1 Introduction to Feedback Systems and Noise budgeting
+1 Introduction to Feedback Systems and Noise budgeting
-In this section, we first introduce some basics of feedback systems (Section 1.1). +In this section, we first introduce some basics of feedback systems (Section 1.1). This should highlight the challenges in terms of combined performance and robustness.
-In Section 1.2 is introduced the dynamic error budgeting which is a powerful tool that allows to derive the total error in a dynamic system from multiple disturbance sources. +In Section 1.2 is introduced the dynamic error budgeting which is a powerful tool that allows to derive the total error in a dynamic system from multiple disturbance sources. This tool will be widely used throughout this study to both predict the performances and identify the effects that do limit the performances.
1.1 Feedback System
+1.1 Feedback System
The use of feedback control as several advantages and pitfalls that are listed below (taken from schmidt14_desig_high_perfor_mechat_revis_edition): @@ -203,11 +214,11 @@ Thus the robustness properties of the feedback system must be carefully g
1.1.1 Simplified Feedback Control Diagram for the NASS
+1.1.1 Simplified Feedback Control Diagram for the NASS
-Let’s consider the block diagram shown in Figure 1 where the signals are: +Let’s consider the block diagram shown in Figure 1 where the signals are:
- \(y\): the relative position of the sample with respect to the granite (the quantity we wish to control) @@ -227,7 +238,7 @@ And the dynamical blocks are:
Figure 1: Block Diagram of a simple feedback system
@@ -249,11 +260,11 @@ In the next section, we see how the use of the feedback system permits to lower1.1.2 How does the feedback loop is modifying the system behavior?
+1.1.2 How does the feedback loop is modifying the system behavior?
-If we write down the position error signal \(\epsilon = r - y\) as a function of the reference signal \(r\), the disturbances \(d\) and the measurement noise \(n\) (using the feedback diagram in Figure 1), we obtain: +If we write down the position error signal \(\epsilon = r - y\) as a function of the reference signal \(r\), the disturbances \(d\) and the measurement noise \(n\) (using the feedback diagram in Figure 1), we obtain: \[ \epsilon = \frac{1}{1 + GK} r + \frac{GK}{1 + GK} n - \frac{G_d}{1 + GK} d \]
@@ -298,8 +309,8 @@ As shown in the next section, there is a trade-off between the disturbance reduc1.1.3 Trade off: Disturbance Reduction / Noise Injection
+1.1.3 Trade off: Disturbance Reduction / Noise Injection
We have from the definition of \(S\) and \(T\) that: @@ -317,7 +328,7 @@ There is therefore a trade-off between the disturbance rejection and the meas
-Typical shapes of \(|S|\) and \(|T|\) as a function of frequency are shown in Figure 2. +Typical shapes of \(|S|\) and \(|T|\) as a function of frequency are shown in Figure 2. We can observe that \(|S|\) and \(|T|\) exhibit different behaviors depending on the frequency band:
-
@@ -338,7 +349,7 @@ We can observe that \(|S|\) and \(|T|\) exhibit different behaviors depending on
Figure 2: Typical shapes and constrain of the Sensibility and Transmibility closed-loop transfer functions
@@ -346,11 +357,11 @@ We can observe that \(|S|\) and \(|T|\) exhibit different behaviors depending on1.1.4 Trade off: Robustness / Performance
+1.1.4 Trade off: Robustness / Performance
@@ -371,11 +382,11 @@ The main issue it that for stability reasons, the behavior of the mechanical
-For mechanical systems, this generally means that control bandwidth should take place before any appearing of flexible dynamics (Right part of Figure 3). +For mechanical systems, this generally means that control bandwidth should take place before any appearing of flexible dynamics (Right part of Figure 3).
-
Figure 3: Envisaged developments in motion systems. In traditional motion systems, the control bandwidth takes place in the rigid-body region. In the next generation systemes, flexible dynamics are foreseen to occur within the control bandwidth. oomen18_advan_motion_contr_precis_mechat
@@ -405,11 +416,11 @@ This problem of robustness represent one of the main challenge for the de1.2 Dynamic error budgeting
+1.2 Dynamic error budgeting
The dynamic error budgeting is a powerful tool to study the effect of multiple error sources and to see how the feedback system does reduce the effect @@ -425,8 +436,8 @@ Finally,
1.2.1 Power Spectral Density
+1.2.1 Power Spectral Density
The Power Spectral Density (PSD) \(S_{xx}(f)\) of the time domain signal \(x(t)\) is defined as the Fourier transform of the autocorrelation function: @@ -453,8 +464,8 @@ One can also integrate the infinitesimal power \(S_{xx}(\omega)d\omega\) over a
1.2.2 Cumulative Power Spectrum
+1.2.2 Cumulative Power Spectrum
The Cumulative Power Spectrum is the cumulative integral of the Power Spectral Density starting from \(0\ \text{Hz}\) with increasing frequency: @@ -484,15 +495,15 @@ The Cumulative Power Spectrum will be used to determine in which frequency band
1.2.3 Modification of a signal’s PSD when going through an LTI system
+1.2.3 Modification of a signal’s PSD when going through an LTI system
-Let’s consider a signal \(u\) with a PSD \(S_{uu}\) going through a LTI system \(G(s)\) that outputs a signal \(y\) with a PSD (Figure 4). +Let’s consider a signal \(u\) with a PSD \(S_{uu}\) going through a LTI system \(G(s)\) that outputs a signal \(y\) with a PSD (Figure 4).
