A Model-free Cross-coupling Control Method for Parallel Kinematic Mechanisms with Symmetric Structure

doi:10.3969/j.issn.1004-132X.2025.08.005

Abstract

Abstract:

The dynamic coupling effectiveness between the driving joints of parallel mechanisms was a key challenge affecting their trajectory tracking performance. Existing multi-motor cooperative control methods for parallel platforms often ignored the inter-joint coupling relationships and their coordination， which affected the motion accuracy of the end-effector. To address this issue， a model-free cross-coupling control method was proposed to improve the tracking accuracy of the mechanisms in the task space. By introducing the motor cooperative erros that represented the relationship between the motions of neighboring motors，the method employed the TDE technique to indirectly perform the closed-loop control of the mechanism dynamics， solving the uncertainty problems of the dynamics model. Additionally， the stability of the proposed method was analyzed and proven using the Lyapunov theorem， and experimental validations were performed on a symmetric parallel kinematic mechanism. The results show that， compared to the existing TDC method， the proposed method may effectively reduce the end-effector’s motion tracking errors through efficient position tracking and stable operations of the parallel mechanisms.

Key words: symmetric parallel kinematic mechanism, model-free controller, cooperative control, time delay estimation（TDE）, trajectory tracking

摘要：

并联机构驱动关节间的动力学耦合效应是影响其轨迹跟踪性能的关键难题，现有并联平台的多电机协同控制方法常常不考虑机构关节间的相互耦合关系及其关节协同关系，影响了机构末端的运动精度。针对上述问题，提出一种提高机构任务空间跟踪精度的无模型交叉耦合控制方法。通过引入表示相邻电机运动关系的电机协同误差，采用时延估计（TDE）技术间接地进行机构动力学闭环控制，解决动力学模型不确性问题；同时采用Lyapunov定理分析证明所提方法的稳定性，并在对称并联运动机构上进行所提方法的实验验证。结果表明，相比现有的时延控制（TDC）方法，所提方法可以通过并联机构的高效位置跟踪和稳定操作有效减小其末端运动跟踪误差。

关键词: 对称并联运动机构, 无模型控制器, 协同控制, 时延估计, 轨迹跟踪

CLC Number:

TH112

Zhiwei ZHOU, Jian GAO, Lanyu ZHANG. A Model-free Cross-coupling Control Method for Parallel Kinematic Mechanisms with Symmetric Structure[J]. China Mechanical Engineering, 2025, 36(8): 1691-1699.

周志伟, 高健, 张揽宇. 对称并联运动机构的无模型交叉耦合控制[J]. 中国机械工程, 2025, 36(8): 1691-1699.

Figures/Tables 12

Fig. 1 Structural of a symmetric parallel motion mechanism

Fig. 2 Kinematic model of a symmetric parallel motion mechanism

Fig. 3 Control diagram of the proposed model-free cross-coupling control method

Fig.4 Experimental system of a symmetric parallel motion mechanism

Fig.5 Desired trajectories for the X， Z and B axis

Fig.6 Tracking error in joint space with the proposed method and TDC

Fig.7 Cooperative error in joint space with the proposed method and TDC

Fig.8 Tracking error in task space with the proposed method and TDC

Tab. 1 Comparison of tracking error and cooperative error in joint space between the proposed method and TDC

		RMSE			IAE
		q₁	q₂	q₃	q₁	q₂	q₃
跟踪误差/μm	TDC	7.49	4.91	2.41	28.21	39.25	13.35
	所提方法	6.42	3.69	1.61	24.61	29.85	9.94
	减幅/%	14.3	24.8	33.2	12.8	23.9	25.5
协同误差/μm	TDC	9.31	5.65	8.07	62.54	45.55	37.82
	所提方法	7.57	4.21	6.13	49.16	34.54	28.46
	减幅/%	18.7	25.5	23.9	21.4	24.2	24.7

Tab.2 Comparison of tracking error in task space between the proposed method and TDC

