From 422ac2073331889cfc5d68ab68433440829c8deb Mon Sep 17 00:00:00 2001 From: Thomas Dehaeze Date: Wed, 29 Apr 2020 17:12:35 +0200 Subject: [PATCH] Re-read the introduction and feedback section --- index.html | 758 ++++++++++++++++++++++++++++------------------------- index.org | 114 ++++---- 2 files changed, 467 insertions(+), 405 deletions(-) diff --git a/index.html b/index.html index 88ba525..d7d92c8 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 @@ -35,93 +35,94 @@

Table of Contents

@@ -129,12 +130,13 @@

-The overall objective is to design a nano-hexapod an the associated control architecture that allows the stabilization of samples down to \(\approx 10nm\) in presence of disturbances and system variability. +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.

-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.

@@ -143,67 +145,73 @@ The mathematical tools (Power Spectral Density, Noise Budgeting, …) that To be able to develop both the nano-hexapod and the control architecture in an optimal way, we need a good estimation of:

-We then develop a model of the system that must represent all the important physical effects in play. -Such model is presented in 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.

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

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).

-
-

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). -This should highlight the challenges in terms of combined performance and robustness. +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. +

+

The use of feedback control as several advantages and pitfalls that are listed below (taken from schmidt14_desig_high_perfor_mechat_revis_edition):

-
    -
  • Advantages: +

    +Advantages: +

    • Reduction of the effect of disturbances: -Disturbances affecting the sample vibrations are observed by the sensor signal, and therefore the feedback controller can compensate for them
      • +Disturbances inducing vibrations are observed by the sensor signal, and therefore the feedback controller can compensate for them
    • Handling of uncertainties: Feedback controlled systems can also be designed for robustness, which means that the stability and performance requirements are guaranteed even for parameter variation of the controller mechatronics system
      • -
    • -
  • Pitfalls: +
+ +

+Pitfalls: +

  • Limited reaction speed: A feedback controller reacts on the difference between the reference signal (wanted motion) and the measurement (actual motion), which means that the error has to occur first before the controller can correct for it. @@ -213,26 +221,25 @@ By closing the loop, the sensor noise is also fed back and will induce positioni
  • Can introduce instability: Feedback control can destabilize a stable plant. Thus the robustness properties of the feedback system must be carefully guaranteed
    • -
-
-

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)
    • -
  • \(d\): the disturbances affecting \(y\) (ground motion, vibration of stages)
    • +
  • \(y\): the relative position of the sample with respect to the granite (the quantity to be controlled)
    • +
  • \(d\): the disturbances affecting \(y\) (ground motion, stages’ vibrations)
  • \(n\): the noise of the sensor measuring \(y\)
  • \(r\): the reference signal, corresponding to the wanted \(y\)
  • \(\epsilon = r - y\): the position error

-And the dynamical blocks are: +The dynamical blocks are:

  • \(G\): representing the dynamics from forces/torques applied by the nano-hexapod to the relative position sample/granite \(y\)
    • @@ -241,20 +248,20 @@ And the dynamical blocks are:
-
+

classical_feedback_small.png

Figure 1: Block Diagram of a simple feedback system

-Without the use of feedback (i.e. nano-hexapod), the disturbances will induce a sample motion error equal to: +Without the use of feedback (i.e. without the nano-hexapod), the disturbances will induce a sample motion error equal to:

\begin{equation} y = G_d d \label{eq:open_loop_error} \end{equation}

-which is out of the specifications (micro-meter range compare to the required \(\approx 10nm\)). +which is, in the case of the NASS out of the specifications (micro-meter range compare to the required \(\approx 10nm\)).

@@ -263,11 +270,11 @@ In the next section, we see how the use of the feedback system permits to lower

-
-

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?

-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: \[ \epsilon = \frac{1}{1 + GK} r + \frac{GK}{1 + GK} n - \frac{G_d}{1 + GK} d \]

@@ -302,18 +309,14 @@ From Eq. \eqref{eq:closed_loop_error} representing the closed-loop system behavi Ideally, we would like to design the controller \(K\) such that:

    -
  • \(|S|\) is small to limit the effect of disturbances
    • -
  • \(|T|\) is small to limit the injection of sensor noise
    • +
  • \(|S|\) is small to reduce the effect of disturbances
    • +
  • \(|T|\) is small to limit the injection of sensor noise
- -

-As shown in the next section, there is a trade-off between the disturbance reduction and the noise injection. -

-
-

1.1.3 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: @@ -331,28 +334,33 @@ 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:

-
    -
  • At low frequency (inside the control bandwidth): + +

    +At low frequency (inside the control bandwidth): +

    • \(|S|\) can be made small and thus the effect of disturbances is reduced
    • \(|T| \approx 1\) and all the sensor noise is transmitted
      • -
    • -
  • At high frequency (outside the control bandwidth): +
+

+At high frequency (outside the control bandwidth): +

  • \(|S| \approx 1\) and the feedback system does not reduce the effect of disturbances
  • \(|T|\) is small and thus the sensor noise is filtered
    • -
-
  • Near the crossover frequency (between the two frequency bands): + +

    +Near the crossover frequency (between the two frequency bands): +

    • The effect of disturbances is increased
      • -
    • -
    +

    h-infinity-2-blocs-constrains.png

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

    @@ -360,36 +368,36 @@ We can observe that \(|S|\) and \(|T|\) exhibit different behaviors depending on
    -
    -

    1.1.4 Trade off: Robustness / Performance

    +
    +

    1.1.4 Trade off: Robustness / Performance

    - +

    -As shown in the previous section, the effect of disturbances is reduced inside the control bandwidth. +As shown in the previous section, the effect of disturbances is reduced inside the control bandwidth.

    -Moreover, the slope of \(|S(j\omega)|\) is limited for stability reasons (not explained here), and therefore a large control bandwidth is required to obtain sufficient disturbance rejection at lower frequencies (where the disturbances have large effects). +Moreover, the slope of \(|S(j\omega)|\) is limited for stability reasons (not explained here), and therefore a large control bandwidth is required to obtain sufficient disturbance rejection at lower frequencies (where the disturbances have usually large effects).

    -The next important question is what effects do limit the attainable control bandwidth? +The next important question is therefore what limits the attainable control bandwidth?

    -The main issue it that for stability reasons, the behavior of the mechanical system must be known with only small uncertainty in the vicinity of the crossover frequency. +The main issue it that for stability reasons, the system dynamics must be known with only small uncertainty in the vicinity of the crossover frequency.

    -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 the control bandwidth should take place before any appearing of flexible dynamics (right part of Figure 3).

    -
    +

    oomen18_next_gen_loop_gain.png

    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

    @@ -409,7 +417,7 @@ For the NASS, the possible changes in the system are:

    -The nano-hexapod and the control architecture have to be developed such that the feedback system remains stable and exhibit acceptable performance for all these possible changes in the system. +The nano-hexapod and the control architecture have to be developed in such a way that the feedback system remains stable and exhibit acceptable performance for all these possible changes in the system.

    @@ -419,29 +427,36 @@ 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 effect of multiple error sources and to see how the feedback system does reduce the effect +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.

    -To understand how to use and understand it, the Power Spectral Density and the Cumulative Power Spectrum are first introduced. -Then, is shown how does multiple error sources are combined and modified by dynamical systems. +The dynamic error budgeting uses two important mathematical functions: the Power Spectral Density and the Cumulative Power Spectrum.