-
Figure 4: LTI dynamical system \(G(s)\) with input signal \(u\) and output signal \(y\)
@@ -507,11 +518,11 @@ The Power Spectral Density of the output signal \(y\) can be computed using:1.2.4 PSD of combined signals
+1.2.4 PSD of combined signals
-Let’s consider a signal \(y\) that is the sum of two uncorrelated signals \(u\) and \(v\) (Figure 5). +Let’s consider a signal \(y\) that is the sum of two uncorrelated signals \(u\) and \(v\) (Figure 5).
@@ -520,7 +531,7 @@ We have that the PSD of \(y\) is equal to sum of the PSD and \(u\) and the PSD o
-
Figure 5: \(y\) as the sum of two signals \(u\) and \(v\)
@@ -528,11 +539,11 @@ We have that the PSD of \(y\) is equal to sum of the PSD and \(u\) and the PSD o1.2.5 Dynamic Noise Budgeting
+1.2.5 Dynamic Noise Budgeting
-Let’s consider the Feedback architecture in Figure 1 where the position error \(\epsilon\) is equal to: +Let’s consider the Feedback architecture in Figure 1 where the position error \(\epsilon\) is equal to: \[ \epsilon = S r + T n - G_d S d \]
@@ -556,24 +567,24 @@ To estimate the PSD of the position error \(\epsilon\) and thus the RMS residual- The Power Spectral Densities of the signals affecting the system:
-
-
- \(S_{dd}\): disturbances, this will be done in Section 3 +
- \(S_{dd}\): disturbances, this will be done in Section 3
- \(S_{nn}\): sensor noise, this can be estimated from the sensor data-sheet
- \(S_{rr}\): which is a deterministic signal that we choose. For simple tomography experiment, we can consider that it is equal to \(0\)
- The dynamics of the complete system comprising the micro-station and the nano-hexapod: \(G\), \(G_d\). -To do so, we need to identify the dynamics of the micro-station (Section 2), include this dynamics in a model (Section 4) and add a model of the nano-hexapod to the model (Section 5) -
- The controller \(K\) that will be designed in Section 6 +To do so, we need to identify the dynamics of the micro-station (Section 2), include this dynamics in a model (Section 4) and add a model of the nano-hexapod to the model (Section 5) +
- The controller \(K\) that will be designed in Section 6
2 Identification of the Micro-Station Dynamics
+2 Identification of the Micro-Station Dynamics
As explained before, it is very important to have a good estimation of the micro-station dynamics as it will be coupled with the dynamics of the nano-hexapod and thus is very important for both the design of the nano-hexapod and controller. @@ -589,7 +600,7 @@ All the measurements performed on the micro-station are detailed in 6. +The general procedure to identify the dynamics of the micro-station is shown in Figure 6.
@@ -602,7 +613,7 @@ The steps are: -
Figure 6: Vibration Analysis Procedure
@@ -613,11 +624,11 @@ The extraction of the Spatial Model (3rd step) was not performed as it requires2.1 Setup
+2.1 Setup
@@ -643,13 +654,13 @@ In order to perform the Modal Analysis, the following devices were used: The measurement thus consists of:
-
-
- Exciting the structure at the same location with the Hammer (Figure 7) +
- Exciting the structure at the same location with the Hammer (Figure 7)
- Move the accelerometers to measure all the DOF of the structure.
The position of the accelerometers are:
- 4 on the first granite
- 4 on the second granite -
- 4 on top of the translation stage (figure 8) +
- 4 on top of the translation stage (figure 8)
- 4 on top of the tilt stage
- 3 on top of the spindle
- 4 on top of the hexapod @@ -665,14 +676,14 @@ We chose to have some redundancy in the measurement to be able to verify that th -
- First, we need to determine the actual wanted pose (3 translations and 3 rotations) of the sample with respect to the granite. @@ -1195,7 +1210,7 @@ Both computation are performed.
- the effect of disturbances is minimized (Section 5.1) -
- the plant uncertainty due to a change of payload mass and experimental conditions is minimized (Section 5.2) +
- the effect of disturbances is minimized (Section 5.1) +
- the plant uncertainty due to a change of payload mass and experimental conditions is minimized (Section 5.2)
- the first resonance (suspension mode of the nano-hexapod) is lowered with an increase of the sample’s mass. +This first resonance corresponds to \(\omega = \sqrt{\frac{k_n}{m_n + m_s}}\) where \(k_n\) is the vertical nano-hexapod stiffness, \(m_n\) the mass of the nano-hexapod’s top platform, and \(m_s\) the sample’s mass +
- the gain after the first resonance and up until the anti-resonance at 100Hz is changing with the sample’s mass +
- https://tdehaeze.github.io/nass-simscape/uncertainty_optimal_stiffness.html +
- the phase stays between 0 and -180deg which is a very nice property for control +
- the dynamical change up until the resonance of the payload is mostly a change of gain
- the dynamics is unchanged until the first resonance which is around 25Hz-35Hz +
- above that frequency, the change of dynamics is quite chaotic (we will see in the next section that this is due to the micro-station dynamics) +
- a change in some mechanical elements +
- a change in the position of one stage. +For instance, a large displacement of the micro-hexapod can change the micro-station compliance +
- a change in a control loop +
- the control could be develop on top on another support and then added to the micro-station without changing the controller +
- the nano-hexapod could be use on top of any other station much more easily +
- without the micro-station (solid curves) +
- with the micro-station dynamics (dashed curves) +
- the payload’s mass influence the plant dynamics above the first resonance of the nano-hexapod. +Thus a high nano-hexapod stiffness helps reducing the effect of a change of the payload’s mass +
- the payload’s first resonance is seen as an anti-resonance in the plant dynamics. +As this effect will largely be variable from one payload to the other, the payload’s first resonance should be maximized (above 300Hz if possible) for all used payloads +
