Distributed Reactive Collision Avoidance - University of Washington

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deconfliction group, D, it performs its deconfliction maneuver, broadcasting a conflict-free
velocity to the group. Since it is this broadcasted velocity that is used by the deconfliction
maintenance algorithm until the new vehicle really is conflict-free, the deconfliction group
never sees any new conflicts.
Theorem 4. The deconfliction maintenance controller described above, when implemented
on n vehicles with dynamics (2.1) and inputs constrained by C i , will keep the system collision
free for all time if the system starts conflict-free.
Proof. To measure the distance to a collision, define m as a signed version of ‖e‖ (in terms
of ṽ from the geometry in Fig. 4.7):
⎧
⎪⎨ ‖ṽ‖ , ĉ T ṽ ≤ 0
m =
⎪⎩ ‖ṽ‖ sin(|β| − α), ĉ T ṽ > 0.
Note that m is negative during conflict and positive during no conflict.
(4.25)
To ensure that a conflicted state is never reached (i.e. m is always greater than zero), it
is sufficient to show that for every pair of vehicles there is a neighborhood on the positive
side of m = 0 where ṁ ≥ 0. This condition implies that as the boundary of a collision cone
approaches, it will either stop approaching or recede before a conflict is formed. The fact
that this is a one-sided neighborhood is important because ṁ does not exist at m = 0 for
the same reason that d ‖ṽ‖ does not exist at ṽ = 0. However, if this condition is satisfied,
dt
then m > 0 for all time, ensuring that m ≠ 0.
For ĉ T ṽ ≤ 0 (using the ij notation again briefly for clarity),
ṁ = eT ˙ṽ
m = êT ˙ṽ, (4.26)
where ê = e
‖e‖ . Expanding u i into its components yields u i = [u ti t i + u ni n i + u bi b i ]. Next
substitute for ṽ ij in (4.26) to get
ê T ij ˙ṽ ij = u ti ê T ijˆt i + u ni ê T ijˆn i + u bi ê T ijˆb i − u tj ê T ijˆt j − u nj ê T ijˆn j − u bj ê T ijˆb j . (4.27)