Numerical Estimation of Balanced and Falling States for Constrained Legged Systems

Instability and risk of fall during standing and walking are common challenges for biped robots. While existing criteria from state-space dynamical systems approach or ground reference points are useful in some applications, complete system models and constraints have not been taken into account for prediction and indication of fall for general legged robots. In this study, a general numerical framework that estimates the balanced and falling states of legged systems is introduced. The overall approach is based on the integration of joint-space and Cartesian-space dynamics of a legged system model. The full-body constrained joint-space dynamics includes the contact forces and moments term due to current foot (or feet) support and another term due to altered contact configuration. According to the refined notions of balanced, falling, and fallen, the system parameters, physical constraints, and initial/final/boundary conditions for balancing are incorporated into constrained nonlinear optimization problems to solve for the velocity extrema (representing the maximum perturbation allowed to maintain balance without changing contacts) in the Cartesian space at each center-of-mass (COM) position within its workspace. The iterative algorithm constructs the stability boundary as a COM state-space partition between balanced and falling states. Inclusion in the resulting six-dimensional manifold is a necessary condition for a state of the given system to be balanced under the given contact configuration, while exclusion is a sufficient condition for falling. The framework is used to analyze the balance stability of example systems with various degrees of complexities. The manifold for a 1-degree-of-freedom (DOF) legged system is consistent with the experimental and simulation results in the existing studies for specific controller designs. The results for a 2-DOF system demonstrate the dependency of the COM state-space partition upon joint-space configuration (elbow-up vs. elbow-down). For both 1- and 2-DOF systems, the results are validated in simulation environments. Finally, the manifold for a biped walking robot is constructed and illustrated against its single-support walking trajectories. The manifold identified by the proposed framework for any given legged system can be evaluated beforehand as a system property and serves as a map for either a specified state or a specific controller's performance.