- the dynamics of the nano-hexapod is not affected by the micro-station dynamics (compliance) when \(k < 10^6\,[N/m]\) +
- the spindle’s rotating speed has no significant influence on the plant dynamics for nano-hexapods with a stiffness \(k > 10^5\,[N/m]\) +
- The multi-body model of the micro-station has been tuned based on actual dynamical measurements +
- Ground motion and stage vibrations are included in the model with realistic measured values +
- The optimal nano-hexapod stiffness has been determined such that is minimized the effect of disturbances and at the same time reduces the plant dynamic uncertainty +
- the plant dynamics does depend quite a lot on the payload’s mass +
- as there is a trade-off robustness/performance, the bigger the plant dynamic spread, the lower the simultaneous attainable performance is for all the plants +
- 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) +
- reducing the effect of uncertainty close to the resonance of the system +
- makes the plant dynamics simpler to control for the High Authority Controller +
- the use of force sensors is to be avoided as it could induce instability in the system due to rotation of the nano-hexapod +
- the use of inertial sensor should not be used as it would tends to decouple the motion of the sample to the motion of the granite (which is not wanted) and it would be difficult to apply in a robust way +
- relative motion sensors can be used to damped the nano-hexapod’s modes in a robust way +
- \(\bm{\tau}\): 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 sensibility is reduced within the controller bandwidth (here \(\approx 100\,[Hz]\)) +
- the sensibility is increase around the crossover frequency and mostly unchanged outside of the control bandwidth +
- The sample’s vibrations are reduced within the control bandwidth +
+An increase in the rotational vibrations is observed. +This is due to the fact that: +
+-
+
- no perturbations inducing rotations are included in the simulation: the vibrations in rotation are very small +
- the feedback control induces some coupling between the translations and rotations. +This means that it introduces some rotations due to translation vibrations +
+This increase in rotation is still very small and is not foreseen to be a problem +
+- The obtained performances for all the three considered masses are very similar. +That shows the robustness of the system +
- The vertical rotation plot is meaningless as the spindle rotation was considered to be perfect and no attempt is made to compensate vertical rotation by the nano-hexapod +
- From the Cumulative Amplitude Spectrum, we see that Z motion is reduced down to \(\approx 30\,nm\,[rms]\) and the Y motion down to \(\approx 25\,nm\,[rms]\) +
- Force Sensor -
- Relative Velocity Sensors -
- Inertial Sensor -
- [schmidt14_desig_high_perfor_mechat_revis_edition] Schmidt, Schitter & Rankers, The Design of High Performance Mechatronics - 2nd Revised Edition, Ios Press (2014).
- [oomen18_advan_motion_contr_precis_mechat] Tom Oomen, Advanced Motion Control for Precision Mechatronics: Control, Identification, and Learning of Complex Systems, IEEJ Journal of Industry Applications, 7(2), 127-140 (2018). link. doi. +
- [preumont18_vibrat_contr_activ_struc_fourt_edition] Andre Preumont, Vibration Control of Active Structures - Fourth Edition, Springer International Publishing (2018).
+-
Figure 7: Example of one hammer impact
+-
Figure 8: 3 tri axis accelerometers fixed to the translation stage
@@ -680,11 +691,11 @@ We chose to have some redundancy in the measurement to be able to verify that th-2.2 Results
++2.2 Results
@@ -692,18 +703,18 @@ From the measurements, we obtain all the transfer functions from forces applied
-Modal shapes and natural frequencies are then computed. Example of mode shapes are shown in Figures 9 10. +Modal shapes and natural frequencies are then computed. Example of mode shapes are shown in Figures 9 10.
-+-
Figure 9: First mode that shows a suspension mode, probably due to bad leveling of one Airloc
+
Figure 10: Sixth mode
@@ -729,12 +740,12 @@ This thus means that a multi-body model can be used to represent the dynamics ofMany Frequency Response Functions (FRF) are obtained from the measurements. -Examples of FRF are shown in Figure 11. +Examples of FRF are shown in Figure 11. These FRF will be used to compare the dynamics of the multi-body model with the micro-station dynamics.
-+-
Figure 11: Frequency Response Function from forces applied by the Hammer in the X direction to the acceleration of each solid body in the X direction
@@ -742,13 +753,13 @@ These FRF will be used to compare the dynamics of the multi-body model with the-2.3 Conclusion
++-2.3 Conclusion
@@ -756,11 +767,11 @@ In Section 4, the obtained Frequency Response FunctionThe modal analysis of the micro-station confirmed the fact that a multi-body model should be able to correctly represents the micro-station dynamics. -In Section 4, the obtained Frequency Response Functions will be used to compare the model dynamics with the micro-station dynamics. +In Section 4, the obtained Frequency Response Functions will be used to compare the model dynamics with the micro-station dynamics.
-3 Identification of the Disturbances
++3 Identification of the Disturbances
In this section, we wish to list and identify all the disturbances affecting the system. @@ -787,11 +798,11 @@ Open Loop Noise budget: -
3.1 Ground Motion
++3.1 Ground Motion
@@ -799,11 +810,11 @@ The ground motion can easily be estimated using an inertial sensor with sufficie
-To verify that the inertial sensors are sensitive enough, a Huddle test has been performed (Figure 12). +To verify that the inertial sensors are sensitive enough, a Huddle test has been performed (Figure 12).