    -Finally, +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.

    -
    -

    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: \[ S_{xx}(\omega) = \int_{-\infty}^{\infty} R_{xx}(\tau) e^{-j \omega \tau} d\tau \ \frac{[\text{unit of } x]^2}{\text{Hz}} \] @@ -467,9 +482,13 @@ 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:

    @@ -498,15 +517,19 @@ 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 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 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).

    -
    +

    psd_lti_system.png

    Figure 4: LTI dynamical system \(G(s)\) with input signal \(u\) and output signal \(y\)

    @@ -521,11 +544,15 @@ 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 5). + +

    + +

    +Let’s consider a signal \(y\) that is the sum of two uncorrelated signals \(u\) and \(v\) (Figure 5).

    @@ -534,7 +561,7 @@ We have that the PSD of \(y\) is equal to sum of the PSD and \(u\) and the PSD o

    -
    +

    psd_sum.png

    Figure 5: \(y\) as the sum of two signals \(u\) and \(v\)

    @@ -542,11 +569,15 @@ We have that the PSD of \(y\) is equal to sum of the PSD and \(u\) and the PSD o
    -
    -

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

    @@ -570,24 +601,24 @@ To estimate the PSD of the position error \(\epsilon\) and thus the RMS residual
    • The Power Spectral Densities of the signals affecting the system:
        -
      • \(S_{dd}\): disturbances, this will be done in Section 3
        • +
      • \(S_{dd}\): disturbances, this will be done in Section 3
      • \(S_{nn}\): sensor noise, this can be estimated from the sensor data-sheet
      • \(S_{rr}\): which is a deterministic signal that we choose. For simple tomography experiment, we can consider that it is equal to \(0\)
    • The dynamics of the complete system comprising the micro-station and the nano-hexapod: \(G\), \(G_d\). -To do so, we need to identify the dynamics of the micro-station (Section 2), include this dynamics in a model (Section 4) and add a model of the nano-hexapod to the model (Section 5)
      • -
    • The controller \(K\) that will be designed in Section 6
      • +To do so, we need to identify the dynamics of the micro-station (Section 2), include this dynamics in a model (Section 4) and add a model of the nano-hexapod to the model (Section 5) +
    • The controller \(K\) that will be designed in Section 6
    -
    -

    2 Identification of the Micro-Station Dynamics

    +
    +

    2 Identification of the Micro-Station Dynamics

    - +

    As explained before, it is very important to have a good estimation of the micro-station dynamics as it will be coupled with the dynamics of the nano-hexapod and thus is very important for both the design of the nano-hexapod and controller. @@ -603,7 +634,7 @@ All the measurements performed on the micro-station are detailed in 6. +The general procedure to identify the dynamics of the micro-station is shown in Figure 6.

    @@ -616,7 +647,7 @@ The steps are: -

    +

    vibration_analysis_procedure.png

    Figure 6: Vibration Analysis Procedure

    @@ -627,11 +658,11 @@ The extraction of the Spatial Model (3rd step) was not performed as it requires

    -
    -

    2.1 Setup

    +
    +

    2.1 Setup

    - +

    @@ -657,13 +688,13 @@ In order to perform the Modal Analysis, the following devices were used: The measurement thus consists of:

      -
    • Exciting the structure at the same location with the Hammer (Figure 7)
      • +
    • Exciting the structure at the same location with the Hammer (Figure 7)
    • Move the accelerometers to measure all the DOF of the structure. The position of the accelerometers are:
      • 4 on the first granite
      • 4 on the second granite
        • -
      • 4 on top of the translation stage (Figure 8)
        • +
      • 4 on top of the translation stage (Figure 8)
      • 4 on top of the tilt stage
      • 3 on top of the spindle
      • 4 on top of the hexapod
        • @@ -679,14 +710,14 @@ We chose to have some redundancy in the measurement to be able to verify that th

        -
        +

        hammer_z.gif

        Figure 7: Example of one hammer impact

        -
        +

        accelerometers_ty_overview.jpg

        Figure 8: 3 tri axis accelerometers fixed to the translation stage

        @@ -694,11 +725,11 @@ We chose to have some redundancy in the measurement to be able to verify that th
        -
        -

        2.2 Results

        +
        +

        2.2 Results

        - +

        @@ -706,18 +737,18 @@ From the measurements, we obtain all the transfer functions from forces applied

        -Modal shapes and natural frequencies are then computed. Example of mode shapes are shown in Figures 9 10. +Modal shapes and natural frequencies are then computed. Example of mode shapes are shown in Figures 9 10.

        -
        +

        mode1.gif

        Figure 9: First mode that shows a suspension mode, probably due to bad leveling of one Airloc

        -
        +

        mode6.gif

        Figure 10: Sixth mode

        @@ -743,12 +774,12 @@ This thus means that a multi-body model can be used to represent the dynamics of

        Many Frequency Response Functions (FRF) are obtained from the measurements. -Examples of FRF are shown in Figure 11. +Examples of FRF are shown in Figure 11. These FRF will be used to compare the dynamics of the multi-body model with the micro-station dynamics.

        -
        +

        frf_all_bodies_one_direction.png

        Figure 11: Frequency Response Function from forces applied by the Hammer in the X direction to the acceleration of each solid body in the X direction

        @@ -756,13 +787,13 @@ These FRF will be used to compare the dynamics of the multi-body model with the
        -
        -

        2.3 Conclusion

        +
        +

        2.3 Conclusion

        The modal analysis of the micro-station confirmed the fact that a multi-body model should be able to correctly represents the micro-station dynamics. -In Section 4, the obtained Frequency Response Functions will be used to compare the model dynamics with the micro-station dynamics. +In Section 4, the obtained Frequency Response Functions will be used to compare the model dynamics with the micro-station dynamics.

        @@ -770,11 +801,11 @@ In Section 4, the obtained Frequency Response Function
        -
        -

        3 Identification of the Disturbances

        +
        +

        3 Identification of the Disturbances

        - +

        In this section, we wish to list and identify all the disturbances affecting the system. @@ -801,11 +832,11 @@ Open Loop Noise budget: -

        3.1 Ground Motion

        +
        +

        3.1 Ground Motion

        - +

        @@ -813,11 +844,11 @@ The ground motion can easily be estimated using an inertial sensor with sufficie

        -To verify that the inertial sensors are sensitive enough, a Huddle test has been performed (Figure 12). +To verify that the inertial sensors are sensitive enough, a Huddle test has been performed (Figure 12).

        -
        +

        geophones.jpg

        Figure 12: Huddle Test Setup

        @@ -829,7 +860,7 @@ The low frequency differences between the ground motion at ID31 and ID09 is just

        -
        +

        ground_motion_compare.png

        Figure 13: Comparison of the PSD of the ground motion measured at different location

        @@ -837,11 +868,11 @@ 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

        - +

        @@ -864,11 +895,11 @@ Complete reports on these measurements are accessible -

        3.3 Stage Vibration - Effect of Motion

        +
        +

        3.3 Stage Vibration - Effect of Motion

        - +

        @@ -880,15 +911,15 @@ Details reports are accessible -

        Spindle and Slip-Ring

        -
        +
        +

        Spindle and Slip-Ring

        +

        -The setup for the measurement of vibrations induced by rotation of the Spindle and Slip-ring is shown in Figure 14. +The setup for the measurement of vibrations induced by rotation of the Spindle and Slip-ring is shown in Figure 14.

