Distributed Reactive Collision Avoidance - University of Washington

28
Theorem 1. Let there be a set of vehicles, D, which are not in conflict with each other.
When another vehicle, i, is in conflict with some or all members of D and performs the
constant-speed maneuver, the system will be conflict-free in time t, where
t ≤ π ‖v i‖
u i,max
, (4.6)
and no vehicles will collide during the maneuver. The vehicles are all modeled by a planar
version of (2.1), have speed constraints ‖v j ‖ ≤ v j,max and vehicle i has the input constraint
‖u i ‖ ≤ u i,max . It is assumed that a feasible solution to the optimization problem, v ′ i, exists
and that the vehicles in D maintain a conflict-free state with v ′ i, using a cone with width
defined by (4.3) and (4.4).
Proof. If a feasible point exists for the optimization problem, then the optimal solution is
guaranteed to be found and this point will satisfy
( )
|∠ṽ ′ dsep + δ
− ∠˜r| ≥ arcsin
. (4.7)
‖˜r‖
The maximum amount of time required for vehicle i to get from its initial v i to v ′ i is t =
π ‖v i ‖ /u i,max , and during this time ∆˜r ≤ δ from Lemma 2. Therefore, once the desired
velocities have been attained, one still has
|∠ṽ ′ − ∠(˜r + ∆˜r)| ≥ arcsin
( )
dsep
, (4.8)
‖˜r‖
meaning the vehicles are not in conflict. The vehicles cannot collide during this time because
as stated earlier, the pairs must be initially separated by at least ‖˜r‖ ≥ d sep + δ,
which means after the maneuver, they still must be outside of collision because ‖˜r + ∆˜r‖ ≥
d sep .
Of course, all of this is for naught if a feasible solution does not exist for the optimization.
The following theorem gives a conditional bound, χ i (D), on initial separation that is
sufficient to guarantee the existence of a solution.