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Thomas Dehaeze 2025-04-03 10:35:31 +02:00
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figs/detail_kinematics_cubic_architecture_example.pdf
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matlab/detail_kinematics_2_cubic.m
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@ -1,3 +1,20 @@
%% Clear Workspace and Close figures
clear; close all; clc;
%% Intialize Laplace variable
s = zpk('s');
%% Path for functions, data and scripts
addpath('./mat/'); % Path for data
addpath('./src/'); % Path for functions
addpath('./subsystems/'); % Path for Subsystems Simulink files
% Simulink Model name
mdl = 'nano_hexapod_model';
%% Colors for the figures
colors = colororder;
%% Example of a typical "cubic" architecture %% Example of a typical "cubic" architecture
MO_B = -50e-3; % Position {B} with respect to {M} [m] MO_B = -50e-3; % Position {B} with respect to {M} [m]
@ -20,7 +37,8 @@ stewart = computeJacobian(stewart);
stewart = initializeStewartPose(stewart); stewart = initializeStewartPose(stewart);
displayArchitecture(stewart, 'labels', false, 'frames', false); displayArchitecture(stewart, 'labels', false, 'frames', false);
plotCube(stewart, 'Hc', Hc, 'FOc', FOc, 'color', [0,0,0,0.5], 'link_to_struts', true); plotCube(stewart, 'Hc', Hc, 'FOc', FOc, 'color', [0,0,0,0.2], 'link_to_struts', false);
view([40, 5]);
%% Example of a typical "cubic" architecture %% Example of a typical "cubic" architecture
MO_B = -20e-3; % Position {B} with respect to {M} [m] MO_B = -20e-3; % Position {B} with respect to {M} [m]
@ -41,24 +59,8 @@ stewart = computeJacobian(stewart);
stewart = initializeCylindricalPlatforms(stewart, 'Fpr', 150e-3, 'Mpr', 150e-3); stewart = initializeCylindricalPlatforms(stewart, 'Fpr', 150e-3, 'Mpr', 150e-3);
displayArchitecture(stewart, 'labels', false, 'frames', false); displayArchitecture(stewart, 'labels', false, 'frames', false);
plotCube(stewart, 'Hc', Hc, 'FOc', FOc, 'color', [0,0,0,0.5], 'link_to_struts', true); plotCube(stewart, 'Hc', Hc, 'FOc', FOc, 'color', [0,0,0,0.2], 'link_to_struts', false);
view([40, 5]);
%% Clear Workspace and Close figures
clear; close all; clc;
%% Intialize Laplace variable
s = zpk('s');
%% Path for functions, data and scripts
addpath('./mat/'); % Path for data
addpath('./src/'); % Path for functions
addpath('./subsystems/'); % Path for Subsystems Simulink files
% Simulink Model name
mdl = 'nano_hexapod_model';
%% Colors for the figures
colors = colororder;
%% Analytical formula for Stiffness matrix of the Cubic Stewart platform %% Analytical formula for Stiffness matrix of the Cubic Stewart platform
% Define symbolic variables % Define symbolic variables

2
matlab/detail_kinematics_3_nano_hexapod.m
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@ -34,7 +34,7 @@ nano_hexapod = computeJacobian(nano_hexapod);
nano_hexapod = initializeCylindricalPlatforms(nano_hexapod, 'Fpr', 125e-3, 'Mpr', 115e-3); nano_hexapod = initializeCylindricalPlatforms(nano_hexapod, 'Fpr', 125e-3, 'Mpr', 115e-3);
%% Obtained architecture for the Nano Hexapod %% Obtained architecture for the Nano Hexapod
displayArchitecture(nano_hexapod, 'labels', true, 'frames', false); displayArchitecture(nano_hexapod, 'labels', true, 'frames', true);
% Bottom circle % Bottom circle
h = 15e-3; h = 15e-3;
r = 120e-3; r = 120e-3;

265
nass-geometry.org
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@ -961,13 +961,11 @@ In this chapter, the nano-hexapod geometry is optimized through careful analysis
The chapter begins with a comprehensive review of existing Stewart platform designs in Section ref:sec:detail_kinematics_stewart_review, surveying various approaches to geometry, actuation, sensing, and joint design from the literature. The chapter begins with a comprehensive review of existing Stewart platform designs in Section ref:sec:detail_kinematics_stewart_review, surveying various approaches to geometry, actuation, sensing, and joint design from the literature.
Section ref:sec:detail_kinematics_geometry develops the analytical framework that connects geometric parameters to performance characteristics, establishing quantitative relationships that guide the optimization process. Section ref:sec:detail_kinematics_geometry develops the analytical framework that connects geometric parameters to performance characteristics, establishing quantitative relationships that guide the optimization process.
Section ref:sec:detail_kinematics_cubic examines the cubic configuration a specialized architecture that has garnered significant attention for its purported advantages—to evaluate its suitability for the nano-hexapod application. Section ref:sec:detail_kinematics_cubic examines the cubic configuration, a specialized architecture that has gathered significant attention, to evaluate its suitability for the nano-hexapod application.
Finally, Section ref:sec:detail_kinematics_nano_hexapod presents the optimized nano-hexapod geometry derived from these analyses and demonstrates how it addresses the specific requirements of the NASS. Finally, Section ref:sec:detail_kinematics_nano_hexapod presents the optimized nano-hexapod geometry derived from these analyses and demonstrates how it addresses the specific requirements of the NASS.
* Review of Stewart platforms * Review of Stewart platforms
<<sec:detail_kinematics_stewart_review>> <<sec:detail_kinematics_stewart_review>>
** Introduction :ignore:
As was explained in the conceptual phase, Stewart platforms have the following key elements: two plates connected by six struts, with each strut composed of a joint at each end, an actuator, and one or several sensors. As was explained in the conceptual phase, Stewart platforms have the following key elements: two plates connected by six struts, with each strut composed of a joint at each end, an actuator, and one or several sensors.
# Section ref:sec:nhexa_stewart_platform # Section ref:sec:nhexa_stewart_platform
@ -977,8 +975,8 @@ Long stroke Stewart platforms are not considered here as their design imposes ot
Some Stewart platforms found in the literature are listed in Table ref:tab:detail_kinematics_stewart_review. Some Stewart platforms found in the literature are listed in Table ref:tab:detail_kinematics_stewart_review.
#+name: tab:detail_kinematics_stewart_review #+name: tab:detail_kinematics_stewart_review
#+caption: 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 #+caption: 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 order of appearance in the literature
#+attr_latex: :environment tabularx :width 0.8\linewidth :align Xcccc #+attr_latex: :environment tabularx :width 0.9\linewidth :align Xcccc
#+attr_latex: :center t :booktabs t :font \scriptsize #+attr_latex: :center t :booktabs t :font \scriptsize
| | *Geometry* | *Actuators* | *Sensors* | *Reference* | | | *Geometry* | *Actuators* | *Sensors* | *Reference* |
|------------------------------------------+--------------+------------------------------+------------------------+---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| |------------------------------------------+--------------+------------------------------+------------------------+---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|
@ -1002,7 +1000,6 @@ Some Stewart platforms found in the literature are listed in Table ref:tab:detai
| Figure ref:fig:detail_kinematics_yang19 | Almost cubic | Piezoelectric | Force, Strain gauge | [[cite:&yang19_dynam_model_decoup_contr_flexib]] | | 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]] | | Figure ref:fig:detail_kinematics_naves | Non-Cubic | 3-phase rotary motor | Rotary Encoder | [[cite:&naves20_desig;&naves20_t_flex]] |
All presented Stewart platforms utilize flexible joints, as this is a prerequisite for nano-positioning capabilities. All presented Stewart platforms utilize flexible joints, as this is a prerequisite for nano-positioning capabilities.
Flexible joints can have various implementations, which will be discussed when designing the nano-hexapod flexible joints. Flexible joints can have various implementations, which will be discussed when designing the nano-hexapod flexible joints.
# TODO - ref:sec:detail_fem_joint # TODO - ref:sec:detail_fem_joint
@ -1014,16 +1011,16 @@ The actuator and sensor selection for the nano-hexapod will also be described in
# TODO - Add reference to the section # TODO - Add reference to the section
#+name: fig:detail_kinematics_stewart_examples_cubic #+name: fig:detail_kinematics_stewart_examples_cubic
#+caption: Some examples of developped Stewart platform with Cubic geometry. (\subref{fig:detail_kinematics_jpl}), (\subref{fig:detail_kinematics_uw_gsp}), (\subref{fig:detail_kinematics_ulb_pz}), (\subref{fig:detail_kinematics_uqp}) #+caption: Some examples of developped Stewart platform with Cubic geometry
#+attr_latex: :options [htbp] #+attr_latex: :options [htbp]
#+begin_figure #+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:detail_kinematics_jpl}California Institute of Technology - USA} #+attr_latex: :caption \subcaption{\label{fig:detail_kinematics_jpl}California Institute of Technology - USA \cite{spanos95_soft_activ_vibrat_isolat}}
#+attr_latex: :options {0.48\textwidth} #+attr_latex: :options {0.48\textwidth}
#+begin_subfigure #+begin_subfigure
#+attr_latex: :width 0.95\linewidth #+attr_latex: :width 0.95\linewidth
[[file:figs/detail_kinematics_jpl.jpg]] [[file:figs/detail_kinematics_jpl.jpg]]
#+end_subfigure #+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:detail_kinematics_uw_gsp}University of Wyoming - USA} #+attr_latex: :caption \subcaption{\label{fig:detail_kinematics_uw_gsp}University of Wyoming - USA \cite{mcinroy99_dynam}}
#+attr_latex: :options {0.48\textwidth} #+attr_latex: :options {0.48\textwidth}
#+begin_subfigure #+begin_subfigure
#+attr_latex: :width 0.95\linewidth #+attr_latex: :width 0.95\linewidth
@ -1031,13 +1028,13 @@ The actuator and sensor selection for the nano-hexapod will also be described in
#+end_subfigure #+end_subfigure
\bigskip \bigskip
#+attr_latex: :caption \subcaption{\label{fig:detail_kinematics_ulb_pz}ULB - Belgium} #+attr_latex: :caption \subcaption{\label{fig:detail_kinematics_ulb_pz}ULB - Belgium \cite{abu02_stiff_soft_stewar_platf_activ}}
#+attr_latex: :options {0.53\textwidth} #+attr_latex: :options {0.53\textwidth}
#+begin_subfigure #+begin_subfigure
#+attr_latex: :width 0.95\linewidth #+attr_latex: :width 0.95\linewidth
[[file:figs/detail_kinematics_ulb_pz.jpg]] [[file:figs/detail_kinematics_ulb_pz.jpg]]
#+end_subfigure #+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:detail_kinematics_uqp}Naval Postgraduate School - USA} #+attr_latex: :caption \subcaption{\label{fig:detail_kinematics_uqp}Naval Postgraduate School - USA \cite{agrawal04_algor_activ_vibrat_isolat_spacec}}
#+attr_latex: :options {0.43\textwidth} #+attr_latex: :options {0.43\textwidth}
#+begin_subfigure #+begin_subfigure
#+attr_latex: :width 0.95\linewidth #+attr_latex: :width 0.95\linewidth
@ -1048,20 +1045,20 @@ The actuator and sensor selection for the nano-hexapod will also be described in
There are two main categories of Stewart platform geometry. There are two main categories of Stewart platform geometry.
The first is cubic architecture (some exampled are presented in Figure ref:fig:detail_kinematics_stewart_examples_cubic), where struts are positioned along six sides of a cube (and are therefore orthogonal to each other). The first is cubic architecture (some exampled are presented in Figure ref:fig:detail_kinematics_stewart_examples_cubic), where struts are positioned along six sides of a cube (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. Such specific architecture has some special properties that will be studied in Section ref:sec:detail_kinematics_cubic.
The second is non-cubic architecture (Figure ref:fig:detail_kinematics_stewart_examples_non_cubic), where the orientation of the struts and position of the joints can be optimized based on performance criteria. The second is non-cubic architecture (Figure ref:fig:detail_kinematics_stewart_examples_non_cubic), where the orientation of the struts and position of the joints can be optimized based on defined performance criteria.
The effect of strut orientation and position of the joints on the Stewart platform properties is discussed Section ref:sec:detail_kinematics_geometry. The effect of strut orientation and position of the joints on the Stewart platform properties is discussed in Section ref:sec:detail_kinematics_geometry.
#+name: fig:detail_kinematics_stewart_examples_non_cubic #+name: fig:detail_kinematics_stewart_examples_non_cubic
#+caption: Some examples of developped Stewart platform with non-cubic geometry. (\subref{fig:detail_kinematics_pph}), (\subref{fig:detail_kinematics_zhang11}), (\subref{fig:detail_kinematics_yang19}), (\subref{fig:detail_kinematics_naves}) #+caption: Some examples of developped Stewart platform with non-cubic geometry
#+attr_latex: :options [htbp] #+attr_latex: :options [htbp]
#+begin_figure #+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:detail_kinematics_pph}Naval Postgraduate School - USA} #+attr_latex: :caption \subcaption{\label{fig:detail_kinematics_pph}Naval Postgraduate School - USA \cite{chen03_payload_point_activ_vibrat_isolat}}
#+attr_latex: :options {0.48\textwidth} #+attr_latex: :options {0.48\textwidth}
#+begin_subfigure #+begin_subfigure
#+attr_latex: :height 5cm #+attr_latex: :height 5cm
[[file:figs/detail_kinematics_pph.jpg]] [[file:figs/detail_kinematics_pph.jpg]]
#+end_subfigure #+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:detail_kinematics_zhang11}Beihang University - China} #+attr_latex: :caption \subcaption{\label{fig:detail_kinematics_zhang11}Beihang University - China \cite{zhang11_six_dof}}
#+attr_latex: :options {0.48\textwidth} #+attr_latex: :options {0.48\textwidth}
#+begin_subfigure #+begin_subfigure
#+attr_latex: :height 5cm #+attr_latex: :height 5cm
@ -1069,13 +1066,13 @@ The effect of strut orientation and position of the joints on the Stewart platfo
#+end_subfigure #+end_subfigure
\bigskip \bigskip
#+attr_latex: :caption \subcaption{\label{fig:detail_kinematics_yang19}Nanjing University - China} #+attr_latex: :caption \subcaption{\label{fig:detail_kinematics_yang19}Nanjing University - China \cite{yang19_dynam_model_decoup_contr_flexib}}
#+attr_latex: :options {0.43\textwidth} #+attr_latex: :options {0.43\textwidth}
#+begin_subfigure #+begin_subfigure
#+attr_latex: :height 5cm #+attr_latex: :height 5cm
[[file:figs/detail_kinematics_yang19.jpg]] [[file:figs/detail_kinematics_yang19.jpg]]
#+end_subfigure #+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:detail_kinematics_naves}University of Twente - Netherlands} #+attr_latex: :caption \subcaption{\label{fig:detail_kinematics_naves}University of Twente - Netherlands \cite{naves20_desig}}
#+attr_latex: :options {0.53\textwidth} #+attr_latex: :options {0.53\textwidth}
#+begin_subfigure #+begin_subfigure
#+attr_latex: :height 5cm #+attr_latex: :height 5cm
@ -1091,11 +1088,11 @@ The effect of strut orientation and position of the joints on the Stewart platfo
** Introduction :ignore: ** Introduction :ignore:
# TODO - Section ref:sec:nhexa_stewart_platform (page pageref:sec:nhexa_stewart_platform), # TODO - Section ref:sec:nhexa_stewart_platform (page pageref:sec:nhexa_stewart_platform),
As was demonstrated during the conceptual phase, the geometry of the Stewart platform impacts the stiffness and compliance characteristics, the mobility or workspace, the force authority, and the dynamics of the manipulator. As was demonstrated during the conceptual phase, the geometry of the Stewart platform impacts the stiffness and compliance characteristics, the mobility (or workspace), the force authority, and the dynamics of the manipulator.
It is therefore essential to understand how the geometry impacts these properties, and to develop methodologies for optimizing the geometry for specific applications. It is therefore essential to understand how the geometry impacts these properties, and to develop methodologies for optimizing the geometry for specific applications.
An important analytical tool for this study is the Jacobian matrix, which depends on $\bm{b}_i$ (joint position with respect to the top platform) and $\hat{\bm{s}}_i$ (orientation of struts). An important analytical tool for this study is the Jacobian matrix, which depends on $\bm{b}_i$ (joints' position with respect to the top platform) and $\hat{\bm{s}}_i$ (struts' orientation).
The choice of frames ($\{A\}$ and $\{B\}$), independently of the physical Stewart platform geometry, impacts the obtained kinematics and stiffness matrix, as these are defined for forces and motion evaluated at the chosen frame. The choice of $\{A\}$ and $\{B\}$ frames, independently of the physical Stewart platform geometry, impacts the obtained kinematics and stiffness matrix, as these are defined for forces and motion evaluated at the chosen frame.
** Matlab Init :noexport:ignore: ** Matlab Init :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) #+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
@ -1136,22 +1133,22 @@ Mobility of parallel manipulators is inherently difficult to study as the transl
The analysis is significantly simplified when considering small motions, as the Jacobian matrix can be used to link the strut motion to the motion of frame $\{B\}$ with respect to $\{A\}$ through eqref:eq:detail_kinematics_jacobian, which is a linear equation. The analysis is significantly simplified when considering small motions, as the Jacobian matrix can be used to link the strut motion to the motion of frame $\{B\}$ with respect to $\{A\}$ through eqref:eq:detail_kinematics_jacobian, which is a linear equation.
\begin{equation}\label{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} \begin{bmatrix} \delta l_1 \\ \delta l_2 \\ \delta l_3 \\ \delta l_4 \\ \delta l_5 \\ \delta l_6 \end{bmatrix} = \underbrace{\begin{bmatrix}
{{}^A\hat{\bm{s}}_1}^T & ({}^A\bm{b}_1 \times {}^A\hat{\bm{s}}_1)^T \\ {{}^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}}_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}}_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}}_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}}_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 {{}^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{bmatrix}}_{\bm{J}} \begin{bmatrix} \delta x \\ \delta y \\ \delta z \\ \delta \theta_x \\ \delta \theta_y \\ \delta \theta_z \end{bmatrix}
\end{equation} \end{equation}
Therefore, the mobility of the Stewart platform (defined as the set of achievable $[\delta x\ \delta y\ \delta z\ \delta \theta_x\ \delta \theta_y\ \delta \theta_z]$) depends on two key factors: the stroke of each strut and the geometry of the Stewart platform (embodied in the Jacobian matrix). Therefore, the mobility of the Stewart platform (defined as the set of achievable $[\delta x\ \delta y\ \delta z\ \delta \theta_x\ \delta \theta_y\ \delta \theta_z]$) depends on two key factors: the stroke of each strut and the geometry of the Stewart platform (embodied in the Jacobian matrix).
