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Research Papers

An Ordinary Differential Equation Formulation for Multibody Dynamics: Nonholonomic Constraints

[+] Author and Article Information
Edward J. Haug

Carver Distinguished Professor Emeritus
Department of Mechanical Engineering,
The University of Iowa,
Iowa City, IA 52242
e-mail: echaug@gmail.com

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTING AND INFORMATION SCIENCE IN ENGINEERING. Manuscript received June 7, 2016; final manuscript received August 4, 2016; published online November 7, 2016. Editor: Bahram Ravani.

J. Comput. Inf. Sci. Eng 17(1), 011009 (Nov 07, 2016) (14 pages) Paper No: JCISE-16-1974; doi: 10.1115/1.4034435 History: Received June 07, 2016; Revised August 04, 2016

A method is presented for formulating and numerically integrating ordinary differential equations of motion for nonholonomically constrained multibody systems. Tangent space coordinates are defined in configuration and velocity spaces as independent generalized coordinates that serve as state variables in the formulation, yielding ordinary differential equations of motion. Orthogonal-dependent coordinates and velocities are used to enforce constraints at position, velocity, and acceleration levels. Criteria that assure accuracy of constraint satisfaction and well conditioning of the reduced mass matrix in the equations of motion are used as the basis for updating local coordinates on configuration and velocity constraint manifolds, transparent to the user and at minimal computational cost. The formulation is developed for multibody systems with nonlinear holonomic constraints and nonholonomic constraints that are linear in velocity coordinates and nonlinear in configuration coordinates. A computational algorithm for implementing the approach is presented and used in the solution of three examples: one planar and two spatial. Numerical results using a fifth-order Runge–Kutta–Fehlberg explicit integrator verify that accurate results are obtained, satisfying all the three forms of kinematic constraint, to within error tolerances that are embedded in the formulation.

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References

Haug, E. J. , 2016, “ An Ordinary Differential Equation Formulation for Multibody Dynamics: Holonomic Constraints,” ASME J. Comput. Inf. Sci. Eng., 16(2), p. 021007. [CrossRef]
Haug, E. J. , 1989, Computer-Aided Kinematics and Dynamics of Mechanical Systems, Allyn and Bacon, Boston, MA.
Neimark, J. I. , and Fufaev, N. A. , 1972, Dynamics of Nonholonomic Systems, American Mathematical Society, Providence, RI.
Saha, S. K. , and Angeles, J. , 1991, “ Dynamics of Nonholonomic Mechanical Systems Using a Natural Orthogonal Complement,” ASME J. Appl. Mech., 58(1), pp. 238–243. [CrossRef]
Liang, C. G. , and Lance, G. M. , 1987, “ A Differentiable Null Space Method for Constrained Dynamic Analysis,” ASME J. Mech. Transm. Autom. Des., 109(3), pp. 405–411. [CrossRef]
Betsch, P. , 2004, “ A Unified Approach to the Energy-Consistent Numerical Integration of Nonholonomic Mechanical Systems and Flexible Multibody Dynamics,” GAMM-Mitt., 27(1), p 66. [CrossRef]
Rabier, P. J. , and Rheinboldt, W. C. , 2000, Nonholonomic Motion of Rigid Mechanical Systems From a DAE Viewpoint, SIAM, Philadelphia, PA.
Rabier, P. J. , and Rheinboldt, W. C. , 2002, “ Theoretical and Numerical Analysis of Differential-Algebraic Equations,” Handbook of Numerical Analysis, Vol. 8, P. G. Ciarlet and J. L. Lions , eds., Elsevier Science B.V., Amsterdam, The Netherlands, pp. 183–540.
Schiehlen, W. , and Eismann, W. , 1994, Reduction of Nonholonomic Systems, P. S. Theocari and A. N. Kounadis , eds., National Technical University of Athens, Athens, Greece, pp. 207–220.
Hairer, E. , Norsett, S. P. , and Wanner, G. , 1993, Solving Ordinary Differential Equations I: Nonstiff Problems, 2nd ed., Springer-Verlag, Berlin.
Hairer, E. , Lubich, C. , and Wanner, G. , 2006, Geometric Numerical Integration, 2nd ed., Springer-Verlag, Berlin.
Coddington, E. A. , and Levinson, N. , 1955, Theory of Ordinary Differential Equations, McGraw-Hill, New York.
Serban, R. , and Haug, E. J. , 1998, “ Kinematic and Kinetic Derivatives in Multibody System Analysis,” Mech. Struct. Mach., 26(2), pp. 145–173. [CrossRef]
Garcia-Naranjo, L. C. , and Marrero, J. C. , 2013, “ Non-Existence of an Invariant Measure for a Homogeneous Ellipsoid Rolling on the Plane,” Regular Chaotic Dyn., 18(4), p. 372. [CrossRef]
Bloch, A. M. , 2003, Nonholonomic Mechanics and Control, Springer, New York.

Figures

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Fig. 1

Three-wheel transporter

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Fig. 2

ryrx trajectory of three-wheel transporter

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Fig. 3

Ellipsoid rolling on moving surface

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Fig. 4

Simulations with asymmetric ellipsoid: α=0.5,β=0.4, and δ=0.3 m

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Fig. 5

Simulations with sphere: α=β=δ=0.4 m

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Fig. 6

Geometry of three-wheel motorcycle model

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Fig. 7

Lane change maneuvers

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Fig. 8

Step steer maneuvers

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