A Practically Linear Relation Between Time Delay and the Optimal Settling Time of a Haptic Device

Abstract

Haptic rendering denotes the process of computing and displaying forces from a virtual environment to a human operator via a haptic device. From the control point of view, the complete haptic system comprising virtual environment, haptic device, and human operator is a hybrid control system that contains both discrete- and continuous-time elements. This letter investigates the influence of time delay on the theoretical optimal performance during haptic rendering with regard to minimal settling time, which belongs to the most frequently used design criteria in control engineering. It is shown that both continuous-time stiffness and damping of human operator or haptic device improve the optimal performance of the device. For the worst case, i.e., without such physical elements, the influence of delay on the optimal settling time becomes almost linear. This observation leads to an easy-to-remember rule of thumb for the optimal settling time that a haptic device should theoretically be able to reach. It states that each sampling period of additional time delay causes the optimal settling time (for a $2\%$ position threshold) to increase by approximately five sampling periods. In this linear relation, the effect of discrete-time sampling appears to correspond to a delay of one whole sampling period. The theoretical investigations are accompanied by a series of experiments on a DLR/KUKA light-weight robot, which shows that the newly introduced rule of thumb also applies for single joints of complex robotic systems.