diff --git a/index.html b/index.html index 9d2f00e..c291eea 100644 --- a/index.html +++ b/index.html @@ -4,7 +4,7 @@ "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> - + Design of the Nano-Hexapod and associated Control Architectures - Summary @@ -37,77 +37,80 @@

Table of Contents

+
+

This report is also available as a pdf.

+
-
-

Introduction

-
+
+

Introduction

+

In this document are gathered and summarized all the developments done for the design of the Nano Active Stabilization System. This consists of a nano-hexapod and an associated control architecture that are used to stabilize samples down to the nano-meter level in presence of disturbances. @@ -115,7 +118,7 @@ This consists of a nano-hexapod and an associated control architecture that are

-To understand the design challenges of such system, a short introduction to Feedback control is provided in Section 1. +To understand the design challenges of such system, a short introduction to Feedback control is provided in Section 1. The mathematical tools (Power Spectral Density, Noise Budgeting, …) that will be used throughout this study are also introduced.

@@ -124,55 +127,53 @@ The mathematical tools (Power Spectral Density, Noise Budgeting, …) that To develop both the nano-hexapod and the control architecture in an optimal way, precise estimation of the following is required:

    -
  • micro-station dynamics (Section 2)
    • -
  • frequency content of the sources of disturbances such as vibrations induced by the micro-station’s stages and ground motion (Section 3)
    • +
  • micro-station dynamics (Section 2)
    • +
  • frequency content of the sources of disturbances such as vibrations induced by the micro-station’s stages and ground motion (Section 3)

-A model of the micro-station is then developed and tuned using the previous estimations (Section 4). +A model of the micro-station is then developed and tuned using the previous estimations (Section 4). The nano-hexapod is further included in the model.

The effects of the nano-hexapod characteristics on the system dynamics are then studied. -Based on that, an optimal choice of the nano-hexapod stiffness is made (Section 5). +Based on that, an optimal choice of the nano-hexapod stiffness is made (Section 5).

Finally, using the optimally designed nano-hexapod, a robust control architecture is developed. -Simulations are performed to show that this design gives acceptable performance and the required robustness (Section 6). +Simulations are performed to show that this design gives acceptable performance and the required robustness (Section 6).

- -This report is also available as a pdf.
-
-

1 Introduction to Feedback Systems and Noise budgeting

+
+

1 Introduction to Feedback Systems and Noise budgeting

- +

-In this section, some basics of feedback systems are first introduced (Section 1.1). +In this section, some basics of feedback systems are first introduced (Section 1.1). This should highlight the challenges of the required 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 in a motion system required to use some sensors to monitor the actual status of the system and actuators to modifies this status. @@ -207,11 +208,11 @@ 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 to be controlled)
    • @@ -231,7 +232,7 @@ The dynamical blocks are:
-
+

classical_feedback_small.png

Figure 1: Block Diagram of a simple feedback system

@@ -253,11 +254,11 @@ In the next section, is explained how the use of the feedback lowers the effect
-
-

1.1.2 How does the feedback loop is modifying the system behavior?

+
+

1.1.2 How does the feedback loop is modifying the system behavior?

-From the feedback diagram in Figure 1, the position error signal \(\epsilon = r - y\) can be written as a function of the reference signal \(r\), the disturbances \(d\) and the measurement noise \(n\): +From the feedback diagram in Figure 1, the position error signal \(\epsilon = r - y\) can be written as a function of the reference signal \(r\), the disturbances \(d\) and the measurement noise \(n\): \[ \epsilon = \frac{1}{1 + GK} r + \frac{GK}{1 + GK} n - \frac{G_d}{1 + GK} d \]

@@ -298,8 +299,8 @@ Ideally, it is desired to design the controller \(K\) such that:
-
-

1.1.3 Trade off: Disturbance Reduction / Noise Injection

+
+

1.1.3 Trade off: Disturbance Reduction / Noise Injection

From the definition of \(S\) and \(T\): @@ -317,7 +318,7 @@ 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. It is shown that \(|S|\) and \(|T|\) exhibit different behaviors depending on the frequency band:

@@ -339,7 +340,7 @@ It is shown that \(|S|\) and \(|T|\) exhibit different behaviors depending on th -
+

h-infinity-2-blocs-constrains.png

Figure 2: Typical shapes and constrain of the Sensibility and Transmibility closed-loop transfer functions

@@ -347,11 +348,11 @@ It is shown that \(|S|\) and \(|T|\) exhibit different behaviors depending on th
-
-

1.1.4 Trade off: Robustness / Performance

+
+

1.1.4 Trade off: Robustness / Performance

- +

@@ -372,11 +373,11 @@ The main issue it that for stability reasons, the system dynamics must be kno

-For mechanical systems, this generally means that the control bandwidth should take place before any appearing of flexible dynamics (right part of Figure 3). +For mechanical systems, this generally means that the control bandwidth should take place before any appearing of flexible dynamics (right part of Figure 3).

-
+

oomen18_next_gen_loop_gain.jpg

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

@@ -406,11 +407,11 @@ This problem of robustness represent one of the main challenge for the de
-
-

1.2 Dynamic error budgeting

+
+

1.2 Dynamic error budgeting

- +

The dynamic error budgeting is a powerful tool to study the effects of multiple error sources (i.e. disturbances and measurement noise) and to predict how much these effects are reduced by a feedback system. @@ -421,19 +422,19 @@ The dynamic error budgeting uses two important mathematical functions: the Po

-After these two functions are introduced (in Sections 1.2.1 and 1.2.2), is shown how do multiple error sources are combined and modified by dynamical systems (in Section 1.2.3 and 1.2.4). +After these two functions are introduced (in Sections 1.2.1 and 1.2.2), is shown how do multiple error sources are combined and modified by dynamical systems (in Section 1.2.3 and 1.2.4).

-Finally, the dynamic noise budgeting for the NASS is derived in Section 1.2.5. +Finally, the dynamic noise budgeting for the NASS is derived in Section 1.2.5.

-
-

1.2.1 Power Spectral Density

+
+

1.2.1 Power Spectral Density

- +

@@ -462,11 +463,11 @@ 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

- +

@@ -496,11 +497,11 @@ It can also helps to determine at which frequencies the effect of disturbances m

-A typical Cumulative Power Spectrum is shown in figure 4. +A typical Cumulative Power Spectrum is shown in figure 4.

-
+

preumont18_cas_plot.jpg

Figure 4: Cumulative Power Spectrum in open-loop and closed-loop for increasing gains (taken from preumont18_vibrat_contr_activ_struc_fourt_edition)

@@ -508,19 +509,19 @@ A typical Cumulative Power Spectrum is shown in figure 4
-
-

1.2.3 Modification of a signal’s PSD when going through a dynamical system

+
+

1.2.3 Modification of a signal’s PSD when going through a dynamical 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 5). +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 5).

-
+

psd_lti_system.png

Figure 5: LTI dynamical system \(G(s)\) with input signal \(u\) and output signal \(y\)

@@ -535,15 +536,15 @@ 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 6). +Let’s consider a signal \(y\) that is the sum of two uncorrelated signals \(u\) and \(v\) (Figure 6).

@@ -552,7 +553,7 @@ The PSD of \(y\) is equal to sum of the PSD and \(u\) and the PSD of \(v\) (can

-
+

psd_sum.png

Figure 6: \(y\) as the sum of two signals \(u\) and \(v\)

@@ -560,15 +561,15 @@ The PSD of \(y\) is equal to sum of the PSD and \(u\) and the PSD of \(v\) (can
-
-

1.2.5 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: \[ \epsilon = S r + T n - G_d S d \]

@@ -592,25 +593,25 @@ To estimate the PSD of the position error \(\epsilon\) and thus the RMS residual
  • The Power Spectral Densities of the signals affecting the system:
      -
    • The disturbances \(S_{dd}\): this will be done in Section 3
      • +
    • The disturbances \(S_{dd}\): this will be done in Section 3
    • The sensor noise \(S_{nn}\): this can be estimated from the sensor data-sheet
    • The wanted sample’s motion \(S_{rr}\): this is a deterministic signal that is chosen by the “user”. For a simple tomography experiment, the wanted sample’s motion can consider to be equal to \(0\) (the point of interest should stay on the focus X-ray)
  • The dynamics of the complete system comprising the micro-station and the nano-hexapod: \(G\), \(G_d\). -To do so, the dynamics of the micro-station (Section 2) should be identified and then included in a model (Section 4). Then a model of the nano-hexapod is merged with the micro-station model (Section 5)
    • -
  • The controller \(K\) that will be designed in Section 6
    • +To do so, the dynamics of the micro-station (Section 2) should be identified and then included in a model (Section 4). Then a model of the nano-hexapod is merged with the micro-station 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 used: @@ -628,7 +629,7 @@ All the measurements performed on the micro-station are detailed in 7. +The general procedure to identify the dynamics of the micro-station is shown in Figure 7. The steps are:

