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): @@ -188,11 +192,11 @@ 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) @@ -212,7 +216,7 @@ And the dynamical blocks are:
Figure 1: Block Diagram of a simple feedback system
@@ -234,11 +238,11 @@ 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: \[ \epsilon = \frac{1}{1 + GK} r + \frac{GK}{1 + GK} n - \frac{G_d}{1 + GK} d \]
@@ -283,8 +287,8 @@ 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: @@ -302,7 +306,7 @@ 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:
-
@@ -323,7 +327,7 @@ 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
@@ -331,11 +335,11 @@ 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
@@ -356,11 +360,11 @@ 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
@@ -390,11 +394,11 @@ 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 @@ -410,8 +414,8 @@ 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: @@ -438,8 +442,8 @@ 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: @@ -469,15 +473,15 @@ 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 \(y\)
@@ -492,11 +496,11 @@ 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).
@@ -505,7 +509,7 @@ 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 \(v\)
@@ -513,11 +517,11 @@ 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: \[ \epsilon = S r + T n - G_d S d \]
@@ -541,43 +545,54 @@ 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 -
- \(S_{rr}\): +
- \(S_{rr}\): 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
-https://tdehaeze.github.io/meas-analysis/ +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 the controller. +The estimated dynamics will also be used to tune the developed multi-body model of the micro-station with which the simulations will be performed.
-Modal Analysis: https://tdehaeze.github.io/meas-analysis/modal-analysis/index.html -
- --The obtained dynamics will allows us to compare the dynamics of the model. +All the measurements performed on the micro-station are detailed in this document and summarized in the following sections.
2.1 Setup
+2.1 Setup
-In order to perform to Modal Analysis and to obtain first a response model, the following devices were used: + +
+ ++To measure the dynamics of such complicated system, it as been chosen to perform a full modal analysis. +
+ ++To limit the number of degrees of freedom to be measured, we suppose that in the frequency range of interest (DC-300Hz), each of the positioning stage is behaving as a solid body. +Thus, to fully describe the dynamics of the station, we (only) need to measure 6 degrees of freedom on each of the positioning stage (that is 36 degrees of freedom for the 6 solid bodies). +
+ + ++In order to perform the Modal Analysis, the following devices were used:
- An acquisition system (OROS) with 24bits ADCs @@ -589,13 +604,13 @@ In order to perform to Modal Analysis and to obtain first a response mode 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 6)
- 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 6) +
- 4 on top of the translation stage (figure 7)
- 4 on top of the tilt stage
- 3 on top of the spindle
- 4 on top of the hexapod @@ -603,119 +618,161 @@ The position of the accelerometers are:
-In total, 69 degrees of freedom are measured (23 tri axis accelerometers). +In total, 69 degrees of freedom are measured (23 tri axis accelerometers) which is way more that what was required. +
+ ++We chose to have some redundancy in the measurement to be able to verify that the solid-body assumption was correct for each of the stage.
--- - - -
-
-Figure 6: Figure caption
-++ + +
-
Figure 7: Figure caption
+Figure 6: Example of one hammer impact
++
+
+Figure 7: 3 tri axis accelerometers fixed to the translation stage
-
-
2.2 Results
+2.2 Results
-From the measurements, we obtain +
--
-
- Reduction of the -
- solid body assumption -
- verification of the assumption => ok -
+From the measurements, we obtain all the transfer functions from forces applied at the location of the hammer impacts to the x-y-z acceleration of each solid body at the location of each accelerometer. +
+ ++Modal shapes and natural frequencies are then computed. Example of mode shapes are shown in Figures 8 9. +
-
Figure 8: Figure caption
+Figure 8: First mode
Figure 9: Figure caption
-Figure 9: Sixth mode
2.3 Conclusion
--The reduction of the number of degrees of freedom from 69 (23 accelerometers with each 3DOF) to 36 (6 solid bodies with 6 DOF) seems to work well. +We then reduce the number of degrees of freedom from 69 (23 accelerometers with each 3DOF) to 36 (6 solid bodies with 6 DOF).
