digital-brain/content/article/hauge04_sensor_contr_space_based_six.md

+++ title = "Sensors and control of a space-based six-axis vibration isolation system" author = ["Thomas Dehaeze"] draft = false +++

Tags

[Stewart Platforms]({{< relref "stewart_platforms" >}}), [Vibration Isolation]({{< relref "vibration_isolation" >}}), [Cubic Architecture]({{< relref "cubic_architecture" >}})

Reference

(Hauge and 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 src="/ox-hugo/hauge04_stewart_platform.png" caption="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 src="/ox-hugo/hauge05_struts.png" caption="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 src="/ox-hugo/hauge04_strut_model.png" caption="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 src="/ox-hugo/hauge04_obtained_transmissibility.png" caption="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.S., and M.E. Campbell. 2004. “Sensors and Control of a Space-Based Six-Axis Vibration Isolation System.” Journal of Sound and Vibration 269 (3-5):913–31. https://doi.org/10.1016/s0022-460x(03)00206-2.