        -
        +

        rz_meas_errors.gif

        Figure 14: Measurement of the sample’s vertical motion when rotating at 6rpm

        @@ -904,7 +935,7 @@ A geophone is fixed at the location of the sample and we measure the motion:

      -The obtained Power Spectral Density of the sample’s absolute velocity are shown in Figure 15. +The obtained Power Spectral Density of the sample’s absolute velocity are shown in Figure 15.

      @@ -920,7 +951,7 @@ However, when rotating with the Spindle (normal functioning mode):

    -
    +

    sr_sp_psd_sample_compare.png

    Figure 15: Comparison of the ASD of the measured voltage from the Geophone at the sample location

    @@ -935,19 +966,19 @@ Some investigation should be performed on the Spindle to determine where does th
    -
    -

    Translation Stage

    -
    +
    +

    Translation Stage

    +

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

    -We impose a 1Hz triangle motion with an amplitude of \(\pm 2.5mm\) on the translation stage (Figure 16), and we measure the absolute velocity of both the sample and the granite. +We impose a 1Hz triangle motion with an amplitude of \(\pm 2.5mm\) on the translation stage (Figure 16), and we measure the absolute velocity of both the sample and the granite.

    -
    +

    ty_position_time.png

    Figure 16: Y position of the translation stage measured by the encoders

    @@ -955,25 +986,25 @@ We impose a 1Hz triangle motion with an amplitude of \(\pm 2.5mm\) on the transl

    -The time domain absolute vertical velocity of the sample and granite are shown in Figure 17. +The time domain absolute vertical velocity of the sample and granite are shown in Figure 17. It is shown that quite large motion of the granite is induced by the translation stage scans. This could be a problem if this is shown to excite the metrology frame of the nano-focusing lens position stage.

    -
    +

    ty_z_time.png

    Figure 17: Vertical velocity of the sample and marble when scanning with the translation stage

    -The Amplitude Spectral Densities of the measured absolute velocities are shown in Figure 18. +The Amplitude Spectral Densities of the measured absolute velocities are shown in Figure 18. We can see many peaks starting from 1Hz showing the large spectral content probably due to the triangular reference of the translation stage.

    -
    +

    asd_z_direction.png

    Figure 18: Amplitude spectral density of the measure velocity corresponding to the geophone in the vertical direction located on the granite and at the sample location when the translation stage is scanning at 1Hz

    @@ -989,15 +1020,15 @@ A smoother motion for the translation stage (such as a sinus motion) could proba
    -
    -

    3.4 Sum of all disturbances

    +
    +

    3.4 Sum of all disturbances

    We can now compare the effect of all the disturbance sources on the position error (relative motion of the sample with respect to the granite).

    -The Power Spectral Density of the motion error due to the ground motion, translation stage scans and spindle rotation are shown in Figure 19. +The Power Spectral Density of the motion error due to the ground motion, translation stage scans and spindle rotation are shown in Figure 19.

    @@ -1005,19 +1036,19 @@ We can see that the ground motion is quite small compare to the translation stag

    -
    +

    dist_effect_relative_motion.png

    Figure 19: Amplitude Spectral Density fo the motion error due to disturbances

    -The Cumulative Amplitude Spectrum is shown in Figure 20. +The Cumulative Amplitude Spectrum is shown in Figure 20. It is shown that the motion induced by translation stage scans and spindle rotation are in the micro-meter range.

    -
    +

    dist_effect_relative_motion_cas.png

    Figure 20: Cumulative Amplitude Spectrum of the motion error due to disturbances

    @@ -1037,8 +1068,8 @@ From that, we can conclude that we will probably need a control bandwidth to aro
    -
    -

    3.5 Better estimation of the disturbances

    +
    +

    3.5 Better estimation of the disturbances

    All the disturbance measurements were made with inertial sensors, and to obtain the relative motion sample/granite, two inertial sensors were used and the signals were subtracted. @@ -1062,8 +1093,8 @@ The detector requirement would be:

    -
    -

    3.6 Conclusion

    +
    +

    3.6 Conclusion

    @@ -1088,14 +1119,14 @@ This should however not change the conclusion of this study nor significantly ch

    -
    -

    4 Multi Body Model

    +
    +

    4 Multi Body Model

    - +

    -As was shown during the modal analysis (Section 2), the micro-station behaves as multiple rigid bodies (granite, translation stage, tilt stage, spindle, hexapod) with some discrete flexibility between those solid bodies. +As was shown during the modal analysis (Section 2), the micro-station behaves as multiple rigid bodies (granite, translation stage, tilt stage, spindle, hexapod) with some discrete flexibility between those solid bodies.

    @@ -1108,8 +1139,8 @@ A small summary of the multi-body Simscape is available -

    4.1 Multi-Body model

    +
    +

    4.1 Multi-Body model

    The mass/inertia of each stage is automatically computed from the imported geometry and the material’s density. @@ -1121,11 +1152,11 @@ Then, the values of the stiffness and damping of each joint is manually tuned un

    -The 3D representation of the simscape model is shown in Figure 21. +The 3D representation of the simscape model is shown in Figure 21.

    -
    +

    simscape_picture.png

    Figure 21: 3D representation of the simscape model

    @@ -1133,15 +1164,15 @@ The 3D representation of the simscape model is shown in Figure -

    4.2 Validity of the model’s dynamics

    +
    +

    4.2 Validity of the model’s dynamics

    It is very difficult the tune the dynamics of such model as there are more than 50 parameters and many curves to compare between the model and the measurements.

    -The comparison of three of the Frequency Response Functions are shown in Figure 22. +The comparison of three of the Frequency Response Functions are shown in Figure 22.

    @@ -1153,7 +1184,7 @@ We believe that the model is representing the micro-station dynamics with suffic

    -
    +

    identification_comp_top_stages.png

    Figure 22: Frequency Response function from Hammer force in the X,Y and Z directions to the X,Y and Z displacements of the micro-hexapod’s top platform. The measurements are shown in blue and the Model in red.

    @@ -1186,11 +1217,11 @@ Then, using the model, we can
    -
    -

    4.3 Wanted position of the sample and position error

    +
    +

    4.3 Wanted position of the sample and position error

    - +

    @@ -1198,7 +1229,7 @@ For the control of the nano-hexapod, we need to now the sample position error (t

    -To do so, we need to perform several computations (summarized in Figure 23): +To do so, we need to perform several computations (summarized in Figure 23):

    • First, we need to determine the actual wanted pose (3 translations and 3 rotations) of the sample with respect to the granite. @@ -1213,7 +1244,7 @@ Both computation are performed.
    -
    +

    control-schematic-nass.png

    Figure 23: Figure caption

    @@ -1225,11 +1256,11 @@ More details about these computations are accessible -

    4.4 Simulation of Experiments

    +
    +

    4.4 Simulation of Experiments

    - + Now that the dynamics of the model is tuned and the disturbances included in the model, we can perform simulation of experiments.