控制方案	RMSE			IAE
控制方案	X/μm	Z/μm	B/μrad	X/μm	Z/μm	B/mrad
TDC	4.91	6.98	89.61	39.25	32.75	0.61
所提方法	3.69	4.51	72.88	29.85	20.95	0.47
减幅/%	24.8	35.4	18.7	23.9	36.1	21.3

Fig.9 Servo commands for each motor of the parallel mechanism

Fig.10 Tracking errors of X and Z axis of the parallel mechanism

References 27

[1]	YAO Suhua， GAO Guoqin， GAO Zhiqiang. On Multi-axis Motion Synchronization： the Cascade Control Structure and Integrated SMC-ADRC Design［J］. ISA Transactions， 2021， 109： 259-268.
[2]	WANG Ke， LI Ju， SHEN Huiping， et al. Inverse Dynamics of a 3-DOF Parallel Mechanism Based on Analytical Forward Kinematics［J］. Chinese Journal of Mechanical Engineering， 2022， 35（1）： 119.
[3]	LUO Yuheng， GAO Jian， ZHANG Lanyu， et al. Kinematic Calibration of a Symmetric Parallel Kinematic Machine Using Sensitivity-based Iterative Planning［J］. Precision Engineering， 2022， 77： 164-178.
[4]	汪满新，李兰彬，李正亮，等. 一类1T2R并联机构拓扑结构综合及优选［J］. 中国机械工程， 2022， 33（20）： 2395-2402.
	WANG Manxin， LI Lanbin， LI Zhengliang， et al. Topological Structure Synthesis and Optimization of 1T2R Parallel Mechanisms［J］. China Mechanical Engineering， 2022， 33（20）： 2395-2402.
[5]	BODIE K， TOGNON M， SIEGWART R. Dynamic End Effector Tracking with an Omnidirectional Parallel Aerial Manipulator［J］. IEEE Robotics and Automation Letters， 2021， 6（4）： 8165-8172.
[6]	LUO Yuheng， GAO Jian， CHEN Disai， et al. Accurate Error Compensation Method for Multi-axis Parallel Machine via Singularized Jacobi Geometric Parameter Correction and Coupling Error Evaluation［J］. Robotics and Computer-Integrated Manufacturing， 2024， 89： 102771.
[7]	YAN Peng， HUANG Hailin， LI Bing， et al. A 5-DOF Redundantly Actuated Parallel Mechanism for Large Tilting Five-face Machining［J］. Mechanism and Machine Theory， 2022， 172： 104785.
[8]	梁旭，苏婷婷，侯增广，等. 基于变阻抗控制的冗余驱动并联机器人多目标内力优化［J］. 自动化学报， 2023， 49（5）： 1099-1115.
	LIANG Xu， SU Tingting， HOU Zengguang， et al. A Multi-objective Internal Preload Optimization Method of Redundantly Actuated Parallel Robots Based on Variable Impedance Control［J］. Acta Automatica Sinica， 2023， 49（5）： 1099-1115.
[9]	HE Yuhang， WU Yaozhong， LI Weijia. Nonlinear Extended State Observer-based Adaptive Higher-order Sliding Mode Control for Parallel Antenna Platform with Input Saturation［J］. Nonlinear Dynamics， 2023， 111（17）： 16111-16132.
[10]	YANG Xu， ZHU Wule. Design， Analysis， and Control of an XY Parallel Nanomanipulator with Multiple Actuation Modes［J］. IEEE Transactions on Industrial Electronics， 2020， 67（9）： 7639-7648.［LinkOut］
[11]	KHALILPOUR S A， KHORRAMBAKHT R， TAGHIRAD H D， et al. Robust Cascade Control of a Deployable Cable-driven Robot［J］. Mechanical Systems and Signal Processing， 2019， 127： 513-530.
[12]	ALTAN A， HACiOĞLU R. Model Predictive Control of Three-axis Gimbal System Mounted on UAV for Real-time Target Tracking under External Disturbances［J］. Mechanical Systems and Signal Processing， 2020， 138： 106548.
[13]	SHANG Weiwei， CONG Shuang， GE Yuan. Coordination Motion Control in the Task Space for Parallel Manipulators with Actuation Redundancy［J］. IEEE Transactions on Automation Science and Engineering， 2013， 10（3）： 665-673.