    @@ -638,7 +639,7 @@ The steps are:
-
+

vibration_analysis_procedure.png

Figure 7: Vibration Analysis Procedure

@@ -650,11 +651,11 @@ Instead, the model will be tuned using both the modal model and the response mod

-
-

2.1 Experimental Setup

+
+

2.1 Experimental Setup

- +

@@ -680,13 +681,13 @@ In order to perform the modal analysis, the following devices were used: The measurement consists of:

    -
  • Exciting the structure at the same location with the instrumented hammer (Figure 8)
    • +
  • Exciting the structure at the same location with the instrumented hammer (Figure 8)
  • Fix the accelerometers on each of the stages 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 9)
      • +
    • 4 on top of the translation stage (Figure 9)
    • 4 on top of the tilt stage
    • 3 on top of the spindle
    • 4 on top of the hexapod
      • @@ -702,14 +703,14 @@ It was chosen to have some redundancy in the measurement to be able to verify th

      -
      +

      hammer_z.gif

      Figure 8: Example of one hammer impact

      -
      +

      accelerometers_ty_overview.jpg

      Figure 9: 3 tri axis accelerometers fixed to the translation stage

      @@ -717,11 +718,11 @@ It was chosen to have some redundancy in the measurement to be able to verify th
      -
      -

      2.2 Results

      +
      +

      2.2 Results

      - +

      @@ -730,18 +731,18 @@ From the measurements are extracted all the transfer functions from forces appli

      Modal shapes and natural frequencies are then computed. -Example of the obtained micro-station’s mode shapes are shown in Figures 10 and 11. +Example of the obtained micro-station’s mode shapes are shown in Figures 10 and 11.

      -
      +

      mode1.gif

      Figure 10: First mode that shows a suspension mode, probably due to bad leveling of one Airloc

      -
      +

      mode6.gif

      Figure 11: Sixth mode

      @@ -770,12 +771,12 @@ This thus means that a multi-body model can be used to correctly represent th

      Many Frequency Response Functions (FRF) are obtained from the measurements. -Examples of FRF are shown in Figure 12. +Examples of FRF are shown in Figure 12. These FRF will be used to compare the dynamics of the multi-body model with the micro-station dynamics.

      -
      +

      frf_all_bodies_one_direction.png

      Figure 12: Frequency Response Function from forces applied by the Hammer in the X direction to the acceleration of each solid body in the X direction

      @@ -783,8 +784,8 @@ These FRF will be used to compare the dynamics of the multi-body model with the
      -
      -

      2.3 Conclusion

      +
      +

      2.3 Conclusion

      @@ -792,7 +793,7 @@ The dynamical measurements made on the micro-station confirmed the fact that a m

      -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.

      @@ -800,11 +801,11 @@ In Section 4, the obtained Frequency Response Function
      -
      -

      3 Identification of the Disturbances

      +
      +

      3 Identification of the Disturbances

      - +

      In this section, all the disturbances affecting the system are identified and quantified. @@ -819,13 +820,13 @@ These are however very important for the evaluation of the required nano-hexapod The main challenge is to reduce the disturbances containing high frequencies, and thus efforts are made to identify these high frequency disturbances such as:

        -
      • Ground motion (Section 3.1)
        • -
      • Vibration introduced by control systems (Section 3.2)
        • -
      • Vibration introduced by the motion of the spindle and of the translation stage (Section 3.3)
        • +
      • Ground motion (Section 3.1)
        • +
      • Vibration introduced by control systems (Section 3.2)
        • +
      • Vibration introduced by the motion of the spindle and of the translation stage (Section 3.3)

      -A noise budgeting is performed in Section 3.4, the vibrations induced by the disturbances are compared and the required control bandwidth is estimated. +A noise budgeting is performed in Section 3.4, the vibrations induced by the disturbances are compared and the required control bandwidth is estimated.

      @@ -833,11 +834,11 @@ The measurements are presented in more detail in -

      3.1 Ground Motion

      +
      +

      3.1 Ground Motion

      - +

      @@ -845,12 +846,12 @@ Ground motion can easily be estimated using an inertial sensor with sufficient s

      -To verify that the inertial sensors are sensitive enough, a Huddle test has been performed (Figure 13). +To verify that the inertial sensors are sensitive enough, a Huddle test has been performed (Figure 13). The details of the Huddle Test can be found here.

      -
      +

      geophones.jpg

      Figure 13: Huddle Test Setup

      @@ -862,7 +863,7 @@ The low frequency differences between the ground motion at ID31 and ID09 is just

      -
      +

      ground_motion_compare.png

      Figure 14: Comparison of the PSD of the ground motion measured at different location

      @@ -870,11 +871,11 @@ 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

      - +

      @@ -897,11 +898,11 @@ Complete reports on these measurements are accessible -

      3.3 Stage Vibration - Effect of Motion

      +
      +

      3.3 Stage Vibration - Effect of Motion

      - +

      In this section, the vibrations induced by scans of the translation stage and rotation of the spindle and studied. @@ -912,15 +913,15 @@ 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 15. +The setup for the measurement of vibrations induced by rotation of the Spindle and Slip-ring is shown in Figure 15.

      -
      +

      rz_meas_errors.gif

      Figure 15: Measurement of the sample’s vertical motion when rotating at 6rpm

      @@ -936,7 +937,7 @@ A geophone is fixed at the location of the sample and the motion is measured:

    -The obtained Power Spectral Densities of the sample’s absolute velocity are shown in Figure 16. +The obtained Power Spectral Densities of the sample’s absolute velocity are shown in Figure 16.

    @@ -953,7 +954,7 @@ Its cause has not been identified yet

-
+

sr_sp_psd_sample_compare.png

Figure 16: Comparison of the ASD of the measured voltage from the Geophone at the sample location

@@ -968,19 +969,19 @@ Some investigation should be performed to determine where does this 23Hz motion
-
-

Translation Stage

-
+
+

Translation Stage

+

The same setup is used: a geophone is located at the sample’s location and another on the granite.

-A 1Hz triangle motion with an amplitude of \(\pm 2.5mm\) is sent to the translation stage (Figure 17), and the absolute velocities of the sample and the granite are measured. +A 1Hz triangle motion with an amplitude of \(\pm 2.5mm\) is sent to the translation stage (Figure 17), and the absolute velocities of the sample and the granite are measured.

-
+

ty_position_time.png

Figure 17: Y position of the translation stage measured by the encoders

@@ -988,20 +989,20 @@ A 1Hz triangle motion with an amplitude of \(\pm 2.5mm\) is sent to the translat

-The time domain absolute vertical velocity of the sample and granite are shown in Figure 18. +The time domain absolute vertical velocity of the sample and granite are shown in Figure 18. 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.

-
+

ty_z_time.png

Figure 18: 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 19. +The Amplitude Spectral Densities of the measured absolute velocities are shown in Figure 19. The ASD contains any peaks starting from 1Hz showing the large spectral content of the motion which is probably due to the triangular reference of the translation stage.

@@ -1019,7 +1020,7 @@ Thus, if the detector is only used in between the triangular peaks, the vibratio
-
+

asd_z_direction.png

Figure 19: 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

@@ -1028,11 +1029,11 @@ Thus, if the detector is only used in between the triangular peaks, the vibratio
-
-

3.4 Open Loop noise budgeting

+
+

3.4 Open Loop noise budgeting

- +

@@ -1040,7 +1041,7 @@ The effect of all the disturbance sources on the position error (relative motion

-The Power Spectral Density of the motion error due to the ground motion, translation stage scans and spindle rotation are shown in Figure 20. +The Power Spectral Density of the motion error due to the ground motion, translation stage scans and spindle rotation are shown in Figure 20.

@@ -1048,26 +1049,26 @@ It can be seen that the ground motion is quite small compare to the translation

-
+

dist_effect_relative_motion.png

Figure 20: Amplitude Spectral Density fo the motion error due to disturbances

-The Cumulative Amplitude Spectrum is shown in Figure 21. +The Cumulative Amplitude Spectrum is shown in Figure 21. It is shown that the motion induced by translation stage scans and spindle rotation are in the micro-meter range for frequencies above 1Hz.

