Optimized 3D stable walking of a bipedal robot with line-shaped massless feet and sagittal underactuation

• We propose a 3D robot model with massless line feet and a corresponding contact model.
• We obtain 3D stable walking with one degree of underactuation using a previous 2D method.
• We generate walking trajectories and optimize them to reduce the energy cost.
• The optimized gaits with smooth energy cost curves are obtained from 0.2 m/s to 1.5 m/s.

This paper seeks to provide a method for optimizing 3D bipedal walking with satisfactory energy efficiency, provable stability and one degree of underactuation. Following the studies of the planar biped RABBIT, we propose a 3D robot with massless line feet which serve as sagittal rotation axes when lying flat on the ground. Using this configuration and the control method based on virtual constraints and feedback linearization, the walking stability can be verified by the restricted scalar Poincaré map of the zero dynamics of the system, and periodic gaits can be obtained analytically as in the case of planar bipeds with point feet. A line-foot contact model is adopted to ensure one degree of underactuation. Unlike most published contact models of bipedal walking, this model also prevents possible yaw movements of the feet on the ground. In addition, we adopt an output function named “symmetry outputs” with the desired outputs parameterized by Bézier coefficients and postural parameters, and the gait optimization is performed using a SQP algorithm. According to the results obtained from a bipedal model with a height of 1.5 m and a weight of 39 kg, the optimization program is capable of calculating stable periodic gaits with a speed between 0.2 m/s and 1.5 m/s.