Comparison of solution strategies for multibody dynamics equations

In the last decades, several different formalisms have been proposed in the literature in order to simulate the dynamics of multibody systems. First, the choice of the coordinates leads to very different formalisms. The set of constraint equations appears very different in complexity or in size depe...

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Vydané v:	International journal for numerical methods in engineering Ročník 88; číslo 7; s. 637 - 656
Hlavní autori:	Mariti, L., Belfiore, N. P., Pennestrì, E., Valentini, P. P.
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	Chichester, UK John Wiley & Sons, Ltd 18.11.2011 Wiley
Predmet:	Computer simulation DAE system Dynamics Exact sciences and technology Formalism Fundamental areas of phenomenology (including applications) Mathematical analysis Mathematical models Mathematics multibody dynamics Multibody systems Numerical analysis Numerical analysis. Scientific computation Numerical approximation numerical methods Physics Redundant Sciences and techniques of general use Solid dynamics (ballistics, collision, multibody system, stabilization...) Solid mechanics Strategy numerical methods multibody dynamics Constraint Redundancy Solid dynamic Pseudoinverse N body system Cartesian coordinate Dynamical system Real time Modeling Complexity DAE system Computation time Orthogonalization Least squares method
ISSN:	0029-5981, 1097-0207, 1097-0207
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Abstract	In the last decades, several different formalisms have been proposed in the literature in order to simulate the dynamics of multibody systems. First, the choice of the coordinates leads to very different formalisms. The set of constraint equations appears very different in complexity or in size depending on whether either relative, natural or reference point Cartesian coordinates are adopted. Second, once such a choice has been made, different solution strategies can be followed. It seems that formalisms based on redundant coordinates are often used in commercial software, whereas those based on a minimum number of coordinates are usually preferred in real‐time computations. In this paper, with reference to models with redundant absolute coordinates, the problem of coordinate reduction will be discussed and a group of 11 different methods will be compared on the basis of their computational efficiency. In particular, methods based on constraint orthogonalization and on the use of pseudoinverse matrices have been selected, together with other methods based on least‐squares block solution. The methods have been implemented on three different test cases. The main purpose is to provide hints and guidelines on the choice and availability of solution strategies during simulation of moderate size multibody systems. Copyright © 2011 John Wiley & Sons, Ltd.
AbstractList	In the last decades, several different formalisms have been proposed in the literature in order to simulate the dynamics of multibody systems. First, the choice of the coordinates leads to very different formalisms. The set of constraint equations appears very different in complexity or in size depending on whether either relative , natural or reference point Cartesian coordinates are adopted. Second, once such a choice has been made, different solution strategies can be followed. It seems that formalisms based on redundant coordinates are often used in commercial software, whereas those based on a minimum number of coordinates are usually preferred in real‐time computations. In this paper, with reference to models with redundant absolute coordinates, the problem of coordinate reduction will be discussed and a group of 11 different methods will be compared on the basis of their computational efficiency. In particular, methods based on constraint orthogonalization and on the use of pseudoinverse matrices have been selected, together with other methods based on least‐squares block solution. The methods have been implemented on three different test cases. The main purpose is to provide hints and guidelines on the choice and availability of solution strategies during simulation of moderate size multibody systems. Copyright © 2011 John Wiley & Sons, Ltd. In the last decades, several different formalisms have been proposed in the literature in order to simulate the dynamics of multibody systems. First, the choice of the coordinates leads to very different formalisms. The set of constraint equations appears very different in complexity or in size depending on whether either relative, natural or reference point Cartesian coordinates are adopted. Second, once such a choice has been made, different solution strategies can be followed. It seems that formalisms based on redundant coordinates are often used in commercial software, whereas those based on a minimum number of coordinates are usually preferred in real‐time computations. In this paper, with reference to models with redundant absolute coordinates, the problem of coordinate reduction will be discussed and a group of 11 different methods will be compared on the basis of their computational efficiency. In particular, methods based on constraint orthogonalization and on the use of pseudoinverse matrices have been selected, together with other methods based on least‐squares block solution. The methods have been implemented on three different test cases. The main purpose is to provide hints and guidelines on the choice and availability of solution strategies during simulation of moderate size multibody systems. Copyright © 2011 John Wiley & Sons, Ltd. In the last decades, several different formalisms have been proposed in the literature in order to simulate the dynamics of multibody systems. First, the choice of the coordinates leads to very different formalisms. The set of constraint equations appears very different in complexity or in size depending on whether either relative, natural or reference point Cartesian coordinates are adopted. Second, once such a choice has been made, different solution strategies can be followed. It seems that formalisms based on redundant coordinates are often used in commercial software, whereas those based on a minimum number of coordinates are usually preferred in real-time computations. In this paper, with reference to models with redundant absolute coordinates, the problem of coordinate reduction will be discussed and a group of 11 different methods will be compared on the basis of their computational efficiency. In particular, methods based on constraint orthogonalization and on the use of pseudoinverse matrices have been selected, together with other methods based on least-squares block solution. The methods have been implemented on three different test cases. The main purpose is to provide hints and guidelines on the choice and availability of solution strategies during simulation of moderate size multibody systems.
