diff --git a/.gitignore b/.gitignore index 4d52511..4ad6859 100644 --- a/.gitignore +++ b/.gitignore @@ -1 +1,2 @@ figs/*.svg +.auctex-auto/ diff --git a/figs/control_architecture_dvf.png b/figs/control_architecture_dvf.png new file mode 100644 index 0000000..e72ae08 Binary files /dev/null and b/figs/control_architecture_dvf.png differ diff --git a/figs/control_architecture_hac_dvf_pos_L.png b/figs/control_architecture_hac_dvf_pos_L.png new file mode 100644 index 0000000..5aef2c7 Binary files /dev/null and b/figs/control_architecture_hac_dvf_pos_L.png differ diff --git a/figs/control_architecture_hac_lac_one_input.png b/figs/control_architecture_hac_lac_one_input.png new file mode 100644 index 0000000..39948f0 Binary files /dev/null and b/figs/control_architecture_hac_lac_one_input.png differ diff --git a/figs/opt_stiff_primary_control_L_senbility_dist.png b/figs/opt_stiff_primary_control_L_senbility_dist.png new file mode 100644 index 0000000..e90cb19 Binary files /dev/null and b/figs/opt_stiff_primary_control_L_senbility_dist.png differ diff --git a/figs/opt_stiff_primary_loop_gain_L.png b/figs/opt_stiff_primary_loop_gain_L.png new file mode 100644 index 0000000..a9d9780 Binary files /dev/null and b/figs/opt_stiff_primary_loop_gain_L.png differ diff --git a/figs/opt_stiff_primary_plant_L.png b/figs/opt_stiff_primary_plant_L.png new file mode 100644 index 0000000..e1dfb6d Binary files /dev/null and b/figs/opt_stiff_primary_plant_L.png differ diff --git a/figs/opt_stiff_primary_plant_damped_L.png b/figs/opt_stiff_primary_plant_damped_L.png new file mode 100644 index 0000000..2996459 Binary files /dev/null and b/figs/opt_stiff_primary_plant_damped_L.png differ diff --git a/figs/opt_stiff_sensibility_dist_dvf.png b/figs/opt_stiff_sensibility_dist_dvf.png new file mode 100644 index 0000000..a089342 Binary files /dev/null and b/figs/opt_stiff_sensibility_dist_dvf.png differ diff --git a/figs/opt_stiff_soft_granite_Dw.png b/figs/opt_stiff_soft_granite_Dw.png new file mode 100644 index 0000000..6053c27 Binary files /dev/null and b/figs/opt_stiff_soft_granite_Dw.png differ diff --git a/index.org b/index.org index e5e49e5..d52a2be 100644 --- a/index.org +++ b/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 +<> + 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 - Section [[sec:optimal_stiff_plant]] + 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. + 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. + 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. + + 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 - -#+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]]. - * 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: -- Force Sensor +Active Damping can help with two things + +#+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 <> +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 +<> + +** Using soft mounts for the +<> + +#+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? diff --git a/ref.bib b/ref.bib index e16e88c..95e2173 100644 --- a/ref.bib +++ b/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}, +}