Compute figures for effect of geometry

nass-geometry.tex

@@ -1,4 +1,4 @@
-% Created 2025-03-19 Wed 19:08
+% Created 2025-03-30 Sun 18:15
 % Intended LaTeX compiler: pdflatex
 \documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt}
 
@@ -25,41 +25,50 @@
 \clearpage
 
 \begin{itemize}
-\item In the conceptual design phase, the geometry of the Stewart platform was not optimized
+\item In the conceptual design phase, the geometry of the Stewart platform was chosen arbitrarily and not optimized
 \item In the detail design phase, we want to see if the geometry can be optimized to improve the overall performances
 \item Optimization criteria: mobility, stiffness, dynamical decoupling, more performance / bandwidth
 \end{itemize}
 
 Outline:
 \begin{itemize}
-\item Review of Stewart platform: Section \ref{sec:detail_kinematics_stewart_review}
+\item Review of Stewart platform (Section \ref{sec:detail_kinematics_stewart_review})
 Geometry, Actuators, Sensors, Joints
-\item Effect of geometry on the Stewart platform characteristics \ref{sec:detail_kinematics_geometry}
-\item Cubic configuration: benefits? \ref{sec:detail_kinematics_cubic}
+\item Effect of geometry on the Stewart platform characteristics (Section \ref{sec:detail_kinematics_geometry})
+\item Cubic configuration: benefits? (Section \ref{sec:detail_kinematics_cubic})
+\item Obtained geometry for the nano hexapod (Section \ref{sec:detail_kinematics_nano_hexapod})
 \end{itemize}
 
 \chapter{Review of Stewart platforms}
 \label{sec:detail_kinematics_stewart_review}
 \begin{itemize}
-\item as was explained in the conceptual phase, Stewart platform have the following key elements:
+\item As was explained in the conceptual phase, Stewart platform have the following key elements:
 \begin{itemize}
-\item two plates
-\item flexible joints
-\item actuators
-\item sensors
+\item Two plates connected by six struts
+\item Each strut is composed of:
+\begin{itemize}
+\item a flexible joint at each end
+\item an actuator
+\item one or several sensors
 \end{itemize}
-\item the geometry
-\item This results in various designs as shown in Table \ref{tab:detail_kinematics_stewart_review}
-\item The focus is here made on Stewart platforms for nano-positioning of vibration control.
-Not on long stroke stewart platforms.
+\end{itemize}
+\item The exact geometry (i.e. position of joints and orientation of the struts) can be chosen freely depending on the application.
+\item This results in many different designs found in the literature.
+\item The focus is here made on Stewart platforms for nano-positioning and vibration control.
+Long stroke stewart platforms are not considered here as their design impose other challenges.
+Some Stewart platforms found in the literature are listed in Table \ref{tab:detail_kinematics_stewart_review}
 \item All presented Stewart platforms are using flexible joints, as it is a prerequisites for nano-positioning capabilities.
 \item Most of stewart platforms are using voice coil actuators or piezoelectric actuators.
 The actuators used for the Stewart platform will be chosen in the next section.
 \item Depending on the application, various sensors are integrated in the struts or on the plates.
 The choice of sensor for the nano-hexapod will be described in the next section.
-
-\item[{$\square$}] Only keep integrated sensor and not external metrology
-\item[{$\square$}] Check for missing information
+\item There are two categories of Stewart platform geometry:
+\begin{itemize}
+\item Cubic architecture (Figure \ref{fig:detail_kinematics_stewart_examples_cubic}).
+Struts are positioned along 6 sides of a cubic (and are therefore orthogonal to each other).
+Such specific architecture has some special properties that will be studied in Section \ref{sec:detail_kinematics_cubic}.
+\item Non-cubic architecture (Figure \ref{fig:detail_kinematics_stewart_examples_non_cubic})
+\end{itemize}
 \end{itemize}
 
 \begin{figure}[htbp]
@@ -123,134 +132,345 @@ The choice of sensor for the nano-hexapod will be described in the next section.
 \end{figure}
 