    @@ -1238,8 +1269,8 @@ We first do a simulation where the nano-hexapod is considered to be a solid-body

    -An animation of the obtained motion is shown in Figure 24. -A zoom in the micro-meter ranger on the sample’s location is shown in Figure 25. +An animation of the obtained motion is shown in Figure 24. +A zoom in the micro-meter ranger on the sample’s location is shown in Figure 25.

    @@ -1252,7 +1283,7 @@ Note that here this frame is moving with the granite. -

    +

    open_loop_sim.gif

    Figure 24: Tomography Experiment using the Simscape Model

    @@ -1260,14 +1291,14 @@ Note that here this frame is moving with the granite. -
    +

    open_loop_sim_zoom.gif

    Figure 25: Tomography Experiment using the Simscape Model - Zoom on the sample’s position (the full vertical scale is \(\approx 10 \mu m\))

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

    @@ -1281,7 +1312,7 @@ For the vertical rotation, this is due to the fact that we suppose perfect rotat

    -
    +

    exp_scans_rz_dist.png

    Figure 26: Position error of the Sample with respect to the granite during a Tomography Experiment with included disturbances

    @@ -1289,8 +1320,8 @@ For the vertical rotation, this is due to the fact that we suppose perfect rotat
    -
    -

    4.5 Conclusion

    +
    +

    4.5 Conclusion

    @@ -1315,11 +1346,11 @@ This model will be used in the next sections to help the design of the nano-hexa

    -
    -

    5 Optimal Nano-Hexapod Design

    +
    +

    5 Optimal Nano-Hexapod Design

    - +

    As explain before, the nano-hexapod properties (mass, stiffness, architecture, …) will influence: @@ -1333,8 +1364,8 @@ As explain before, the nano-hexapod properties (mass, stiffness, architecture, & Thus, we here wish to find the optimal nano-hexapod properties such that:

      -
    • the effect of disturbances is minimized (Section 5.1)
      • -
    • the plant uncertainty due to a change of payload mass and experimental conditions is minimized (Section 5.2)
      • +
    • the effect of disturbances is minimized (Section 5.1)
      • +
    • the plant uncertainty due to a change of payload mass and experimental conditions is minimized (Section 5.2)

    @@ -1346,11 +1377,11 @@ The study presented here only consider changes in the nano-hexapod stiffness<

    -
    -

    5.1 Optimal Stiffness to reduce the effect of disturbances

    +
    +

    5.1 Optimal Stiffness to reduce the effect of disturbances

    - +

    As will be seen, the nano-hexapod stiffness have a large influence on the sensibility to disturbance (the norm of \(G_d\)). @@ -1362,11 +1393,11 @@ A complete study of the optimal nano-hexapod stiffness for the minimization of d

    -
    -

    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 27. +The sensibility to the spindle’s vibration for all the considered nano-hexapod stiffnesses (from \(10^3\,[N/m]\) to \(10^9\,[N/m]\)) is shown in Figure 27. It is shown that a softer nano-hexapod it better to filter out vertical vibrations of the spindle. More precisely, is start to filters the vibration at the first suspension mode of the payload on top of the nano-hexapod.

    @@ -1376,7 +1407,7 @@ The same conclusion is made for vibrations of the translation stage.

    -
    +

    opt_stiff_sensitivity_Frz.png

    Figure 27: Sensitivity to Spindle vertical motion error to the vertical error position of the sample

    @@ -1384,21 +1415,21 @@ The same conclusion is made for vibrations of the translation stage.
    -
    -

    Sensibility to ground motion

    -
    +
    +

    Sensibility to ground motion

    +

    -The sensibilities to ground motion in the Y and Z directions are shown in Figure 28. +The sensibilities to ground motion in the Y and Z directions are shown in Figure 28. We can see that above the suspension mode of the nano-hexapod, the norm of the transmissibility is close to one until the suspension mode of the granite. Thus, a stiff nano-hexapod is better for reducing the effect of ground motion at low frequency.

    -It will be suggested in Section 7.3 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 28: Sensitivity to Ground motion to the position error of the sample

    @@ -1406,13 +1437,13 @@ It will be suggested in Section 7.3 that using soft mo
    -
    -

    Dynamic Noise Budgeting considering all the disturbances

    -
    +
    +

    Dynamic Noise Budgeting considering all the disturbances

    +

    However, lowering the sensibility to some disturbance at a frequency where its effect is already small compare to the other disturbances sources is not really interesting. What is more important than comparing the sensitivity to disturbances, is thus to compare the obtain power spectral density of the sample’s position error. -From the Power Spectral Density of all the sources of disturbances identified in Section 3, we compute what would be the Power Spectral Density of the vertical motion error for all the considered nano-hexapod stiffnesses (Figure 29). +From the Power Spectral Density of all the sources of disturbances identified in Section 3, we compute what would be the Power Spectral Density of the vertical motion error for all the considered nano-hexapod stiffnesses (Figure 29).

    @@ -1420,7 +1451,7 @@ We can see that the most important change is in the frequency range 30Hz to 300H

    -
    +

    opt_stiff_psd_dz_tot.png

    Figure 29: Amplitude Spectral Density of the Sample vertical position error due to Vertical vibration of the Spindle for multiple nano-hexapod stiffnesses

    @@ -1428,13 +1459,13 @@ We can see that the most important change is in the frequency range 30Hz to 300H

    -If we look at the Cumulative amplitude spectrum of the vertical error motion in Figure 30, we can observe that a soft hexapod (\(k < 10^5 - 10^6\,[N/m]\)) helps reducing the high frequency disturbances, and thus a smaller control bandwidth will suffice to obtain the wanted performance. +If we look at the Cumulative amplitude spectrum of the vertical error motion in Figure 30, we can observe that a soft hexapod (\(k < 10^5 - 10^6\,[N/m]\)) helps reducing the high frequency disturbances, and thus a smaller control bandwidth will suffice to obtain the wanted performance.

    -
    +

    opt_stiff_cas_dz_tot.png

    Figure 30: Cumulative Amplitude Spectrum of the Sample vertical position error due to all considered perturbations for multiple nano-hexapod stiffnesses

    @@ -1443,11 +1474,11 @@ If we look at the Cumulative amplitude spectrum of the vertical error motion in
    -
    -

    5.2 Optimal Stiffness to reduce the plant uncertainty

    +
    +

    5.2 Optimal Stiffness to reduce the plant uncertainty

    - +

    One of the most important design goal is to obtain a system that is robust to all changes in the system. @@ -1479,15 +1510,15 @@ 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 31 the dynamics is shown for payloads having a first resonance mode at 100Hz and a mass equal to 1kg, 20kg and 50kg. +In Figure 31 the dynamics is shown for payloads having a first resonance mode at 100Hz and a mass equal to 1kg, 20kg and 50kg. On the left side, the change of dynamics is computed for a very soft nano-hexapod, while on the right side, it is computed for a very stiff nano-hexapod.

    @@ -1505,14 +1536,14 @@ For the stiff-nano-hexapod, the change of payload mass has very little effect (t

    -
    +

    opt_stiffness_payload_mass_fz_dz.png

    Figure 31: Dynamics from \(\mathcal{F}_z\) to \(\mathcal{X}_z\) for varying payload mass, both for a soft nano-hexapod (left) and a stiff nano-hexapod (right)

    -In Figure 32 is shown the effect of a change of payload dynamics. +In Figure 32 is shown the effect of a change of payload dynamics. The mass of the payload is fixed and its resonance frequency is changing from 50Hz to 500Hz.