-+
Figure 12: Huddle Test Setup
@@ -815,7 +826,7 @@ The low frequency differences between the ground motion at ID31 and ID09 is just -+-
Figure 13: Comparison of the PSD of the ground motion measured at different location
@@ -823,11 +834,11 @@ The low frequency differences between the ground motion at ID31 and ID09 is just-3.2 Stage Vibration - Effect of Control systems
++3.2 Stage Vibration - Effect of Control systems
@@ -850,11 +861,11 @@ Complete reports on these measurements are accessible -
3.3 Stage Vibration - Effect of Motion
++3.3 Stage Vibration - Effect of Motion
@@ -866,15 +877,15 @@ Details reports are accessible -
Spindle and Slip-Ring
-++Spindle and Slip-Ring
+-The setup for the measurement of vibrations induced by rotation of the Spindle and Slip-ring is shown in Figure 14. +The setup for the measurement of vibrations induced by rotation of the Spindle and Slip-ring is shown in Figure 14.
-+
Figure 14: Measurement of the sample’s vertical motion when rotating at 6rpm
@@ -890,7 +901,7 @@ A geophone is fixed at the location of the sample and we measure the motion:-The obtained Power Spectral Density of the sample’s absolute velocity are shown in Figure 15. +The obtained Power Spectral Density of the sample’s absolute velocity are shown in Figure 15.
@@ -906,7 +917,7 @@ However, when rotating with the Spindle (normal functioning mode): -
+-
Figure 15: Comparison of the ASD of the measured voltage from the Geophone at the sample location
@@ -921,19 +932,19 @@ Some investigation should be performed on the Spindle to determine where does th-Translation Stage
-++Translation Stage
+The same setup is used (a geophone is located at the sample’s location and another on the granite).
-We impose a 1Hz triangle motion with an amplitude of \(\pm 2.5mm\) on the translation stage (Figure 16), and we measure the absolute velocity of both the sample and the granite. +We impose a 1Hz triangle motion with an amplitude of \(\pm 2.5mm\) on the translation stage (Figure 16), and we measure the absolute velocity of both the sample and the granite.
-+
Figure 16: Y position of the translation stage measured by the encoders
@@ -941,25 +952,25 @@ We impose a 1Hz triangle motion with an amplitude of \(\pm 2.5mm\) on the transl-The time domain absolute vertical velocity of the sample and granite are shown in Figure 17. +The time domain absolute vertical velocity of the sample and granite are shown in Figure 17. It is shown that quite large motion of the granite is induced by the translation stage scans. This could be a problem if this is shown to excite the metrology frame of the nano-focusing lens position stage.
-+
Figure 17: Vertical velocity of the sample and marble when scanning with the translation stage
-The Amplitude Spectral Densities of the measured absolute velocities are shown in Figure 18. +The Amplitude Spectral Densities of the measured absolute velocities are shown in Figure 18. We can see many peaks starting from 1Hz showing the large spectral content probably due to the triangular reference of the translation stage.
-+-
Figure 18: Amplitude spectral density of the measure velocity corresponding to the geophone in the vertical direction located on the granite and at the sample location when the translation stage is scanning at 1Hz
@@ -975,15 +986,15 @@ A smoother motion for the translation stage (such as a sinus motion) could proba-3.4 Sum of all disturbances
++3.4 Sum of all disturbances
We can now compare the effect of all the disturbance sources on the position error (relative motion of the sample with respect to the granite).
-The Power Spectral Density of the motion error due to the ground motion, translation stage scans and spindle rotation are shown in Figure 19. +The Power Spectral Density of the motion error due to the ground motion, translation stage scans and spindle rotation are shown in Figure 19.
@@ -991,19 +1002,19 @@ We can see that the ground motion is quite small compare to the translation stag
-+
Figure 19: Amplitude Spectral Density fo the motion error due to disturbances
-The Cumulative Amplitude Spectrum is shown in Figure 20. +The Cumulative Amplitude Spectrum is shown in Figure 20. It is shown that the motion induced by translation stage scans and spindle rotation are in the micro-meter range.
-+-
Figure 20: Cumulative Amplitude Spectrum of the motion error due to disturbances
@@ -1023,8 +1034,8 @@ From that, we can conclude that we will probably need a control bandwidth to aro-3.5 Better estimation of the disturbances
++-3.5 Better estimation of the disturbances
All the disturbance measurements were made with inertial sensors, and to obtain the relative motion sample/granite, two inertial sensors were used and the signals were subtracted. @@ -1048,8 +1059,8 @@ The detector requirement would be:
-3.6 Conclusion
++3.6 Conclusion
-@@ -1074,14 +1085,14 @@ This should however not change the conclusion of this study nor significantly ch
-4 Multi Body Model
++4 Multi Body Model
-As was shown during the modal analysis (Section 2), the micro-station behaves as multiple rigid bodies (granite, translation stage, tilt stage, spindle, hexapod) with some discrete flexibility between those solid bodies. +As was shown during the modal analysis (Section 2), the micro-station behaves as multiple rigid bodies (granite, translation stage, tilt stage, spindle, hexapod) with some discrete flexibility between those solid bodies.
@@ -1094,8 +1105,8 @@ A small summary of the multi-body Simscape is available -
4.1 Multi-Body model
++4.1 Multi-Body model
The mass/inertia of each stage is automatically computed from the imported geometry and the material’s density. @@ -1107,11 +1118,11 @@ Then, the values of the stiffness and damping of each joint is manually tuned un
-The 3D representation of the simscape model is shown in Figure 21. +The 3D representation of the simscape model is shown in Figure 21.
-+
Figure 21: 3D representation of the simscape model
@@ -1119,15 +1130,15 @@ The 3D representation of the simscape model is shown in Figure -4.2 Validity of the model’s dynamics
++4.2 Validity of the model’s dynamics
It is very difficult the tune the dynamics of such model as there are more than 50 parameters and many curves to compare between the model and the measurements.
