Start re-reading part about control

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@@ -1418,32 +1418,33 @@ For instance, the flexible joint used for the ID16 nano-hexapod have the followi
 Before designing the control system, let's summarize what have been done:
 - The multi-body model of the micro-station has been tuned based on actual dynamical measurements
 - Ground motion and stage vibrations have been estimated and included in the model
-- The optimal nano-hexapod stiffness has been determined such that is minimizes the effect of disturbances and at the same time reduces the plant dynamic uncertainty
+- The optimal nano-hexapod stiffness has been determined such that it minimizes the effect of disturbances and at the same time reduces the plant dynamic uncertainty
 
 The optimal nano-hexapod is now included in the model, and a robust control architecture is developed.
 
 
-Note that it is preferred to design *one* controller that gives acceptable performance for *all* the changes in the system (payload masses, spindle's rotation speeds, etc), this control property is called *robust performance*.
+Note that it is preferred to design *one* controller that gives acceptable performance for *all* the changes in the system (payload masses, spindle's rotation speeds, etc).
+This control property is referred to as *robust performance*.
 This is however quite challenging as the plant dynamics changes a lot with experimental conditions such as a change of payload's mass.
 
 If it turns out that no robust controller can give acceptable performance, an alternative would be to develop an *adaptive* controller that depends on the payload mass/inertia.
-This would require to measure the mass/inertia of each used payload and manually choose the controller that was design for that particular mass/inertia.
+This would however require to measure the mass/inertia of each used payload and to manually choose the controller that was designed for that particular mass/inertia.
 
 
+This part is divided in the following sections:
 - Section [[sec:hac_lac]]: the High Authority Control / Low Authority Control Architecture is described and the reasons of its use are explained
 - Section [[sec:lac_control]]: the active damping strategy is implemented and its effects on the system are described
 - Section [[sec:hac_control]]: the high authority control is developed and the control robustness is studied
 - Section [[sec:tomography_experiment]]: tomography experiments are simulated and the performances are estimated
-- Section [[sec:more_simulations]]: more complex simulations are performed
+- Section [[sec:more_simulations]]: more complex simulations are performed to further validate this control architecture
 
 ** High Authority Control / Low Authority Control Architecture
 <<sec:hac_lac>>
 
-Many control architecture for the control of Stewart platform have been developed and presented in the scientific literature.
+There exist many control architectures that could be used on Stewart platforms.
+One of the them that seems the most adapted for the NASS is called the *High Authority Control / Low Authority Control* (HAC-LAC) architecture.
 
-One of the control architectures that seems adapted for the current problem is the *High Authority Control / Low Authority Control* (HAC-LAC) architecture.
+Some interesting properties of the HAC-LAC architecture are summarized below (taken from cite:preumont18_vibrat_contr_activ_struc_fourt_edition):
-
-Some properties of the HAC-LAC architecture are explained below (taken from cite:preumont18_vibrat_contr_activ_struc_fourt_edition):
 #+begin_quote
 The HAC/LAC approach consist of combining the two approached in a dual-loop control as shown in Figure [[fig:control_architecture_hac_lac_one_input]].
 The inner loop uses a set of collocated actuator/sensor pairs for decentralized active damping with guaranteed stability ; the outer loop consists of a non-collocated HAC based on a model of the actively damped structure.
@@ -1465,25 +1466,23 @@ The HAC-LAC architecture thus consists of two cascade controllers:
 <<sec:lac_control>>
 
 *** Introduction                                                    :ignore:
-Three main active damping techniques can usually be applied:
+
+Three active damping techniques could be applied for the Low Authority Control:
 - Integral Force Feedback
 - Direct Velocity Feedback with relative motion sensors
 - Direct Velocity Feedback with inertial sensors
 
-These active damping techniques are compared based on (reports accessible [[https://tdehaeze.github.io/rotating-frame/index.html][here]] and [[https://tdehaeze.github.io/nass-simscape/control_active_damping.html][here]]):
+To determine the most suited active damping technique, they are compared based on their ability to (reports accessible [[https://tdehaeze.github.io/rotating-frame/index.html][here]] and [[https://tdehaeze.github.io/nass-simscape/control_active_damping.html][here]]):
-- the obtained reduction of the effect of disturbances close to the resonance of the system
+- reduce the effect of disturbances
-- their ability to render the plant dynamics simpler to control for the High Authority Controller
+- render the plant dynamics simpler to control for the High Authority Controller
-- their robustness properties
+- remains stable for all changes in the system
 
