Analytical Dynamic Modelling of Heel-off and Toe-off Motions for a 2D Humanoid Robot

The main objective of this article is to optimize the walking pattern of a 2D humanoid robot with heel-off and toe-off motions in order to minimize the energy consumption and maximize the stability margin. To this end, at first, a gait planning method is introduced based on the ankle and hip joint position trajectories. Then, using these trajectories and the inverse kinematics, the position trajectories of the knee joint and all joint angles are determined. Afterwards, the dynamic model of the 2D humanoid robot is derived using Lagrange and Kane methods. The dynamic model equations are obtained for different phases of motion and the unknowns, including ground reactions, and joint torques are also calculated. Next, the derived dynamic model is verified by comparing the position of the ZMP point based on the robot kinematics and the ground reactions. Then, the obtained trajectories have been optimized to determine the optimal heel-off and toe-off angles using a genetic algorithm (GA) by two different objective functions: minimum energy consumption and maximum stability margin. After optimization, a parametric analysis has been adopted to inspect the effects of heel-off and toe-off motions on the selected objective functions. Finally, it is concluded that to have more stable walking in high velocities, small angles of heel-off and toe-off motions are needed. Consequently, in low velocities, walking patterns with large angles of heel-off and toe-off motions are more stable. On the contrary, large heel-off and toe-off motions lead to less energy consumption in high velocities, while small heel-off and toe-off motions are suitable for low velocities. Another important point is that for the maximum stability optimization, compared to minimum energy consumption optimization, more heel-off and toe-off motions are needed.