Distributed Reactive Collision Avoidance - University of Washington

26
Figure 3.1):
where R is the 2 × 2 rotation matrix. Now v ′ i = v j − aĉ where
a = ĉ T v j ±
ĉ = R(±α) ˜r
‖˜r‖ , (4.1)
√
(ĉ T v j ) 2 − v T j v j + v T i v i, (4.2)
and only solutions with real and positive values of a are valid. The list of v is ′ are ordered
by increasing ∆v i and checked consecutively for conflicts with the other vehicles. Because
of the ordering, as soon as a point is found which is conflict free for all j, it is the optimal
solution and the algorithm terminates.
There are a finite number of points to check as possible optima, which bounds the
maximum possible time the deconfliction maneuver will take to compute. For a vehicle
deconflicting with n other members of D, there are a maximum of 4n points to check. Each
of these much be checked against a maximum of n − 1 other collision cones. Therefore
the maximum computation time is upper-bounded by cn 2 , where c is related to the time
each type of computation requires. In general, the computation times are less than this
bound because as soon as a feasible point is found, the algorithm terminates, so most points
are never computed or checked. This bound is better than a general “polynomial time”
guarantee (as with convex optimization, for instance), since in this case the polynomial is
known to be quadratic.
This analysis would guarantee a conflict-free solution if the vehicle could attain its
desired velocity vector instantaneously. However, the limited control authority available
makes this impossible. Instead, it takes a finite amount of time for the vehicle to attain
its desired velocity, and during that time it and the other vehicles move, which causes the
collision cones to move. In order to ensure that the system is still conflict-free after this
motion, the initial collision cones must be enlarged to the point of enclosing all possible
movements.
To bound the collision cone, one must simply bound ‖∆˜r‖ ≤ δ, or how much the vehicles
can change position before the maneuver is complete. Then the width of the collision