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<p>Among the numerous studies devoted to simulating human walking, very few take into account a non-instantaneous double support phase, i.e., one spread over a non-zero time interval. And yet this double support phase plays an important role, if only for the overall stability of the movement. A new periodic walking motion is proposed for a planar biped with quasi-ballistic single-support phases, where only the ankle of the supporting foot is actuated, and double-support phases with synchronized rotations of the feet, one around the heel and the other around the toes. The biped consists of a torso, two identical legs with knees, and massless feet. The synthesis of this walking period is based on a step that includes a single support phase and a double support phase. The quasi-ballistic movement is calculated by numerically solving a boundary value problem, taking two velocities into account. The initial velocities of the biped and the initial and final orientations of the foot of the pendulum leg are the unknowns in this boundary value problem. A velocity constraint is applied to the ankle of the foot just before it leaves the ground to prevent any unwanted contact rebound, and a velocity constraint is applied to the same ankle at the end of the movement to prevent the heel from impacting the ground. The double support phase begins as soon as the heel of the front foot touches the ground. The foot then rotates around its heel until it is flat on the ground. The rear foot begins to rotate around its toes in preparation for lifting off the ground. During this support phase, the ankle, knee, and hip joints are actuated by six motors. The six torques are piecewise constant functions whose values are the unknowns in an optimization problem designed to ensure the periodicity of the walking. Numerical results from the walking synthesis illustrate the approach. It is possible to see a video with animation of the design of bipedal walking in this paper, using reference <a href="https://uncloud.univ-nantes.fr/index.php/s/BmCMbnpaRZK8E2Z">here</a>. These numerical results are also compared with human walking data, freely available.</p>

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Bipedal walking: quasi-ballistic single-support phase, double-support under finite joint torques

  • 期刊论文

  • 发表日期：

    2026年3月10日

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  • Yannick Aoustin,
  • Alexander Formalskii

摘要

Among the numerous studies devoted to simulating human walking, very few take into account a non-instantaneous double support phase, i.e., one spread over a non-zero time interval. And yet this double support phase plays an important role, if only for the overall stability of the movement. A new periodic walking motion is proposed for a planar biped with quasi-ballistic single-support phases, where only the ankle of the supporting foot is actuated, and double-support phases with synchronized rotations of the feet, one around the heel and the other around the toes. The biped consists of a torso, two identical legs with knees, and massless feet. The synthesis of this walking period is based on a step that includes a single support phase and a double support phase. The quasi-ballistic movement is calculated by numerically solving a boundary value problem, taking two velocities into account. The initial velocities of the biped and the initial and final orientations of the foot of the pendulum leg are the unknowns in this boundary value problem. A velocity constraint is applied to the ankle of the foot just before it leaves the ground to prevent any unwanted contact rebound, and a velocity constraint is applied to the same ankle at the end of the movement to prevent the heel from impacting the ground. The double support phase begins as soon as the heel of the front foot touches the ground. The foot then rotates around its heel until it is flat on the ground. The rear foot begins to rotate around its toes in preparation for lifting off the ground. During this support phase, the ankle, knee, and hip joints are actuated by six motors. The six torques are piecewise constant functions whose values are the unknowns in an optimization problem designed to ensure the periodicity of the walking. Numerical results from the walking synthesis illustrate the approach. It is possible to see a video with animation of the design of bipedal walking in this paper, using reference here. These numerical results are also compared with human walking data, freely available.