Leg-adjustment strategies for stable running in three dimensions

The dynamics of the center of mass (CoM) in the sagittal plane in humans and animals during running is well described by the spring-loaded inverted pendulum (SLIP). With appropriate parameters, SLIP running patterns are stable, and these models can recover from perturbations without the need for corrective strategies, such as the application of additional forces. Rather, it is sufficient to adjust the leg to a fixed angle relative to the ground. In this work, we consider the extension of the SLIP to three dimensions (3D SLIP) and investigate feed-forward strategies for leg adjustment during the flight phase. As in the SLIP model, the leg is placed at a fixed angle. We extend the scope of possible reference axes from only fixed horizontal and vertical axes to include the CoM velocity vector as a movement-related reference, resulting in six leg-adjustment strategies. Only leg-adjustment strategies that include the CoM velocity vector produced stable running and large parameter domains of stability. The ability of the model to recover from perturbations along the direction of motion (directional stability) depended on the strategy for lateral leg adjustment. Specifically, asymptotic and neutral directional stability was observed for strategies based on the global reference axis and the velocity vector, respectively. Additional features of velocity-based leg adjustment are running at arbitrary low speed (kinetic energy) and the emergence of large domains of stable 3D running that are smoothly transferred to 2D SLIP stability and even to 1D SLIP hopping. One of the additional leg-adjustment strategies represented a large convex region of parameters where stable and robust hopping and running patterns exist. Therefore, this strategy is a promising candidate for implementation into engineering applications, such as robots, for instance. In a preliminary comparison, the model predictions were in good agreement with the experimental data, suggesting that the 3D SLIP is an appropriate model to describe human running in three dimensions. The prediction of stable running based on movement-related leg-adjustment strategies indicates that both humans and robots may not require external targets directing the movement to run in three dimensions based on compliant leg function. This new movement-based reference enables the control of 3D running because leg adjustment is less sensitive and gait stability is separated from directional stability.