More specifically, the XYZ mobility only depends on the $s_i$ (orientation of struts), while the mobility in rotation also depends on $b_i$ (position of top joints). More specifically, the XYZ mobility only depends on the $\hat{\bm{s}}_i$ (orientation of struts), while the mobility in rotation also depends on $\bm{b}_i$ (position of top joints).
**** Mobility in translation **** Mobility in translation
For simplicity, only translations are first considered (i.e., fixed orientation of the Stewart platform). For simplicity, only translations are first considered (i.e., the Stewart platform is considered to have fixed orientation).
In the general case, the translational mobility can be represented by a 3D shape having 12 faces, where each actuator limits the stroke along its axis in positive and negative directions. In the general case, the translational mobility can be represented by a 3D shape having 12 faces, where each actuator limits the stroke along its axis in positive and negative directions.
The faces are therefore perpendicular to the strut direction. The faces are therefore perpendicular to the strut direction.
The obtained mobility for the Stewart platform geometry shown in Figure ref:fig:detail_kinematics_mobility_trans_arch is computed and represented in Figure ref:fig:detail_kinematics_mobility_trans_result. The obtained mobility for the Stewart platform geometry shown in Figure ref:fig:detail_kinematics_mobility_trans_arch is computed and represented in Figure ref:fig:detail_kinematics_mobility_trans_result.
@ -1224,7 +1221,7 @@ exportFig('figs/detail_kinematics_mobility_trans_result.pdf', 'width', 'normal',
#+end_src #+end_src
#+name: fig:detail_kinematics_mobility_trans #+name: fig:detail_kinematics_mobility_trans
#+caption: Example of one Stewart platform (\subref{fig:detail_kinematics_mobility_trans_arch}) and associated translational mobility (\subref{fig:detail_kinematics_mobility_trans_result}) #+caption: One Stewart platform geometry (\subref{fig:detail_kinematics_mobility_trans_arch}) and its associated translational mobility (\subref{fig:detail_kinematics_mobility_trans_result}). A sphere with radius equal to the strut stroke is contained in the translational mobility shape.
#+attr_latex: :options [htbp] #+attr_latex: :options [htbp]
#+begin_figure #+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:detail_kinematics_mobility_trans_arch}Stewart platform geometry} #+attr_latex: :caption \subcaption{\label{fig:detail_kinematics_mobility_trans_arch}Stewart platform geometry}
@ -1241,16 +1238,16 @@ exportFig('figs/detail_kinematics_mobility_trans_result.pdf', 'width', 'normal',
#+end_subfigure #+end_subfigure
#+end_figure #+end_figure
With the previous interpretations of the 12 faces making the translational mobility 3D shape, it can be concluded that for a strut stroke of $\pm d$, a sphere with radius $d$ is contained in the 3D shape and touches it along the six lines defined by the strut axes, as illustrated in Figure ref:fig:detail_kinematics_mobility_trans_result. With the previous interpretations of the 12 faces making the translational mobility 3D shape, it can be concluded that for a strut stroke of $\pm d$, a sphere with radius $d$ is contained in the 3D shape and touches it in directions defined by the strut axes, as illustrated in Figure ref:fig:detail_kinematics_mobility_trans_result.
This means that the mobile platform can be translated in any direction with a stroke of $d$. This means that the mobile platform can be translated in any direction with a stroke equal to the strut stroke.
To better understand how the geometry of the Stewart platform impacts the translational mobility, two configurations are compared with struts oriented vertically (Figure ref:fig:detail_kinematics_stewart_mobility_vert_struts) and struts oriented horizontally (Figure ref:fig:detail_kinematics_stewart_mobility_hori_struts). To better understand how the geometry of the Stewart platform impacts the translational mobility, two configurations are compared with struts oriented vertically (Figure ref:fig:detail_kinematics_stewart_mobility_vert_struts) and struts oriented horizontally (Figure ref:fig:detail_kinematics_stewart_mobility_hori_struts).
The vertically oriented struts lead to greater stroke in the horizontal direction and reduced stroke in the vertical direction (Figure ref:fig:detail_kinematics_mobility_translation_strut_orientation). The vertically oriented struts configuration leads to greater stroke in the horizontal direction and reduced stroke in the vertical direction (Figure ref:fig:detail_kinematics_mobility_translation_strut_orientation).
Conversely, horizontal oriented struts provide more stroke in the vertical direction. Conversely, horizontal oriented struts configuration provides more stroke in the vertical direction.
It may seem counterintuitive that less stroke is available in the direction of the struts. It may seem counterintuitive that less stroke is available in the direction of the struts.
This phenomenon occurs because the struts form a lever mechanism that amplifies the motion. This phenomenon occurs because the struts form a lever mechanism that amplifies the motion.
The amplification factor increases when the struts have a high angle with the direction of motion and equals one when aligned with the direction of motion. The amplification factor increases when the struts have a high angle with the direction of motion and equals one (i.e. is minimal) when aligned with the direction of motion.
#+begin_src matlab :exports none :results none #+begin_src matlab :exports none :results none
%% Stewart platform with Vertically oriented struts %% Stewart platform with Vertically oriented struts
@ -1515,15 +1512,16 @@ exportFig('figs/detail_kinematics_mobility_angle_strut_distance.pdf', 'width', '
It is possible to consider combined translations and rotations, although displaying such mobility becomes more complex. It is possible to consider combined translations and rotations, although displaying such mobility becomes more complex.
For a fixed geometry and a desired mobility (combined translations and rotations), it is possible to estimate the required minimum actuator stroke. For a fixed geometry and a desired mobility (combined translations and rotations), it is possible to estimate the required minimum actuator stroke.
This analysis was conducted in Section ref:sec:detail_kinematics_nano_hexapod to estimate the required actuator stroke for the nano-hexapod geometry. This analysis is conducted in Section ref:sec:detail_kinematics_nano_hexapod to estimate the required actuator stroke for the nano-hexapod geometry.
** Stiffness ** Stiffness
<<ssec:detail_kinematics_geometry_stiffness>> <<ssec:detail_kinematics_geometry_stiffness>>
**** Introduction :ignore: **** Introduction :ignore:
The stiffness matrix defines how the nano-hexapod deforms (frame $\{B\}$ with respect to frame $\{A\}$) due to static forces/torques applied on $\{B\}$. 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. 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 there were optimized after the geometry was fixed, but several work were done to quantify the impact of the flexible joint stiffness [[cite:&mcinroy00_desig_contr_flexur_joint_hexap;&mcinroy02_model_desig_flexur_joint_stewar]]. 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]].
\begin{equation}\label{eq:detail_kinematics_stiffness_matrix} \begin{equation}\label{eq:detail_kinematics_stiffness_matrix}
\bm{K} = \bm{J}^T \bm{\mathcal{K}} \bm{J} \bm{K} = \bm{J}^T \bm{\mathcal{K}} \bm{J}
@ -1546,25 +1544,25 @@ In that case, the obtained stiffness matrix linearly depends on the strut stiffn
**** Translation Stiffness **** Translation Stiffness
As shown by equation eqref:eq:detail_kinematics_stiffness_matrix_simplified, the translation stiffnesses (the $3 \times 3$ top left terms of the stiffness matrix) only depend on the orientation of the struts and not their location: $\hat{\bm{s}}_i \cdot \hat{\bm{s}}_i^T$. As shown by equation eqref:eq:detail_kinematics_stiffness_matrix_simplified, the translation stiffnesses (the $3 \times 3$ top left terms of the stiffness matrix) only depend on the orientation of the struts and not their location: $\hat{\bm{s}}_i \cdot \hat{\bm{s}}_i^T$.
In the extreme case where all struts are vertical with $s_i = [0,\ 0,\ 1]$, a vertical stiffness of $6k$ is achieved, but with null stiffness in the X and Y directions. In the extreme case where all struts are vertical ($s_i = [0\ 0\ 1]$), a vertical stiffness of $6k$ is achieved, but with null stiffness in the horizontal directions.
If two struts are aligned along the X axis, two struts along the Y axis, and two struts along the Z axis, then $\hat{\bm{s}}_i \cdot \hat{\bm{s}}_i^T = 2 \bm{I}_3$, resulting in well-distributed stiffness along all directions. If two struts are aligned with the X axis, two struts with the Y axis, and two struts with the Z axis, then $\hat{\bm{s}}_i \cdot \hat{\bm{s}}_i^T = 2 \bm{I}_3$, resulting in well-distributed stiffness along all directions.
This configuration corresponds to the cubic architecture presented in Section ref:sec:detail_kinematics_cubic. This configuration corresponds to the cubic architecture presented in Section ref:sec:detail_kinematics_cubic.
When struts are oriented more vertically (Figure ref:fig:detail_kinematics_stewart_mobility_vert_struts), vertical stiffness increases while horizontal stiffness decreases. When the struts are oriented more vertically, as shown in Figure ref:fig:detail_kinematics_stewart_mobility_vert_struts, the vertical stiffness increases while the horizontal stiffness decreases.
Additionally, $R_x$ and $R_y$ stiffness increases while $R_z$ stiffness decreases. Additionally, $R_x$ and $R_y$ stiffness increases while $R_z$ stiffness decreases.
The opposite conclusions apply if struts are oriented more horizontally (Figure ref:fig:detail_kinematics_stewart_mobility_hori_struts). The opposite conclusions apply if struts are oriented more horizontally, illustrated in Figure ref:fig:detail_kinematics_stewart_mobility_hori_struts.
**** Rotational Stiffness **** Rotational Stiffness
The rotational stiffnesses depend both on the orientation of the struts and on the location of the top joints (with respect to the considered center of rotation, i.e., the location of frame $\{B\}$). The rotational stiffnesses depend both on the orientation of the struts and on the location of the top joints with respect to the considered center of rotation (i.e., the location of frame $\{A\}$).
With the same orientation but increased distances ($b_i$) by a factor of 2, the rotational stiffness is increased by a factor of 4. With the same orientation but increased distances to the frame $\{A\}$ by a factor of 2, the rotational stiffness is increased by a factor of 4.
Therefore, the compact Stewart platform depicted in Figure ref:fig:detail_kinematics_stewart_mobility_close_struts has less rotational stiffness than the Stewart platform shown in Figure ref:fig:detail_kinematics_stewart_mobility_space_struts. Therefore, the compact Stewart platform depicted in Figure ref:fig:detail_kinematics_stewart_mobility_close_struts has less rotational stiffness than the Stewart platform shown in Figure ref:fig:detail_kinematics_stewart_mobility_space_struts.
**** Diagonal Stiffness Matrix **** Diagonal Stiffness Matrix
Having a diagonal stiffness matrix $\bm{K}$ can be beneficial for control purposes as it would make the plant in the Cartesian frame decoupled at low frequency. Having a diagonal stiffness matrix $\bm{K}$ can be beneficial for control purposes as it would make the plant in the Cartesian frame decoupled at low frequency.
This property depends on both the geometry and the chosen $\{B\}$ frame. This property depends on both the geometry and the chosen $\{A\}$ frame.
For specific geometry and choice of $\{B\}$ frame, it is possible to achieve a diagonal $K$ matrix. For specific geometry and choice of $\{A\}$ frame, it is possible to achieve a diagonal $K$ matrix.
This is discussed in Section ref:ssec:detail_kinematics_cubic_static. This is discussed in Section ref:ssec:detail_kinematics_cubic_static.
** Dynamical properties ** Dynamical properties
@ -1581,14 +1579,14 @@ These conditions are studied in Section ref:ssec:detail_kinematics_cubic_dynamic
\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} \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} \end{equation}
In the frame of the struts, the equations of motion given by equation eqref:eq:detail_kinematics_transfer_function_struts are well decoupled at low frequency. In the frame of the struts, the equations of motion eqref:eq:detail_kinematics_transfer_function_struts are well decoupled at low frequency.
This is why most Stewart platforms are controlled in the frame of the struts: below the resonance frequency, the system is decoupled and SISO control may be applied for each strut, independently of the payload being used. This is why most Stewart platforms are controlled in the frame of the struts: below the resonance frequency, the system is well decoupled and SISO control may be applied for each strut, independently of the payload being used.
\begin{equation}\label{eq:detail_kinematics_transfer_function_struts} \begin{equation}\label{eq:detail_kinematics_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} \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} \end{equation}
Coupling between sensors (force sensors, relative position sensors, inertial sensors) in different struts may also be important for decentralized control. Coupling between sensors (force sensors, relative position sensors or inertial sensors) in different struts may also be important for decentralized control.
In section ref:ssec:detail_kinematics_decentralized_control, it will be studied whether the Stewart platform geometry can be optimized to have lower coupling between the struts. In section ref:ssec:detail_kinematics_decentralized_control, it will be studied whether the Stewart platform geometry can be optimized to have lower coupling between the struts.
** Conclusion ** Conclusion
@ -1596,14 +1594,14 @@ In section ref:ssec:detail_kinematics_decentralized_control, it will be studied
:UNNUMBERED: t :UNNUMBERED: t
:END: :END:
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. The effects of two changes in the manipulator's geometry, namely the position and orientation of the struts, are summarized in Table ref:tab:detail_kinematics_geometry.
These results could have been easily deduced based on mechanical principles, but thanks to the kinematic analysis, they can be quantified. These results could have been easily deduced based on mechanical principles, but thanks to the kinematic analysis, they can be quantified.
These trade-offs provide important guidelines when choosing the Stewart platform geometry. These trade-offs provide important guidelines when choosing the Stewart platform geometry.
#+name: tab:detail_kinematics_geometry #+name: tab:detail_kinematics_geometry
#+attr_latex: :environment tabularx :width 0.8\linewidth :align Xcc #+attr_latex: :environment tabularx :width 0.8\linewidth :align Xcc
#+attr_latex: :center t :booktabs t :float t :font \small #+attr_latex: :center t :booktabs t :float t :font \small
#+caption: Effect of a change in geometry on the manipulator's stiffness, force authority and stroke #+caption: Effect of a change in geometry on the manipulator's stiffness and mobility
| *Struts* | *Vertically Oriented* | *Increased separation* | | *Struts* | *Vertically Oriented* | *Increased separation* |
|-------------------------------+-----------------------+------------------------| |-------------------------------+-----------------------+------------------------|
| Vertical stiffness | $\nearrow$ | $=$ | | Vertical stiffness | $\nearrow$ | $=$ |
@ -1611,16 +1609,41 @@ These trade-offs provide important guidelines when choosing the Stewart platform
| Vertical rotation stiffness | $\searrow$ | $\nearrow$ | | Vertical rotation stiffness | $\searrow$ | $\nearrow$ |
| Horizontal rotation stiffness | $\nearrow$ | $\nearrow$ | | Horizontal rotation stiffness | $\nearrow$ | $\nearrow$ |
|-------------------------------+-----------------------+------------------------| |-------------------------------+-----------------------+------------------------|
| Vertical stroke | $\searrow$ | $=$ | | Vertical mobility | $\searrow$ | $=$ |
| Horizontal stroke | $\nearrow$ | $=$ | | Horizontal mobility | $\nearrow$ | $=$ |
| Vertical rotation stroke | $\nearrow$ | $\searrow$ | | Vertical rotation mobility | $\nearrow$ | $\searrow$ |
| Horizontal rotation stroke | $\searrow$ | $\searrow$ | | Horizontal rotation mobility | $\searrow$ | $\searrow$ |
* The Cubic Architecture * The Cubic Architecture
:PROPERTIES: :PROPERTIES:
:HEADER-ARGS:matlab+: :tangle matlab/detail_kinematics_2_cubic.m :HEADER-ARGS:matlab+: :tangle matlab/detail_kinematics_2_cubic.m
:END: :END:
<<sec:detail_kinematics_cubic>> <<sec:detail_kinematics_cubic>>
** Matlab Init :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
<<matlab-dir>>
#+end_src
#+begin_src matlab :exports none :results silent :noweb yes
<<matlab-init>>
#+end_src
#+begin_src matlab :tangle no :noweb yes
<<m-init-path>>
#+end_src
#+begin_src matlab :eval no :noweb yes
<<m-init-path-tangle>>
#+end_src
#+begin_src matlab :noweb yes
<<m-init-simscape>>
#+end_src
#+begin_src matlab :noweb yes
<<m-init-other>>
#+end_src
** Introduction :ignore: ** Introduction :ignore:
The Cubic configuration for the Stewart platform was first proposed in [[cite:&geng94_six_degree_of_freed_activ]]. The Cubic configuration for the Stewart platform was first proposed in [[cite:&geng94_six_degree_of_freed_activ]].
