2. Houston 3/9/01 A TALK IN TWO PARTS PART I DEDICATED to BOYD PEARSON on the occasion of his 70 & 2/3 birthday

3. VIRTUAL SHELLS FOR AVOIDING COLLISIONS Yale University A. S. Morse Collaborators: Jin Lee {Archer} Shawn Walker Mustafa Unel Bijoy Ghosh

6. FISH SCHOOLING

7. 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.

8. Princeton Experimental Test-bed The Tank: 21’ 6.5” Dia, 8’ Deep 23000 Gallon Capacity 2 Windows 10’ Perimeter Platform

9. THE GROUPER

10. Band Formation Project Spacing Problem: Given a weighted, connected, planer graph and a set of n agents with an arbitrary initial distribution on the graph, develop a “distributed” control strategy which causes the agents to “space themselves evenly” over the graph.

12. THE OVERRIDING PROBLEM COLLISION AVOIDANCE

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

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

15. 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 A key component of large group leaderless 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:

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

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

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

19. Personal Rapid Transit A dedicated guideway network on which small driverless vehicles move between stations under computer control. At most one vehicle can occupy one slot at one time. 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.

20. 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

21. 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”.

22. 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.

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

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

25. Issues Shell shape Impact rules Impact detection Tracking controllers Virtual force fields 2. Reflection rule: Impacting shells each change the sign of their velocity vector’s normal components at impact point. 1. Elastic collision rule: Impacting shells interchange normal component of velocity vectors at impact point.

26. 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 ?

27. Detecting Intersecting EllipsoidsDetecting Impacting Ellipsoids

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

30. Issues Shell shape Impact rules Impact detection Tracking controllers Virtual force fields ….for a three-wheel nonholonomic cart

31. LL RR  x y Nonholonomic Cart

32. Brockett nonholonomic cart model nonholonomic integrator Do not exist continuous time-invariant control laws u = f(x, y, f ) v = g(x, y, f ) which asymptotically stabilize the origin x =y = f = 0 !

33. nonholonomic cart model Tracking Problem: Devise a feedback controller which causes the nonholonomic cart to “track” a given reference trajectory {x r, y r,  r }. Model Following

34. nonholonomic cart model nonholonomic cart reference Find tracking controls to cause x ! x r, y ! y r, and  !  r Implement

35. Nonholonomic Cart Tracking Nonholonomic Cart

36. Shell-Contained Nonholonomic Cart {with rate constraints} Holonomic Shell Nonholonomic Shell

37. 12 Agents Surrounding Target obstacle target

38. PART II of the TALK

39. PURDUE UNIVERSITY 1964

48. Synthesis of Linear Servomechanisms

50. Gantmacher

54. LINEAR PROCESS DECENTRALIZED SPECTRUM ASSIGNMENT u1u1 u2u2 u3u3 u4u4 umum y1y1 y2y2 y3y3 y4y4 ymym

55. leader, president …   theorist 1 theorist linear geometric control theorist teacher good guy BOYD PEARSON multi-variable control theorist …. and beer-lover