Like Playing a Video Game: Spatial-Temporal Optimization of Foot Trajectories for Controlled Football Kicking in Bipedal Robots

TL;DR

The paper tackles the challenge of precise and dynamic kicking in humanoid robot soccer while maintaining stability during aggressive motions. It introduces the STOFT Planner, a spatial-temporal trajectory optimization framework based on MINCO, integrated with MPC-based balance control and an adaptive gait scheduler to generate natural backswing trajectories for kicking. The approach enables real-time planning (under 1 ms) and self-collision avoidance, demonstrated on the PEARL humanoid robot in both simulation and hardware with high accuracy across kick angles. This work advances autonomous, precise shooting in robot soccer by providing a generalizable, real-time kicking planner that integrates perception, control, and motion planning for robust performance.

Abstract

Humanoid robot soccer presents 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 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, hindering 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 and hardware 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°.

Figures (7)

• Figure 1: Real-world and simulated kicking experiments with the PEARL biped robot. The robot has 5 degrees of freedom per leg, weighs 40 kg, and stands 1.5 m tall.
• Figure 2: 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.
• Figure 3: (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.
• Figure 4: Gait Schedule. The upper part shows the regular gait. The lower section shows the adaptive gait during kicking, where red boxes mark the swing phase extended by the STOFT planner compared to regular swing. The first two rows indicate the FSM states of the legs, and the third row shows the overall gait. The dashed box means the stance-hold phase has exceeded the threshold and is waiting for the other leg to exit the swing phase. "Single support" and "double support" denote whether one or both legs support the robot.
• Figure 5: Two sets of simulation experiments. The first set features the toe kick, while the second set demonstrates the robot's inside foot kick. In part (a), the snapshots illustrate the experiments, where the blue trajectory represents the predicted path, and the green trajectory indicates the actual path. Part (b) presents the experimental data in three rows: the top row compares the foot position reference with the actual data in the hip frame, the middle row displays foot velocity tracking in the hip frame and the bottom row showcases the robot's body velocity along with the foot Euler angles for both the toe kick and inside foot kick experiments, respectively. The orange dashed line indicates the timing of the hit.