diff --git a/figs/nano_hexapod_size.png b/figs/nano_hexapod_size.png new file mode 100644 index 0000000..8831152 Binary files /dev/null and b/figs/nano_hexapod_size.png differ diff --git a/figs/preumont18_cas_plot.png b/figs/preumont18_cas_plot.png new file mode 100644 index 0000000..df794dd Binary files /dev/null and b/figs/preumont18_cas_plot.png differ diff --git a/figs/taghirad13_stewart_schematic.png b/figs/taghirad13_stewart_schematic.png new file mode 100644 index 0000000..d9f46c4 Binary files /dev/null and b/figs/taghirad13_stewart_schematic.png differ diff --git a/index.html b/index.html index 951d391..7d7d860 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 @@ -35,119 +35,128 @@

Table of Contents

@@ -161,7 +170,7 @@ 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.

@@ -170,51 +179,51 @@ 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:

-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).

-
-

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. @@ -249,11 +258,11 @@ 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)
    • @@ -273,7 +282,7 @@ The dynamical blocks are:
-
+

classical_feedback_small.png

Figure 1: Block Diagram of a simple feedback system

@@ -295,11 +304,11 @@ 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: \[ \epsilon = \frac{1}{1 + GK} r + \frac{GK}{1 + GK} n - \frac{G_d}{1 + GK} d \]

@@ -340,8 +349,8 @@ Ideally, we would like 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

We have from the definition of \(S\) and \(T\) that: @@ -359,7 +368,7 @@ There is therefore a trade-off between the disturbance rejection and the meas

-Typical shapes of \(|S|\) and \(|T|\) as a function of frequency are shown in Figure 2. +Typical shapes of \(|S|\) and \(|T|\) as a function of frequency are shown in Figure 2. We can observe that \(|S|\) and \(|T|\) exhibit different behaviors depending on the frequency band:

@@ -385,7 +394,7 @@ We can observe that \(|S|\) and \(|T|\) exhibit different behaviors depending on -
+

h-infinity-2-blocs-constrains.png

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

@@ -393,11 +402,11 @@ 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

- +

@@ -418,11 +427,11 @@ 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.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

@@ -452,11 +461,11 @@ 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. @@ -467,19 +476,19 @@ 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

- +

@@ -508,11 +517,11 @@ 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

- +

@@ -537,31 +546,39 @@ And thus \(CPS_x(f)\) represents the power in the signal \(x\) for frequencies a

-The cumulative +The Cumulative Power Spectrum is generally shown as a function of frequency, and is used to identify the critical modes in a design, at which the effort should be targeted. +It can also helps to determine at which frequencies the effect of disturbances must 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. +A typical Cumulative Power Spectrum is shown in figure 4.

+ + +
+

preumont18_cas_plot.png +

+

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

+
-
-

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 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 5).

-
+

psd_lti_system.png

-

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

+

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

@@ -573,15 +590,15 @@ 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 6).

@@ -590,23 +607,23 @@ 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 \(v\)

+

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

-
-

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

@@ -630,25 +647,25 @@ 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 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 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 used: @@ -666,7 +683,7 @@ 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 7. The steps are:

    @@ -676,10 +693,10 @@ The steps are:
-
+

vibration_analysis_procedure.png

-

Figure 6: Vibration Analysis Procedure

+

Figure 7: Vibration Analysis Procedure

@@ -688,11 +705,11 @@ Instead, the model will be tuned using both the modal model and the response mod

-
-

2.1 Experimental Setup

+
+

2.1 Experimental Setup

- +

@@ -718,13 +735,13 @@ 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 7)
    • +
  • 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 8)
      • +
    • 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
      • @@ -740,26 +757,26 @@ It was chosen to have some redundancy in the measurement to be able to verify th

      -
      +

      hammer_z.gif

      -

      Figure 7: Example of one hammer impact

      +

      Figure 8: Example of one hammer impact

      -
      +

      accelerometers_ty_overview.jpg

      -

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

      +

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

      -
      -

      2.2 Results

      +
      +

      2.2 Results

      - +

      @@ -768,21 +785,21 @@ 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 9 and 10. +Example of the obtained micro-station’s mode shapes are shown in Figures 10 and 11.

      -
      +

      mode1.gif

      -

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

      +

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

      -
      +

      mode6.gif

      -

      Figure 10: Sixth mode

      +

      Figure 11: Sixth mode

      @@ -807,21 +824,21 @@ 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 11. +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 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

      +

      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

      -
      -

      2.3 Conclusion

      +
      +

      2.3 Conclusion

      @@ -829,7 +846,7 @@ 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.

      @@ -837,11 +854,11 @@ 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. @@ -855,13 +872,13 @@ Note that the low frequency disturbances such as static guiding errors and therm 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.

      @@ -869,11 +886,11 @@ The measurements are presented in more detail in -

      3.1 Ground Motion

      +
      +

      3.1 Ground Motion

      - +

      @@ -881,15 +898,15 @@ 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 12). +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 12: Huddle Test Setup

      +

      Figure 13: Huddle Test Setup

      @@ -898,19 +915,19 @@ 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

      +

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

      -
      -

      3.2 Stage Vibration - Effect of Control systems

      +
      +

      3.2 Stage Vibration - Effect of Control systems

      - +

      @@ -933,11 +950,11 @@ Complete reports on these measurements are accessible -

      3.3 Stage Vibration - Effect of Motion

      +
      +

      3.3 Stage Vibration - Effect of Motion

      - +

      We consider here the vibrations induced by scans of the translation stage and rotation of the spindle. @@ -948,18 +965,18 @@ 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 15.

      -
      +

      rz_meas_errors.gif

      -

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

      +

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

      @@ -972,7 +989,7 @@ 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 15. +The obtained Power Spectral Densities of the sample’s absolute velocity are shown in Figure 16.

    @@ -989,10 +1006,10 @@ Its cause has not been identified yet

-
+

sr_sp_psd_sample_compare.png

-

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

+

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

@@ -1004,40 +1021,40 @@ 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 16), 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 16: Y position of the translation stage measured by the encoders

+

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

-The time domain absolute vertical velocity of the sample and granite are shown in Figure 17. +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 17: Vertical velocity of the sample and marble when scanning with the translation stage

+

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 18. +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.

@@ -1055,20 +1072,20 @@ Thus, if the detector is only used in between the triangular peaks, the vibratio
-
+

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

+

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

-
-

3.4 Open Loop noise budgeting

+
+

3.4 Open Loop noise budgeting

- +

@@ -1076,7 +1093,7 @@ We can now compare the effect of all the disturbance sources on the position err

-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 20.

@@ -1084,26 +1101,26 @@ 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

+

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

+

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

-From Figure 20, 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.

@@ -1116,8 +1133,8 @@ 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. @@ -1137,8 +1154,8 @@ The detector requirement would need to have a sample frequency above \(400Hz\) a

-
-

3.6 Conclusion

+
+

3.6 Conclusion

@@ -1163,14 +1180,14 @@ 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).

@@ -1183,11 +1200,11 @@ A small summary of the multi-body Simscape is available -

4.1 Multi-Body model

+
+

4.1 Multi-Body model

- +

@@ -1211,23 +1228,23 @@ 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 21. +The 3D representation of the simscape model is shown in Figure 22.

-
+

simscape_picture.png

-

Figure 21: 3D representation of the simscape model

+

Figure 22: 3D representation of the simscape model

-
-

4.2 Validity of the model’s dynamics

+
+

4.2 Validity of the model’s dynamics

- +

@@ -1235,7 +1252,7 @@ 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 22. +The comparison of three of the Frequency Response Functions are shown in Figure 23.

@@ -1247,10 +1264,10 @@ 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.

+

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.

@@ -1280,11 +1297,11 @@ 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

- +

@@ -1292,7 +1309,7 @@ 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 23): +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. @@ -1306,10 +1323,10 @@ Both computation are performed
-
+

control-schematic-nass.png

-

Figure 23: Figure caption

+

Figure 24: Figure caption

@@ -1318,11 +1335,11 @@ More details about these computations are accessible -

4.4 Simulation of Experiments

+
+

4.4 Simulation of a Tomography Experiment

- +

@@ -1332,16 +1349,16 @@ 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 24. +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 25 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. @@ -1355,22 +1372,22 @@ The motion of the sample follows the wanted motion but with vibrations in the mi

    -
    +

    open_loop_sim.gif

    -

    Figure 24: Tomography Experiment using the Simscape Model

    +

    Figure 25: Tomography Experiment using the Simscape Model

    -
    +

    open_loop_sim_zoom.gif

    -

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

    +

    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 26. +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.

    @@ -1388,16 +1405,16 @@ The vertical rotation error is meaningless for two reasons:
-
+

exp_scans_rz_dist.png

-

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

+

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

-
-

4.5 Conclusion

+
+

4.5 Conclusion

@@ -1422,47 +1439,109 @@ 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:

    -
  • the effect of disturbances \(G_d\) (important for the rejection of disturbances)
    • -
  • the plant dynamics \(G\) (important for the control robustness properties)
    • +
  • the effect of disturbances
    • +
  • the plant dynamics

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 plant has nice dynamical properties for control (Section 5.3)
    • +
  • 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)

-In this study, the effect of the nano-hexapod’s mass characteristics is not taken into account because: +In this study, the effect of the nano-hexapod’s mass characteristics is not taken into account because it cannot be changed a lot and it is quite negligible compare the to metrology reflector and the payload’s masses that are fixed to nano-hexapod’s top platform.

-
    -
  1. it cannot be changed a lot
    2. -
  2. it is quite negligible compare the to metrology reflector and the payload’s masses that is fixed to nano-hexapod’s top platform
    3. -

Also, the effect of the nano-hexapod’s damping properties will be studied when applying active damping techniques.

-
-

5.1 Optimal Stiffness to reduce the effect of disturbances

+
+

5.1 A brief introduction to Stewart Platforms

- + +

+ +

+A typical Stewart platform is composed of two platforms connected by six identical struts (or legs) composed of: +

+
    +
  • a universal joint at one end
    • +
  • a spherical joint at the other end
    • +
  • a prismatic joint with an associated actuator
    • +
+ +

+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. +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. +

+ + +
+

stewart_architecture_example_pose.png +

+

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

+
+ +

+The Stewart Platform is very adapted for the NASS application for the following reasons: +

+
    +
  • it is a fully parallel manipulator, thus all the motions errors can be compensated
    • +
  • it is very compact compared to a serial manipulator
    • +
  • it has high stiffness and good dynamic performances
    • +
+ +

+The main disadvantage of Stewart platforms is the small workspace when compare the serial manipulators which is not a problem here. +

+ +

+A Matlab toolbox to study and design Stewart Platforms has been developed and used for the design of the nano-hexapod. +The source code is accessible here and the documentation here. +

+
+
+ +
+

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 \(G_d\)). @@ -1470,17 +1549,17 @@ For instance, it is quite obvious that a stiff nano-hexapod is better than a sof

-A complete study of the optimal nano-hexapod stiffness for the minimization of disturbance sensibility here and summarized below. +A study of the optimal nano-hexapod stiffness for the minimization of disturbance sensibility is accessible here and summarized below.

-
-

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. -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. +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.

@@ -1488,43 +1567,56 @@ 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

+

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

-
-

Sensibility to ground motion

-
+
+

Sensibility to ground motion

+

-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. +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 7.6 that using soft mounts for the granite can greatly lower the sensibility to ground motion. +It will be suggested in Section 7.3 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

+

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

-
-

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). +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. +

+ +

+To do so, the power spectral density of the disturbances should be taken into account, as the sensibility of one disturbance should be reduced only where the PSD of the considered disturbance is large compared to the other disturbances. +

+ +

+What is more important than comparing the sensitivity to disturbances, is to compare the resulting open-loop power spectral density of the sample’s position error with the change of the nano-hexapod’s stiffness. +This is the dynamic noise budgeting. +

+ + +

+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 32).