@ -1653,7 +1676,8 @@ stewart = computeJacobian(stewart);
stewart = initializeStewartPose(stewart); stewart = initializeStewartPose(stewart);
displayArchitecture(stewart, 'labels', false, 'frames', false); displayArchitecture(stewart, 'labels', false, 'frames', false);
plotCube(stewart, 'Hc', Hc, 'FOc', FOc, 'color', [0,0,0,0.5], 'link_to_struts', true); plotCube(stewart, 'Hc', Hc, 'FOc', FOc, 'color', [0,0,0,0.2], 'link_to_struts', false);
view([40, 5]);
#+end_src #+end_src
#+begin_src matlab :tangle no :exports results :results file none #+begin_src matlab :tangle no :exports results :results file none
@ -1680,7 +1704,8 @@ stewart = computeJacobian(stewart);
stewart = initializeCylindricalPlatforms(stewart, 'Fpr', 150e-3, 'Mpr', 150e-3); stewart = initializeCylindricalPlatforms(stewart, 'Fpr', 150e-3, 'Mpr', 150e-3);
displayArchitecture(stewart, 'labels', false, 'frames', false); displayArchitecture(stewart, 'labels', false, 'frames', false);
plotCube(stewart, 'Hc', Hc, 'FOc', FOc, 'color', [0,0,0,0.5], 'link_to_struts', true); plotCube(stewart, 'Hc', Hc, 'FOc', FOc, 'color', [0,0,0,0.2], 'link_to_struts', false);
view([40, 5]);
#+end_src #+end_src
#+begin_src matlab :tangle no :exports results :results file none #+begin_src matlab :tangle no :exports results :results file none
@ -1688,16 +1713,16 @@ exportFig('figs/detail_kinematics_cubic_architecture_example_small.pdf', 'width'
#+end_src #+end_src
#+name: fig:detail_kinematics_cubic_architecture_examples #+name: fig:detail_kinematics_cubic_architecture_examples
#+caption: Typical Stewart platform cubic architectures. (\subref{fig:detail_kinematics_cubic_architecture_example}) (\subref{fig:detail_kinematics_cubic_architecture_example_small}) #+caption: Typical Stewart platform cubic architectures in which struts' length is similar to the cube edges's length (\subref{fig:detail_kinematics_cubic_architecture_example}) or is taking just a portion of the edge (\subref{fig:detail_kinematics_cubic_architecture_example_small}).
#+attr_latex: :options [htbp] #+attr_latex: :options [htbp]
#+begin_figure #+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:detail_kinematics_cubic_architecture_example}sub caption a} #+attr_latex: :caption \subcaption{\label{fig:detail_kinematics_cubic_architecture_example}Classical Cubic architecture}
#+attr_latex: :options {0.49\textwidth} #+attr_latex: :options {0.49\textwidth}
#+begin_subfigure #+begin_subfigure
#+attr_latex: :scale 1 #+attr_latex: :scale 1
[[file:figs/detail_kinematics_cubic_architecture_example.png]] [[file:figs/detail_kinematics_cubic_architecture_example.png]]
#+end_subfigure #+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:detail_kinematics_cubic_architecture_example_small}sub caption b} #+attr_latex: :caption \subcaption{\label{fig:detail_kinematics_cubic_architecture_example_small}Alternative configuration}
#+attr_latex: :options {0.49\textwidth} #+attr_latex: :options {0.49\textwidth}
#+begin_subfigure #+begin_subfigure
#+attr_latex: :scale 1 #+attr_latex: :scale 1
@ -1715,46 +1740,20 @@ Dynamical decoupling is investigated in Section ref:ssec:detail_kinematics_cubic
Given that the cubic architecture imposes strict geometric constraints, alternative designs are proposed in Section ref:ssec:detail_kinematics_cubic_design. Given that the cubic architecture imposes strict geometric constraints, alternative designs are proposed in Section ref:ssec:detail_kinematics_cubic_design.
The ultimate objective is to determine the suitability of the cubic architecture for the nano-hexapod. The ultimate objective is to determine the suitability of the cubic architecture for the nano-hexapod.
** Matlab Init :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
<<matlab-dir>>
#+end_src
#+begin_src matlab :exports none :results silent :noweb yes
<<matlab-init>>
#+end_src
#+begin_src matlab :tangle no :noweb yes
<<m-init-path>>
#+end_src
#+begin_src matlab :eval no :noweb yes
<<m-init-path-tangle>>
#+end_src
#+begin_src matlab :noweb yes
<<m-init-simscape>>
#+end_src
#+begin_src matlab :noweb yes
<<m-init-other>>
#+end_src
** Static Properties ** Static Properties
<<ssec:detail_kinematics_cubic_static>> <<ssec:detail_kinematics_cubic_static>>
**** Stiffness matrix for the Cubic architecture **** Stiffness matrix for the Cubic architecture
Consider the cubic architecture depicted in Figure ref:fig:detail_kinematics_cubic_schematic_full.
Consider the cubic architecture shown in Figure ref:fig:detail_kinematics_cubic_schematic_full. Consider the cubic architecture shown in Figure ref:fig:detail_kinematics_cubic_schematic_full.
The unit vectors corresponding to the edges of the cube are described by equation eqref:eq:detail_kinematics_cubic_s. The unit vectors corresponding to the edges of the cube are described by equation eqref:eq:detail_kinematics_cubic_s.
\begin{equation}\label{eq:detail_kinematics_cubic_s} \begin{equation}\label{eq:detail_kinematics_cubic_s}
\hat{\bm{s}}_1 = \begin{bmatrix} \sqrt{2}/\sqrt{3} \\ 0 \\ 1/\sqrt{3} \end{bmatrix} \quad \hat{\bm{s}}_1 = \begin{bmatrix} \frac{\sqrt{2}}{\sqrt{3}} \\ 0 \\ \frac{1}{\sqrt{3}} \end{bmatrix} \quad
\hat{\bm{s}}_2 = \begin{bmatrix} -1/\sqrt{6} \\ -1/\sqrt{2} \\ 1/\sqrt{3} \end{bmatrix} \quad \hat{\bm{s}}_2 = \begin{bmatrix} \frac{-1}{\sqrt{6}} \\ \frac{-1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} \end{bmatrix} \quad
\hat{\bm{s}}_3 = \begin{bmatrix} -1/\sqrt{6} \\ 1/\sqrt{2} \\ 1/\sqrt{3} \end{bmatrix} \quad \hat{\bm{s}}_3 = \begin{bmatrix} \frac{-1}{\sqrt{6}} \\ \frac{ 1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} \end{bmatrix} \quad
\hat{\bm{s}}_4 = \begin{bmatrix} \sqrt{2}/\sqrt{3} \\ 0 \\ 1/\sqrt{3} \end{bmatrix} \quad \hat{\bm{s}}_4 = \begin{bmatrix} \frac{\sqrt{2}}{\sqrt{3}} \\ 0 \\ \frac{1}{\sqrt{3}} \end{bmatrix} \quad
\hat{\bm{s}}_5 = \begin{bmatrix} -1/\sqrt{6} \\ -1/\sqrt{2} \\ 1/\sqrt{3} \end{bmatrix} \quad \hat{\bm{s}}_5 = \begin{bmatrix} \frac{-1}{\sqrt{6}} \\ \frac{-1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} \end{bmatrix} \quad
\hat{\bm{s}}_6 = \begin{bmatrix} -1/\sqrt{6} \\ 1/\sqrt{2} \\ 1/\sqrt{3} \end{bmatrix} \hat{\bm{s}}_6 = \begin{bmatrix} \frac{-1}{\sqrt{6}} \\ \frac{ 1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} \end{bmatrix}
\end{equation} \end{equation}
#+begin_src latex :file detail_kinematics_cubic_schematic_full.pdf :results file #+begin_src latex :file detail_kinematics_cubic_schematic_full.pdf :results file
@ -1952,7 +1951,7 @@ In that case, the location of the top joints can be expressed by equation eqref:
The stiffness matrix is therefore diagonal when the considered $\{B\}$ frame is located at the center of the cube (shown by frame $\{C\}$). The stiffness matrix is therefore diagonal when the considered $\{B\}$ frame is located at the center of the cube (shown by frame $\{C\}$).
This means that static forces (or torques) applied at the cube's center will induce pure translations (or rotations around the cube's center). This means that static forces (resp torques) applied at the cube's center will induce pure translations (resp rotations around the cube's center).
This specific location where the stiffness matrix is diagonal is referred to as the "Center of Stiffness" (analogous to the "Center of Mass" where the mass matrix is diagonal). This specific location where the stiffness matrix is diagonal is referred to as the "Center of Stiffness" (analogous to the "Center of Mass" where the mass matrix is diagonal).
#+begin_src matlab #+begin_src matlab
@ -2043,10 +2042,6 @@ This stiffness matrix structure is characteristic of Stewart platforms exhibitin
Therefore, the stiffness characteristics of the cubic architecture are only distinctive when considering a reference frame located at the cube's center. Therefore, the stiffness characteristics of the cubic architecture are only distinctive when considering a reference frame located at the cube's center.
This poses a practical limitation, as in most applications, the relevant frame (where motion is of interest and forces are applied) is located above the top platform. This poses a practical limitation, as in most applications, the relevant frame (where motion is of interest and forces are applied) is located above the top platform.
It should be noted that the cube's center need not be at the "center" of the Stewart platform.
This can lead to interesting architectures shown in Section ref:ssec:detail_kinematics_cubic_design.
It should be noted that for the stiffness matrix to be diagonal, the cube's center need not coincide with the geometric center of the Stewart platform. It should be noted that for the stiffness matrix to be diagonal, the cube's center need not coincide with the geometric center of the Stewart platform.
This observation leads to the interesting alternative architectures presented in Section ref:ssec:detail_kinematics_cubic_design. This observation leads to the interesting alternative architectures presented in Section ref:ssec:detail_kinematics_cubic_design.
@ -2059,9 +2054,8 @@ The resulting mobility in X, Y, and Z directions for the cubic architecture is i
The translational workspace analysis reveals that for the cubic architecture, the achievable positions form a cube whose axes align with the struts, with the cube's edge length corresponding to the strut axial stroke. The translational workspace analysis reveals that for the cubic architecture, the achievable positions form a cube whose axes align with the struts, with the cube's edge length corresponding to the strut axial stroke.
This findings suggest that the mobility pattern is more nuanced than sometimes described in the literature [[cite:&mcinroy00_desig_contr_flexur_joint_hexap]], exhibiting uniformity primarily along directions aligned with the cube's edges rather than uniform spherical distribution in all XYZ directions. This findings suggest that the mobility pattern is more nuanced than sometimes described in the literature [[cite:&mcinroy00_desig_contr_flexur_joint_hexap]], exhibiting uniformity primarily along directions aligned with the cube's edges rather than uniform spherical distribution in all XYZ directions.
This configuration still offers more consistent mobility characteristics compared to alternative architectures illustrated in Figure ref:fig:detail_kinematics_mobility_trans. This configuration still offers more consistent mobility characteristics compared to alternative architectures illustrated in Figure ref:fig:detail_kinematics_mobility_trans.
It is worth noting that the translational mobility properties remain independent of the cube's size.
The rotational mobility, illustrated in Figure ref:fig:detail_kinematics_cubic_mobility_rotations, exhibit greater achievable angular displacements in the $R_x$ and $R_y$ directions compared to the $R_z$ direction. The rotational mobility, illustrated in Figure ref:fig:detail_kinematics_cubic_mobility_rotations, exhibit greater achievable angular stroke in the $R_x$ and $R_y$ directions compared to the $R_z$ direction.
Furthermore, an inverse relationship exists between the cube's dimension and rotational mobility, with larger cube sizes corresponding to more limited angular displacement capabilities. Furthermore, an inverse relationship exists between the cube's dimension and rotational mobility, with larger cube sizes corresponding to more limited angular displacement capabilities.
#+begin_src matlab #+begin_src matlab
@ -2193,12 +2187,9 @@ exportFig('figs/detail_kinematics_cubic_mobility_rotations.pdf', 'width', 'norma
<<ssec:detail_kinematics_cubic_dynamic>> <<ssec:detail_kinematics_cubic_dynamic>>
**** Introduction :ignore: **** Introduction :ignore:
This section examines the dynamics of the cubic architecture in the Cartesian frame. This section examines the dynamics of the cubic architecture in the Cartesian frame which corresponds to the transfer function from forces and torques $\bm{\mathcal{F}}$ to translations and rotations $\bm{\mathcal{X}}$ of the top platform.
This corresponds to the transfer function from forces and torques $\bm{\mathcal{F}}$ to translations and rotations $\bm{\mathcal{X}}$ of the top platform.
When relative motion sensors are integrated in each strut (measuring $\bm{\mathcal{L}}$), the pose $\bm{\mathcal{X}}$ is computed using the Jacobian matrix as shown in Figure ref:fig:detail_kinematics_centralized_control. When relative motion sensors are integrated in each strut (measuring $\bm{\mathcal{L}}$), the pose $\bm{\mathcal{X}}$ is computed using the Jacobian matrix as shown in Figure ref:fig:detail_kinematics_centralized_control.
The analysis aims to identify whether the cubic configuration exhibits special properties for control in the Cartesian frame.
#+begin_src latex :file detail_kinematics_centralized_control.pdf #+begin_src latex :file detail_kinematics_centralized_control.pdf
\begin{tikzpicture} \begin{tikzpicture}
\node[block] (Jt) at (0, 0) {$\bm{J}^{-T}$}; \node[block] (Jt) at (0, 0) {$\bm{J}^{-T}$};
@ -2220,14 +2211,14 @@ The analysis aims to identify whether the cubic configuration exhibits special p
#+end_src #+end_src
#+name: fig:detail_kinematics_centralized_control #+name: fig:detail_kinematics_centralized_control
#+caption: From Strut coordinate to Cartesian coordinate using the Jacobian matrix #+caption: Typical control architecture in the cartesian frame
#+RESULTS: #+RESULTS:
[[file:figs/detail_kinematics_centralized_control.png]] [[file:figs/detail_kinematics_centralized_control.png]]
**** Low frequency and High frequency coupling **** Low frequency and High frequency coupling
As derived during the conceptual design phase, the dynamics from $\bm{\mathcal{F}}$ to $\bm{\mathcal{X}}$ is described by Equation eqref:eq:detail_kinematics_transfer_function_cart. As derived during the conceptual design phase, the dynamics from $\bm{\mathcal{F}}$ to $\bm{\mathcal{X}}$ is described by Equation eqref:eq:detail_kinematics_transfer_function_cart.
At low frequency, the static behavior of the platform depends on the stiffness matrix eqref:eq:detail_kinematics_transfer_function_cart_low_freq. At low frequency, the behavior of the platform depends on the stiffness matrix eqref:eq:detail_kinematics_transfer_function_cart_low_freq.
\begin{equation}\label{eq:detail_kinematics_transfer_function_cart_low_freq} \begin{equation}\label{eq:detail_kinematics_transfer_function_cart_low_freq}
\frac{{\mathcal{X}}}{\bm{\mathcal{F}}}(j \omega) \xrightarrow[\omega \to 0]{} \bm{K}^{-1} \frac{{\mathcal{X}}}{\bm{\mathcal{F}}}(j \omega) \xrightarrow[\omega \to 0]{} \bm{K}^{-1}
@ -2248,7 +2239,7 @@ To achieve a diagonal mass matrix, the center of mass of the mobile components m
#+attr_latex: :width 0.6\linewidth #+attr_latex: :width 0.6\linewidth
[[file:figs/detail_kinematics_cubic_payload.png]] [[file:figs/detail_kinematics_cubic_payload.png]]
To verify these properties, a cubic Stewart platform with a cylindrical payload on top (Figure ref:fig:detail_kinematics_cubic_payload) was analyzed. To verify these properties, a cubic Stewart platform with a cylindrical payload was analyzed (Figure ref:fig:detail_kinematics_cubic_payload).
Transfer functions from $\bm{\mathcal{F}}$ to $\bm{\mathcal{X}}$ were computed for two specific locations of the $\{B\}$ frames. Transfer functions from $\bm{\mathcal{F}}$ to $\bm{\mathcal{X}}$ were computed for two specific locations of the $\{B\}$ frames.
When the $\{B\}$ frame was positioned at the center of mass, coupling at low frequency was observed due to the non-diagonal stiffness matrix (Figure ref:fig:detail_kinematics_cubic_cart_coupling_com). When the $\{B\}$ frame was positioned at the center of mass, coupling at low frequency was observed due to the non-diagonal stiffness matrix (Figure ref:fig:detail_kinematics_cubic_cart_coupling_com).
Conversely, when positioned at the center of stiffness, coupling occurred at high frequency due to the non-diagonal mass matrix (Figure ref:fig:detail_kinematics_cubic_cart_coupling_cok). Conversely, when positioned at the center of stiffness, coupling occurred at high frequency due to the non-diagonal mass matrix (Figure ref:fig:detail_kinematics_cubic_cart_coupling_cok).
@ -2417,7 +2408,7 @@ exportFig('figs/detail_kinematics_cubic_cart_coupling_cok.pdf', 'width', 'half',
#+end_src #+end_src
#+name: fig:detail_kinematics_cubic_cart_coupling #+name: fig:detail_kinematics_cubic_cart_coupling
#+caption: Transfer functions for a Cubic Stewart platform expressed in the Cartesian frame. Two locations of the $\{B\}$ frame are considered: at the cube's center (\subref{fig:detail_kinematics_cubic_cart_coupling_cok}) and at the center of mass of the moving body (\subref{fig:detail_kinematics_cubic_cart_coupling_com}). #+caption: Transfer functions for a Cubic Stewart platform expressed in the Cartesian frame. Two locations of the $\{B\}$ frame are considered: at the center of mass of the moving body (\subref{fig:detail_kinematics_cubic_cart_coupling_com}) and at the cube's center (\subref{fig:detail_kinematics_cubic_cart_coupling_cok}).
#+attr_latex: :options [htbp] #+attr_latex: :options [htbp]
#+begin_figure #+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:detail_kinematics_cubic_cart_coupling_com}$\{B\}$ at the center of mass} #+attr_latex: :caption \subcaption{\label{fig:detail_kinematics_cubic_cart_coupling_com}$\{B\}$ at the center of mass}
@ -2540,18 +2531,15 @@ exportFig('figs/detail_kinematics_cubic_cart_coupling_com_cok.pdf', 'width', 'ha
The analysis of dynamical properties of the cubic architecture yields several important conclusions. The analysis of dynamical properties of the cubic architecture yields several important conclusions.