 \begin{table}[htbp]
-\caption{\label{tab:detail_kinematics_stewart_review}Examples of Stewart platform developed. When not specifically indicated, sensors are included in the struts. All presented Stewart platforms are using flexible joints. The table is sorted by ``date''}
+\caption{\label{tab:detail_kinematics_stewart_review}Examples of Stewart platform developed. When not specifically indicated, sensors are included in the struts. All presented Stewart platforms are using flexible joints. The table is ordered by appearance in the literature}
 \centering
 \scriptsize
 \begin{tabularx}{\linewidth}{llllX}
 \toprule
  & \textbf{Geometry} & \textbf{Actuators} & \textbf{Sensors} & \textbf{Reference}\\
 \midrule
- & Cubic (6-UPU) & Magnetostrictive & Force (collocated), Accelerometers & \cite{geng93_six_degree_of_freed_activ,geng94_six_degree_of_freed_activ,geng95_intel_contr_system_multip_degree}\\
-Figure \ref{fig:detail_kinematics_jpl} & Cubic & Voice Coil (0.5 mm) & Force (collocated) & \cite{spanos95_soft_activ_vibrat_isolat,rahman98_multiax}\\
+ & Cubic & Magnetostrictive & Force, Accelerometers & \cite{geng93_six_degree_of_freed_activ,geng94_six_degree_of_freed_activ,geng95_intel_contr_system_multip_degree}\\
+Figure \ref{fig:detail_kinematics_jpl} & Cubic & Voice Coil (0.5 mm) & Force & \cite{spanos95_soft_activ_vibrat_isolat,rahman98_multiax}\\
  & Cubic & Voice Coil (10 mm) & Force, LVDT, Geophones & \cite{thayer98_stewar,thayer02_six_axis_vibrat_isolat_system,hauge04_sensor_contr_space_based_six}\\
-Figure \ref{fig:detail_kinematics_uw_gsp} & Cubic (CoM=CoK) & Voice Coil & Force & \cite{mcinroy99_dynam,mcinroy99_precis_fault_toler_point_using_stewar_platf,mcinroy00_desig_contr_flexur_joint_hexap,li01_simul_vibrat_isolat_point_contr,jafari03_orthog_gough_stewar_platf_microm}\\
- & Cubic & Piezoelectric (\(25\,\mu m\)) & Piezo force sensors & \cite{defendini00_techn}\\
-Figure \ref{fig:detail_kinematics_ulb_pz} & Cubic & APA  (\(50\,\mu m\)) & Force sensor & \cite{abu02_stiff_soft_stewar_platf_activ}\\
+Figure \ref{fig:detail_kinematics_uw_gsp} & Cubic & Voice Coil & Force & \cite{mcinroy99_dynam,mcinroy99_precis_fault_toler_point_using_stewar_platf,mcinroy00_desig_contr_flexur_joint_hexap,li01_simul_vibrat_isolat_point_contr,jafari03_orthog_gough_stewar_platf_microm}\\
+ & Cubic & Piezoelectric (\(25\,\mu m\)) & Force & \cite{defendini00_techn}\\
+Figure \ref{fig:detail_kinematics_ulb_pz} & Cubic & APA  (\(50\,\mu m\)) & Force & \cite{abu02_stiff_soft_stewar_platf_activ}\\
 Figure \ref{fig:detail_kinematics_pph} & Non-Cubic & Voice Coil & Accelerometers & \cite{chen03_payload_point_activ_vibrat_isolat}\\
  & Cubic & Voice Coil & Force & \cite{hanieh03_activ_stewar,preumont07_six_axis_singl_stage_activ}\\
-Figure \ref{fig:detail_kinematics_uqp} & Cubic & Piezoelectric (\(50\,\mu m\)) & Geophone aligned with the strut & \cite{agrawal04_algor_activ_vibrat_isolat_spacec}\\
+Figure \ref{fig:detail_kinematics_uqp} & Cubic & Piezoelectric (\(50\,\mu m\)) & Geophone & \cite{agrawal04_algor_activ_vibrat_isolat_spacec}\\
  & Non-Cubic & Piezoelectric (\(16\,\mu m\)) & Eddy Current & \cite{furutani04_nanom_cuttin_machin_using_stewar}\\
- & Cubic & Piezoelectric (\(120\,\mu m\)) & External capacitive & \cite{ting06_desig_stewar_nanos_platf,ting13_compos_contr_desig_stewar_nanos_platf}\\
- & Non-Cubic & Piezoelectric (\(160\,\mu m\)) & External capacitive (LION) & \cite{ting07_measur_calib_stewar_microm_system}\\
-Figure \ref{fig:detail_kinematics_zhang11} & Non-cubic & Magnetostrictive & Inertial & \cite{zhang11_six_dof}\\
- & 6-SPS (Optimized) & Piezoelectric & Strain Gauge & \cite{du14_piezo_actuat_high_precis_flexib}\\
- & Cubic & Voice Coil & Accelerometer in each leg & \cite{chi15_desig_exper_study_vcm_based,tang18_decen_vibrat_contr_voice_coil,jiao18_dynam_model_exper_analy_stewar}\\
- & Cubic & Piezoelectric & Force Sensor + Accelerometer & \cite{wang16_inves_activ_vibrat_isolat_stewar}\\
- & Almost cubic & Voice Coil & Force Sensor + Accelerometer & \cite{beijen18_self_tunin_mimo_distur_feedf,tjepkema12_activ_ph}\\
-Figure \ref{fig:detail_kinematics_yang19} & 6-UPS (Cubic?) & Piezoelectric & Force, Position & \cite{yang19_dynam_model_decoup_contr_flexib}\\
-Figure \ref{fig:detail_kinematics_naves} & Non-Cubic & 3-phase rotary motor & Rotary Encoders & \cite{naves20_desig,naves20_t_flex}\\
+ & Cubic & Piezoelectric (\(120\,\mu m\)) & (External) Capacitive & \cite{ting06_desig_stewar_nanos_platf,ting13_compos_contr_desig_stewar_nanos_platf}\\
+ & Non-Cubic & Piezoelectric (\(160\,\mu m\)) & (External) Capacitive & \cite{ting07_measur_calib_stewar_microm_system}\\
+Figure \ref{fig:detail_kinematics_zhang11} & Non-cubic & Magnetostrictive & Accelerometer & \cite{zhang11_six_dof}\\
+ & Non-Cubic & Piezoelectric & Strain Gauge & \cite{du14_piezo_actuat_high_precis_flexib}\\
+ & Cubic & Voice Coil & Accelerometer & \cite{chi15_desig_exper_study_vcm_based,tang18_decen_vibrat_contr_voice_coil,jiao18_dynam_model_exper_analy_stewar}\\
+ & Cubic & Piezoelectric & Force & \cite{wang16_inves_activ_vibrat_isolat_stewar}\\
+ & Almost cubic & Voice Coil & Force, Accelerometer & \cite{beijen18_self_tunin_mimo_distur_feedf,tjepkema12_activ_ph}\\
+Figure \ref{fig:detail_kinematics_yang19} & Almost cubic & Piezoelectric & Force, Strain gauge & \cite{yang19_dynam_model_decoup_contr_flexib}\\
+Figure \ref{fig:detail_kinematics_naves} & Non-Cubic & 3-phase rotary motor & Rotary Encoder & \cite{naves20_desig,naves20_t_flex}\\
 \bottomrule
 \end{tabularx}
 \end{table}
 
