San Diego

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San Diego 7/11/01
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VIRTUAL SHELLS FOR AVOIDING COLLISIONS Y
ale University A. S. Morse
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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.
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THE MAIN ISSUE
COLLISION AVOIDANCE
Virtual Shells
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The concept of a virtual shell stems from two
ideas

• Neighbors can cooperate
• The block or moving slot protocol

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The concept of a virtual shell stems from two
ideas

• Neighbors can cooperate
• The block or moving slot protocol

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Behaviors which affect group coordination

• 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.
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• Neighbors can cooperate
• The block or moving slot protocol

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• Neighbors can cooperate
• The block or moving slot protocol

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RAILROAD
BLOCK CONTROL
At most one train in one block at one time
Generalization
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Personal Rapid Transit
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.
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Personal Rapid Transit
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

Induces a natural hierarchy
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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.
Shells can gently hit each other, but such
collisions are always lossless.
A swarm or school or flock of virtual shells
thus admits the model of a hybrid dynamical
system.
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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.
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Issues

• Shell shape
• Impact rules
• Impact detection
• Tracking controllers
• Virtual force fields

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Issues

• Shell shape
• Impact rules
• Impact detection
• Tracking controllers
• Virtual force fields

2D circles or ellipses 3D spheres or ellipsoids
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Issues

• Shell shape
• Impact rules
• Impact detection
• Tracking controllers
• Virtual force fields

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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 vectors at impact point.
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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 ?
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Detecting Intersecting Ellipsoids
Detecting Impacting Ellipsoids
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Detecting Impacting Ellipsoids
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Issues

• Shell shape
• Impact rules
• Impact detection
• Tracking controllers
• Virtual force fields

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Issues

• Shell shape
• Impact rules
• Impact detection
• Tracking controllers
• Virtual force fields

.for a three wheeled, nonholonomic cart
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Nonholonomic Cart
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Do not exist continuous time-invariant control
laws u f(x, y, f) v g(x, y, f) which
stabilize the origin x y f 0 !
nonholonomic cart model
Brockett
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nonholonomic cart model
Tracking Problem Devise a feedback controller
which causes the nonholonomic cart to track a
given reference trajectory xr, yr, fr.
Sussmann and Liu Brockett and Morgansen DeLuca
et. al.
Model Following
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nonholonomic cart model
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nonholonomic cart model
nonholonomic cart reference
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closed-loop
error system in new coordinates
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closed-loop
error system in new coordinates
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closed-loop
error system in new coordinates
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Whats it take to keep a vehicle in its shell?
Suppose a vehicle is initially centered on its
shell and that at most n reflection rule
determined collisions can occur before the
preceding tracking control can re-center the
vehicle within its shell. Then the maximum
distance from center cannot exceed np
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r 35
distance of vehicle from center of its shell
number of impacts
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Each vehicle must continuously compute its own
shells position.
Since each vehicles shell position depends on
impacts with the shells of nearest neighbors,
nearest neighbor shell position and
orientation must be available to each vehicle.
Communication between nearest neighbors is thus
required.
One possible way to avoid the need for this
communication, it for each vehicle to assume that
its neighbors are centered within their shells at
impact this is the centering protocol.
Worst case vehicle spacing shell radius
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