacobian matrix, the dynamics in the
Although this simplified model provides useful insights, real Stewart platforms exhibit more complex behaviors.
Several factors can significantly increase the model complexity, such as:
\begin{itemize}
\item Strut dynamics, including mass distribution and internal resonances~\cite{afzali-far16_inert_matrix_hexap_strut_joint_space,chen04_decoup_contr_flexur_joint_hexap}
\item Strut dynamics, including mass distribution and internal resonances~\cite{chen04_decoup_contr_flexur_joint_hexap}
\item Joint compliance and friction effects~\cite{mcinroy00_desig_contr_flexur_joint_hexap,mcinroy02_model_desig_flexur_joint_stewar}
\item Supporting structure dynamics and payload dynamics, which are both very critical for NASS
\end{itemize}
@@ -5903,7 +5903,7 @@ This analysis is conducted in Section~\ref{sec:detail_kinematics_nano_hexapod} t
The stiffness matrix defines how the top platform of the Stewart platform (i.e. frame \(\{B\}\)) deforms with respect to its fixed base (i.e. frame \(\{A\}\)) due to static forces/torques applied between frames \(\{A\}\) and \(\{B\}\).
It depends on the Jacobian matrix (i.e., the geometry) and the strut axial stiffness as shown in equation~\eqref{eq:detail_kinematics_stiffness_matrix}.
The contribution of joints stiffness is not considered here, as the joints were optimized after the geometry was fixed.
However, theoretical frameworks for evaluating flexible joint contribution to the stiffness matrix have been established in the literature ~\cite{mcinroy00_desig_contr_flexur_joint_hexap,mcinroy02_model_desig_flexur_joint_stewar}.
However, theoretical frameworks for evaluating flexible joint contribution to the stiffness matrix have been established in the literature~\cite{mcinroy00_desig_contr_flexur_joint_hexap,mcinroy02_model_desig_flexur_joint_stewar}.
\begin{equation}\label{eq:detail_kinematics_stiffness_matrix}
\bm{K} = \bm{J}^{\intercal} \bm{\mathcal{K}} \bm{J}
@@ -7372,7 +7372,7 @@ One way to overcome these limitations is to combine several sensors using a tech
Fortunately, a wide variety of sensors exists, each with different characteristics.
By carefully selecting the sensors to be fused, a ``super sensor'' is obtained that combines the benefits of the individual sensors.
In some applications, sensor fusion is employed to increase measurement bandwidth~\cite{shaw90_bandw_enhan_posit_measur_using_measur_accel,zimmermann92_high_bandw_orien_measur_contr,min15_compl_filter_desig_angle_estim}.
In some applications, sensor fusion is employed to increase measurement bandwidth~\cite{shaw90_bandw_enhan_posit_measur_using_measur_accel,zimmermann92_high_bandw_orien_measur_contr}.
For instance, in~\cite{shaw90_bandw_enhan_posit_measur_using_measur_accel}, the bandwidth of a position sensor is extended by fusing it with an accelerometer that provides high-frequency motion information.
In other applications, sensor fusion is used to obtain an estimate of the measured quantity with reduced noise~\cite{hua05_low_ligo,hua04_polyp_fir_compl_filter_contr_system,plummer06_optim_compl_filter_their_applic_motion_measur,robert12_introd_random_signal_applied_kalman}.
More recently, the fusion of sensors measuring different physical quantities has been proposed to enhance control properties~\cite{collette15_sensor_fusion_method_high_perfor,yong16_high_speed_vertic_posit_stage}.
@@ -7380,12 +7380,12 @@ In~\cite{collette15_sensor_fusion_method_high_perfor}, an inertial sensor used f
Beyond Stewart platforms, practical applications of sensor fusion are numerous.
It is widely implemented for attitude estimation in autonomous vehicles such as unmanned aerial vehicles~\cite{baerveldt97_low_cost_low_weigh_attit,corke04_inert_visual_sensin_system_small_auton_helic,jensen13_basic_uas} and underwater vehicles~\cite{pascoal99_navig_system_desig_using_time,batista10_optim_posit_veloc_navig_filter_auton_vehic}.
Sensor fusion offers significant benefits for high-performance positioning control as demonstrated in~\cite{shaw90_bandw_enhan_posit_measur_using_measur_accel,zimmermann92_high_bandw_orien_measur_contr,min15_compl_filter_desig_angle_estim,yong16_high_speed_vertic_posit_stage}.
Sensor fusion offers significant benefits for high-performance positioning control as demonstrated in~\cite{shaw90_bandw_enhan_posit_measur_using_measur_accel,zimmermann92_high_bandw_orien_measur_contr,yong16_high_speed_vertic_posit_stage}.
It has also been identified as a key technology for improving the performance of active vibration isolation systems~\cite{tjepkema12_sensor_fusion_activ_vibrat_isolat_precis_equip}.
Emblematic examples include the isolation stages of gravitational wave detectors~\cite{collette15_sensor_fusion_method_high_perfor,heijningen18_low} such as those employed at LIGO~\cite{hua05_low_ligo,hua04_polyp_fir_compl_filter_contr_system} and Virgo~\cite{lucia18_low_frequen_optim_perfor_advan}.
Two principal methods are employed to perform sensor fusion: using complementary filters~\cite{anderson53_instr_approac_system_steer_comput} or using Kalman filtering~\cite{brown72_integ_navig_system_kalman_filter}.
For sensor fusion applications, these methods share many relationships~\cite{brown72_integ_navig_system_kalman_filter,higgins75_compar_compl_kalman_filter,robert12_introd_random_signal_applied_kalman,fonseca15_compl}.