    @@ -1521,7 +1552,7 @@ We can see (more easily for the soft nano-hexapod), that resonance of the payloa

    -
    +

    opt_stiffness_payload_freq_fz_dz.png

    Figure 32: Dynamics from \(\mathcal{F}_z\) to \(\mathcal{X}_z\) for varying payload resonance frequency, both for a soft nano-hexapod and a stiff nano-hexapod

    @@ -1529,7 +1560,7 @@ We can see (more easily for the soft nano-hexapod), that resonance of the payloa

    -The dynamics for all the payloads (mass from 1kg to 50kg and first resonance from 50Hz to 500Hz) and all the considered nano-hexapod stiffnesses are display in Figure 33. +The dynamics for all the payloads (mass from 1kg to 50kg and first resonance from 50Hz to 500Hz) and all the considered nano-hexapod stiffnesses are display in Figure 33.

    @@ -1550,7 +1581,7 @@ For nano-hexapod stiffnesses above \(10^7\,[N/m]\): -

    +

    opt_stiffness_payload_impedance_all_fz_dz.png

    Figure 33: Dynamics from \(\mathcal{F}_z\) to \(\mathcal{X}_z\) for varying payload dynamics, both for a soft nano-hexapod and a stiff nano-hexapod

    @@ -1579,11 +1610,11 @@ Heavy samples with low first resonance mode will be very 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
      • @@ -1604,7 +1635,7 @@ This as several other advantages:

      -To identify the effect of the micro-station compliance on the system dynamics, for each nano-hexapod stiffness, we identify the plant dynamics in two different case (Figure 34): +To identify the effect of the micro-station compliance on the system dynamics, for each nano-hexapod stiffness, we identify the plant dynamics in two different case (Figure 34):

      • without the micro-station (solid curves)
        • @@ -1620,7 +1651,7 @@ For nano-hexapod stiffnesses above \(10^7\,[N/m]\), the micro-station compliance

        -
        +

        opt_stiffness_micro_station_fx_dx.png

        Figure 34: Change of dynamics from force \(\mathcal{F}_x\) to displacement \(\mathcal{X}_x\) due to the micro-station compliance

        @@ -1640,15 +1671,15 @@ If a stiff nano-hexapod is used, the control bandwidth should probably be limite
        -
        -

        Effect of Spindle Rotating Speed

        -
        +
        +

        Effect of Spindle Rotating Speed

        +

        Let’s now consider the rotation of the Spindle.

        -The plant dynamics for spindle rotation speed from 0rpm up to 60rpm are shown in Figure 35. +The plant dynamics for spindle rotation speed from 0rpm up to 60rpm are shown in Figure 35.

        @@ -1660,7 +1691,7 @@ For very soft nano-hexapods, the main resonance is split into two resonances and

        -
        +

        opt_stiffness_wz_fx_dx.png

        Figure 35: Change of dynamics from force \(\mathcal{F}_x\) to displacement \(\mathcal{X}_x\) for a spindle rotation speed from 0rpm to 60rpm

        @@ -1679,16 +1710,16 @@ A very soft (\(k < 10^4\,[N/m]\)) nano-hexapod should not be used due to the eff
        -
        -

        Total Plant Uncertainty

        -
        +
        +

        Total Plant Uncertainty

        +

        -Finally, let’s combined all the uncertainties and display the plant dynamics “spread” for all the nano-hexapod stiffnesses (Figure 36). +Finally, let’s combined all the uncertainties and display the plant dynamics “spread” for all the nano-hexapod stiffnesses (Figure 36). This show how the dynamics evolves with the stiffness and how different effects enters the plant dynamics.

        -
        +

        opt_stiffness_plant_dynamics_task_space.gif

        Figure 36: Variability of the dynamics from \(\bm{\mathcal{F}}_x\) to \(\bm{\mathcal{X}}_x\) with varying nano-hexapod stiffness

        @@ -1721,22 +1752,22 @@ In such case, the main limitation will be heavy samples with small stiffnesses.
        -
        -

        5.3 Conclusion

        +
        +

        5.3 Conclusion

        -In Section 5.1, it has been concluded that a nano-hexapod stiffness below \(10^5-10^6\,[N/m]\) helps reducing the high frequency vibrations induced by all sources of disturbances considered. +In Section 5.1, it has been concluded that a nano-hexapod stiffness below \(10^5-10^6\,[N/m]\) helps reducing the high frequency vibrations induced by all sources of disturbances considered. As the high frequency vibrations are the most difficult to compensate for when using feedback control, a soft hexapod will most certainly helps improving the performances.

        -In Section 5.2, we concluded that a nano-hexapod leg stiffness in the range \(10^5 - 10^6\,[N/m]\) is a good compromise between the uncertainty induced by the micro-station dynamics and by the rotating speed. +In Section 5.2, we concluded that a nano-hexapod leg stiffness in the range \(10^5 - 10^6\,[N/m]\) is a good compromise between the uncertainty induced by the micro-station dynamics and by the rotating speed. Provided that the samples used have a first mode that is sufficiently high in frequency, the total plant dynamic uncertainty should be manageable.

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

        @@ -1748,11 +1779,11 @@ A more detailed study of the determination of the optimal stiffness based on all

        -
        -

        6 Robust Control Architecture

        +
        +

        6 Robust Control Architecture

        - +

        Before designing the control system, let’s summarize what has been done: @@ -1779,8 +1810,8 @@ This would require to measure the mass/inertia of each used payload and manually

        -
        -

        6.1 High Authority Control / Low Authority Control Architecture

        +
        +

        6.1 High Authority Control / Low Authority Control Architecture

        Many control architecture could be used for the control of the nano-hexapod. @@ -1796,7 +1827,7 @@ Some properties of the HAC-LAC architecture are explained below (taken from

        -The HAC/LAC approach consist of combining the two approached in a dual-loop control as shown in Figure 37. +The HAC/LAC approach consist of combining the two approached in a dual-loop control as shown in Figure 37. The inner loop uses a set of collocated actuator/sensor pairs for decentralized active damping with guaranteed stability ; the outer loop consists of a non-collocated HAC based on a model of the actively damped structure. This approach has the following advantages:

        @@ -1808,7 +1839,7 @@ This approach has the following advantages:
        -
        +

        control_architecture_hac_lac_one_input.png

        Figure 37: HAC-LAC Architecture with a system having only one input

        @@ -1818,17 +1849,17 @@ This approach has the following advantages: The HAC-LAC architecture thus consisted 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

        - +

        @@ -1856,11 +1887,11 @@ It would also be difficult to apply in a robust way due to the non-collocation w

        -Relative motion sensors are then included in each of the nano-hexapod’s leg and a decentralized direct velocity feedback control architecture is applied (Figure 38). +Relative motion sensors are then included in each of the nano-hexapod’s leg and a decentralized direct velocity feedback control architecture is applied (Figure 38).

        -The signals shown in Figure 38 are: +The signals shown in Figure 38 are:

        • \(\bm{\tau}\): Actuator forces applied in each leg
          • @@ -1876,7 +1907,7 @@ The force applied in each leg being proportional to the relative velocity of the

          -
          +

          control_architecture_dvf.png

          Figure 38: Low Authority Control: Decentralized Direct Velocity Feedback

          @@ -1889,20 +1920,20 @@ This may not be the optimal choice as will be further explained.

