Instantaneous Walkability Determination Method for Almost Linear Passive Dynamic Walker with Nontrivial Limit Cycle Stability

Abstract

This paper proposes a novel passive dynamic walker with a body shape similar to an eight-legged rimless wheel that performs a natural swinging motion of the swing leg through storage and release of elastic energy. The generated motion is period-1 and asymptotically stable, but the inherent limit cycle stability is nontrivial because it does not achieve constraint on impact posture. Since it has almost linear dynamics, however, its walkability can be instantaneously determined using a linearized model without numerical integration. With the equations of linearized motion and exact collision, the step period and the state at the next collision can be obtained numerically and instantaneously using a bisection method based on the geometric constraint condition at impact. Then, by updating the state for each collision and repeating the same calculation, it is possible to instantaneously determine whether or not the walking motion continues stably for a long period of time. By comparing the results of this calculation with those of the numerical integration of the nonlinear and linearized models, the effectiveness of the proposed method is confirmed. Furthermore, using the proposed method, we analyze the period-doubling bifurcation phenomenon and the change in the singular values of the Poincaré map that occurs with the change in the elastic modulus.