Add some notes on stewart architectures

index.org

@@ -1048,12 +1048,31 @@ This show how the dynamics evolves with the stiffness and how different effects
   In such case, the main limitation will be heavy samples with small stiffnesses.
 #+end_important
 
-** Nano-Hexapod Architecture
+** Optimal Nano-Hexapod Geometry
 <<sec:nano_hexapod_architecture>>
 
 *** Introduction                                                    :ignore:
+As will be shown in this section, the Nano-Hexapod geometry has an influence on:
+- the overall stiffness/compliance
+- the mobility
+- the dynamics and coupling
 
-*** Kinematic Analysis - Jacobian Matrix
+A typical Stewart platform is composed of six identical legs:
+- a universal joint
+- a spherical joint
+- a prismatic joint with an integrated actuator
+
+#+name: fig:stewart_architecture_example
+#+caption: Figure caption
+[[file:figs/stewart_architecture_example.png]]
+
+
+#+name: fig:stewart_architecture_example_pose
+#+caption: Display of the Stewart platform architecture at some defined pose
+[[file:figs/stewart_architecture_example_pose.png]]
+
+
+*** Kinematic Analysis and the Jacobian Matrix
 :PROPERTIES:
 :UNNUMBERED: t
 :END:
@@ -1065,33 +1084,55 @@ In this analysis, the relation between the geometrical parameters of the manipul
 #+end_quote
 
 
-From cite:taghirad13_paral:
-#+begin_quote
-The Jacobian matrix not only reveals the *relation between the joint variable velocities of a parallel manipulator to the moving platform linear and angular velocities*, it also constructs the transformation needed to find the *actuator forces from the forces and moments acting on the moving platform*.
-#+end_quote
+One of the main analysis tool for the Kinematic analysis is the *Jacobian Matrix* that not only reveals the *relation between the joint variable velocities of a parallel manipulator to the moving platform linear and angular velocities*, but also constructs the transformation needed to find the *actuator forces from the forces and moments acting on the moving platform*.
 
-The Jacobian matrix $\bm{\mathcal{J}}$ can be computed form the orientation of the legs and the position of the flexible joints.
 
-If we note:
+The Jacobian matrix $\bm{J}$ can be computed form the orientation of the legs (describes by the unit vectors ${}^A\hat{\bm{s}}_i$) and the position of the flexible joints (described by the position vectors ${}^A\bm{b}_i$):
+\begin{equation*}
+  \bm{J} = \begin{bmatrix}
+    {\hat{\bm{s}}_1}^T & (\bm{b}_1 \times \hat{\bm{s}}_1)^T \\
+    {\hat{\bm{s}}_2}^T & (\bm{b}_2 \times \hat{\bm{s}}_2)^T \\
+    {\hat{\bm{s}}_3}^T & (\bm{b}_3 \times \hat{\bm{s}}_3)^T \\
+    {\hat{\bm{s}}_4}^T & (\bm{b}_4 \times \hat{\bm{s}}_4)^T \\
+    {\hat{\bm{s}}_5}^T & (\bm{b}_5 \times \hat{\bm{s}}_5)^T \\
+    {\hat{\bm{s}}_6}^T & (\bm{b}_6 \times \hat{\bm{s}}_6)^T
+  \end{bmatrix}
+\end{equation*}
+
+It can be easily shown that:
+\begin{equation}
+  \delta\bm{\mathcal{L}} = \bm{J} \delta\bm{\mathcal{X}}, \quad \delta\bm{\mathcal{X}} = \bm{J}^{-1} \delta\bm{\mathcal{L}} \label{eq:jacobian_L}
+\end{equation}
+with:
 - $\delta\bm{\mathcal{L}} = [ \delta l_1, \delta l_2, \delta l_3, \delta l_4, \delta l_5, \delta l_6 ]^T$: the vector of small legs' displacements
 - $\delta \bm{\mathcal{X}} = [\delta x, \delta y, \delta z, \delta \theta_x, \delta \theta_y, \delta \theta_z ]^T$: the vector of small mobile platform displacements
 
-The Jacobian matrix links the two vectors:
-\begin{align*}
-  \delta\bm{\mathcal{L}} &= \bm{J} \delta\bm{\mathcal{X}} \\
-  \delta\bm{\mathcal{X}} &= \bm{J}^{-1} \delta\bm{\mathcal{L}}
-\end{align*}
+Thus, from a wanted small displacement $\delta \bm{\mathcal{X}}$, it is easy to compute the required displacement of the legs $\delta \bm{\mathcal{L}}$.
+Similarly, from a measurement of the legs' displacement, it is easy to compute the resulting platform's motion.
+
+This will be used to estimate the platform's mobility from the stroke of the legs, or inversely, to estimate the required stroke of the legs from the wanted platform's mobility.
+
+Note that Eq. eqref:eq:jacobian_L is an approximation and is only valid for leg's displacement less than $1\%$ of the leg's length which is the case for the nano-hexapod.
 