          -The plant dynamics before (solid curves) and after (dashed curves) the Low Authority Control implementation are compared in Figure 39. +The plant dynamics before (solid curves) and after (dashed curves) the Low Authority Control implementation are compared in Figure 39. It is clear that the use of the DVF reduces the dynamical spread of the plant dynamics between 5Hz up too 100Hz. This will make the primary controller more robust and easier to develop.

          -
          +

          opt_stiff_primary_plant_damped_L.png

          Figure 39: Primary plant in the space of the legs with (dashed) and without (solid) Direct Velocity Feedback

          -The change of sensibility to disturbances with the use of DVF is shown in Figure 40. +The change of sensibility to disturbances with the use of DVF is shown in Figure 40. It is shown that the DVF control lowers the sensibility to disturbances in the vicinity of the nano-hexapod resonance but increases the sensibility at higher frequencies.

          @@ -1911,7 +1942,7 @@ This is probably not the optimal gain that could have been used, and further ana

          -
          +

          opt_stiff_sensibility_dist_dvf.png

          Figure 40: Norm of the transfer function from vertical disturbances to vertical position error with (dashed) and without (solid) Direct Velocity Feedback applied

          @@ -1919,20 +1950,20 @@ This is probably not the optimal gain that could have been used, and further ana
          -
          -

          6.3 High Authority Control

          +
          +

          6.3 High Authority Control

          - +

          -The complete HAC-LAC architecture is shown in Figure 41 where an outer loop is added to the decentralized direct velocity feedback loop. +The complete HAC-LAC architecture is shown in Figure 41 where an outer loop is added to the decentralized direct velocity feedback loop.

          The block Compute Position 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.

          @@ -1945,7 +1976,7 @@ Then, a diagonal controller \(\bm{K}_\mathcal{L}\) generates the required force

          -
          +

          control_architecture_hac_dvf_pos_L.png

          Figure 41: Cascade Control Architecture. The inner loop consist of a decentralized Direct Velocity Feedback. The outer loop consist of position control in the leg’s space

          @@ -1956,12 +1987,12 @@ Some alternative to this control architecture have been studied, but this is the

          -The plant dynamics for each of the six legs and for the three payload’s masses is shown in Figure 42. +The plant dynamics for each of the six legs and for the three payload’s masses is shown in Figure 42. The dynamical spread is kept reasonably small thanks to both the optimal nano-hexapod design and the Low Authority Controller.

          -
          +

          opt_stiff_primary_plant_L.png

          Figure 42: Diagonal elements of the transfer function matrix from \(\bm{\tau}^\prime\) to \(\bm{\epsilon}_{\mathcal{X}_n}\) for the three considered masses

          @@ -1970,18 +2001,18 @@ The dynamical spread is kept reasonably small thanks to both the optimal nano-he

          The diagonal controller \(\bm{K}_\mathcal{L}\) is then tuned in such a way that the control bandwidth is around 100Hz and such that enough stability margins are obtained for all the payload’s masses used. -The obtained loop gain is shown in Figure 43. +The obtained loop gain is shown in Figure 43.

          -
          +

          opt_stiff_primary_loop_gain_L.png

          Figure 43: Loop gain for the primary plant

          -The sensibility to disturbance after the use of HAC-LAC control is shown in Figure 44. +The sensibility to disturbance after the use of HAC-LAC control is shown in Figure 44. The change of sensibility is very typical for feedback system:

            @@ -1999,7 +2030,7 @@ This should gives slightly better performance and robustness, but should not cha

            -
            +

            opt_stiff_primary_control_L_senbility_dist.png

            Figure 44: Sensibility to disturbances when the HAC-LAC control is applied (dashed) and when it is not (solid)

            @@ -2007,22 +2038,22 @@ This should gives slightly better performance and robustness, but should not cha
            -
            -

            6.4 Simulation of Tomography Experiments

            +
            +

            6.4 Simulation of Tomography Experiments

            - +

            A new simulation of a tomography is performed with the optimal nano-hexapod and the HAC-LAC architecture implemented in the model. -The results of this simulation will be compare to the simulation performed in Section 4.4 without the nano-hexapod. +The results of this simulation will be compare 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.

            -The Power Spectral Density of the sample’s position error is plotted in Figure 45 and the Cumulative Amplitude Spectrum is shown in Figure 46. +The Power Spectral Density of the sample’s position error is plotted in Figure 45 and the Cumulative Amplitude Spectrum is shown in Figure 46. The top three plots corresponds to the X, Y and Z translations and the bottom three plots corresponds to the X,Y and Z rotations.

            @@ -2051,7 +2082,7 @@ 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 45: Amplitude Spectral Density of the position error in Open Loop and with the HAC-LAC controller

          @@ -2059,7 +2090,7 @@ This increase in rotation is still very small and is not foreseen to be a proble -
          +

          opt_stiff_hac_dvf_L_cas_disp_error.png

          Figure 46: Cumulative Amplitude Spectrum of the position error in Open Loop and with the HAC-LAC controller

          @@ -2067,23 +2098,23 @@ This increase in rotation is still very small and is not foreseen to be a proble

          -The time domain sample’s vibrations are shown in Figure 47. +The time domain sample’s vibrations are shown in Figure 47. The use of the nano-hexapod combined with the HAC-LAC architecture is shown to considerably reduce the sample’s vibrations.

          -An animation of the experiment is shown in Figure 48 and we can see that the actual sample’s position is more closely following the ideal position compared to the simulation of the micro-station alone in Figure 25 (same scale was used for both animations). +An animation of the experiment is shown in Figure 48 and we can see that the actual sample’s position is more closely following the ideal position compared to the simulation of the micro-station alone in Figure 25 (same scale was used for both animations).

          -
          +

          opt_stiff_hac_dvf_L_pos_error.png

          Figure 47: 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 48: Tomography Experiment using the Simscape Model in Closed Loop with the HAC-LAC Control - Zoom on the sample’s position (the full vertical scale is \(\approx 10 \mu m\))

          @@ -2091,8 +2122,8 @@ An animation of the experiment is shown in Figure 48 a
          -
          -

          6.5 Conclusion

          +
          +

          6.5 Conclusion

          @@ -2127,33 +2158,44 @@ A more complete study of the control of the NASS is performed -

          7 General Conclusion and Further notes

          +
          +

          7 General Conclusion and Further notes

          - +

          -
          -

          7.1 General Conclusion

          +
          +

          7.1 General Conclusion

          -
          -

          7.2 Further Work

          -
          - - -
          -

          7.3 Using soft mounts for the

          -
          +
          +

          7.2 Sensor Noise introduced by the Metrology

          +

          - +Say that is will introduce noise inside the bandwidth (100Hz) +This should not be significant. +

          +
          +
          + + +
          +

          7.3 Further Work

          +
          + + +
          +

          7.4 Using soft mounts for the

          +
          +

          +

          -
          +

          opt_stiff_soft_granite_Dw.png

          Figure 49: Change of sensibility to Ground motion when using a stiff Granite (solid curves) and a soft Granite (dashed curves)

          @@ -2169,9 +2211,9 @@ Sensible to detector motion?
          -
          -

          7.4 Others

          -
          +
          +

          7.5 Others

          +

          Common metrology frame for the nano-focusing optics and the measurement of the sample position?

