digital-brain/content/article/hauge04_sensor_contr_space_based_six.md

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

Tags
[Stewart Platforms]({{< relref "stewart_platforms.md" >}}), [Vibration Isolation]({{< relref "vibration_isolation.md" >}}), [Cubic Architecture]({{< relref "cubic_architecture.md" >}})
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="<span class="figure-number">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="<span class="figure-number">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="<span class="figure-number">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="<span class="figure-number">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): 91331. doi:10.1016/s0022-460x(03)00206-2.