The effects of displacement quantization and zero-order hold on the limit cycle behavior of haptic knobs

Haptic feedback devices measure human motion using displacement sensors such as optical encoders, and use actuators to apply computer-programmable forces to the user. Haptic devices act as both displays and input devices, and haptic system control loops include human operator dynamics. Limit cycle oscillations upon contact with a virtual barrier are a common problem in haptic devices, contributed to by factors such as discrete sampling effects and displacement quantization. Several investigators have examined discrete sampling effects, but a knowledge gap exists regarding the interaction of these two factors, particularly for coarse-resolution displacement sensors common in commercial haptic systems. As a prerequisite to simulations and theoretical analysis of a haptic knob in contact with a virtual barrier (unilateral spring) for various combinations of sample rate and displacement quantization, the present dissertation includes system identification experiments for the human hand grasping a haptic knob. A second-order rotational model fits the data for mild to moderate grasp forces, and a fourth-order model that treats the fingerpad and finger separately is suggested for strong grip forces. Simulations reveal that increasing displacement quantization increases limit cycle oscillation magnitude but has relatively little effect on limit cycle frequency. Increasing sample rate decreases oscillation magnitude but can easily increase limit cycle frequencies into ranges more easily perceptible by the sense of touch or hearing. Empirical data gathered with a haptic knob and human user support the simulation results. Analysis shows that displacement quantization exacerbates the well-known energy-instilling effects of the zero-order hold inherent in a computer-driven haptic system, especially upon entry and exit from the virtual barrier. The product of encoder resolution and sample period is shown to be a good predictor of the sensitivity of limit cycle oscillation magnitude to sample rate and displacement quantization. Describing function analysis is used to provide accurate predictions of limit cycle magnitude and frequency. Methods for mitigating limit cycle oscillations in systems with coarse displacement sensing, and other implications for design, are discussed, emphasizing the desirability of reducing limit cycle oscillation magnitude without increasing oscillation frequency into ranges of heightened sensitivity for the senses of touch or hearing.