Dynamic Self-Organization & Computation by Natural and

Artificial Potential Fields

John H ReifDuke University

Download: www.cs.duke.edu/~reif/paper/DynamicSelfOrganization

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. 162-176.

- 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 Easyl

Near linear in number of particles n:

• Can use multipole algorithms of Greengard and Rokhlin (1985).

• Also speeded up byJohn 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 byJohn 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. 431-459. • Published in Robotics and Autonomous Systems, Vol. 27, no.3, pp.171-194, (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 Xi

• Has a current velocity Vi

• Is subject to external forces on it depending on the arrangement of the other particles

• Has mass mk

• 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 Fi 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 Fi on particle i is the sum of the forces between particle and each other particle j:

Potential Fields induced by Particle Attraction and Repulsion

• ri,j = ||Xi-Xj|| is the distance between particle i and particle j

• There is a inverse power force law Fi,j (Xi, Xj) between particle i and particle j that depends on distance ri,j .

• The inverse power force laws between particles is defined by parameters ci,j and σi,j

Potential Fields induced by Particle Attraction and Repulsion

Although the inverse power laws can be complex,the force Fi 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 arbitraryDistribution 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

① Artificial Potential Fields [Reif&Wang94] provide a powerful method for programming complex behavior in autonomous systems

② Even 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.