Author	Pennestrì, E. Mariti, L. Valentini, P. P. Belfiore, N. P.
Author_xml	– sequence: 1 givenname: L. surname: Mariti fullname: Mariti, L. organization: Department of Mechanical Engineering, University of Rome 'Tor Vergata', via del Politecnico, 1, 00133 Rome, Italy – sequence: 2 givenname: N. P. surname: Belfiore fullname: Belfiore, N. P. organization: Department of Mechanical and Aerospace Engineering, 'Sapienza' University of Rome, via Eudossiana, 18, 00184 Rome, Italy – sequence: 3 givenname: E. surname: Pennestrì fullname: Pennestrì, E. email: pennestri@mec.uniroma2.it organization: Department of Mechanical Engineering, University of Rome 'Tor Vergata', via del Politecnico, 1, 00133 Rome, Italy – sequence: 4 givenname: P. P. surname: Valentini fullname: Valentini, P. P. organization: Department of Mechanical Engineering, University of Rome 'Tor Vergata', via del Politecnico, 1, 00133 Rome, Italy
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Keywords	numerical methods multibody dynamics Constraint Redundancy Solid dynamic Pseudoinverse N body system Cartesian coordinate Dynamical system Real time Modeling Complexity DAE system Computation time Orthogonalization Least squares method
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References	Wang JT, Huston RL. A comparison of analysis methods of redundant multibody systems. Mechanics Research Communications 1989; 16:175-182. Kamman RL, Huston JW. Dynamics of constrained multibody systems. ASME Journal of Applied Mechanics 1985; 51:899-903. Kim SS, Vanderploeg MJ. QR decomposition for state space representation of constrained mechanical dynamic systems. ASME Journal of Mechanisms, Transmissions, and Automation in Design 1986; 108:176-182. Udwadia FE, Kalaba RE. Analytical Dynamics a New Approach. Cambridge University Press: Cambridge, 1996. Eich-Soellner E, Führer C. Numerical Methods in Multibody Dynamics. B.G. Teubner: Stuttgart, 1998. Haug EJ. Computer-aided Kinematics and Dynamics of Mechanical Systems. Allyn and Bacon: Newton, MA, 1989. Udwadia FE, Kalaba R. Explicit equations of motion for mechanical systems with nonideal constraints. ASME Journal of Applied Mechanics 2001; 68:462-467. Campbell SL. Generalized Inverses of Linear Transformations. Dover Publications, Inc.: New York, 1979. Amirouche FML, Ider SK. Coordinate reduction in the dynamics of constrained multibody systems-a new approach. ASME Journal of Applied Mechanics 1988; 55:899-904. Shabana AA. Computational Dynamics. Wiley: New York, 1994. Pennestrì E, Vita L, de Falco D. An investigation of the influence of pseudoinverse matrix calculations on multibody dynamics simulations by means of the Udwadia-Kalaba formulation. Journal of Aerospace Engineering 2008; 22:365-372. Shabana AA. Dynamics of Multibody Systems. Wiley: New York, 1989. Nikravesh PE. Initial condition correction in multibody dynamics. Multibody System Dynamics 2007; 18:107-115. Singh RP, Likins PW. Singular value decomposition for constrained mechanical systems. ASME Journal of Applied Mechanics 1985; 52:943-948. Kurdila A, Papastravidis JG. Kamat MP. Role of Maggi's equations in computational methods for constrained multibody systems. Journal of Guidance, Control, and Dynamics 1990; 13:113-120. Unda J, de Jalon JG, Losantos F, Emparantza R. A comparative study of different formulations of the dynamic equations of constrained mechanical systems. ASME Journal of Mechanisms, Transmissions, and Automation in Design 1987; 109:466-474. Borri M, Bottasso CL, Mantegazza P. Equivalence of Kane's and Maggi's equations. Meccanica 1990; 25:272-274. Baumgarte JW. A new method of stabilization for holonomic constraints. ASME Journal of Applied Mechanics 1983; 50:869-870. Wehage RA, Haug EJ. Generalized coordinate partitioning for dimension reduction in analysis of constrained dynamic systems. ASME Journal of Mechanical Design 1982; 134:247-255. Cheli F, Pennestrì E. Cinematica e Dinamica dei Sistemi Multibody, vol. 1. Casa Editrice Ambrosiana: Milano, 2006. Bauchau O, Laulusa A. Review of classical approaches for constraints enforcement in multibody systems. Journal of Computational and Non Linear Dynamics 2008; 3:1-8. Mani NK, Haug EJ, Atkinson KE. Singular value decomposition for analysis of mechanical system dynamics. ASME Journal of Mechanisms, Transmissions, and Automation in Design 1985; 107:82-87. Erberhard P, Schielen W. Computational dynamics of multibody systems: history, formalisms, and applications. ASME Journal of Computational and Nonlinear Dynamics 2006; 1:3-12. Laulusa A, Bauchau OA. Review of classical approaches for constraint enforcement in multibody systems. Journal of Computational and Nonlinear Dynamics 2008; 3. http://dx.doi.org/10.1115/1.2803257. Golub G, van Loan C. Matrix Computations (3rd edn). The Johns Hopkins University Press: London, 1996. Vlasenko D, Kasper R. Implementation of consequent stabilization method for simulation of multibodies described in absolute coordinates. Multibody System Dynamics 2009; 22:297-315. Baumgarte JW. Stabilization of constraints and integrals of motion in dynamical systems. Computer Methods in Applied Mechanics and Engineering 1972; 1:1-16. Wang JT, Huston RL. Computational methods in constrained multibody dynamics: matrix formalisms. Computers and Structures 1988; 29:331-338. Amirouche F. Fundamentals of Multibody Dynamics. Birkhhäuser: Basel, 2004. Pennestrì E, Valentini PP, Vita L. Multibody dynamics simulation of planar linkages with Dahl friction. Multibody System Dynamics 2007; 17:321-347. Udwadia FE, Phohomsiri P. Explicit equations of motion for constrained mechanical systems with singular mass matrices and applications to multi-body dynamics. Proceedings of the Royal Society, Series A 2006; 462:2097-2117. Maggi GA. Di alcune nuove forme della dinamica applicabili ai sistemi anolonomi. Rendiconti della Regia Accademia dei Lincei-Serie V 1901; X:287-291. Arabyan A, Wu F. An improved formulation for constrained mechanical systems. Multibody System Dynamics 1998; 2:49-69. 2007; 17 2009; 22 2007; 18 1990; 13 1987; 109 1998 1988; 55 1996 1994 2004 1983; 50 2006; 1 2008; 3 1985; 107 1991 2001; 68 1972; 1 1979 1986; 108 1990; 25 1988; 29 1982; 134 2006; 462 1985; 51 2008; 22 1985; 52 1998; 2 1989; 16 1901; X 1969 1989 e_1_2_10_22_2 e_1_2_10_23_2 e_1_2_10_21_2 Shabana AA (e_1_2_10_26_2) 1994 Golub G (e_1_2_10_30_2) 1996 Pennestrì E (e_1_2_10_18_2) 2008; 22 Haug EJ (e_1_2_10_7_2) 1989 Bauchau O (e_1_2_10_4_2) 2008; 3 e_1_2_10_19_2 e_1_2_10_3_2 e_1_2_10_17_2 e_1_2_10_2_2 e_1_2_10_5_2 e_1_2_10_15_2 e_1_2_10_16_2 e_1_2_10_13_2 e_1_2_10_6_2 e_1_2_10_14_2 e_1_2_10_9_2 e_1_2_10_11_2 e_1_2_10_34_2 e_1_2_10_8_2 e_1_2_10_12_2 e_1_2_10_33_2 e_1_2_10_32_2 e_1_2_10_10_2 e_1_2_10_31_2 Campbell SL (e_1_2_10_35_2) 1979 Cheli F (e_1_2_10_36_2) 2006 Shabana AA (e_1_2_10_37_2) 1989 Maggi GA (e_1_2_10_20_2) 1901 e_1_2_10_28_2 e_1_2_10_29_2 Amirouche F (e_1_2_10_27_2) 2004 e_1_2_10_24_2 e_1_2_10_25_2
References_xml	– reference: Kim SS, Vanderploeg MJ. QR decomposition for state space representation of constrained mechanical dynamic systems. ASME Journal of Mechanisms, Transmissions, and Automation in Design 1986; 108:176-182. – reference: Cheli F, Pennestrì E. Cinematica e Dinamica dei Sistemi Multibody, vol. 1. Casa Editrice Ambrosiana: Milano, 2006. – reference: Udwadia FE, Kalaba RE. Analytical Dynamics a New Approach. Cambridge University Press: Cambridge, 1996. – reference: Udwadia FE, Kalaba R. Explicit equations of motion for mechanical systems with nonideal constraints. ASME Journal of Applied Mechanics 2001; 68:462-467. – reference: Laulusa A, Bauchau OA. Review of classical approaches for constraint enforcement in multibody systems. Journal of Computational and Nonlinear Dynamics 2008; 3. http://dx.doi.org/10.1115/1.2803257. – reference: Wang JT, Huston RL. Computational methods in constrained multibody dynamics: matrix formalisms. Computers and Structures 1988; 29:331-338. – reference: Wang JT, Huston RL. A comparison of analysis methods of redundant multibody systems. Mechanics Research