-The comparison of three of the Frequency Response Functions are shown in Figure 22. +The comparison of three of the Frequency Response Functions are shown in Figure 22.
@@ -1139,7 +1150,7 @@ We believe that the model is representing the micro-station dynamics with suffic
-+-
Figure 22: Frequency Response function from Hammer force in the X,Y and Z directions to the X,Y and Z displacements of the micro-hexapod’s top platform. The measurements are shown in blue and the Model in red.
@@ -1172,15 +1183,19 @@ Then, using the model, we can-4.3 Wanted position of the sample and position error
++4.3 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 23): +To do so, we need to perform several computations (summarized in Figure 23):
+
Figure 23: Figure caption
@@ -1207,10 +1222,11 @@ More details about these computations are accessible -4.4 Simulation of Experiments
++4.4 Simulation of Experiments
+ Now that the dynamics of the model is tuned and the disturbances included in the model, we can perform simulation of experiments.
@@ -1219,8 +1235,8 @@ We first do a simulation where the nano-hexapod is considered to be a solid-body-An animation of the obtained motion is shown in Figure 24. -A zoom in the micro-meter ranger on the sample’s location is shown in Figure 25. +An animation of the obtained motion is shown in Figure 24. +A zoom in the micro-meter ranger on the sample’s location is shown in Figure 25.
@@ -1233,7 +1249,7 @@ Note that here this frame is moving with the granite. -
+
Figure 24: Tomography Experiment using the Simscape Model
@@ -1241,14 +1257,14 @@ Note that here this frame is moving with the granite. -+
Figure 25: Tomography Experiment using the Simscape Model - Zoom on the sample’s position (the full vertical scale is \(\approx 10 \mu m\))
-The position error of the sample with respect to the granite are shown in Figure 26. +The position error of the sample with respect to the granite are shown in Figure 26. It is shown that the X-Y-Z position errors are in the micro-meter range.
@@ -1262,7 +1278,7 @@ For the vertical rotation, this is due to the fact that we suppose perfect rotat -+-
Figure 26: Position error of the Sample with respect to the granite during a Tomography Experiment with included disturbances
@@ -1270,8 +1286,8 @@ For the vertical rotation, this is due to the fact that we suppose perfect rotat-4.5 Conclusion
++4.5 Conclusion
-@@ -1296,11 +1312,11 @@ This model will be used in the next sections to help the design of the nano-hexa
-5 Optimal Nano-Hexapod Design
++5 Optimal Nano-Hexapod Design
-As explain before, the nano-hexapod properties (mass, stiffness, architecture, …) will influence: @@ -1314,8 +1330,8 @@ As explain before, the nano-hexapod properties (mass, stiffness, architecture, & Thus, we here wish to find the optimal nano-hexapod properties such that:
-
-
@@ -1327,13 +1343,12 @@ The study presented here only consider changes in the nano-hexapod stiffness<
-5.1 Optimal Stiffness to reduce the effect of disturbances
++5.1 Optimal Stiffness to reduce the effect of disturbances
-- +As will be seen, the nano-hexapod stiffness have a large influence on the sensibility to disturbance (the norm of \(G_d\)). For instance, it is quite obvious that a stiff nano-hexapod is better than a soft one when it comes to direct forces applied to the sample such as cable forces. @@ -1342,10 +1357,13 @@ For instance, it is quite obvious that a stiff nano-hexapod is better than a sof
A complete study of the optimal nano-hexapod stiffness for the minimization of disturbance sensibility here and summarized below.
++Sensibility to stage vibrations
+- +-The sensibility to the spindle’s vibration for all the considered nano-hexapod stiffnesses (from \(10^3\,[N/m]\) to \(10^9\,[N/m]\)) is shown in Figure 27. +The sensibility to the spindle’s vibration for all the considered nano-hexapod stiffnesses (from \(10^3\,[N/m]\) to \(10^9\,[N/m]\)) is shown in Figure 27. It is shown that a softer nano-hexapod it better to filter out vertical vibrations of the spindle. More precisely, is start to filters the vibration at the first suspension mode of the payload on top of the nano-hexapod.
@@ -1355,34 +1373,43 @@ The same conclusion is made for vibrations of the translation stage. -+++
Figure 27: Sensitivity to Spindle vertical motion error to the vertical error position of the sample
+Sensibility to ground motion
++-The sensibilities to ground motion in the Y and Z directions are shown in Figure 28. +The sensibilities to ground motion in the Y and Z directions are shown in Figure 28. 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 7.1 that using soft mounts for the granite can greatly lower the sensibility to ground motion.
-+++
Figure 28: Sensitivity to Ground motion to the position error of the sample
+Dynamic Noise Budgeting considering all the disturbances
+-However, lowering the sensibility to some disturbance at a frequency where its effect is already small compare to the other disturbances sources is not really interesting. What is more important than comparing the sensitivity to disturbances, is thus to compare the obtain power spectral density of the sample’s position error. -From the Power Spectral Density of all the sources of disturbances identified in Section 3, we compute what would be the Power Spectral Density of the vertical motion error for all the considered nano-hexapod stiffnesses (Figure 29). +From the Power Spectral Density of all the sources of disturbances identified in Section 3, we compute what would be the Power Spectral Density of the vertical motion error for all the considered nano-hexapod stiffnesses (Figure 29).