+Conclusion:
 \begin{itemize}
-\item[{$\square$}] \url{https://research.tdehaeze.xyz/stewart-simscape/docs/bibliography.html}
-\item[{$\square$}] Joints and actuators are optimized in the next section
+\item Various Stewart platform designs:
+\begin{itemize}
+\item geometry, sizes, orientation of struts
+\item Lot's have a ``cubic'' architecture that will be discussed in Section \ref{sec:detail_kinematics_cubic}
+\item actuator types
+\item various sensors
+\item flexible joints (discussed in next chapter)
+\end{itemize}
+\item The effect of geometry on the properties of the Stewart platform is studied in section \ref{sec:detail_kinematics_geometry}
+\item It is determined what is the optimal geometry for the NASS
 \end{itemize}
 
 \chapter{Effect of geometry on Stewart platform properties}
 \label{sec:detail_kinematics_geometry}
 \begin{itemize}
-\item Remind that the choice of frames (independently of the physical geometry) impacts the obtained stiffness matrix (as it is defined as forces/motion evaluated at the chosen frame)
-\item Important: bi (join position w.r.t top platform) and si (orientation of struts)
+\item As was shown during the conceptual phase, the geometry of the Stewart platform influences:
+\begin{itemize}
+\item the stiffness and compliance properties
+\item the mobility
+\item the force authority
+\item the dynamics of the manipulator
+\end{itemize}
+\item It is therefore important to understand how the geometry impact these properties, and to be able to optimize the geometry for a specific application.
 \end{itemize}
 
