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