> 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.
- 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
| 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:
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:
- 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
{{<figuresrc="/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)">}}
<aid="org186272b"></a>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.