@@ -1390,31 +1417,31 @@ We can see that the most important change is in the frequency range 30Hz to 300H
-++-
Figure 29: Amplitude Spectral Density of the Sample vertical position error due to Vertical vibration of the Spindle for multiple nano-hexapod stiffnesses
-If we look at the Cumulative amplitude spectrum of the vertical error motion in Figure 30, we can observe that a soft hexapod (\(k < 10^5 - 10^6\,[N/m]\)) helps reducing the high frequency disturbances, and thus a smaller control bandwidth will suffice to obtain the wanted performance. +If we look at the Cumulative amplitude spectrum of the vertical error motion in Figure 30, we can observe that a soft hexapod (\(k < 10^5 - 10^6\,[N/m]\)) helps reducing the high frequency disturbances, and thus a smaller control bandwidth will suffice to obtain the wanted performance.
-+
Figure 30: Cumulative Amplitude Spectrum of the Sample vertical position error due to all considered perturbations for multiple nano-hexapod stiffnesses
-5.2 Optimal Stiffness to reduce the plant uncertainty
++5.2 Optimal Stiffness to reduce the plant uncertainty
-One of the most important design goal is to obtain a system that is robust to all changes in the system. @@ -1435,236 +1462,646 @@ All these uncertainties will limit the attainable bandwidth and hence the obtain
-In the next sections, the effect of each change on the obtained uncertainty is quantified and conclusions are made on the optimal stiffness for robustness properties. +In the next sections, the effect the considered changes on the plant dynamics is quantified and conclusions are made on the optimal stiffness for robustness properties.
-Note that only the dynamics from forces applied by the nano-hexapod to the measured sample’s position by the metrology are compared. -This is because it is the most important dynamics for robustness and performance properties. -However, the dynamics from forces to sensors located in the nano-hexapod legs, such as force and relative motion sensors, have also been considered. +In the following study, when we refer to plant dynamics, this means the dynamics from forces applied by the nano-hexapod to the measured sample’s position by the metrology. +We will only compare the plant dynamics as it is the most important dynamics for robustness and performance properties. +However, the dynamics from forces to sensors located in the nano-hexapod legs, such as force and relative motion sensors, have also been considered in a separate study.
-Effect of Payload
-++Effect of Payload
+The most obvious change in the system is the change of payload.
-In Figure +In Figure 31 the dynamics is shown for payloads having a first resonance mode at 100Hz and a mass equal to 1kg, 20kg and 50kg. +On the left side, the change of dynamics is computed for a very soft nano-hexapod, while on the right side, it is computed for a very stiff nano-hexapod. +
+ ++One can see that for the soft nano-hexapod: +
+-
+
+For the stiff-nano-hexapod, the change of payload mass has very little effect (the vertical scale for the amplitude is quite small).
- -+- -+
-
Figure 31: Dynamics from \(\mathcal{F}_z\) to \(\mathcal{X}_z\) for varying payload mass, both for a soft nano-hexapod and a stiff nano-hexapod
+Figure 31: Dynamics from \(\mathcal{F}_z\) to \(\mathcal{X}_z\) for varying payload mass, both for a soft nano-hexapod (left) and a stiff nano-hexapod (right)
+In Figure 32 is shown the effect of a change of payload dynamics. +The mass of the payload is fixed and its resonance frequency is changing from 50Hz to 500Hz. +
-+-+We can see (more easily for the soft nano-hexapod), that resonance of the payload produces an anti-resonance for the considered dynamics. +
+ + +-
Figure 32: Dynamics from \(\mathcal{F}_z\) to \(\mathcal{X}_z\) for varying payload resonance frequency, both for a soft nano-hexapod and a stiff nano-hexapod
-- - - -Effect of Micro-Station Compliance
-- ----
-
-Figure 33: Change of dynamics from force \(\mathcal{F}_x\) to displacement \(\mathcal{X}_x\) due to the micro-station compliance
--- - -Effect of Spindle Rotating Speed
-- ----
-
-Figure 34: Change of dynamics from force \(\mathcal{F}_x\) to displacement \(\mathcal{X}_x\) for a spindle rotation speed from 0rpm to 60rpm
--Total Uncertainty
-- ---- - -
-
-Figure 35: Variability of the dynamics from \(\bm{\mathcal{F}}_x\) to \(\bm{\mathcal{X}}_x\) with varying nano-hexapod stiffness
--- - --The leg stiffness should be at higher than \(k = 10^4\,[N/m]\) such that the main resonance frequency does not shift too much when rotating. -
- --- - - --It is usually a good idea to maximize the mass, damping and stiffness of the isolation platform in order to be less sensible to the payload dynamics. -The best thing to do is to have a stiff isolation platform. -
-The dynamics of the nano-hexapod is not affected by the micro-station dynamics (compliance) when the stiffness of the legs is less than \(10^6\,[N/m]\). When the nano-hexapod is stiff (\(k > 10^7\,[N/m]\)), the compliance of the micro-station appears in the primary plant. +The dynamics for all the payloads (mass from 1kg to 50kg and first resonance from 50Hz to 500Hz) and all the considered nano-hexapod stiffnesses are display in Figure 33.
--Determination of the optimal stiffness based on all the effects: +For nano-hexapod stiffnesses below \(10^6\,[N/m]\):
-
-
-The main performance limitation are payload variability -
---Main problem: heavy samples with small stiffness. -The first resonance frequency of the sample will limit the performance. +For nano-hexapod stiffnesses above \(10^7\,[N/m]\):
+-
+
+-
+
+Figure 33: Dynamics from \(\mathcal{F}_z\) to \(\mathcal{X}_z\) for varying payload dynamics, both for a soft nano-hexapod and a stiff nano-hexapod
+ +-- +For soft nano-hexapods, the payload has an important impact on the dynamics. +This will have to be carefully taken into account for the controller design.
--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. +For stiff nano-hexapod, the dynamics doe not change with the payload until the first resonance frequency of the nano-hexapod or of the payload.
