This thesis deals with the problem of finding trajectories of objects through space which do not result in collisions. Specifically, a method is presented which calculates such trajectories in the case of a robot attempting to grasp a certain body in a,,cluttered environment, i.e. where there are other bodies nearby.' We restrict ourselves to bodies with planar faces, robots with certain physical characteristics, and a specific class of trajectories. We present a method of describing bodies in terms of the shape of their faces. The robot's trajectories are not predetermined, but instead are calculated on the basis of the present configuration of bodies in the robot's world. Once the body to be grasped has been chosen, we examine all nearby objects to determine the region of space which is inaccessible to the robot. This region is projected onto a suitable plane, and making use of algorithms to compute the intersection and union of two-dimensional figures, we are able to find a region of the surface of the body to be grasped which is sufficiently distant from neighbouring objects, assuming such a region exists. This region in turn allows us to calculate a set of feasible robot trajectories for grasping the body. We also suggest a possible approach to computing more general trajectories involving both the robot and a body.