A Fast Algorithm for Incremental Distance Calculation Paper by Ming C. Ling and John F. Canny Presented by Denise Jones

Algorithm Concept A method for calculating the closest features on two convex polyhedra Algorithm is complete (will always find closest features between 2 polyhedra) Can be used for: –Collision detection –Motion planning –Distance between objects in 3-D space

Algorithm Concept Apply applicability criteria to features (vertices, edges, faces) of each polyhedron Without any initialization, running time of the algorithm is linear for the number of vertices With initialization, running time is constant Can detect collision –Returns an error and features that have collided or intersected

Efficiency Once the closest features are determined, these will change infrequently When a change does occur, the new closest features will usually be on a boundary of the previous closest features.

Efficiency Exceptions Initial features are parallel and on opposite sides (only on initialization) Exceptions after initialization Parallel faces Before RotationAfter Rotation

Applicability Criterion: Point-Vertex Determine planes that are perpendicular to coboundary (edges) of the vertex Point must be contained within the boundary of these planes in order to be the closest point If outside one of the boundaries, indicates that edge is closer and will perform the test on that edge.

Applicability Criterion: Point-Vertex Voronoi Region Vertex Point

Applicability Criterion: Point-Edge Determine region created by planes perpendicular to the head and tail of the edge and perpendicular to the coboundaries (faces) Point must be contained within the boundary of these planes in order for edge to be the closest feature If outside one of the boundaries, indicates corresponding feature is closer and will “walk” to the next feature and apply the appropriate test.

Applicability Criterion: Point-Edge Point Edge Voronoi Space

Applicability Criterion: Point-Face Determine planes that are perpendicular to each edge of the face Point must be contained within the applicability prism (region comprised of these planes and the edges of the face) If outside one of the boundaries, indicates that edge is closer and performs the test for the corresponding edge. If point lies below the face, two possibilities –Collision –Another feature is closer than face or any of edges Algorithm will return closest feature

Applicability Criterion: Point-Face Point Face Voronoi Space

Algorithm Cases Vertex-Vertex Vertex-Edge Vertex-Face Edge-Edge Edge-Face Face-Face

Algorithm Example Determine closest features on the following polyhedra Randomly choose 2 features

Algorithm Example: Vertex-Vertex Both vertices must meet point-vertex applicability criteria If either fails, will return vertex and corresponding edge

Algorithm Example: Vertex-Vertex Test fails Voronoi Region Vertex Point

Algorithm Example: Vertex-Edge Edge must meet point-edge criterion Vertex must meet point-vertex criterion for closest point on edge to vertex If either fails, will return new feature pair based on failed test

Algorithm Example: Vertex-Edge Test fails Point Edge Voronoi Space Closest point to vertex on edge

Algorithm Example: Vertex-Face Face must meet point-face criterion Vertex must meet point-vertex criterion for closest point on face to vertex If either fails, will return new feature pair based on failed test

Algorithm Example: Vertex-Face Test passes for both criteria As polyhedra continue along their trajectories, the necessary tests are reapplied starting with the last one performed. Usually only one test will be required. Vertex Face Voronoi Space Closest point to vertex on face

Motion Planning Discretization a factor in collision-avoidance. Location 2 Location 1

Conclusion Algorithm calculates closest features on 2 convex polyhedra Algorithm is relatively simple Algorithm is efficient –Runs in constant time once initialized –Runs linearly in proportion to number of vertices when initializing Algorithm is complete

Data Structures and Concavity Polyhedron: faces, edges, vertices, position, and orientation Face: outward normal, distance from origin, vertices, edges, and coboundary Edge: head, tail, right face, and left face Vertex: x, y, z, and coboundary Polyhedron must be convex A concave polyhedron must be converted to multiple convex polyhedra –May effect efficiency (quadratic)

Algorithm: Edge-Face Determine if they are parallel If parallel, they are closest features if both: –The edge passes through the applicability prism created by the face –The normal of the face being evaluated is between the normals of each face bounding the edge being evaluated

Algorithm: Edge-Face If not parallel –One of the vertices of the edge is closer to the face if it meets the point-face applicability criterion and the edge points into the face, and this pair is returned –If this criterion is not met, the edge-edge component of the algorithm is applied to the edge bounding the face that is closest to the edge and the edge under examination.

Algorithm: Edge-Face Applicability Prism Edge (Parallel to Face)

Algorithm: Edge-Face Test Failure Vertex closer to face than edge Edge (Not Parallel to Face)

Algorithm: Edge-Face Test Failure Test Failure Edge closer to edge than face Edge (Not Parallel to Face)

Algorithm: Edge-Edge Determine closest points on the edges Apply point-edge applicability criteria to each pair If either fails, will return new feature pair based on failed test

Algorithm: Edge-Edge Nearest Points

Algorithm: Face-Face Determine if faces are parallel If parallel –Check to see if overlapping, if so, they are closest features If not parallel or not overlapping –Return first face and nearest edge of second face to first face

Algorithm: Face-Face Parallel Faces