Control of Equilibrium Position and Stiffness Through Postural Modules

If muscles are viewed as spring-like torque generators, then the integral of torque with respect to joint angle is the potential energy of that muscle. An energy function for the musculoskeletal system can be defined by summing the energy contribution of each muscle and the potential energy stored in the limb. Any local minimum in this energy landscape is a possible equilibrium position for the limb. The gradient of this function with respect to joint angles is a torque field, and the task of postural control is to find a set of muscle activations to produce a desired field. We consider one technique by which this approximation may be achieved: A postural module is defined as a synergy of muscles that produce a class of torque functions that converge at a constant equilibrium position, but whose stiffness at this position varies as a function of activation of the postural module. For a single-joint system, we show that through control of two such modules it is possible to produce any stiffness at any desired equilibrium position. To extend this scheme to a multijoint system, we initially derive the mechanical constraints on the shape of the restoring force field when a multijoint limb is displaced from equilibrium. Next, we consider voluntary control of the force field when the human arm is displaced from equilibrium: Mussa-Ivaldi, Hogan, and Bizzi (1985) have suggested that subjects are unable to voluntarily change the shape and orientation of the field, although they can readily scale it. This suggests existence of a limitation on the independent recruitment of arm muscles. We show, through simulation, that the inability to voluntarily control the shape and orientation of the restoring force field can be attributed to an organization of postural modules that act as local stiffness controllers. We propose that through coactivation, postural modules coarsely encode the work space and serve as an intermediate control system in the motor control hierarchy.