Energy-Consistent Force Feedback Laws for Virtual Environments

When digitally realized, virtual environments (VEs) do not perfectly match the physical environments they are supposed to emulate. This paper deals with energy aspects of such a mismatch, i.e., artificial energy leaks. A methodology is developed that employs smooth correction (SC) and leak dissipation (LD) to achieve a stable interconnection of the VE with the haptic device. The SC-LD naturally blends with the original laws for rendering the VE and gives rise to modified force feedback laws. These laws can be regarded as energy-consistent discretizations of their continuous-time counterparts. For some fundamental examples including virtual springs and masses, these laws are analytically reduced to simple closed-form equations. The methodology is then generalized to the multivariable case. Several experiments are conducted including a 2-DOF coupled nonlinear VE example, and a scenario leading to a sequence of contacts with a virtual object. Besides the conceptual advantage, simulation and experimental results demonstrate some other advantages of the SC-LD over well-known time-domain passivity methods. These advantages include improved fidelity, simpler implementation, and less susceptibility to produce impulsive/chattering response.