+ ++If possible, the first resonance frequency of the payload should be maximized (stiff fixation). +
+ ++Heavy samples with low first resonance mode will be very problematic. +
+-+ +5.3 Conclusion
+++Effect of Micro-Station Compliance
+++The micro-station dynamics is quite complex as was shown in Section 2, moreover, its dynamics can change due to: +
+-
+
+Thus, it would be much more robust if the plant dynamics were not depending on the micro-station dynamics. +This as several other advantages: +
+-
+
+To identify the effect of the micro-station compliance on the system dynamics, for each nano-hexapod stiffness, we identify the plant dynamics in two different case (Figure 34): +
+-
+
+One can see that for nano-hexapod stiffnesses below \(10^6\,[N/m]\), the plant dynamics does not significantly changed due to the micro station dynamics (the solid and dashed curves are superimposed). +
+ ++For nano-hexapod stiffnesses above \(10^7\,[N/m]\), the micro-station compliance appears in the plant dynamics starting at about 45Hz. +
+ + +++ +
+
+Figure 34: Change of dynamics from force \(\mathcal{F}_x\) to displacement \(\mathcal{X}_x\) due to the micro-station compliance
+++If the resonance of the nano-hexapod is below the first resonance of the micro-station, then the micro-station dynamics if “filtered out” and does not appears in the dynamics to be controlled. +This renders the system robust to any possible change of the micro-station dynamics. +
+ ++If a stiff nano-hexapod is used, the control bandwidth should probably be limited to around the first micro-station’s mode (\(\approx 45\,[Hz]\)) which will likely no give acceptable performance. +
+ +++ +Effect of Spindle Rotating Speed
+++ ++Let’s now consider the rotation of the Spindle. +
--+6 Robust Control Architecture
++The plant dynamics for spindle rotation speed from 0rpm up to 60rpm are shown in Figure 35. +
+ ++One can see that for nano-hexapods with a stiffness above \(10^5\,[N/m]\), the dynamics is mostly not changing with the spindle’s rotating speed. +
+ ++For very soft nano-hexapods, the main resonance is split into two resonances and one anti-resonance that are all moving at a function of the rotating speed. +
+ + +++ +
+
+Figure 35: Change of dynamics from force \(\mathcal{F}_x\) to displacement \(\mathcal{X}_x\) for a spindle rotation speed from 0rpm to 60rpm
++++If the resonance of the nano-hexapod is (say a factor 5) above the maximum rotation speed, then the plant dynamics will be mostly not impacted by the rotation. +
+ ++A very soft (\(k < 10^4\,[N/m]\)) nano-hexapod should not be used due to the effect of the spindle’s rotation. +
+ +++Total Plant Uncertainty
++++Finally, let’s combined all the uncertainties and display the plant dynamics “spread” for all the nano-hexapod stiffnesses (Figure 36). +This show how the dynamics evolves with the stiffness and how different effects enters the plant dynamics. +
+ + +++ +
+
+Figure 36: Variability of the dynamics from \(\bm{\mathcal{F}}_x\) to \(\bm{\mathcal{X}}_x\) with varying nano-hexapod stiffness
++++Let’s summarize the findings: +
+-
+
+Concerning the plant dynamic uncertainty, the resonance frequency of the nano-hexapod should be between 5Hz (way above the maximum rotating speed) and 50Hz (before the first micro-station resonance) for all the considered payloads. +This corresponds to an optimal nano-hexapod leg stiffness in the range \(10^5 - 10^6\,[N/m]\). +
+ ++In such case, the main limitation will be heavy samples with small stiffnesses. +
+ +++5.3 Conclusion
++++++In Section 5.1, 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. +
+ ++In Section 5.2, 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 6 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 here. +
+ ++6 Robust Control Architecture
+ ++Before designing the control system, let’s summarize what has been done: +
+-
+
+The optimal nano-hexapod is now included in the model, and a robust control architecture that minimizes the vibrations of the sample is developed. +
+ + ++It is preferred to design one controller that gives acceptable performance for all the payloads that will be used. +This is quite challenging as: +
+-
+
+If it 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 manually choose the controller that was design for that particular mass/inertia. +
+ + ++For such system, the High-Authority-Control/Low-Authority-Control (HAC-LAC) architecture +
+ ++from the following reasons explained in preumont18_vibrat_contr_activ_struc_fourt_edition: +
++
+ + ++The HAC/LAC approach consist of combining the two approached in a dual-loop control as shown in Figure 37. +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: +
+-
+
++
+
+Figure 37: HAC-LAC Architecture with a system having only one input
+++ +6.1 Active Damping and Sensors to be included
++ + +++Active Damping can help with two things: +
+-
+
+There are different ways to actively damp a system depending on the sensor used : either force sensor, relative motion sensor or inertial sensor. +
+ ++A separate study (accessible here) for all three sensor type have been done, the conclusions are: +
+-
+
+Relative motion sensors are then included in each of the nano-hexapod’s leg and a decentralized direct velocity feedback control architecture as shown in figure 38 is applied. +
+ ++The signals shown in Figure 38 are: +
+-
+
+\(\bm{K}_{\text{DVF}}\) is a diagonal controller that consists of applying a force in each actuator proportional to the relative velocity of the associated leg. +This adds damping to the nano-hexapod’s modes. +
+ + +++ +
+
+Figure 38: Low Authority Control: Decentralized Direct Velocity Feedback
++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 explained below. +
