State-Space Characterization of Balance Capabilities in Biped Systems with Segmented Feet

The human ability of keeping balance during various locomotion tasks is attributed to our capability of withstanding complex interactions with the environment and coordinating whole-body movements. Despite this, several stability analysis methods are limited by the use of overly simplified biped and foot structures and corresponding contact models. As a result, existing stability criteria tend to be overly restrictive and do not represent the full balance capabilities of complex biped systems. The proposed methodology allows for the characterization of the balance capabilities of general biped models (ranging from reduced-order to whole-body) with segmented feet. Limits of dynamic balance are evaluated by the Boundary of Balance (BoB) and the associated novel balance indicators, both formulated in the Center of Mass (COM) state space. Intermittent heel, flat, and toe contacts are enabled by a contact model that maps discrete contact modes into corresponding center of pressure constraints. For demonstration purposes, the BoB and balance indicators are evaluated for a whole-body biped model with segmented feet representative of the human-like standing posture in the sagittal plane. The BoB is numerically constructed as the set of maximum allowable COM perturbations that the biped can sustain along a prescribed direction. For each point of the BoB, a constrained trajectory optimization algorithm generates the biped's whole-body trajectory as it recovers from extreme COM velocity perturbations in the anterior-posterior direction. Balance capabilities for the cases of flat and segmented feet are compared, demonstrating the functional role the foot model plays in the limits of postural balance. The state-space evaluation of the BoB and balance indicators allows for a direct comparison between the proposed balance benchmark and existing stability criteria based on reduced-order models [e.g., Linear Inverted Pendulum (LIP)] and their associated stability metrics [e.g., Margin of Stability (MOS)]. The proposed characterization of balance capabilities provides an important benchmarking framework for the stability of general biped/foot systems.