Jur van den Berg, Stephen J. Guy, Ming Lin, Dinesh Manocha

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Optimal Reciprocal Collision Avoidance (ORCA)

• Jur van den Berg, Stephen J. Guy, Ming Lin,
  Dinesh Manocha
• University of North Carolina at Chapel Hill

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Motivation

• Robots are becoming cheaper, more mobile, and
  better sensing
• Several mobile robots sharing space is becoming
  increasingly practical
• Our Goal
• Allow robots to share physical space
• Encourage smooth, goal directed navigation
• Guaranteed collision avoidance

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Overview

• Our Goals
• Background Previous Work
• Algorithm Overview
• Implementation Details
• Performance Results
• Conclusions Future Work

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Background Previous Work
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Collision Avoidance Static Dynamic Obstacles

• Collision Avoidance is a well studied problem
• Velocity Obstacles Fiorini Shillier, 98
• Inevitable Collision StatesFraichard Asama,
  98
• Dynamic Window Fox, Burgard, Thrun, 97
• Focused on one robot avoiding static and moving
  obstacles
• Inappropriate for responsive obstacles

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Collision Avoidance Responsive Obstacles

• Reciprocal Velocity Obstacles(RVO) Berg et al,
  08
• Extends Velocity Obstacle concept
• Oscillation free, guaranteed avoidance (2 agents)
• Limitations
• Guarantees limited to 2 agents

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ORCA

• A new algorithm for collision avoidance
• A linear programming based formulation
• Extends Velocity Obstacle concepts
• Velocity Based
• Provides sufficient conditions for avoiding
  collisions
• Decisions are made independently, w/o
  communication
• Guaranteed avoidance

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ORCA Algorithmic Details
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Problem overview

• Inputs
• Independent Robots
• Current Velocity of all
• Own Desired Velocity (Vpref)
• Outputs
• New collision-free velocity (Vout)
• Description Each Robot
• Determines permitted (collision free) velocities
• Chooses velocity closest to Vpref which is
  permitted

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Velocity Space Forbidden Regions

• Forbidden Regions
• Potentially colliding velocities
• An obstacle in velocity space
• VO Velocity Obstacle Fiorini Shiller 98
• Assumes other agent is unresponsive
• Appropriate for static unresponsive obstacles
• RVO Reciprocal VO van den Berg et al., 08
• Assumes other agent is mutually cooperating

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Velocity Obstacle

• Time horizon t
• Relative velocities AB
• Relative velocities BA symmetric in O

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Permitted Velocities

• If velocity of B is vB
• A should choose velocity outside VOAB ? vB.
• If velocity of B is in set VB
• permitted velocities PVAB(VB) for A are
  outside VOAB ? VB

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Reciprocally Permitted Velocities

• Set VA of velocities for A and set VB of
  velocities for B are reciprocally permitted if
• VA ? PVAB(VB) and VB ? PVBA(VA)
• Set VA of velocities for A and set VB of
  velocities for B are reciprocally maximal if
• VA PVAB(VB) and VB PVBA(VA)

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ORCA

• u Vector which escapes VOtAB
• Each robot is responsible for ½u
• ORCAtAB
• The set of velocities allowed to A
• Sufficient condition for collision avoidance if B
  chooses from ORCAtAB

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Optimality

• Infinitely many half plane pairs reciprocally
  permitted
• ORCA chooses plans to
• Maximize velocities near current velocities
• Fairly distribute permitted velocities between A
  and B
• For any radius r

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Multi-Robot Navigation

• Choose a velocity inside ALL pair-wise ORCAs
• Efficient O(n) implementation w/ Linear
  Programming

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Performance Results
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Small Scale Simulation (1)

• Two robots are asked to swap positions
• Generated Path is
• Smooth
• Collision free

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Small Scale Simulation (2)

• 5 Robots moving to antipodal points
• Smooth, Collision paths result

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Performance - Scaling

• Our performance sales nearly linearly w.r.t.
• Number of Cores
• Number of Agents

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Large Scale Simulations

• 1,000 Virtual robots move across a circle
• Collision Avoidance is a major component of Crowd
  Sims.
• ORCA can be applied to virtual agents to produce
  believable motion

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Conclusion Future Work

• ORCA
• Efficient, decentralized, guaranteed collision
  avoidance
• 3-5µs per robot
• No explicit communication required
• Fast running time smooth, convincing behavior
• Future Work
• Incorporating kinematic dynamic constraints
• Implement in 3D environments

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Acknowledgments

• Funding Support
• ARO (Contract W911NF-04-1-0088)
• DARPA/RDECOM (Contracts N61339-04-C-0043
  WR91CRB-08-C-0137)
• Intel
• Intel fellowship
• Microsoft
• National Science Foundation (Award 0636208)

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Questions?

• ?

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Backup Slides
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Choosing Vopt

• Vopt impacts the robot behavior
• Vopt Vpref
• Vpref may not be know
• No solution guaranteed to exist
• Vopt 0
• Deadlock likely in dense scenarios
• Vopt Vcur
• Nice balance
• Vcur Vperf in low density
• Vcur 0 in high density

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Densely Packed Conditions

• If Vopt ! 0, solution may not exist
• Find the least bad velocity
• Efficient implementation possible with 3D linear
  programming

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Static Obstacles

• ORCAs can also be created for obstacles in the
  environment
• ORCA is half-plane tangent to VO t AO