Add figures for the control section

.gitignore

@@ -1 +1,2 @@
 figs/*.svg
+.auctex-auto/

index.org

@@ -690,6 +690,8 @@ Then, using the model, we can
 - include a multi-body model of the nano-hexapod and closed-loop simulations
 
 ** Wanted position of the sample and position error
+<<sec:pos_error_nass>>
+
 For the control of the nano-hexapod, we need to now the sample position error (the motion to be compensated) in the frame of the nano-hexapod.
 
 To do so, we need to perform several computations (summarized in Figure [[fig:control-schematic-nass]]):
@@ -804,7 +806,7 @@ The sensibilities to ground motion in the Y and Z directions are shown in Figure
 We can see that above the suspension mode of the nano-hexapod, the norm of the transmissibility is close to one until the suspension mode of the granite.
 Thus, a stiff nano-hexapod is better for reducing the effect of ground motion at low frequency.
 
-It will be further suggested that using soft mounts for the granite can greatly lower the sensibility to ground motion.
+It will be suggested in Section [[sec:soft_granite]] that using soft mounts for the granite can greatly lower the sensibility to ground motion.
 
 #+name: fig:opt_stiff_sensitivity_Dw
 #+caption: Sensitivity to Ground motion to the position error of the sample
@@ -998,44 +1000,139 @@ This show how the dynamics evolves with the stiffness and how different effects
 
 ** Conclusion
 #+begin_important
-  In Section [[sec:optimal_stiff_dist]], it has been concluded that a nano-hexapod stiffness
+  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.
-  Section [[sec:optimal_stiff_plant]]
+  As the high frequency vibrations are the most difficult to compensate for when using feedback control, a soft hexapod will most certainly helps improving the performances.
 
-  A stiffness of $10^5\,[N/m]$ will be used.
+  In Section [[sec:optimal_stiff_plant]], we concluded that a nano-hexapod leg stiffness in the range $10^5 - 10^6\,[N/m]$ is a good compromise between the uncertainty induced by the micro-station dynamics and by the rotating speed.
-#+end_important
+  Provided that the samples used have a first mode that is sufficiently high in frequency, the total plant dynamic uncertainty should be manageable.
 
-
+  Thus, a stiffness of $10^5\,[N/m]$ will be used in Section [[sec:robust_control_architecture]] to develop the robust control architecture and to perform simulations.
-#+begin_important
-It is preferred that *one* controller is working for all the payloads.
-If not possible, the alternative would be to develop an adaptive controller that depends on the payload mass/inertia.
-#+end_important
 
   A more detailed study of the determination of the optimal stiffness based on all the effects is available [[https://tdehaeze.github.io/nass-simscape/uncertainty_optimal_stiffness.html][here]].
+#+end_important
 
 * Robust Control Architecture
 <<sec:robust_control_architecture>>
 
 ** Introduction                                                      :ignore:
+
+
 https://tdehaeze.github.io/nass-simscape/optimal_stiffness_control.html
 
 stiffness 10^5
 
+
+It is preferred that *one* controller is designed such that it will give acceptable performance for all the payloads that will be used.
+
+This is quite challenging as the plant dynamics does depend quite a lot on the payload's mass.
+
+It is difficult to design a
+
+As there is a trade-off robustness/performance, the bigger the plant dynamic change, the lower the attainable performance.
+
+
+If not possible to develop a robust controller that gives 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
+adaptive control is generally difficult to use in practice.
+
+
+HAC-LAC
+
+#+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. This approach has the following advantages:
+- The active damping extends outside the bandwidth of the HAC and reduces the settling time of the modes which are outsite the bandwidth
+- The active damping makes it easier to gain-stabilize the modes outside the bandwidth of the output loop (improved gain margin)
+- The larger damping of the modes within the controller bandwidth makes them more robust to the parmetric uncertainty (improved phase margin)
+#+end_quote
+
+#+name: fig:control_architecture_hac_lac_one_input
+#+caption: HAC-LAC Architecture with a system having only one input
+[[file:figs/control_architecture_hac_lac_one_input.png]]
+
 ** Active Damping and Sensors to be included
-Ways to damp:
+Active Damping can help with two things
-- Force Sensor
+
+#+begin_quote
+Active damping is very effective in reducing the settling time of transient disturbances and the effect of steady state disturbances near the resonance frequencies of the system; however, away from the resonances, the active damping is completely ineffective and leaves the closed-loop response essentially unchanged.
+Such low-gain controllers are often called Low Authority Controllers (LAC), because they modify the poles of the system only slightly.
+#+end_quote
+
+There are three main ways to actively damp a system:
+- force Sensor
 - Relative Velocity Sensors
 - Inertial Sensor
 
