High Precision Methods for Solving Multi-Body Systems With Close Loop Constraints

Zhao Tianjiao; Qi Zhaohui; Wang Tianyu; Xu Jinshuai

doi:10.21656/1000-0887.450230

Volume 47 Issue 8

Aug. 2026

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Article Navigation > Applied Mathematics and Mechanics > 2026 > 47(8): 1068-1085

Zhao Tianjiao, Qi Zhaohui, Wang Tianyu, Xu Jinshuai. High Precision Methods for Solving Multi-Body Systems With Close Loop Constraints[J]. Applied Mathematics and Mechanics, 2026, 47(8): 1068-1085. doi: 10.21656/1000-0887.450230

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High Precision Methods for Solving Multi-Body Systems With Close Loop Constraints

doi: 10.21656/1000-0887.450230

1. Dalian Institute of Chemical Physics, Chinese Academy of Sciences,Dalian, Liaoning 116024, P.R. China;

Funds:

The National Science Foundation of China(11872137)

Received Date: 2024-08-14
Rev Recd Date: 2025-02-11

Available Online: 2026-07-30

Publish Date: 2026-08-01

Abstract

Abstract

The dynamics equations for multi-body systems with closed-loop constraints are a set of differential algebraic mixed equations. The main difficulties in their numerical solution are as follow: it is difficult to satisfy the displacement, velocity and acceleration constraint equations with high precision at the same time, which will cause large defaults with the accumulation of errors; there are often redundant constraints and singular configuration problems, which seriously affect the numerical behavior of the solution process and the accuracy of the results. Herein, a numerical method for solving singular configuration closed-loop multibody systems was presented, without the traditional constraint independence assumption. The correction of position and velocity were done to satisfy the constraint equations before the motion equation formulation, rather than the correction of the constraint default at the end of each integration step. This method works with any standard ODE solver. According to the geometric characteristics of constraints, the components of velocity and acceleration in the hypersurface constraint space were all determined by constraints. The orthogonal basis vectors of the constrained tangent space were obtained through QR decomposition of the column permutation. The relationship between the generalized velocity of the system and the independent generalized velocity was established, and the default values of the generalized velocity and the generalized acceleration were filtered out in the dynamic calculation process, and the differential algebraic mixed equation was transformed into a pure differential equation satisfying the velocity and acceleration constraints ( with the number of equations equal to the generalized coordinate number of the system). Then, the orthogonal projection default correction method was applied to the system, and the default magnitude of the system position and speed was effectively suppressed by several iterations. At the same time, according to the principle of virtual power equivalence, the physical significance of the Laplace multiplier in solving the constraint reaction torque of the cut hinge was further revealed. Numerical examples show that, the precision of displacement, velocity and acceleration constraints of the proposed method is higher than that of the traditional augmented method and the penalty function method. The proposed method can deal with redundant constraints and singular configuration problems, and can be programmed to solve multi-body systems with closed-loop constraints with high precision. The proposed method can identify the independent constraints dynamically and modify the motion equation accordingly. Numerical examples demonstrate the effectiveness of the proposed method.
- closed-loop constraint,
- multi-body system,
- mixed differential algebraic equation,
- joint constraint force,
- correction of default

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References(35)

References

刘延柱, 潘振宽, 戈新生. 多体系统动力学[M]. 2版. 北京: 高等教育出版社, 2014.(Liu Yanzhu, Pan Zhenkuan, Ge Xinsheng. Dynamics of Multibody Systems[M]. 2nd ed. Beijing: Higher Education Press, 2014. (in Chinese))

[2]富立. 非光滑多体系统动力学LCP方法[M]. 北京: 清华大学出版社, 2016.(Fu Li. Dynamics of Non-Smooth Multibody Systems: LCP Method[M]. Beijing: Tsinghua University Press, 2016. (in Chinese))

[3]齐朝晖. 多体系统动力学[M]. 北京: 科学出版社, 2008.(Qi Zhaohui. Dynamics of Multibody Systems[M]. Beijing: Science Press, 2008. (in Chinese))

[4]Blajer W, Koodziejczyk K. A geometric approach to solving problems of control constraints: theory and a DAE framework[J]. Multibody System Dynamics,2004,11(4): 343-364.

[5]Pappalardo C M, Guida D. On the computational methods for solving the differential-algebraic equations of motion of multibody systems[J]. Machines,2018,6(2): 20.

[6]Bottasso C L, Dopico D, Trainelli L. On the optimal scaling of index three DAEs in multibody dynamics[J]. Multibody System Dynamics,2008,19(1): 3-20.

[7]Sun W. Numerical algorithms for differential-algebraic equations of multibody dynamics[C]//2016 16th International Conference on Control, Automation and Systems (ICCAS). Gyeongju, Korea (South), 2016: 786-791.

[8]Hussein B A, Shabana A A. Sparse matrix implicit numerical integration of the stiff differential/algebraic equations: implementation[J]. Nonlinear Dynamics,2011,65(4): 369-382.

[9]Hu J, Wang T. An efficient high-precision recursive dynamic algorithm for closed-loop multibody systems[J]. International Journal for Numerical Methods in Engineering,2019,118(4): 181-208.

[10]赵维加, 潘振宽, 王艺兵. 多体系统动力学微分/代数方程约束误差小扰动自我稳定方法[J]. 应用数学和力学, 2000,21(1): 94-98.(Zhao Weijia, Pan Zhenkuan, Wang Yibing. An automatic constraint violation stabilization method for differential/algebraic equations of motion multibody system dynamics[J]. Applied Mathematics and Mechanics,2000,21(1): 94-98. (in Chinese))

[11]刘颖, 马建敏, 苏芳, 等. 多体系统动力学方程的无违约数值计算方法[J]. 计算力学学报, 2010,27(5): 942-947.(Liu Ying, Ma Jianmin, Su Fang, et al. Precise numerical solution for multi-body system’s equations of motion based on algorithm without constraint violation[J]. Chinese Journal of Computational Mechanics,2010,27(5): 942-947. (in Chinese))

[12]付士慧, 王琪. 多体系统动力学方程违约修正的数值计算方法[J]. 计算力学学报, 2007,24(1): 44-49.(Fu Shihui, Wang Qi. A numerical method for constraint stabilization of dynamic equations of multi-body systems[J]. Chinese Journal of Computational Mechanics,2007,24(1): 44-49. (in Chinese))

[13]Kikuuwe R, Brogliato B. A new representation of systems with frictional unilateral constraints and its Baumgarte-like relaxation[J]. Multibody System Dynamics,2017,39(3): 267-290.

[14]Pogorelov D. Differential-algebraic equations in multibody system modeling[J]. Numerical Algorithms,1998,19(1): 183-194.

[15]潘振宽, 赵维加, 洪嘉振, 等. 多体系统动力学微分/代数方程组数值方法[J]. 青岛大学学报, 1996,9(1): 83-96. （Pan Zhenkuan, Zhao Weijia, Hong Jiazhen, et al. Numerical algorithms for differential/algebraic equations of motion of multibody system[J]. Journal of Qingdao University,1996,9(1): 83-96.（in Chinese））

[16]Pennestrì E, Vita L. Strategies for the numerical integration of DAE systems in multibody dynamics[J]. Computer Applications in Engineering Education,2004,12(2): 106-116.

[17]Bayo E, Avello A. Singularity-free augmented Lagrangian algorithms for constrained multibody dynamics[J]. Nonlinear Dynamics,1994,5(2): 209-231.

[18]Uchida T, Mcphee J. Triangularizing kinematic constraint equations using Grbner bases for real-time dynamic simulation[J]. Multibody System Dynamics,2011,25(3): 335-356.

[19]Blajer W, Schiehlen W, Schirm W. A projective criterion to the coordinate partitioning method for multibody dynamics[J]. Archive of Applied Mechanics,1994,64(2): 86-98.

[20]Carpinelli M, Gubitosa M, Mundo D, et al. Automated independent coordinates’ switching for the solution of stiff DAEs with the linearly implicit Euler method[J]. Multibody System Dynamics,2016,36(1): 67-85.

[21]Wehage K T, Wehage R A, Ravani B. Generalized coordinate partitioning for complex mechanisms based on kinematic substructuring[J]. Mechanism and Machine Theory,2015,92: 464-483.

[22]Bauchau O A, Laulusa A. Review of contemporary approaches for constraint enforcement in multibody systems[J]. Journal of Computational and Nonlinear Dynamics,2008,3(1): 011005.

[23]Kurdila A J, Junkins J L, Hsu S. Lyapunov stable penalty methods for imposing holonomic constraints in multibody system dynamics[J]. Nonlinear Dynamics,1993,4(1): 51-58.

[24]Yu Q, Chen I M. A direct violation correction method in numerical simulation of constrained multibody systems[J]. Computational Mechanics,2000,26(1): 52-57.

[25]Blajer W. A geometric unification of constrained system dynamics[J]. Multibody System Dynamics,1997,1(1): 3-21.

[26]Zhang J, Liu D, Liu Y. A constraint violation suppressing formulation for spatial multibody dynamics with singular mass matrix[J]. Multibody System Dynamics,2016,36(1): 87-110.

[27]Eich E. Convergence results for a coordinate projection method applied to mechanical systems with algebraic constraints[J]. SIAM Journal on Numerical Analysis,1993,30(5): 1467-1482.

[28]Yoon S, Howe R M, Greenwood D T. Geometric elimination of constraint violations in numerical simulation of Lagrangian equations[J]. Journal of Mechanical Design,1994,116(4): 1058-1064.

[29]Blajer W. Elimination of constraint violation and accuracy aspects in numerical simulation of multibody systems[J]. Multibody System Dynamics, 2002,7: 265-284.

[30]Aghili F, Piedbceuf J C. Simulation of motion of constrained multibody systems based on projection operator[J]. Multibody System Dynamics,2003,10(1): 3-16.

[31]Marques F, Souto A P, Flores P. On the constraints violation in forward dynamics of multibody systems[J]. Multibody System Dynamics,2017,39(4): 385-419.

[32]Bayo E, Garcia De Jalon J, Serna M A. A modified Lagrangian formulation for the dynamic analysis of constrained mechanical systems[J]. Computer Methods in Applied Mechanics and Engineering,1988,71(2): 183-195.

[33]Blajer W. Augmented Lagrangian formulation: geometrical interpretation and application to systems with singularities and redundancy[J]. Multibody System Dynamics,2002,8(2): 141-159.

[34]Zhang J, Liu D, Liu Y. A constraint violation suppressing formulation for spatial multibody dynamics with singular mass matrix[J]. Multibody System Dynamics,2016,36(1): 87-110.

[35]Neto M A, Ambrósio J. Stabilization methods for the integration of DAE in the presence of redundant constraints[J]. Multibody System Dynamics,2003,10: 81-105.

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