Simulation of aperiodic bipedal sprinting.

Synthesis of legged locomotion through dynamic simulation is useful for exploration of the mechanical and control variables that contribute to efficient gait. Most previous simulations have made use of periodicity constraints, a sensible choice for investigations of steady-state walking or running. Sprinting from rest, however, is aperiodic by nature and this aperiodicity is central to the goal of the movement, as performance is determined in large part by a rapid acceleration phase early in the race. The purpose of this study was to create a novel simulation of aperiodic sprinting using a modified spring-loaded inverted pendulum (SLIP) biped model. The optimal control problem was to find the set of controls that minimized the time for the model to run 20 m, and this problem was solved using a direct multiple shooting algorithm that converts the original continuous time problem into piecewise discrete subproblems. The resulting nonlinear programming problem was solved iteratively using a sequential quadratic programming method. The starting point for the optimizer was an initial guess simulation that was a slow alternating-gait "jogging" simulation developed using proportional-derivative feedback to control trunk attitude, swing leg angle, and leg retraction and extension. The optimized aperiodic sprint simulation solution yielded a substantial improvement in locomotion time over the initial guess (2.79 s versus 6.64 s). Following optimization, the model produced forward impulses at the start of the sprint that were four times greater than those of the initial guess simulation, producing more rapid acceleration. Several gait features demonstrated in the optimized sprint simulation correspond to behaviors of human sprinters: forward trunk lean at the start; straightening of the trunk during acceleration; and a dive at the finish. Optimization resulted in reduced foot contact times (0.065 s versus 0.210 s), but contact times early in the optimized simulation were longer to facilitate acceleration. The present study represents the first simulation of multistep aperiodic sprinting with optimal controls. Although the minimized objective function was simple, the model replicated several complex behaviors such as modulation of the foot contact and executing a forward dive at the finish line. None of these observed behaviors were imposed explicitly by constraints but rather were "discovered" by the optimizer. These methods will be extended by addition of musculotendon actuators and joints in order to gain understanding of the influence of musculoskeletal mechanics on gait speed.