An important research motivation in Applied Computational Geometry is that many geometric algorithms, for instance used in Computer Aided Design systems, are actually not robust, e.g., due to use of fast but inexact floating point arithmetic. Within this general setting we are interested in the robust solution of general geometric problems that steam from the context of robot motion planing. The initial proposal for this project was to investigate the Voronoi Diagram of polyhedra in three-dimensional space as the diagram encodes the set of positions that keep maximal clearance from the obstacles. Important milestones where the improvement of the basic algebraic tools as well as a guaranteed logarithmic planar point location within planar subdivisions of general (possibly unbounded) algebraic curves. The second milestone was a unexpected success as we fully achieved the milestone and in addition also improved the runtime of the general algorithm from O(nlog2^n) to O(nlogn). This was unexpected as planar point location is very fundamental and well studied field. For the first milestone, we where able to extend ideas for lazy construction for algebraic numbers of low algebraic degree, but we where not able to generalize ideas to algebraic numbers of arbitrary (i.e. the required) degree. As a consequence the runtimes for an input of around 25 lines in generic position remained above several hours, which is not an acceptable runtime for a practical application of the approach. Therefore, I had to draw my attention to several other topics that are directly or indirectly motivated by exact computing and robot motion planing. Together with Alon Braham I worked on exact planar Minkowski sum computation for simple polygons. The work covers different convolution methods and provides concrete proofs of several theoretical aspects. Up to a few pathological examples the implementation computes Minkowski sums significantly faster than the old exact implementation within CGAL. Working with Oren Salzman on robot motion planing we suggest to take samples of the robots configuration space that are entire low-dimensional manifolds. These samples capture the connectivity of the configuration space much better than isolated point samples. Moreover, together with Andreas von Dziegielewski and Elmar Schomer I introduce a novel and efficient technique to generate a high quality mesh that approximates the outer boundary of a swept volume (SV) with guarantees.