Numerical construction of balanced state manifold for single-support legged mechanism in sagittal plane

Legged mechanisms can walk and run but sometimes encounter a risk of falling. In this article, a general numerical framework of balance criteria for a single-support legged mechanism is proposed and demonstrated. Explicit forms of necessary and sufficient conditions for balancing are identified and the balanced state manifold is constructed accordingly in the extended phase space of joint angle, joint velocity, and actuation limit. Within the iteration loops for partitioned joint angle and actuation limit, a nonlinear constrained optimization problem is formulated where the dynamic models of the legged mechanism are incorporated. The necessary conditions for balancing, such as the Zero-Moment Point, positive normal reaction, friction, and the ability to end up at a final static equilibrium, are implemented along with the system parameters for generality. The sequential quadratic programming method numerically solves for the velocity extrema within the complete feasible domain to construct the balanced state manifold as a viability kernel, which is a reachable superset of all possible controller-based domains. The balanced state manifold, along with its demonstration using the proposed optimal balancing motion for minimum energy and biped walking motions in sagittal plane, shows valid features that are physically consistent. The framework can be extended to systems in three-dimension with higher complexity, both in single and double support phases, for the development and stability analysis of walking robots and humans.