dynamic self-organization & computation by natural and artificial potential fields john h reif...
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Dynamic Self-Organization & Computation by Natural and
Artificial Potential Fields
John H ReifDuke University
Download: www.cs.duke.edu/~reif/paper/DynamicSelfOrganization
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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)
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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)
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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.
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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.
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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.
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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).
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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).
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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)
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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
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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
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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.
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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
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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
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Inverse Power Force Laws
Example:power law force law:
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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:
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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:
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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
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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:
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Clustering using an Artificial Potential Field
Initial State:An arbitrary
Distribution of Point Robots
The final resulting Equilibrium State:
Uniform Clustering of the Robots
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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
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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
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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
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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)
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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.