-
+

dist_effect_relative_motion_cas.png

Figure 21: Cumulative Amplitude Spectrum of the motion error due to disturbances

-From Figure 21, required bandwidth can be estimated by seeing that \(10\ nm [rms]\) motion is induced by the perturbations above 100Hz. +From Figure 21, required bandwidth can be estimated by seeing that \(10\ nm [rms]\) motion is induced by the perturbations above 100Hz.

@@ -1080,8 +1081,8 @@ From that, it can be concluded that control bandwidth will have to be around 100

-
-

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. @@ -1101,8 +1102,8 @@ The detector requirement would need to have a sample frequency above \(400Hz\) a

-
-

3.6 Conclusion

+
+

3.6 Conclusion

@@ -1127,14 +1128,14 @@ 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) connected with some discrete flexibility (stiffnesses and dampers). +As was shown during the modal analysis (Section 2), the micro-station behaves as multiple rigid bodies (granite, translation stage, tilt stage, spindle, hexapod) connected with some discrete flexibility (stiffnesses and dampers).

@@ -1147,11 +1148,11 @@ A small summary of the multi-body Simscape is available -

4.1 Multi-Body model

+
+

4.1 Multi-Body model

- +

@@ -1175,11 +1176,11 @@ Then, the values of the stiffnesses and damping properties of each joint is manu

-The 3D representation of the simscape model is shown in Figure 22. +The 3D representation of the simscape model is shown in Figure 22.

-
+

simscape_picture.jpg

Figure 22: 3D representation of the simscape model

@@ -1187,11 +1188,11 @@ 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

- +

@@ -1199,7 +1200,7 @@ Tuning the dynamics of such model is very difficult as there are more than 50 pa

-The comparison of three of the Frequency Response Functions are shown in Figure 23. +The comparison of three of the Frequency Response Functions are shown in Figure 23.

@@ -1211,7 +1212,7 @@ We believe that the model is representing the micro-station dynamics sufficient

-
+

identification_comp_top_stages.png

Figure 23: 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.

@@ -1244,11 +1245,11 @@ Then, using the model, it is possible to:
-
-

4.3 Wanted position of the sample and position error

+
+

4.3 Wanted position of the sample and position error

- +

@@ -1256,7 +1257,7 @@ For the control of the nano-hexapod, the sample position error (the motion to be

-To do so, several computations are performed (summarized in Figure 24): +To do so, several computations are performed (summarized in Figure 24):

  • First, the wanted pose (3 translations and 3 rotations) of the sample with respect to the granite is computed. @@ -1270,7 +1271,7 @@ Both computation are performed
-
+

control-schematic-nass.png

Figure 24: Schematic of how the elements are interacting with the Speedgoat

@@ -1282,11 +1283,11 @@ More details about these computations are accessible -

4.4 Simulation of a Tomography Experiment

+
+

4.4 Simulation of a Tomography Experiment

- +

@@ -1296,16 +1297,16 @@ Now that the dynamics of the model is tuned and the disturbances included in the

A first simulation is done with the nano-hexapod modeled as a rigid-body. This does represent the system without the NASS and permits to estimate the sample’s vibrations using the micro-station alone. -The results of this simulation will be compared to simulations using the NASS in Section 6.4. +The results of this simulation will be compared to simulations using the NASS in Section 6.4.

-An 3D animation of the simulation is shown in Figure 25. +An 3D animation of the simulation is shown in Figure 25.

-A zoom in the micro-meter ranger on the sample’s location is shown in Figure 26 with two frames: +A zoom in the micro-meter ranger on the sample’s location is shown in Figure 26 with two frames:

  • a non-rotating frame corresponding to the focusing point of the X-ray. @@ -1319,7 +1320,7 @@ The motion of the sample follows the wanted motion but with vibrations in the mi

    -
    +

    open_loop_sim.gif

    Figure 25: Tomography Experiment using the Simscape Model

    @@ -1327,14 +1328,14 @@ The motion of the sample follows the wanted motion but with vibrations in the mi -
    +

    open_loop_sim_zoom.gif

    Figure 26: 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 27. +The position error of the sample with respect to the granite are shown in Figure 27. It is confirmed that the X-Y-Z position errors are in the micro-meter range.

    @@ -1352,7 +1353,7 @@ The vertical rotation error is meaningless for two reasons:
-
+

exp_scans_rz_dist.png

Figure 27: Position error of the Sample with respect to the granite during a Tomography Experiment with included disturbances

@@ -1360,8 +1361,8 @@ The vertical rotation error is meaningless for two reasons:
-
-

4.5 Conclusion

+
+

4.5 Conclusion

@@ -1386,11 +1387,11 @@ In the next sections, it will allows to optimally design the nano-hexapod, to de

-
-

5 Optimal Nano-Hexapod Design

+
+

5 Optimal Nano-Hexapod Design

- +

As explain before, the nano-hexapod properties (mass, stiffness, legs’ orientation, …) will influence: @@ -1404,9 +1405,9 @@ As explain before, the nano-hexapod properties (mass, stiffness, legs’ ori The objective is here to find the optimal nano-hexapod properties such that:

    -
  • the effect of disturbances is minimized (Section 5.2)
    • -
  • the plant uncertainty due to a change of payload mass and experimental conditions is minimized (Section 5.3)
    • -
  • the plant has nice dynamical properties for control (Section 5.4)
    • +
  • the effect of disturbances is minimized (Section 5.2)
    • +
  • the plant uncertainty due to a change of payload mass and experimental conditions is minimized (Section 5.3)
    • +
  • the plant has nice dynamical properties for control (Section 5.4)

@@ -1418,11 +1419,11 @@ Also, the nano-hexapod’s damping is not studied here as it is supposed to

-
-

5.1 A brief introduction to Stewart Platforms

+
+

5.1 A brief introduction to Stewart Platforms

- +

@@ -1435,30 +1436,30 @@ A typical Stewart platform is composed of two platforms connected by six identic

-This is very schematically shown in Figure 28 where the \(a_i\) are the location of the joints connected to the fixed platform and the \(b_i\) are the joints connected to the mobile platform. +This is very schematically shown in Figure 28 where the \(a_i\) are the location of the joints connected to the fixed platform and the \(b_i\) are the joints connected to the mobile platform.

-
+

stewart_architecture_example.png

Figure 28: Schematic representation of a Stewart platform

-As shows in Figure 28, two frames \(\{A\}\) and \(\{B\}\) are virtually fixed to respectively the bottom and the top platforms. +As shows in Figure 28, two frames \(\{A\}\) and \(\{B\}\) are virtually fixed to respectively the bottom and the top platforms. These frames are used to describe the relative motion of the two platforms through the position vector \({}^A\bm{P}_B\) of \(\{B\}\) expressed in \(\{A\}\) and the rotation matrix \({}^A\bm{R}_B\) expressing the orientation of \(\{B\}\) with respect to \(\{A\}\). For the nano-hexapod, these frames are chosen to be located at the theoretical center of the spherical metrology reflector.

Since the Stewart platform has six-degrees-of-freedom and six actuators, it is called a fully parallel manipulator. -A change in the length of the legs \(\bm{\mathcal{L}} = \left[ l_1, l_2, l_3, l_4, l_5, l_6 \right]^T\) will induce a motion of the mobile platform with respect to the fixed platform as shown in Figure 29. -The relation between a change in length of the legs and the relative motion of the platforms is studied thanks to the kinematic analysis, which is explained in Section 5.4. +A change in the length of the legs \(\bm{\mathcal{L}} = \left[ l_1, l_2, l_3, l_4, l_5, l_6 \right]^T\) will induce a motion of the mobile platform with respect to the fixed platform as shown in Figure 29. +The relation between a change in length of the legs and the relative motion of the platforms is studied thanks to the kinematic analysis, which is explained in Section 5.4.

-
+

stewart_architecture_example_pose.png

Figure 29: Display of the Stewart platform architecture at some defined pose

@@ -1484,11 +1485,11 @@ The source code is accessible -

5.2 Optimal Stiffness to reduce the effect of disturbances

+
+

5.2 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 \(G_d\)). @@ -1500,11 +1501,11 @@ A study of the optimal nano-hexapod stiffness for the minimization of disturbanc

-
-

Sensibility to stage vibrations

-
+
+

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 30. +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 30. It is shown that a softer nano-hexapod is better to filter out vertical vibrations of the spindle. More precisely, the nano-hexapod filters out the vibration starting at the first suspension mode of the payload on top of the nano-hexapod.

