Add figs about cubic architecture, flexible joints

index.org

@@ -604,6 +604,7 @@ The ASD contains any peaks starting from 1Hz showing the large spectral content
 
 #+begin_important
   A smoother motion for the translation stage (such as a sinus motion, of a filtered triangular signal) could help reducing much of the vibrations.
+  The goal is to inject no motion outside the control bandwidth.
 
   We should also note that away from the rapid change of velocity, the sample's vibrations are much reduced.
   Thus, if the detector is only used in between the triangular peaks, the vibrations are expected to be much lower than those estimated.
@@ -758,8 +759,9 @@ The results of this simulation will be compared to simulations using the NASS in
 An 3D animation of the simulation is shown in Figure [[fig:open_loop_sim]].
 
 A zoom in the micro-meter ranger on the sample's location is shown in Figure [[fig:open_loop_sim_zoom]] with two frames:
-- a non-rotating frame corresponding to the wanted position of the sample.
+- a non-rotating frame corresponding to the focusing point of the X-ray.
-  Note that this frame is moving with the granite.
+  It does in that case correspond to the wanted position of the sample.
+  Note that this frame is moving with the granite as the nano-focusing optics are fixed to the granite.
 - a rotating frame that corresponds to the actual pose of the sample
 
 The motion of the sample follows the wanted motion but with vibrations in the micro-meter range as was expected.
@@ -803,17 +805,20 @@ In the next sections, it will allows to optimally design the nano-hexapod, to de
 <<sec:nano_hexapod_design>>
 
 ** Introduction                                                      :ignore:
-As explain before, the nano-hexapod properties (mass, stiffness, architecture, ...) will influence:
+As explain before, the nano-hexapod properties (mass, stiffness, legs' orientation, ...) will influence:
 - the effect of disturbances $G_d$ (important for the rejection of disturbances)
 - the plant dynamics $G$ (important for the control robustness properties)
 
 Thus, we here wish to find the optimal nano-hexapod properties such that:
 - the effect of disturbances is minimized (Section [[sec:optimal_stiff_dist]])
 - the plant uncertainty due to a change of payload mass and experimental conditions is minimized (Section [[sec:optimal_stiff_plant]])
+- the plant has nice dynamical properties for control (Section [[sec:nano_hexapod_architecture]])
 
-The study presented here only consider changes in the nano-hexapod *stiffness* for two reasons:
+In this study, the effect of the nano-hexapod's mass characteristics is not taken into account because:
-- the nano-hexapod mass cannot be change too much, and will anyway be negligible compare to the metrology reflector and the payload masses
+1. it cannot be changed a lot
-- the choice of the nano-hexapod architecture (e.g. orientations of the actuators and implementation of sensors) will be further studied in accord with the control architecture
+2. it is quite negligible compare the to metrology reflector and the payload's masses that is fixed to nano-hexapod's top platform
+
+Also, the effect of the nano-hexapod's damping properties will be studied when applying active damping techniques.
 
