Effect of Virtual Mass and Time Delay on the Stability of Haptic Rendering

Virtual mass simulation is one of the recent topics in the field of haptic devices (HDs), which can alter the apparent mass of the HD. Simulating negative values of virtual mass leads to a decrease in the apparent effective mass, improving transparency but weakening stability. Positive virtual mass rendering increases the apparent mass, reduces transparency, and enhances stability. This paper analyzes the stability of a haptic device while simulating a virtual environment consisting of a mass, spring, and damper in the presence of a constant time delay. The results are closed-form equations that can predict the stability boundary for small and even large values of virtual damping and time delay. These closed-form equations demonstrate that the maximum renderable virtual mass is twice the physical mass of the HD, and the minimum value equals its negative; both occur in the case of zero time delay. Increasing the time delay reduces both the minimum and maximum values of the renderable virtual mass. The study also shows that using virtual mass can improve the maximum value of a renderable virtual spring. The equations show that, in the absence of delay, properly tuning the virtual mass and virtual damping can enlarge the maximum renderable stiffness by up to 5.8 times in theory. In the experiments under time delay, the stiffness increased by a factor of 3.5, compared to the theoretical prediction of 4.1 times. The results further reveal situations where a nonzero minimum stiffness is required for stability. All findings are validated via simulations and experiments on a dedicated test bed.