Heuristics

Asymptotically optimal algorithm

Pure mathematics

Dimension (graph theory)

Mathematics

Economic growth

Algorithm

Euclidean distance

Filter (signal processing)

Mathematical optimization

Euclidean geometry

Curse of dimensionality

Sampling (signal processing)

Political science

Artificial intelligence

Norm (philosophy)

Economics

Programming language

Geometry

Convergence (economics)

Path (computing)

Computer science

Computer vision

State (computer science)

Motion planning

Robot

IEEE Transactions on Robotics

Anytime almost-surely asymptotically optimal planners, such as RRT*, incrementally find paths to every state in the search domain. This is inefficient once an initial solution is found, as then only states that can provide a better solution need to be considered. Exact knowledge of these states requires solving the problem but can be approximated with heuristics. This paper formally defines these sets of states and demonstrates how they can be used to analyze arbitrary planning problems. It uses the well-known L2 norm (i.e., Euclidean distance) to analyze minimum-path-length problems and shows that existing approaches decrease in effectiveness factorially (i.e., faster than exponentially) with state dimension. It presents a method to address this curse of dimensionality by directly sampling the prolate hyperspheroids (i.e., symmetric ndimensional ellipses) that define the L2 informed set. The importance of this direct informed sampling technique is demonstrated with Informed RRT*. This extension of RRT* has less theoretical dependence on state dimension and problem size than existing techniques and allows for linear convergence on some problems. It is shown experimentally to find better solutions faster than existing techniques on both abstract planning problems and HERB, a two-arm manipulation robot.

Informed Sampling for Asymptotically Optimal Path Planning