 ** Optimal Stiffness to reduce the effect of disturbances
 <<sec:optimal_stiff_dist>>
@@ -1043,6 +1048,127 @@ 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
+<<sec:nano_hexapod_architecture>>
+
+*** Introduction                                                    :ignore:
+
+*** Kinematic Analysis - Jacobian Matrix
+:PROPERTIES:
+:UNNUMBERED: t
+:END:
+
+The kinematic analysis of a parallel manipulator is well described in cite:taghirad13_paral:
+#+begin_quote
+Kinematic analysis refers to the study of the geometry of motion of a robot, without considering the forces an torques that cause the motion.
+In this analysis, the relation between the geometrical parameters of the manipulator with the final motion of the moving platform is derived and analyzed.
+#+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
+
+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:
+- $\delta\bm{\mathcal{L}} = [ \delta l_1, \delta l_2, \delta l_3, \delta l_4, \delta l_5, \delta l_6 ]^T$: 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*}
+
+
+If we note:
+- $\bm{\tau} = [\tau_1, \tau_2, \cdots, \tau_6]^T$: 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*}
+
+
+
+\begin{equation*}
+  \bm{\mathcal{F}} = \bm{K} \delta \bm{\mathcal{X}}
+\end{equation*}
+
+\begin{equation*}
+  \bm{K} = \bm{J}^T \mathcal{K} \bm{J}
+\end{equation*}
+\begin{equation*}
+  \bm{C} = \bm{K}^{-1} = (\bm{J}^T \mathcal{K} \bm{J})^{-1}
+\end{equation*}
+
+
+
+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
+:PROPERTIES:
+:UNNUMBERED: t
+:END:
+
+
+#+name: fig:mobility_translations_null_rotation
+#+caption: Figure caption
+[[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
+Flexible Joint stiffness => not problematic for the chosen active damping technique
+
+#+name: tab:yang19_stiffness_flexible_joints
+#+caption: Stiffness of flexible joints
+| $k_{\theta u},\ k_{\psi u}$ | $72 Nm/rad$ |
+| $k_{\theta s}$              | $51 Nm/rad$ |
+| $k_{\psi s}$                | $62 Nm/rad$ |
+| $k_{\gamma s}$              | $64 Nm/rad$ |
+
+#+name: fig:preumont07_flexible_joints
+#+caption: Figure caption cite:preumont07_six_axis_singl_stage_activ
+[[file:figs/preumont07_flexible_joints.png]]
+
+
+#+name: fig:yang19_flexible_joints
+#+caption: Figure caption
+[[file:figs/yang19_flexible_joints.png]]
+
+
+*** Cubic Architecture
+:PROPERTIES:
+:UNNUMBERED: t
+:END:
+
+Study of cubic architecture https://tdehaeze.github.io/stewart-simscape/cubic-configuration.html
+Has some advantages such as uniform stiffness and uniform mobility.
+It can have very nice properties in specific conditions that will not be the case for this application.
+The cubic configuration also puts much restriction on the position and orientation of each leg.
+This configuration is such not recommended.
+
+#+name: fig:3d-cubic-stewart-aligned
+#+caption: Figure caption
+[[file:figs/3d-cubic-stewart-aligned.png]]
+
 ** 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.
@@ -1102,19 +1228,39 @@ The HAC-LAC architecture thus consisted of two cascade controllers:
 ** Active Damping and Sensors to be included in the nano-hexapod
 <<sec:lac_control>>
 
-Active Damping can help:
+*** Introduction                                                    :ignore:
-- by reducing the effect of disturbances close to the resonance of the system
-- by making the plant dynamics simpler to control for the High Authority Controller
-
- 
 Depending on the chosen active damping technique, either force sensors, relative motion sensors or inertial sensors should be included in each of the nano-hexapod's legs.
 
+Because of the rotation of the hexapod,
+
 A separate study (accessible [[https://tdehaeze.github.io/rotating-frame/index.html][here]]) about the use of all three sensors types have been done, the conclusions are:
 - the use of force sensors is to be avoided as it could introduce instability in the system due to the nano-hexapod's rotation
 - the use of inertial sensor should not be used as it would tends to decouple the motion of the sample from the motion of the granite (which is not wanted).
   It would also be difficult to apply in a robust way due to the non-collocation with the actuators
 - relative motion sensors can be used to damped the nano-hexapod's modes in a robust way but may increase the sensibility to stages vibrations
 
+*** Effect of the Spindle's Rotation
+:PROPERTIES:
+:UNNUMBERED: t
+:END:
+
+#+name: fig:dvf_root_locus_ws
+#+caption: Figure caption
+[[file:figs/dvf_root_locus_ws.png]]
+
+#+name: fig:iff_root_locus_ws
+#+caption: Figure caption
+[[file:figs/iff_root_locus_ws.png]]
+
+*** Relative Direct Velocity Feedback Architecture
+:PROPERTIES:
+:UNNUMBERED: t
+:END:
+
+Active Damping can help:
+- by reducing the effect of disturbances close to the resonance of the system
+- by making the plant dynamics simpler to control for the High Authority Controller
+
 
 *Relative motion sensors* are then included in each of the nano-hexapod's leg and a decentralized direct velocity feedback control architecture is applied (Figure [[fig:control_architecture_dvf]]).
 