Static decoupling, characterized by a diagonal stiffness matrix, is achieved when reference frames $\{A\}$ and $\{B\}$ are positioned at the cube's center. Static decoupling, characterized by a diagonal stiffness matrix, is achieved when reference frames $\{A\}$ and $\{B\}$ are positioned at the cube's center.
This property can also be obtained with non-cubic architectures that exhibit symmetrical strut arrangements. Note that this property can also be obtained with non-cubic architectures that exhibit symmetrical strut arrangements.
Dynamic decoupling requires both static decoupling and coincidence of the mobile platform's center of mass with reference frame $\{B\}$. Dynamic decoupling requires both static decoupling and coincidence of the mobile platform's center of mass with reference frame $\{B\}$.
While this configuration offers powerful control advantages, it requires positioning the payload at the cube's center, which is highly restrictive and often impractical. While this configuration offers powerful control advantages, it requires positioning the payload at the cube's center, which is highly restrictive and often impractical.
Additionally, the cubic architecture provides uniform stiffness in XYZ directions, which may be advantageous for certain applications.
** Decentralized Control ** Decentralized Control
<<ssec:detail_kinematics_decentralized_control>> <<ssec:detail_kinematics_decentralized_control>>
**** Introduction :ignore: **** Introduction :ignore:
The orthogonal arrangement of struts in the cubic architecture suggests a potential minimization of inter-strut coupling, which could theoretically create favorable conditions for decentralized control. The orthogonal arrangement of struts in the cubic architecture suggests a potential minimization of inter-strut coupling, which could theoretically create favorable conditions for decentralized control.
This section examines whether the cubic architecture actually demonstrates advantageous properties for decentralized control in the frame of the struts.
Two sensor types integrated in the struts are considered: displacement sensors and force sensors. Two sensor types integrated in the struts are considered: displacement sensors and force sensors.
The control architecture is illustrated in Figure ref:fig:detail_kinematics_decentralized_control, where $\bm{K}_{\mathcal{L}}$ represents a diagonal transfer function matrix. The control architecture is illustrated in Figure ref:fig:detail_kinematics_decentralized_control, where $\bm{K}_{\mathcal{L}}$ represents a diagonal transfer function matrix.
@ -2573,7 +2561,7 @@ The control architecture is illustrated in Figure ref:fig:detail_kinematics_dece
#+end_src #+end_src
#+name: fig:detail_kinematics_decentralized_control #+name: fig:detail_kinematics_decentralized_control
#+caption: From Strut coordinate to Cartesian coordinate using the Jacobian matrix #+caption: Decentralized control in the frame of the struts.
#+RESULTS: #+RESULTS:
[[file:figs/detail_kinematics_decentralized_control.png]] [[file:figs/detail_kinematics_decentralized_control.png]]
@ -2834,7 +2822,7 @@ exportFig('figs/detail_kinematics_cubic_decentralized_fn.pdf', 'width', 'half',
**** Conclusion **** Conclusion
The presented results do not demonstrate the pronounced decoupling advantages often associated with cubic architectures in the literature. The presented results do not demonstrate the pronounced decoupling advantages often associated with cubic architectures in the literature.
Both the cubic and non-cubic configurations exhibited similar coupling characteristics in our modeling scenarios, suggesting that the benefits of orthogonal strut arrangement may be more nuanced than commonly described for decentralized control. Both the cubic and non-cubic configurations exhibited similar coupling characteristics, suggesting that the benefits of orthogonal strut arrangement may be more nuanced than commonly described for decentralized control.
** Cubic architecture with Cube's center above the top platform ** Cubic architecture with Cube's center above the top platform
<<ssec:detail_kinematics_cubic_design>> <<ssec:detail_kinematics_cubic_design>>
@ -3167,7 +3155,6 @@ disp('Radius for mobile platform:');
simplify(sqrt(bi_z_H(1,1).^2 + bi_z_H(1,2).^2)) simplify(sqrt(bi_z_H(1,1).^2 + bi_z_H(1,2).^2))
#+end_src #+end_src
In order to determine the approximate size of the platform as a function of
For the proposed configuration, the top joints $\bm{b}_i$ (resp. the bottom joints $\bm{a}_i$) and are positioned on a circle with radius $R_{b_i}$ (resp. $R_{a_i}$) described by Equation eqref:eq:detail_kinematics_cube_joints. For the proposed configuration, the top joints $\bm{b}_i$ (resp. the bottom joints $\bm{a}_i$) and are positioned on a circle with radius $R_{b_i}$ (resp. $R_{a_i}$) described by Equation eqref:eq:detail_kinematics_cube_joints.
\begin{subequations}\label{eq:detail_kinematics_cube_joints} \begin{subequations}\label{eq:detail_kinematics_cube_joints}
@ -3186,9 +3173,6 @@ The configurations proposed in this analysis represent derivations from the clas
Three distinct configurations have been identified, each with different geometric arrangements but sharing the common characteristic that the cube's center is positioned above the top platform. Three distinct configurations have been identified, each with different geometric arrangements but sharing the common characteristic that the cube's center is positioned above the top platform.
This structural modification enables the alignment of the moving body's center of mass with the center of stiffness, resulting in beneficial decoupling properties in the Cartesian frame. This structural modification enables the alignment of the moving body's center of mass with the center of stiffness, resulting in beneficial decoupling properties in the Cartesian frame.
These proposed architectures maintain the fundamental advantages inherent to the cubic configuration, such as uniform stiffness and uniform mobility, while providing favorable dynamical properties when payloads are placed on top of the mobile platform.
This approach allows for practical payload positioning while preserving the desirable control characteristics associated with the cubic architecture, making these configurations potentially useful for applications requiring both specific payload placement and good dynamic performance.
** Conclusion ** Conclusion
:PROPERTIES: :PROPERTIES:
:UNNUMBERED: t :UNNUMBERED: t
@ -3202,10 +3186,10 @@ Regarding mobility, the translational capabilities of the cubic configuration ex
This understanding refines the characterization of cubic architecture mobility commonly presented in literature. This understanding refines the characterization of cubic architecture mobility commonly presented in literature.
The analysis of decentralized control in the frame of the struts revealed more nuanced results than expected. The analysis of decentralized control in the frame of the struts revealed more nuanced results than expected.
While cubic architectures are frequently associated with reduced coupling between actuators and sensors, our comparative study showed that these benefits may be more subtle or context-dependent than commonly described. While cubic architectures are frequently associated with reduced coupling between actuators and sensors, this study showed that these benefits may be more subtle or context-dependent than commonly described.
Under the conditions analyzed, the coupling characteristics of cubic and non-cubic configurations appeared similar. Under the conditions analyzed, the coupling characteristics of cubic and non-cubic configurations, in the frame of the struts, appeared similar.
Fully decoupled dynamics can be achieved when the center of mass of the moving body coincides with the cube's center. Fully decoupled dynamics in the Cartesian frame can be achieved when the center of mass of the moving body coincides with the cube's center.
However, this arrangement presents practical challenges, as the cube's center is traditionally located between the top and bottom platforms, making payload placement problematic for many applications. However, this arrangement presents practical challenges, as the cube's center is traditionally located between the top and bottom platforms, making payload placement problematic for many applications.
To address this limitation, modified cubic architectures have been proposed with the cube's center positioned above the top platform. To address this limitation, modified cubic architectures have been proposed with the cube's center positioned above the top platform.
@ -3218,8 +3202,7 @@ These configurations maintain the fundamental advantages of the cubic architectu
<<sec:detail_kinematics_nano_hexapod>> <<sec:detail_kinematics_nano_hexapod>>
** Introduction :ignore: ** Introduction :ignore:
Based on previous analysis, this section aims to determine the nano-hexapod geometry. Based on previous analysis, this section aims to determine the nano-hexapod optimal geometry.
For the NASS, the chosen reference frames $\{A\}$ and $\{B\}$ coincide with the sample's point of interest, which is positioned $150\,mm$ above the top platform. For the NASS, the chosen reference frames $\{A\}$ and $\{B\}$ coincide with the sample's point of interest, which is positioned $150\,mm$ above the top platform.
This is the location where precise control of the sample's position is required, as it is where the x-ray beam is focused. This is the location where precise control of the sample's position is required, as it is where the x-ray beam is focused.
@ -3258,24 +3241,21 @@ At any position, the system should be capable of performing $R_x$ and $R_y$ rota
Regarding stiffness, the resonance frequencies should be well above the maximum rotational velocity of $2\pi\,\text{rad/s}$ to minimize gyroscopic effects, while remaining below the problematic modes of the micro-station to ensure decoupling from its complex dynamics. Regarding stiffness, the resonance frequencies should be well above the maximum rotational velocity of $2\pi\,\text{rad/s}$ to minimize gyroscopic effects, while remaining below the problematic modes of the micro-station to ensure decoupling from its complex dynamics.
In terms of dynamics, the design should facilitate implementation of Integral Force Feedback (IFF) in a decentralized manner, and provide good decoupling for the high authority controller in the frame of the struts. In terms of dynamics, the design should facilitate implementation of Integral Force Feedback (IFF) in a decentralized manner, and provide good decoupling for the high authority controller in the frame of the struts.
A significant challenge in optimizing the nano-hexapod design arises from the variety of payloads that will be used, with masses ranging from 1 to 50kg.
This variation in payload characteristics makes it impossible to develop a single geometry that provides optimal dynamical properties for all possible configurations.
** Obtained Geometry ** Obtained Geometry
<<ssec:detail_kinematics_nano_hexapod_geometry>> <<ssec:detail_kinematics_nano_hexapod_geometry>>
Based on the previous analysis of Stewart platform configurations, while the geometry can be optimized to achieve the desired trade-off between stiffness and mobility in different directions, the wide range of potential payloads complicates the optimization process for obtaining consistent dynamical properties across all usage scenarios. Based on the previous analysis of Stewart platform configurations, while the geometry can be optimized to achieve the desired trade-off between stiffness and mobility in different directions, the wide range of potential payloads, with masses ranging from 1kg to 50kg, makes it impossible to develop a single geometry that provides optimal dynamical properties for all possible configurations.
For the nano-hexapod design, the struts were oriented more vertically compared to a cubic architecture due to several important considerations. For the nano-hexapod design, the struts were oriented more vertically compared to a cubic architecture due to several considerations.
First, the requirements in the vertical direction are more stringent than in the horizontal direction. First, the performance requirements in the vertical direction are more stringent than in the horizontal direction.
This vertical strut orientation decreases the amplification factor in the vertical direction, providing greater resolution and reducing the effects of actuator noise. This vertical strut orientation decreases the amplification factor in the vertical direction, providing greater resolution and reducing the effects of actuator noise.
Second, the micro-station's vertical modes exhibit higher frequencies than its lateral modes. Second, the micro-station's vertical modes exhibit higher frequencies than its lateral modes.
Therefore, higher resonance frequencies of the nano-hexapod in the vertical direction compared to the horizontal direction enhance the decoupling properties between the micro-station and the nano-hexapod. Therefore, higher resonance frequencies of the nano-hexapod in the vertical direction compared to the horizontal direction enhance the decoupling properties between the micro-station and the nano-hexapod.
Regarding dynamic properties, particularly for control in the frame of the struts, no specific optimization was implemented since the analysis revealed that the particular geometry has minimal impact on the resulting coupling characteristics. Regarding dynamical properties, particularly for control in the frame of the struts, no specific optimization was implemented since the analysis revealed that strut orientation has minimal impact on the resulting coupling characteristics.
Consequently, the geometry was selected according to practical constraints. Consequently, the geometry was selected according to practical constraints.
The height between the two plates is set at $95\,mm$. The height between the two plates is maximized and set at $95\,mm$.
Both platforms utilize the maximum available size, with joints offset by $15\,mm$ from the plate surfaces and positioned along circles with radii of $120\,mm$ for the fixed joints and $110\,mm$ for the mobile joints. Both platforms utilize the maximum available size, with joints offset by $15\,mm$ from the plate surfaces and positioned along circles with radii of $120\,mm$ for the fixed joints and $110\,mm$ for the mobile joints.
The positioning angles, as shown in Figure ref:fig:detail_kinematics_nano_hexapod_top, are $[255,\ 285,\ 15,\ 45,\ 135,\ 165]$ degrees for the top joints and $[220,\ 320,\ 340,\ 80,\ 100,\ 200]$ degrees for the bottom joints. The positioning angles, as shown in Figure ref:fig:detail_kinematics_nano_hexapod_top, are $[255,\ 285,\ 15,\ 45,\ 135,\ 165]$ degrees for the top joints and $[220,\ 320,\ 340,\ 80,\ 100,\ 200]$ degrees for the bottom joints.
@ -3301,7 +3281,7 @@ nano_hexapod = initializeCylindricalPlatforms(nano_hexapod, 'Fpr', 125e-3, 'Mpr'
#+begin_src matlab :exports none :results none #+begin_src matlab :exports none :results none
%% Obtained architecture for the Nano Hexapod %% Obtained architecture for the Nano Hexapod
displayArchitecture(nano_hexapod, 'labels', true, 'frames', false); displayArchitecture(nano_hexapod, 'labels', true, 'frames', true);
% Bottom circle % Bottom circle
h = 15e-3; h = 15e-3;
r = 120e-3; r = 120e-3;
@ -3330,8 +3310,35 @@ end
exportFig('figs/detail_kinematics_nano_hexapod_iso.pdf', 'width', 'normal', 'height', 'tall'); exportFig('figs/detail_kinematics_nano_hexapod_iso.pdf', 'width', 'normal', 'height', 'tall');
#+end_src #+end_src
#+begin_src matlab :tangle no :exports results :results file none #+begin_src matlab :exports none :results none :tangle no
%% Obtained architecture for the Nano Hexapod
displayArchitecture(nano_hexapod, 'labels', false, 'frames', true);
% Bottom circle
h = 15e-3;
r = 120e-3;
theta = linspace(0, 2*pi, 100);
x = r * cos(theta);
y = r * sin(theta);
z = h * ones(size(theta)); % All points at same height
plot3(x, y, z, '--', 'color', [colors(1,:)], 'LineWidth', 0.5);
for i = 1:6
plot3([0, nano_hexapod.platform_F.Fa(1,i)], [0, nano_hexapod.platform_F.Fa(2,i)], [h,h], '--', 'color', [colors(1,:)], 'LineWidth', 0.5);
end
% Top circle
h = 95e-3 - 15e-3;
r = 110e-3;
theta = linspace(0, 2*pi, 100);
x = r * cos(theta);
y = r * sin(theta);
z = h * ones(size(theta)); % All points at same height
plot3(x, y, z, '--', 'color', [colors(2,:)], 'LineWidth', 0.5);
for i = 1:6
plot3([0, nano_hexapod.platform_M.Mb(1,i)], [0, nano_hexapod.platform_M.Mb(2,i)], [h,h], '--', 'color', [colors(2,:)], 'LineWidth', 0.5);
end
view([0,90]) view([0,90])
#+end_src
#+begin_src matlab :tangle no :exports results :results file none
exportFig('figs/detail_kinematics_nano_hexapod_top.pdf', 'width', 500, 'height', 'tall'); exportFig('figs/detail_kinematics_nano_hexapod_top.pdf', 'width', 500, 'height', 'tall');
#+end_src #+end_src
@ -3598,7 +3605,7 @@ For the cubic configuration, complete dynamical decoupling in the Cartesian fram
Modified cubic architectures with the cube's center positioned above the top platform were proposed as a potential solution, but proved unsuitable for the nano-hexapod due to size constraints and the impracticality of ensuring that different payloads' centers of mass would consistently align with the cube's center. Modified cubic architectures with the cube's center positioned above the top platform were proposed as a potential solution, but proved unsuitable for the nano-hexapod due to size constraints and the impracticality of ensuring that different payloads' centers of mass would consistently align with the cube's center.
For the nano-hexapod design, a key challenge was addressing the wide range of potential payloads (1 to 50kg), which made it impossible to optimize the geometry for consistent dynamic performance across all usage scenarios. For the nano-hexapod design, a key challenge was addressing the wide range of potential payloads (1 to 50kg), which made it impossible to optimize the geometry for consistent dynamic performance across all usage scenarios.
This led to a practical design approach where struts were oriented more vertically than in cubic configurations to address several application-specific needs: achieving higher resolution in the vertical direction by reducing amplification factors, better matching the micro-station's modal characteristics with higher vertical resonance frequencies, and accommodating the stringent vertical positioning requirements. This led to a practical design approach where struts were oriented more vertically than in cubic configurations to address several application-specific needs: achieving higher resolution in the vertical direction by reducing amplification factors and better matching the micro-station's modal characteristics with higher vertical resonance frequencies.
* Bibliography :ignore: * Bibliography :ignore:
#+latex: \printbibliography[heading=bibintoc,title={Bibliography}] #+latex: \printbibliography[heading=bibintoc,title={Bibliography}]

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@ -1,4 +1,4 @@
% Created 2025-04-02 Wed 17:04 % Created 2025-04-03 Thu 10:35
% Intended LaTeX compiler: pdflatex % Intended LaTeX compiler: pdflatex
\documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt} \documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt}
@ -13,7 +13,7 @@
pdftitle={Nano Hexapod - Optimal Geometry}, pdftitle={Nano Hexapod - Optimal Geometry},
pdfkeywords={}, pdfkeywords={},
pdfsubject={}, pdfsubject={},
pdfcreator={Emacs 29.4 (Org mode 9.6)}, pdfcreator={Emacs 30.1 (Org mode 9.7.26)},
pdflang={English}} pdflang={English}}
\usepackage{biblatex} \usepackage{biblatex}
@ -23,7 +23,6 @@
\tableofcontents \tableofcontents
\clearpage \clearpage
The performance of a Stewart platform depends on its geometric configuration, especially the orientation of its struts and the positioning of its joints. The performance of a Stewart platform depends on its geometric configuration, especially the orientation of its struts and the positioning of its joints.
During the conceptual design phase of the nano-hexapod, a preliminary geometry was selected based on general principles without detailed optimization. During the conceptual design phase of the nano-hexapod, a preliminary geometry was selected based on general principles without detailed optimization.