@@ -1532,34 +1624,39 @@ 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

+

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

+
+
+
+

Conclusion

+

-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 33, 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 be required 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

+

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

-
-

5.2 Optimal Stiffness to reduce the plant uncertainty

-
+
+

5.3 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. @@ -1570,36 +1667,35 @@ Therefore, we have to identify all changes that might occurs in the system and c The uncertainty in the system can be caused by:

    -
  • A change in the Support’s compliance (complete analysis here): if the micro-station dynamics is changing due to the change of parts or just because of aging effects, the feedback system should remains stable and the obtained performance should not change
    • +
  • A change in the Support’s compliance (complete analysis here): if the micro-station dynamics is changing due to the change of mechanical parts or just because of aging effects, the feedback system should remains stable and the obtained performance should not change
  • A change in the Payload mass/dynamics (complete analysis here): the sample’s mass is ranging from \(1\,kg\) to \(50\,kg\)
    • -
  • A change of experimental condition such as the micro-station’s pose or the spindle rotation (complete analysis here)
    • +
  • A change of experimental condition such as the micro-station’s pose or the spindle rotation speed (complete analysis here)

-Because of the trade-off between robustness and performance, the bigger the plant dynamic uncertainty, the lower the simultaneous attainable performance is for all the plants. -Thus, all these uncertainties will limit the attainable bandwidth and hence the obtained performance. +Because of the trade-off between robustness and performance, the bigger the plant dynamic uncertainty, the lower the attainable performance will be for all the system changes.

-In the next sections, the effect the considered changes on the plant dynamics is quantified and conclusions are made on the optimal stiffness for robustness properties. +In the next sections, the effect the considered changes on the plant dynamics is quantified and conclusions are drawn on the optimal stiffness for robustness properties.

-In the following study, when we refer to plant dynamics, this means the dynamics from forces applied by the nano-hexapod to the measured sample’s position by the metrology. +In the following study, when we refer to plant dynamics, this means the dynamics from forces applied by the nano-hexapod to the measured sample’s position by the metrology. We will only compare the plant dynamics as it is the most important dynamics for robustness and performance properties. However, the dynamics from forces to sensors located in the nano-hexapod legs, such as force and relative motion sensors, have also been considered in a separate study.

-
-

Effect of Payload

-
+
+

Effect of Payload

+

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

-In Figure 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 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.

@@ -1608,23 +1704,29 @@ One can see that for the soft nano-hexapod:

  • the first resonance (suspension mode of the nano-hexapod) is lowered with an increase of the sample’s mass. -This first resonance corresponds to \(\omega = \sqrt{\frac{k_n}{m_n + m_s}}\) where \(k_n\) is the vertical nano-hexapod stiffness, \(m_n\) the mass of the nano-hexapod’s top platform, and \(m_s\) the sample’s mass
    • -
  • the gain after the first resonance and up until the anti-resonance at 100Hz is changing with the sample’s mass
    • +This is very logical as the first resonance corresponds to \(\omega = \sqrt{\frac{k_n}{m_n + m_s}}\) where \(k_n\) is the vertical nano-hexapod stiffness, \(m_n\) the mass of the nano-hexapod’s top platform, and \(m_s\) the sample’s mass +
  • the gain after the first resonance and up until the anti-resonance at 100Hz is changing with the sample’s mass. +It is indeed equal to \(\frac{1}{(m_n + m_s) \omega^2}\)

For the stiff-nano-hexapod, the change of payload mass has very little effect (the vertical scale for the amplitude is quite small).

+

+To minimize the uncertainty to the payload’s mass, the mass of the nano-hexapod’s top platform plus the metrology reflector should be maximized, and ideally close to the maximum payload’s mass. +As the maximum payload’s mass is \(50\,kg\), this may however not be practical, and thus the control architecture must be developed to be robust to a change of the payload’s mass. +

-
+ +

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)

+

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 32 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.

@@ -1633,15 +1735,14 @@ 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

+

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 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 36.

@@ -1649,7 +1750,7 @@ For nano-hexapod stiffnesses below \(10^6\,[N/m]\):

  • the phase stays between 0 and -180deg which is a very nice property for control
    • -
  • the dynamical change up until the resonance of the payload is mostly a change of gain
    • +
  • the dynamical change up until the resonance of the payload can be considered as a change of gain
@@ -1658,21 +1759,20 @@ For nano-hexapod stiffnesses above \(10^7\,[N/m]\):

  • the dynamics is unchanged until the first resonance which is around 25Hz-35Hz
    • -
  • above that frequency, the change of dynamics is quite chaotic (we will see in the next section that this is due to the micro-station dynamics)
    • +
  • above that frequency, the change of dynamics is quite chaotic (we will see in the next section that this is due to the micro-station dynamics) and it would be difficult to have a controller with high bandwidth which is robust to such change of dynamics
-
+

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

+

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

-For soft nano-hexapods, the payload has an important impact on the dynamics. -This will have to be carefully taken into account for the controller design. +For soft nano-hexapods, the payload has an important impact on the dynamics that will have to be carefully taken into account for the controller design.

@@ -1684,24 +1784,24 @@ If possible, the first resonance frequency of the payload should be maximized (s

-Heavy samples with low first resonance mode will be very problematic. +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
  • a change in the position of one stage. For instance, a large displacement of the micro-hexapod can change the micro-station compliance
    • -
  • a change in a control loop
    • +
  • a change in a control loop of some of its stage
@@ -1716,10 +1816,10 @@ 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, the plant dynamics is identified in two different cases (Figure 37):

    -
  • without the micro-station (solid curves)
    • +
  • with a micro-station considered as a solid body (solid curves)
  • with the micro-station dynamics (dashed curves)
@@ -1732,10 +1832,10 @@ 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

+

Figure 37: Change of dynamics from force \(\mathcal{F}_x\) to displacement \(\mathcal{X}_x\) due to the micro-station compliance

@@ -1752,15 +1852,15 @@ 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 varying from 0rpm up to 60rpm are identified and shown in Figure 38.

@@ -1771,11 +1871,16 @@ One can see that for nano-hexapods with a stiffness above \(10^5\,[N/m]\), the d For very soft nano-hexapods, the main resonance is split into two resonances and one anti-resonance that are all moving at a function of the rotating speed.

+

+The change of dynamics is due to both centrifugal forces and Coriolis forces. +This effect has been studied in details in this document. +

-
+ +

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

+

Figure 38: Change of dynamics from force \(\mathcal{F}_x\) to displacement \(\mathcal{X}_x\) for a spindle rotation speed from 0rpm to 60rpm

@@ -1791,41 +1896,42 @@ 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 “spread” of the plant dynamics for all the nano-hexapod stiffnesses (Figure 39). 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

+

Figure 39: Variability of the dynamics from \(\bm{\mathcal{F}}_x\) to \(\bm{\mathcal{X}}_x\) with varying nano-hexapod stiffness

+
+
+
+

Conclusion

+

-Let’s summarize the findings: +Let’s summarize the findings about the effect of the nano-hexapod’s stiffness on the plant uncertainty:

    -
  • the payload’s mass influence the plant dynamics above the first resonance of the nano-hexapod. +
  • The payload’s mass influence the plant dynamics above the first resonance of the nano-hexapod. Thus a high nano-hexapod stiffness helps reducing the effect of a change of the payload’s mass
    • -
  • the payload’s first resonance is seen as an anti-resonance in the plant dynamics. -As this effect will largely be variable from one payload to the other, the payload’s first resonance should be maximized (above 300Hz if possible) for all used payloads
    • -
  • the dynamics of the nano-hexapod is not affected by the micro-station dynamics (compliance) when \(k < 10^6\,[N/m]\)
    • -
  • the spindle’s rotating speed has no significant influence on the plant dynamics for nano-hexapods with a stiffness \(k > 10^5\,[N/m]\)
    • +
  • The payload’s first resonance is seen as an anti-resonance in the plant dynamics. +As this effect will largely be variable from one payload to the other, thus the payload’s first resonance should be maximized (above 300Hz if possible) for all used payloads
    • +
  • The dynamics of the nano-hexapod is not affected by the micro-station dynamics (compliance) if \(k < 10^6\,[N/m]\)
    • +
  • The spindle’s rotating speed has no significant influence on the plant dynamics for nano-hexapod’s stiffnesses \(k > 10^5\,[N/m]\)

-Concerning the plant dynamic uncertainty, the resonance frequency of the nano-hexapod should be between 5Hz (way above the maximum rotating speed) and 50Hz (before the first micro-station resonance) for all the considered payloads. -This corresponds to an optimal nano-hexapod leg stiffness in the range \(10^5 - 10^6\,[N/m]\). -

- -

-In such case, the main limitation will be heavy samples with small stiffnesses. +The resonance frequency of the nano-hexapod should be between 5Hz (way above the maximum rotating speed) and 50Hz (before the first micro-station resonance) for all the considered payloads. +This corresponds to an optimal nano-hexapod leg stiffness in the range \(10^5 - 10^6\,[N/m]\).

@@ -1833,45 +1939,97 @@ In such case, the main limitation will be heavy samples with small stiffnesses.
-
-

5.3 Optimal Nano-Hexapod Geometry

-
+
+

5.4 Optimal Nano-Hexapod Geometry

+

- +

-As will be shown in this section, the Nano-Hexapod geometry has an influence on: +Stewart platforms can be studied with:

    -
  • the overall stiffness/compliance
    • +
  • Kinematic analysis: study of the geometry of motion without considering the forces and torques that cause the motion
    • +
  • Jacobian analysis: reveals the relation between the actuators velocities and the moving platform linear and angular velocities. It also constructs the transformation between actuator forces and task space forces
    • +
+

+and moments acting on the moving platform. +

+
    +
  • Dynamic analysis: consists of the equations of motion for the manipulator which are quite complex to derive
    • +
+ +

+Kinematic and Jacobian analysis are briefly introduced in this section, however the dynamic analysis is not performed analytically but rather studied using the Simscape model. +

+ +

+As will be shown, the Nano-Hexapod geometry has an influence on: +

+
    +
  • the stiffness and compliance properties
  • the mobility
    • -
  • the dynamics and coupling
    • +
  • the force authority
    • +
  • the dynamics of the manipulator
-
-

Kinematic Analysis and the Jacobian Matrix

-
+
+

Kinematic Analysis

+

-The kinematic analysis of a parallel manipulator is well described in taghirad13_paral: +The Kinematic analysis of the Stewart platform can be divided into two problems: the inverse kinematics and the forward kinematics.

+

-Kinematic analysis refers to the study of the geometry of motion of a robot, without considering the forces an torques that cause the motion. -In this analysis, the relation between the geometrical parameters of the manipulator with the final motion of the moving platform is derived and analyzed. +For inverse kinematic analysis, it is assumed that the position \({}^A\bm{P}\) and orientation of the moving platform \({}^A\bm{R}_B\) are given and the problem is to obtain the joint variables \(\bm{\mathcal{L}} = \left[ l_1, l_2, l_3, l_4, l_5, l_6 \right]^T\).

-

-One of the main analysis tool for the Kinematic analysis is the Jacobian Matrix that not only reveals the relation between the joint variable velocities of a parallel manipulator to the moving platform linear and angular velocities, but also constructs the transformation needed to find the actuator forces from the forces and moments acting on the moving platform. -

- - -

-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 flexible joints (described by the position vectors \({}^A\bm{b}_i\)): +This problem can be easily solved, and the obtain joint variables are:

\begin{equation*} + \begin{aligned} + l_i = &\Big[ {}^A\bm{P}^T {}^A\bm{P} + {}^B\bm{b}_i^T {}^B\bm{b}_i + {}^A\bm{a}_i^T {}^A\bm{a}_i - 2 {}^A\bm{P}^T {}^A\bm{a}_i + \dots\\ + &2 {}^A\bm{P}^T \left[{}^A\bm{R}_B {}^B\bm{b}_i\right] - 2 \left[{}^A\bm{R}_B {}^B\bm{b}_i\right]^T {}^A\bm{a}_i \Big]^{1/2} + \end{aligned} +\end{equation*} + +

+If the position and orientation of the platform lie in the feasible workspace, the solution is unique. +Otherwise, the solution gives complex numbers. +

+ +

+This means that from the wanted position of the nano-hexapod’s mobile platform with respect to the fixed platform (described by \({}^A\bm{P}\) and \({}^A\bm{R}_B\)) and for a specific geometry (position of the top joints \(^{B}\bm{b}\) and bottom joints \({}^A\bm{a}\)), the required motion of each leg can easily be determined. +

+ + +
+

+In forward kinematic analysis, it is assumed that the vector of limb lengths \(\bm{L}\) is given and the problem is to find the position \({}^A\bm{P}\) and the orientation \({}^A\bm{R}_B\). +

+
+ +

+This is a difficult problem that requires to solve nonlinear equations. +

+ +

+However, as will be shown in the next section, approximate solution of the forward kinematic analysis can be obtained thanks to the 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 \(\{A\}\): +

+\begin{equation} \bm{J} = \begin{bmatrix} {\hat{\bm{s}}_1}^T & (\bm{b}_1 \times \hat{\bm{s}}_1)^T \\ {\hat{\bm{s}}_2}^T & (\bm{b}_2 \times \hat{\bm{s}}_2)^T \\ @@ -1880,29 +2038,36 @@ The Jacobian matrix \(\bm{J}\) can be computed form the orientation of the legs {\hat{\bm{s}}_5}^T & (\bm{b}_5 \times \hat{\bm{s}}_5)^T \\ {\hat{\bm{s}}_6}^T & (\bm{b}_6 \times \hat{\bm{s}}_6)^T \end{bmatrix} -\end{equation*} +\end{equation} +

-It can be easily shown that: +It can be shown that the Jacobian matrix reveals the relation between the legs’ velocities to the moving platform linear and angular velocities:

\begin{equation} - \delta\bm{\mathcal{L}} = \bm{J} \delta\bm{\mathcal{X}}, \quad \delta\bm{\mathcal{X}} = \bm{J}^{-1} \delta\bm{\mathcal{L}} \label{eq:jacobian_L} + \dot{\bm{\mathcal{L}}} = \bm{J} \dot{\bm{\mathcal{X}}} \label{eq:jacobian_velocity} \end{equation}

with:

    -
  • \(\delta\bm{\mathcal{L}} = [ \delta l_1, \delta l_2, \delta l_3, \delta l_4, \delta l_5, \delta l_6 ]^T\): the vector of small legs’ displacements
    • -
  • \(\delta \bm{\mathcal{X}} = [\delta x, \delta y, \delta z, \delta \theta_x, \delta \theta_y, \delta \theta_z ]^T\): the vector of small mobile platform displacements
    • +
  • \(\dot{\bm{\mathcal{L}}} = [ \dot{l}_1, \dot{l}_2, \dot{l}_3, \dot{l}_4, \dot{l}_5, \dot{l}_6 ]^T\): relative velocity of each strut
    • +
  • \(\dot{\bm{X}} = [^A\bm{v}_p, {}^A\bm{\omega}]^T\): output twist vector of the mobile platform

-Thus, from a wanted small displacement \(\delta \bm{\mathcal{X}}\), it is easy to compute the required displacement of the legs \(\delta \bm{\mathcal{L}}\). -Similarly, from a measurement of the legs’ displacement, it is easy to compute the resulting platform’s motion. +For small legs motions \(\delta\bm{\mathcal{L}}\) and small mobile platform motion \(\delta \bm{\mathcal{X}}\), the following approximation can be computed from Eq. \eqref{eq:jacobian_velocity}: +

+\begin{equation} + \delta\bm{\mathcal{L}} \approx \bm{J} \delta\bm{\mathcal{X}}, \quad \delta\bm{\mathcal{X}} \approx \bm{J}^{-1} \delta\bm{\mathcal{L}} \label{eq:jacobian_L} +\end{equation} + +

+Equations \eqref{eq:jacobian_L} can be used to approximate the forward and inverse kinematic problems for small displacements.

-This will be used to estimate the platform’s mobility from the stroke of the legs, or inversely, to estimate the required stroke of the legs from the wanted platform’s mobility. +This approximation will be used to estimate the platform’s mobility from the legs’ stroke, or inversely, to estimate the required stroke of the legs from the wanted platform’s mobility.

@@ -1910,8 +2075,9 @@ Note that Eq. \eqref{eq:jacobian_L} is an approximation and is only valid for le

+

-It can also be shown that: +It can also be shown that the Jacobian matrix links the actuator forces to forces and moments acting on the moving platform:

\begin{equation} \bm{\mathcal{F}} = \bm{J}^T \bm{\tau}, \quad \bm{\tau} = \bm{J}^{-T} \bm{\mathcal{F}} \label{eq:jacobian_F} @@ -1925,188 +2091,410 @@ with:

-And thus the Jacobian matrix can be used to compute the forces that should be applied on each leg from forces and torques that we want to apply on the top platform. +And thus the Jacobian matrix can be used to compute the forces that should be applied on each leg from forces and torques that we want to apply on the nano-hexapod’s top platform.

-

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

-
-

Stiffness and Compliance matrices

-
-\begin{equation*} - \bm{\mathcal{F}} = \bm{K} \delta \bm{\mathcal{X}} -\end{equation*} - -\begin{equation*} - \bm{K} = \bm{J}^T \mathcal{K} \bm{J} -\end{equation*} -\begin{equation*} - \bm{C} = \bm{K}^{-1} = (\bm{J}^T \mathcal{K} \bm{J})^{-1} -\end{equation*} - +
+

Mobility of the Stewart Platform

+

-Stiffness properties is estimated from the architecture and leg’s stiffness -

- - - - -

-Kinematic Study https://tdehaeze.github.io/stewart-simscape/kinematic-study.html -

-
-
- - -
-

Mobility of the Stewart Platform

-
-

-For a specified geometry and actuator stroke, the mobility of the Stewart platform can be estimated. +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 37. +An example of the mobility considering only pure translations is shown in Figure 40.

-
+

mobility_translations_null_rotation.png

-

Figure 37: Obtained mobility of a Stewart platform for pure translations (the platform’s orientation is fixed)

-
-
+

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

-
-

Flexible Joints

-
+

-Active Damping Study https://tdehaeze.github.io/stewart-simscape/control-active-damping.html +From a wanted mobility and a specific geometry, the required actuator stroke can be estimated.

+

+Suppose we want the following mobility: +

    -
  • Advantages compared to conventional joints
    • -
  • Simulations will help determine the required rotational stroke and will help with the design
    • -
  • Typical joint stiffness is included in the model
    • +
  • x, y and z translations up to \(50\,\mu m\)
    • +
  • x and y rotation up to \(30\,\mu rad\)
    • +
  • no z rotation

-Flexible Joint stiffness => not problematic for the chosen active damping technique +The geometry is chosen arbitrary and corresponds to the wanted nano-hexapod size.

- - +

+If only pure translations and pure rotations are considered, the required actuator stroke is \(76 \mu m\), whereas if combined translations and rotations are considered, the required actuator stroke is \(177 \mu m\). +

+ +

+This gives an idea of the relation between the mobility and the actuator stroke. +

+ + + +
+

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 \(\tau_i\). +

+ +

+The stiffness of the actuator \(k_i\) links the applied (constant) actuator force \(\delta \tau_i\) and the corresponding small deflection \(\delta l_i\): +

+\begin{equation*} + \tau_i = k_i \delta l_i, \quad i = 1,\ \dots,\ 6 +\end{equation*} +

+If we combine these 6 relations: +

+\begin{equation*} + \bm{\tau} = \mathcal{K} \delta \bm{\mathcal{L}} \quad \mathcal{K} = \text{diag}\left[ k_1,\ \dots,\ k_6 \right] +\end{equation*} + +

+Equations \eqref{eq:jacobian_L} and \eqref{eq:jacobian_F} are used to obtain: +

+\begin{equation} + \bm{\mathcal{F}} = \bm{K} \delta \bm{\mathcal{X}} +\end{equation} +

+with the stiffness matrix +

+\begin{equation} + \bm{K} = \bm{J}^T \mathcal{K} \bm{J} \label{eq:jacobian_K} +\end{equation} + +

+And the compliance matrix can be computed by taking the inverse of the Stiffness matrix: +

+\begin{equation*} + \bm{C} = \bm{K}^{-1} = (\bm{J}^T \mathcal{K} \bm{J})^{-1} +\end{equation*} +

+The compliance matrix of a manipulator shows the mapping of the moving platform wrench applied at \(\bm{O}_B\) to its small deflection by +

+\begin{equation*} + \delta \bm{\mathcal{X}} = \bm{C} \cdot \bm{\mathcal{F}} +\end{equation*} + +

+Stiffness properties of the Stewart platform can then be estimated from the architecture (through the Jacobian matrix) and leg’s stiffness. +

+
+
+ +
+

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. +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. +

+ +

+The nano-hexapod geometry and further be optimized in terms of stiffness and stroke using the presented tools. +

+ +
Table 1: Stiffness of unversal and sperical flexible joints yang19_dynam_model_decoup_contr_flexib
+++ + + + + + + + - - + + + - - + + + - - + + + - - + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Table 1: Effect of a change in geometry on the manipulator’s stiffness, force authority and stroke
 legs pointing more verticallylegs further apart
\(k_{\theta u},\ k_{\psi u}\)\(72 Nm/rad\)Vertical stiffness\(\nearrow\)\(=\)
\(k_{\theta s}\)\(51 Nm/rad\)Horizontal stiffness\(\searrow\)\(=\)
\(k_{\psi s}\)\(62 Nm/rad\)Vertical rotation stiffness\(\searrow\)\(\nearrow\)
\(k_{\gamma s}\)\(64 Nm/rad\)Horizontal rotation stiffness\(\nearrow\)\(\nearrow\)
Vertical force authority\(\nearrow\)\(=\)
Horizontal force authority\(\searrow\)\(=\)
Vertical torque authority\(\searrow\)\(\nearrow\)
Horizontal torque authority\(\nearrow\)\(\nearrow\)
Vertical stroke\(\searrow\)\(=\)
Horizontal stroke\(\nearrow\)\(=\)
Vertical rotation stroke\(\nearrow\)\(\searrow\)
Horizontal rotation stroke\(\searrow\)\(\searrow\)
- - -
-

preumont07_flexible_joints.png -

-

Figure 38: Flexible joints used in preumont07_six_axis_singl_stage_activ

-
- - -
-

yang19_flexible_joints.png -

-

Figure 39: An alternative type of flexible joints that has been used for Stewart platforms yang19_dynam_model_decoup_contr_flexib

-
- -
-

Cubic Architecture

-
+
+

Cubic Architecture

+

-Study of cubic architecture https://tdehaeze.github.io/stewart-simscape/cubic-configuration.html -Has some advantages such as uniform stiffness and uniform mobility. -It can have very nice properties in specific conditions that will not be the case for this application. -The cubic configuration also puts much restriction on the position and orientation of each leg. -This configuration is such not recommended. +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).

-
+

3d-cubic-stewart-aligned.png

-

Figure 40: Figure caption

-
-
+

Figure 41: Schematic representation of the Cubic architecture

-
-

Conclusion

-
-
-
-
-
-

5.4 Conclusion

-
+

+This architecture provides some advantages such as uniform stiffness and uniform mobility in all directions. +It can also have very nice dynamical properties in specific conditions (center of mass of the payload located at the cube’s center) that are not met for the NASS. +

+ +

+The cubic configuration also puts much restriction on the position and orientation of the legs as there is only one design variable: the size of the cube. +

+

-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. +For these reasons, the cubic configuration is not recommended for the nano-hexapod. +

+ +
+ +

+Separate study of the cubic architecture is performed here. +

+
+
+ +
+

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. +

+ +

+When only small stroke is required, flexible joints can be used: material is bend to achieve motion, rather than relying on sliding or rolling across two surfaces. +

+ +

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

+ + +
+

preumont07_flexible_joints.png +

+

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

+
+ +

+The flexible joints have few advantages compared to conventional joints such as the absence of wear, friction and backlash which allows extremely high-precision (predictable) motion. +

+ +

+The parasitic bending and torsional stiffness of these joints usually induce some limitation on the control performance. +

+ +

+This has been studied using the Simscape model (report available here), and conclusions on the required characteristics of the flexible joints are summarized below. +

+ + +

+The bending and torsional stiffnesses somehow adds a parasitic stiffness in parallel with the struts. +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). +

+ + +

+The finite axial stiffness of the flexible joints can however be very problematic for control. +Small values of the axial stiffness are shown to limit the achievable damping using Direct Velocity Feedback. +The axial stiffness should therefore be maximized and taken into account in the model of the nano-hexapod. +

+ + +

+For the identified optimal actuator stiffness \(k = 10^5\,[N/m]\), the flexible joint should have the following stiffness properties: +

+
    +
  • Axial Stiffness: \(K_a > 10^7\,[N/m]\)
    • +
  • Bending Stiffness: \(K_b < 50\,[Nm/rad]\)
    • +
  • Torsion Stiffness: \(K_t < 50\,[Nm/rad]\)
    • +
+ +

+These requirements are easily obtained in practice. +For instance, the flexible joint used for the ID16 nano-hexapod have the following stiffness properties: +

+
    +
  • Axial Stiffness: \(K_a = 6 \cdot 10^7\,[N/m]\)
    • +
  • Bending Stiffness: \(K_b = 15\,[Nm/rad]\)
    • +
  • Torsion Stiffness: \(K_t = 20\,[Nm/rad]\)
    • +
+ +
+

+Even though much attention should be paid on the proper design of the flexible joints, they should not impose limitation on the performance of the system. +

+ +

+Simulations will help determine the required rotational stroke of the flexible joints and will help with the design. +

+ +
+
+
+ +
+

Conclusion

+
+
+

+Relations between the geometry of the Stewart platform and its characteristics such as stiffness, maneuverability and force authority have been derived. +

+ +

+These relations can help optimize the Stewart platform’s geometry, however, the choice of the geometry is quite constrained by the limited size of the hexapod, the size of the flexible joints and of the included actuators and sensors. +

+ +

+The effects of flexible joints stiffness on the dynamics have been studied and requirements on the flexible joints have been derived. +

+ +
+
+
+
+ +
+

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. As the high frequency vibrations are the most difficult to compensate for when using feedback control, a soft hexapod will most certainly helps improving the performances.

-In Section 5.2, we concluded that a nano-hexapod leg stiffness in the range \(10^5 - 10^6\,[N/m]\) is a good compromise between the uncertainty induced by the micro-station dynamics and by the rotating speed. -Provided that the samples used have a first mode that is sufficiently high in frequency, the total plant dynamic uncertainty should be manageable. +In Section 5.3, 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 by the control.

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

A more detailed study of the determination of the optimal stiffness based on all the effects is available here.

+

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

+
-
-

6 Robust Control Architecture

+
+

6 Robust Control Architecture

- +

Before designing the control system, let’s summarize what has been done: @@ -2133,8 +2521,8 @@ 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. @@ -2150,7 +2538,7 @@ 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 41. +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:

@@ -2162,27 +2550,27 @@ This approach has the following advantages:
-
+

control_architecture_hac_lac_one_input.png

-

Figure 41: HAC-LAC Architecture with a system having only one input

+

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

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

- +

Depending on the chosen active damping technique, either force sensors, relative motion sensors or inertial sensors should be included in each of the nano-hexapod’s legs. @@ -2203,28 +2591,28 @@ It would also be difficult to apply in a robust way due to the non-collocation w

-
-

Effect of the Spindle’s Rotation

-
+
+

Effect of the Spindle’s Rotation

+
-
+

dvf_root_locus_ws.png

-

Figure 42: Figure caption

+

Figure 45: Figure caption

-
+

iff_root_locus_ws.png

-

Figure 43: Figure caption

+

Figure 46: Figure caption

-
-

Relative Direct Velocity Feedback Architecture

-
+
+

Relative Direct Velocity Feedback Architecture

+

Active Damping can help:

@@ -2235,11 +2623,11 @@ Active Damping can help:

-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 44). +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 47).

-The signals shown in Figure 44 are: +The signals shown in Figure 47 are:

  • \(\bm{\tau}\): Actuator forces applied in each leg
    • @@ -2255,10 +2643,10 @@ The force applied in each leg being proportional to the relative velocity of the

    -
    +

    control_architecture_dvf.png

    -

    Figure 44: Low Authority Control: Decentralized Direct Velocity Feedback

    +

    Figure 47: Low Authority Control: Decentralized Direct Velocity Feedback

    @@ -2268,29 +2656,29 @@ This may not be the optimal choice as will be further explained.

-
-

Effect of Active Damping on the Primary Plant Dynamics

-
+
+

Effect of Active Damping on the Primary Plant Dynamics

+

-The plant dynamics before (solid curves) and after (dashed curves) the Low Authority Control implementation are compared in Figure 45. +The plant dynamics before (solid curves) and after (dashed curves) the Low Authority Control implementation are compared in Figure 48. 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 45: Primary plant in the space of the legs with (dashed) and without (solid) Direct Velocity Feedback

+

Figure 48: Primary plant in the space of the legs with (dashed) and without (solid) Direct Velocity Feedback

-
-

Effect of Active Damping on the Sensibility to Disturbances

-
+
+

Effect of Active Damping on the Sensibility to Disturbances

+

-The change of sensibility to disturbances with the use of DVF is shown in Figure 46. +The change of sensibility to disturbances with the use of DVF is shown in Figure 49. 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.

@@ -2299,29 +2687,29 @@ This is probably not the optimal gain that could have been used, and further ana

-
+

opt_stiff_sensibility_dist_dvf.png

-

Figure 46: Norm of the transfer function from vertical disturbances to vertical position error with (dashed) and without (solid) Direct Velocity Feedback applied

+

Figure 49: Norm of the transfer function from vertical disturbances to vertical position error with (dashed) and without (solid) Direct Velocity Feedback applied

-
-

6.3 High Authority Control

+
+

6.3 High Authority Control

- +

-The complete HAC-LAC architecture is shown in Figure 47 where an outer loop is added to the decentralized direct velocity feedback loop. +The complete HAC-LAC architecture is shown in Figure 50 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.

@@ -2334,10 +2722,10 @@ Then, a diagonal controller \(\bm{K}_\mathcal{L}\) generates the required force

-
+

control_architecture_hac_dvf_pos_L.png

-

Figure 47: 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

+

Figure 50: 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

@@ -2345,32 +2733,32 @@ 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 48. +The plant dynamics for each of the six legs and for the three payload’s masses is shown in Figure 51. 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 48: Diagonal elements of the transfer function matrix from \(\bm{\tau}^\prime\) to \(\bm{\epsilon}_{\mathcal{X}_n}\) for the three considered masses

+

Figure 51: Diagonal elements of the transfer function matrix from \(\bm{\tau}^\prime\) to \(\bm{\epsilon}_{\mathcal{X}_n}\) for the three considered masses

The diagonal controller \(\bm{K}_\mathcal{L}\) is then tuned in such a way that the control bandwidth is around 100Hz and such that enough stability margins are obtained for all the payload’s masses used. -The obtained loop gain is shown in Figure 49. +The obtained loop gain is shown in Figure 52.

-
+

opt_stiff_primary_loop_gain_L.png

-

Figure 49: Loop gain for the primary plant

+

Figure 52: Loop gain for the primary plant

-The sensibility to disturbance after the use of HAC-LAC control is shown in Figure 50. +The sensibility to disturbance after the use of HAC-LAC control is shown in Figure 53. The change of sensibility is very typical for feedback system:

    @@ -2388,30 +2776,30 @@ This should gives slightly better performance and robustness, but should not cha

    -
    +

    opt_stiff_primary_control_L_senbility_dist.png

    -

    Figure 50: Sensibility to disturbances when the HAC-LAC control is applied (dashed) and when it is not (solid)

    +

    Figure 53: Sensibility to disturbances when the HAC-LAC control is applied (dashed) and when it is not (solid)

-
-

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 51 and the Cumulative Amplitude Spectrum is shown in Figure 52. +The Power Spectral Density of the sample’s position error is plotted in Figure 54 and the Cumulative Amplitude Spectrum is shown in Figure 55. 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.

@@ -2440,77 +2828,85 @@ 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 51: Amplitude Spectral Density of the position error in Open Loop and with the HAC-LAC controller

+

Figure 54: Amplitude Spectral Density of the position error in Open Loop and with the HAC-LAC controller

-
+

opt_stiff_hac_dvf_L_cas_disp_error.png

-

Figure 52: Cumulative Amplitude Spectrum of the position error in Open Loop and with the HAC-LAC controller

+

Figure 55: Cumulative Amplitude Spectrum of the position error in Open Loop and with the HAC-LAC controller

-The time domain sample’s vibrations are shown in Figure 53. +The time domain sample’s vibrations are shown in Figure 56. 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 54 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 57 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 26 (same scale was used for both animations).

-
+

opt_stiff_hac_dvf_L_pos_error.png

-

Figure 53: Position Error of the sample during a tomography experiment when no control is applied and with the HAC-DVF control architecture

+

Figure 56: 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 54: 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 \(\approx 10 \mu m\))

+

Figure 57: Tomography Experiment using the Simscape Model in Closed Loop with the HAC-LAC Control - Zoom on the sample’s position (the full vertical scale is \(\approx 10 \mu m\))

-
-

6.5 Simulation of More Complex Experiments

+
+

6.5 Simulation of More Complex Experiments

-
-

Micro-Hexapod offset

-
+
+

Micro-Hexapod offset

+
-
+

tomography_dh_offset.gif

-

Figure 55: Top View of a tomography experiment with a 10mm offset imposed by the micro-hexapod

+

Figure 58: Top View of a tomography experiment with a 10mm offset imposed by the micro-hexapod

+ +

+The force applied by the actuators are compensating the centripetal force induced by the spindle’s rotation. +

+ +

+This centrifugal force 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 \(F = m r \omega^2 \approx 12 \cdot 0.01 \cdot (2\pi)^2 \approx 5\,[N]\) +

-
-

Simultaneous Translation Scans

-
+
+

Simultaneous Translation Scans

+
-
+

ty_scans.gif

-

Figure 56: Top View of a tomography experiment combined with translation scans

+

Figure 59: Top View of a tomography experiment combined with translation scans

-
-

6.6 Conclusion

+
+

6.6 Conclusion

@@ -2545,26 +2941,67 @@ 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 Nano-Hexapod Specifications

+
+

7.1 Nano-Hexapod Specifications

+
+

+Table summarizing the nano-hexapod wanted characteristics: +

+ + +
    +
  • Dimensions (Figure 60)
    • +
  • Stiffness: +
      +
    • Resonances should be between 5Hz and 50Hz
      • +
    • +
    • +
  • Flexible joints: +
      +
    • To be optimized such that the above resonances are in the required range
      • +
    • + +
    • Required angular stroke: can be estimated with simulations
      • +
    • +
  • Force +
      +
    • Weight: \(60\,kg \rightarrow 600\,N \rightarrow 60\,N\) on each actuator
      • +
    • Dynamic: few Newtons
      • +
    • +
  • Estimation of the required stroke: +
      +
    • From simulation (i.e. for disturbance rejection alone), \(\pm 5 \mu m\) is good
      • +
    • However, the required stroke probably depends on two other factors: +
        +
      • Static positioning errors of the stages
        • +
      • Maximum tracking errors of the stages (mainly translation stage and tilt stage). +This is probably more difficult to obtain. +However, by limiting the acceleration of these stages, we may limit the dynamic tracking errors to acceptable levels
        • +
      • +
    • +
  • Sensors to be included
    • +
+ + +
+

nano_hexapod_size.png +

+

Figure 60: First implementation of the nano-hexapod / deflector and coolsed sample plate support

+
+
-
-

7.2 General Conclusion

-
- - -
-

7.3 Sensor Noise introduced by the Metrology

-
+
+

7.2 Sensor Noise introduced by the Metrology

+

Say that is will introduce noise inside the bandwidth (100Hz) This should not be significant. @@ -2573,27 +3010,18 @@ This should not be significant.

-
-

7.4 Further Work

-
- -
-

7.5 Cable Forces

-
- - -
-

7.6 Using soft mounts for the Granite

-
+
+

7.3 Using soft mounts for the Granite

+

- +

-
+

opt_stiff_soft_granite_Dw.png

-

Figure 57: Change of sensibility to Ground motion when using a stiff Granite (solid curves) and a soft Granite (dashed curves)

+

Figure 61: Change of sensibility to Ground motion when using a stiff Granite (solid curves) and a soft Granite (dashed curves)

@@ -2606,13 +3034,9 @@ Sensible to detector motion?

-
-

7.7 Others

-
-

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

- +
+

7.4 Others Factors that may limit the performances

+

Cable forces?

@@ -2620,26 +3044,77 @@ Cable forces?

Slip-Ring noise?

+ +

+Metrology Reflector: +

+
    +
  • it is very important that the first resonance of the metrology frame is high enough
    • +
  • it should be above 200Hz
    • +
+ + +

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

+
+
+ +
+

7.5 Other Notes

+
+
+
+

7.5.1 Modification of the Granite

+
+

+Possible to lower the granite? +

+ +

+This can help the mechanical design and make the nano-hexapod stiffer. +

+ +

+The problem is that the Tilt stage rotation axis will not be position correctly anymore. +