-This confirms the fact that the stages are indeed behaving as a solid body in the frequency band of interest. This valid the fact that a multi-body model can be used to represent the dynamics of the micro-station. +From the reduced transfer function matrix, we can re-synthesize the response at the 69 measured degrees of freedom and we find that we have an exact match. +
+ ++This confirms the fact that the stages are indeed behaving as a solid body in the frequency band of interest. +This thus means that a multi-body model can be used to represent the dynamics of the micro-station.
3 Identification of the Disturbances
+3 Identification of the Disturbances
-https://tdehaeze.github.io/meas-analysis/ +In this section, we wish to list and identify all the disturbances affecting the system.
-Open Loop Noise budget: https://tdehaeze.github.io/nass-simscape/disturbances.html +Note that here we are not much interested by low frequency disturbances such as thermal effects and static guiding errors of each positioning stage. +This is because the frequency content of these errors will be located in the controller bandwidth and thus will be easily compensated by the nano-hexapod.
--Static Guiding errors: +The problem are on the high frequency disturbances. +In the following sections, we consider:
-
-
- measured at the PEL -
- low frequency errors, will thus be compensated +
- the ground motion +
- vibrations of each stage, due either to their control systems or their motion
-The problem are on the high frequency disturbances +https://tdehaeze.github.io/meas-analysis/ +Open Loop Noise budget: https://tdehaeze.github.io/nass-simscape/disturbances.html
3.1 Ground Motion
+3.1 Ground Motion
+The ground motion can easily be estimated using an inertial sensor with sufficient sensitivity. +
+ ++To verify that the inertial sensors are sensitive enough, a Huddle test has been performed (Figure 10). +
+ + +
+
Figure 10: Huddle Test Setup
++The measured Power Spectral Density of the ground motion at the ID31 floor is compared with other measurements performed at ID09 and at CERN. +The low frequency differences between the ground motion at ID31 and ID09 is just due to the fact that for the later measurement, the low frequency sensitivity of the inertial sensor was not taken into account. +
+ + +
+
Figure 11: Comparison of the PSD of the ground motion measured at different location
+3.2 Stage Vibration - Effect of Control systems
+3.2 Stage Vibration - Effect of Control systems
Control system of each stage has been tested -https://tdehaeze.github.io/meas-analysis/disturbance-control-system/index.html -https://tdehaeze.github.io/meas-analysis/2018-10-15%20-%20Marc/index.html +
+ ++Each motor are turn off and then on. +The goal is to see what noise is injected in the system due to the regulation loop of each stage. +
+ + ++Complete reports on these measurements are accessible here and here.
@@ -724,87 +781,232 @@ Control system of each stage has been tested
3.3 Stage Vibration - Effect of Motion
+3.3 Stage Vibration - Effect of Motion
-We consider: +We consider here the vibrations induced by the scans of the translation stage and rotation of the spindle. +
+ ++Details reports are accessible here for the translation stage and here for the spindle/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 12. +
+ + +
+
Figure 12: Measurement of the sample’s vertical motion when rotating at 6rpm
++A geophone is fixed at the location of the sample and we measure the motion:
-
-
- The rotation of the Spindle -
- The translation of the Translation Stage +
- without rotation +
- when rotating at 6rpm using the slip-ring motor +
- when rotating at 6rpm using the spindle motor synchronized with the slip-ring motor
+The obtained Power Spectral Density of the sample’s absolute velocity are shown in Figure 13. +
+ ++We can see that when using the Slip-ring motor to rotate the sample, only a little increase of the motion is observed above 100Hz. +
+ ++However, when rotating with the Spindle (normal functioning mode): +
+-
+
- a very sharp peak at 23Hz is observed. Its cause has not been identified yet +
- a general large increase in motion above 30Hz +
+
Figure 13: Comparison of the ASD of the measured voltage from the Geophone at the sample location
++Some investigation should be performed on the Spindle to determine where does this 23Hz motion comes from. +
+ +3.4 Sum of all disturbances
+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 14), and we measure the absolute velocity of both the sample and the granite. +
+ + +
+
Figure 14: Y position of the translation stage measured by the encoders