-For the nano-hexapod:
+One important tool to study this is the Jacobian matrix which depends on the \(\bm{b}_i\) (join position w.r.t top platform) and \(\hat{\bm{s}}_i\) (orientation of struts).
+The choice of frames (\(\{A\}\) and \(\{B\}\)), independently of the physical Stewart platform geometry, impacts the obtained kinematics and stiffness matrix, as it is defined for forces and motion evaluated at the chosen frame.
+\section{Platform Mobility}
+The mobility of the Stewart platform (or any manipulator) is here defined as the range of motion that it can perform.
+It corresponds to the set of possible pose (i.e. combined translation and rotation) of frame \{B\} with respect to frame \{A\}.
+It should therefore be represented in a six dimensional space.
+
+As was shown during the conceptual phase, for small displacements, the Jacobian matrix can be used to link the strut motion to the motion of frame B with respect to A through equation \eqref{eq:detail_kinematics_jacobian}.
+\begin{equation}\label{eq:detail_kinematics_jacobian}
+  \begin{bmatrix} \delta l_1 \\ \delta l_2 \\ \delta l_3 \\ \delta l_4 \\ \delta l_5 \\ \delta l_6 \end{bmatrix} = \begin{bmatrix}
+    {{}^A\hat{\bm{s}}_1}^T & ({}^A\bm{b}_1 \times {}^A\hat{\bm{s}}_1)^T \\
+    {{}^A\hat{\bm{s}}_2}^T & ({}^A\bm{b}_2 \times {}^A\hat{\bm{s}}_2)^T \\
+    {{}^A\hat{\bm{s}}_3}^T & ({}^A\bm{b}_3 \times {}^A\hat{\bm{s}}_3)^T \\
+    {{}^A\hat{\bm{s}}_4}^T & ({}^A\bm{b}_4 \times {}^A\hat{\bm{s}}_4)^T \\
+    {{}^A\hat{\bm{s}}_5}^T & ({}^A\bm{b}_5 \times {}^A\hat{\bm{s}}_5)^T \\
+    {{}^A\hat{\bm{s}}_6}^T & ({}^A\bm{b}_6 \times {}^A\hat{\bm{s}}_6)^T
+  \end{bmatrix} \begin{bmatrix} \delta x \\ \delta y \\ \delta z \\ \delta \theta_x \\ \delta \theta_y \\ \delta \theta_z \end{bmatrix}
+\end{equation}
+
+Therefore, the mobility of the Stewart platform (set of \([\delta x\ \delta y\ \delta z\ \delta \theta_x\ \delta \theta_y\ \delta \theta_z]\)) depends on:
 \begin{itemize}
-\item Size requirements: Maximum height, maximum radius
+\item the stroke of each strut
+\item the geometry of the Stewart platform (embodied in the Jacobian matrix)
 \end{itemize}
+
+More specifically:
+\begin{itemize}
+\item the XYZ mobility only depends on the si (orientation of struts)
+\item the mobility in rotation depends on bi (position of top joints)
+\end{itemize}
+
+As will be shown in Section \ref{sec:detail_kinematics_cubic}, there are some geometry that gives same stroke in X, Y and Z directions.
+
+As the mobility is of dimension six, it is difficult to represent.
+Depending on the applications, only the translation mobility or the rotation mobility may be represented.
+
+\paragraph{Mobility in translation}
+
+Here, for simplicity, only translations are first considered:
+\begin{itemize}
+\item Let's consider a general Stewart platform geometry shown in Figure \ref{fig:detail_kinematics_mobility_trans_arch}.
+\item In the general case: the translational mobility can be represented by a 3D shape with 12 faces (each actuator limits the stroke along its orientation in positive and negative directions).
+The faces are therefore perpendicular to the strut direction.
+The obtained mobility is shown in Figure \ref{fig:detail_kinematics_mobility_trans_result}.
+\item Considering an actuator stroke of \(\pm d\), the mobile platform can be translated in any direction with a stroke of \(d\)
+A circle with radius \(d\) can be contained in the general shape.
+It will touch the shape along six lines defined by the strut axes.
+The sphere with radius \(d\) is shown in Figure \ref{fig:detail_kinematics_mobility_trans_result}.
+\item Therefore, for any (small stroke) Stewart platform with actuator stroke \(\pm d\), it is possible to move the top platform in any direction by at least a distance \(d\).
+Note that no platform angular motion is here considered. When combining angular motion, the linear stroke decreases.
+\item When considering some symmetry in the system (as typically the case), the shape becomes a Trigonal trapezohedron whose height and width depends on the orientation of the struts.
+We only get 6 faces as usually the Stewart platform consists of 3 sets of 2 parallels struts.
+\end{itemize}
+
+\begin{figure}[htbp]
+\begin{subfigure}{0.48\textwidth}
+\begin{center}
+\includegraphics[scale=1,scale=1]{figs/detail_kinematics_mobility_trans_arch.png}
+\end{center}
+\subcaption{\label{fig:detail_kinematics_mobility_trans_arch}Stewart platform geometry}
+\end{subfigure}
+\begin{subfigure}{0.48\textwidth}
+\begin{center}
+\includegraphics[scale=1,scale=1]{figs/detail_kinematics_mobility_trans_result.png}
+\end{center}
+\subcaption{\label{fig:detail_kinematics_mobility_trans_result}Translational mobility}
+\end{subfigure}
+\caption{\label{fig:detail_kinematics_mobility_trans}Example of one Stewart platform (\subref{fig:detail_kinematics_mobility_trans_arch}) and associated translational mobility (\subref{fig:detail_kinematics_mobility_trans_result})}
+\end{figure}
+
+To better understand how the geometry of the Stewart platform impacts the translational mobility, two configurations are compared:
+\begin{itemize}
+\item Struts oriented horizontally (Figure \ref{fig:detail_kinematics_stewart_mobility_vert_struts}) => more stroke in horizontal direction
+\item Struts oriented vertically (Figure \ref{fig:detail_kinematics_stewart_mobility_hori_struts}) => more stroke in vertical direction
+\item Corresponding mobility shown in Figure \ref{fig:detail_kinematics_mobility_translation_strut_orientation}
+\end{itemize}
+
+\begin{figure}[htbp]
+\begin{subfigure}{0.48\textwidth}
+\begin{center}
