Validated Modeling of Mechanical Systems with SmartMOBILE: Improvement of Performance by ValEncIA-IVP

Computer simulations of real life processes can generate erroneous results, in many cases due to the use of finite precision arithmetic. To ensure correctness of the results obtained with the help of a computer, various kinds of validating arithmetic and algorithms were developed. Their purpose is to provide bounds in which the exact result is guaranteed to be contained. Verified modeling of kinematics and dynamics of multibody systems is a challenging application field for such methods, largely because of possible overestimation of the guaranteed bounds, leading to meaningless results. In this paper, we discuss approaches to validated modeling of multibody systems and present a template-based tool SmartMOBILE , which features the possibility to choose an appropriate kind of arithmetic according to the modeling task. We consider different strategies for obtaining tight state enclosures in SmartMOBILE including improvements in the underlying data types (Taylor models), modeling elements (rotation error reduction), and focus on enhancement through the choice of initial value problem solvers ( ValEncIA-IVP ).

[1]  Youdong Lin,et al.  Validated Solution of Initial Value Problems for ODEs with Interval Parameters , 2006 .

[2]  O. Knüppel,et al.  PROFIL/BIAS—A fast interval library , 1994, Computing.

[3]  Rudolf J. Lohner,et al.  On the Ubiquity of the Wrapping Effect in the Computation of Error Bounds , 2001, Perspectives on Enclosure Methods.

[4]  Douglas Gregor,et al.  C++ Templates: The Complete Guide , 2002 .

[5]  Andreas Rauh,et al.  A novel interval method for validating state enclosures of the solution of initial value problems , 2008 .

[6]  Luc Jaulin,et al.  Applied Interval Analysis , 2001, Springer London.

[7]  N.S. Nedialkov,et al.  Rigorous simulation of hybrid dynamic systems with symbolic and interval methods , 2002, Proceedings of the 2002 American Control Conference (IEEE Cat. No.CH37301).

[8]  Nedialko S. Nedialkov,et al.  An Interval Hermite-Obreschkoff Method for Computing Rigorous Bounds on the Solution of an Initial Value Problem for an Ordinary Differential Equation , 1998, SCAN.

[9]  Henrietta Hardy Mrs Hammond.  The complete guide , 1883 .

[10]  Ulrich W. Kulisch,et al.  Perspectives on Enclosure Methods , 2001 .

[11]  Andrés Kecskeméthy,et al.  Interval Algorithms in Modeling of Multibody Systems , 2003, Numerical Software with Result Verification.

[12]  Nathalie Revol,et al.  Taylor models and floating-point arithmetic: proof that arithmetic operations are validated in COSY , 2005, J. Log. Algebraic Methods Program..

[13]  David Vandevoorde,et al.  C++ Templates , 2002 .

[14]  Martin Berz,et al.  Verified Integration of ODEs and Flows Using Differential Algebraic Methods on High-Order Taylor Models , 1998, Reliab. Comput..

[15]  Wolfram Luther,et al.  Numerical Software with Result Verification , 2004, Lecture Notes in Computer Science.

[16]  Pascal Van Hentenryck,et al.  Consistency Techniques in Ordinary Differential Equations , 1998, Constraints.

[17]  T. Csendes.  Developments in Reliable Computing , 2000 .

[18]  Ansi Ieee,et al.  IEEE Standard for Binary Floating Point Arithmetic , 1985 .

[19]  Ekaterina Auer.  Interval modeling of dynamics for multibody systems , 2007 .

[20]  Siegfried M. Rump,et al.  INTLAB - INTerval LABoratory , 1998, SCAN.

[21]  Arnold Neumaier,et al.  Taylor Forms—Use and Limits , 2003, Reliab. Comput..

[22]  Andreas Rauh,et al.  Reduction of overestimation in interval arithmetic simulation of biological wastewater treatment processes , 2007 .

[23]  E. Hofer,et al.  VALENCIA-IVP: A Comparison with Other Initial Value Problem Solvers , 2006, 12th GAMM - IMACS International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics (SCAN 2006).

[24]  Andreas Griewank,et al.  Evaluating derivatives - principles and techniques of algorithmic differentiation, Second Edition , 2000, Frontiers in applied mathematics.

[25]  H. Aschemann,et al.  Interval methods for simulation of dynamical systems with state-dependent switching characteristics , 2006, 2006 IEEE Conference on Computer Aided Control System Design, 2006 IEEE International Conference on Control Applications, 2006 IEEE International Symposium on Intelligent Control.

[26]  N. Nedialkov,et al.  Computing rigorous bounds on the solution of an initial value problem for an ordinary differential equation , 1999 .

[27]  Martin Berz,et al.  COSY INFINITY version 8 , 1999 .