@@ -1514,7 +1515,7 @@ The same conclusion is made for vibrations of the translation stage.

-
+

opt_stiff_sensitivity_Frz.png

Figure 30: Sensitivity to Spindle vertical motion error to the vertical error position of the sample

@@ -1522,21 +1523,21 @@ The same conclusion is made for vibrations of the translation stage.
-
-

Sensibility to ground motion

-
+
+

Sensibility to ground motion

+

-The sensibility to ground motion in the Y and Z directions is shown in Figure 31. +The sensibility to ground motion in the Y and Z directions is shown in Figure 31. 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 (\(k>10^5\,[N/m]\)) is better for reducing the effect of ground motion at low frequency.

-It will be suggested in Section 7.4 that using soft mounts for the granite can greatly lower the sensibility to ground motion. +It will be suggested in Section 7.4 that using soft mounts for the granite can greatly lower the sensibility to ground motion.

-
+

opt_stiff_sensitivity_Dw.png

Figure 31: Sensitivity to Ground motion to the position error of the sample

@@ -1544,9 +1545,9 @@ It will be suggested in Section 7.4 that using soft mo
-
-

Dynamic Noise Budgeting considering all the disturbances

-
+
+

Dynamic Noise Budgeting considering all the disturbances

+

Looking at the change of sensibility with the nano-hexapod’s stiffness helps understand the physics of the system. It however, does not permit to estimate the optimal stiffness that will lower the motion error due to disturbances. @@ -1563,7 +1564,7 @@ This is the dynamic noise budgeting.

-From the Power Spectral Density of all the sources of disturbances identified in Section 3 is computed the Power Spectral Density of the vertical motion error for all the considered nano-hexapod stiffnesses (Figure 32). +From the Power Spectral Density of all the sources of disturbances identified in Section 3 is computed the Power Spectral Density of the vertical motion error for all the considered nano-hexapod stiffnesses (Figure 32).

@@ -1571,7 +1572,7 @@ It can be seen that the most important change is in the frequency range 30Hz to

-
+

opt_stiff_psd_dz_tot.png

Figure 32: Amplitude Spectral Density of the Sample vertical position error due to Vertical vibration of the Spindle for multiple nano-hexapod stiffnesses

@@ -1579,18 +1580,18 @@ It can be seen that the most important change is in the frequency range 30Hz to
-
-

Conclusion

-
+
+

Conclusion

+

-It can be observe on the Cumulative amplitude spectrum of the vertical error motion in Figure 33, that a soft hexapod (\(k < 10^5 - 10^6\,[N/m]\)) helps reducing the high frequency disturbances, and thus a smaller control bandwidth will be required to obtain the wanted performance. +It can be observe on the Cumulative amplitude spectrum of the vertical error motion in Figure 33, that a soft hexapod (\(k < 10^5 - 10^6\,[N/m]\)) helps reducing the high frequency disturbances, and thus a smaller control bandwidth will be required to obtain the wanted performance.

-
+

opt_stiff_cas_dz_tot.png

Figure 33: Cumulative Amplitude Spectrum of the Sample vertical position error due to all considered perturbations for multiple nano-hexapod stiffnesses

@@ -1599,11 +1600,11 @@ It can be observe on the Cumulative amplitude spectrum of the vertical error mot
-
-

5.3 Optimal Stiffness to reduce the plant uncertainty

+
+

5.3 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. @@ -1634,15 +1635,15 @@ However, the dynamics from forces to sensors located in the nano-hexapod legs, s

-
-

Effect of Payload

-
+
+

Effect of Payload

+

The most obvious change in the system is the change of payload.

-In Figure 34 the dynamics is shown for payloads with a mass equal to 1kg, 20kg and 50kg (the resonance of the payload is fixed to 100Hz). +In Figure 34 the dynamics is shown for payloads with a mass equal to 1kg, 20kg and 50kg (the resonance of the payload is fixed to 100Hz). 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.

@@ -1666,14 +1667,14 @@ As the maximum payload’s mass is \(50\,kg\), this may however not be pract

-
+

opt_stiffness_payload_mass_fz_dz.png

Figure 34: 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 35 is shown the effect of a change of payload dynamics. +In Figure 35 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.

@@ -1682,14 +1683,14 @@ It can be seen (more easily for the soft nano-hexapod), that resonance of the pa

-
+

opt_stiffness_payload_freq_fz_dz.png

Figure 35: 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

-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 36. +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 36.

@@ -1710,7 +1711,7 @@ For nano-hexapod stiffnesses above \(10^7\,[N/m]\): -

+

opt_stiffness_payload_impedance_all_fz_dz.png

Figure 36: Dynamics from \(\mathcal{F}_z\) to \(\mathcal{X}_z\) for varying payload dynamics, both for a soft nano-hexapod and a stiff nano-hexapod

@@ -1738,11 +1739,11 @@ Heavy samples with low first resonance mode will be the most problematic.
-
-

Effect of Micro-Station Compliance

-
+
+

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: +The micro-station dynamics is quite complex as was shown in Section 2, moreover, its dynamics can change due to:

  • a change in some mechanical elements
    • @@ -1763,7 +1764,7 @@ This as several other advantages:

    -To identify the effect of the micro-station compliance on the system dynamics, the plant dynamics is identified in two different cases (Figure 37): +To identify the effect of the micro-station compliance on the system dynamics, the plant dynamics is identified in two different cases (Figure 37):

-
-

Kinematic Analysis

-
+
+

Kinematic Analysis

+

The Kinematic analysis of the Stewart platform can be divided into two problems: the inverse kinematics and the forward kinematics.

@@ -1970,9 +1971,9 @@ However, as will be shown in the next section, approximate solution of the forwa
-
-

Jacobian Analysis

-
+
+

Jacobian Analysis

+

The Jacobian matrix \(\bm{J}\) can be computed form the orientation of the legs (describes by the unit vectors \({}^A\hat{\bm{s}}_i\)) and the position of the top joints (described by the position vectors \({}^A\bm{b}_i\)) both expressed in the frame \(\{A\}\):

@@ -2042,24 +2043,24 @@ And thus the Jacobian matrix can be used to compute the forces that should be

-Linear transformations in Eq. \eqref{eq:jacobian_L} and \eqref{eq:jacobian_F} will be widely in the developed control architectures in Section 6. +Linear transformations in Eq. \eqref{eq:jacobian_L} and \eqref{eq:jacobian_F} will be widely in the developed control architectures in Section 6.

-
-

Mobility of the Stewart Platform

-
+
+

Mobility of the Stewart Platform

+

For a specified geometry and actuator stroke, the mobility of the Stewart platform can be estimated thanks to the approximate forward kinematic analysis.

-An example of the mobility considering only pure translations is shown in Figure 40. +An example of the mobility considering only pure translations is shown in Figure 40.

-
+

mobility_translations_null_rotation.png

Figure 40: Obtained mobility of a Stewart platform for pure translations (the platform’s orientation is fixed)

@@ -2093,9 +2094,9 @@ This gives an idea of the relation between the mobility and the actuator stroke.
-
-

Stiffness and Compliance matrices

-
+
+

Stiffness and Compliance matrices

+

In order to determine the stiffness and compliance matrices of the Stewart platform, let’s model the actuators by a spring with a stiffness \(k_i\) in parallel with a force source \(\tau_i\).

@@ -2145,15 +2146,15 @@ Stiffness properties of the Stewart platform can then be estimated from the arch
-
-

Effect of a change of geometry

-
+
+

Effect of a change of geometry

+

Equations \eqref{eq:jacobian_L}, \eqref{eq:jacobian_F} and \eqref{eq:jacobian_K} can be used to see how the maneuverability, the force authority and the stiffness of the Stewart platform are changing with a the geometry (position of the joints and orientation of the legs).

-The effects of two changes in the manipulator’s geometry are summarized in Table 1. +The effects of two changes in the manipulator’s geometry are summarized in Table 1. These results could have been easily deduced with some basics of mechanics, but they can be easily quantified thanks to the Kinematic and Jacobian analysis.

@@ -2161,7 +2162,7 @@ These results could have been easily deduced with some basics of mechanics, but The nano-hexapod geometry and further be optimized in terms of stiffness and stroke using the presented tools.

- +
@@ -2257,19 +2258,19 @@ The nano-hexapod geometry and further be optimized in terms of stiffness and str -
-

Cubic Architecture

-
+
+

Cubic Architecture

+

A very popular choice of Stewart platform architecture, especially for vibration isolation, is the Cubic architecture.