+\includegraphics[scale=1,scale=1]{figs/detail_kinematics_stewart_mobility_vert_struts.png}
+\end{center}
+\subcaption{\label{fig:detail_kinematics_stewart_mobility_vert_struts}Struts oriented vertically}
+\end{subfigure}
+\begin{subfigure}{0.48\textwidth}
+\begin{center}
+\includegraphics[scale=1,scale=1]{figs/detail_kinematics_stewart_mobility_hori_struts.png}
+\end{center}
+\subcaption{\label{fig:detail_kinematics_stewart_mobility_hori_struts}Struts oriented horizontally}
+\end{subfigure}
+
+\bigskip
+\begin{subfigure}{0.95\textwidth}
+\begin{center}
+\includegraphics[scale=1,scale=1]{figs/detail_kinematics_mobility_translation_strut_orientation.png}
+\end{center}
+\subcaption{\label{fig:detail_kinematics_mobility_translation_strut_orientation}Translational mobility of the two configurations}
+\end{subfigure}
+\caption{\label{fig:detail_kinematics_stewart_mobility_examples}Effect of strut orientation on the obtained mobility in translation. Two Stewart platform geometry are considered: struts oriented vertically (\subref{fig:detail_kinematics_stewart_mobility_vert_struts}) and struts oriented vertically (\subref{fig:detail_kinematics_stewart_mobility_hori_struts}). Obtained mobility for both geometry are shown in (\subref{fig:detail_kinematics_mobility_translation_strut_orientation}).}
+\end{figure}
+
+\paragraph{Mobility in rotation}
+
+As shown by equation \eqref{eq:detail_kinematics_jacobian}, the rotational mobility depends both on the orientation of the struts and on the location of the top joints.
+
+Similarly to the translational case, to increase the rotational mobility in one direction, it is advantageous to have the struts more perpendicular to the rotational direction.
+
+For instance, having the struts more vertical (Figure \ref{fig:detail_kinematics_stewart_mobility_vert_struts}) gives less rotational stroke along the vertical direction than having the struts oriented more horizontally (Figure \ref{fig:detail_kinematics_stewart_mobility_hori_struts}).
+
+Two cases are considered with same strut orientation but with different top joints positions:
+\begin{itemize}
+\item struts close to each other (Figure \ref{fig:detail_kinematics_stewart_mobility_close_struts})
+\item struts further apart (Figure \ref{fig:detail_kinematics_stewart_mobility_space_struts})
+\end{itemize}
+
+The mobility for pure rotations are compared in Figure \ref{fig:detail_kinematics_mobility_angle_strut_distance}.
+Note that the same strut stroke are considered in both cases to evaluate the mobility.
+Having struts further apart decreases the ``level arm'' and therefore the rotational mobility is reduced.
+
+For rotations and translations, having more mobility also means increasing the effect of actuator noise on the considering degree of freedom.
+Somehow, the level arm is increased, so any strut vibration gets amplified.
+Therefore, the designed Stewart platform should just have the necessary mobility.
+
+\begin{figure}[htbp]
+\begin{subfigure}{0.48\textwidth}
+\begin{center}
+\includegraphics[scale=1,scale=1]{figs/detail_kinematics_stewart_mobility_close_struts.png}
+\end{center}
+\subcaption{\label{fig:detail_kinematics_stewart_mobility_close_struts}Struts oriented closeically}
+\end{subfigure}
+\begin{subfigure}{0.48\textwidth}
+\begin{center}
+\includegraphics[scale=1,scale=1]{figs/detail_kinematics_stewart_mobility_space_struts.png}
+\end{center}
+\subcaption{\label{fig:detail_kinematics_stewart_mobility_space_struts}Struts oriented spacezontally}
+\end{subfigure}
+
+\bigskip
+\begin{subfigure}{0.95\textwidth}
+\begin{center}
+\includegraphics[scale=1,scale=1]{figs/detail_kinematics_mobility_angle_strut_distance.png}
+\end{center}
+\subcaption{\label{fig:detail_kinematics_mobility_angle_strut_distance}Translational mobility of the two configurations}
+\end{subfigure}
+\caption{\label{fig:detail_kinematics_stewart_mobility_examples}Effect of strut position on the obtained mobility in rotation. Two Stewart platform geometry are considered: struts close to each other (\subref{fig:detail_kinematics_stewart_mobility_close_struts}) and struts further appart (\subref{fig:detail_kinematics_stewart_mobility_space_struts}). Obtained mobility for both geometry are shown in (\subref{fig:detail_kinematics_mobility_angle_strut_distance}).}
+\end{figure}
+
+\paragraph{Combined translations and rotations}
+
+It is possible to consider combined translations and rotations.
+Displaying such mobility is more complex.
+It will be used for the nano-hexapod to verify that the obtained design has the necessary mobility.
+
+For a fixed geometry and a wanted mobility (combined translations and rotations), it is possible to estimate the required minimum actuator stroke.
+It will be done in Section \ref{sec:detail_kinematics_nano_hexapod} to estimate the required actuator stroke for the nano-hexapod geometry.
+
 \section{Stiffness}
-
+Stiffness matrix:
 \begin{itemize}
-\item Give some examples:
-\begin{itemize}
-\item struts further apart: higher angular stiffness, same linear stiffness
-\item orientation: more vertical => increase vertical stiffness, decrease horizontal stiffness
-\end{itemize}
+\item defines how the nano-hexapod deforms (frame \(\{B\}\) with respect to frame \(\{A\}\)) due to static forces/torques applied on \(\{B\}\).
+\item Depends on the Jacobian matrix (i.e. the geometry) and the strut axial stiffness \eqref{eq:detail_kinematics_stiffness_matrix}
+\item Contribution of joints stiffness is here not considered \cite{mcinroy00_desig_contr_flexur_joint_hexap,mcinroy02_model_desig_flexur_joint_stewar}
 \end{itemize}
 