          @@ -2199,7 +2241,7 @@ Slip-Ring noise?

          Date: 05-2020

          Author: Thomas Dehaeze

          -

          Created: 2020-04-29 mer. 15:23

          +

          Created: 2020-04-29 mer. 17:04

          diff --git a/index.org b/index.org index 55e7b52..55f9447 100644 --- a/index.org +++ b/index.org @@ -34,7 +34,8 @@ :END: * Introduction :ignore: -The overall objective is to design a nano-hexapod an the associated control architecture that allows the stabilization of samples down to $\approx 10nm$ in presence of disturbances and system variability. +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. To understand the design challenges of such system, a short introduction to Feedback control is provided in Section [[sec:feedback_introduction]]. @@ -43,15 +44,14 @@ The mathematical tools (Power Spectral Density, Noise Budgeting, ...) that will To be able to develop both the nano-hexapod and the control architecture in an optimal way, we need a good estimation of: - the micro-station dynamics (Section [[sec:micro_station_dynamics]]) -- the frequency content of the important source of disturbances in play such as vibration of stages and ground motion (Section [[sec:identification_disturbances]]) +- the frequency content of the sources of disturbances such as vibrations induced by the micro-station's stages and ground motion (Section [[sec:identification_disturbances]]) -We then develop a model of the system that must represent all the important physical effects in play. -Such model is presented in Section [[sec:multi_body_model]]. +A model of the micro-station is then developed and tuned using the previous estimations (Section [[sec:multi_body_model]]). +The nano-hexapod is further included in the model. -A modular model of the nano-hexapod is then included in the system. -The effects of the nano-hexapod characteristics on the dynamics are then studied. +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 [[sec:nano_hexapod_design]]). @@ -61,8 +61,9 @@ Simulations are performed to show that this design gives acceptable performance * Introduction to Feedback Systems and Noise budgeting <> -In this section, we first introduce some basics of *feedback systems* (Section [[sec:feedback]]). -This should highlight the challenges in terms of combined performance and robustness. +** Introduction :ignore: +In this section, some basics of *feedback systems* are first introduced (Section [[sec:feedback]]). +This should highlight the challenges of the required combined performance and robustness. In Section [[sec:noise_budget]] 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. @@ -72,14 +73,17 @@ This tool will be widely used throughout this study to both predict the performa <> *** Introduction :ignore: +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. + The use of feedback control as several advantages and pitfalls that are listed below (taken from cite:schmidt14_desig_high_perfor_mechat_revis_edition): -- *Advantages*: +*Advantages*: - *Reduction of the effect of disturbances*: - Disturbances affecting the sample vibrations are observed by the sensor signal, and therefore the feedback controller can compensate for them + Disturbances inducing vibrations are observed by the sensor signal, and therefore the feedback controller can compensate for them - *Handling of uncertainties*: Feedback controlled systems can also be designed for /robustness/, which means that the stability and performance requirements are guaranteed even for parameter variation of the controller mechatronics system -- *Pitfalls*: + +*Pitfalls*: - *Limited reaction speed*: A feedback controller reacts on the difference between the reference signal (wanted motion) and the measurement (actual motion), which means that the error has to occur first /before/ the controller can correct for it. The limited reaction speed means that the controller will be able to compensate the positioning errors only in some frequency band, called the controller /bandwidth/ @@ -91,13 +95,13 @@ The use of feedback control as several advantages and pitfalls that are listed b *** Simplified Feedback Control Diagram for the NASS Let's consider the block diagram shown in Figure [[fig:classical_feedback_small]] where the signals are: -- $y$: the relative position of the sample with respect to the granite (the quantity we wish to control) -- $d$: the disturbances affecting $y$ (ground motion, vibration of stages) +- $y$: the relative position of the sample with respect to the granite (the quantity to be controlled) +- $d$: the disturbances affecting $y$ (ground motion, stages' vibrations) - $n$: the noise of the sensor measuring $y$ - $r$: the reference signal, corresponding to the wanted $y$ - $\epsilon = r - y$: the position error -And the dynamical blocks are: +The /dynamical/ blocks are: - $G$: representing the dynamics from forces/torques applied by the nano-hexapod to the relative position sample/granite $y$ - $G_d$: representing how the disturbances (e.g. ground motion) are affecting the relative position sample/granite $y$ - $K$: representing the controller (to be designed) @@ -129,11 +133,11 @@ And the dynamical blocks are: #+RESULTS: [[file:figs/classical_feedback_small.png]] -Without the use of feedback (i.e. nano-hexapod), the disturbances will induce a sample motion error equal to: +Without the use of feedback (i.e. without the nano-hexapod), the disturbances will induce a sample motion error equal to: \begin{equation} y = G_d d \label{eq:open_loop_error} \end{equation} -which is out of the specifications (micro-meter range compare to the required $\approx 10nm$). +which is, in the case of the NASS out of the specifications (micro-meter range compare to the required $\approx 10nm$). In the next section, we see how the use of the feedback system permits to lower the effect of the disturbances $d$ on the sample motion error. @@ -159,10 +163,8 @@ From Eq. eqref:eq:closed_loop_error representing the closed-loop system behavior - the measurement noise $n$ is injected and multiplied by a factor $T$ Ideally, we would like to design the controller $K$ such that: -- $|S|$ is small to limit the effect of disturbances -- $|T|$ is small to limit the injection of sensor noise - -As shown in the next section, there is a trade-off between the disturbance reduction and the noise injection. +- $|S|$ is small to *reduce the effect of disturbances* +- $|T|$ is small to *limit the injection of sensor noise* *** Trade off: Disturbance Reduction / Noise Injection We have from the definition of $S$ and $T$ that: @@ -176,14 +178,15 @@ There is therefore a *trade-off between the disturbance rejection and the measur Typical shapes of $|S|$ and $|T|$ as a function of frequency are shown in Figure [[fig:h-infinity-2-blocs-constrains]]. We can observe that $|S|$ and $|T|$ exhibit different behaviors depending on the frequency band: -- *At low frequency* (inside the control bandwidth): - - $|S|$ can be made small and thus the effect of disturbances is reduced - - $|T| \approx 1$ and all the sensor noise is transmitted -- *At high frequency* (outside the control bandwidth): - - $|S| \approx 1$ and the feedback system does not reduce the effect of disturbances - - $|T|$ is small and thus the sensor noise is filtered -- *Near the crossover frequency* (between the two frequency bands): - - The effect of disturbances is increased + +*At low frequency* (inside the control bandwidth): +- $|S|$ can be made small and thus the effect of disturbances is reduced +- $|T| \approx 1$ and all the sensor noise is transmitted +*At high frequency* (outside the control bandwidth): +- $|S| \approx 1$ and the feedback system does not reduce the effect of disturbances +- $|T|$ is small and thus the sensor noise is filtered +*Near the crossover frequency* (between the two frequency bands): +- The effect of disturbances is increased #+begin_src latex :file h-infinity-2-blocs-constrains.pdf \begin{tikzpicture} @@ -228,16 +231,16 @@ We can observe that $|S|$ and $|T|$ exhibit different behaviors depending on the *** Trade off: Robustness / Performance <> -As shown in the previous section, the effect of disturbances is reduced /inside/ the control bandwidth. +As shown in the previous section, the effect of disturbances is reduced *inside* the control bandwidth. -Moreover, the slope of $|S(j\omega)|$ is limited for stability reasons (not explained here), and therefore a large control bandwidth is required to obtain sufficient disturbance rejection at lower frequencies (where the disturbances have large effects). +Moreover, the slope of $|S(j\omega)|$ is limited for stability reasons (not explained here), and therefore a large control bandwidth is required to obtain sufficient disturbance rejection at lower frequencies (where the disturbances have usually large effects). -The next important question is *what effects do limit the attainable control bandwidth?