-The cubic architecture is quite specific in the sense that the active struts are arranged in a mutually orthogonal configuration connecting the corners of a cube (Figure 41). +The cubic architecture is quite specific in the sense that the active struts are arranged in a mutually orthogonal configuration connecting the corners of a cube (Figure 41).

-
+

3d-cubic-stewart-aligned.png

Figure 41: Schematic representation of the Cubic architecture

@@ -2298,9 +2299,9 @@ Separate study of the cubic architecture is performed -

Effect of Flexible Joints

-
+
+

Effect of Flexible Joints

+

Each of the nano-hexapod legs has a universal joint at one end and a spherical joint at the other end.

@@ -2310,18 +2311,18 @@ When only small stroke is required, flexible joints can be used: material

-Example of flexible joints used for Stewart platforms are shown in Figures 42 and 43. +Example of flexible joints used for Stewart platforms are shown in Figures 42 and 43.

-
+

preumont07_flexible_joints.jpg

Figure 42: Flexible joints used in preumont07_six_axis_singl_stage_activ

-
+

yang19_flexible_joints.png

Figure 43: An alternative type of flexible joints that has been used for Stewart platforms yang19_dynam_model_decoup_contr_flexib

@@ -2342,7 +2343,7 @@ This has been studied using the Simscape model (report available 6 (it is however, if Integral Force Feedback is to be used, explained here). +This is not found to be problematic for the control architecture that will be developed in Section 6 (it is however, if Integral Force Feedback is to be used, explained here).

@@ -2385,9 +2386,9 @@ Simulations will help determine the required rotational stroke of the flexible j
-
-

Conclusion

-
+
+

Conclusion

+

Relations between the geometry of the Stewart platform and its characteristics such as stiffness, maneuverability and force authority have been derived. @@ -2406,24 +2407,24 @@ The effects of flexible joints stiffness on the dynamics have been studied and r

-
-

5.5 Conclusion

+
+

5.5 Conclusion

-In Section 5.2, 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. +In Section 5.2, 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 5.3, it has been 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. +In Section 5.3, it has been 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 by the control.

-Thus, a leg’s stiffness of \(10^5\,[N/m]\) will be used in Section 6 to develop the robust control architecture and to perform simulations. +Thus, a leg’s stiffness of \(10^5\,[N/m]\) will be used in Section 6 to develop the robust control architecture and to perform simulations.

@@ -2433,7 +2434,7 @@ A more detailed study of the determination of the optimal stiffness based on all

-Finally, in section 5.4 some insights on the wanted nano-hexapod geometry are given. +Finally, in section 5.4 some insights on the wanted nano-hexapod geometry are given.

@@ -2441,11 +2442,11 @@ Finally, in section 5.4 some insights on the wanted na
-
-

6 Robust Control Architecture

+
+

6 Robust Control Architecture

- +

Before designing the control system, let’s summarize what have been done: @@ -2477,19 +2478,19 @@ This would however require to measure the mass/inertia of each used payload and This part is divided in the following sections:

    -
  • Section 6.1: the High Authority Control / Low Authority Control Architecture is described and the reasons of its use are explained
    • -
  • Section 6.2: the active damping strategy is implemented and its effects on the system are described
    • -
  • Section 6.3: the high authority control is developed and the control robustness is studied
    • -
  • Section 6.4: tomography experiments are simulated and the performances are estimated
    • -
  • Section 6.5: more complex simulations are performed to further validate this control architecture
    • +
  • Section 6.1: the High Authority Control / Low Authority Control Architecture is described and the reasons of its use are explained
    • +
  • Section 6.2: the active damping strategy is implemented and its effects on the system are described
    • +
  • Section 6.3: the high authority control is developed and the control robustness is studied
    • +
  • Section 6.4: tomography experiments are simulated and the performances are estimated
    • +
  • Section 6.5: more complex simulations are performed to further validate this control architecture
-
-

6.1 High Authority Control / Low Authority Control Architecture

+
+

6.1 High Authority Control / Low Authority Control Architecture

- +

@@ -2502,7 +2503,7 @@ Some interesting properties of the HAC-LAC architecture are summarized below (ta

-The HAC/LAC approach consist of combining the two approached in a dual-loop control as shown in Figure 44. +The HAC/LAC approach consist of combining the two approached in a dual-loop control as shown in Figure 44. 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:

@@ -2514,7 +2515,7 @@ This approach has the following advantages:
-
+

control_architecture_hac_lac_one_input.png

Figure 44: HAC-LAC Architecture with a system having only one input

@@ -2524,17 +2525,17 @@ This approach has the following advantages: The HAC-LAC architecture thus consists of two cascade controllers:

    -
  • a Low Authority Controller that is used to damp the system (Section 6.2)
    • -
  • a High Authority Controller used to suppress the sample’s vibration in a wide frequency range (Section 6.3)
    • +
  • a Low Authority Controller that is used to damp the system (Section 6.2)
    • +
  • a High Authority Controller used to suppress the sample’s vibration in a wide frequency range (Section 6.3)
-
-

6.2 Active Damping and Sensors to be included in the nano-hexapod

+
+

6.2 Active Damping and Sensors to be included in the nano-hexapod

- +

Three active damping techniques could be applied for the Low Authority Control: @@ -2555,7 +2556,7 @@ To determine the most suited active damping technique, they are compared based o

-The conclusions are (summarized in Table 2): +The conclusions are (summarized in Table 2):

  • Integral Force Feedback is to be avoided as it renders the system unstable when the nano-hexapod’s is rotating (effect explained in the next section)
    • @@ -2565,7 +2566,7 @@ It also does not give the wanted robustness properties It however may increases the sensibility to stages vibrations at higher frequency
-
Table 1: Effect of a change in geometry on the manipulator’s stiffness, force authority and stroke
+
@@ -2634,9 +2635,9 @@ Therefore, relative motion sensors must be integrated in the six nano-hex -
-

Effect of the Spindle’s Rotation - Guaranteed Stability

-
+
+

Effect of the Spindle’s Rotation - Guaranteed Stability

+

To see why Integral Force Feedback should not be applied to damp the nano-hexapod’s modes, a simple model of a rotating positioning platform integration force sensors has been developed (described in details here).

@@ -2646,11 +2647,11 @@ The platform main resonance frequency is \(\omega_0\) and the rotation speed is

-Root Locus plots for Integral Force Feedback and Direct Velocity Feedback are shown in Table 3. +Root Locus plots for Integral Force Feedback and Direct Velocity Feedback are shown in Table 3. These plots show the evolution of the system’s poles in the complex plane as a function of the control gain.

-
Table 2: Comparison of the three main active damping techniques that could be applied to the nano-hexapod
+
@@ -2673,7 +2674,7 @@ These plots show the evolution of the system’s poles in the complex plane

-To understand what the root locus means, consider Figure 45 where two resonant systems are compared: +To understand what the root locus means, consider Figure 45 where two resonant systems are compared:

  • The first one (represented in blue) is undamped. @@ -2692,7 +2693,7 @@ A pole with a positive real part corresponds to an unstable system, and thus the

    -
    +

    preumont18_effect_damping.jpg

    Figure 45: Role of damping (preumont18_vibrat_contr_activ_struc_fourt_edition). (a) Pole position in the complex plane. (b) Change of dynamic amplification (\(1/2\xi\))

    @@ -2700,7 +2701,7 @@ A pole with a positive real part corresponds to an unstable system, and thus the

    -Coming back to the Root Locus in Table 3, it can be seen that: +Coming back to the Root Locus in Table 3, it can be seen that:

    • For Direct Velocity Feedback: @@ -2721,15 +2722,15 @@ Similar observations are made using the Simscape model of the NASS, and this sho
    -
    -

    Relative Direct Velocity Feedback Architecture

    -
    +
    +

    Relative Direct Velocity Feedback Architecture

    +

    -Relative motion sensors are included in each of the nano-hexapod’s leg and a decentralized direct velocity feedback control architecture is applied (Figure 46). +Relative motion sensors are included in each of the nano-hexapod’s leg and a decentralized direct velocity feedback control architecture is applied (Figure 46).