-\section{Mobility and required joint and actuator stroke}
+\begin{equation}\label{eq:detail_kinematics_stiffness_matrix}
+  \bm{K} = \bm{J}^T \bm{\mathcal{K}} \bm{J}
+\end{equation}
+
+It is assumed that the stiffness of all strut is the same: \(\bm{\mathcal{K}} = k \cdot \mathbf{I}_6\).
+Obtained stiffness matrix linearly depends on the strut stiffness \(k\) \eqref{eq:detail_kinematics_stiffness_matrix_simplified}.
+
+\begin{equation}\label{eq:detail_kinematics_stiffness_matrix_simplified}
+  \bm{K} = k \bm{J}^T \bm{J} =
+  k \left[
+  \begin{array}{c|c}
+      \Sigma_{i = 0}^{6} \hat{\bm{s}}_i \cdot \hat{\bm{s}}_i^T & \Sigma_{i = 0}^{6} \bm{\hat{s}}_i \cdot ({}^A\bm{b}_i \times {}^A\hat{\bm{s}}_i)^T \\
+      \hline
+      \Sigma_{i = 0}^{6} ({}^A\bm{b}_i \times {}^A\hat{\bm{s}}_i) \cdot \hat{\bm{s}}_i^T & \Sigma_{i = 0}^{6} ({}^A\bm{b}_i \times {}^A\hat{\bm{s}}_i) \cdot ({}^A\bm{b}_i \times {}^A\hat{\bm{s}}_i)^T\\
+  \end{array}
+  \right]
+\end{equation}
+
+\paragraph{Translation Stiffness}
+
+XYZ stiffnesses:
+\begin{itemize}
+\item Only depends on the orientation of the struts and not their location: \(\hat{\bm{s}}_i \cdot \hat{\bm{s}}_i^T\)
+\item Extreme case: all struts are vertical \(s_i = [0,\ 0,\ 1]\) => vertical stiffness of \(6 k\), but null stiffness in X and Y directions
+\item If two struts along X, two struts along Y, and two struts along Z => \(\hat{\bm{s}}_i \cdot \hat{\bm{s}}_i^T = 2 \bm{I}_3\)
+Stiffness is well distributed along directions.
+This corresponds to the cubic architecture.
+\end{itemize}
+
+If struts more vertical (Figure \ref{fig:detail_kinematics_stewart_mobility_vert_struts}):
+\begin{itemize}
+\item increase vertical stiffness
+\item decrease horizontal stiffness
+\item increase Rx,Ry stiffness
+\item decrease Rz stiffness
+\end{itemize}
+
+Opposite conclusions if struts are not horizontal (Figure \ref{fig:detail_kinematics_stewart_mobility_hori_struts}).
+
+\paragraph{Rotational Stiffness}
+
+Rotational stiffnesses:
+\begin{itemize}
+\item Same orientation but increased distances (bi) by a factor 2 => rotational stiffness increased by factor 4
+Figure \ref{fig:detail_kinematics_stewart_mobility_close_struts}
+Figure \ref{fig:detail_kinematics_stewart_mobility_space_struts}
+\end{itemize}
+
+Struts further apart:
+\begin{itemize}
+\item no change to XYZ
+\item increase in rotational stiffness (by the square of the distance)
+\end{itemize}
+
+\paragraph{Diagonal Stiffness Matrix}
+
+
+\section{Dynamics ?}
+
+Dynamical equations (both in the cartesian frame and in the frame of the struts) for the Stewart platform were derived during the conceptual phase with simplifying assumptions (massless struts and perfect joints).
+The dynamics depends both on the geometry (Jacobian matrix) but also on the payload being placed on top of the platform.
+
+Under very specific conditions, the equations of motion can be decoupled in the Cartesian space.
+These are studied in Section \ref{ssec:detail_kinematics_cubic_dynamic}.
+
+\begin{equation}\label{eq:nhexa_transfer_function_cart}
+    \frac{{\mathcal{X}}}{\bm{\mathcal{F}}}(s) = ( \bm{M} s^2 + \bm{J}^{T} \bm{\mathcal{C}} \bm{J} s + \bm{J}^{T} \bm{\mathcal{K}} \bm{J} )^{-1}
+\end{equation}
+
+In the frame of the struts, the equations of motion are well decoupled at low frequency.
+This is why most of Stewart platforms are controlled in the frame of the struts: bellow the resonance frequency, the system is decoupled and SISO control may be applied for each strut.
+
+\begin{equation}\label{eq:nhexa_transfer_function_struts}
+    \frac{\bm{\mathcal{L}}}{\bm{f}}(s) = ( \bm{J}^{-T} \bm{M} \bm{J}^{-1} s^2 + \bm{\mathcal{C}} + \bm{\mathcal{K}} )^{-1}
+\end{equation}
+
+For the NASS, the payloads can have various inertia, with masses ranging from 1 to 50kg.
+It is therefore not possible to have one geometry that gives good dynamical properties for all the payloads.
+
+
+Coupling between force sensors in different struts may also be important.
 