++The time domain absolute vertical velocity of the sample and granite are shown in Figure 15. +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 15: 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 16. +We can see many peaks starting from 1Hz showing the large spectral content probably due to the triangular reference of the translation stage. +
+ + +
+
Figure 16: 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
++A smoother motion for the translation stage (such as a sinus motion) could probably help reducing much of the vibrations produced. +
+ +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 17. +
+ ++We can see that the ground motion is quite small compare to the translation stage and spindle induced motions. +
+ + +
Figure 10: Amplitude Spectral Density fo the motion error due to disturbances
+Figure 17: Amplitude Spectral Density fo the motion error due to disturbances
+The Cumulative Amplitude Spectrum is shown in Figure 18. +It is shown that the motion induced by translation stage scans and spindle rotation are in the micro-meter range. +
-
Figure 11: Cumulative Amplitude Spectrum of the motion error due to disturbances
+Figure 18: Cumulative Amplitude Spectrum of the motion error due to disturbances
+We can also estimate the required bandwidth by seeing that \(10\ nm [rms]\) motion is induced by the perturbations above 100Hz. +
-Expected required bandwidth +This means that if the controller compensate all the motion errors below 100Hz (ideal case), 10nm [rms] of motion will still remain. +
+ ++From that, we can conclude that we will probably need a control bandwidth to around 100Hz.
3.5 Better measurement of the effect of disturbances
+3.5 Better estimation of the disturbances
-Here, the measurement were made with inertial sensors. -However, we are interested in the relative motion of the sample with respect to the granite and not the absolute motion. -
- - --The best measurement of the disturbances would be to have the metrology already functioning. -
- - --We could perform a measurement using the X-ray. +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.
-Detector Requirement: +This is not perfect as using only one geophone on the sample and one on the granite do not permit to separate the translations and the rotations. +
+ ++An alternative could be to position a reference object at the sample location and to use the X-ray to measure its motion. +
+ ++The detector requirement would be:
- Sample frequency above \(400Hz\) -
- Resolution of \(\approx 20nm\) +
- Resolution of \(\approx 100nm\) (to be discussed)
3.6 Conclusion
+3.6 Conclusion
++Main disturbance sources have been identified. +These disturbances will then be included in the multi-body model. +
+ + ++Other disturbance sources were not estimated such as cable forces and acoustic disturbances. +If heavy/stiff cables are to be fixed to the sample, this should be quantified and included in the model. +
+ + ++Having better estimation of the disturbances would allows to more precisely estimate the attainable performances. +This should however not change the conclusion of this study nor significantly change the nano-hexapod design. +
+4 Multi Body Model
+4 Multi Body Model
https://tdehaeze.github.io/nass-simscape/ @@ -815,8 +1017,8 @@ Multi-Body model
4.1 Validity of the model
+4.1 Validity of the model
The mass/inertia of each stage is automatically computed from the geometry and the density of the materials.
@@ -840,16 +1042,16 @@ Comparison model - measurements :
+ Figure 12: Figure caption Figure 19: Figure caption
4.2 Wanted position of the sample and position error
+4.2 Wanted position of the sample and position error
From the reference position of each stage, we can compute the wanted pose of the sample with respect to the granite. @@ -861,10 +1063,10 @@ Then, from the measurement of the metrology corresponding to the position of the
-
Figure 13: Figure caption
+Figure 20: Figure caption
@@ -873,8 +1075,8 @@ Measurement of the sample’s position - conversion of positioning error in
4.3 Simulation of Experiments
+4.3 Simulation of Experiments
Now that the @@ -893,20 +1095,20 @@ We can perform simulation of experiments.
-
Figure 14: Position error of the Sample with respect to the granite during a Tomography Experiment with included disturbances
+Figure 21: Position error of the Sample with respect to the granite during a Tomography Experiment with included disturbances
4.4 Conclusion
+4.4 Conclusion
@@ -923,11 +1125,11 @@ Simulation of experiments to validate performance.
5 Optimal Nano-Hexapod Design
+5 Optimal Nano-Hexapod Design
As explain before, the nano-hexapod properties (mass, stiffness, architecture, …) will influence: @@ -947,12 +1149,12 @@ We which here to choose the nano-hexapod properties such that: