# Sensors and control of a space-based six-axis vibration isolation system - My digital brain
Tags Stewart Platforms, Vibration Isolation, Cubic Architecture Reference (Hauge &amp; Campbell, 2004) Author(s) Hauge, G., &amp; Campbell, M. Year 2004 Discusses: Choice of sensors and control architecture Predictability and limitations of the system dynamics Two-Sensor control architecture Vibration isolation using a Stewart platform Experimental comparison of Force sensor and Inertial Sensor and associated control architecture for vibration isolation Figure 1: Hexapod for active vibration isolation
Author: — Thomas Dehaeze

# Sensors and control of a space-based six-axis vibration isolation system

Tags
  Stewart Platforms, Vibration Isolation, Cubic Architecture
Reference
  (Hauge & Campbell, 2004)
Author(s)
  Hauge, G., & Campbell, M.
Year
  2004

Discusses:

- Choice of sensors and control architecture
- Predictability and limitations of the system dynamics
- Two-Sensor control architecture
- Vibration isolation using a Stewart platform
- Experimental comparison of Force sensor and Inertial Sensor and associated control architecture for vibration isolation

Figure 1: Hexapod for active vibration isolation

Stewart platform (Figure 1):

- Low corner frequency
- Large actuator stroke (\(\pm5mm\))
- Sensors in each strut (Figure 2):
  - three-axis load cell
  - base and payload geophone in parallel with the struts
  - LVDT

Figure 2: Strut

> Force sensors typically work well because they are not as sensitive to payload and base dynamics, but are limited in performance by a low-frequency zero pair resulting from the cross-axial stiffness.

Performance Objective (frequency domain metric):

- The transmissibility should be close to 1 between 0-1.5Hz \(-3dB < |T(\omega)| < 3db\)
- The transmissibility should be below -20dB in the 5-20Hz range \(|T(\omega)| < -20db\)

With \(|T(\omega)|\) is the Frobenius norm of the transmissibility matrix and is used to obtain a scalar performance metric.

Challenge:

- small frequency separation between the two requirements

Robustness:

- minimization of the transmissibility amplification (Bode’s “pop”) outside the performance region

Model:

- single strut axis as the cubic Stewart platform can be decomposed into 6 single-axis systems

Figure 3: Strut model

Zero Pair when using a Force Sensor:

- The frequency of the zero pair corresponds to the resonance frequency of the payload mass and the “parasitic” stiffness (sum of the cross-axial, suspension, wiring stiffnesses)
- This zero pair is usually not predictable nor repeatable
- In this Stewart platform, this zero pair uncertainty is due to the internal wiring of the struts

Control:

- Single-axis controllers => combine them into a full six-axis controller => evaluate the full controller in terms of stability and robustness
- Sensitivity weighted LQG controller (SWLQG) => address robustness in flexible dynamic systems
- Three type of controller:
  - Force feedback (cell-based)
  - Inertial feedback (geophone-based)
  - Combined force/velocity feedback (load cell/geophone based)

> The use of multivariable and robust control on the full 6x6 hexapod does not improve performance over single-axis designs.

Table 1: Typical characteristics of sensors used for isolation in hexapod systems

| Load cell | Geophone
--- | --- | ---
Type | Relative | Inertial
Relationship with voice coil | Collocated and Dual | Non-Collocated and non-Dual
Open loop transfer function | (+) Alternating poles/zeros | (-) Large phase drop
Limitation from low-frequency zero pair | (-) Yes | (+) No
Sensitive to payload/base dynamics | (+) No | (-) Yes
Best frequency range | High (low-freq zero limitation) | Low (high-freq toll-off limitation)

Ability of a sensor-actuator pair to improve performance: General system with input \(u\), performance \(z\), output \(y\) disturbance \(u\).

Given a sensor \(u\) and actuator \(y\) and a controller \(u = -K(s) y\), the closed loop disturbance to performance transfer function can be written as:

\[ \left[ \frac{z}{w} \right]_\text{CL} = \frac{G(s)_{zw} + K(G(s)_{zw} G(s)_{yu} - G(s)_{zu} G(s)_{yw})}{1 + K G(s)_{yu}} \]

In order to obtain a significant performance improvement is to use a high gain controller, provided the term \(G(s)_{zw} + K(G(s)_{zw} G(s)_{yu} - G(s)_{zu} G(s)_{yw})\) is small.

We can compare the transfer function from \(w\) to \(z\) with and without a high gain controller. And we find that for \(u\) and \(y\) to be an acceptable pair for high gain control: \[ \left| \frac{G(j\omega)_{zw} G(j\omega)_{yu} - G(j\omega)_{zu} G(j\omega)_{yw}}{K G(j\omega)_{yu}} \right| \ll |G_{zw}(j\omega)| \]

Controllers:

Force feedback:

- Performance limited by the low frequency zero-pair
- It is desirable to separate the zero-pair and first most are separated by at least a decade in frequency
- This can be achieve by reducing the cross-axis stiffness
- If the low frequency zero pair is inverted, robustness is lost
- Thus, the force feedback controller should be designed to have combined performance and robustness at frequencies at least a decade above the zero pair
- The presented controller as a high pass filter at to reduce the gain below the zero-pair, a lag at low frequency to improve phase margin, and a low pass filter for roll off

Inertial feedback:

- Non-Collocated => multiple phase drops that limit the bandwidth of the controller
- Good performance, but the transmissibility “pops” due to low phase margin and thus this indicates robustness problems

Combined force/velocity feedback:

- Use the low frequency performance advantages of geophone sensor with the high robustness advantages of the load cell sensor
- A Single-Input-Multiple-Outputs (SIMO) controller is found using LQG
- The performance requirements are met
- Good robustness

Figure 4: Experimental open loop (solid) and closed loop six-axis transmissibility using the geophone only controller (dotted), and combined geophone/load cell controller (dashed)

# Bibliography

Hauge, G., & Campbell, M., Sensors and control of a space-based six-axis vibration isolation system, Journal of Sound and Vibration, 269(3-5), 913–931 (2004). http://dx.doi.org/10.1016/s0022-460x(03)00206-2 ↩