As the project advanced to the detailed design phase, a rigorous analysis of how geometry influences system performance became essential to ensure that the final design would meet the demanding requirements of the Nano Active Stabilization System (NASS). As the project advanced to the detailed design phase, a rigorous analysis of how geometry influences system performance became essential to ensure that the final design would meet the demanding requirements of the Nano Active Stabilization System (NASS).
@ -32,9 +31,8 @@ In this chapter, the nano-hexapod geometry is optimized through careful analysis
The chapter begins with a comprehensive review of existing Stewart platform designs in Section \ref{sec:detail_kinematics_stewart_review}, surveying various approaches to geometry, actuation, sensing, and joint design from the literature. The chapter begins with a comprehensive review of existing Stewart platform designs in Section \ref{sec:detail_kinematics_stewart_review}, surveying various approaches to geometry, actuation, sensing, and joint design from the literature.
Section \ref{sec:detail_kinematics_geometry} develops the analytical framework that connects geometric parameters to performance characteristics, establishing quantitative relationships that guide the optimization process. Section \ref{sec:detail_kinematics_geometry} develops the analytical framework that connects geometric parameters to performance characteristics, establishing quantitative relationships that guide the optimization process.
Section \ref{sec:detail_kinematics_cubic} examines the cubic configuration a specialized architecture that has garnered significant attention for its purported advantages—to evaluate its suitability for the nano-hexapod application. Section \ref{sec:detail_kinematics_cubic} examines the cubic configuration, a specialized architecture that has gathered significant attention, to evaluate its suitability for the nano-hexapod application.
Finally, Section \ref{sec:detail_kinematics_nano_hexapod} presents the optimized nano-hexapod geometry derived from these analyses and demonstrates how it addresses the specific requirements of the NASS. Finally, Section \ref{sec:detail_kinematics_nano_hexapod} presents the optimized nano-hexapod geometry derived from these analyses and demonstrates how it addresses the specific requirements of the NASS.
\chapter{Review of Stewart platforms} \chapter{Review of Stewart platforms}
\label{sec:detail_kinematics_stewart_review} \label{sec:detail_kinematics_stewart_review}
As was explained in the conceptual phase, Stewart platforms have the following key elements: two plates connected by six struts, with each strut composed of a joint at each end, an actuator, and one or several sensors. As was explained in the conceptual phase, Stewart platforms have the following key elements: two plates connected by six struts, with each strut composed of a joint at each end, an actuator, and one or several sensors.
@ -44,10 +42,10 @@ Long stroke Stewart platforms are not considered here as their design imposes ot
Some Stewart platforms found in the literature are listed in Table \ref{tab:detail_kinematics_stewart_review}. Some Stewart platforms found in the literature are listed in Table \ref{tab:detail_kinematics_stewart_review}.
\begin{table}[htbp] \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 ordered by appearance in the literature} \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 order of appearance in the literature}
\centering \centering
\scriptsize \scriptsize
\begin{tabularx}{0.8\linewidth}{Xcccc} \begin{tabularx}{0.9\linewidth}{Xcccc}
\toprule \toprule
& \textbf{Geometry} & \textbf{Actuators} & \textbf{Sensors} & \textbf{Reference}\\ & \textbf{Geometry} & \textbf{Actuators} & \textbf{Sensors} & \textbf{Reference}\\
\midrule \midrule
@ -74,7 +72,6 @@ Figure \ref{fig:detail_kinematics_naves} & Non-Cubic & 3-phase rotary motor & Ro
\end{tabularx} \end{tabularx}
\end{table} \end{table}
All presented Stewart platforms utilize flexible joints, as this is a prerequisite for nano-positioning capabilities. All presented Stewart platforms utilize flexible joints, as this is a prerequisite for nano-positioning capabilities.
Flexible joints can have various implementations, which will be discussed when designing the nano-hexapod flexible joints. Flexible joints can have various implementations, which will be discussed when designing the nano-hexapod flexible joints.
In terms of actuation, most Stewart platforms employ either voice coil actuators (such as the ones shown in Figures \ref{fig:detail_kinematics_jpl}, \ref{fig:detail_kinematics_uw_gsp} and \ref{fig:detail_kinematics_pph}) or piezoelectric actuators (such as the ones shown in Figures \ref{fig:detail_kinematics_ulb_pz}, \ref{fig:detail_kinematics_uqp} and \ref{fig:detail_kinematics_yang19}). In terms of actuation, most Stewart platforms employ either voice coil actuators (such as the ones shown in Figures \ref{fig:detail_kinematics_jpl}, \ref{fig:detail_kinematics_uw_gsp} and \ref{fig:detail_kinematics_pph}) or piezoelectric actuators (such as the ones shown in Figures \ref{fig:detail_kinematics_ulb_pz}, \ref{fig:detail_kinematics_uqp} and \ref{fig:detail_kinematics_yang19}).
@ -86,13 +83,13 @@ The actuator and sensor selection for the nano-hexapod will also be described in
\begin{center} \begin{center}
\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_kinematics_jpl.jpg} \includegraphics[scale=1,width=0.95\linewidth]{figs/detail_kinematics_jpl.jpg}
\end{center} \end{center}
\subcaption{\label{fig:detail_kinematics_jpl}California Institute of Technology - USA} \subcaption{\label{fig:detail_kinematics_jpl}California Institute of Technology - USA \cite{spanos95_soft_activ_vibrat_isolat}}
\end{subfigure} \end{subfigure}
\begin{subfigure}{0.48\textwidth} \begin{subfigure}{0.48\textwidth}
\begin{center} \begin{center}
\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_kinematics_uw_gsp.jpg} \includegraphics[scale=1,width=0.95\linewidth]{figs/detail_kinematics_uw_gsp.jpg}
\end{center} \end{center}
\subcaption{\label{fig:detail_kinematics_uw_gsp}University of Wyoming - USA} \subcaption{\label{fig:detail_kinematics_uw_gsp}University of Wyoming - USA \cite{mcinroy99_dynam}}
\end{subfigure} \end{subfigure}
\bigskip \bigskip
@ -100,35 +97,35 @@ The actuator and sensor selection for the nano-hexapod will also be described in
\begin{center} \begin{center}
\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_kinematics_ulb_pz.jpg} \includegraphics[scale=1,width=0.95\linewidth]{figs/detail_kinematics_ulb_pz.jpg}
\end{center} \end{center}
\subcaption{\label{fig:detail_kinematics_ulb_pz}ULB - Belgium} \subcaption{\label{fig:detail_kinematics_ulb_pz}ULB - Belgium \cite{abu02_stiff_soft_stewar_platf_activ}}
\end{subfigure} \end{subfigure}
\begin{subfigure}{0.43\textwidth} \begin{subfigure}{0.43\textwidth}
\begin{center} \begin{center}
\includegraphics[scale=1,width=0.95\linewidth]{figs/detail_kinematics_uqp.jpg} \includegraphics[scale=1,width=0.95\linewidth]{figs/detail_kinematics_uqp.jpg}
\end{center} \end{center}
\subcaption{\label{fig:detail_kinematics_uqp}Naval Postgraduate School - USA} \subcaption{\label{fig:detail_kinematics_uqp}Naval Postgraduate School - USA \cite{agrawal04_algor_activ_vibrat_isolat_spacec}}
\end{subfigure} \end{subfigure}
\caption{\label{fig:detail_kinematics_stewart_examples_cubic}Some examples of developped Stewart platform with Cubic geometry. (\subref{fig:detail_kinematics_jpl}), (\subref{fig:detail_kinematics_uw_gsp}), (\subref{fig:detail_kinematics_ulb_pz}), (\subref{fig:detail_kinematics_uqp})} \caption{\label{fig:detail_kinematics_stewart_examples_cubic}Some examples of developped Stewart platform with Cubic geometry}
\end{figure} \end{figure}
There are two main categories of Stewart platform geometry. There are two main categories of Stewart platform geometry.
The first is cubic architecture (some exampled are presented in Figure \ref{fig:detail_kinematics_stewart_examples_cubic}), where struts are positioned along six sides of a cube (and are therefore orthogonal to each other). The first is cubic architecture (some exampled are presented in Figure \ref{fig:detail_kinematics_stewart_examples_cubic}), where struts are positioned along six sides of a cube (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}. Such specific architecture has some special properties that will be studied in Section \ref{sec:detail_kinematics_cubic}.
The second is non-cubic architecture (Figure \ref{fig:detail_kinematics_stewart_examples_non_cubic}), where the orientation of the struts and position of the joints can be optimized based on performance criteria. The second is non-cubic architecture (Figure \ref{fig:detail_kinematics_stewart_examples_non_cubic}), where the orientation of the struts and position of the joints can be optimized based on defined performance criteria.
The effect of strut orientation and position of the joints on the Stewart platform properties is discussed Section \ref{sec:detail_kinematics_geometry}. The effect of strut orientation and position of the joints on the Stewart platform properties is discussed in Section \ref{sec:detail_kinematics_geometry}.
\begin{figure}[htbp] \begin{figure}[htbp]
\begin{subfigure}{0.48\textwidth} \begin{subfigure}{0.48\textwidth}
\begin{center} \begin{center}
\includegraphics[scale=1,height=5cm]{figs/detail_kinematics_pph.jpg} \includegraphics[scale=1,height=5cm]{figs/detail_kinematics_pph.jpg}
\end{center} \end{center}
\subcaption{\label{fig:detail_kinematics_pph}Naval Postgraduate School - USA} \subcaption{\label{fig:detail_kinematics_pph}Naval Postgraduate School - USA \cite{chen03_payload_point_activ_vibrat_isolat}}
\end{subfigure} \end{subfigure}
\begin{subfigure}{0.48\textwidth} \begin{subfigure}{0.48\textwidth}
\begin{center} \begin{center}
\includegraphics[scale=1,height=5cm]{figs/detail_kinematics_zhang11.jpg} \includegraphics[scale=1,height=5cm]{figs/detail_kinematics_zhang11.jpg}
\end{center} \end{center}
\subcaption{\label{fig:detail_kinematics_zhang11}Beihang University - China} \subcaption{\label{fig:detail_kinematics_zhang11}Beihang University - China \cite{zhang11_six_dof}}
\end{subfigure} \end{subfigure}
\bigskip \bigskip
@ -136,24 +133,23 @@ The effect of strut orientation and position of the joints on the Stewart platfo
\begin{center} \begin{center}
\includegraphics[scale=1,height=5cm]{figs/detail_kinematics_yang19.jpg} \includegraphics[scale=1,height=5cm]{figs/detail_kinematics_yang19.jpg}
\end{center} \end{center}
\subcaption{\label{fig:detail_kinematics_yang19}Nanjing University - China} \subcaption{\label{fig:detail_kinematics_yang19}Nanjing University - China \cite{yang19_dynam_model_decoup_contr_flexib}}
\end{subfigure} \end{subfigure}
\begin{subfigure}{0.53\textwidth} \begin{subfigure}{0.53\textwidth}
\begin{center} \begin{center}
\includegraphics[scale=1,height=5cm]{figs/detail_kinematics_naves.jpg} \includegraphics[scale=1,height=5cm]{figs/detail_kinematics_naves.jpg}
\end{center} \end{center}
\subcaption{\label{fig:detail_kinematics_naves}University of Twente - Netherlands} \subcaption{\label{fig:detail_kinematics_naves}University of Twente - Netherlands \cite{naves20_desig}}
\end{subfigure} \end{subfigure}
\caption{\label{fig:detail_kinematics_stewart_examples_non_cubic}Some examples of developped Stewart platform with non-cubic geometry. (\subref{fig:detail_kinematics_pph}), (\subref{fig:detail_kinematics_zhang11}), (\subref{fig:detail_kinematics_yang19}), (\subref{fig:detail_kinematics_naves})} \caption{\label{fig:detail_kinematics_stewart_examples_non_cubic}Some examples of developped Stewart platform with non-cubic geometry}
\end{figure} \end{figure}
\chapter{Effect of geometry on Stewart platform properties} \chapter{Effect of geometry on Stewart platform properties}
\label{sec:detail_kinematics_geometry} \label{sec:detail_kinematics_geometry}
As was demonstrated during the conceptual phase, the geometry of the Stewart platform impacts the stiffness and compliance characteristics, the mobility or workspace, the force authority, and the dynamics of the manipulator. As was demonstrated during the conceptual phase, the geometry of the Stewart platform impacts the stiffness and compliance characteristics, the mobility (or workspace), the force authority, and the dynamics of the manipulator.
It is therefore essential to understand how the geometry impacts these properties, and to develop methodologies for optimizing the geometry for specific applications. It is therefore essential to understand how the geometry impacts these properties, and to develop methodologies for optimizing the geometry for specific applications.
An important analytical tool for this study is the Jacobian matrix, which depends on \(\bm{b}_i\) (joint position with respect to the top platform) and \(\hat{\bm{s}}_i\) (orientation of struts). An important analytical tool for this study is the Jacobian matrix, which depends on \(\bm{b}_i\) (joints' position with respect to the top platform) and \(\hat{\bm{s}}_i\) (struts' orientation).
The choice of frames (\(\{A\}\) and \(\{B\}\)), independently of the physical Stewart platform geometry, impacts the obtained kinematics and stiffness matrix, as these are defined for forces and motion evaluated at the chosen frame. The choice of \(\{A\}\) and \(\{B\}\) frames, independently of the physical Stewart platform geometry, impacts the obtained kinematics and stiffness matrix, as these are defined for forces and motion evaluated at the chosen frame.
\section{Platform Mobility / Workspace} \section{Platform Mobility / Workspace}
\label{ssec:detail_kinematics_geometry_mobility} \label{ssec:detail_kinematics_geometry_mobility}
The mobility of the Stewart platform (or any manipulator) is defined as the range of motion that it can perform. The mobility of the Stewart platform (or any manipulator) is defined as the range of motion that it can perform.
@ -166,21 +162,21 @@ Mobility of parallel manipulators is inherently difficult to study as the transl
The analysis is significantly simplified when considering small motions, as the Jacobian matrix can be used to link the strut motion to the motion of frame \(\{B\}\) with respect to \(\{A\}\) through \eqref{eq:detail_kinematics_jacobian}, which is a linear equation. The analysis is significantly simplified when considering small motions, as the Jacobian matrix can be used to link the strut motion to the motion of frame \(\{B\}\) with respect to \(\{A\}\) through \eqref{eq:detail_kinematics_jacobian}, which is a linear equation.
\begin{equation}\label{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} \begin{bmatrix} \delta l_1 \\ \delta l_2 \\ \delta l_3 \\ \delta l_4 \\ \delta l_5 \\ \delta l_6 \end{bmatrix} = \underbrace{\begin{bmatrix}
{{}^A\hat{\bm{s}}_1}^T & ({}^A\bm{b}_1 \times {}^A\hat{\bm{s}}_1)^T \\ {{}^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}}_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}}_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}}_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}}_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 {{}^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{bmatrix}}_{\bm{J}} \begin{bmatrix} \delta x \\ \delta y \\ \delta z \\ \delta \theta_x \\ \delta \theta_y \\ \delta \theta_z \end{bmatrix}
\end{equation} \end{equation}
Therefore, the mobility of the Stewart platform (defined as the set of achievable \([\delta x\ \delta y\ \delta z\ \delta \theta_x\ \delta \theta_y\ \delta \theta_z]\)) depends on two key factors: the stroke of each strut and the geometry of the Stewart platform (embodied in the Jacobian matrix). Therefore, the mobility of the Stewart platform (defined as the set of achievable \([\delta x\ \delta y\ \delta z\ \delta \theta_x\ \delta \theta_y\ \delta \theta_z]\)) depends on two key factors: the stroke of each strut and the geometry of the Stewart platform (embodied in the Jacobian matrix).
More specifically, the XYZ mobility only depends on the \(s_i\) (orientation of struts), while the mobility in rotation also depends on \(b_i\) (position of top joints). More specifically, the XYZ mobility only depends on the \(\hat{\bm{s}}_i\) (orientation of struts), while the mobility in rotation also depends on \(\bm{b}_i\) (position of top joints).
\paragraph{Mobility in translation} \paragraph{Mobility in translation}
For simplicity, only translations are first considered (i.e., fixed orientation of the Stewart platform). For simplicity, only translations are first considered (i.e., the Stewart platform is considered to have fixed orientation).
In the general case, the translational mobility can be represented by a 3D shape having 12 faces, where each actuator limits the stroke along its axis in positive and negative directions. In the general case, the translational mobility can be represented by a 3D shape having 12 faces, where each actuator limits the stroke along its axis in positive and negative directions.
The faces are therefore perpendicular to the strut direction. The faces are therefore perpendicular to the strut direction.
The obtained mobility for the Stewart platform geometry shown in Figure \ref{fig:detail_kinematics_mobility_trans_arch} is computed and represented in Figure \ref{fig:detail_kinematics_mobility_trans_result}. The obtained mobility for the Stewart platform geometry shown in Figure \ref{fig:detail_kinematics_mobility_trans_arch} is computed and represented in Figure \ref{fig:detail_kinematics_mobility_trans_result}.
@ -198,19 +194,19 @@ The obtained mobility for the Stewart platform geometry shown in Figure \ref{fig
\end{center} \end{center}
\subcaption{\label{fig:detail_kinematics_mobility_trans_result}Translational mobility} \subcaption{\label{fig:detail_kinematics_mobility_trans_result}Translational mobility}
\end{subfigure} \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})} \caption{\label{fig:detail_kinematics_mobility_trans}One Stewart platform geometry (\subref{fig:detail_kinematics_mobility_trans_arch}) and its associated translational mobility (\subref{fig:detail_kinematics_mobility_trans_result}). A sphere with radius equal to the strut stroke is contained in the translational mobility shape.}
\end{figure} \end{figure}
With the previous interpretations of the 12 faces making the translational mobility 3D shape, it can be concluded that for a strut stroke of \(\pm d\), a sphere with radius \(d\) is contained in the 3D shape and touches it along the six lines defined by the strut axes, as illustrated in Figure \ref{fig:detail_kinematics_mobility_trans_result}. With the previous interpretations of the 12 faces making the translational mobility 3D shape, it can be concluded that for a strut stroke of \(\pm d\), a sphere with radius \(d\) is contained in the 3D shape and touches it in directions defined by the strut axes, as illustrated in Figure \ref{fig:detail_kinematics_mobility_trans_result}.