 \begin{itemize}
-\item Comparison of the XYZ mobility (fixed orientation) for two geometry (or maybe only in the XY or YZ plane to see more clearly the differences)
-
-\item[{$\square$}] \href{file:///home/thomas/Cloud/work-projects/ID31-NASS/matlab/stewart-simscape/org/kinematic-study.org}{Estimated required actuator stroke from specified platform mobility}
-Will be useful to choose the actuators
-\item[{$\square$}] \href{file:///home/thomas/Cloud/work-projects/ID31-NASS/matlab/stewart-simscape/org/kinematic-study.org}{Estimation of the Joint required Stroke}
-Will be useful to design the flexible joints
+\item[{$\square$}] Maybe study that for the cubic architecture, and then say that except for very specific conditions, coupling is similar for different geometries
 \end{itemize}
 
 \section*{Conclusion}
-\begin{itemize}
-\item[{$\square$}] Table that summarize the findings
-\href{file:///home/thomas/Cloud/work-projects/ID31-NASS/documents/state-of-thesis-2020/index.org}{Optimal Nano-Hexapod Geometry}
-\end{itemize}
+The effects of two changes in the manipulator's geometry, namely the position and orientation of the legs, are summarized in Table \ref{tab:detail_kinematics_geometry}.
+These results could have been easily deduced based on some mechanical principles, but thanks to the kinematic analysis, they can be quantified.
 
