Constructive Solid Geometry with Projection: An Approach to Piano Movers’ Problem

Configuration space obstacles are regions in a configuration space which represent forbidden configuration of the object due to the presence of other objects. They can be regarded as geometric objects which are representable using Boolean combinations of equations and inequalities. Moreover, they can also be represented using existential quantifiers which correspond to geometric projections. If the projected variables only occur algebraically then it is possible to eliminate quantifiers and represent configuration space obstacles in the semi-algebraic form. However, no matter how it is done, the quantifier elimination is computationally hard and the output in semi-algebraic representation is often large and cumbersome. Therefore, we are looking for a possibility to work directly with the quantified representation of the configuration space obstacles. We investigate combination of tools which can be used in conjunction with this quantified representation. The main idea is the use of interval evaluation on equations and inequalities involving some transcendental functions. We suggested an extended version of Constructive Solid Geometry system which has trigonometric functions as well as the usual Boolean operators. The combination of this system together with interval arithmetic allows simplification and spatial subdivision and pruning to be done in a natural way. We also raise the possibility of an extended Constructive Solid Geometry system which would have projection and boundary formation as operators. This would allow compact representation of the configuration space, but presents computational problems which are, as yet, unsolved. ? Supported by the scholarship Technolgiestipendien Südostasien 2005/2006 (Postdoc) awarded by the Austrian Exchange Service – Agency for International Cooperation in Education and Research (ÖAD), Academic Cooperation and Mobility (ACM).