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This paper generalizes the concept of velocity obstacles given by Fiorini et al. (1998) to obstacles moving along arbitrary trajectories. We introduce the nonlinear velocity obstacle, which takes into account the shape, velocity and path curvature of the moving obstacle. The nonlinear v-obstacle allows selecting a single avoidance maneuver (if one exists) that avoids any number of obstacles moving on any known trajectories. For unknown trajectories, the nonlinear v-obstacles can be used to generate local avoidance maneuvers based on the current velocity and path curvature of the moving obstacle. This elevates the planning strategy to a second order method, compared to the first order avoidance using the linear v-obstacle, and zero order avoidance using only position information. Analytic expressions for the nonlinear v-obstacle are derived for general trajectories in the plane. The nonlinear v-obstacles are demonstrated in a complex traffic example.