基于广义Udwadia-Kalaba方程的移动机器人横纵向耦合轨迹跟踪控制

doi:10.3901/JME.2025.19.078

摘要/Abstract

摘要： 在运动控制领域，移动机器人系统通常需要满足一系列等式(包括完整性和非完整性)与不等式约束，以完成目标任务。然而，目前的研究主要集中在移动机器人轨迹跟踪性能方面，缺乏对其运动过程中横纵耦合控制的深入研究。为此，提出了一种基于广义Udwadia-Kalaba方程的横纵耦合约束跟随控制方法。首先，针对纵向跟踪任务，设计基于伺服等式约束理论的纵向轨迹跟踪控制器，并利用李雅普诺夫稳定性理论证明了其稳定性。其次，根据不等式约束信息，分别建立了y方向的固定约束以及直线和圆轨迹的边界约束。在此基础上，提出了一种能够满足边界不等式约束的横纵耦合控制器τ。最后，通过Matlab仿真软件对所提方法进行了数值仿真验证。仿真结果表明，与传统的PD控制和模型预测控制相比，所提控制器在满足不等式约束的同时，能高效完成目标任务，具有更快的收敛速度和优越的控制性能。同时，基于ROS移动机器人物理实验平台对所提控制方法的实际有效性进行验证，实现了不等式约束条件下的定点、直线及圆形轨迹跟踪控制，为移动机器人横纵耦合控制领域提供了新的研究思路。

关键词: 机械系统, 等式/不等式约束, 广义Udwadia-Kalaba方程, 李雅普诺夫稳定性, 横纵耦合控制

Abstract: In the field of motion control, mobile robot systems usually need to satisfy a series of equations (including completeness and non-completeness) with inequality constraints in order to accomplish the target task. However, the current research mainly focuses on the trajectory tracking performance of mobile robots and lacks an in-depth study on the transverse-longitudinal coupling control during their motion. To this end, a transverse-longitudinal coupled constraint following control method based on the generalized Udwadia-Kalaba equation is proposed. First, for the longitudinal tracking task, a longitudinal trajectory tracking controller based on the servo-equivalent constraint theory is designed and its stability is proved by using the Lyapunov stability theory. Secondly, based on the inequality constraint information, the fixed constraints in the y-direction and the boundary constraints for linear and circular trajectories are established respectively. On this basis, a transverse-longitudinal coupling controller τ that can satisfy the boundary inequality constraints is proposed. Finally, the proposed method is verified by numerical simulation through Matlab simulation software. The simulation results show that, compared with the traditional PD control and model predictive control, the proposed controller can efficiently accomplish the target task with faster convergence speed and superior control performance while satisfying the inequality constraints. Meanwhile, the practical effectiveness of the proposed control method is verified based on the ROS mobile robot physics experimental platform, which realizes the fixed-point, linear and circular trajectory tracking control under the inequality constraints, and provides a new research idea in the field of transverse-longitudinal coupling control of mobile robots.

Key words: mechanical system, equation/inequality constraints, generalized Udwadia-Kalaba equation, Lyapunov stability, transverse-longitudinal coupled control

中图分类号:

TP242

张佳乐, 李志勇, 张传伟, 魏孟, 焦生杰. 基于广义Udwadia-Kalaba方程的移动机器人横纵向耦合轨迹跟踪控制[J]. 机械工程学报, 2025, 61(19): 78-90.

ZHANG Jiale, LI Zhiyong, ZHANG Chuanwei, WEI Meng, JIAO Shengjie. Transverse-longitudinal Coupled Trajectory Tracking Control of Mobile Robots Based on Generalized Udwadia-Kalaba Equation[J]. Journal of Mechanical Engineering, 2025, 61(19): 78-90.