    -The signals shown in Figure 46 are: +The signals shown in Figure 46 are:

    • \(\bm{\tau}\): Actuator forces applied in each leg
      • @@ -2745,7 +2746,7 @@ The force applied in each leg being proportional to the relative velocity of the

      -
      +

      control_architecture_dvf.png

      Figure 46: Low Authority Control: Decentralized Direct Velocity Feedback

      @@ -2753,29 +2754,29 @@ The force applied in each leg being proportional to the relative velocity of the
      -
      -

      Dynamics and Root Locus

      -
      +
      +

      Dynamics and Root Locus

      +

      -The dynamics from \(\tau_i\) to \(d\mathcal{L}_i\) for three payload masses is shown in Figure 47. +The dynamics from \(\tau_i\) to \(d\mathcal{L}_i\) for three payload masses is shown in Figure 47. It is shown that for all the payload masses, the dynamics shows an alternation of poles and zeros which makes the direct velocity feedback loop robust.

      -This is confirmed by the Root Locus in Figure 48 where all the poles are staying in the left half plane. +This is confirmed by the Root Locus in Figure 48 where all the poles are staying in the left half plane. Moreover, it is seen that arbitrary damping can be applied to the nano-hexapod’s modes.

      -
      +

      opt_stiff_dvf_plant.png

      Figure 47: Dynamics from actuator force \(\tau_i\) to the relative displacement of the corresponding leg \(d\mathcal{L}_i\) for three payload masses

      -
      +

      opt_stiff_dvf_root_locus.png

      Figure 48: Root Locus (zoomed on the nano-hexapod modes) corresponding to the Direct Velocity Feedback control for three payload masses

      @@ -2791,11 +2792,11 @@ This may not be the optimal choice as will be further explained.
      -
      -

      Effect of Active Damping on the Sensibility to Disturbances

      -
      +
      +

      Effect of Active Damping on the Sensibility to Disturbances

      +

      -One objective of the active damping technique is to lower the sensibility to disturbances which are shown in Figure 49 without active damping (solid) and with the use of DVF (dashed). +One objective of the active damping technique is to lower the sensibility to disturbances which are shown in Figure 49 without active damping (solid) and with the use of DVF (dashed).

      @@ -2808,7 +2809,7 @@ Further optimization of the gain should then be performed.

      -
      +

      opt_stiff_sensibility_dist_dvf.png

      Figure 49: Norm of the transfer function from vertical disturbances to vertical position error with (dashed) and without (solid) Direct Velocity Feedback applied. Disturbances are: ground motion (top left), direct forces (top right), translation stage vibration (bottom left) and spindle vibrations (bottom right)

      @@ -2816,21 +2817,21 @@ Further optimization of the gain should then be performed.
      -
      -

      Effect of Active Damping on the Primary Plant Dynamics

      -
      +
      +

      Effect of Active Damping on the Primary Plant Dynamics

      +

      Another control objective for the LAC is to render the plant dynamics simpler to control for the High Authority Controller.

      -The plant dynamics before (solid curves) and after (dashed curves) the Low Authority Control implementation are compared in Figure 50. +The plant dynamics before (solid curves) and after (dashed curves) the Low Authority Control implementation are compared in Figure 50. It is clear that the use of the DVF reduces the dynamical spread of the plant dynamics between 5Hz and 100Hz. This will make the primary controller more robust and easier to develop.

      -
      +

      opt_stiff_primary_plant_damped_L.png

      Figure 50: Primary plant in the space of the legs with (dashed) and without (solid) Direct Velocity Feedback

      @@ -2838,9 +2839,9 @@ This will make the primary controller more robust and easier to develop.
      -
      -

      Conclusion

      -
      +
      +

      Conclusion

      +

      It has been shown that Direct Velocity Feedback using relative motion sensors is the most adapted active damping technique to be applied to the nano-hexapod. @@ -2859,11 +2860,11 @@ Thus, further improvements and optimization will be applied to this control arch

      -
      -

      6.3 High Authority Control

      +
      +

      6.3 High Authority Control

      - +

      The High Authority Controller objective is to stabilize the position of the sample with respect to the granite. @@ -2878,11 +2879,11 @@ Its proper design will most likely determine the performance of the system.

      -
      -

      Control in the Task space or in the Leg Space?

      -
      +
      +

      Control in the Task space or in the Leg Space?

      +

      -Let’s consider the two HAC-LAC control architectures shown in Figures 51 and 52 where an outer control loop is added to the already damped plant. +Let’s consider the two HAC-LAC control architectures shown in Figures 51 and 52 where an outer control loop is added to the already damped plant.

      @@ -2894,7 +2895,7 @@ The control objective for the High Authority Controller \(\bm{K}\) is to

      To do so, the block Compute Pos. Error is used to compute the position error \(\bm{\epsilon}_{\mathcal{X}_n}\) of the sample with respect to the nano-hexapod’s base platform 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. +The computation done in such block was briefly explained in Section 4.3.

      @@ -2903,13 +2904,13 @@ The two proposed control architectures are very similar in the sense that their The difference between the two architectures relies in the way the controllers are designed:

        -
      • For the architecture shown in Figure 51: +
      • For the architecture shown in Figure 51:
        • The controller \(\bm{K}_\mathcal{X}\) is designed in the task space: from the position/orientation error \(\bm{\epsilon}_{\mathcal{X}_n}\), it generates a force/torque \(\bm{\mathcal{F}}\) to be applied to sample
        • The forces/torques are then further converted to actuators forces \(\bm{\tau}^\prime\) with the use of the Jacobian matrix \(\bm{J}^{-T}\)
        • The full controller is \(\bm{K} = \bm{J}^{-T} \bm{K}_\mathcal{X}\)
        • -
      • For the architecture shown in Figure 52: +
      • For the architecture shown in Figure 52:
        • The sample’s position error \(\bm{\epsilon}_{\mathcal{X}_n}\) is first converted to the corresponding length errors of the six nano-hexapod’s legs \(\bm{\epsilon}_\mathcal{L}\) with the approximate inverse kinematics using the Jacobian matrix \(\bm{J}\)
        • The controller \(\bm{K}_\mathcal{L}\) then computes the actuator forces \(\bm{\tau}^\prime\) such that each of the legs have the wanted displacement
          • @@ -2918,7 +2919,7 @@ The difference between the two architectures relies in the way the controllers a
        -
        +

        control_architecture_hac_dvf_pos_X.png

        Figure 51: HAC-LAC architecture. The inner loop consist of a decentralized Direct Velocity Feedback. The outer loop consist of position control in the task space

        @@ -2926,7 +2927,7 @@ The difference between the two architectures relies in the way the controllers a -
        +

        control_architecture_hac_dvf_pos_L.png

        Figure 52: HAC-LAC architecture. The inner loop consist of a decentralized Direct Velocity Feedback. The outer loop consist of position control in the leg’s space

        @@ -2936,12 +2937,12 @@ The difference between the two architectures relies in the way the controllers a The choice of whether the controller should be designed in the leg space or in the task space does however makes some differences, that can be seen by looking at the dynamics to be controlled:

          -
        • Typical dynamics from \(\bm{\mathcal{F}}\) to \(\bm{\epsilon}_{\mathcal{X}_n}\) is shown in Figure 53: +
        • Typical dynamics from \(\bm{\mathcal{F}}\) to \(\bm{\epsilon}_{\mathcal{X}_n}\) is shown in Figure 53:
          • The suspension modes of the Stewart platform are separated, and the direct (diagonal) dynamical terms are different
          • The coupling is very small except for the dynamics from \(\mathcal{F}_{x,y}\) to \(R_{y,x}\) and from \(\mathcal{M}_{x,y}\) to \(D_{y,x}\) which is due to an non-diagonal stiffness and mass matrices
          • -
        • Typical dynamics from \(\bm{\tau}\) to \(\bm{\epsilon}_\mathcal{L}\) is shown in Figure 54: +
        • Typical dynamics from \(\bm{\tau}\) to \(\bm{\epsilon}_\mathcal{L}\) is shown in Figure 54:
          • The dynamics from \(\tau_i\) to \(d\mathcal{L}_i\) are all identical and contains all the Stewart platform modes, and thus only one controller has to be designed
          • The coupling is small at low frequency, quite high near the suspension modes of the Stewart platform and then small again at high frequency
            • @@ -2949,7 +2950,7 @@ The choice of whether the controller should be designed in the leg space or in t
          -
          +

          plant_centralized_X.png

          Figure 53: Direct (diagonal) dynamical terms (left) and coupled terms (right, shown in black) for the plant in the task space

          @@ -2957,7 +2958,7 @@ The choice of whether the controller should be designed in the leg space or in t -
          +

          plant_centralized_L.png

          Figure 54: Direct (diagonal) dynamical terms (left) and coupled terms (right, shown in black) for the plant in the leg space

          @@ -2965,10 +2966,10 @@ The choice of whether the controller should be designed in the leg space or in t

          -The differences of a control in the leg space and in the task space are summarized in Table 4. +The differences of a control in the leg space and in the task space are summarized in Table 4.

