1. Jur van den Berg, Stephen J. Guy, Ming Lin, Dinesh Manocha University of North Carolina at Chapel Hill Optimal Reciprocal Collision Avoidance (ORCA)

2. 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

3. http://gamma.cs.unc.edu/CA 3 Overview  Our Goals  Background & Previous Work  Algorithm Overview  Implementation Details Performance Results  Conclusions & Future Work

4. http://gamma.cs.unc.edu/CA Background & Previous Work

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

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

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

8. http://gamma.cs.unc.edu/CA ORCA Algorithmic Details

9. http://gamma.cs.unc.edu/CA 9  Inputs: Independent Robots Current Velocity of all Own Desired Velocity ( V pref )  Outputs: New collision-free velocity ( V out )  Description – Each Robot: Determines permitted (collision free) velocities Chooses velocity closest to V pref which is permitted Problem overview

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

11. http://gamma.cs.unc.edu/CA 11 Velocity Obstacle  Time horizon τ  Relative velocities A–B  Relative velocities B–A symmetric in O

12. http://gamma.cs.unc.edu/CA 12 Permitted Velocities  If velocity of B is v B A should choose velocity outside VO A|B  {v B }.  If velocity of B is in set V B permitted velocities PV A|B (V B ) for A are outside VO A|B  V B

13. http://gamma.cs.unc.edu/CA 13 Reciprocally Permitted Velocities  Set V A of velocities for A and set V B of velocities for B are reciprocally permitted if V A  PV A|B (V B ) and V B  PV B|A (V A )  Set V A of velocities for A and set V B of velocities for B are reciprocally maximal if V A  PV A|B (V B ) and V B  PV B|A (V A )

14. http://gamma.cs.unc.edu/CA 14 ORCA  u – Vector which escapes VO τ A|B Each robot is responsible for ½u  ORCA τ A|B The set of velocities allowed to A Sufficient condition for collision avoidance if B chooses from ORCA τ A|B

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

16. http://gamma.cs.unc.edu/CA 16 Multi-Robot Navigation  Choose a velocity inside ALL pair-wise ORCAs  Efficient O(n) implementation w/ Linear Programming

17. http://gamma.cs.unc.edu/CA Performance Results

18. http://gamma.cs.unc.edu/CA 18 Small Scale Simulation (1)  Two robots are asked to swap positions  Generated Path is: Smooth Collision free

19. http://gamma.cs.unc.edu/CA 19 Small Scale Simulation (2)  5 Robots moving to antipodal points  Smooth, Collision paths result

20. http://gamma.cs.unc.edu/CA 20 Performance - Scaling  Our performance sales nearly linearly w.r.t. Number of Cores Number of Agents

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

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

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

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

25. http://gamma.cs.unc.edu/CA Backup Slides

26. http://gamma.cs.unc.edu/CA 26 Choosing V opt  V opt impacts the robot behavior  V opt = V pref Vpref may not be know No solution guaranteed to exist  V opt = 0 Deadlock likely in dense scenarios  V opt = V cur Nice balance V cur ~= V perf in low density V cur ~= 0 in high density

27. http://gamma.cs.unc.edu/CA 27 Densely Packed Conditions  If V opt != 0, solution may not exist Find the “least bad” velocity Efficient implementation possible with 3D linear programming

28. http://gamma.cs.unc.edu/CA 28 Static Obstacles  ORCAs can also be created for obstacles in the environment  ORCA is half-plane tangent to VO τ A|O