Curve Shortening and the Rendezvous Problem for Mobile Autonomous Robots

If a smooth, closed, and embedded curve is deformed along its normal vector field at a rate proportional to its curvature, it shrinks to a circular point. This curve evolution is called Euclidean curve shortening and the result is known as the Gage-Hamilton-Grayson theorem. Motivated by the rendezvous problem for mobile autonomous robots, we address the problem of creating a polygon shortening flow. A linear scheme is proposed that exhibits several analogues to Euclidean curve shortening: The polygon shrinks to an elliptical point, convex polygons remain convex, and the perimeter of the polygon is monotonically decreasing.

• Publication year

  2006

• Venue

  IEEE Transactions on Automatic Control

• Publication date

  2006-05-16

• Fields of study

  Mathematics, Computer Science, Engineering

• Identifiers
  DOI 10.1109/TAC.2007.899024arXiv cs/0605070
• External record

  Open on Semantic Scholar

• Source metadata

  Semantic Scholar

• No claims are published for this paper.

• No concepts are published for this paper.

• Nonlinear Control Systems

  2008cited by this paper
• Robust rendezvous for mobile autonomous agents via proximity graphs in arbitrary dimensions

  2006cited by this paper
• Eventually consistent compasses for robust gathering of asynchronous mobile robots with limited visibility

  2005cited by this paper
• Curve Shortening and its Application to Multi-Agent Systems

  2005cited by this paper
• A hierarchical cyclic pursuit scheme for vehicle networks

  2005cited by this paper
• Stability of multiagent systems with time-dependent communication links

  2005cited by this paper
• Gathering of asynchronous robots with limited visibility

  2005cited by this paper
• Formations of vehicles in cyclic pursuit

  2004cited by this paper
• Performance recovery under output feedback sampled-data stabilization of a class of nonlinear systems

  2004cited by this paper
• Leaderless coordination via bidirectional and unidirectional time-dependent communication

  2003cited by this paper
• Local control strategies for groups of mobile autonomous agents

  2003influential reference
• The multi-agent rendezvous problem

  2003cited by this paper
• Coordination of groups of mobile autonomous agents using nearest neighbor rules

  2002cited by this paper
• POLYGON SHORTENING MAKES (MOST) QUADRILATERALS CIRCULAR

  2002cited by this paper
• Hierarchy Among Automata on Linear Orderings

  2002cited by this paper
• The Curve Shortening Problem

  2001cited by this paper
• Distributed memoryless point convergence algorithm for mobile robots with limited visibility

  1999influential reference
• A discrete curve-shortening equation

  1997cited by this paper
• Asymptotic controllability implies feedback stabilization

  1997cited by this paper
• Geometric expansion of convex plane curves

  1996cited by this paper
• Analytic capacity: discrete approach and curvature of measure

  1995cited by this paper
• Evolutions of Planar Polygons

  1995cited by this paper
• Nonlinear control via approximate input-output linearization: the ball and beam example

  1989cited by this paper
• Shortening embedded curves

  1989cited by this paper
• The heat equation shrinks embedded plane curves to round points

  1987cited by this paper
• The heat equation shrinking convex plane curves

  1986cited by this paper
• Remarks on smooth feedback stabilization of nonlinear systems

  1986cited by this paper
• Stabilization of a class of nonlinear systems by a smooth feedback control

  1985cited by this paper
• Curve shortening makes convex curves circular

  1984cited by this paper
• Approximate linearization by state feedback and coordinate change

  1984cited by this paper
• An isoperimetric inequality with applications to curve shortening

  1983influential reference
• Finite dimensional linear systems

  1972cited by this paper

• The manifold of polygons degenerated to segments

  2024cites this paper
• Distance‐targeted competitive follower‐attraction containment control for multi‐agent systems with weighted directed graphs

  2023cites this paper
• Optimal control of robotic systems and biased Riemannian splines

  2023cites this paper
• Moduli space of simple polygons

  2023cites this paper
• Rolling Maps and Nonlinear Data

  2020cites this paper
• Reachable Set of Unmanned Aerial Vehicles in Cyclic Pursuit with Heterogeneous Controller and Damping Gains

  2019cites this paper
• A Topological Obstruction to Almost Global Synchronization on Riemannian Manifolds

  2018cites this paper
• A Geometric Obstruction to Almost Global Synchronization on Riemannian Manifolds

  2018cites this paper
• Multi-Agent Cooperative Control Consensus: A Comparative Review

  2018cites this paper
• Minimal time splines on the sphere

  2018cites this paper
• Approximation Algorithms for Geometric Clustering and Touring Problems

  2018influential citation
• Exploring the Rendezvous of Agents in Cyclic Pursuit with Possible Negative Controller Gain and Homogeneous Input Delay

  2018cites this paper
• The Designated Convergence Rate Problem of Consensus or Flocking of Double-Integrator Agents With General Non-Equal Velocity and Position Couplings

  2017cites this paper
• Minimal time splines on the sphere

  2017cites this paper
• The Designated Convergence Rate Problems of Consensus or Flocking of Double-Integrator Agents With General Nonequal Velocity and Position Couplings: Further Results and Patterns of Convergence Rate Contours

  2017cites this paper
• About simple variational splines from the Hamiltonian viewpoint