Communications 1989; 16:175-182. – reference: Pennestrì E, Valentini PP, Vita L. Multibody dynamics simulation of planar linkages with Dahl friction. Multibody System Dynamics 2007; 17:321-347. – reference: Nikravesh PE. Initial condition correction in multibody dynamics. Multibody System Dynamics 2007; 18:107-115. – reference: Kurdila A, Papastravidis JG. Kamat MP. Role of Maggi's equations in computational methods for constrained multibody systems. Journal of Guidance, Control, and Dynamics 1990; 13:113-120. – reference: Maggi GA. Di alcune nuove forme della dinamica applicabili ai sistemi anolonomi. Rendiconti della Regia Accademia dei Lincei-Serie V 1901; X:287-291. – reference: Baumgarte JW. Stabilization of constraints and integrals of motion in dynamical systems. Computer Methods in Applied Mechanics and Engineering 1972; 1:1-16. – reference: Amirouche FML, Ider SK. Coordinate reduction in the dynamics of constrained multibody systems-a new approach. ASME Journal of Applied Mechanics 1988; 55:899-904. – reference: Mani NK, Haug EJ, Atkinson KE. Singular value decomposition for analysis of mechanical system dynamics. ASME Journal of Mechanisms, Transmissions, and Automation in Design 1985; 107:82-87. – reference: Singh RP, Likins PW. Singular value decomposition for constrained mechanical systems. ASME Journal of Applied Mechanics 1985; 52:943-948. – reference: Wehage RA, Haug EJ. Generalized coordinate partitioning for dimension reduction in analysis of constrained dynamic systems. ASME Journal of Mechanical Design 1982; 134:247-255. – reference: Udwadia FE, Phohomsiri P. Explicit equations of motion for constrained mechanical systems with singular mass matrices and applications to multi-body dynamics. Proceedings of the Royal Society, Series A 2006; 462:2097-2117. – reference: Campbell SL. Generalized Inverses of Linear Transformations. Dover Publications, Inc.: New York, 1979. – reference: Haug EJ. Computer-aided Kinematics and Dynamics of Mechanical Systems. Allyn and Bacon: Newton, MA, 1989. – reference: Borri M, Bottasso CL, Mantegazza P. Equivalence of Kane's and Maggi's equations. Meccanica 1990; 25:272-274. – reference: Amirouche F. Fundamentals of Multibody Dynamics. Birkhhäuser: Basel, 2004. – reference: Golub G, van Loan C. Matrix Computations (3rd edn). The Johns Hopkins University Press: London, 1996. – reference: Pennestrì E, Vita L, de Falco D. An investigation of the influence of pseudoinverse matrix calculations on multibody dynamics simulations by means of the Udwadia-Kalaba formulation. Journal of Aerospace Engineering 2008; 22:365-372. – reference: Unda J, de Jalon JG, Losantos F, Emparantza R. 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Wiley: New York, 1994. – reference: Eich-Soellner E, Führer C. Numerical Methods in Multibody Dynamics. B.G. Teubner: Stuttgart, 1998. – reference: Kamman RL, Huston JW. Dynamics of constrained multibody systems. ASME Journal of Applied Mechanics 1985; 51:899-903. – reference: Vlasenko D, Kasper R. Implementation of consequent stabilization method for simulation of multibodies described in absolute coordinates. 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Snippet	In the last decades, several different formalisms have been proposed in the literature in order to simulate the dynamics of multibody systems. First, the...
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SubjectTerms	Computer simulation DAE system Dynamics Exact sciences and technology Formalism Fundamental areas of phenomenology (including applications) Mathematical analysis Mathematical models Mathematics multibody dynamics Multibody systems Numerical analysis Numerical analysis. Scientific computation Numerical approximation numerical methods Physics Redundant Sciences and techniques of general use Solid dynamics (ballistics, collision, multibody system, stabilization...) Solid mechanics Strategy
Title	Comparison of solution strategies for multibody dynamics equations
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