7.5 KiB
+++ 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.
| 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)" >}}