 
-If we note:
+It can also be shown that:
+\begin{equation}
+  \bm{\mathcal{F}} = \bm{J}^T \bm{\tau}, \quad \bm{\tau} = \bm{J}^{-T} \bm{\mathcal{F}} \label{eq:jacobian_F}
+\end{equation}
+with:
 - $\bm{\tau} = [\tau_1, \tau_2, \cdots, \tau_6]^T$: vector of actuator forces applied in each strut
 - $\bm{\mathcal{F}} = [\bm{f}, \bm{n}]^T$: external force/torque action on the mobile platform
 
-\begin{equation*}
-  \bm{\mathcal{F}} = \bm{J}^T \bm{\tau}
-\end{equation*}
+And thus the Jacobian matrix can be used to compute the forces that should be applied on each leg from forces and torques that we want to apply on the top platform.
 
 
+Transformations in Eq. eqref:eq:jacobian_L and eqref:eq:jacobian_F will be widely in the developed control architectures.
+
+*** Stiffness and Compliance matrices
+:PROPERTIES:
+:UNNUMBERED: t
+:END:
+
 
 \begin{equation*}
   \bm{\mathcal{F}} = \bm{K} \delta \bm{\mathcal{X}}
@@ -1104,53 +1145,55 @@ If we note:
   \bm{C} = \bm{K}^{-1} = (\bm{J}^T \mathcal{K} \bm{J})^{-1}
 \end{equation*}
 
+Stiffness properties is estimated from the architecture and leg's stiffness
+
+
 
 
 Kinematic Study https://tdehaeze.github.io/stewart-simscape/kinematic-study.html
 
 
-Mobility can be estimated from the architecture of the Stewart platform and the leg's stroke.
-
-
-Stiffness properties is estimated from the architecture and leg's stiffness
-
-*** Kinematic Analysis - Mobility
+*** Mobility of the Stewart Platform
 :PROPERTIES:
 :UNNUMBERED: t
 :END:
 
+For a specified geometry and actuator stroke, the mobility of the Stewart platform can be estimated.
+
+An example of the mobility considering only pure translations is shown in Figure [[fig:mobility_translations_null_rotation]].
 
 #+name: fig:mobility_translations_null_rotation
-#+caption: Figure caption
+#+caption: Obtained mobility of a Stewart platform for pure translations (the platform's orientation is fixed)
 [[file:figs/mobility_translations_null_rotation.png]]
 
-*** Kinematic Study
-:PROPERTIES:
-:UNNUMBERED: t
-:END:
-
 *** Flexible Joints
 :PROPERTIES:
 :UNNUMBERED: t
 :END:
 
 Active Damping Study https://tdehaeze.github.io/stewart-simscape/control-active-damping.html
+
+- Advantages compared to conventional joints
+- Simulations will help determine the required rotational stroke and will help with the design
+- Typical joint stiffness is included in the model
+
 Flexible Joint stiffness => not problematic for the chosen active damping technique
 
+Example of flexible joints used [[fig:preumont07_flexible_joints]], [[fig:yang19_flexible_joints]]
+
 #+name: tab:yang19_stiffness_flexible_joints
-#+caption: Stiffness of flexible joints
+#+caption: Stiffness of unversal and spherical flexible joints cite:yang19_dynam_model_decoup_contr_flexib
 | $k_{\theta u},\ k_{\psi u}$ | $72 Nm/rad$ |
 | $k_{\theta s}$              | $51 Nm/rad$ |
 | $k_{\psi s}$                | $62 Nm/rad$ |
 | $k_{\gamma s}$              | $64 Nm/rad$ |
 
 #+name: fig:preumont07_flexible_joints
-#+caption: Figure caption cite:preumont07_six_axis_singl_stage_activ
+#+caption: Flexible joints used in cite:preumont07_six_axis_singl_stage_activ
 [[file:figs/preumont07_flexible_joints.png]]
 
-
 #+name: fig:yang19_flexible_joints
-#+caption: Figure caption
+#+caption: An alternative type of flexible joints that has been used for Stewart platforms cite:yang19_dynam_model_decoup_contr_flexib
 [[file:figs/yang19_flexible_joints.png]]
 
 
@@ -1169,6 +1212,11 @@ This configuration is such not recommended.
 #+caption: Figure caption
 [[file:figs/3d-cubic-stewart-aligned.png]]
 
+*** Conclusion
+:PROPERTIES:
+:UNNUMBERED: t
+:END:
+
 ** Conclusion
 #+begin_important
   In Section [[sec:optimal_stiff_dist]], it has been concluded that a nano-hexapod stiffness below $10^5-10^6\,[N/m]$ helps reducing the high frequency vibrations induced by all sources of disturbances considered.