Near-Minimum-Energy Paths on a Vertical-Axis Cone with Anisotropic Friction and Gravity Effects

We determine near-optimal paths with respect to work against friction and gravity on the surface of a vertical-axis ideal cone, assuming a moving agent of negligible size, friction proportional to the normal force, and power-maximum and stability limitations on the agent. This can provide good paths across hilly terrain for mobile robots. Our previous work required difficult-to-obtain polyhedral terrain models; cone surface patches permit easier and better models. We prove that our near-optimal paths on a vertical-axis cone surface are not much more complex than on polyhedra: There are qualitatively 22 kinds of path behavior (as opposed to four on a polyhedral face), and the area of the surface optimally reachable from a fixed start point by a given qualitative behavior has mathematically simple boundaries (only line segments, circle arcs, and arcs of logarithmic spirals in azimuth projection). We examine the possible cases based on agent characteristics and cone steepness, and provide "behavior maps" for quickly finding near-optimal paths between any two points on the cone surface. Our models incorporate discontinuous effects with respect to traversal heading. Comparisons with a program using uniform-grid path planning show our methods run considerably faster in less space, and our paths are much simpler to describe and easier to follow in the real world.