Distributed Reactive Collision Avoidance - University of Washington

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Figure 4.5: Example of a borderline case for six vehicles where no conflict-free variablespeed
maneuver exists. The circle is the edge of the allowable disk (radius v i,max ) and the
blue regions are the collision cones, shown with all equal velocities, of magnitude greater
than v i,max .
Note that for large n, the small angle approximation can be used to simplify (4.15) to
⎧
⎪⎨
‖˜r‖ ≥
⎪⎩
(n−1)(d
„ sep+δ)
vi,max
«,
arcsin v j,max
v i,max < v j,max
2(n−1)
π
(d sep + δ), v i,max ≥ v j,max ,
(4.16)
however the result is not actually conservative, so (4.15) should still be used for safety
purposes. It is interesting to note though, that as n gets large, the bound approaches a linear
relationship with n − 1. This makes intuitive sense in that as more vehicles are in a space,
they must be spaced out more to ensure a safe trajectory exists between them.
4.1.3 Heuristic Performance
The guarantees provided by the previous maneuvers are ideal if one can ensure the vehicles
always have enough warning of each other’s presence. However, some systems may not
always be that capable. We know from the previous analysis that these requirements are
conservative, so it may still be possible to deconflict the system, though not in a guaranteed