Extremal Collective Behavior

Abstract

Curves and natural frames can be used for describing and controlling motion in both biological and engineering contexts (e.g., pursuit and formation control). The geometry of curves and frames leads naturally to a Lie group formulation where coordinated motion is represented by interacting particles on Lie groups - specifically, SE(2) or SE(3). Here we consider a particular type of optimal control problem in which the interactions between particles arise from a cost function dependent on each particle's steering, and which penalizes steering differences between the particles (expressed via the graph Laplacian). With this choice of cost function we are able to perform Lie-Poisson reduction. Furthermore we are able to derive a closed-form expression (using Jacobi elliptic functions) for certain special solutions of the coupled multi-particle problem on SE(2).

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Document Details

Document Type
Technical Report
Publication Date
Dec 01, 2010
Accession Number
ADA544233

Entities

People

  • E. W. Justh
  • P.S.Krishnaprasad

Organizations

  • United States Naval Research Laboratory

Tags

Communities of Interest

  • Air Platforms
  • Energy and Power Technologies
  • Materials and Manufacturing Processes

DTIC Thesaurus Topics

  • Air Force
  • Curvature
  • Differential Geometry
  • Dynamics
  • Engineering
  • Equations
  • Formation Flight
  • Fuel Consumption
  • Geometry
  • Ground Vehicles
  • Lie Groups
  • Military Research
  • Particles
  • Steering
  • Three Dimensional
  • Trajectories
  • Vehicles

Fields of Study

  • Mathematics

Readers

  • Calculus or Mathematical Analysis
  • Control Systems Engineering.
  • Robotics and Automation.