+Because of the rotation
  
 https://tdehaeze.github.io/rotating-frame/index.html
 
-Sensors to be included:
+Thus, relative motion sensors should be included in each of the nano-hexapod's leg.
+
+The decentralized direct velocity feedback control architecture is shown in figure [[fig:control_architecture_dvf]] where:
+- $\bm{\tau}$: Forces applied in each leg
+- $\bm{\tau}_m$: Force sensor located in each leg
+- $\bm{\mathcal{X}}$: Measurement of the payload position with respect to the granite
+- $d\bm{\mathcal{L}}$: Measurement of the (small) relative motion of each leg
+
+The controller $\bm{K}_{\text{DVF}}$ is a diagonal
+
+#+name: fig:control_architecture_dvf
+#+caption: Low Authority Control: Decentralized Direct Velocity Feedback
+[[file:figs/control_architecture_dvf.png]]
+
+
+
+#+name: fig:opt_stiff_primary_plant_damped_L
+#+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]]
+
+As shown in Figure [[fig:opt_stiff_sensibility_dist_dvf]], the use of the DVF 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 be used, and further analysis and optimization will 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
+[[file:figs/opt_stiff_sensibility_dist_dvf.png]]
 
 ** Motion Control
+The complete control architecture is shown in Figure [[fig:control_architecture_hac_dvf_pos_L]] where an outer loop is added to the decentralized direct velocity feedback loop.
+
+The block =Compute Position Error= is used to compute the position error of the sample with respect to the nano-hexapod's base platform $\bm{\epsilon}_{\mathcal{X}_n}$ from the actual measurement of the sample's pose $\bm{\mathcal{X}}$ and the wanted pose $\bm{r}_\mathcal{X}$.
+The computation done in such block was explained briefly in Section [[sec:pos_error_nass]].
+
+From the position error express in the frame of the nano-hexapod, $\bm{J}$
+
+$\bm{\epsilon}_\mathcal{L}$ thus express the length error of each of the nano hexapod's leg such that it position the sample at the correct position.
+
+Then, a diagonal controller $\bm{K}_\mathcal{L}$ generates the required force in each leg such that
+
+#+name: fig:control_architecture_hac_dvf_pos_L
+#+caption: Cascade Control Architecture. The inner loop consist of a decentralized Direct Velocity Feedback. The outer loop consist of position control in the leg's space
+[[file:figs/control_architecture_hac_dvf_pos_L.png]]
+
+
+
+#+name: fig:opt_stiff_primary_plant_L
+#+caption: Diagonal elements of the transfer function matrix from $\bm{\tau}^\prime$ to $\bm{\epsilon}_{\mathcal{X}_n}$ for the three considered masses
+[[file:figs/opt_stiff_primary_plant_L.png]]
+
+
+
+#+name: fig:opt_stiff_primary_loop_gain_L
+#+caption: Loop gain for the primary plant
+[[file:figs/opt_stiff_primary_loop_gain_L.png]]
+
+
+
+#+name: fig:opt_stiff_primary_control_L_senbility_dist
+#+caption: Sensibility to disturbances when the HAC-LAC control is applied
+[[file:figs/opt_stiff_primary_control_L_senbility_dist.png]]
 
 ** Simulation of Tomography Experiments
 <<sec:tomography_experiment>>
 
+The obtained performances for all the three considered masses are very similar.
+That shows the robustness of the system.
+
 #+name: fig:opt_stiff_hac_dvf_L_psd_disp_error
 #+caption: Amplitude Spectral Density of the position error in Open Loop and with the HAC-LAC controller
 [[file:figs/opt_stiff_hac_dvf_L_psd_disp_error.png]]
@@ -1053,11 +1150,22 @@ Sensors to be included:
 [[file:figs/closed_loop_sim_zoom.gif]]
 
 ** Conclusion
+
 * Further notes
-Soft granite
+<<sec:further_notes>>
+
+** Using soft mounts for the
+<<sec:soft_granite>>
+
+#+name: fig:opt_stiff_soft_granite_Dw
+#+caption: Change of sensibility to Ground motion when using a stiff Granite (solid curves) and a soft Granite (dashed curves)
+[[file:figs/opt_stiff_soft_granite_Dw.png]]
+
+This means that above the suspension mode of the granite (here around 2Hz), the granite
 
 Sensible to detector motion?
 
+** Others
 Common metrology frame for the nano-focusing optics and the measurement of the sample position?
 
 Cable forces?

ref.bib

@@ -18,3 +18,15 @@
   url = {https://doi.org/10.1541/ieejjia.7.127},
   tags = {favorite},
 }
+
+@book{preumont18_vibrat_contr_activ_struc_fourt_edition,
+  author = {Andre Preumont},
+  title = {Vibration Control of Active Structures - Fourth Edition},
+  year = {2018},
+  publisher = {Springer International Publishing},
+  url = {https://doi.org/10.1007/978-3-319-72296-2},
+  doi = {10.1007/978-3-319-72296-2},
+  pages = {nil},
+  series = {Solid Mechanics and Its Applications},
+  tags = {favorite, parallel robot},
+}