# Velocity obstacle

The velocity obstacle VOAB for a robot A, with position xA, induced by another robot B, with position xB and velocity vB.

In robotics and motion planning, a velocity obstacle, commonly abbreviated VO, is the set of all velocities of a robot that will result in a collision with another robot at some moment in time, assuming that the other robot maintains its current velocity.[1] If the robot chooses a velocity inside the velocity obstacle then the two robots will eventually collide, if it chooses a velocity outside the velocity obstacle, such a collision is guaranteed not to occur.[1]

This algorithm for robot collision avoidance has been repeatedly rediscovered and published under different names: in 1989 as a maneuvering-board approach,[2] in 1993 it was first introduced as the "velocity obstacle",[3] in 1998 as collision cones,[4] and in 2009 as forbidden velocity maps.[5] The same algorithm has been used in maritime port navigation since at least 1903.[6]

The velocity obstacle for a robot ${\displaystyle A}$ induced by a robot ${\displaystyle B}$ may be formally written as

${\displaystyle VO_{A|B}=\{{\mathbf {v} }\,|\,\exists t>0:({\mathbf {v} }-{\mathbf {v} }_{B})t\in D({\mathbf {x} }_{B}-{\mathbf {x} }_{A},r_{A}+r_{B})\}}$

Variations include common velocity obstacles (CVO),[7] finite-time-interval velocity obstacles (FVO),[8] generalized velocity obstacles (GVO),[9] hybrid reciprocal velocity obstacles (HRVO),[10] nonlinear velocity obstacles (NLVO),[11] reciprocal velocity obstacles (RVO),[12] and recursive probabilistic velocity obstacles (PVO).[13]

## References

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