          -
Table 3: Variation of the Root Locus for DVF and IFF in presence of rotation. \(\omega\) is the spindle rotation speed, and \(\omega_0\) is the resonance frequency of the considered rotating system.
+
@@ -3028,16 +3029,16 @@ An alternative that could increase the control performance and robustness would -
-

Plant Dynamics in the leg space

-
+
+

Plant Dynamics in the leg space

+

-The plant dynamics from \(\tau_i\) to \(\epsilon_{\mathcal{L}_i}\) for each of the six legs and for the three payload’s masses is shown in Figure 55. +The plant dynamics from \(\tau_i\) to \(\epsilon_{\mathcal{L}_i}\) for each of the six legs and for the three payload’s masses is shown in Figure 55. The dynamical spread is kept reasonably small thanks to both the optimal nano-hexapod design and the Low Authority Controller.

-
+

opt_stiff_primary_plant_L.png

Figure 55: Diagonal elements of the transfer function matrix from \(\bm{\tau}^\prime\) to \(\bm{\epsilon}_{\mathcal{X}_n}\) for the three considered masses

@@ -3046,16 +3047,16 @@ The dynamical spread is kept reasonably small thanks to both the optimal nano-he
-
-

Controller Design

-
+
+

Controller Design

+

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. -The obtained loop gain is shown in Figure 56. +The obtained loop gain is shown in Figure 56.

-
+

opt_stiff_primary_loop_gain_L.png

Figure 56: Loop gain for the primary plant

@@ -3063,11 +3064,11 @@ The obtained loop gain is shown in Figure 56.
-
-

Noise Budgeting

-
+
+

Noise Budgeting

+

-The sensibility to disturbance after the use of HAC-LAC control is shown in Figure 57. +The sensibility to disturbance after the use of HAC-LAC control is shown in Figure 57. The change of sensibility is very typical for feedback system:

    @@ -3080,7 +3081,7 @@ The large increase at around 250Hz when using a mass of either 1kg or 10kg is pr

    -
    +

    opt_stiff_primary_control_L_senbility_dist.png

    Figure 57: Sensibility to disturbances when the HAC-LAC control is applied (dashed) and when it is not (solid)

    @@ -3089,30 +3090,30 @@ The large increase at around 250Hz when using a mass of either 1kg or 10kg is pr
    -
    -

    6.4 Simulation of Tomography Experiments

    +
    +

    6.4 Simulation of Tomography Experiments

    - +

    -
    -

    Simulation Setup

    -
    +
    +

    Simulation Setup

    +

    A simulation of a tomography is performed with the optimal nano-hexapod and the HAC-LAC architecture implemented. -The results of this simulation are compared to the simulation performed in Section 4.4 without the nano-hexapod. +The results of this simulation are compared to the simulation performed in Section 4.4 without the nano-hexapod. All the disturbances are included such as ground motion, spindle and translation stage vibrations.

    -
    -

    Frequency Analysis

    -
    +
    +

    Frequency Analysis

    +

    -The Power Spectral Density of the sample’s position error is plotted in Figure 58 and the Cumulative Amplitude Spectrum is shown in Figure 59. +The Power Spectral Density of the sample’s position error is plotted in Figure 58 and the Cumulative Amplitude Spectrum is shown in Figure 59. 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.

    @@ -3124,7 +3125,7 @@ Several observations can be made:
  • The sample’s vibrations are reduced within the control bandwidth as was expected
  • The obtained performances for all the three considered masses are very similar. This is an indication of the good system’s robustness
    • -
  • From the Cumulative Amplitude Spectrum (Figure 59), it can be seen that Z motion is reduced down to \(\approx 30\,nm\,[rms]\) and the Y motion down to \(\approx 25\,nm\,[rms]\)
    • +
  • From the Cumulative Amplitude Spectrum (Figure 59), it can be seen that Z motion is reduced down to \(\approx 30\,nm\,[rms]\) and the Y motion down to \(\approx 25\,nm\,[rms]\)

  • An increase in the rotational vibrations is observed. This is due to the fact that: @@ -3141,14 +3142,14 @@ This increase in rotation is still very small and is not foreseen to be a proble

-
+

opt_stiff_hac_dvf_L_psd_disp_error.png

Figure 58: Amplitude Spectral Density of the position error in Open Loop (black) and with the HAC-LAC controller for three payload masses

-
+

opt_stiff_hac_dvf_L_cas_disp_error.png

Figure 59: Cumulative Amplitude Spectrum of the position error in Open Loop (black) and with the HAC-LAC controller for three payload masses

@@ -3156,27 +3157,27 @@ This increase in rotation is still very small and is not foreseen to be a proble
-
-

Time Domain Analysis

-
+
+

Time Domain Analysis

+

-The time domain sample’s vibrations are shown in Figure 60. +The time domain sample’s vibrations are shown in Figure 60. 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 61 and it can be seen that the actual sample’s position is more closely following the ideal position compared to the simulation of the micro-station alone in Figure 26 (same scale was used for both animations). +An animation of the experiment is shown in Figure 61 and it can be seen that the actual sample’s position is more closely following the ideal position compared to the simulation of the micro-station alone in Figure 26 (same scale was used for both animations).

-
+

opt_stiff_hac_dvf_L_pos_error.png

Figure 60: Position Error of the sample during a tomography experiment when no control is applied and with the HAC-DVF control architecture

-
+

closed_loop_sim_zoom.gif

Figure 61: 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 \(\approx 10 \mu m\))

@@ -3185,11 +3186,11 @@ An animation of the experiment is shown in Figure 61 a
-
-

6.5 Simulation of More Complex Experiments

+
+

6.5 Simulation of More Complex Experiments

- +

Two additional simulations of experiments are performed: @@ -3210,9 +3211,9 @@ For both simulations, the following values are saved during the simulation:

-
-

Position offset introduced by the Micro-Hexapod

-
+
+

Position offset introduced by the Micro-Hexapod

+

Let’s consider that the micro-hexapod introduces a 10mm offset on the sample’s position such that the X-ray is focus on an interesting part of the sample.

@@ -3226,24 +3227,24 @@ The control objective is to keep the point of interest on the focused X-ray.

-An animation showing the simulation is shown in Figure 62. +An animation showing the simulation is shown in Figure 62.

-
+

tomography_dh_offset.gif

Figure 62: Top View of a tomography experiment with a 10mm offset imposed by the micro-hexapod

-One can see that the forces applied by the actuator are fluctuating around a constant value (Figure 63). +One can see that the forces applied by the actuator are fluctuating around a constant value (Figure 63). This is because the controller generates the actuator forces such that they counteracts the disturbances affecting the sample’s position. The disturbance causing this constant force is the centrifugal force induced by the spindle’s rotation which is a constant force in the frame of the nano-hexapod (provided the rotation speed is constant), directed radially outwards the rotation spindle’s axis, and is equal to \(F = m r \omega^2 \approx 12 \cdot 0.01 \cdot (2\pi)^2 \approx 5\,[N]\).

-
+

opt_stiff_hac_dvf_Dh_offset_F.png

Figure 63: Forces applied by the six nano-hexapod’s actuators

@@ -3251,23 +3252,23 @@ The disturbance causing this constant force is the centrifugal force induced by

-The relative motions of the nano-hexapod’s legs is shown in Figure 64 and are in the micro-meter range. +The relative motions of the nano-hexapod’s legs is shown in Figure 64 and are in the micro-meter range.

-
+

opt_stiff_hac_dvf_Dh_offset_dL.png

Figure 64: Relative displacement of the nano-hexapod’s legs

-Finally, the position/orientation error of the sample is shown in Figure 65. +Finally, the position/orientation error of the sample is shown in Figure 65. The root mean square value of the x-y-z error motions is around \(30\,nm\) which is very similar than for the “simple” tomography experiment.

-
+

opt_stiff_hac_dvf_Dh_offset_disp_error.png

Figure 65: Position/orientation error of the sample during the simulation

@@ -3275,9 +3276,9 @@ The root mean square value of the x-y-z error motions is around \(30\,nm\) which
-
-

Simultaneous Translation Scans and Spindle’s rotation

-
+
+

Simultaneous Translation Scans and Spindle’s rotation

+

In this simulation:

@@ -3288,46 +3289,46 @@ In this simulation:

-The obtained sample’s motion during the simulation is shown in Figure 66. +The obtained sample’s motion during the simulation is shown in Figure 66.