This means that the mobile platform can be translated in any direction with a stroke of \(d\). This means that the mobile platform can be translated in any direction with a stroke equal to the strut stroke.
To better understand how the geometry of the Stewart platform impacts the translational mobility, two configurations are compared with struts oriented vertically (Figure \ref{fig:detail_kinematics_stewart_mobility_vert_struts}) and struts oriented horizontally (Figure \ref{fig:detail_kinematics_stewart_mobility_hori_struts}). To better understand how the geometry of the Stewart platform impacts the translational mobility, two configurations are compared with struts oriented vertically (Figure \ref{fig:detail_kinematics_stewart_mobility_vert_struts}) and struts oriented horizontally (Figure \ref{fig:detail_kinematics_stewart_mobility_hori_struts}).
The vertically oriented struts lead to greater stroke in the horizontal direction and reduced stroke in the vertical direction (Figure \ref{fig:detail_kinematics_mobility_translation_strut_orientation}). The vertically oriented struts configuration leads to greater stroke in the horizontal direction and reduced stroke in the vertical direction (Figure \ref{fig:detail_kinematics_mobility_translation_strut_orientation}).
Conversely, horizontal oriented struts provide more stroke in the vertical direction. Conversely, horizontal oriented struts configuration provides more stroke in the vertical direction.
It may seem counterintuitive that less stroke is available in the direction of the struts. It may seem counterintuitive that less stroke is available in the direction of the struts.
This phenomenon occurs because the struts form a lever mechanism that amplifies the motion. This phenomenon occurs because the struts form a lever mechanism that amplifies the motion.
The amplification factor increases when the struts have a high angle with the direction of motion and equals one when aligned with the direction of motion. The amplification factor increases when the struts have a high angle with the direction of motion and equals one (i.e. is minimal) when aligned with the direction of motion.
\begin{figure}[htbp] \begin{figure}[htbp]
\begin{subfigure}{0.25\textwidth} \begin{subfigure}{0.25\textwidth}
@ -233,7 +229,6 @@ The amplification factor increases when the struts have a high angle with the di
\end{subfigure} \end{subfigure}
\caption{\label{fig:detail_kinematics_stewart_mobility_translation_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}).} \caption{\label{fig:detail_kinematics_stewart_mobility_translation_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} \end{figure}
\paragraph{Mobility in rotation} \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. 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.
@ -266,18 +261,17 @@ Having struts further apart decreases the ``lever arm'' and therefore reduces th
\end{subfigure} \end{subfigure}
\caption{\label{fig:detail_kinematics_stewart_mobility_rotation_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}).} \caption{\label{fig:detail_kinematics_stewart_mobility_rotation_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} \end{figure}
\paragraph{Combined translations and rotations} \paragraph{Combined translations and rotations}
It is possible to consider combined translations and rotations, although displaying such mobility becomes more complex. It is possible to consider combined translations and rotations, although displaying such mobility becomes more complex.
For a fixed geometry and a desired mobility (combined translations and rotations), it is possible to estimate the required minimum actuator stroke. For a fixed geometry and a desired mobility (combined translations and rotations), it is possible to estimate the required minimum actuator stroke.
This analysis was conducted in Section \ref{sec:detail_kinematics_nano_hexapod} to estimate the required actuator stroke for the nano-hexapod geometry. This analysis is conducted in Section \ref{sec:detail_kinematics_nano_hexapod} to estimate the required actuator stroke for the nano-hexapod geometry.
\section{Stiffness} \section{Stiffness}
\label{ssec:detail_kinematics_geometry_stiffness} \label{ssec:detail_kinematics_geometry_stiffness}
The stiffness matrix defines how the nano-hexapod deforms (frame \(\{B\}\) with respect to frame \(\{A\}\)) due to static forces/torques applied on \(\{B\}\). 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}. 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 there were optimized after the geometry was fixed, but several work were done to quantify the impact of the flexible joint stiffness \cite{mcinroy00_desig_contr_flexur_joint_hexap,mcinroy02_model_desig_flexur_joint_stewar}. 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}.
\begin{equation}\label{eq:detail_kinematics_stiffness_matrix} \begin{equation}\label{eq:detail_kinematics_stiffness_matrix}
\bm{K} = \bm{J}^T \bm{\mathcal{K}} \bm{J} \bm{K} = \bm{J}^T \bm{\mathcal{K}} \bm{J}
@ -299,27 +293,24 @@ In that case, the obtained stiffness matrix linearly depends on the strut stiffn
\paragraph{Translation Stiffness} \paragraph{Translation Stiffness}
As shown by equation \eqref{eq:detail_kinematics_stiffness_matrix_simplified}, the translation stiffnesses (the \(3 \times 3\) top left terms of the stiffness matrix) only depend on the orientation of the struts and not their location: \(\hat{\bm{s}}_i \cdot \hat{\bm{s}}_i^T\). As shown by equation \eqref{eq:detail_kinematics_stiffness_matrix_simplified}, the translation stiffnesses (the \(3 \times 3\) top left terms of the stiffness matrix) only depend on the orientation of the struts and not their location: \(\hat{\bm{s}}_i \cdot \hat{\bm{s}}_i^T\).
In the extreme case where all struts are vertical with \(s_i = [0,\ 0,\ 1]\), a vertical stiffness of \(6k\) is achieved, but with null stiffness in the X and Y directions. In the extreme case where all struts are vertical (\(s_i = [0\ 0\ 1]\)), a vertical stiffness of \(6k\) is achieved, but with null stiffness in the horizontal directions.
If two struts are aligned along the X axis, two struts along the Y axis, and two struts along the Z axis, then \(\hat{\bm{s}}_i \cdot \hat{\bm{s}}_i^T = 2 \bm{I}_3\), resulting in well-distributed stiffness along all directions. If two struts are aligned with the X axis, two struts with the Y axis, and two struts with the Z axis, then \(\hat{\bm{s}}_i \cdot \hat{\bm{s}}_i^T = 2 \bm{I}_3\), resulting in well-distributed stiffness along all directions.
This configuration corresponds to the cubic architecture presented in Section \ref{sec:detail_kinematics_cubic}. This configuration corresponds to the cubic architecture presented in Section \ref{sec:detail_kinematics_cubic}.
When struts are oriented more vertically (Figure \ref{fig:detail_kinematics_stewart_mobility_vert_struts}), vertical stiffness increases while horizontal stiffness decreases. When the struts are oriented more vertically, as shown in Figure \ref{fig:detail_kinematics_stewart_mobility_vert_struts}, the vertical stiffness increases while the horizontal stiffness decreases.
Additionally, \(R_x\) and \(R_y\) stiffness increases while \(R_z\) stiffness decreases. Additionally, \(R_x\) and \(R_y\) stiffness increases while \(R_z\) stiffness decreases.
The opposite conclusions apply if struts are oriented more horizontally (Figure \ref{fig:detail_kinematics_stewart_mobility_hori_struts}). The opposite conclusions apply if struts are oriented more horizontally, illustrated in Figure \ref{fig:detail_kinematics_stewart_mobility_hori_struts}.
\paragraph{Rotational Stiffness} \paragraph{Rotational Stiffness}
The rotational stiffnesses depend both on the orientation of the struts and on the location of the top joints (with respect to the considered center of rotation, i.e., the location of frame \(\{B\}\)). The rotational stiffnesses depend both on the orientation of the struts and on the location of the top joints with respect to the considered center of rotation (i.e., the location of frame \(\{A\}\)).
With the same orientation but increased distances (\(b_i\)) by a factor of 2, the rotational stiffness is increased by a factor of 4. With the same orientation but increased distances to the frame \(\{A\}\) by a factor of 2, the rotational stiffness is increased by a factor of 4.
Therefore, the compact Stewart platform depicted in Figure \ref{fig:detail_kinematics_stewart_mobility_close_struts} has less rotational stiffness than the Stewart platform shown in Figure \ref{fig:detail_kinematics_stewart_mobility_space_struts}. Therefore, the compact Stewart platform depicted in Figure \ref{fig:detail_kinematics_stewart_mobility_close_struts} has less rotational stiffness than the Stewart platform shown in Figure \ref{fig:detail_kinematics_stewart_mobility_space_struts}.
\paragraph{Diagonal Stiffness Matrix} \paragraph{Diagonal Stiffness Matrix}
Having a diagonal stiffness matrix \(\bm{K}\) can be beneficial for control purposes as it would make the plant in the Cartesian frame decoupled at low frequency. Having a diagonal stiffness matrix \(\bm{K}\) can be beneficial for control purposes as it would make the plant in the Cartesian frame decoupled at low frequency.
This property depends on both the geometry and the chosen \(\{B\}\) frame. This property depends on both the geometry and the chosen \(\{A\}\) frame.
For specific geometry and choice of \(\{B\}\) frame, it is possible to achieve a diagonal \(K\) matrix. For specific geometry and choice of \(\{A\}\) frame, it is possible to achieve a diagonal \(K\) matrix.
This is discussed in Section \ref{ssec:detail_kinematics_cubic_static}. This is discussed in Section \ref{ssec:detail_kinematics_cubic_static}.
\section{Dynamical properties} \section{Dynamical properties}
\label{ssec:detail_kinematics_geometry_dynamics} \label{ssec:detail_kinematics_geometry_dynamics}
@ -332,23 +323,22 @@ These conditions are studied in Section \ref{ssec:detail_kinematics_cubic_dynami
\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} \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} \end{equation}
In the frame of the struts, the equations of motion given by equation \eqref{eq:detail_kinematics_transfer_function_struts} are well decoupled at low frequency. In the frame of the struts, the equations of motion \eqref{eq:detail_kinematics_transfer_function_struts} are well decoupled at low frequency.
This is why most Stewart platforms are controlled in the frame of the struts: below the resonance frequency, the system is decoupled and SISO control may be applied for each strut, independently of the payload being used. This is why most Stewart platforms are controlled in the frame of the struts: below the resonance frequency, the system is well decoupled and SISO control may be applied for each strut, independently of the payload being used.
\begin{equation}\label{eq:detail_kinematics_transfer_function_struts} \begin{equation}\label{eq:detail_kinematics_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} \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} \end{equation}
Coupling between sensors (force sensors, relative position sensors, inertial sensors) in different struts may also be important for decentralized control. Coupling between sensors (force sensors, relative position sensors or inertial sensors) in different struts may also be important for decentralized control.
In section \ref{ssec:detail_kinematics_decentralized_control}, it will be studied whether the Stewart platform geometry can be optimized to have lower coupling between the struts. In section \ref{ssec:detail_kinematics_decentralized_control}, it will be studied whether the Stewart platform geometry can be optimized to have lower coupling between the struts.
\section*{Conclusion} \section*{Conclusion}
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}. The effects of two changes in the manipulator's geometry, namely the position and orientation of the struts, are summarized in Table \ref{tab:detail_kinematics_geometry}.
These results could have been easily deduced based on mechanical principles, but thanks to the kinematic analysis, they can be quantified. These results could have been easily deduced based on mechanical principles, but thanks to the kinematic analysis, they can be quantified.
These trade-offs provide important guidelines when choosing the Stewart platform geometry. These trade-offs provide important guidelines when choosing the Stewart platform geometry.
\begin{table}[htbp] \begin{table}[htbp]
\caption{\label{tab:detail_kinematics_geometry}Effect of a change in geometry on the manipulator's stiffness, force authority and stroke} \caption{\label{tab:detail_kinematics_geometry}Effect of a change in geometry on the manipulator's stiffness and mobility}
\centering \centering
\small \small
\begin{tabularx}{0.8\linewidth}{Xcc} \begin{tabularx}{0.8\linewidth}{Xcc}
@ -360,14 +350,13 @@ Horizontal stiffness & \(\searrow\) & \(=\)\\
Vertical rotation stiffness & \(\searrow\) & \(\nearrow\)\\ Vertical rotation stiffness & \(\searrow\) & \(\nearrow\)\\
Horizontal rotation stiffness & \(\nearrow\) & \(\nearrow\)\\ Horizontal rotation stiffness & \(\nearrow\) & \(\nearrow\)\\
\midrule \midrule
Vertical stroke & \(\searrow\) & \(=\)\\ Vertical mobility & \(\searrow\) & \(=\)\\
Horizontal stroke & \(\nearrow\) & \(=\)\\ Horizontal mobility & \(\nearrow\) & \(=\)\\
Vertical rotation stroke & \(\nearrow\) & \(\searrow\)\\ Vertical rotation mobility & \(\nearrow\) & \(\searrow\)\\
Horizontal rotation stroke & \(\searrow\) & \(\searrow\)\\ Horizontal rotation mobility & \(\searrow\) & \(\searrow\)\\
\bottomrule \bottomrule
\end{tabularx} \end{tabularx}
\end{table} \end{table}
\chapter{The Cubic Architecture} \chapter{The Cubic Architecture}
\label{sec:detail_kinematics_cubic} \label{sec:detail_kinematics_cubic}
The Cubic configuration for the Stewart platform was first proposed in \cite{geng94_six_degree_of_freed_activ}. The Cubic configuration for the Stewart platform was first proposed in \cite{geng94_six_degree_of_freed_activ}.
@ -382,15 +371,15 @@ It is also possible to implement designs with strut lengths smaller than the cub
\begin{center} \begin{center}
\includegraphics[scale=1,scale=1]{figs/detail_kinematics_cubic_architecture_example.png} \includegraphics[scale=1,scale=1]{figs/detail_kinematics_cubic_architecture_example.png}
\end{center} \end{center}
\subcaption{\label{fig:detail_kinematics_cubic_architecture_example}sub caption a} \subcaption{\label{fig:detail_kinematics_cubic_architecture_example}Classical Cubic architecture}
\end{subfigure} \end{subfigure}
\begin{subfigure}{0.49\textwidth} \begin{subfigure}{0.49\textwidth}
\begin{center} \begin{center}
\includegraphics[scale=1,scale=1]{figs/detail_kinematics_cubic_architecture_example_small.png} \includegraphics[scale=1,scale=1]{figs/detail_kinematics_cubic_architecture_example_small.png}
\end{center} \end{center}
\subcaption{\label{fig:detail_kinematics_cubic_architecture_example_small}sub caption b} \subcaption{\label{fig:detail_kinematics_cubic_architecture_example_small}Alternative configuration}
\end{subfigure} \end{subfigure}
\caption{\label{fig:detail_kinematics_cubic_architecture_examples}Typical Stewart platform cubic architectures. (\subref{fig:detail_kinematics_cubic_architecture_example}) (\subref{fig:detail_kinematics_cubic_architecture_example_small})} \caption{\label{fig:detail_kinematics_cubic_architecture_examples}Typical Stewart platform cubic architectures in which struts' length is similar to the cube edges's length (\subref{fig:detail_kinematics_cubic_architecture_example}) or is taking just a portion of the edge (\subref{fig:detail_kinematics_cubic_architecture_example_small}).}
\end{figure} \end{figure}
@ -406,17 +395,16 @@ The ultimate objective is to determine the suitability of the cubic architecture
\label{ssec:detail_kinematics_cubic_static} \label{ssec:detail_kinematics_cubic_static}
\paragraph{Stiffness matrix for the Cubic architecture} \paragraph{Stiffness matrix for the Cubic architecture}
Consider the cubic architecture depicted in Figure \ref{fig:detail_kinematics_cubic_schematic_full}.
Consider the cubic architecture shown in Figure \ref{fig:detail_kinematics_cubic_schematic_full}. Consider the cubic architecture shown in Figure \ref{fig:detail_kinematics_cubic_schematic_full}.
The unit vectors corresponding to the edges of the cube are described by equation \eqref{eq:detail_kinematics_cubic_s}. The unit vectors corresponding to the edges of the cube are described by equation \eqref{eq:detail_kinematics_cubic_s}.
\begin{equation}\label{eq:detail_kinematics_cubic_s} \begin{equation}\label{eq:detail_kinematics_cubic_s}
\hat{\bm{s}}_1 = \begin{bmatrix} \sqrt{2}/\sqrt{3} \\ 0 \\ 1/\sqrt{3} \end{bmatrix} \quad \hat{\bm{s}}_1 = \begin{bmatrix} \frac{\sqrt{2}}{\sqrt{3}} \\ 0 \\ \frac{1}{\sqrt{3}} \end{bmatrix} \quad
\hat{\bm{s}}_2 = \begin{bmatrix} -1/\sqrt{6} \\ -1/\sqrt{2} \\ 1/\sqrt{3} \end{bmatrix} \quad \hat{\bm{s}}_2 = \begin{bmatrix} \frac{-1}{\sqrt{6}} \\ \frac{-1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} \end{bmatrix} \quad
\hat{\bm{s}}_3 = \begin{bmatrix} -1/\sqrt{6} \\ 1/\sqrt{2} \\ 1/\sqrt{3} \end{bmatrix} \quad \hat{\bm{s}}_3 = \begin{bmatrix} \frac{-1}{\sqrt{6}} \\ \frac{ 1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} \end{bmatrix} \quad
\hat{\bm{s}}_4 = \begin{bmatrix} \sqrt{2}/\sqrt{3} \\ 0 \\ 1/\sqrt{3} \end{bmatrix} \quad \hat{\bm{s}}_4 = \begin{bmatrix} \frac{\sqrt{2}}{\sqrt{3}} \\ 0 \\ \frac{1}{\sqrt{3}} \end{bmatrix} \quad
\hat{\bm{s}}_5 = \begin{bmatrix} -1/\sqrt{6} \\ -1/\sqrt{2} \\ 1/\sqrt{3} \end{bmatrix} \quad \hat{\bm{s}}_5 = \begin{bmatrix} \frac{-1}{\sqrt{6}} \\ \frac{-1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} \end{bmatrix} \quad
\hat{\bm{s}}_6 = \begin{bmatrix} -1/\sqrt{6} \\ 1/\sqrt{2} \\ 1/\sqrt{3} \end{bmatrix} \hat{\bm{s}}_6 = \begin{bmatrix} \frac{-1}{\sqrt{6}} \\ \frac{ 1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} \end{bmatrix}
\end{equation} \end{equation}
\begin{figure}[htbp] \begin{figure}[htbp]
@ -466,9 +454,8 @@ In that case, the location of the top joints can be expressed by equation \eqref
The stiffness matrix is therefore diagonal when the considered \(\{B\}\) frame is located at the center of the cube (shown by frame \(\{C\}\)). The stiffness matrix is therefore diagonal when the considered \(\{B\}\) frame is located at the center of the cube (shown by frame \(\{C\}\)).
