Distributed Reactive Collision Avoidance - University of Washington

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Once again the time required to attain this new velocity vector must be accounted for
in order to ensure the system actually comes to a conflict-free state and that there are no
collisions. The following lemma parallels Lemma 2, but obtains a smaller bound by use of
the now 2D acceleration vector.
Lemma 3. Let there be two vehicles (i and j), each modeled by a planar version of (2.1).
Vehicle j is subject to the maximum speed constraint ‖v j ‖ ≤ v j,max . Vehicle i is subject to
‖u i ‖ ≤ u i,max and ‖v i ‖ ≤ v i,max and is accelerating as quickly as possible from its initial
velocity, v i , to its desired velocity, v ′ i. The relative motion between the vehicles in the time
it takes vehicle i to attain its desired velocity is bounded by ‖∆˜r‖ ≤ δ, where
δ = v i,max
u i,max
(v i,max + 2v j,max ) . (4.12)
Proof. Let the angle between v i and v ′ i be 2γ and let t be the time required for the maneuver.
Then ‖∆v i ‖ = u i,max t, ‖∆r i ‖ = t ‖v i + v ′ i‖ /2, and ‖∆r j ‖ ≤ v j,max t. Also,
‖v i + v ′ i‖ ≤ 2v i,max cos γ and ‖∆v i ‖ ≤ 2v i,max |sin γ|. Therefore,
‖∆˜r‖ ≤ ‖∆r i ‖ + ‖∆r j ‖ ≤ 2v2 i,max |cos γ sin γ|
u i,max
+ 2v i,maxv j,max |sin γ|
u i,max
≤ v i,max
u i,max
(v i,max + 2v j,max ) . (4.13)
This bound can in turn be used in (4.3) to size the enlarged collision cone. Note that for
a homogeneous group of vehicles this bound can be written in terms of the deconfliction
difficulty factor as δ = 3ηd sep . Also note that if one decides to use the constant-speed
maneuver on a vehicle that fits the assumptions of Lemma 3, then one can still use the δ
bound of (4.12), since the proof is only based on the vehicle’s capabilities.
Theorem 1 still applies to the variable-speed maneuver, with the slight adjustment of
(4.6) to t = 2v i,max /u i,max (see above). However, Theorem 2 requires some adjustments to
the bound χ i (D):