Representing sets of orientations as convex cones

Abstract

In a wide range of applications the orientation of a rigid body does not need to be restricted to one given orientation, but can be given as a continuous set of frames. We address the problem of defining such sets and to find simple tests to verify if an orientation lies within a given set. The unit quaternion is used to represent the orientation of the rigid body and we develop three different sets of orientations that can easily be described by simple constraints in quaternion space. The three sets discussed can also be described as convex cones in Ropf <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> defined by different norms. By describing the sets as convex cones and using certain properties of dual cones, we are able find simpler representations for the set of orientations and computationally faster and more accurate tests to verify if a quaternion lies within the given set.