-The conclusions are:
+The conclusions are (summarized in Table [[tab:comp_active_damping]]):
-- the use of force sensors is to be avoided as it could introduce instability in the system due to the nano-hexapod's rotation (effect explained in the next section)
+- Integral Force Feedback is to be avoided as it renders the system unstable when the nano-hexapod's is rotating (effect explained in the next section)
-- 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).
+- Direct velocity feedback with inertial sensor should also be avoided as it would tends to decouple the motion of the sample from the motion of the granite (which is not wanted).
-  It also does not give the wanted robust properties
+  It also does not give the wanted robustness properties
-- relative motion sensors can be used to damp the nano-hexapod's modes in a robust way but may increase the sensibility to stages vibrations
+- Direct velocity feedback with relative motion sensors can be used to damp the nano-hexapod's modes in a robust way.
-
+  It however may increases the sensibility to stages vibrations at higher frequency
-*Direct velocity feedback* is then the active damping techniques that will be applied to the nano-hexapod, and *relative motion sensor* are integrated in the six nano-hexapod's legs.
-
-Main advantages and disadvantage of these active damping techniques are summarized in Table [[tab:comp_active_damping]].
 
 #+name: tab:comp_active_damping
 #+caption: Comparison of the three main active damping techniques that could be applied to the nano-hexapod
@@ -1497,14 +1496,19 @@ Main advantages and disadvantage of these active damping techniques are summariz
 | Disadvantages          | Reduces the compliance at low frequency         | Increases the transmissibility at high frequency | No guaranteed stability                                  |
 |                        | No guaranteed stability in presence of rotation |                                                  | Increases the sensibility to ground motion               |
 
+#+begin_important
+ *Relative Direct Velocity Feedback* is found to be the most suitable active damping techniques for the nano-hexapod.
+  Therefore, *relative motion sensors* must be integrated in the six nano-hexapod's legs.
+#+end_important
+
 *** Effect of the Spindle's Rotation - Guaranteed Stability
 :PROPERTIES:
 :UNNUMBERED: t
 :END:
 
-To see why Integral Force Feedback should not be applied for active damping of the nano-hexapod, a simple model of a rotating position platform integration force sensors has been developed (described in details [[https://tdehaeze.github.io/rotating-frame/index.html][here]]).
+To see why Integral Force Feedback should not be applied to damp the nano-hexapod's modes, a simple model of a rotating positioning platform integration force sensors has been developed (described in details [[https://tdehaeze.github.io/rotating-frame/index.html][here]]).
 
-The position platform main resonance frequency is $\omega_0$ and its rotating speed is $\omega$.
+The platform main resonance frequency is $\omega_0$ and the rotation speed is $\omega$.
 
 Root Locus plots for Integral Force Feedback and Direct Velocity Feedback are shown in Table [[fig:root_locus_rotation_active_damping]].
 These plots show the evolution of the system's poles in the complex plane as a function of the control gain.
@@ -1515,12 +1519,13 @@ These plots show the evolution of the system's poles in the complex plane as a f
 | Direct Velocity Feedback        | Integral Force Feedback         |
 
 
-To understand what the root locus means, consider Figure [[fig:preumont18_effect_damping]] where two resonant systems are compared.
+To understand what the root locus means, consider Figure [[fig:preumont18_effect_damping]] where two resonant systems are compared:
+- The first one (represented in blue) is undamped.
+  The corresponding poles of this system is located on the imaginary axis.
+- The second one (represented in red) has some added damping which can be easily see by the reduction of the amplitude near the resonance.
+  On the complex plane, this is manifested by the being "moved" such that the angle $\phi$ it makes with the imaginary axis is increase.
 
-The first one in blue is undamped and its pole in the complex plane is located on the imaginary axis.
+As a matter of fact, the damping ratio of a system resonance can be approximated by $\xi \approx \sin \phi$.
-The second one in red has some added damping which can be easily see by the maximum value of its amplitude.
-
-On the complex plane, the damping ratio $\xi$ of a system resonance can be estimated by the angle its corresponding pole makes with the imaginary axis: $\xi = \sin \phi$.
 
 To damp a system, it is thus wanted to move the poles to the left part of the complex plane.
 A pole with a positive real part corresponds to an unstable system, and thus the right part of the complex should be avoided.
@@ -1532,20 +1537,20 @@ A pole with a positive real part corresponds to an unstable system, and thus the
 
 Coming back to the Root Locus in Table [[fig:root_locus_rotation_active_damping]], it can be seen that:
 - For Direct Velocity Feedback:
-  - The system's poles are staying in the left half plane which shows the guaranteed stability
+  - The system's poles are staying in the left half plane which means guaranteed stability
   - Arbitrary damping can be added to the system's resonances
 - For Integral Force Feedback:
   - For non null rotation speed, and whatever the control gain is, a pole is located in the right part of the complex plane, showing that the closed-loop system is unstable
   - Limited damping can be added to the system
 
-Same observations are made on the Simscape model of the NASS, and this shows why Direct Velocity Feedback is the chosen method to apply active damping.
+Similar observations are made using the Simscape model of the NASS, and this shows why Direct Velocity Feedback is the most suitable active damping technique for the NASS.
 
