One of the more commonly used position sensors in haptic displays is the optical encoder. Encoders are reasonably rugged and easy to interface, and are extremely linear and free of dynamics. Unfortunately, the output of an encoder is quantized, and it is well-known that quantization can lead to limit cycles in digital control systems [8]. Of course, the angle quanta are typically quite small and therefore cause little practical problem, unless the quantized signal is differentiated (e.g., to obtain a velocity signal for virtual damping).
Differentiation is notorious for amplification of high frequency noise. In
the context of an encoder-based, sampled-data control system, the consequences
of differentiation are easily understood. Suppose that a quantum is
radians and the sampling period is T seconds. Then the resolution of a
finite difference differentiator is
rad/sec. If an 8000 count/rev encoder is used and T = 0.001 sec, then the
velocity resolution is
!
If we further assume a 0.1 m lever arm, then the smallest measurable
translational velocity at the tip of this lever arm is 7.8 cm/sec. Clearly,
this is an extremely high velocity compared to that which would be desirable
when contacting a wall.
How can a better velocity estimate be obtained? One way to improve resolution is to sample more slowly! Unfortunately, this runs contrary to the goal of high stiffness as discussed above. Another approach is to filter the velocity estimate digitally. This will be discussed briefly below. A third approach is to use higher resolution encoders. Simulation experience suggests that encoder resolution has little effect on the existence of limit cycles, but considerable effect on the amplitude of limit cycles [10]. A fourth approach is to use analog sensors (for position, velocity, or both), although these sensors suffer from noise as well.
As an aid in understanding the effects of filtering, consider a virtual wall
implementation in which the differentiator is cascaded with a first order low
pass filter of time constant
.
Details of the derivation are shown in
Appendix A.
If a backwards difference
mapping is used, the transfer function of the wall is:
|
(6) |
It is easily shown that the resolution of the filtered differentiator is
.
Thus, the slower the filter, the better the velocity resolution, as might be
expected. In practice, we generally find that better than an order of
magnitude improvement may be obtained in resolution with no obvious
performance cost (the experiments described below provide a quantitative
assessment).
One might also expect that the cost of filtering would be that the haptic display becomes less passive. In general, this is true because filters introduce delay. For the first order filter considered here, however, the condition for passivity is considerably less restrictive than without a filter (compare to Equation 5):
|
|
(7a) |
|
|
(7b) |
In this section, we have discussed the features of tool simulation, particularly impedance range and unilateral constraints. We have also reviewed aspects of passivity that relate to human manipulation. We have suggested that passivity plays a role in physical interaction, and that it provides a useful means to evaluate the performance of a haptic display (Z-width). Finally, we have reviewed the some of the factors affecting the Z-width of a haptic display (inherent mechanism dynamics, sampling rate and sensor resolution). To verify these effects, particularly that of mechanism damping, we conducted human subject experiments using a 1 DOF haptic display in our laboratory. In the next section, we will cover the experimental protocol, results and conclusions of these experiments.