For sensor fusion applications, these methods share many relationships~\cite{brown72_integ_navig_system_kalman_filter,higgins75_compar_compl_kalman_filter,robert12_introd_random_signal_applied_kalman,carreira15_compl_filter_desig_three_frequen_bands}.
However, Kalman filtering requires assumptions about the probabilistic characteristics of sensor noise~\cite{robert12_introd_random_signal_applied_kalman}, whereas complementary filters do not impose such requirements.
Furthermore, complementary filters offer advantages over Kalman filtering for sensor fusion through their general applicability, low computational cost~\cite{higgins75_compar_compl_kalman_filter}, and intuitive nature, as their effects can be readily interpreted in the frequency domain.
@@ -7395,7 +7395,7 @@ While analog complementary filters remain in use today~\cite{yong16_high_speed_v
Various design methods have been developed to optimize complementary filters.
The most straightforward approach is based on analytical formulas, which depending on the application may be first order~\cite{corke04_inert_visual_sensin_system_small_auton_helic,yeh05_model_contr_hydraul_actuat_two,yong16_high_speed_vertic_posit_stage}, second order~\cite{baerveldt97_low_cost_low_weigh_attit,stoten01_fusion_kinet_data_using_compos_filter,jensen13_basic_uas}, or higher orders~\cite{shaw90_bandw_enhan_posit_measur_using_measur_accel,zimmermann92_high_bandw_orien_measur_contr,stoten01_fusion_kinet_data_using_compos_filter,collette15_sensor_fusion_method_high_perfor,matichard15_seism_isolat_advan_ligo}.
Since the characteristics of the super sensor depend on proper complementary filter design~\cite{dehaeze19_compl_filter_shapin_using_synth}, several optimization techniques have emerged—ranging from optimizing parameters for analytical formulas~\cite{jensen13_basic_uas,min15_compl_filter_desig_angle_estim,fonseca15_compl} to employing convex optimization tools~\cite{hua04_polyp_fir_compl_filter_contr_system,hua05_low_ligo} such as linear matrix inequalities~\cite{pascoal99_navig_system_desig_using_time}.
Since the characteristics of the super sensor depend on proper complementary filter design~\cite{dehaeze19_compl_filter_shapin_using_synth}, several optimization techniques have emerged—ranging from optimizing parameters for analytical formulas~\cite{jensen13_basic_uas,carreira15_compl_filter_desig_three_frequen_bands} to employing convex optimization tools~\cite{hua04_polyp_fir_compl_filter_contr_system,hua05_low_ligo} such as linear matrix inequalities~\cite{pascoal99_navig_system_desig_using_time}.
As demonstrated in~\cite{plummer06_optim_compl_filter_their_applic_motion_measur}, complementary filter design can be linked to the standard mixed-sensitivity control problem, allowing powerful classical control theory tools to be applied.
For example, in~\cite{jensen13_basic_uas}, two gains of a Proportional Integral (PI) controller are optimized to minimize super sensor noise.
@@ -7693,14 +7693,14 @@ This straightforward example demonstrates that the proposed methodology for shap
\subsubsection{Synthesis of a set of three complementary filters}
\label{ssec:detail_control_sensor_hinf_three_comp_filters}
Certain applications necessitate the fusion of more than two sensors~\cite{stoten01_fusion_kinet_data_using_compos_filter,fonseca15_compl}.
Certain applications necessitate the fusion of more than two sensors~\cite{stoten01_fusion_kinet_data_using_compos_filter,carreira15_compl_filter_desig_three_frequen_bands}.
At LIGO, for example, a super sensor is formed by merging three distinct sensors: an LVDT, a seismometer, and a geophone~\cite{matichard15_seism_isolat_advan_ligo}.
For merging \(n>2\) sensors with complementary filters, two architectural approaches are possible, as illustrated in Figure~\ref{fig:detail_control_sensor_fusion_three}.
Fusion can be implemented either ``sequentially,'' using \(n-1\) sets of two complementary filters (Figure~\ref{fig:detail_control_sensor_fusion_three_sequential}), or ``in parallel,'' employing a single set of \(n\) complementary filters (Figure~\ref{fig:detail_control_sensor_fusion_three_parallel}).
While conventional sensor fusion synthesis techniques can be applied to the sequential approach, parallel architecture implementation requires a novel synthesis method for multiple complementary filters.
Previous literature has offered only simple analytical formulas for this purpose~\cite{stoten01_fusion_kinet_data_using_compos_filter,fonseca15_compl}.
Previous literature has offered only simple analytical formulas for this purpose~\cite{stoten01_fusion_kinet_data_using_compos_filter,carreira15_compl_filter_desig_three_frequen_bands}.
This section presents a generalization of the proposed complementary filter synthesis method to address this gap.
\begin{figure}[htbp]
@@ -10038,7 +10038,7 @@ The transfer function from the ``damped'' plant input \(u\prime\) to the encoder
\caption{\label{fig:test_apa_iff_schematic}Implementation of Integral Force Feedback in the Speedgoat. The damped plant has a new input \(u\prime\)}
\end{figure}
The identified dynamics were then fitted by second order transfer functions\footnote{The transfer function fitting was computed using the \texttt{vectfit3} routine, see ~\cite{gustavsen99_ration_approx_frequen_domain_respon}}.
The identified dynamics were then fitted by second order transfer functions\footnote{The transfer function fitting was computed using the \texttt{vectfit3} routine, see~\cite{gustavsen99_ration_approx_frequen_domain_respon}}.
A comparison between the identified damped dynamics and the fitted second-order transfer functions is shown in Figure~\ref{fig:test_apa_identified_damped_plants} for different gains \(g\).
It is clear that a large amount of damping is added when the gain is increased and that the frequency of the pole is shifted to lower frequencies.