+
+
+ +
+

7.5.2 Sample trajectories

+
+

+More precise definition of the sample trajectories? +

+
    +
  • Simultaneous Ty and Rz scans at what speed?
    • +
+
+

7.6 General Conclusion

+
+
+ +

Bibliography

Date: 05-2020

Author: Thomas Dehaeze

-

Created: 2020-04-30 jeu. 15:44

+

Created: 2020-05-05 mar. 17:50

diff --git a/index.org b/index.org index e02983e..a26d1a2 100644 --- a/index.org +++ b/index.org @@ -304,9 +304,14 @@ 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 is generally shown as a function of frequency, and is used to identify the critical modes in a design, at which the effort should be targeted. +It can also helps to determine at which frequencies the effect of disturbances must 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. +A typical Cumulative Power Spectrum is shown in figure [[fig:preumont18_cas_plot]]. + +#+name: fig:preumont18_cas_plot +#+caption: Cumulative Power Spectrum in open-loop and closed-loop for increasing gains (taken from cite:preumont18_vibrat_contr_activ_struc_fourt_edition) +[[file:figs/preumont18_cas_plot.png]] *** Modification of a signal's PSD when going through a dynamical system <> @@ -746,7 +751,7 @@ To do so, several computations are performed (summarized in Figure [[fig:control More details about these computations are accessible [[https://tdehaeze.github.io/nass-simscape/positioning_error.html][here]]. -** Simulation of Experiments +** Simulation of a Tomography Experiment <> Now that the dynamics of the model is tuned and the disturbances included in the model, simulations of experiments can be performed. @@ -806,20 +811,54 @@ In the next sections, it will allows to optimally design the nano-hexapod, to de ** Introduction :ignore: As explain before, the nano-hexapod properties (mass, stiffness, legs' orientation, ...) will influence: -- the effect of disturbances $G_d$ (important for the rejection of disturbances) -- the plant dynamics $G$ (important for the control robustness properties) +- the effect of disturbances +- the plant dynamics Thus, we here wish to find the optimal nano-hexapod properties such that: - the effect of disturbances is minimized (Section [[sec:optimal_stiff_dist]]) - the plant uncertainty due to a change of payload mass and experimental conditions is minimized (Section [[sec:optimal_stiff_plant]]) - the plant has nice dynamical properties for control (Section [[sec:nano_hexapod_architecture]]) -In this study, the effect of the nano-hexapod's mass characteristics is not taken into account because: -1. it cannot be changed a lot -2. it is quite negligible compare the to metrology reflector and the payload's masses that is fixed to nano-hexapod's top platform +In this study, the effect of the nano-hexapod's mass characteristics is not taken into account because it cannot be changed a lot and it is quite negligible compare the to metrology reflector and the payload's masses that are fixed to nano-hexapod's top platform. Also, the effect of the nano-hexapod's damping properties will be studied when applying active damping techniques. +** A brief introduction to Stewart Platforms +<> + +A typical Stewart platform is composed of two platforms connected by six identical struts (or legs) composed of: +- a universal joint at one end +- a spherical joint at the other end +- a prismatic joint with an associated actuator + +This is very schematically shown in Figure [[fig:stewart_architecture_example]] 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. + +#+name: fig:stewart_architecture_example +#+caption: Schematic representation of a Stewart platform +[[file:figs/stewart_architecture_example.png]] + +As shows in Figure [[fig:stewart_architecture_example]], 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 [[fig:stewart_architecture_example_pose]]. +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 [[sec:nano_hexapod_architecture]]. + +#+name: fig:stewart_architecture_example_pose +#+caption: Display of the Stewart platform architecture at some defined pose +[[file:figs/stewart_architecture_example_pose.png]] + +The Stewart Platform is very adapted for the NASS application for the following reasons: +- it is a fully parallel manipulator, thus all the motions errors can be compensated +- it is very compact compared to a serial manipulator +- it has high stiffness and good dynamic performances + +The main disadvantage of Stewart platforms is the small workspace when compare the serial manipulators which is not a problem here. + +A Matlab toolbox to study and design Stewart Platforms has been developed and used for the design of the nano-hexapod. +The source code is accessible [[https://github.com/tdehaeze/stewart-simscape][here]] and the documentation [[https://tdehaeze.github.io/stewart-simscape/][here]]. + ** Optimal Stiffness to reduce the effect of disturbances <> @@ -827,7 +866,7 @@ Also, the effect of the nano-hexapod's damping properties will be studied when a As will be seen, the nano-hexapod stiffness have a large influence on the sensibility to disturbance (the norm of $G_d$). For instance, it is quite obvious that a stiff nano-hexapod is better than a soft one when it comes to direct forces applied to the sample such as cable forces. -A complete study of the optimal nano-hexapod stiffness for the minimization of disturbance sensibility [[https://tdehaeze.github.io/nass-simscape/optimal_stiffness_disturbances.html][here]] and summarized below. +A study of the optimal nano-hexapod stiffness for the minimization of disturbance sensibility is accessible [[https://tdehaeze.github.io/nass-simscape/optimal_stiffness_disturbances.html][here]] and summarized below. *** Sensibility to stage vibrations :PROPERTIES: @@ -835,8 +874,8 @@ A complete study of the optimal nano-hexapod stiffness for the minimization of d :END: 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 [[fig:opt_stiff_sensitivity_Frz]]. -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. +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. The same conclusion is made for vibrations of the translation stage. @@ -849,9 +888,9 @@ The same conclusion is made for vibrations of the translation stage. :UNNUMBERED: t :END: -The sensibilities to ground motion in the Y and Z directions are shown in Figure [[fig:opt_stiff_sensitivity_Dw]]. -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. +The sensibility to ground motion in the Y and Z directions is shown in Figure [[fig:opt_stiff_sensitivity_Dw]]. +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 [[sec:soft_granite]] that using soft mounts for the granite can greatly lower the sensibility to ground motion. @@ -864,8 +903,15 @@ It will be suggested in Section [[sec:soft_granite]] that using soft mounts for :UNNUMBERED: t :END: -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. +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. + +To do so, the power spectral density of the disturbances should be taken into account, as the sensibility of one disturbance should be reduced only where the PSD of the considered disturbance is large compared to the other disturbances. + +What is more important than comparing the sensitivity to disturbances, is to compare the resulting open-loop power spectral density of the sample's position error with the change of the nano-hexapod's stiffness. +This is the *dynamic noise budgeting*. + + From the Power Spectral Density of all the sources of disturbances identified in Section [[sec:identification_disturbances]], we compute what would be the Power Spectral Density of the vertical motion error for all the considered nano-hexapod stiffnesses (Figure [[fig:opt_stiff_psd_dz_tot]]). We can see that the most important change is in the frequency range 30Hz to 300Hz where a stiffness smaller than $10^5\,[N/m]$ greatly reduces the sample's vibrations. @@ -874,8 +920,13 @@ We can see that the most important change is in the frequency range 30Hz to 300H #+caption: Amplitude Spectral Density of the Sample vertical position error due to Vertical vibration of the Spindle for multiple nano-hexapod stiffnesses [[file:figs/opt_stiff_psd_dz_tot.png]] +*** Conclusion +:PROPERTIES: +:UNNUMBERED: t +:END: + #+begin_important -If we look at the Cumulative amplitude spectrum of the vertical error motion in Figure [[fig:opt_stiff_cas_dz_tot]], 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 [[fig:opt_stiff_cas_dz_tot]], 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 be required to obtain the wanted performance. #+end_important #+name: fig:opt_stiff_cas_dz_tot @@ -890,16 +941,15 @@ One of the most important design goal is to obtain a system that is *robust* to Therefore, we have to identify all changes that might occurs in the system and choose the nano-hexapod stiffness such that the uncertainty to these changes is minimized. The uncertainty in the system can be caused by: -- A change in the *Support's compliance* (complete analysis [[https://tdehaeze.github.io/nass-simscape/uncertainty_support.html][here]]): if the micro-station dynamics is changing due to the change of parts or just because of aging effects, the feedback system should remains stable and the obtained performance should not change +- A change in the *Support's compliance* (complete analysis [[https://tdehaeze.github.io/nass-simscape/uncertainty_support.html][here]]): if the micro-station dynamics is changing due to the change of mechanical parts or just because of aging effects, the feedback system should remains stable and the obtained performance should not change - A change in the *Payload mass/dynamics* (complete analysis [[https://tdehaeze.github.io/nass-simscape/uncertainty_payload.html][here]]): the sample's mass is ranging from $1\,kg$ to $50\,kg$ -- A change of *experimental condition* such as the micro-station's pose or the spindle rotation (complete analysis [[https://tdehaeze.github.io/nass-simscape/uncertainty_experiment.html][here]]) +- A change of *experimental condition* such as the micro-station's pose or the spindle rotation speed (complete analysis [[https://tdehaeze.github.io/nass-simscape/uncertainty_experiment.html][here]]) -Because of the trade-off between robustness and performance, the bigger the plant dynamic uncertainty, the lower the simultaneous attainable performance is for all the plants. -Thus, all these uncertainties will limit the attainable bandwidth and hence the obtained performance. +Because of the trade-off between robustness and performance, *the bigger the plant dynamic uncertainty, the lower the attainable performance will be for all the system changes*. -In the next sections, the effect the considered changes on the *plant dynamics* is quantified and conclusions are made on the optimal stiffness for robustness properties. +In the next sections, the effect the considered changes on the *plant dynamics* is quantified and conclusions are drawn on the optimal stiffness for robustness properties. -In the following study, when we refer to plant dynamics, this means the dynamics from forces applied by the nano-hexapod to the measured sample's position by the metrology. +In the following study, when we refer to /plant dynamics/, this means the dynamics from forces applied by the nano-hexapod to the measured sample's position by the metrology. We will only compare the plant dynamics as it is the most important dynamics for robustness and performance properties. However, the dynamics from forces to sensors located in the nano-hexapod legs, such as force and relative motion sensors, have also been considered in a separate study. @@ -910,16 +960,20 @@ However, the dynamics from forces to sensors located in the nano-hexapod legs, s The most obvious change in the system is the change of payload. -In Figure [[fig:opt_stiffness_payload_mass_fz_dz]] the dynamics is shown for payloads having a first resonance mode at 100Hz and a mass equal to 1kg, 20kg and 50kg. +In Figure [[fig:opt_stiffness_payload_mass_fz_dz]] 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. One can see that for the soft nano-hexapod: - the first resonance (suspension mode of the nano-hexapod) is lowered with an increase of the sample's mass. - This first resonance corresponds to $\omega = \sqrt{\frac{k_n}{m_n + m_s}}$ where $k_n$ is the vertical nano-hexapod stiffness, $m_n$ the mass of the nano-hexapod's top platform, and $m_s$ the sample's mass -- the gain after the first resonance and up until the anti-resonance at 100Hz is changing with the sample's mass + This is very logical as the first resonance corresponds to $\omega = \sqrt{\frac{k_n}{m_n + m_s}}$ where $k_n$ is the vertical nano-hexapod stiffness, $m_n$ the mass of the nano-hexapod's top platform, and $m_s$ the sample's mass +- the gain after the first resonance and up until the anti-resonance at 100Hz is changing with the sample's mass. + It is indeed equal to $\frac{1}{(m_n + m_s) \omega^2}$ For the stiff-nano-hexapod, the change of payload mass has very little effect (the vertical scale for the amplitude is quite small). +To minimize the uncertainty to the payload's mass, the mass of the nano-hexapod's top platform plus the metrology reflector should be maximized, and ideally close to the maximum payload's mass. +As the maximum payload's mass is $50\,kg$, this may however not be practical, and thus the control architecture must be developed to be robust to a change of the payload's mass. + #+name: fig:opt_stiffness_payload_mass_fz_dz #+caption: 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) [[file:figs/opt_stiffness_payload_mass_fz_dz.png]] @@ -933,17 +987,16 @@ We can see (more easily for the soft nano-hexapod), that resonance of the payloa #+caption: 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 [[file:figs/opt_stiffness_payload_freq_fz_dz.png]] - 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 [[fig:opt_stiffness_payload_impedance_all_fz_dz]]. For nano-hexapod stiffnesses below $10^6\,[N/m]$: - the phase stays between 0 and -180deg which is a very nice property for control -- the dynamical change up until the resonance of the payload is mostly a change of gain +- the dynamical change up until the resonance of the payload can be considered as a change of gain For nano-hexapod stiffnesses above $10^7\,[N/m]$: - the dynamics is unchanged until the first resonance which is around 25Hz-35Hz -- above that frequency, the change of dynamics is quite chaotic (we will see in the next section that this is due to the micro-station dynamics) +- above that frequency, the change of dynamics is quite chaotic (we will see in the next section that this is due to the micro-station dynamics) and it would be difficult to have a controller with high bandwidth which is robust to such change of dynamics #+name: fig:opt_stiffness_payload_impedance_all_fz_dz #+caption: Dynamics from $\mathcal{F}_z$ to $\mathcal{X}_z$ for varying payload dynamics, both for a soft nano-hexapod and a stiff nano-hexapod @@ -951,14 +1004,13 @@ For nano-hexapod stiffnesses above $10^7\,[N/m]$: #+begin_important -For soft nano-hexapods, the payload has an important impact on the dynamics. -This will have to be carefully taken into