参考文献

[1] 宋荆洲，宫兴龙，段嘉辰，等. 可跳跃移动机器人机构设计与跳跃过程控制研究综述[J]. 机械工程学报，2024，60(15)：1-17. SONG Jingzhou，GONG Xinglong，DUAN Jiachen，et al. Review on mechanism design and jumping process control of jumpable mobile robots[J]. Journal of Mechanical Engineering，2024，60(15)：1-17.
[2] 周竞烨，李家旺，邸青，等. 非完整轮式机器人的鲁棒同步编队跟踪及镇定控制[J]. 控制理论与应用，2020，37(7)：1461-1470. ZHOU Jingye，LI Jiawang，DI Qing，et al. Robust simultaneous formation tracking and stabilization of nonholonomic wheeled mobile robots[J]. Control Theory & Applications，2020，37(7)：1461-1470.
[3] TAKAHIRO M，JOONHO L，JEMIN H，et al. Learning robust perceptive locomotion for quadrupedal robots in the wild[J]. Science Robots，2022，62(7)：1-14.
[4] 刘江涛，周乐来，李贻斌. 复杂地形六轮独立驱动与转向机器人轨迹跟踪与避障控制[J]. 兵工学报，2024，45(1)：166-183. LIU Jiangtao，ZHOU Lelai，LI Yibin. Trajectory tracking and obstacle avoidance control of a six-wheel independent drive and steering robot in complex terrain[J]. Acta Armamentarii，2024，45(1)：166-183.
[5] 周卫琪，赵羿寒，刘擎超，等. 基于改进LQR的车辆路径跟踪横向控制策略[J]. 华中科技大学学报，2024，52(3)：135-141. ZHOU Weiqi，ZHAO Yihan，LIU Qingchao，et al. Lateral control strategy for vehicle path tracking based on improved LQR[J]. Journal of Huazhong University of Science and Technology，2024，52(3)：135-141.
[6] 庞辉，刘楠，刘敏豪，等. 基于线性时变模型预测控制的自主车辆轨迹跟踪控制器设计与验证[J]. 机械工程学报，2022，58(24)：264-274. PANG Hui，LIU Nan，LIU Minhao，et al. Design and validation of trajectory tracking controller for autonomous vehicle based on linear time-varying MPC Method[J]. Journal of Mechanical Engineering，2022，58(24)：264-274.
[7] AHMADI S M，BEHNAM T M，HAQSHENAS M A R. A state augmented adaptive backstepping control of wheeled mobile robots[J]. Transactions of the Institute of Measurement and Control，2021，43(2)：434-450.
[8] 马梓竣，张欣刚，张树翠，等. 轮式移动机器人的反演二次滑模优化轨迹跟踪控制策略研究[J]. 北京大学学报(自然科学版)，2024，60(4)：597-606. MA Zijun，ZHANG Xingang，ZHANG Shucui，et al. Research on trajectory tracking control strategy of backstepping optimized double sliding mode for two-wheeled robot[J]. Acta Scientiarum Naturalium Universitatis Pekinensis，2024，60(4)：597-606.
[9] 张佳乐，赵睿英，焦生杰，等. 基于Udwadia-Kalaba方程的交通锥机器人编队摆放控制方法[J]. 中国公路学报，2023，36(7)：236-248. ZHANG Jiale，ZHAO Ruiying，JIAO Shengjie，et al. Research on the Udwadia-Kalaba equation-based cone formation placement control method for traffic cone robots[J]. China Journal of Highway and Transport，2023，36(7)：236-248.
[10] UDWADIA F E，KALABA R E. Analytical dynamics：A new approach[M]. Cambridge：University Press，2007.
[11] CHEN Y H. Constraint-following servo control design for mechanical systems[J]. Journal of Vibration and Control，2009，15(3)：369-389.
[12] CHEN Y H，ZHANG X. Adaptive robust approximate constraint-following control for mechanical systems[J]. Journal of the Franklin Institute，2010，347(1)：69-86.
[13] ZHAO X，CHEN Y H，ZHAO H，et al. Udwadia-Kalaba equation for constrained mechanical systems：formulation and applications[J]. Chinese Journal of Mechanical Engineering，2018，31(6)：11-24.
[14] ZHANG B W，CHEN Y H，JIA T F，et al. Accuracy and safety：Tracking control of heavy-duty cooperative transportation systems using constraint-following method[J]. IEEE Transactions on Intelligent Transportation Systems，2024，25(5)：3628-3641.
[15] 李旻，熊亮，尹辉，等. 含不等式约束的欠驱动系统约束跟随控制[J]. 控制理论与应用，2020，37(9)：1971-1982. LI Min，XIONG Liang，YIN Hui，et al. Constraint-following control for underactuated systems with inequality constraints[J]. Control Theory & Applications，2020，37(9)：1971-1982.
[16] SUN H，TU L，YANG L，et al. Adaptive robust control for nonlinear mechanical systems with inequality constraints and uncertainties[J]. IEEE Transactions on Systems，Man，and Cybernetics：Systems，2023，53(3)：1761-1772.
[17] ZHAO H，ZHEN S，CHEN Y H. Dynamic modeling and simulation of multi-body systems using the Udwadia-Kalaba theory[J]. Chinese Journal of Mechanical Engineering，2013，26(5)：839-850.
[18] ZHEN S C，HAO J J，LIU X L，et al. A practical robust bounded control for permanent magnet linear motor with inequality constraints[J]. Control Engineering Practice，2022，122：105086.
[19] 王宗范，杨国来. 计及不等式约束的坦克炮稳定系统约束跟随控制[J]. 南京理工大学学报，2023，47(4)：461-467. WANG Zongfan，YANG Guolai. Constrained following control for tank gun stabilization systems with inequality constraints[J]. Journal of Nanjing University of Science and Technology，2023，47(4)：461-467.
[20] ZHANG J L，ZHAO R Y，JIAO S J. Optimization of robust formation tracking control for traffic cone robots with matching and mismatching uncertainties：A fuzzy-set theory-based approach[J]. Expert Systems with Applications，2023，227：120233.
[21] ZHANG X R，ZHAO R Y，CHEN Y H，et al. A novel modeling and control approach considering equality and inequality constraints based on generalized Udwadia-Kalaba equation[J]. Nonlinear Dynamics，2023，111：17109-17122.
[22] ZHANG D Y，ZHANG Y，ZHANG H，et al. A generalized Udwadia-Kalaba control design for uncertain belt conveyor systems with inequality constraints[J]. ISA Transactions，2024，151：409-422.
[23] 王明明，朱莹莹，张磊，等. 麦克纳姆轮驱动的移动机器人自适应滑模控制器设计[J]. 西北工业大学学报，2018，36(4)：627-635. WANG Mingming，ZHU Yingying，ZHANG Lei，et al. Design of adaptive sliding mode controller for mobile robot driven by McNamee wheel[J]. Journal of Northwestern Polytechnical University，2018，36(4)：627-635.
[24] 辛喆，陈海亮，林子钰，等. 智能汽车的路面附着极限横向轨迹跟踪控制[J]. 机械工程学报，2020，56(14)：138-145. XIN Zhe，CHEN Hailiang，LIN Ziyu，et al. Lateral trajectory following for automated vehicles at handling limits[J]. Journal of Mechanical Engineering，2020，56(14)：138-145.