2. Virtual Shells for Avoiding Collisions A. S. Morse Yale University Gif – sur - Yvette May 24, 2012 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A A A A A A AA A A AAAA A Supelec EECI Graduate School in Control

3. OVERALL PROBLEM Develop local control concepts to enable a large grouping of mobile autonomous agents to perform biologically inspired group maneuvers such as schooling, swarming, flocking in a safe and purposeful manner.

4. THE MAIN ISSUE COLLISION AVOIDANCE Virtual Shells

5. The concept of a virtual shell stems from two ideas: Neighbors can cooperate The ``block’’ or ``moving slot’’ protocol

6. The concept of a virtual shell stems from two ideas: Neighbors can cooperate The ``block’’ or ``moving slot’’ protocol

7. Schooling fish sometimes rub against each other Flocking birds sometimes gently hit each other Individuals sometimes maneuver through a crowd by pushing Crowds reform by gently nudging to pass through a portal Children can successfully maneuver bumper cars at amusement parks Football players sometimes guide teammates motion by pushing Cocktail Party Problem A key component of large group coordination seems to be the ability of agents to cause nearest neighbors to cooperatively react to their wishes in order to effectively maneuver. Behaviors which affect group coordination:

8. Neighbors can cooperate The ``block’’ or ``moving slot’’ protocol

9. Neighbors can cooperate The ``block’’ or ``moving slot’’ protocol

10. RAILROAD block 1block 4block 3block 2 At most one train in one block at one time BLOCK CONTROL Generalization

11. Personal Rapid Transit A dedicated guideway network on which small driverless vehicles move between stations under computer control. Slot Concept: Contiguous streams of computer generated virtual blocks or slots move along each segment of guideway with variable temporal and physical spacings defined in such a way so that the slot flow is the same throughout the network. Slots merge at merges and diverge at diverges. At most one vehicle can occupy one slot at one time.

12. Personal Rapid Transit A dedicated guideway network on which small driverless vehicles move between stations under computer control. Induces a natural hierarchy: Slot Concept: Contiguous streams of computer generated virtual blocks or slots move along each segment of guideway with variable temporal and physical spacings defined in such a way so that the slot flow is the same throughout the network. Slots merge at merges and diverge at diverges. At most one vehicle can occupy one slot at one time. Vehicle slot-tracking controllers Slot assignment based on real-time network flows Slot slipping or vehicle maneuvering control

13. The Virtual Shell Concept By a virtual shell is meant a closed non-deformable surface of appropriate shape For planning purposes, shells are regarded as rigid dynamical bodies which move through 2d or 3d space and are subject to force fields. Force fields are typically determined by potential functions designed to accomplish particular tasks. A swarm or school or flock of virtual shells thus admits the model of a hybrid dynamical system. Shells can gently hit each other, but such collisions are always “lossless”.

14. Collision avoidance is achieved by requiring each vehicle to remain within its own shell for all time This is accomplished by “conventional” tracking control applied to each vehicle. For this to be possible, each vehicle must know the trajectory of the shell it is tracking. Since shell trajectories are determined by force fields and collisions with nearest neighbors, nearest neighbor shell position and orientation must be available to each vehicle. Communication between nearest neighbors is thus required.

15. Issues Shell shape Impact rules Impact detection Tracking controllers Virtual force fields

16. Issues Shell shape Impact rules Impact detection Tracking controllers Virtual force fields 2D: circles or ellipses 3D: spheres or ellipsoids

17. Issues Shell shape Impact rules Impact detection Tracking controllers Virtual force fields

18. 1. Elastic collision rule: Impacting shells interchange normal components of velocity vectors at impact point. 2. Reflection rule: Impacting shells each change the sign of its normal component of its velocity vector’s at impact point.

19. Issues Shell shape Impact rules Impact detection Tracking controllers Virtual force fields Easy for circles and spheres: Impact occurs just when distance between centers equals sum of radii. What about ellipsoids ?

20. Detecting Impacting Ellipsoids