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 = ["Thomas Dehaeze"]
draft = false
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Tags
: [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Vibration Isolation]({{< relref "vibration_isolation" >}}), [Cubic Architecture]({{< relref "cubic_architecture" >}})
Reference
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: <sup id="f9698a1741fe7492aa9b7b42c7724670"><a class="reference-link" href="#hauge04_sensor_contr_space_based_six" title="Hauge \&amp; Campbell, Sensors and Control of a Space-Based Six-Axis Vibration Isolation System, {Journal of Sound and Vibration}, v(3-5), 913-931 (2004).">(Hauge \& Campbell, 2004)</a></sup>
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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
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<a id="org342e642"></a>
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{{< figure src="/ox-hugo/hauge04_stewart_platform.png" caption="Figure 1: Hexapod for active vibration isolation" >}}
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**Stewart platform** (Figure [1](#org342e642)):
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- Low corner frequency
- Large actuator stroke (\\(\pm5mm\\))
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- Sensors in each strut (Figure [2](#orge1d3dcd)):
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- three-axis load cell
- base and payload geophone in parallel with the struts
- LVDT
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<a id="orge1d3dcd"></a>
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{{< 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
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<a id="org5bf1a1a"></a>
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{{< 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.
<a id="table--tab:hauge05-comp-load-cell-geophone"></a>
<div class="table-caption">
<span class="table-number"><a href="#table--tab:hauge05-comp-load-cell-geophone">Table 1</a></span>:
Typical characteristics of sensors used for isolation in hexapod systems
</div>
| | **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
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<a id="org52ac01d"></a>
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{{< 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
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<a class="bibtex-entry" id="hauge04_sensor_contr_space_based_six">Hauge, G., & Campbell, M., *Sensors and control of a space-based six-axis vibration isolation system*, Journal of Sound and Vibration, *269(3-5)*, 913931 (2004). http://dx.doi.org/10.1016/s0022-460x(03)00206-2</a> [](#f9698a1741fe7492aa9b7b42c7724670)