Adaptive LIPM-based trajectory planning for energy-efficient bipedal locomotion on inclined terrains

Achieving stable and energy-efficient bipedal locomotion, particularly on inclined terrain, is challenging due to gravitational disturbances and the limitations of simplified models like the Linear Inverted Pendulum Model (LIPM). In the present work, a Slope Adaptive LIPM (SA-LIPM) is used for trajectory planning and control of a 12-DOF lower-limb bipedal robot on level, 5°, and 10° slopes. The control outputs are analyzed through detailed high-end dynamic simulations to perform stability analysis using Zero Moment Point (ZMP), trajectory analysis of Centre of Mass (COM) and Ground Reaction Forces (GRF) distribution for each step. Moreover, the total energy consumption and joint level energy audit is also carried out. The obtained results show that the ZMP stability criterion was successfully met across all terrains. The COM trajectory analysis indicates that inclines require a quantified adjustment of up to 12% to the planned trajectory for 10° slopes. In case of energy consumption, the total energy expenditure increased by 7% to 10% on the 10° incline compared to level ground, with the knee joint being the primary source of increased energy cost. The results confirm the effectiveness of a slope adaptive LIPM for both flat and sloped walking and provide a critical joint-level energy breakdown, which is essential for designing efficient hardware and advanced torque controllers for future humanoid platforms.