account for the controller design. +For soft nano-hexapods, the payload has an important impact on the dynamics that will have to be carefully taken into account for the controller design. For stiff nano-hexapod, the dynamics doe not change with the payload until the first resonance frequency of the nano-hexapod or of the payload. If possible, the first resonance frequency of the payload should be maximized (stiff fixation). -Heavy samples with low first resonance mode will be very problematic. +Heavy samples with low first resonance mode will be the most problematic. #+end_important *** Effect of Micro-Station Compliance @@ -970,7 +1022,7 @@ The micro-station dynamics is quite complex as was shown in Section [[sec:micro_ - a change in some mechanical elements - a change in the position of one stage. For instance, a large displacement of the micro-hexapod can change the micro-station compliance -- a change in a control loop +- a change in a control loop of some of its stage Thus, it would be much more robust if the plant dynamics were not depending on the micro-station dynamics. @@ -979,8 +1031,8 @@ This as several other advantages: - the nano-hexapod could be use on top of any other station much more easily -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 [[fig:opt_stiffness_micro_station_fx_dx]]): -- without the micro-station (solid curves) +To identify the effect of the micro-station compliance on the system dynamics, the plant dynamics is identified in two different cases (Figure [[fig:opt_stiffness_micro_station_fx_dx]]): +- with a micro-station considered as a solid body (solid curves) - with the micro-station dynamics (dashed curves) One can see that for nano-hexapod stiffnesses below $10^6\,[N/m]$, the plant dynamics does not significantly changed due to the micro station dynamics (the solid and dashed curves are superimposed). @@ -1005,12 +1057,15 @@ If a stiff nano-hexapod is used, the control bandwidth should probably be limite 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 [[fig:opt_stiffness_wz_fx_dx]]. +The plant dynamics for spindle rotation speed varying from 0rpm up to 60rpm are identified and shown in Figure [[fig:opt_stiffness_wz_fx_dx]]. One can see that for nano-hexapods with a stiffness above $10^5\,[N/m]$, the dynamics is mostly not changing with the spindle's rotating speed. For very soft nano-hexapods, the main resonance is split into two resonances and one anti-resonance that are all moving at a function of the rotating speed. +The change of dynamics is due to both centrifugal forces and Coriolis forces. +This effect has been studied in details in [[https://tdehaeze.github.io/rotating-frame/index.html][this]] document. + #+name: fig:opt_stiffness_wz_fx_dx #+caption: Change of dynamics from force $\mathcal{F}_x$ to displacement $\mathcal{X}_x$ for a spindle rotation speed from 0rpm to 60rpm [[file:figs/opt_stiffness_wz_fx_dx.png]] @@ -1026,69 +1081,89 @@ A very soft ($k < 10^4\,[N/m]$) nano-hexapod should not be used due to the effec :UNNUMBERED: t :END: -Finally, let's combined all the uncertainties and display the plant dynamics "spread" for all the nano-hexapod stiffnesses (Figure [[fig:opt_stiffness_plant_dynamics_task_space]]). +Finally, let's combined all the uncertainties and display the "spread" of the plant dynamics for all the nano-hexapod stiffnesses (Figure [[fig:opt_stiffness_plant_dynamics_task_space]]). This show how the dynamics evolves with the stiffness and how different effects enters the plant dynamics. #+name: fig:opt_stiffness_plant_dynamics_task_space #+caption: Variability of the dynamics from $\bm{\mathcal{F}}_x$ to $\bm{\mathcal{X}}_x$ with varying nano-hexapod stiffness [[file:figs/opt_stiffness_plant_dynamics_task_space.gif]] +*** Conclusion +:PROPERTIES: +:UNNUMBERED: t +:END: + #+begin_important - Let's summarize the findings: - - the payload's mass influence the plant dynamics above the first resonance of the nano-hexapod. + Let's summarize the findings about the effect of the nano-hexapod's stiffness on the plant uncertainty: + - The payload's mass influence the plant dynamics above the first resonance of the nano-hexapod. Thus a high nano-hexapod stiffness helps reducing the effect of a change of the payload's mass - - the payload's first resonance is seen as an anti-resonance in the plant dynamics. - As this effect will largely be variable from one payload to the other, *the payload's first resonance should be maximized* (above 300Hz if possible) for all used payloads - - the dynamics of the nano-hexapod is not affected by the micro-station dynamics (compliance) when $k < 10^6\,[N/m]$ - - the spindle's rotating speed has no significant influence on the plant dynamics for nano-hexapods with a stiffness $k > 10^5\,[N/m]$ + - The payload's first resonance is seen as an anti-resonance in the plant dynamics. + As this effect will largely be variable from one payload to the other, thus *the payload's first resonance should be maximized* (above 300Hz if possible) for all used payloads + - The dynamics of the nano-hexapod is not affected by the micro-station dynamics (compliance) if $k < 10^6\,[N/m]$ + - The spindle's rotating speed has no significant influence on the plant dynamics for nano-hexapod's stiffnesses $k > 10^5\,[N/m]$ - Concerning the plant dynamic uncertainty, the resonance frequency of the nano-hexapod should be between 5Hz (way above the maximum rotating speed) and 50Hz (before the first micro-station resonance) for all the considered payloads. - This corresponds to an optimal nano-hexapod leg stiffness in the range $10^5 - 10^6\,[N/m]$. - - In such case, the main limitation will be heavy samples with small stiffnesses. + The *resonance frequency of the nano-hexapod should be between 5Hz* (way above the maximum rotating speed) *and 50Hz* (before the first micro-station resonance) for all the considered payloads. + This corresponds to an *optimal nano-hexapod leg stiffness in the range* $10^5 - 10^6\,[N/m]$. #+end_important ** Optimal Nano-Hexapod Geometry <> *** Introduction :ignore: -As will be shown in this section, the Nano-Hexapod geometry has an influence on: -- the overall stiffness/compliance +Stewart platforms can be studied with: +- *Kinematic analysis*: study of the geometry of motion without considering the forces and torques that cause the motion +- *Jacobian analysis*: reveals the relation between the actuators velocities and the moving platform linear and angular velocities. It also constructs the transformation between actuator forces and task space forces +and moments acting on the moving platform. +- *Dynamic analysis*: consists of the equations of motion for the manipulator which are quite complex to derive + +Kinematic and Jacobian analysis are briefly introduced in this section, however the dynamic analysis is not performed analytically but rather studied using the Simscape model. + +As will be shown, the Nano-Hexapod geometry has an influence on: +- the stiffness and compliance properties - the mobility -- the dynamics and coupling +- the force authority +- the dynamics of the manipulator -A typical Stewart platform is composed of six identical legs: -- a universal joint -- a spherical joint -- a prismatic joint with an integrated actuator - -#+name: fig:stewart_architecture_example -#+caption: Figure caption -[[file:figs/stewart_architecture_example.png]] - - -#+name: fig:stewart_architecture_example_pose -#+caption: Display of the Stewart platform architecture at some defined pose -[[file:figs/stewart_architecture_example_pose.png]] - - -*** Kinematic Analysis and the Jacobian Matrix +*** Kinematic Analysis :PROPERTIES: :UNNUMBERED: t :END: -The kinematic analysis of a parallel manipulator is well described in cite:taghirad13_paral: +The Kinematic analysis of the Stewart platform can be divided into two problems: the inverse kinematics and the forward kinematics. + #+begin_quote -Kinematic analysis refers to the study of the geometry of motion of a robot, without considering the forces an torques that cause the motion. -In this analysis, the relation between the geometrical parameters of the manipulator with the final motion of the moving platform is derived and analyzed. +For *inverse kinematic analysis*, it is assumed that the position ${}^A\bm{P}$ and orientation of the moving platform ${}^A\bm{R}_B$ are given and the problem is to obtain the joint variables $\bm{\mathcal{L}} = \left[ l_1, l_2, l_3, l_4, l_5, l_6 \right]^T$. #+end_quote - -One of the main analysis tool for the Kinematic analysis is the *Jacobian Matrix* that not only reveals the *relation between the joint variable velocities of a parallel manipulator to the moving platform linear and angular velocities*, but also constructs the transformation needed to find the *actuator forces from the forces and moments acting on the moving platform*. - - -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 flexible joints (described by the position vectors ${}^A\bm{b}_i$): +This problem can be easily solved, and the obtain joint variables are: \begin{equation*} + \begin{aligned} + l_i = &\Big[ {}^A\bm{P}^T {}^A\bm{P} + {}^B\bm{b}_i^T {}^B\bm{b}_i + {}^A\bm{a}_i^T {}^A\bm{a}_i - 2 {}^A\bm{P}^T {}^A\bm{a}_i + \dots\\ + &2 {}^A\bm{P}^T \left[{}^A\bm{R}_B {}^B\bm{b}_i\right] - 2 \left[{}^A\bm{R}_B {}^B\bm{b}_i\right]^T {}^A\bm{a}_i \Big]^{1/2} + \end{aligned} +\end{equation*} + +If the position and orientation of the platform lie in the feasible workspace, the solution is unique. +Otherwise, the solution gives complex numbers. + +This means that from the wanted position of the nano-hexapod's mobile platform with respect to the fixed platform (described by ${}^A\bm{P}$ and ${}^A\bm{R}_B$) and for a specific geometry (position of the top joints $^{B}\bm{b}$ and bottom joints ${}^A\bm{a}$), the required motion of each leg can easily be determined. + + +#+begin_quote +In *forward kinematic analysis*, it is assumed that the vector of limb lengths $\bm{L}$ is given and the problem is to find the position ${}^A\bm{P}$ and the orientation ${}^A\bm{R}_B$. +#+end_quote + +This is a difficult problem that requires to solve nonlinear equations. + +However, as will be shown in the next section, approximate solution of the forward kinematic analysis can be obtained thanks to the Jacobian analysis. + +*** Jacobian Analysis +:PROPERTIES: +:UNNUMBERED: t +:END: + +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 $\{A\}$: +\begin{equation} \bm{J} = \begin{bmatrix} {\hat{\bm{s}}_1}^T & (\bm{b}_1 \times \hat{\bm{s}}_1)^T \\ {\hat{\bm{s}}_2}^T & (\bm{b}_2 \times \hat{\bm{s}}_2)^T \\ @@ -1097,25 +1172,31 @@ The Jacobian matrix $\bm{J}$ can be computed form the orientation of the legs (d {\hat{\bm{s}}_5}^T & (\bm{b}_5 \times \hat{\bm{s}}_5)^T \\ {\hat{\bm{s}}_6}^T & (\bm{b}_6 \times \hat{\bm{s}}_6)^T \end{bmatrix} -\end{equation*} +\end{equation} -It can be easily shown that: + +It can be shown that the Jacobian matrix reveals the *relation between the legs' velocities to the moving platform linear and angular velocities*: \begin{equation} - \delta\bm{\mathcal{L}} = \bm{J} \delta\bm{\mathcal{X}}, \quad \delta\bm{\mathcal{X}} = \bm{J}^{-1} \delta\bm{\mathcal{L}} \label{eq:jacobian_L} + \dot{\bm{\mathcal{L}}} = \bm{J} \dot{\bm{\mathcal{X}}} \label{eq:jacobian_velocity} \end{equation} with: -- $\delta\bm{\mathcal{L}} = [ \delta l_1, \delta l_2, \delta l_3, \delta l_4, \delta l_5, \delta l_6 ]^T$: the vector of small legs' displacements -- $\delta \bm{\mathcal{X}} = [\delta x, \delta y, \delta z, \delta \theta_x, \delta \theta_y, \delta \theta_z ]^T$: the vector of small mobile platform displacements +- $\dot{\bm{\mathcal{L}}} = [ \dot{l}_1, \dot{l}_2, \dot{l}_3, \dot{l}_4, \dot{l}_5, \dot{l}_6 ]^T$: relative velocity of each strut +- $\dot{\bm{X}} = [^A\bm{v}_p, {}^A\bm{\omega}]^T$: output twist vector of the mobile platform -Thus, from a wanted small displacement $\delta \bm{\mathcal{X}}$, it is easy to compute the required displacement of the legs $\delta \bm{\mathcal{L}}$. -Similarly, from a measurement of the legs' displacement, it is easy to compute the resulting platform's motion. +For small legs motions $\delta\bm{\mathcal{L}}$ and small mobile platform motion $\delta \bm{\mathcal{X}}$, the following approximation can be computed from Eq. eqref:eq:jacobian_velocity: +\begin{equation} + \delta\bm{\mathcal{L}} \approx \bm{J} \delta\bm{\mathcal{X}}, \quad \delta\bm{\mathcal{X}} \approx \bm{J}^{-1} \delta\bm{\mathcal{L}} \label{eq:jacobian_L} +\end{equation} -This will be used to estimate the platform's mobility from the stroke of the legs, or inversely, to estimate the required stroke of the legs from the wanted platform's mobility. +Equations eqref:eq:jacobian_L can be used to *approximate the forward and inverse kinematic* problems for small displacements. + +This approximation will be used to estimate the platform's mobility from the legs' stroke, or inversely, to estimate the required stroke of the legs from the wanted platform's mobility. Note that Eq. eqref:eq:jacobian_L is an approximation and is only valid for leg's displacement less than $1\%$ of the leg's length which is the case for the nano-hexapod. -It can also be shown that: + +It can also be shown that the Jacobian matrix links the *actuator forces to forces and moments acting on the moving platform*: \begin{equation} \bm{\mathcal{F}} = \bm{J}^T \bm{\tau}, \quad \bm{\tau} = \bm{J}^{-T} \bm{\mathcal{F}} \label{eq:jacobian_F} \end{equation} @@ -1123,42 +1204,16 @@ with: - $\bm{\tau} = [\tau_1, \tau_2, \cdots, \tau_6]^T$: vector of actuator forces applied in each strut - $\bm{\mathcal{F}} = [\bm{f}, \bm{n}]^T$: external force/torque action on the mobile platform -And thus the Jacobian matrix can be used to compute the forces that should be applied on each leg from forces and torques that we want to apply on the top platform. - - -Transformations in Eq. eqref:eq:jacobian_L and eqref:eq:jacobian_F will be widely in the developed control architectures. - -*** Stiffness and Compliance matrices -:PROPERTIES: -:UNNUMBERED: t -:END: - - -\begin{equation*} - \bm{\mathcal{F}} = \bm{K} \delta \bm{\mathcal{X}} -\end{equation*} - -\begin{equation*} - \bm{K} = \bm{J}^T \mathcal{K} \bm{J} -\end{equation*} -\begin{equation*} - \bm{C} = \bm{K}^{-1} = (\bm{J}^T \mathcal{K} \bm{J})^{-1} -\end{equation*} - -Stiffness properties is estimated from the architecture and leg's stiffness - - - - -Kinematic Study https://tdehaeze.github.io/stewart-simscape/kinematic-study.html +And thus *the Jacobian matrix can be used to compute the forces that should be applied on each leg from forces and torques that we want to apply on the nano-hexapod's top platform*. +Linear transformations in Eq. eqref:eq:jacobian_L and eqref:eq:jacobian_F will be widely in the developed control architectures in Section [[sec:robust_control_architecture]]. *** Mobility of the Stewart Platform :PROPERTIES: :UNNUMBERED: t :END: -For a specified geometry and actuator stroke, the mobility of the Stewart platform can be estimated. +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 [[fig:mobility_translations_null_rotation]]. @@ -1166,27 +1221,122 @@ An example of the mobility considering only pure translations is shown in Figure #+caption: Obtained mobility of a Stewart platform for pure translations (the platform's orientation is fixed) [[file:figs/mobility_translations_null_rotation.png]] -*** Flexible Joints + +From a wanted mobility and a specific geometry, the required actuator stroke can be estimated. + +Suppose we want the following mobility: +- x, y and z translations up to $50\,\mu m$ +- x and y rotation up to $30\,\mu rad$ +- no z rotation + +The geometry is chosen arbitrary and corresponds to the wanted nano-hexapod size. + +If only pure translations and pure rotations are considered, the required actuator stroke is $76 \mu m$, whereas if *combined* translations and rotations are considered, the required actuator stroke is $177 \mu m$. + +This gives an idea of the relation between the mobility and the actuator stroke. + +*** Stiffness and Compliance matrices :PROPERTIES: :UNNUMBERED: t :END: -Active Damping Study https://tdehaeze.github.io/stewart-simscape/control-active-damping.html +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 $\tau_i$. -- Advantages compared to conventional joints -- Simulations will help determine the required rotational stroke and will help with the design -- Typical joint stiffness is included in the model +The stiffness of the actuator $k_i$ links the applied (constant) actuator force $\delta \tau_i$ and the corresponding small deflection $\delta l_i$: +\begin{equation*} + \tau_i = k_i \delta l_i, \quad i = 1,\ \dots,\ 6 +\end{equation*} +If we combine these 6 relations: +\begin{equation*} + \bm{\tau} = \mathcal{K} \delta \bm{\mathcal{L}} \quad \mathcal{K} = \text{diag}\left[ k_1,\ \dots,\ k_6 \right] +\end{equation*} -Flexible Joint stiffness => not problematic for the chosen active damping technique +Equations eqref:eq:jacobian_L and eqref:eq:jacobian_F are used to obtain: +\begin{equation} + \bm{\mathcal{F}} = \bm{K} \delta \bm{\mathcal{X}} +\end{equation} +with the stiffness matrix +\begin{equation} + \bm{K} = \bm{J}^T \mathcal{K} \bm{J} \label{eq:jacobian_K} +\end{equation} -Example of flexible joints used [[fig:preumont07_flexible_joints]], [[fig:yang19_flexible_joints]] +And the compliance matrix can be computed by taking the inverse of the Stiffness matrix: +\begin{equation*} + \bm{C} = \bm{K}^{-1} = (\bm{J}^T \mathcal{K} \bm{J})^{-1} +\end{equation*} +The compliance matrix of a manipulator shows the mapping of the moving platform wrench applied at $\bm{O}_B$ to its small deflection by +\begin{equation*} + \delta \bm{\mathcal{X}} = \bm{C} \cdot \bm{\mathcal{F}} +\end{equation*} -#+name: tab:yang19_stiffness_flexible_joints -#+caption: Stiffness of unversal and spherical flexible joints cite:yang19_dynam_model_decoup_contr_flexib -| $k_{\theta u},\ k_{\psi u}$ | $72 Nm/rad$ | -| $k_{\theta s}$ | $51 Nm/rad$ | -| $k_{\psi s}$ | $62 Nm/rad$ | -| $k_{\gamma s}$ | $64 Nm/rad$ | +Stiffness properties of the Stewart platform can then be estimated from the architecture (through the Jacobian matrix) and leg's stiffness. + +*** Effect of a change of geometry +:PROPERTIES: +:UNNUMBERED: t +:END: + +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 [[tab:effect_legs_jacobian]]. +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. + +The nano-hexapod geometry and further be optimized in terms of stiffness and stroke using the presented tools. + +#+name: tab:effect_legs_jacobian +#+caption: Effect of a change in geometry on the manipulator's stiffness, force authority and stroke +| | *legs pointing more vertically* | *legs further apart* | +|-------------------------------+---------------------------------+----------------------| +| Vertical stiffness | $\nearrow$ | $=$ | +| Horizontal stiffness | $\searrow$ | $=$ | +| Vertical rotation stiffness | $\searrow$ | $\nearrow$ | +| Horizontal rotation stiffness | $\nearrow$ | $\nearrow$ | +|-------------------------------+---------------------------------+----------------------| +| Vertical force authority | $\nearrow$ | $=$ | +| Horizontal force authority | $\searrow$ | $=$ | +| Vertical torque authority | $\searrow$ | $\nearrow$ | +| Horizontal torque authority | $\nearrow$ | $\nearrow$ | +|-------------------------------+---------------------------------+----------------------| +| Vertical stroke | $\searrow$ | $=$ | +| Horizontal stroke | $\nearrow$ | $=$ | +| Vertical rotation stroke | $\nearrow$ | $\searrow$ | +| Horizontal rotation stroke | $\searrow$ | $\searrow$ | + +*** Cubic Architecture +:PROPERTIES: +:UNNUMBERED: t +:END: + +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 [[fig:3d-cubic-stewart-aligned]]). + +#+name: fig:3d-cubic-stewart-aligned +#+caption: Schematic representation of the Cubic architecture +[[file:figs/3d-cubic-stewart-aligned.png]] + + +This architecture provides some advantages such as *uniform stiffness* and *uniform mobility* in all directions. +It can also have very nice dynamical properties in specific conditions (center of mass of the payload located at the cube's center) that are not met for the NASS. + +The cubic configuration also puts much restriction on the position and orientation of the legs as there is only one design variable: the size of the cube. + +#+begin_important +For these reasons, the cubic configuration is not recommended for the nano-hexapod. +#+end_important + +Separate study of the cubic architecture is performed [[https://tdehaeze.github.io/stewart-simscape/cubic-configuration.html][here]]. + +*** Effect of Flexible Joints +:PROPERTIES: +:UNNUMBERED: t +:END: + +Each of the nano-hexapod legs has a universal joint at one end and a spherical joint at the other end. + +When only small stroke is required, *flexible* joints can be used: material is bend to achieve motion, rather than relying on sliding or rolling across two surfaces. + +Example of flexible joints used for Stewart platforms are shown in Figures [[fig:preumont07_flexible_joints]] and [[fig:yang19_flexible_joints]]. #+name: fig:preumont07_flexible_joints #+caption: Flexible joints used in cite:preumont07_six_axis_singl_stage_activ @@ -1196,38 +1346,65 @@ Example of flexible joints used [[fig:preumont07_flexible_joints]], [[fig:yang19 #+caption: An alternative type of flexible joints that has been used for Stewart platforms cite:yang19_dynam_model_decoup_contr_flexib [[file:figs/yang19_flexible_joints.png]] +The flexible joints have few advantages compared to conventional joints such as the *absence of wear, friction and backlash* which allows extremely high-precision (predictable) motion. -*** Cubic Architecture -:PROPERTIES: -:UNNUMBERED: t -:END: +The parasitic bending and torsional stiffness of these joints usually induce some *limitation on the control performance*. -Study of cubic architecture https://tdehaeze.github.io/stewart-simscape/cubic-configuration.html -Has some advantages such as uniform stiffness and uniform mobility. -It can have very nice properties in specific conditions that will not be the case for this application. -The cubic configuration also puts much restriction on the position and orientation of each leg. -This configuration is such not recommended. +This has been studied using the Simscape model (report available [[https://tdehaeze.github.io/nass-simscape/flexible_joints_study.html][here]]), and conclusions on the required characteristics of the flexible joints are summarized below. -#+name: fig:3d-cubic-stewart-aligned -#+caption: Figure caption -[[file:figs/3d-cubic-stewart-aligned.png]] + +The bending and torsional stiffnesses somehow adds a parasitic stiffness in parallel with the struts. +This is not found to be problematic for the control architecture that will be developed in Section [[sec:robust_control_architecture]] (it is however, if Integral Force Feedback is to be used, explained [[https://tdehaeze.github.io/stewart-simscape/control-active-damping.html][here]]). + + +The finite axial stiffness of the flexible joints can however be very problematic for control. +Small values of the axial stiffness are shown to limit the achievable damping using Direct Velocity Feedback. +The axial stiffness should therefore be maximized and taken into account in the model of the nano-hexapod. + + +For the identified optimal actuator stiffness $k = 10^5\,[N/m]$, the flexible joint should have the following stiffness properties: +- Axial Stiffness: $K_a > 10^7\,[N/m]$ +- Bending Stiffness: $K_b < 50\,[Nm/rad]$ +- Torsion Stiffness: $K_t < 50\,[Nm/rad]$ + +These requirements are easily obtained in practice. +For instance, the flexible joint used for the ID16 nano-hexapod have the following stiffness properties: +- Axial Stiffness: $K_a = 6 \cdot 10^7\,[N/m]$ +- Bending Stiffness: $K_b = 15\,[Nm/rad]$ +- Torsion Stiffness: $K_t = 20\,[Nm/rad]$ + +#+begin_important + Even though much attention should be paid on the proper design of the flexible joints, they should not impose limitation on the performance of the system. + + Simulations will help determine the required rotational stroke of the flexible joints and will help with the design. +#+end_important *** Conclusion :PROPERTIES: :UNNUMBERED: t :END: +#+begin_important + Relations between the geometry of the Stewart platform and its characteristics such as stiffness, maneuverability and force authority have been derived. + + These relations can help optimize the Stewart platform's geometry, however, the choice of the geometry is quite constrained by the limited size of the hexapod, the size of the flexible joints and of the included actuators and sensors. + + The effects of flexible joints stiffness on the dynamics have been studied and requirements on the flexible joints have been derived. +#+end_important + ** Conclusion #+begin_important In Section [[sec:optimal_stiff_dist]], it has been concluded that a nano-hexapod stiffness below $10^5-10^6\,[N/m]$ helps reducing the high frequency vibrations induced by all sources of disturbances considered. As the high frequency vibrations are the most difficult to compensate for when using feedback control, a soft hexapod will most certainly helps improving the performances. In Section [[sec:optimal_stiff_plant]], we concluded that a nano-hexapod leg stiffness in the range $10^5 - 10^6\,[N/m]$ is a good compromise between the uncertainty induced by the micro-station dynamics and by the rotating speed. - Provided that the samples used have a first mode that is sufficiently high in frequency, the total plant dynamic uncertainty should be manageable. + 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 stiffness of $10^5\,[N/m]$ will be used in Section [[sec:robust_control_architecture]] to develop the robust control architecture and to perform simulations. + Thus, a leg's stiffness of $10^5\,[N/m]$ will be used in Section [[sec:robust_control_architecture]] to develop the robust control architecture and to perform simulations. A more detailed study of the determination of the optimal stiffness based on all the effects is available [[https://tdehaeze.github.io/nass-simscape/uncertainty_optimal_stiffness.html][here]]. + + Finally, in section [[sec:nano_hexapod_architecture]] some insights on the wanted nano-hexapod geometry are given. #+end_important * Robust Control Architecture @@ -1464,6 +1641,10 @@ An animation of the experiment is shown in Figure [[fig:closed_loop_sim_zoom]] a #+caption: Top View of a tomography experiment with a 10mm offset imposed by the micro-hexapod [[file:figs/tomography_dh_offset.gif]] +The force applied by the actuators are compensating the centripetal force induced by the spindle's rotation. + +This centrifugal force 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 $F = m r \omega^2 \approx 12 \cdot 0.01 \cdot (2\pi)^2 \approx 5\,[N]$ + *** Simultaneous Translation Scans :PROPERTIES: :UNNUMBERED: t @@ -1495,20 +1676,38 @@ A more complete study of the control of the NASS is performed [[https://tdehaeze <> ** Nano-Hexapod Specifications +Table summarizing the nano-hexapod wanted characteristics: -** General Conclusion +- Dimensions (Figure [[fig:nano_hexapod_size]]) +- Stiffness: + - Resonances should be between 5Hz and 50Hz + - +- Flexible joints: + - To be optimized such that the above resonances are in the required range + - + - Required angular stroke: can be estimated with simulations +- Force + - Weight: $60\,kg \rightarrow 600\,N \rightarrow 60\,N$ on each actuator + - Dynamic: few Newtons +- Estimation of the required stroke: + - From simulation (i.e. for disturbance rejection alone), $\pm 5 \mu m$ is good + - However, the required stroke probably depends on two other factors: + - Static positioning errors of the stages + - Maximum tracking errors of the stages (mainly translation stage and tilt stage). + This is probably more difficult to obtain. + However, by limiting the acceleration of these stages, we may limit the dynamic tracking errors to acceptable levels +- Sensors to be included + +#+name: fig:nano_hexapod_size +#+caption: First implementation of the nano-hexapod / deflector and coolsed sample plate support +[[file:figs/nano_hexapod_size.png]] ** Sensor Noise introduced by the Metrology Say that is will introduce noise inside the bandwidth (100Hz) This should not be significant. -** Further Work - -** Cable Forces - - ** Using soft mounts for the Granite <> @@ -1520,13 +1719,33 @@ This means that above the suspension mode of the granite (here around 2Hz), the Sensible to detector motion? -** Others -Common metrology frame for the nano-focusing optics and the measurement of the sample position? - +** Others Factors that may limit the performances Cable forces? Slip-Ring noise? +Metrology Reflector: +- it is very important that the first resonance of the metrology frame is high enough +- it should be above 200Hz + + +Common metrology frame for the nano-focusing optics and the measurement of the sample position? + +** Other Notes +*** Modification of the Granite +Possible to lower the granite? + +This can help the mechanical design and make the nano-hexapod stiffer. + +The problem is that the Tilt stage rotation axis will not be position correctly anymore. + +*** Sample trajectories +More precise definition of the sample trajectories? +- Simultaneous Ty and Rz scans at what speed? + +** General Conclusion + + * Bibliography :ignore: bibliographystyle:unsrt bibliography:ref.bib diff --git a/opt_stiff_hac_dvf_Dh_offset_F.png b/opt_stiff_hac_dvf_Dh_offset_F.png new file mode 100644 index 0000000..38358ef Binary files /dev/null and b/opt_stiff_hac_dvf_Dh_offset_F.png differ diff --git a/opt_stiff_hac_dvf_Dh_offset_dL.png b/opt_stiff_hac_dvf_Dh_offset_dL.png new file mode 100644 index 0000000..f38e7f1 Binary files /dev/null and b/opt_stiff_hac_dvf_Dh_offset_dL.png differ diff --git a/opt_stiff_hac_dvf_Dy_scans_F.png b/opt_stiff_hac_dvf_Dy_scans_F.png new file mode 100644 index 0000000..0f8e5e6 Binary files /dev/null and b/opt_stiff_hac_dvf_Dy_scans_F.png differ diff --git a/opt_stiff_hac_dvf_Dy_scans_dL.png b/opt_stiff_hac_dvf_Dy_scans_dL.png new file mode 100644 index 0000000..1faa6ed Binary files /dev/null and b/opt_stiff_hac_dvf_Dy_scans_dL.png differ