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Optimal Control and Emergence of Kinematic Synergies in Underactuated Biped Locomotion

Authors :

Mahdi Alipoor¹ Masoud Yousefi² Farzam Farahmand³

1- دانشگاه صنعتی شریف 2- دانشگاه صنعتی شریف 3- دانشگاه صنعتی شریف

Keywords :

Trajectory optimization،bipedal locomotion،kinematic synergy،Non-Negative Matrix Factorization (NNMF)،optimal control

Abstract :

This study examines the relationship between cost function selection in trajectory optimization and the resulting joint coordination patterns in a planar five-link biped model. The model represents a simplified humanoid with two legs and a torso connected by revolute joints, and a pointed foot that makes the system underactuated. Four cost functions—sum of squared joint torques, sum of squared hip–knee torques, sum of squared joint powers, and squared head acceleration—were optimized using direct collocation combined with a global search strategy in the OptimTraj framework. The resulting optimal kinematics were analyzed using Non-Negative Matrix Factorization (NNMF) to extract kinematic synergies and evaluate coordination through the Variance Accounted For (VAF). Results indicate that the power-squared and head acceleration-squared cost functions produced two synergies, achieving global VAF values of 94% and 88%, respectively, while the torque squared and hip–knee torque-squared cost functions required three. Across all cost functions, the torso joint exhibited the lowest local VAF, likely due to its periodic motion, which could not be coupled with the motions of other joints using NNMF. The power-squared cost function demonstrated the highest coordination (global VAF 94% and local VAF 85%), suggesting that energy-based optimization encourages the emergence of more coherent synergy structures. These findings highlight both the influence of optimization objectives on coordination and the need for nonlinear approaches to more accurately model joint synergies in underactuated locomotion.