STOFT Planner

Humanoid robot soccer poses several challenges, particularly in maintaining system stability during aggressive kicking motions while achieving precise ball trajectory control. Current solutions, whether traditional position-based control methods or reinforcement learning (RL) approaches, exhibit significant limitations. Model predictive control (MPC) is a prevalent approach for ordinary quadruped and biped robots. While MPC has demonstrated advantages in dynamic motion control for legged robots, existing studies often oversimplify the leg swing progress, relying merely on simple trajectory interpolation methods. This severely constrains the foot's environmental interaction capability, which is particularly detrimental for tasks such as ball kicking. This study innovatively adapts the spatial-temporal trajectory planning method, which has been successful in drone applications, to bipedal robotic systems. The proposed approach autonomously generates foot trajectories that satisfy constraints on target kicking position, velocity, and acceleration while simultaneously optimizing swing phase duration. Experimental results demonstrate that the optimized trajectories closely mimic human kicking behavior, featuring a backswing motion. Simulation experiments confirm the algorithm's efficiency, with trajectory planning times under 1 ms, and its reliability, achieving nearly 100 \% task completion accuracy when the soccer goal is within the range of -90° to 90°.

Control Framework: The robot operator provides control commands, including the desired robot velocity \((\boldsymbol{v}^{des}, \boldsymbol{\omega}^{des})\) and the kicking foot target \((\boldsymbol{p}_{kick}^{des}, \boldsymbol{v}_{kick}^{des})\). The state estimator calculates the current system states \((\boldsymbol{p}^{cur}, \boldsymbol{\dot{p}}^{cur}, \boldsymbol{\Theta}^{cur}, \boldsymbol{\omega}^{cur})\) and foot states \((\boldsymbol{p}^{cur}_{f}, \boldsymbol{\dot{p}}^{cur}_{f})\). The gait generator then produces an adaptive gait schedule \(\boldsymbol{\Upsilon}\). Using the control commands, current states, and gait schedule, the MPC module solves an OCP to compute the optimal GRFs and GRTs \((\boldsymbol{f}^{des}, \boldsymbol{\tau}^{des})\). Simultaneously, the foot planner generates a reference trajectory \(\boldsymbol{\phi}(t)\) for either executing a kick or regular walking. Finally, based on the desired reference trajectory, GRFs, and GRTs, the joint torques \(\boldsymbol{\tau}^{des}_{joint}\) are computed for precise actuation.

(a) Side view showing the foot trajectory. (b) Top-down view illustrating the foot orientation during the kicking task. Points A, B, and C represent the initial position, intermediate target, and desired foothold, respectively. The orange arrow indicates the kicking velocity, and the red trajectory is optimized to satisfy position, velocity, and dynamic constraints at points A, B, and C. In (a), (1) and (2) illustrate regular walking, while (3) and (4) demonstrate a high-velocity kick. In (b), (1) shows a straight kick, and (2) depicts a side kick.

Accuracy test for STOFT. The left figure illustrates the testing scenario where kicking angles \(\theta\) vary from -90° to 90° with the robot positioned \(z_d = 3 \ m\) from the soccer goal, while the right figure presents the success rates of both the STOFT planner and the baseline methods across different angles. The STOFT planner consistently achieves a higher scoring success rate than the baseline methods.