[14]	AFLAKIAN A， SAFARYAZDI A， TALE MASOULEH M， et al. Experimental Study on the Kinematic Control of a Cable Suspended Parallel Robot for Object Tracking Purpose［J］. Mechatronics， 2018， 50： 160-176.
[15]	SHANG Weiwei， ZHANG Bin， CONG Shuang， et al. Dual-space Adaptive Synchronization Control of Redundantly-actuated Cable-driven Parallel Robots［J］. Mechanism and Machine Theory， 2020， 152： 103954.
[16]	MERLET J P. Simulation of Discrete-time Controlled Cable-driven Parallel Robots on a Trajectory［J］. IEEE Transactions on Robotics， 2017， 33（3）： 675-688.
[17]	PI Yangjun， WANG Xuanyin. Observer-based Cascade Control of a 6-DOF Parallel Hydraulic Manipulator in Joint Space Coordinate［J］. Mechatronics， 2010， 20（6）： 648-655.
[18]	ZHANG Gang， WU Jianhua， LIU Pinkuan， et al. Dynamic Analysis and Model-based Feedforward Control of a 2-DOF Translational Parallel Manipulator Driven by Linear Motors［J］. Industrial Robot， 2013， 40（6）： 597-609.
[19]	LIU Xiaofei， YAO Jiantao， LI Qi， et al. Coordination Dynamics and Model-based Neural Network Synchronous Controls for Redundantly Full-actuated Parallel Manipulator［J］. Mechanism and Machine Theory， 2021， 160： 104284.
[20]	ZHAN Boyu， ZHANG Lanyu， LIU Yachao， et al. Model Predictive and Compensated ADRC for Permanent Magnet Synchronous Linear Motors［J］. ISA Transactions， 2023， 136： 605-621.
[21]	商德勇，黄欣怡，黄云山，等. 基于Kane方程的Delta并联机器人刚柔耦合动力学研究［J］. 机械工程学报， 2024， 60（7）： 124-133.
	SHANG Deyong， HUANG Xinyi， HUANG Yunshan， et al. Research on Rigid-flexible Coupling Dynamics of Delta Parallel Robot Based on Kane Equation［J］. Journal of Mechanical Engineering， 2024， 60（7）： 124-133.
[22]	ZHU Baojun， LI Erwei， ZHAO Tieshi， et al. Dynamic Characteristics of Series-parallel Hybrid Rigid-flexible Coupling Double-mass Underactuated System on Floating Platform［J］. Mechanism and Machine Theory， 2023， 179： 105132.［LinkOut］
[23]	SHI Pengwei， SUN Weichao， YANG Xuebo， et al. Master-slave Synchronous Control of Dual-drive Gantry Stage with Cogging Force Compensation［J］. IEEE Transactions on Systems， Man， and Cybernetics： Systems， 2022， 53（1）： 216-225.
[24]	ZHONG Yongbin， GAO Jian， ZHANG Lanyu. Fractional-order Feedforward Control Method for Permanent Magnet Linear Synchronous Motor Based on Frequency-domain Adjustment Theory［J］. Mechanical Systems and Signal Processing， 2023， 190： 110115.
[25]	LIU Zhitai， LIN Weiyang， YU Xinghu， et al. Approximation-free Robust Synchronization Control for Dual-linear-motors-driven Systems with Uncertainties and Disturbances［J］. IEEE Transactions on Industrial Electronics， 2021， 69（10）： 10500-10509.
[26]	XIE Bin， LEI Li， YAO Yuan， et al. Advanced Synchronous Control of Dual Parallel Motion Systems［J］. IEEE Transactions on Industrial Electronics， 2022， 70（2）： 1270-1281.
[27]	张鑫，刘金国. 基于时延估计的空间机器人无模型姿态解耦控制［J］. 宇航学报， 2021， 42（9）： 1150-1161.
	ZHANG Xin， LIU Jinguo. Model-free Attitude Decoupling Control of Space Robots Based on Time- delay Estimation［J］. Journal of Astronautics， 2021， 42（9）： 1150-1161.