Solving a generalized constrained optimization problem with both logic AND and OR relationships by a mathematical transformation and its application to robot motion planning

The logic relationship among the equality and inequality constraints in a standard constrained optimization problem (SCOP) is the logical AND. Various efficient, convergent and robust algorithms have been developed for such a SCOP. Motivated by a practical application, a more general constrained optimization problem (GCOP) with not only logic AND but also OR relationships is proposed in this paper. In order to solve such a generalized problem, a mathematical transformation which can transfer a set of inequalities with logic OR into one inequality is developed. This transformation provides a necessary and sufficient condition which enables us to use the algorithms developed for SCOP's to solve the generalized optimization problems. The research is motivated by the requirements of developing an efficient, robust, and reliable navigation algorithm for a mobile robot such as an Autonomous Underwater Vehicle (AUV). The original contributions of the paper include threefold: First, from the viewpoint of optimization theory, this paper, to our best knowledge, is the first one to propose such a GCOP. Second, a method is developed to solve such a GCOP. Third, from the viewpoint of robot path planning, this paper presents a new way of using classical optimization approach to solve robot path planning. It is our belief that some other engineering problems may also be cast as such a generalized form of optimization problem and benefit from the solution given in this paper. Results are presented for path planning in three dimensions, showing its effectiveness, efficiency and the approach is being implemented for European Community MAST project CT97-0083.