-\chapter{The Cubic Architecture}
-\label{sec:detail_kinematics_cubic}
-Cubic configuration \url{file:///home/thomas/Cloud/work-projects/ID31-NASS/matlab/stewart-simscape/org/cubic-configuration.org}
-\section{The Cubic Architecture}
+These trade-offs give some guidelines when choosing the Stewart platform geometry.
 
-From \cite{geng94_six_degree_of_freed_activ}, 7 properties of cubic configuration:
-\begin{enumerate}
-\item Uniformity in control capability in all directions
-\item Uniformity in stiffness in all directions
-\item Minimum cross coupling force effect among actuators
-\item Facilitate collocated sensor-actuator control system design
-\item Simple kinematics relationships
-\item Simple dynamic analysis
-\item Simple mechanical design
-\end{enumerate}
+\begin{table}[htbp]
+\caption{\label{tab:detail_kinematics_geometry}Effect of a change in geometry on the manipulator's stiffness, force authority and stroke}
+\centering
+\begin{tabularx}{\linewidth}{lXX}
+\toprule
+ & \textbf{legs pointing more vertically} & \textbf{legs further apart}\\
+\midrule
+Vertical stiffness & \(\nearrow\) & \(=\)\\
+Horizontal stiffness & \(\searrow\) & \(=\)\\
+Vertical rotation stiffness & \(\searrow\) & \(\nearrow\)\\
+Horizontal rotation stiffness & \(\nearrow\) & \(\nearrow\)\\
+\midrule
+Vertical force authority & \(\nearrow\) & \(=\)\\
+Horizontal force authority & \(\searrow\) & \(=\)\\
+Vertical torque authority & \(\searrow\) & \(\nearrow\)\\
+Horizontal torque authority & \(\nearrow\) & \(\nearrow\)\\
+\midrule
+Vertical stroke & \(\searrow\) & \(=\)\\
+Horizontal stroke & \(\nearrow\) & \(=\)\\
+Vertical rotation stroke & \(\nearrow\) & \(\searrow\)\\
+Horizontal rotation stroke & \(\searrow\) & \(\searrow\)\\
+\bottomrule
+\end{tabularx}
+\end{table}
 
 
-
-\begin{itemize}
-\item Principle
-\item Examples of Stewart platform with Cubic architecture
-\item Different options?
-Center of the cube above the top platform?
-Where to mention that ? With examples
-\end{itemize}
-
-
-
-\section{Static Properties}
-
-Explain that we get diagonal K matrix => static decoupling in the cartesian frame.
-Uniform mobility in X,Y,Z directions
-
-\section{Dynamical Properties?}
-
-\cite{mcinroy00_desig_contr_flexur_joint_hexap}
-
-\cite{afzali-far16_vibrat_dynam_isotr_hexap_analy_studies}:
-\begin{itemize}
-\item proposes an architecture where the CoM can be above the top platform
-\item ``\textbf{Dynamic isotropy}, leading to equal eigenfrequencies, is a powerful optimization measure.''
-\end{itemize}
-
-
-
-\begin{itemize}
-\item Show examples where the dynamics can indeed be decoupled in the cartesian frame (i.e. decoupled K and M matrices)
-\item Better decoupling between the struts? not sure\ldots{}
-Compute the coupling between the struts for a cubic and non-cubic architecture
-\item Same resonance frequencies for suspension modes?
-Maybe in one case: sphere at the CoM?
-Could be nice to show that.
-Say that this can be nice for optimal damping for instance (link to paper explaining that)
-\end{itemize}
-
 \chapter{Conclusion}
 \label{sec:detail_kinematics_conclusion}
 

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+\usepackage{amssymb}
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