This means that static forces (or torques) applied at the cube's center will induce pure translations (or rotations around the cube's center). This means that static forces (resp torques) applied at the cube's center will induce pure translations (resp rotations around the cube's center).
This specific location where the stiffness matrix is diagonal is referred to as the ``Center of Stiffness'' (analogous to the ``Center of Mass'' where the mass matrix is diagonal). This specific location where the stiffness matrix is diagonal is referred to as the ``Center of Stiffness'' (analogous to the ``Center of Mass'' where the mass matrix is diagonal).
\paragraph{Effect of having frame \(\{B\}\) off-centered} \paragraph{Effect of having frame \(\{B\}\) off-centered}
When the reference frames \(\{A\}\) and \(\{B\}\) are shifted from the cube's center, off-diagonal elements emerge in the stiffness matrix. When the reference frames \(\{A\}\) and \(\{B\}\) are shifted from the cube's center, off-diagonal elements emerge in the stiffness matrix.
@ -491,13 +478,8 @@ This stiffness matrix structure is characteristic of Stewart platforms exhibitin
Therefore, the stiffness characteristics of the cubic architecture are only distinctive when considering a reference frame located at the cube's center. Therefore, the stiffness characteristics of the cubic architecture are only distinctive when considering a reference frame located at the cube's center.
This poses a practical limitation, as in most applications, the relevant frame (where motion is of interest and forces are applied) is located above the top platform. This poses a practical limitation, as in most applications, the relevant frame (where motion is of interest and forces are applied) is located above the top platform.
It should be noted that the cube's center need not be at the ``center'' of the Stewart platform.
This can lead to interesting architectures shown in Section \ref{ssec:detail_kinematics_cubic_design}.
It should be noted that for the stiffness matrix to be diagonal, the cube's center need not coincide with the geometric center of the Stewart platform. It should be noted that for the stiffness matrix to be diagonal, the cube's center need not coincide with the geometric center of the Stewart platform.
This observation leads to the interesting alternative architectures presented in Section \ref{ssec:detail_kinematics_cubic_design}. This observation leads to the interesting alternative architectures presented in Section \ref{ssec:detail_kinematics_cubic_design}.
\paragraph{Uniform Mobility} \paragraph{Uniform Mobility}
The translational mobility of the Stewart platform with constant orientation was analyzed. The translational mobility of the Stewart platform with constant orientation was analyzed.
@ -507,9 +489,8 @@ The resulting mobility in X, Y, and Z directions for the cubic architecture is i
The translational workspace analysis reveals that for the cubic architecture, the achievable positions form a cube whose axes align with the struts, with the cube's edge length corresponding to the strut axial stroke. The translational workspace analysis reveals that for the cubic architecture, the achievable positions form a cube whose axes align with the struts, with the cube's edge length corresponding to the strut axial stroke.
This findings suggest that the mobility pattern is more nuanced than sometimes described in the literature \cite{mcinroy00_desig_contr_flexur_joint_hexap}, exhibiting uniformity primarily along directions aligned with the cube's edges rather than uniform spherical distribution in all XYZ directions. This findings suggest that the mobility pattern is more nuanced than sometimes described in the literature \cite{mcinroy00_desig_contr_flexur_joint_hexap}, exhibiting uniformity primarily along directions aligned with the cube's edges rather than uniform spherical distribution in all XYZ directions.
This configuration still offers more consistent mobility characteristics compared to alternative architectures illustrated in Figure \ref{fig:detail_kinematics_mobility_trans}. This configuration still offers more consistent mobility characteristics compared to alternative architectures illustrated in Figure \ref{fig:detail_kinematics_mobility_trans}.
It is worth noting that the translational mobility properties remain independent of the cube's size.
The rotational mobility, illustrated in Figure \ref{fig:detail_kinematics_cubic_mobility_rotations}, exhibit greater achievable angular displacements in the \(R_x\) and \(R_y\) directions compared to the \(R_z\) direction. The rotational mobility, illustrated in Figure \ref{fig:detail_kinematics_cubic_mobility_rotations}, exhibit greater achievable angular stroke in the \(R_x\) and \(R_y\) directions compared to the \(R_z\) direction.
Furthermore, an inverse relationship exists between the cube's dimension and rotational mobility, with larger cube sizes corresponding to more limited angular displacement capabilities. Furthermore, an inverse relationship exists between the cube's dimension and rotational mobility, with larger cube sizes corresponding to more limited angular displacement capabilities.
\begin{figure}[htbp] \begin{figure}[htbp]
@ -527,24 +508,20 @@ Furthermore, an inverse relationship exists between the cube's dimension and rot
\end{subfigure} \end{subfigure}
\caption{\label{fig:detail_kinematics_cubic_mobility}Mobility of a Stewart platform with Cubic architecture. Both for translations (\subref{fig:detail_kinematics_cubic_mobility_translations}) and rotations (\subref{fig:detail_kinematics_cubic_mobility_rotations})} \caption{\label{fig:detail_kinematics_cubic_mobility}Mobility of a Stewart platform with Cubic architecture. Both for translations (\subref{fig:detail_kinematics_cubic_mobility_translations}) and rotations (\subref{fig:detail_kinematics_cubic_mobility_rotations})}
\end{figure} \end{figure}
\section{Dynamical Decoupling} \section{Dynamical Decoupling}
\label{ssec:detail_kinematics_cubic_dynamic} \label{ssec:detail_kinematics_cubic_dynamic}
This section examines the dynamics of the cubic architecture in the Cartesian frame. This section examines the dynamics of the cubic architecture in the Cartesian frame which corresponds to the transfer function from forces and torques \(\bm{\mathcal{F}}\) to translations and rotations \(\bm{\mathcal{X}}\) of the top platform.
This corresponds to the transfer function from forces and torques \(\bm{\mathcal{F}}\) to translations and rotations \(\bm{\mathcal{X}}\) of the top platform.
When relative motion sensors are integrated in each strut (measuring \(\bm{\mathcal{L}}\)), the pose \(\bm{\mathcal{X}}\) is computed using the Jacobian matrix as shown in Figure \ref{fig:detail_kinematics_centralized_control}. When relative motion sensors are integrated in each strut (measuring \(\bm{\mathcal{L}}\)), the pose \(\bm{\mathcal{X}}\) is computed using the Jacobian matrix as shown in Figure \ref{fig:detail_kinematics_centralized_control}.
The analysis aims to identify whether the cubic configuration exhibits special properties for control in the Cartesian frame.
\begin{figure}[htbp] \begin{figure}[htbp]
\centering \centering
\includegraphics[scale=1]{figs/detail_kinematics_centralized_control.png} \includegraphics[scale=1]{figs/detail_kinematics_centralized_control.png}
\caption{\label{fig:detail_kinematics_centralized_control}From Strut coordinate to Cartesian coordinate using the Jacobian matrix} \caption{\label{fig:detail_kinematics_centralized_control}Typical control architecture in the cartesian frame}
\end{figure} \end{figure}
\paragraph{Low frequency and High frequency coupling} \paragraph{Low frequency and High frequency coupling}
As derived during the conceptual design phase, the dynamics from \(\bm{\mathcal{F}}\) to \(\bm{\mathcal{X}}\) is described by Equation \eqref{eq:detail_kinematics_transfer_function_cart}. As derived during the conceptual design phase, the dynamics from \(\bm{\mathcal{F}}\) to \(\bm{\mathcal{X}}\) is described by Equation \eqref{eq:detail_kinematics_transfer_function_cart}.
At low frequency, the static behavior of the platform depends on the stiffness matrix \eqref{eq:detail_kinematics_transfer_function_cart_low_freq}. At low frequency, the behavior of the platform depends on the stiffness matrix \eqref{eq:detail_kinematics_transfer_function_cart_low_freq}.
\begin{equation}\label{eq:detail_kinematics_transfer_function_cart_low_freq} \begin{equation}\label{eq:detail_kinematics_transfer_function_cart_low_freq}
\frac{{\mathcal{X}}}{\bm{\mathcal{F}}}(j \omega) \xrightarrow[\omega \to 0]{} \bm{K}^{-1} \frac{{\mathcal{X}}}{\bm{\mathcal{F}}}(j \omega) \xrightarrow[\omega \to 0]{} \bm{K}^{-1}
@ -566,7 +543,7 @@ To achieve a diagonal mass matrix, the center of mass of the mobile components m
\caption{\label{fig:detail_kinematics_cubic_payload}Cubic stewart platform with top cylindrical payload} \caption{\label{fig:detail_kinematics_cubic_payload}Cubic stewart platform with top cylindrical payload}
\end{figure} \end{figure}
To verify these properties, a cubic Stewart platform with a cylindrical payload on top (Figure \ref{fig:detail_kinematics_cubic_payload}) was analyzed. To verify these properties, a cubic Stewart platform with a cylindrical payload was analyzed (Figure \ref{fig:detail_kinematics_cubic_payload}).
Transfer functions from \(\bm{\mathcal{F}}\) to \(\bm{\mathcal{X}}\) were computed for two specific locations of the \(\{B\}\) frames. Transfer functions from \(\bm{\mathcal{F}}\) to \(\bm{\mathcal{X}}\) were computed for two specific locations of the \(\{B\}\) frames.
When the \(\{B\}\) frame was positioned at the center of mass, coupling at low frequency was observed due to the non-diagonal stiffness matrix (Figure \ref{fig:detail_kinematics_cubic_cart_coupling_com}). When the \(\{B\}\) frame was positioned at the center of mass, coupling at low frequency was observed due to the non-diagonal stiffness matrix (Figure \ref{fig:detail_kinematics_cubic_cart_coupling_com}).
Conversely, when positioned at the center of stiffness, coupling occurred at high frequency due to the non-diagonal mass matrix (Figure \ref{fig:detail_kinematics_cubic_cart_coupling_cok}). Conversely, when positioned at the center of stiffness, coupling occurred at high frequency due to the non-diagonal mass matrix (Figure \ref{fig:detail_kinematics_cubic_cart_coupling_cok}).
@ -584,9 +561,8 @@ Conversely, when positioned at the center of stiffness, coupling occurred at hig
\end{center} \end{center}
\subcaption{\label{fig:detail_kinematics_cubic_cart_coupling_cok}$\{B\}$ at the cube's center} \subcaption{\label{fig:detail_kinematics_cubic_cart_coupling_cok}$\{B\}$ at the cube's center}
\end{subfigure} \end{subfigure}
\caption{\label{fig:detail_kinematics_cubic_cart_coupling}Transfer functions for a Cubic Stewart platform expressed in the Cartesian frame. Two locations of the \(\{B\}\) frame are considered: at the cube's center (\subref{fig:detail_kinematics_cubic_cart_coupling_cok}) and at the center of mass of the moving body (\subref{fig:detail_kinematics_cubic_cart_coupling_com}).} \caption{\label{fig:detail_kinematics_cubic_cart_coupling}Transfer functions for a Cubic Stewart platform expressed in the Cartesian frame. Two locations of the \(\{B\}\) frame are considered: at the center of mass of the moving body (\subref{fig:detail_kinematics_cubic_cart_coupling_com}) and at the cube's center (\subref{fig:detail_kinematics_cubic_cart_coupling_cok}).}
\end{figure} \end{figure}
\paragraph{Payload's CoM at the cube's center} \paragraph{Payload's CoM at the cube's center}
An effective strategy for improving dynamical performances involves aligning the cube's center (center of stiffness) with the center of mass of the moving components \cite{li01_simul_fault_vibrat_isolat_point}. An effective strategy for improving dynamical performances involves aligning the cube's center (center of stiffness) with the center of mass of the moving components \cite{li01_simul_fault_vibrat_isolat_point}.
@ -611,28 +587,23 @@ If a design similar to Figure \ref{fig:detail_kinematics_cubic_centered_payload}
\end{subfigure} \end{subfigure}
\caption{\label{fig:detail_kinematics_cubic_com_cok}Cubic Stewart platform with payload at the cube's center (\subref{fig:detail_kinematics_cubic_centered_payload}). Obtained cartesian plant is fully decoupled (\subref{fig:detail_kinematics_cubic_cart_coupling_com_cok})} \caption{\label{fig:detail_kinematics_cubic_com_cok}Cubic Stewart platform with payload at the cube's center (\subref{fig:detail_kinematics_cubic_centered_payload}). Obtained cartesian plant is fully decoupled (\subref{fig:detail_kinematics_cubic_cart_coupling_com_cok})}
\end{figure} \end{figure}
\paragraph{Conclusion} \paragraph{Conclusion}
The analysis of dynamical properties of the cubic architecture yields several important conclusions. The analysis of dynamical properties of the cubic architecture yields several important conclusions.
Static decoupling, characterized by a diagonal stiffness matrix, is achieved when reference frames \(\{A\}\) and \(\{B\}\) are positioned at the cube's center. Static decoupling, characterized by a diagonal stiffness matrix, is achieved when reference frames \(\{A\}\) and \(\{B\}\) are positioned at the cube's center.
This property can also be obtained with non-cubic architectures that exhibit symmetrical strut arrangements. Note that this property can also be obtained with non-cubic architectures that exhibit symmetrical strut arrangements.
Dynamic decoupling requires both static decoupling and coincidence of the mobile platform's center of mass with reference frame \(\{B\}\). Dynamic decoupling requires both static decoupling and coincidence of the mobile platform's center of mass with reference frame \(\{B\}\).
While this configuration offers powerful control advantages, it requires positioning the payload at the cube's center, which is highly restrictive and often impractical. While this configuration offers powerful control advantages, it requires positioning the payload at the cube's center, which is highly restrictive and often impractical.
Additionally, the cubic architecture provides uniform stiffness in XYZ directions, which may be advantageous for certain applications.
\section{Decentralized Control} \section{Decentralized Control}
\label{ssec:detail_kinematics_decentralized_control} \label{ssec:detail_kinematics_decentralized_control}
The orthogonal arrangement of struts in the cubic architecture suggests a potential minimization of inter-strut coupling, which could theoretically create favorable conditions for decentralized control. The orthogonal arrangement of struts in the cubic architecture suggests a potential minimization of inter-strut coupling, which could theoretically create favorable conditions for decentralized control.
This section examines whether the cubic architecture actually demonstrates advantageous properties for decentralized control in the frame of the struts.
Two sensor types integrated in the struts are considered: displacement sensors and force sensors. Two sensor types integrated in the struts are considered: displacement sensors and force sensors.
The control architecture is illustrated in Figure \ref{fig:detail_kinematics_decentralized_control}, where \(\bm{K}_{\mathcal{L}}\) represents a diagonal transfer function matrix. The control architecture is illustrated in Figure \ref{fig:detail_kinematics_decentralized_control}, where \(\bm{K}_{\mathcal{L}}\) represents a diagonal transfer function matrix.
\begin{figure}[htbp] \begin{figure}[htbp]
\centering \centering
\includegraphics[scale=1]{figs/detail_kinematics_decentralized_control.png} \includegraphics[scale=1]{figs/detail_kinematics_decentralized_control.png}
\caption{\label{fig:detail_kinematics_decentralized_control}From Strut coordinate to Cartesian coordinate using the Jacobian matrix} \caption{\label{fig:detail_kinematics_decentralized_control}Decentralized control in the frame of the struts.}
\end{figure} \end{figure}
The obtained plant dynamics in the frame of the struts are compared for two Stewart platforms. The obtained plant dynamics in the frame of the struts are compared for two Stewart platforms.
@ -668,7 +639,6 @@ The resonance frequencies differ between the two cases because the more vertical
\end{subfigure} \end{subfigure}
\caption{\label{fig:detail_kinematics_decentralized_dL}Bode plot of the transfer functions from actuator force to relative displacement sensor in each strut. Both for a non-cubic architecture (\subref{fig:detail_kinematics_non_cubic_decentralized_dL}) and for a cubic architecture (\subref{fig:detail_kinematics_cubic_decentralized_dL})} \caption{\label{fig:detail_kinematics_decentralized_dL}Bode plot of the transfer functions from actuator force to relative displacement sensor in each strut. Both for a non-cubic architecture (\subref{fig:detail_kinematics_non_cubic_decentralized_dL}) and for a cubic architecture (\subref{fig:detail_kinematics_cubic_decentralized_dL})}
\end{figure} \end{figure}
\paragraph{Force Sensors} \paragraph{Force Sensors}
Similarly, the transfer functions from actuator force to force sensors in each strut were analyzed for both cubic and non-cubic Stewart platforms. Similarly, the transfer functions from actuator force to force sensors in each strut were analyzed for both cubic and non-cubic Stewart platforms.
@ -690,12 +660,10 @@ The system demonstrates good decoupling at high frequency in both cases, with no
\end{subfigure} \end{subfigure}
\caption{\label{fig:detail_kinematics_decentralized_fn}Bode plot of the transfer functions from actuator force to force sensor in each strut. Both for a non-cubic architecture (\subref{fig:detail_kinematics_non_cubic_decentralized_fn}) and for a cubic architecture (\subref{fig:detail_kinematics_cubic_decentralized_fn})} \caption{\label{fig:detail_kinematics_decentralized_fn}Bode plot of the transfer functions from actuator force to force sensor in each strut. Both for a non-cubic architecture (\subref{fig:detail_kinematics_non_cubic_decentralized_fn}) and for a cubic architecture (\subref{fig:detail_kinematics_cubic_decentralized_fn})}
\end{figure} \end{figure}
\paragraph{Conclusion} \paragraph{Conclusion}
The presented results do not demonstrate the pronounced decoupling advantages often associated with cubic architectures in the literature. The presented results do not demonstrate the pronounced decoupling advantages often associated with cubic architectures in the literature.