 *** Relative Direct Velocity Feedback Architecture
 :PROPERTIES:
 :UNNUMBERED: t
 :END:
 
-*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]]).
+*Relative motion sensors* are included in each of the nano-hexapod's leg and a decentralized direct velocity feedback control architecture is applied (Figure [[fig:control_architecture_dvf]]).
 
 The signals shown in Figure [[fig:control_architecture_dvf]] are:
 - $\bm{\tau}$: Actuator forces applied in each leg
@@ -1555,7 +1560,7 @@ The signals shown in Figure [[fig:control_architecture_dvf]] are:
 
 $\bm{K}_{\text{DVF}}$ is a diagonal controller with derivative action.
 This control architecture is equivalent as to have six independent control loops from the relative motion sensor of one leg to the actuator of the same leg (hence the term *decentralized*).
-The force applied in each leg being proportional to the relative velocity of the associated leg (thanks to the derivative action), this adds *damping* to the nano-hexapod's modes.
+The force applied in each leg being proportional to the relative velocity of the associated leg (thanks to the derivative action), this is equivalent as adding *damping* in each of the legs.
 
 #+name: fig:control_architecture_dvf
 #+caption: Low Authority Control: Decentralized Direct Velocity Feedback
@@ -1581,7 +1586,8 @@ Moreover, it is seen that arbitrary damping can be applied to the nano-hexapod's
 #+caption: Root Locus (zoomed on the nano-hexapod modes) corresponding to the Direct Velocity Feedback control for three payload masses
 [[file:figs/opt_stiff_dvf_root_locus.png]]
 
-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.
+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 Sensibility to Disturbances
@@ -1589,15 +1595,15 @@ This may not be the optimal choice as will be further explained.
 :UNNUMBERED: t
 :END:
 
-The first objective of the active damping technique is to lower the sensibility to disturbances.
+One objective of the active damping technique is to lower the sensibility to disturbances which are shown in Figure [[fig:opt_stiff_sensibility_dist_dvf]] without active damping (solid) and with the use of DVF (dashed).
-These are shown without active damping (solid) and with the use of DVF (dashed) is shown in Figure [[fig:opt_stiff_sensibility_dist_dvf]].
 
 The Direct Velocity Feedback control lowers the sensibility to disturbances in the vicinity of the nano-hexapod resonance but increases the sensibility at higher frequencies.
 
-This is probably not the optimal gain that could have been used, and further analysis and optimization should be performed.
+A smaller control gain could probably limit the increase of the sensibility at higher frequencies while still providing sufficient sensibility reduction near the nano-hexapod resonances.
+Further optimization of the gain should then be performed.
 
 #+name: fig:opt_stiff_sensibility_dist_dvf
-#+caption: Norm of the transfer function from vertical disturbances to vertical position error with (dashed) and without (solid) Direct Velocity Feedback applied
+#+caption: Norm of the transfer function from vertical disturbances to vertical position error with (dashed) and without (solid) Direct Velocity Feedback applied. Disturbances are: ground motion (top left), direct forces (top right), translation stage vibration (bottom left) and spindle vibrations (bottom right)
 [[file:figs/opt_stiff_sensibility_dist_dvf.png]]
 
 *** Effect of Active Damping on the Primary Plant Dynamics
@@ -1605,10 +1611,10 @@ This is probably not the optimal gain that could have been used, and further ana
 :UNNUMBERED: t
 :END:
 
-The second objective is to render the plant dynamics simpler to control for the High Authority Controller.
+Another control objective for the LAC is to render the plant dynamics simpler to control for the High Authority Controller.
 
 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.
+It is clear that the use of the DVF reduces the dynamical spread of the plant dynamics between 5Hz and 100Hz.
 This will make the primary controller more robust and easier to develop.
 
 #+name: fig:opt_stiff_primary_plant_damped_L
@@ -1621,7 +1627,7 @@ This will make the primary controller more robust and easier to develop.
 :END:
 
 #+begin_important
-  It has been shown that *Direct Velocity Feedback* using *relative motion sensors* is the most adapted active damping technique that should be applied to the nano-hexapod.
+  It has been shown that *Direct Velocity Feedback* using *relative motion sensors* is the most adapted active damping technique to be applied to the nano-hexapod.
 
   Its main disadvantage is the increase of the sensibility to stage vibrations at high frequency which may limit the performances of the system.
 
@@ -1632,11 +1638,11 @@ This will make the primary controller more robust and easier to develop.
 <<sec:hac_control>>
 
 *** Introduction                                                    :ignore:
-Why this is the most important part
+The High Authority Controller objective is to stabilize the position of the sample with respect to the granite.
 
-The use of the Low Authority Controller was mostly to facilitate the design of the High Authority Controller.
+It might be the most important element of the control architecture as it acts directly based on the actual sample's position.
 
-The bandwidth of the High Authority Controller it what will most likely determine the performance of the system.
+Its proper design will most likely determine the performance of the system.
 
 *** Control in the Task space or in the Leg Space?
 :PROPERTIES: