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標題: Dead-Beat Control of the Constraint Stabilization Method for Numerical Integration of Multibody Mechanical Systems
使用Dead-Beat Control在多體機械系統拘束穩定數值積分法之研究
作者: 陳書愷
Shu-Kai Chen
關鍵字: Numerical Integration
引用: [1] Haug, E. J., 1989, 'Computer Aided Kinematics and Dynamics of Mechanical System,' Vol. I: Basic Methods, Allyn and Bacon, Massachusetts, USA. [2] DADS User's Manual, 1987, Rev. 4.0, Computer Aided Design Software, Inc. Oakdale, Iowa. [3] Bae, D. S., Hwang, R. S., and Haug, E. J., 1988, 'A Recursive Formulation for Real-Time Dynamic Simulation,' Advances in Design Automation, ASME, New York, pp. 499-508. [4] Hwang, R. S., Bae, D. S., Haug. E. J., and Kuhl, J. G., 1998, 'Parallel Processing for Real-Time Dynamic System Simulation,' Advances in Design Automation, ASME, New York, pp. 509-518. [5] Wehage, R. A., and Haug, E. J., 1982, 'Generalized Coordinates Partitioning for Dimension Reduction in Analysis of Constrained Dynamic System,' ASME Journal of Mechanical Design, Vol. 104, pp. 247-255. [6] Baumgarte, J., 1972, 'Stabilization of Constraints and Integrals of Motion in Dynamical Systems,' Computer Methods in Applied Mechanics and Engineering, Vol. 1, pp. 1-16. [7] Chang, C. O., and Nikravesh, P. E., 1985, 'An Adaptive Constraint Violation Stabilization Method for Dynamic Analysis of Mechanical Systems,' ASME Journal of Mechanism, Transmissions and Automation in Design, Vol. 17, pp. 488-492. [8] Lin, S. T. and Hong, M. C., 1998, 'Stabilization Method for Numerical Integration of Multibody Mechanical Systems,' ASME Journal of Mechanical Design, Vol. 120, pp. 565-572. [9] Lin, S. T. and Huang, J. N., 2000, 'Parameters Selection for Baugarte's Constraint Stabilization Method Using the Predictor-Corrector Approach,' AIAA Journal of Guidance, Control , and Dynamics, Vol. 23, No. 3, pp. 566-570. [10] Park, T.,1986, 'A Hybrid Constraint Stabilization-Generalized Coordinate Partitioning Method for Machine Dynamics, ' ASME Journal of Mechanism, Transmissions, and Automation in Design, Vol. 108, No. 2, pp. 211-216. [11] D.J. Braun, M. Goldfarb, 2009, 'Eliminating constraint drift in the numerical simulation of constrained dynamical systems,' Comput. Methods Appl. Mech. Engrg. Vol. 198, pp. 3151–3160. [12] Park, K. C., and Chiou, J. C., 1988, 'Stabilization of Computational Procedures for Constrained Dynamical Systems,' AIAA Journal of Guidance and Control, Vol. 11, pp. 365-370. [13] Bauchau, O. A., Damilano, G., and Theron, N. J., 1995, 'Numerical Integration of Non-Linear Elastic Multi-Body Systems,' International Journal for Numerical Methods in Engineering, Vol. 38, pp. 2727-2751. [14] Rosen, A., and Edelstein, E., 1997, 'Investigation of a New Formulation of the Lagrange Method for Constrained Dynamic Systems,' ASME Journal of Applied Mechanics, Vol. 64, pp. 116-122. [15] Yoon, S., Howe, R. M., and Greenwood, D. T., 1994, 'Geometric Elimination of Constraint Violations in Numerical Simulation of Lagrange Equations,' ASME Journal of Mechanical Design, Vol. 116, pp. 1058-1064. [16] Yoon, S., Howe, R. M., and Greenwood, D. T., 1995, 'Stability and Accuracy Analysis of Baumgarte's Constraint Stabilization Method,' ASME Journal of Mechanical Design, Vol. 117, pp. 446-453. [17] Haug, E. J., and Yen, J., 1992, 'Implicit Numerical Integration of Constrained Equations of motion Via Generalized Coordinate Partitioning,' ASME Journal of Mechanical Design, Vol. 114, pp. 296-304. [18] Wojciech Blajer, 2011, 'Methods for constraint violation suppression in the numerical simulation of constrained multibody systems – A comparative study' Comput. Methods Appl. Mech. Engrg. Vol. 200, pp. 1568-1576 [19] Chung, S., and Haug, E. J., 1993, 'Real-Time Simulation of Multibody Dynamics on Shared Memory Multiprocessors,' ASME, Journal of Dynamic Systems, Measurement, and Control, Vol. 115, pp. 627-637. [20] Petzold, L., 1982, 'Differential/Algebraic Equations are not ODEs,' SIAM J. Sci. Stat. Comput., Vol. 3, No. 3, pp. 367-384. [21] Ogata, K., 1995, Discrete-Time Control Systems, 2nd Edition, Prentice-Hall, Englewood Cliffs, New Jersey, USA. [22] Franklin, G. F., Powell, A. E.-N., 1994, Feedback Control of Dynamic Systems, 3rd Edition, Addision-Wesley. [23] Lambert, J. D., 1991, Numerical Methods for Ordinary Differential Systems, John Wiley & Sons, West Sussex, England.
摘要: The objective of this thesis is to resolve the stability problem for the numerical integration of constrained multibody mechanical systems. The dynamic equations of motion of the constrained multibody mechanical system is a mixed differential-algebraic equation(DAE) which contains external forces, constraint reaction forces as well as acceleration of the generalized coordinates of the system. In applying numerical integration methods to solve the mixed differential-algebraic equation, the constraint equation and its first and second derivatives must be satisfied simultaneously. That is, the generalized coordinates are dependent. Direct integration methods do not consider this dependency and constraint violation occurs. To solve this problem, Baumgarte proposed a constraint stabilization method in which a velocity term and a position term were added in the second derivative of the constraint equation. The disadvantage of this method is that there is no known reliable method for selecting the coefficients of the position and velocity term. Improper selection of these coefficients can lead to erroneous results. In this paper, we will use the so-called pseudo-integration equation to analyse the constraint stabilization method for numerical integration. Dead-beat Control of the stability analysis methods in digital control theory will be used to give correct choice of the control law for the Adams-bashforth and Adams Predictor-Corrector integration methods.
本文的目的在於解決受拘束(constraint)條件下,多體(multibody)機械系統數值積分的穩定性問題。多體系統之運動方程式是一個包含外力、拘束力、加速度的混合微分代數方程式(mixed differential-algebraic equations, 簡稱DAE),在使用數值積分法解微分方程式的同時,其位置及速度,必須滿足滿足拘束運動方程式與速度運動方程式,也就是說積分變數是相關的。然而一般的數值積分法在積分的過程中並不考慮之間的相關性,直接求解進而造成數值的偏差。 為了解決此問題,Baumgarte嘗試將加速度方程式加入位置項與速度項,適當的選擇系數之後,數值積分將會獲得正確的解,稱為拘束穩定法(constraint stabilization method),不過此方法唯一的缺點在於追加參數的選擇並無規則可循。 本文針對拘束穩定法這部分的研究成果,引入假積分(pseudo-integration)的觀念,由於一般文獻中拘束穩定法是在連續時間下討論離散時間的動態系統,這並不符合理論,因此我們在此引用Dead-Beat Control直接在離散時間下進行控制,並對數值積分法中常用的方法Adams-bashforth及Predictor-Corrector法,提出一個選擇控制法則的辦法。
文章公開時間: 2015-08-27
Appears in Collections:機械工程學系所



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