Both the cubic and non-cubic configurations exhibited similar coupling characteristics in our modeling scenarios, suggesting that the benefits of orthogonal strut arrangement may be more nuanced than commonly described for decentralized control. Both the cubic and non-cubic configurations exhibited similar coupling characteristics, suggesting that the benefits of orthogonal strut arrangement may be more nuanced than commonly described for decentralized control.
\section{Cubic architecture with Cube's center above the top platform} \section{Cubic architecture with Cube's center above the top platform}
\label{ssec:detail_kinematics_cubic_design} \label{ssec:detail_kinematics_cubic_design}
As demonstrated in Section \ref{ssec:detail_kinematics_cubic_dynamic}, the cubic architecture can exhibit advantageous dynamical properties when the center of mass of the moving body coincides with the cube's center, resulting in diagonal mass and stiffness matrices. As demonstrated in Section \ref{ssec:detail_kinematics_cubic_dynamic}, the cubic architecture can exhibit advantageous dynamical properties when the center of mass of the moving body coincides with the cube's center, resulting in diagonal mass and stiffness matrices.
@ -742,7 +710,6 @@ This approach yields a compact architecture, but the small cube size may result
\end{subfigure} \end{subfigure}
\caption{\label{fig:detail_kinematics_cubic_above_small}Cubic architecture with cube's center above the top platform. A cube height of 40mm is used.} \caption{\label{fig:detail_kinematics_cubic_above_small}Cubic architecture with cube's center above the top platform. A cube height of 40mm is used.}
\end{figure} \end{figure}
\paragraph{Medium sized cube} \paragraph{Medium sized cube}
Increasing the cube's size such that \eqref{eq:detail_kinematics_cube_medium} is verified produces an architecture with intersecting struts (Figure \ref{fig:detail_kinematics_cubic_above_medium}). Increasing the cube's size such that \eqref{eq:detail_kinematics_cube_medium} is verified produces an architecture with intersecting struts (Figure \ref{fig:detail_kinematics_cubic_above_medium}).
@ -774,7 +741,6 @@ This configuration resembles the design proposed in \cite{yang19_dynam_model_dec
\end{subfigure} \end{subfigure}
\caption{\label{fig:detail_kinematics_cubic_above_medium}Cubic architecture with cube's center above the top platform. A cube height of 140mm is used.} \caption{\label{fig:detail_kinematics_cubic_above_medium}Cubic architecture with cube's center above the top platform. A cube height of 140mm is used.}
\end{figure} \end{figure}
\paragraph{Large cube} \paragraph{Large cube}
When the cube's height exceeds twice the sum of the platform height and CoM height \eqref{eq:detail_kinematics_cube_large}, the architecture shown in Figure \ref{fig:detail_kinematics_cubic_above_large} is obtained. When the cube's height exceeds twice the sum of the platform height and CoM height \eqref{eq:detail_kinematics_cube_large}, the architecture shown in Figure \ref{fig:detail_kinematics_cubic_above_large} is obtained.
@ -804,10 +770,8 @@ When the cube's height exceeds twice the sum of the platform height and CoM heig
\end{subfigure} \end{subfigure}
\caption{\label{fig:detail_kinematics_cubic_above_large}Cubic architecture with cube's center above the top platform. A cube height of 240mm is used.} \caption{\label{fig:detail_kinematics_cubic_above_large}Cubic architecture with cube's center above the top platform. A cube height of 240mm is used.}
\end{figure} \end{figure}
\paragraph{Platform size} \paragraph{Platform size}
In order to determine the approximate size of the platform as a function of
For the proposed configuration, the top joints \(\bm{b}_i\) (resp. the bottom joints \(\bm{a}_i\)) and are positioned on a circle with radius \(R_{b_i}\) (resp. \(R_{a_i}\)) described by Equation \eqref{eq:detail_kinematics_cube_joints}. For the proposed configuration, the top joints \(\bm{b}_i\) (resp. the bottom joints \(\bm{a}_i\)) and are positioned on a circle with radius \(R_{b_i}\) (resp. \(R_{a_i}\)) described by Equation \eqref{eq:detail_kinematics_cube_joints}.
\begin{subequations}\label{eq:detail_kinematics_cube_joints} \begin{subequations}\label{eq:detail_kinematics_cube_joints}
@ -819,16 +783,11 @@ For the proposed configuration, the top joints \(\bm{b}_i\) (resp. the bottom jo
Since the rotational stiffness for the cubic architecture scales with the square of the cube's height \eqref{eq:detail_kinematics_cubic_stiffness}, the cube's size can be determined based on rotational stiffness requirements. Since the rotational stiffness for the cubic architecture scales with the square of the cube's height \eqref{eq:detail_kinematics_cubic_stiffness}, the cube's size can be determined based on rotational stiffness requirements.
Subsequently, using Equation \eqref{eq:detail_kinematics_cube_joints}, the dimensions of the top and bottom platforms can be calculated. Subsequently, using Equation \eqref{eq:detail_kinematics_cube_joints}, the dimensions of the top and bottom platforms can be calculated.
\paragraph{Conclusion} \paragraph{Conclusion}
The configurations proposed in this analysis represent derivations from the classical cubic architecture, wherein the cube's center is typically located at the Stewart platform's center. The configurations proposed in this analysis represent derivations from the classical cubic architecture, wherein the cube's center is typically located at the Stewart platform's center.
Three distinct configurations have been identified, each with different geometric arrangements but sharing the common characteristic that the cube's center is positioned above the top platform. Three distinct configurations have been identified, each with different geometric arrangements but sharing the common characteristic that the cube's center is positioned above the top platform.
This structural modification enables the alignment of the moving body's center of mass with the center of stiffness, resulting in beneficial decoupling properties in the Cartesian frame. This structural modification enables the alignment of the moving body's center of mass with the center of stiffness, resulting in beneficial decoupling properties in the Cartesian frame.
These proposed architectures maintain the fundamental advantages inherent to the cubic configuration, such as uniform stiffness and uniform mobility, while providing favorable dynamical properties when payloads are placed on top of the mobile platform.
This approach allows for practical payload positioning while preserving the desirable control characteristics associated with the cubic architecture, making these configurations potentially useful for applications requiring both specific payload placement and good dynamic performance.
\section*{Conclusion} \section*{Conclusion}
The analysis of the cubic architecture for Stewart platforms has yielded several important findings. The analysis of the cubic architecture for Stewart platforms has yielded several important findings.
While the cubic configuration provides uniform stiffness in the XYZ directions, it stiffness property becomes particularly advantageous when forces and torques are applied at the cube's center. While the cubic configuration provides uniform stiffness in the XYZ directions, it stiffness property becomes particularly advantageous when forces and torques are applied at the cube's center.
@ -838,19 +797,17 @@ Regarding mobility, the translational capabilities of the cubic configuration ex
This understanding refines the characterization of cubic architecture mobility commonly presented in literature. This understanding refines the characterization of cubic architecture mobility commonly presented in literature.
The analysis of decentralized control in the frame of the struts revealed more nuanced results than expected. The analysis of decentralized control in the frame of the struts revealed more nuanced results than expected.
While cubic architectures are frequently associated with reduced coupling between actuators and sensors, our comparative study showed that these benefits may be more subtle or context-dependent than commonly described. While cubic architectures are frequently associated with reduced coupling between actuators and sensors, this study showed that these benefits may be more subtle or context-dependent than commonly described.
Under the conditions analyzed, the coupling characteristics of cubic and non-cubic configurations appeared similar. Under the conditions analyzed, the coupling characteristics of cubic and non-cubic configurations, in the frame of the struts, appeared similar.
Fully decoupled dynamics can be achieved when the center of mass of the moving body coincides with the cube's center. Fully decoupled dynamics in the Cartesian frame can be achieved when the center of mass of the moving body coincides with the cube's center.
However, this arrangement presents practical challenges, as the cube's center is traditionally located between the top and bottom platforms, making payload placement problematic for many applications. However, this arrangement presents practical challenges, as the cube's center is traditionally located between the top and bottom platforms, making payload placement problematic for many applications.
To address this limitation, modified cubic architectures have been proposed with the cube's center positioned above the top platform. To address this limitation, modified cubic architectures have been proposed with the cube's center positioned above the top platform.
These configurations maintain the fundamental advantages of the cubic architecture while enabling practical payload placement. These configurations maintain the fundamental advantages of the cubic architecture while enabling practical payload placement.
\chapter{Nano Hexapod} \chapter{Nano Hexapod}
\label{sec:detail_kinematics_nano_hexapod} \label{sec:detail_kinematics_nano_hexapod}
Based on previous analysis, this section aims to determine the nano-hexapod geometry. Based on previous analysis, this section aims to determine the nano-hexapod optimal geometry.
For the NASS, the chosen reference frames \(\{A\}\) and \(\{B\}\) coincide with the sample's point of interest, which is positioned \(150\,mm\) above the top platform. For the NASS, the chosen reference frames \(\{A\}\) and \(\{B\}\) coincide with the sample's point of interest, which is positioned \(150\,mm\) above the top platform.
This is the location where precise control of the sample's position is required, as it is where the x-ray beam is focused. This is the location where precise control of the sample's position is required, as it is where the x-ray beam is focused.
\section{Requirements} \section{Requirements}
@ -862,25 +819,21 @@ Based on the measured errors of all stages of the micro-stations, and incorporat
At any position, the system should be capable of performing \(R_x\) and \(R_y\) rotations of \(\pm 50\,\mu \text{rad}\). At any position, the system should be capable of performing \(R_x\) and \(R_y\) rotations of \(\pm 50\,\mu \text{rad}\).
Regarding stiffness, the resonance frequencies should be well above the maximum rotational velocity of \(2\pi\,\text{rad/s}\) to minimize gyroscopic effects, while remaining below the problematic modes of the micro-station to ensure decoupling from its complex dynamics. Regarding stiffness, the resonance frequencies should be well above the maximum rotational velocity of \(2\pi\,\text{rad/s}\) to minimize gyroscopic effects, while remaining below the problematic modes of the micro-station to ensure decoupling from its complex dynamics.
In terms of dynamics, the design should facilitate implementation of Integral Force Feedback (IFF) in a decentralized manner, and provide good decoupling for the high authority controller in the frame of the struts. In terms of dynamics, the design should facilitate implementation of Integral Force Feedback (IFF) in a decentralized manner, and provide good decoupling for the high authority controller in the frame of the struts.
A significant challenge in optimizing the nano-hexapod design arises from the variety of payloads that will be used, with masses ranging from 1 to 50kg.
This variation in payload characteristics makes it impossible to develop a single geometry that provides optimal dynamical properties for all possible configurations.
\section{Obtained Geometry} \section{Obtained Geometry}
\label{ssec:detail_kinematics_nano_hexapod_geometry} \label{ssec:detail_kinematics_nano_hexapod_geometry}
Based on the previous analysis of Stewart platform configurations, while the geometry can be optimized to achieve the desired trade-off between stiffness and mobility in different directions, the wide range of potential payloads complicates the optimization process for obtaining consistent dynamical properties across all usage scenarios. Based on the previous analysis of Stewart platform configurations, while the geometry can be optimized to achieve the desired trade-off between stiffness and mobility in different directions, the wide range of potential payloads, with masses ranging from 1kg to 50kg, makes it impossible to develop a single geometry that provides optimal dynamical properties for all possible configurations.
For the nano-hexapod design, the struts were oriented more vertically compared to a cubic architecture due to several important considerations. For the nano-hexapod design, the struts were oriented more vertically compared to a cubic architecture due to several considerations.
First, the requirements in the vertical direction are more stringent than in the horizontal direction. First, the performance requirements in the vertical direction are more stringent than in the horizontal direction.
This vertical strut orientation decreases the amplification factor in the vertical direction, providing greater resolution and reducing the effects of actuator noise. This vertical strut orientation decreases the amplification factor in the vertical direction, providing greater resolution and reducing the effects of actuator noise.
Second, the micro-station's vertical modes exhibit higher frequencies than its lateral modes. Second, the micro-station's vertical modes exhibit higher frequencies than its lateral modes.
Therefore, higher resonance frequencies of the nano-hexapod in the vertical direction compared to the horizontal direction enhance the decoupling properties between the micro-station and the nano-hexapod. Therefore, higher resonance frequencies of the nano-hexapod in the vertical direction compared to the horizontal direction enhance the decoupling properties between the micro-station and the nano-hexapod.
Regarding dynamic properties, particularly for control in the frame of the struts, no specific optimization was implemented since the analysis revealed that the particular geometry has minimal impact on the resulting coupling characteristics. Regarding dynamical properties, particularly for control in the frame of the struts, no specific optimization was implemented since the analysis revealed that strut orientation has minimal impact on the resulting coupling characteristics.
Consequently, the geometry was selected according to practical constraints. Consequently, the geometry was selected according to practical constraints.
The height between the two plates is set at \(95\,mm\). The height between the two plates is maximized and set at \(95\,mm\).
Both platforms utilize the maximum available size, with joints offset by \(15\,mm\) from the plate surfaces and positioned along circles with radii of \(120\,mm\) for the fixed joints and \(110\,mm\) for the mobile joints. Both platforms utilize the maximum available size, with joints offset by \(15\,mm\) from the plate surfaces and positioned along circles with radii of \(120\,mm\) for the fixed joints and \(110\,mm\) for the mobile joints.
The positioning angles, as shown in Figure \ref{fig:detail_kinematics_nano_hexapod_top}, are \([255,\ 285,\ 15,\ 45,\ 135,\ 165]\) degrees for the top joints and \([220,\ 320,\ 340,\ 80,\ 100,\ 200]\) degrees for the bottom joints. The positioning angles, as shown in Figure \ref{fig:detail_kinematics_nano_hexapod_top}, are \([255,\ 285,\ 15,\ 45,\ 135,\ 165]\) degrees for the top joints and \([220,\ 320,\ 340,\ 80,\ 100,\ 200]\) degrees for the bottom joints.
@ -907,7 +860,6 @@ Implementing a cubic architecture as proposed in Section \ref{ssec:detail_kinema
However, positioning the cube's center \(150\,mm\) above the top platform would have resulted in platform dimensions exceeding the maximum available size. However, positioning the cube's center \(150\,mm\) above the top platform would have resulted in platform dimensions exceeding the maximum available size.
Additionally, to benefit from the cubic configuration's dynamical properties, each payload would require careful calibration of inertia before placement on the nano-hexapod, ensuring that its center of mass coincides with the cube's center. Additionally, to benefit from the cubic configuration's dynamical properties, each payload would require careful calibration of inertia before placement on the nano-hexapod, ensuring that its center of mass coincides with the cube's center.
Given the impracticality of consistently aligning the center of mass with the cube's center, the cubic architecture was deemed unsuitable for the nano-hexapod application. Given the impracticality of consistently aligning the center of mass with the cube's center, the cubic architecture was deemed unsuitable for the nano-hexapod application.
\section{Required Actuator stroke} \section{Required Actuator stroke}
\label{ssec:detail_kinematics_nano_hexapod_actuator_stroke} \label{ssec:detail_kinematics_nano_hexapod_actuator_stroke}
@ -926,7 +878,6 @@ The diagram confirms that the required workspace fits within the system's capabi
\includegraphics[scale=1]{figs/detail_kinematics_nano_hexapod_mobility.png} \includegraphics[scale=1]{figs/detail_kinematics_nano_hexapod_mobility.png}
\caption{\label{fig:detail_kinematics_nano_hexapod_mobility}Wanted translation mobility of the Nano-Hexapod (grey cube) and computed Mobility (red volume).} \caption{\label{fig:detail_kinematics_nano_hexapod_mobility}Wanted translation mobility of the Nano-Hexapod (grey cube) and computed Mobility (red volume).}
\end{figure} \end{figure}
\section{Required Joint angular stroke} \section{Required Joint angular stroke}
\label{ssec:detail_kinematics_nano_hexapod_joint_stroke} \label{ssec:detail_kinematics_nano_hexapod_joint_stroke}
@ -951,7 +902,6 @@ For the cubic configuration, complete dynamical decoupling in the Cartesian fram
Modified cubic architectures with the cube's center positioned above the top platform were proposed as a potential solution, but proved unsuitable for the nano-hexapod due to size constraints and the impracticality of ensuring that different payloads' centers of mass would consistently align with the cube's center. Modified cubic architectures with the cube's center positioned above the top platform were proposed as a potential solution, but proved unsuitable for the nano-hexapod due to size constraints and the impracticality of ensuring that different payloads' centers of mass would consistently align with the cube's center.
For the nano-hexapod design, a key challenge was addressing the wide range of potential payloads (1 to 50kg), which made it impossible to optimize the geometry for consistent dynamic performance across all usage scenarios. For the nano-hexapod design, a key challenge was addressing the wide range of potential payloads (1 to 50kg), which made it impossible to optimize the geometry for consistent dynamic performance across all usage scenarios.
This led to a practical design approach where struts were oriented more vertically than in cubic configurations to address several application-specific needs: achieving higher resolution in the vertical direction by reducing amplification factors, better matching the micro-station's modal characteristics with higher vertical resonance frequencies, and accommodating the stringent vertical positioning requirements. This led to a practical design approach where struts were oriented more vertically than in cubic configurations to address several application-specific needs: achieving higher resolution in the vertical direction by reducing amplification factors and better matching the micro-station's modal characteristics with higher vertical resonance frequencies.
\printbibliography[heading=bibintoc,title={Bibliography}] \printbibliography[heading=bibintoc,title={Bibliography}]
\end{document} \end{document}