M-Width: Stability and Accuracy of Haptic Rendering of Virtual Mass

Abstract

Abstract-In many physical human-robot interaction scenarios, such as haptic virtual environments for training and rehabilitation, it is desirable to carefully control the apparent inertia of a robot. Inertia compensation can be used to mitigate forces felt by the user during free-space motion, and rendering of additional inertia is desired for particular rehabilitation and training procedures. Many factors influence the stability and accuracy of rendering for haptic display of a pure mass, including device mechanical properties, sample rate, control structure, and human behavior. Inspired by the "Z-Width" approach to haptic device stability and performance analysis, we introduce "M-width", which we define as the dynamic range of virtual masses renderable in a stable manner. We identify the important parameters for system stability, find stability boundaries, and describe the expected accuracy of the haptic rendering for a canonical haptic system. These results serve as a design tool for creating stable and accurate haptic virtual environments, establish limits of performance, and lay the groundwork for new controllers to improve mass rendering.

Effects of discretization on the K-width of series elastic actuators

“…We then support the derived K-width using a discrete model, with which coupled stability can be numerically evaluated [15]- [18]. Our resultant approach is perhaps most similar to Colonnese and Okamura [16], in that we will consider both continuous and discrete models; however, unlike [16], we here claim to find a closed-form passivity constraint.…”

“…We then plotted these results in Fig. 4, along with the passivity region defined by (16). From the resultant plot, we observed that our passivity region is contained within each of the coupled stability regions, as should be expected.…”

“…Based on this plot, it is clear that as m h increases, the coupled stability region converges to the passivity region regardless of sampling frequency; hence, we conclude that large pure masses (for example, m h /m > 3) are the most destabilizing operator for our SEA system when T > 0. Moreover, the fact that k d,stable converges to k d,passive supports the passivity constraint (16).…”

“…This discrete model will be used to determine the maximum virtual stiffness which results in stability when the SEA is coupled to a human operator of known impedance. Importantly, because the discrete model's stability bounds are the same as the stability bounds of the hybrid system [16], no approximations are necessary here. We will, however, need to choose Z h , the impedance of the human operator.…”

“…k d,stable is defined as the upper bound of (22) when considering Z h,1 , the impedance model for a pure mass attached to the load. k d,passive is defined as the upper bound of (16). With the exception of the independent variable m h , all parameters were taken from Table I.…”