Reach envelope of a 9-degree-of-freedom model of the upper extremity

This paper presents a rigorous mathematical formulation for modelling the upper extremity that is capable of considering a relatively large number of degrees of freedom, thus yielding a realistic model and associated envelope. Kinematic models are used to determine the reach envelope in closed form and to better understand human motion. Joint ranges of motion are taken into account by transforming unilateral inequality constraints into equalities that are included in the formulation. Methods from geometry are implemented to analyze the motion and delineate barriers within the workspace. These barriers are, in fact, observed to be surfaces where one or more joints of the limb are at their limits, but also where the hand's motion has encountered a kinematic singular configuration. Such a configuration is mathematically defined, and is physically associated with two links being parallel at an instant in time or where two joints axes are parallel (e.g., a fully extended arm yields a singular configuration). Barriers to motion can now be characterized in terms of different human performance measures, thus leading to a better understanding of the path trajectories assumed by humans as they execute tasks.