-
+

ty_scans.gif

Figure 66: Top View of a tomography experiment combined with translation scans

-The forces applied by the nano-hexapod’s are shown in Figure 67. +The forces applied by the nano-hexapod’s are shown in Figure 67. Peak values of the forces are appearing when the translation stage changes the direction of the scan.

-
+

opt_stiff_hac_dvf_Dy_scans_F.png

Figure 67: Forces applied by the six nano-hexapod’s actuators

-The relative motions of the nano-hexapod’s legs is shown in Figure 68 and are again in the micro-meter range. +The relative motions of the nano-hexapod’s legs is shown in Figure 68 and are again in the micro-meter range.

-
+

opt_stiff_hac_dvf_Dy_scans_dL.png

Figure 68: Relative displacement of the nano-hexapod’s legs

-The time domain position/orientation error of the sample is shown in Figure 69. +The time domain position/orientation error of the sample is shown in Figure 69. The RMS value of the x-y-z position error is again \(\approx 30\,nm\).

-
+

opt_stiff_hac_dvf_Dy_scans_disp_error.png

Figure 69: Position/orientation error of the sample during the simulation

@@ -3335,9 +3336,9 @@ The RMS value of the x-y-z position error is again \(\approx 30\,nm\).
-
-

Conclusion

-
+
+

Conclusion

+

These two simulations of more complex experiments shows the robustness of the developed system. @@ -3356,8 +3357,8 @@ The required actuator stroke is shown to be around \(\pm 5\,\mu m\) to compensat

-
-

6.6 Conclusion

+
+

6.6 Conclusion

@@ -3394,30 +3395,30 @@ The simulation is considered to be fairly realistic as the model used has been s

-
-

7 General Conclusion and Further notes

+
+

7 General Conclusion and Further notes

- +

-
-

7.1 Nano-Hexapod Specifications

+
+

7.1 Nano-Hexapod Specifications

- +

In this section are gathered all the specifications related to the nano-hexapod.

-
-

Dimensions

-
+
+

Dimensions

+

-The wanted dimension of the nano-hexapod are shown in Figure 70: +The wanted dimension of the nano-hexapod are shown in Figure 70:

  • Diameter of the bottom platform: 300mm
    • @@ -3430,7 +3431,7 @@ The limiting height might be quite problematic for the integration of the flexib

    -
    +

    nano_hexapod_size.jpg

    Figure 70: First implementation of the nano-hexapod / metrology reflector and sample interface

    @@ -3438,9 +3439,9 @@ The limiting height might be quite problematic for the integration of the flexib
    -
    -

    Flexible Joints

    -
    +
    +

    Flexible Joints

    +

    Flexible joints are located at each end of the six struts. These flexible joints should have the following properties: @@ -3459,9 +3460,9 @@ Typical angular stroke for such flexible joints is expected.

    -
    -

    Strut Stiffness

    -
    +
    +

    Strut Stiffness

    +

    The axial stiffness of the struts (between two flexible joints) should be equal to \(\approx 10^5 - 10^6\,[N/m]\).

    @@ -3472,9 +3473,9 @@ If voice coils are used, this corresponds to the axial stiffness of the membrane
    -
    -

    Actuator Force

    -
    +
    +

    Actuator Force

    +

    Based on simulations:

    @@ -3489,9 +3490,9 @@ If static deflection is to be compensated by the actuator, \(\approx 100\,[N]\)
    -
    -

    Actuator Stroke

    -
    +
    +

    Actuator Stroke

    +

    Based on simulations, the required actuator stroke seems to be \(\pm 5\,[\mu m]\).

    @@ -3525,11 +3526,11 @@ A piezo stack with a stroke of \(\pm 50\,[\mu m]\) will have a length size of \(
    -
    -

    Sensors

    -
    +
    +

    Sensors

    +

    -A relative displacement sensor must be included in each of the nano-hexapod’s legs as explained in Section 6. +A relative displacement sensor must be included in each of the nano-hexapod’s legs as explained in Section 6.

    @@ -3552,10 +3553,10 @@ This adds few constrains:

    Several sensor technology could be used for the nano-hexapod. -Characteristics of those sensors are shown in Table 5. +Characteristics of those sensors are shown in Table 5.

    -
Table 4: Comparison of a control in the leg space and in the task space
+
@@ -3623,15 +3624,15 @@ Characteristics of those sensors are shown in Table 5. -
-

Architecture

-
+
+

Architecture

+
-
-

7.2 Problem of Static Deflection?

+
+

7.2 Problem of Static Deflection?

Let’s now consider the problem of static deflection when changing the payload. @@ -3667,8 +3668,8 @@ This will change a little bit the architecture of the nano-hexapod but this shou

-
-

7.3 Sensor Noise introduced by the Metrology

+
+

7.3 Sensor Noise introduced by the Metrology

During all this study, the measurement of the relative position of the sample with respect to the granite was considered to be perfect, that is to say noiseless and with infinite bandwidth. @@ -3686,11 +3687,11 @@ It is then quite simple to predict what will be the effect of the sensor noise o

-
-

7.4 Using soft mounts for the Granite

+
+

7.4 Using soft mounts for the Granite

- +

@@ -3702,12 +3703,12 @@ These soft mounts can topically consists of air cylinders with pistons guided by

-The suspension mode of the granite would then be in the order of few Hertz, and the ground motion would be filtered out above that frequency as shown in Figure 71. +The suspension mode of the granite would then be in the order of few Hertz, and the ground motion would be filtered out above that frequency as shown in Figure 71.

-
+

opt_stiff_soft_granite_Dw.png

Figure 71: Change of sensibility to Ground motion when using stiff Granite mounts (solid curves) and soft Granite mounts (dashed curves)

@@ -3715,8 +3716,8 @@ The suspension mode of the granite would then be in the order of few Hertz, and
-
-

7.5 Micro Station Architecture

+
+

7.5 Micro Station Architecture

https://tdehaeze.github.io/nass-simscape/alternative-micro-station-architecture.html @@ -3724,8 +3725,8 @@ The suspension mode of the granite would then be in the order of few Hertz, and

-
-

7.6 Others Factors that may limit the performances

+
+

7.6 Others Factors that may limit the performances

Cable forces? @@ -3749,12 +3750,12 @@ Common metrology frame for the nano-focusing optics and the measurement of the s

-
-

7.7 Other Notes

+
+

7.7 Other Notes

-
-

7.7.1 Modification of the Granite

+
+

7.7.1 Modification of the Granite

Possible to lower the granite? @@ -3770,8 +3771,8 @@ The problem is that the Tilt stage rotation axis will not be position correctly

-
-

7.7.2 Sample trajectories

+
+

7.7.2 Sample trajectories

More precise definition of the sample trajectories? @@ -3782,8 +3783,8 @@ More precise definition of the sample trajectories?

-
-

7.7.3 Control Improvement

+
+

7.7.3 Control Improvement

Feedforward if the motion error is found to be correlated with the motion of the stages. @@ -3792,8 +3793,8 @@ Feedforward if the motion error is found to be correlated with the motion of the

-
-

7.8 General Conclusion

+
+

7.8 General Conclusion

@@ -3803,7 +3804,7 @@ bibliography:ref.bib

Author: Thomas Dehaeze

-

Created: 2020-05-13 mer. 16:16

+

Created: 2020-05-13 mer. 16:20

diff --git a/index.org b/index.org index e0eedae..17d4cf8 100644 --- a/index.org +++ b/index.org @@ -45,6 +45,12 @@ #+PROPERTY: header-args:latex+ :post pdf2svg(file=*this*, ext="png") :END: +#+begin_export html +
+

This report is also available as a pdf.

+
+#+end_export + * Introduction :PROPERTIES: :UNNUMBERED: t @@ -74,10 +80,6 @@ Based on that, an optimal choice of the nano-hexapod stiffness is made (Section Finally, using the optimally designed nano-hexapod, a robust control architecture is developed. Simulations are performed to show that this design gives acceptable performance and the required robustness (Section [[sec:robust_control_architecture]]). -#+begin_export html - This report is also available as a pdf. -#+end_export - * Introduction to Feedback Systems and Noise budgeting <>
Table 5: Characteristics of relative measurement sensors collette11_review