+ + ++The plant dynamics before (solid curves) and after (dashed curves) the Law-Authority-Control implementation are compared in Figure 39. +It is clear that the use of the DVF reduces the dynamical spread of the plant dynamics between 5Hz up too 100Hz. +This will make the primary controller more robust and easier to develop. +
+ + +++ +
+
+Figure 39: Primary plant in the space of the legs with (dashed) and without (solid) Direct Velocity Feedback
++As shown in Figure 40, 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. +
+ + +++
+
+Figure 40: Norm of the transfer function from vertical disturbances to vertical position error with (dashed) and without (solid) Direct Velocity Feedback applied
+++ +6.2 Motion Control
++ + +++The complete HAC-LAC architecture is shown in Figure 41 where an outer loop is added to the decentralized direct velocity feedback loop. +
+ ++The block
+ + +Compute Position Erroris 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 briefly explained in Section 4.3. ++From the position error \(\bm{\epsilon}_{\mathcal{X}_n}\) expressed in the frame of the nano-hexapod, the nano-hexapod’s Jacobian \(\bm{J}\) (which is a real matrix) is used to compute the corresponding length error of each of the nano hexapod’s leg \(\bm{\epsilon}_\mathcal{L}\). +
+ ++Then, a diagonal controller \(\bm{K}_\mathcal{L}\) generates the required force in each leg to compensate the position error. +
+ + +++ +
+
+Figure 41: 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
++Many alternative to this control architecture has been studied, but this is the one that actually gives to best performance/robustness compromise. +
+ ++The plant dynamics for each of the six legs and for the three payload’s masses is shown in Figure 42. +The dynamical spread is kept reasonably small thanks to both the optimal nano-hexapod design and to Law-Authority-Controller. +
+ + +++ + +
+
+Figure 42: Diagonal elements of the transfer function matrix from \(\bm{\tau}^\prime\) to \(\bm{\epsilon}_{\mathcal{X}_n}\) for the three considered masses
++The diagonal controller \(\bm{K}_\mathcal{L}\) is then tuned in such a way that the control bandwidth is around 100Hz and such that enough stability margins are obtained for all the payload’s masses used. +
+ + +++ +
+
+Figure 43: Loop gain for the primary plant
++The sensibility to disturbance after the use of HAC-LAC control is shown in Figure 44. +When obtain a change of sensibility which is very typical to feedback system: +
+-
+
+The large increase at around 250Hz when using a mass of either 1kg or 10kg is probably caused by insufficient stability margins. +
+ + +++
+
+Figure 44: Sensibility to disturbances when the HAC-LAC control is applied (dashed) and when it is not (solid)
+++ +6.3 Simulation of Tomography Experiments
++ + +++The same simulation of a tomography experiment performed in Section 4.4 is now re-done with the used of the HAC-LAC architecture. +All the disturbances are included such as ground motion, spindle and translation stage vibrations. +
+ ++After the simulation is performed, the Power Spectral Density of the sample’s position error is plotted in Figure 45 and the Cumulative Amplitude Spectrum is shown in Figure 46. +The top three plots corresponds to the X, Y and Z translations and the bottom three plots corresponds to the X,Y and Z rotations. +
+ ++Several observations can be made: +
+-
+
++ + + +
+
+Figure 45: Amplitude Spectral Density of the position error in Open Loop and with the HAC-LAC controller
+++ + +
+
+Figure 46: Cumulative Amplitude Spectrum of the position error in Open Loop and with the HAC-LAC controller
++The time domain sample’s vibrations are shown in Figure 47. +The use of the nano-hexapod combined with the HAC-LAC architecture is shown to considerably reduce the sample’s vibrations. +
+ ++An animation of the experiment is shown in Figure 48 and we can see that the actual sample’s position is more closely following the ideal position as was the case with the simulation of the micro-station alone in Figure 25 (same scale was used for both simulations). +
+ + +++ + +
+
+Figure 47: Position Error of the sample during a tomography experiment when no control is applied and with the HAC-DVF control architecture
+++
+
+Figure 48: Tomography Experiment using the Simscape Model in Closed Loop with the HAC-LAC Control - Zoom on the sample’s position (the full vertical scale is \(\approx 10 \mu m\))
++-6.4 Conclusion
+ - ---6.1 Active Damping and Sensors to be included
---Ways to damp: -
--
-
-https://tdehaeze.github.io/rotating-frame/index.html -
- --Sensors to be included: -
-- -6.2 Motion Control
--- -6.3 Simulation of Tomography Experiments
-- - - ---- - -
-
-Figure 36: Amplitude Spectral Density of the position error in Open Loop and with the HAC-LAC controller
--- - -
-
-Figure 37: Cumulative Amplitude Spectrum of the position error in Open Loop and with the HAC-LAC controller
--- - -
-
-Figure 38: Position Error of the sample during a tomography experiment when no control is applied and with the HAC-DVF control architecture
---
-
-Figure 39: Tomography Experiment using the Simscape Model in Closed Loop with the HAC-LAC Control - Zoom on the sample’s position (the full vertical scale is \(\approx 10 \mu m\))
---6.4 Conclusion
--7 Further notes
+++7 Further notes
+ +++7.1 Using soft mounts for the
+ +++7.2 Others
+Common metrology frame for the nano-focusing optics and the measurement of the sample position?
@@ -1678,18 +2115,21 @@ Slip-Ring noise?Bibliography
-diff --git a/index.org b/index.org index d52a2be..5e6550a 100644 --- a/index.org +++ b/index.org @@ -712,6 +712,7 @@ To do so, we need to perform several computations (summarized in Figure [[fig:co More details about these computations are accessible [[https://tdehaeze.github.io/nass-simscape/positioning_error.html][here]]. ** Simulation of Experiments +<Date: 04-2020
+Date: 05-2020
-Created: 2020-04-28 mar. 14:04
+Created: 2020-04-29 mer. 11:57
+