  2017cites this paper
• Distributed sampled-data control of nonholonomic multi-robot systems with proximity networks

  2016cites this paper
• Smooth Time-Varying Formation Control of Multiple Nonholonomic Agents

  2016cites this paper
• Finite time rendezvous control of multi-agent networks preserving topology connectivity with constrained energy

  2016cites this paper
• Solitons of Discrete Curve Shortening

  2016cites this paper
• Consensus with a reference state for fractional-order multi-agent systems

  2016cites this paper
• Finite-time containment control of multi-agent network with nonlinear dynamics

  2015cites this paper
• Distributed formation control of fractional-order multi-agent systems with absolute damping and communication delay

  2015cites this paper
• Distributed coordination of fractionnal-order multi-agent systems

  2015cites this paper
• Self-Organization of Shortest Spanning Tree and Optimal Sink Node Position for Various Shapes of Large-Scale Wireless Sensor Networks

  2015cites this paper
• Solitons of Discrete Curve Shortening

  2015cites this paper
• Analysis of Flocking of Cooperative Multiple Inertial Agents via A Geometric Decomposition Technique

  2014cites this paper
• Unified Cooperative Control of Multiple Agents on a Sphere for Different Spherical Patterns

  2014cites this paper
• Rendezvous of unicycles: A bearings-only and perimeter shortening approach

  2013cites this paper
• Consensus Recovery from Intentional Attacks in Directed Nonlinear Multi-agent Systems

  2013cites this paper
• Finite time containment control of nonlinear multi-agent networks

  2013cites this paper
• Convergent systems vs. incremental stability

  2013cites this paper
• Shortest path set induced vertex ordering and its application to distributed distance optimal formation path planning and control on graphs

  2013cites this paper
• UNIVERSIDADE FEDERAL DE SANTA CATARINA DEPARTAMENTO DE AUTOMAC˜ AO E SISTEMAS

  2013cites this paper
• Connectivity preservation for multi-agent rendezvous with link failure

  2012cites this paper
• Distance optimal formation control on graphs with a tight convergence time guarantee

  2012cites this paper
• Rendezvous Without Coordinates

  2012cites this paper
• Shortest Path Set Induced Vertex Ordering and its Application to Distributed Distance Optimal Multi-agent Formation Path Planning

  2012cites this paper
• Distributed optimal cooperative tracking control of multiple autonomous robots

  2012cites this paper
• Stochastic consensus in directed networks of agents with non-linear dynamics and repairable actuator failures

  2012cites this paper
• Experimental verification of a multi-robot distributed control algorithm with containment and group dispersion behaviors

  2012cites this paper
• Asynchronous Rendezvous Analysis via Set-valued Consensus Theory

  2012cites this paper
• Fully Distributed Self-Organization of Shortest Spanning Tree and Optimal Sink Node Position for Large-Scale Wireless Sensor Network

  2012cites this paper
• Rendezvous of networked autonomous surface vehicles with distributed neural protocols

  2012cites this paper
• Using spatial embeddedness and physical embodiment for computation in multi-robot systems

  2011cites this paper
• Bayesian rendezvous for distributed robotic systems

  2011cites this paper
• Control of single and multiple agent systems with input and communication delays.

  2011cites this paper
• Combination framework of rendezvous algorithm for multi‐agent systems with limited sensing ranges

  2011cites this paper
• Motion coordination and control in systems of nonholonomic autonomous vehicles

  2010cites this paper
• Constructive Decomposition of the Controllability Lie Algebra for Quantum Systems

  2010cites this paper
• Rendezvous of multiple mobile agents with preserved network connectivity

  2010cites this paper
• Distributed formation control for fractional-order systems: Dynamic interaction and absolute/relative damping

  2010cites this paper
• Coordination of multiple agents via combination algorithms

  2010cites this paper
• Rendezvous of multiple Dubins car agents with minimal sensing and control requirements

  2010cites this paper
• Consensus algorithm for robotic agents over packet dropping links

  2010cites this paper
• Stability of General Coupled Inertial Agents

  2010cites this paper
• A contribution to collision-free trajectory planning for handling systems in varying environments

  2010cites this paper
• Distributed Control of Robotic Networks: A Mathematical Approach to Motion Coordination Algorithms

  2009cites this paper
• Distributed Control of Robotic Networks

  2009cites this paper
• A Distributed Heuristic for Energy-Efficient Multirobot Multiplace Rendezvous

  2009cites this paper
• Solitonic fermions in the confining phase of SU(2) Yang-Mills theory

  2009cites this paper
• Geometry of cyclic pursuit

  2009cites this paper
• Distributed coordination of fractional-order systems with extensions to directed dynamic networks and absolute/relative damping

  2009cites this paper
• Rendezvous without coordinates

  2008cites this paper
• A Embedding equations in Riemannian geometry 66 B Deforming curves and integrability 68 C Resistive diffusion of magnetic fields 69 1

  2008cites this paper
• Autonomous Sustain and Resupply, Phase 1 Report

  2008cites this paper
• Experimental Validation of Consensus Algorithms for Multivehicle Cooperative Control

  2008cites this paper
• Dirichlet sigma models and mean curvature flow

  2007cites this paper
• Tanker based robot recharging

  2007cites this paper
• Energy Efficient Robot Rendezvous

  2007cites this paper