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On the Gröbner basis triangularization of constraint equations in natural coordinates

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Abstract

Efficient dynamic simulation code is essential in many situations (including hardware-in-the-loop and model-predictive control applications), and highly beneficial in others (such as design optimization, sensitivity analysis, parameter identification, and controller tuning tasks). When the number of modeling coordinates n exceeds the degrees-of-freedom of the system f, as is often the case when closed kinematic chains are present, the governing dynamic equations consist of n second-order ordinary differential equations (ODEs) coupled with m=n−f algebraic constraint equations. This set of n+m index-3 differential-algebraic equations can be difficult to solve in an efficient yet accurate manner. Embedding (or generalized coordinate partitioning) can be used to obtain f ODEs (one for each independent acceleration), which are generally more amenable to numerical integration; however, the dependent positions are typically computed from the independent positions at each time step. Newton–Raphson iteration is often used for solving the position-level kinematics, but only provides solutions to within a specified tolerance, and can require several iterations to converge. In this work, Gröbner bases are used to obtain recursively solvable symbolic solutions for the dependent positions, which can then be evaluated to within machine precision using a fixed number of arithmetic operations. Natural coordinates are particularly attractive in this context, since the resulting constraint equations are maximally quadratic polynomials and are, therefore, easily triangularized. The proposed approach is suitable for use in an automated formulation procedure and, as demonstrated by three examples, is capable of generating highly efficient simulation code with minimal additional effort required at the formulation stage.

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Notes

  1. The options shown in Figs. 1(b) and 1(c) lead to position-dependent inertia matrices and velocity-dependent inertial forces.

  2. The constraints are switched at θ i ≈37∘ rather than 45∘, a heuristic that favors the use of the cosine constraint, which is the less computationally expensive of the two.

  3. Maple is a trademark of Waterloo Maple Inc.

  4. Microsoft and Visual C++ are registered trademarks of Microsoft Corporation.

  5. Intel and Intel Core are trademarks of Intel Corporation.

  6. Xeon is a trademark of Intel Corporation.

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Acknowledgements

The authors gratefully acknowledge the financial support provided by the Natural Sciences and Engineering Research Council of Canada (NSERC), the NSERC/Toyota/Maplesoft Industrial Research Chair program, the OPTIVIRTEST Project (reference TRA2009-14513-C02-01) of the Spanish Ministry of Science and Innovation, and the Education Department of the Government of Navarra, Spain.

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Authors and Affiliations

  1. Department of Systems Design Engineering, University of Waterloo, 200 University Avenue West, Waterloo, ON, N2L 3G1, Canada

    Thomas Uchida & John McPhee

  2. Computational Mechanics Group, Instituto Universitario de Investigación del Automóvil, Universidad Politécnica de Madrid, Carretera de Valencia km 7, 28031, Madrid, Spain

    Alfonso Callejo & Javier García de Jalón

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  1. Thomas Uchida
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  2. Alfonso Callejo
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Correspondence to Thomas Uchida.

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Uchida, T., Callejo, A., García de Jalón, J. et al. On the Gröbner basis triangularization of constraint equations in natural coordinates. Multibody Syst Dyn 31, 371–392 (2014). https://doi.org/10.1007/s11044-013-9397-0

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