This paper employs velocity decomposition of underactuated mechanical systems to determine the degree of dynamic coupling in the gaits of a two-link biped model. The degree of coupling between controlled and uncontrolled directions quantifies the control authority the system has over its unactuated degree of freedom. This paper shows that the amount of coupling is directly correlated to gait robustness, as seen through the size of the gait's region of attraction. The analytical measure of coupling is applied in the context of trajectory optimization to generate two-link gaits that maximize or minimize coupling. Simulation studies show that gaits maximizing coupling exhibit significantly superior robustness, as measured by 1) stochastic performance on uneven terrain, 2) ability to maintain desired walking speed under non-vanishing disturbances, 3) size of the region of attraction, and 4) robustness to model uncertainties.

This paper shows how CasADi’s state-of-the-art implementation of algorithmic differentiation can be leveraged to formulate and efficiently solve gait optimization problems, enabling rapid gait design for high-dimensional biped robots. Comparative studies on a 7-DOF planar biped show that CasADi generates optimal gaits 4 times faster than another existing advanced optimization package. The framework is also applied to simultaneously generate a gait and a feedback controller for 2 spatial bipeds: a 12-DOF model and a 20DOF model. Results suggest that CasADi’s unprecedented efficiency could provide a practical path toward real-time gait optimization for high-dimensional biped robots.

This paper introduces a new gait switching approach for underactuated biped walkers. The switching condition relies on a reduced region of attraction - that of the unactuated dynamics - which is shown to be sufficient to predict falls. A gait transition is accordingly stable (as in “the robot does not fall if the biped's state is within the reduced region of attraction of the switched-in gait when switching takes place. Two- and five-link biped models complete a sequence of random gait transitions using the switching logic. The condition is also used to enlarge the full region of stability of the five-link model by embedding feedback-stabilized trajectories from a gait library in a mapping. The mapping stitches stable trajectories to the orbit of the desired gait based on each gait's reduced region of attraction. This improves the robustness of the five-link model walking on uneven terrain without ground perception regardless of the size of the gait library.