GRF-based trajectory planning and application for bipedal motion

  • Qianying Zhu orcid

    Faculty of Architecture and Civil Engineering, Huai’an University, Huai’an 223001, China
  • Aliyu Hamza orcid

    Faculty of Architecture and Civil Engineering, Huai’an University, Huai’an 223001, China
  • Yanan Gao orcid

    Faculty of Architecture and Civil Engineering, Huai’an University, Huai’an 223001, China
  • Jiaqi Gao orcid

    College of Traditional Chinese Medicine, Xinjiang Medical University, Urumqi 830011, China
  • Shouhua Liu orcid

    Faculty of Architecture and Civil Engineering, Huai’an University, Huai’an 223001, China
  • Enping Guo orcid

    School of Civil and Environmental Engineering, Hunan Institute of Science and Engineering, Yongzhou 425199, China
  • Qiuyue Cheng orcid

    School of Architectural Engineering, Chuzhou Polytechnic College, Chuzhou 239000, China
  • Yanxiang Chen orcid

    School of Civil and Environmental Engineering, Hunan Institute of Science and Engineering, Yongzhou 425199, China
  • Yuanyang Chen orcid

    Faculty of Architecture and Civil Engineering, Huai’an University, Huai’an 223001, China
  • Yue Wang orcid

    Jiangsu Huaiyin Power Generation Co., Ltd., Huai’an 223002, China
Article ID: 4582
DOI: https://doi.org/10.59400/adecp4582
Keywords: bipedal gait; ground reaction force; trajectory design; sliding mode control; application verification

Abstract

Trajectory planning for the center of mass (CoM) constitutes a fundamental challenge in bipedal locomotion. Conventional approaches, which predominantly rely on experimental measurements or simplified analytical models, are often constrained by discretization errors, parameter uncertainty, and prohibitive experimental costs. To address these limitations, this paper proposes a numerical framework for generating CoM trajectories directly from ground reaction force (GRF) dynamics. The methodology first determines key gait event positions and formulates a coupled system of governing equations that integrate force and motion. The method then establishes a unified spatio-temporal-GRF coordinate system and incorporates boundary conditions, along with load-sharing mechanisms between the trailing and leading legs, to derive the force and geometric constraint equations. The resulting reference trajectory is represented using Fourier coefficients, which are determined via an Adam-based constrained residual minimization solver. This Fourier-based representation offers notable advantages, including a compact parameter set, high computational efficiency, and the capacity to capture the essential kinematic features of bipedal walking. To validate the proposed reference trajectory, a sliding mode control (SMC) framework is employed for bipedal walking control, in which the walker successfully tracks the reference trajectory with stable gait execution. The results confirm that the generated trajectory faithfully reproduces the characteristic CoM response of bipedal walking, thereby providing a high-fidelity reference for robust motion control. Furthermore, a systematic analysis of the control parameters is conducted to evaluate their influence on walking stability. This work offers a new paradigm that synergistically combines physical interpretability with computational efficiency, holding significant promise for applications in humanoid robot motion control and biomechanical analysis.

Published
2026-08-26
How to Cite
Zhu, Q., Hamza, A., Gao, Y., Gao, J., Liu, S., Guo, E., Cheng, Q., Chen, Y., Chen, Y., & Wang, Y. (2026). GRF-based trajectory planning and application for bipedal motion. Advances in Differential Equations and Control Processes, 33(3). https://doi.org/10.59400/adecp4582
Issue
Vol. 33 No. 3 (2026): In Progress
Section
Article

Copyright (c) 2026 Qianying Zhu, Aliyu Hamza, Yanan Gao, Jiaqi Gao, Shouhua Liu, Enping Guo, Qiuyue Cheng, Yanxiang Chen, Yuanyang Chen, Yue Wang

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

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