Distributed Reactive Collision Avoidance - University of Washington

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ṽ ij
d sep,ij
d sep,ij
β
Vehicle j
α
˜r ij
α
Vehicle i
Figure 3.1: A 2D section of the collision cone along the ˜r ij -ṽ ij plane. The area between
the two dotted lines is the collision cone; a conflict occurs when the relative velocity vector,
ṽ ij , lies within this area.
Lemma 1. Let β = ∠ṽ − ∠˜r 0 , α = arcsin
( )
dsep
‖˜r 0
and ˜r
‖ 0 be the relative position vector
at the time conflict is being checked. A necessary and sufficient condition for no conflict to
occur is
|β| ≥ α. (3.3)
The angle α represents the half-width of the collision cone ([7, 8, 10]), which is depicted
in Fig. 3.1.
Proof. First, define ˜r min as the position vector corresponding to the closest approach of
one vehicle to another in (3.2). By definition, at ˜r min the time derivative of ‖˜r‖ 2 = 0.
Therefore:
Next, note that for constant velocity, ṽ:
d (˜rT˜r ) = 0
dt
ṽ T˜r min = 0. (3.4)
˜r min = ˜r 0 − ṽt, (3.5)