Dynamic Self-Organization & Computation by Natural and Artificial Potential Fields John H Reif Duke University Download:

Example of Natural Potential Fields -Gravitation Force Fields (date to Newton in 1600s) -Electrostatic Force Fields e.g., Coulomb attraction (dates to 1700s) -Magnetic Force Fields (dates to 1800s) -Social Behavior (eg Flocking) by Groups of Animals (dates to 1800s) -Molecular Force Fields (dates to 1900s)

Closed Form Solution of 2 Particle Systems -For 2 particle systems: -Quadratic trajectories definable in closed form -Proof dates at least to Newton’s Philosophiæ Naturalis Principia Mathematica (1676)

Closed Form Solution of 3 Particle Systems -Except in special cases, the motion of three bodies is generally non-repeating -Would like an analytical solution given by algebraic expressions and integrals. -Posed as open problem in Newton’s Philosophiæ Naturalis Principia Mathematica (1676) -Henri Poincaré (1887) proved there is no general analytical solution of the general three- body problem.

n-Body Simulation -Given: the initial positions and velocities of n particles that have pair-wise inverse power force interactions -The n-body simulation problem is to simulate the movement of these particles so as to determine these particles at a future time. -The reachability problem is to determine if a specific particle will reach a certain specified region at some specified target time.

Computational Complexity of n-body Simulation Steve R. Tate and John H. Reif, The Complexity of N-body Simulation, Proceedings of the 20th Annual Colloquium on Automata, Languages and Programming (ICALP'93), Lund, Sweden, July, 1993, pp Proof that the n-body Simulation reachability problem for a set of interacting particles in three dimensions is PSPACE-hard: -Assumes: a polynomial number of bits of accuracy and polynomial target time. -All previous lower bound proofs required either artificial external forces or obstacles.

In Practice Approx. n-body Simulation is Often Easy l Near linear in number of particles n: Can use multipole algorithms of Greengard and Rokhlin (1985). Also speeded up by John H. Reif and Steve R. Tate, "N-body simulation I: Fast algorithms for potential field evaluation and Trummer's problem”. (1992).

In Practice n-body Simulation is Easy Near linear in number of particles n: Can use multipole algorithms of Greengard and Rokhlin (1985). Also speeded up by John H. Reif and Steve R. Tate, "N-body simulation I: Fast algorithms for potential field evaluation and Trummer's problem”. (1992).

Flocking: Natural “Social” Potential Field Guided Clustering of Birds on Ground and in Sky First Flocking models due to Thomas Henry Huxley in the 1800s. Applied to Computer Graphics by Reynolds (1987)

Artificial Potential Fields First Used in robotic motion planning Obstacles: provide a negative force to object to be moved Not always correct solution for robotic motion planning, but of practical use

Artificial Potential Fields John H. Reif and Hongyan Wang, Social Potential Fields: A Distributed Behavioral Control for Autonomous Robots (1994): Workshop on Algorithmic Foundations of Robotics (WAFR'94), San Francisco, California, February, 1994; The Algorithmic Foundations of Robotics, A.K.Peters, Boston, MA. 1995, pp Published in Robotics and Autonomous Systems, Vol. 27, no.3, pp , (May 1999). Use n particles to represent dynamically moving objects Particles may be: Animals Predators and Prey Robots

Artificial Potential Fields: For distributed autonomous control of autonomous robots. We define simple artificial force laws between pairs of robots or robot groups. The force laws are sums of multiple inverse-power force laws, incorporating both attraction & repulsion. The force laws can be distinct for distinct robots - they reflect the 'social relations' among robots. The resulting artificial force imposed by other robots and other components of the system control each individual robot’s motion. The approach is distributed in that the force calculations and motion control can be done in an asynchronous and distributed manner.

Application of Artificial Potential Fields to Autonomous Robotic systems Autonomous Robotic systems can consist of from hundreds to perhaps tens of thousands or more autonomous robots. The costs of robots are going down, and the robots are getting more compact, more capable, and more flexible. Hence, in the near future, we expect to see many industrial and military applications of Autonomous Robotic systems in industrial, social, and military tasks such as: Organizing Group Activities such as Assembling Transporting Hazardous inspection Patrolling, Guarding, and/or Attacking

Particle Systems n Particles named 1,…,n Each particle i=1,…,n is: Positioned in d-dimensional space at position X i Has a current velocity V i Is subject to external forces on it depending on the arrangement of the other particles Has mass m k Obeys Newton’s laws

Inverse Power Force Laws Example: power law force law:

Potential Fields induced by Particle Attraction and Repulsion Although the inverse power laws can be complex, the force F i on particle i is just the sum of the forces between particle and each other particle j:

Potential Fields induced by Particle Attraction and Repulsion The force F i on particle i is the sum of the forces between particle and each other particle j:

Potential Fields induced by Particle Attraction and Repulsion r i,j = ||X i -X j || is the distance between particle i and particle j There is a inverse power force law F i,j (X i, X j ) between particle i and particle j that depends on distance r i,j. The inverse power force laws between particles is defined by parameters c i,j and σ i,j

Potential Fields induced by Particle Attraction and Repulsion Although the inverse power laws can be complex, the force F i on particle i is just the sum of the forces between particle and each other particle j:

Clustering using an Artificial Potential Field Initial State: An arbitrary Distribution of Point Robots The final resulting Equilibrium State: Uniform Clustering of the Robots

Clustering around a “Square Castle” using an Artificial Potential Field Initial State: An arbitrary Distribution of Point Robot Guards around the Green Castle Final Equilibrium State providing Dynamic Guarding Behavior: The guards converge to a guarding ring surrounding the Green Castle

Dynamic Guarding Behavior around a “Square Castle” using an Artificial Potential Field Initial State: An arbitrary Distribution of Point Robot Guards around the Green Castle Dynamic Guarding Behavior: Red Invader is confronted by nearby Point Robot Guards around the Green Castle

Reorganization of two groups (Dark and Light Circles) before and after Bivouacking together Final State: Two separate clusters of point robots Intermediate Bivouacking State: Merged clusters of point robots Initial State: Two separate clusters of point robots Reorganization after Bivouacking

Clustering of Deminers (Squares) around a Mines (Squares) using an Artificial Potential Field Initial State: Separate clusters of mines(disks) and deminer robots (squares) Final State: Clusters of deminers (squares) near mines (disks)

Conclusion ①Artificial Potential Fields [Reif&Wang94] provide a powerful method for programming complex behavior in autonomous systems ②Even though in theory [Reif&Tate93] the simulation can be hard, in practice we can use efficient multipole algorithms [Greengard&Rokhlin,85][Reif&Tate92] for simulating n-body movement and predicting the particle’s long range behavior.