A model-independent definition of attractor behavior applicable to interactive tasks

Both in designing teleoperators or haptic interfaces and in fundamental biological motor control studies, it is important to characterize the motor commands and mechanical impedance responses of the operator (or subject). Although such a characterization is fundamentally impossible for isolated movements when these two aspects of motor behavior have similar time scales (as is the case with humans), it is nonetheless possible, if we are dealing with repeated movements, to measure a trajectory which is analogous to the current source in Norton-equivalent electrical circuits. We define the attractor trajectory to be this equivalent source and show that it rigorously embodies the notion of the attractor point of a time-evolving system. We demonstrate that most previous attempts to test a controversial motor control hypothesis known as the equilibrium point or virtual trajectory hypothesis are based on inadequate models of the neuromuscular system and we propose here a model-independent means of testing the hypothesis based on a comparison of measurable attractor trajectories at different levels of the motor system. We present and demonstrate means of making such measurements experimentally and of assigning error bounds to the estimated trajectories.