Workloop energetics of antagonist muscles.

The capability of muscle to produce mechanical work under periodic motions has been traditionally estimated using the workloop technique. We extend this method by allow- ing a pair of antagonist muscles to interact against a common load with non-zero admittance. We present an experimental approach to measuring the work done by a two-muscle system and show preliminary data. A complimentary problem is to maximize the work produced by the muscle pair in this setting. We formulate this problem in an optimal control framework. We derive conditions for the optimal activation of muscles to produce the maximum work, and show computed solutions. I. INTRODUCTION The capability of muscle to produce mechanical work under periodic motions has been traditionally estimated using the workloop technique. This technique was pioneered by Josephson (1), and has since then been the standard method for estimating the energetics of muscle in different species (2), (3), (4), (5). In this technique, the mechanical power delivered by the muscles is characterized by imposing sinu- soidal motion on a muscle while activating it via electrical stimulation for a brief duration at a particular phase of the cycle. Consequently a contraction force is generated, and a cyclic plot of force versus displacement defines a workloop. The instantaneous mechanical power delivered by the muscle is the product of the force and velocity of the interaction point. Integrating this power over a complete cycle yields an estimate of the work done by the muscle on its load, and is represented by the area enclosed by the workloop. While the workloop technique provides a benchmark test for estimating power output capability of muscle, it also provides a qualitative description of the function of muscle. In (6), it was shown that the shape, volume, and direction of a workloop provide a succinct description of its function in a periodic setting. For example, muscles can act as motors, breaks, springs, or struts, and thereby achieving different functional tasks. In this paper, we extend the workloop method in several ways. First, we allow the muscles to operate against loads that admit interaction. By contrast, in (1) and the ensuing literature, motion is imposed on a muscle with zero ad- mittance. Therefore, in our tests, the muscle forces cause the motion of the load as would be expected in realistic situations. Second, we allow for two muscles to interact against a common (virtual) load in an antagonist manner. This allows for testing how muscle co-contraction affects net workloop energetics and power transmission. Third, we provide a mathematical framework that is based on optimal control theory to maximize the work output of a muscle under loading conditions, instead of searching for the optimal parameters emperically. This problem is of interest for several reasons. First, it provides insight to the energetic interaction of muscles and loads in biomechanical systems. Second, it may provide an energetic interpretation to in vivo observations for muscle co-contracting during periodic cyclic motions. Thirdly, max- imizing a muscle's mechanical work output may be desir- able for certain applications, such as electrically stimulated induced workout (such as cycling) for stroke patients. It what follows, we present the experimenal procedure and show preliminary results. In terms of maximizing mus- cle power output, we present a mathematical formulation, derivation of the optimal control, and simulation results.