# Free Space Computation for a Polyhedron with Two Translations and One Rotation

#### Abstract

We present a robust algorithm for free space construction for a moving polyhedron with three degrees of freedom: two translations and one rotation. The free space is a 3D manifold of all the transformations of a moving polyhedron with respect to an obstacle polyhedron, where there is no overlap between them. The algorithm explicitly constructs a combinatorial representation of the configuration space, where configuration space is the superset of both free space and blocked space. For robust implementation, we use the adaptive control perturbation (ACP) library and we handle identities explicitly. ACP perturbs the input and stores the input as intervals. It computes the signs of the predicate using interval arithmetic. Failed predicates are reevaluated after increasing the precision of its arguments. However algebraically dependent input could yield degenerate predicates and hence any perturbation strategies would fail. We call this type of degeneracy an identity because the predicate is identically zero for all inputs. In our implementation of the free space computation algorithm, we explicitly detect identities by inspecting the input vertices and handle them accordingly. Free space computation algorithms can help in solving other computational geometry problems. The path planning problem, where the goal is to find a path from a start pose of a robot to a goal pose, can be solved using these algorithms. Another application of free space construction algorithms is the maximum clearance problem, which tries to find a path that stays as far away from the obstacle as possible. We validate our free space algorithm by implementing path planning and maximum clearance using our free space construction algorithm.

#### Degree

Ph.D.

#### Advisors

HOFFMANN, Purdue University.

#### Subject Area

Computer science

Off-Campus Purdue Users:

To access this dissertation, please **log in to our
proxy server**.