* +The next important question is therefore *what limits the attainable control bandwidth?* -The main issue it that for stability reasons, *the behavior of the mechanical system must be known with only small uncertainty in the vicinity of the crossover frequency*. +The main issue it that for stability reasons, *the system dynamics must be known with only small uncertainty in the vicinity of the crossover frequency*. -For mechanical systems, this generally means that control bandwidth should take place before any appearing of flexible dynamics (Right part of Figure [[fig:oomen18_next_gen_loop_gain]]). +For mechanical systems, this generally means that the control bandwidth should take place before any appearing of flexible dynamics (right part of Figure [[fig:oomen18_next_gen_loop_gain]]). #+name: fig:oomen18_next_gen_loop_gain #+caption: 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. cite:oomen18_advan_motion_contr_precis_mechat @@ -250,30 +253,32 @@ For the NASS, the possible changes in the system are: - a change of experimental condition: spindle's rotation speed, position of each micro-station's stage - a change in the micro-station dynamics (change of mechanical elements, aging effect, ...) -The nano-hexapod and the control architecture have to be developed such that the feedback system remains stable and exhibit acceptable performance for all these possible changes in the system. +The nano-hexapod and the control architecture have to be developed in such a way that the feedback system remains stable and exhibit acceptable performance for all these possible changes in the system. This problem of *robustness* represent one of the main challenge for the design of the NASS. -# High performance mechatronics systems (e.g. Wafer stages, or Atomic Force Microscopes) are usually developed in such a way that their mechanical behavior is extremely well known up to high frequency and such that the experimental conditions are usually be carefully controlled. - ** Dynamic error budgeting <> *** Introduction :ignore: -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 +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. -To understand how to use and understand it, the Power Spectral Density and the Cumulative Power Spectrum are first introduced. -Then, is shown how does multiple error sources are combined and modified by dynamical systems. +The dynamic error budgeting uses two important mathematical functions: the *Power Spectral Density* and the *Cumulative Power Spectrum*. -Finally, +After these two functions are introduced (in Sections [[sec:psd]] and [[sec:cps]]), is shown how do multiple error sources are combined and modified by dynamical systems (in Section [[sec:psd_lti_system]] and [[sec:psd_combined_signals]]). + +Finally, the dynamic noise budgeting for the NASS is derived in Section [[sec:dynamic_noise_budget]]. *** 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: -\[ S_{xx}(\omega) = \int_{-\infty}^{\infty} R_{xx}(\tau) e^{-j \omega \tau} d\tau \ \frac{[\text{unit of } x]^2}{\text{Hz}} \] +\[ S_{xx}(\omega) = \int_{-\infty}^{\infty} R_{xx}(\tau) e^{-j \omega \tau} d\tau \quad \frac{[\text{unit of } x]^2}{\text{Hz}} \] The PSD $S_{xx}(\omega)$ represents the *distribution of the (average) signal power over frequency*. -Thus, the total power in the signal can be obtained by integrating these infinitesimal contributions, the Root Mean Square (RMS) value of the signal $x(t)$ is then: +Thus, the total power in the signal can be obtained by integrating these infinitesimal contributions. +The Root Mean Square (RMS) value of the signal $x(t)$ is then: \begin{equation} x_{\text{rms}} = \sqrt{\int_{0}^{\infty} S_{xx}(\omega) d\omega} \end{equation} @@ -284,6 +289,8 @@ One can also integrate the infinitesimal power $S_{xx}(\omega)d\omega$ over a fi \end{equation} *** Cumulative Power Spectrum +<> + The *Cumulative Power Spectrum* is the cumulative integral of the Power Spectral Density starting from $0\ \text{Hz}$ with increasing frequency: \begin{equation} CPS_x(f) = \int_0^f S_{xx}(\nu) d\nu \quad [\text{unit of } x]^2 @@ -297,10 +304,13 @@ An alternative definition of the Cumulative Power Spectrum can be used where the \end{equation} And thus $CPS_x(f)$ represents the power in the signal $x$ for frequencies above $f$. +The cumulative -The Cumulative Power Spectrum will be used to determine in which frequency band the effect of disturbances should be reduced, and thus the approximate required control bandwidth. +The Cumulative Power Spectrum is generally shown as a function of frequency, and is used to determine at which frequencies the effect of disturbances must be reduced, and thus the approximate required control bandwidth. + +*** Modification of a signal's PSD when going through a dynamical system +<> -*** 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 [[fig:psd_lti_system]]). #+begin_src latex :file psd_lti_system.pdf @@ -323,6 +333,8 @@ The Power Spectral Density of the output signal $y$ can be computed using: \end{equation} *** PSD of combined signals +<> + Let's consider a signal $y$ that is the sum of two *uncorrelated* signals $u$ and $v$ (Figure [[fig:psd_sum]]). We have that the PSD of $y$ is equal to sum of the PSD and $u$ and the PSD of $v$ (can be easily shown from the definition of the PSD): @@ -345,6 +357,8 @@ We have that the PSD of $y$ is equal to sum of the PSD and $u$ and the PSD of $v [[file:figs/psd_sum.png]] *** Dynamic Noise Budgeting +<> + Let's consider the Feedback architecture in Figure [[fig:classical_feedback_small]] where the position error $\epsilon$ is equal to: \[ \epsilon = S r + T n - G_d S d \] @@ -360,9 +374,10 @@ And we can compute the RMS value of the residual motion using: To estimate the PSD of the position error $\epsilon$ and thus the RMS residual motion (in closed-loop), we need to determine: - The Power Spectral Densities of the signals affecting the system: - - $S_{dd}$: disturbances, this will be done in Section [[sec:identification_disturbances]] - - $S_{nn}$: sensor noise, this can be estimated from the sensor data-sheet - - $S_{rr}$: which is a deterministic signal that we choose. For simple tomography experiment, we can consider that it is equal to $0$ + - The disturbances $S_{dd}$: this will be done in Section [[sec:identification_disturbances]] + - 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 we choose. + For a simple tomography experiment, we can consider that it is equal to $0$ as we only want to compensate all the sample's vibrations - 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 [[sec:micro_station_dynamics]]), include this dynamics in a model (Section [[sec:multi_body_model]]) and add a model of the nano-hexapod to the model (Section [[sec:nano_hexapod_design]]) - The controller $K$ that will be designed in Section [[sec:robust_control_architecture]] @@ -1232,6 +1247,11 @@ A more complete study of the control of the NASS is performed [[https://tdehaeze ** General Conclusion +** Sensor Noise introduced by the Metrology +Say that is will introduce noise inside the bandwidth (100Hz) +This should not be significant. + + ** Further Work