@@ -1135,6 +1281,10 @@ The force applied in each leg being proportional to the relative velocity of the
 The DVF gain is here chosen in such a way that the suspension modes of the nano-hexapod are critically damped whatever the sample mass.
 This may not be the optimal choice as will be further explained.
 
+*** Effect of Active Damping on the Primary Plant Dynamics
+:PROPERTIES:
+:UNNUMBERED: t
+:END:
 
 The plant dynamics before (solid curves) and after (dashed curves) the Low Authority Control implementation are compared in Figure [[fig:opt_stiff_primary_plant_damped_L]].
 It is clear that the use of the DVF reduces the dynamical spread of the plant dynamics between 5Hz up too 100Hz.
@@ -1144,6 +1294,11 @@ This will make the primary controller more robust and easier to develop.
 #+caption: Primary plant in the space of the legs with (dashed) and without (solid) Direct Velocity Feedback
 [[file:figs/opt_stiff_primary_plant_damped_L.png]]
 
+*** Effect of Active Damping on the Sensibility to Disturbances
+:PROPERTIES:
+:UNNUMBERED: t
+:END:
+
 The change of sensibility to disturbances with the use of DVF is shown in Figure [[fig:opt_stiff_sensibility_dist_dvf]].
 It is shown that the DVF control lowers the sensibility to disturbances in the vicinity of the nano-hexapod resonance but increases the sensibility at higher frequencies.
 
@@ -1250,14 +1405,24 @@ An animation of the experiment is shown in Figure [[fig:closed_loop_sim_zoom]] a
 [[file:figs/closed_loop_sim_zoom.gif]]
 
 ** Simulation of More Complex Experiments
+*** Introduction                                                    :ignore:
+
+*** Micro-Hexapod offset
+:PROPERTIES:
+:UNNUMBERED: t
+:END:
 
 #+name: fig:tomography_dh_offset
-#+caption: Figure caption
+#+caption: Top View of a tomography experiment with a 10mm offset imposed by the micro-hexapod
 [[file:figs/tomography_dh_offset.gif]]
 
+*** Simultaneous Translation Scans
+:PROPERTIES:
+:UNNUMBERED: t
+:END:
 
 #+name: fig:ty_scans
-#+caption: Figure caption
+#+caption: Top View of a tomography experiment combined with translation scans
 [[file:figs/ty_scans.gif]]
 
 ** Conclusion
@@ -1281,6 +1446,8 @@ A more complete study of the control of the NASS is performed [[https://tdehaeze
 * General Conclusion and Further notes
 <<sec:conclusion_and_further_notes>>
 
+** Nano-Hexapod Specifications
+
 ** General Conclusion
 
 
@@ -1291,8 +1458,10 @@ This should not be significant.
 
 ** Further Work
 
+** Cable Forces
 
-** Using soft mounts for the
+
+** Using soft mounts for the Granite
 <<sec:soft_granite>>
 
 #+name: fig:opt_stiff_soft_granite_Dw

ref.bib

@@ -3,7 +3,6 @@
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   publisher = {Ios Press},
-  tags = {favorite},
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@@ -16,7 +15,6 @@
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@@ -28,5 +26,40 @@
   doi = {10.1007/978-3-319-72296-2},
   pages = {nil},
   series = {Solid Mechanics and Its Applications},
+}
+
+@book{taghirad13_paral,
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+  publisher = {CRC Press},
+  address = {Boca Raton, FL},
+  isbn = {9781466555778},
   tags = {favorite, parallel robot},
 }
+
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+  publisher = {Elsevier BV},
+}
+
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+}