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Research Papers

Simulation of Aperiodic Bipedal Sprinting

[+] Author and Article Information
Huseyin Celik

Department of Mechanical and Nuclear Engineering,
The Pennsylvania State University,
University Park, PA 16802

Stephen J. Piazza

Department of Kinesiology,
The Pennsylvania State University,
University Park, PA 16802;
Department of Mechanical and Nuclear Engineering,
Department of Orthopaedics and Rehabilitation,
The Pennsylvania State University,
Hershey, PA 17033
e-mail: piazza@psu.edu

1Corresponding author.

Contributed by the Bioengineering Division of ASME for publication in the JOURNAL OF BIOMECHANICAL ENGINEERING. Manuscript received November 19, 2012; final manuscript received April 19, 2013; accepted manuscript posted May 15, 2013; published online June 12, 2013. Assoc. Editor: Richard Neptune.

J Biomech Eng 135(8), 081008 (Jun 12, 2013) (8 pages) Paper No: BIO-12-1567; doi: 10.1115/1.4024577 History: Received November 19, 2012; Revised April 19, 2013; Accepted May 15, 2013

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.

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Figures

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Fig. 1

The simple biped model used to simulate sprinting. Body segment inertial properties shown in the figure are defined in the text, as are the generalized coordinates of the model, hip actuator torques, and leg actuator forces. The right and left legs of the model were identical; labeling of the left leg inertial properties, generalized coordinates, and actuator forces and torques are omitted here for purposes of clarity. The left hip flexion angle θL is positive when the hip is flexed.

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Fig. 2

Stick-figure trajectories for the model (top) completing the 20 m course under PD control that produced a “jog” with duration of 6.64 s; and (middle) sprinting following optimization for which the course was covered in 2.79 s. The sprinting simulation begins with the trunk flexed forward, straightens as the race progresses and dives forward at the finish. The first 5 ms of the sprinting simulation are also shown in detail (bottom). The time between frames represented in these illustrations are 125 ms (top) and 53 ms (middle and bottom).

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Fig. 3

Forward velocity of the hip in for the initial guess jog (gray) and sprinting (black) simulations. Both simulations began from rest. The feedback-controlled jog slowly approached a steady forward velocity of approximately 4 m s−1. The sprinting simulation gains speed quickly over the first few steps, then reaches a steady speed of about 8 m s−1 for much of the race, before diving forward at the end.

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Fig. 4

The horizontal (continuous lines) and vertical (broken lines) ground reaction forces of the initial guess jog simulation (top) and the sprinting simulation (bottom). Ground reaction forces for the left and right feet are shown in gray and black, respectively.

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Fig. 5

The net horizontal impulses of the ground reaction force (GRF) for each step during the initial guess jog simulation (unfilled markers) and the sprinting simulation (filled markers). Impulses for GRFs applied to both the right (diamonds) and left (squares) feet are shown. Large forward impulses were generated in the first few steps of the sprinting simulation and again in the last two steps to generate the terminal dive.

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Fig. 6

The angular position of the trunk in the sprinting simulation. The trunk angle was defined such that negative values of θt corresponded to forward flexion (Fig. 1). The negation of that angle is plotted here, with 90 deg corresponding to the trunk parallel to the ground and 0 deg indicating an upright posture.

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Fig. 7

Temporal foot contact pattern for the initial guess jog simulation (gray) and the sprinting simulation (black). Both simulations resulted in alternating gaits. While the foot contacts in the initial guess simulation were fairly constant in duration, in the sprinting simulation contact times were larger at the start during the acceleration phase and became much shorter for the remainder of the simulation.

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Fig. 8

Flexion of the right hip plotted versus time for the sprinting simulation. Right foot contact (circles), consistently occurred as the hip was beginning to extend following maximum flexion. This “leg retraction” behavior was not present in the initial guess jog simulation, for which foot contact always coincided with maximum hip flexion.

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