LiU Electronic Press

# Conference article

## Using an Adaptive FEM to Determine the Optimal Control of a Vehicle During a Collision Avoidance Manoeuvre

Karin Kraft
Mathematical Sciences, Chalmers University of Technology, Sweden \ Göteborg University, Sweden

Mathematical Sciences, Chalmers University of Technology, Sweden \ Göteborg University, Sweden

Mathias Lidberg
Applied Mechanics, Chalmers University of Technology, Sweden

Linköping Electronic Conference Proceedings 27:15, p. 126-133

Published: 2007-12-21

ISBN:

ISSN: 1650-3686 (print), 1650-3740 (online)

### Abstract

The optimal manoeuvering of a vehicle during a collision avoidance manoeuvre is investigated. A simple model where the vehicle is modelled as point mass and the mathematical formulation of the optimal manoeuvre are presented. The resulting two-point boundary problem is solved by an adaptive finite element method and the theory behind this method is described.

### Keywords

Vehicle dynamics; collision avoidance manoeuvre; optimal control; boundary value problem; adaptive finite element method.

### References

[1] National Highway Traffic Safety Administration; Laboratory Test Instruction for FMVSS 126 Stability Control Systems; http://www.nhtsa.dot.gov.

[2] C. Andersson; Solving optimal control problems us- ing FEM; Master’s thesis; Chalmers University of Technology; 2007.

[3] U. M. Ascher; R. M. M. Mattheij; and R. D. Russell; Numerical Solution of Boundary Value Prob- lems for Ordinary Differential Equations; first ed.; Prentice-Hall Inc.; Englewood Cliffs; New Jersey; 1988.

[4] J. T. Betts; Survey of numerical methods for trajectory optimization; AIAA J. Guidance Control Dynam. 21 (1998); 193–207.

[5] S. C. Brenner and L. R. Scott; The Mathemati- cal Theory of Finite Element Methods; second ed.; Springer-Verlag; New York; 2002.

[6] A. E. Bryson; Jr and Y. Ho; Applied Optimal Con- trol; revised printing ed.; Hemisphere Publishing Corporation; Washington; D.C; 1975.

[7] P. Deuflhard and F. Bornemann; Scientific Computing with Ordinary Differential Equations; Springer-Verlag New York Inc.; New York; 2002.

[8] D. J. Estep; D. H. Hodges; and M.Warner; Computational error estimation and adaptive error control for a finite element solution of launch vehicle trajectory problems; SIAM J. Sci. Comput. 21 (1999); 1609– 1631.

[9] H. B. Keller; Numerical Methods for Two-Point Boundary-Value Problems; Dover Publications; Inc.; New York; 1992.

[10] B. Schmidtbauer; Sv¨ang och bromsa samtidigt (in Swedish); Teknisk Tidskrift (1971).

[11] L. F. Shampine; J. Kierzenka; and M. W.Reichelt; Solving Boundary Value Problems for Ordinary Differential Equations in Matlab using bvp4c; Tech. report; MathWorks; 2000; http://www.mathworks.com. 133