  This paper introduces a novel multi-step preview foot placement planning
algorithm designed to enhance the robustness of bipedal robotic walking across
challenging terrains with restricted footholds. Traditional one-step preview
planning struggles to maintain stability when stepping areas are severely
limited, such as with random stepping stones. In this work, we developed a
discrete-time Model Predictive Control (MPC) based on the step-to-step discrete
evolution of the Divergent Component of Motion (DCM) of bipedal locomotion.
This approach adaptively changes the step duration for optimal foot placement
under constraints, thereby ensuring the robot's operational viability over
multiple future steps and significantly improving its ability to navigate
through environments with tight constraints on possible footholds. The
effectiveness of this planning algorithm is demonstrated through simulations
that include a variety of complex stepping-stone configurations and external
perturbations. These tests underscore the algorithm's improved performance for
navigating foothold-restricted environments, even with the presence of external
disturbances.
