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

Optimal Reciprocal Collision Optimal Reciprocal Collision Avoidance (ORCA)Avoidance (ORCA)

http://gamma.cs.unc.edu/CA 2

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

http://gamma.cs.unc.edu/CA 3

OverviewOverview

Our Goals Background & Previous Work Algorithm Overview Implementation Details

•Performance Results Conclusions & Future Work

http://gamma.cs.unc.edu/CA

Background & Previous WorkBackground & Previous Work

http://gamma.cs.unc.edu/CA 5

Collision Avoidance Static & Dynamic Obstacles

Collision Avoidance Static & Dynamic Obstacles

Collision Avoidance is a well studied problem•Velocity Obstacles [Fiorini & Shillier, 98]

•Inevitable Collision States[Fraichard & Asama, 98]

•Dynamic Window [Fox, Burgard, & Thrun, 97]

Focused on one robot avoiding static and moving obstacles

Inappropriate for “responsive” obstacles

http://gamma.cs.unc.edu/CA 6

Collision Avoidance Responsive Obstacles

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

http://gamma.cs.unc.edu/CA 7

ORCAORCA

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

http://gamma.cs.unc.edu/CA

ORCA Algorithmic DetailsORCA Algorithmic Details

http://gamma.cs.unc.edu/CA 9

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

Problem overviewProblem overview

http://gamma.cs.unc.edu/CA 10

Velocity Space & Forbidden Regions

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

http://gamma.cs.unc.edu/CA 11

Velocity ObstacleVelocity Obstacle

Time horizon τ Relative velocities A–B Relative velocities B–A symmetric in O

http://gamma.cs.unc.edu/CA 12

Permitted VelocitiesPermitted Velocities

If velocity of B is vB

•A should choose velocity outside VOA|B {v

B}.

If velocity of B is in set VB •permitted velocities

PVA|B(VB) for A are outside VOA|B V

B

http://gamma.cs.unc.edu/CA 13

Reciprocally Permitted Velocities

Reciprocally Permitted Velocities

Set VA of velocities for A and set VB of velocities for B are reciprocally permitted if•VA PVA|B(VB) and VB PVB|A(VA)

Set VA of velocities for A and

set VB of velocities for B are

reciprocally maximal if

•VA PVA|B(VB) and VB PVB|A(VA)

http://gamma.cs.unc.edu/CA 14

ORCAORCA

u – Vector which escapes VOτ

A|B •Each robot is

responsible for ½u ORCAτ

A|B •The set of velocities

allowed to A•Sufficient condition

for collision avoidance if B chooses from ORCAτ

A|B

http://gamma.cs.unc.edu/CA 15

OptimalityOptimality

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:

http://gamma.cs.unc.edu/CA 16

Multi-Robot Navigation

Multi-Robot Navigation

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

Programming

http://gamma.cs.unc.edu/CA

Performance ResultsPerformance Results

http://gamma.cs.unc.edu/CA 18

Small Scale Simulation (1)

Small Scale Simulation (1)

Two robots are asked to swap positions

Generated Path is:

•Smooth

•Collision free

http://gamma.cs.unc.edu/CA 19

Small Scale Simulation (2)

Small Scale Simulation (2)

5 Robots moving to antipodal points

Smooth, Collision paths result

http://gamma.cs.unc.edu/CA 20

Performance - ScalingPerformance - Scaling

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

• Number of Agents

http://gamma.cs.unc.edu/CA 21

Large Scale SimulationsLarge 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

http://gamma.cs.unc.edu/CA 22

Conclusion & Future Work

Conclusion & Future Work

ORCA:•Efficient, decentralized, guaranteed

collision avoidance3-5µs per robot

•No explicit communication required•Fast running time & smooth, convincing

behavior Future Work

•Incorporating kinematic & dynamic constraints

•Implement in 3D environments

http://gamma.cs.unc.edu/CA 23

AcknowledgmentsAcknowledgments

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)

http://gamma.cs.unc.edu/CA 24

Questions?Questions?

?

http://gamma.cs.unc.edu/CA

Backup SlidesBackup Slides

http://gamma.cs.unc.edu/CA 26

Choosing VoptChoosing 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

http://gamma.cs.unc.edu/CA 27

Densely Packed Conditions

Densely Packed Conditions

If Vopt != 0, solution may not exist•Find the “least bad” velocity•Efficient implementation possible with

3D linear programming

http://gamma.cs.unc.edu/CA 28

Static ObstaclesStatic Obstacles

ORCAs can also be created for obstacles in the environment

ORCA is half-plane tangent to VO τ A|O