Numerical optimization-based methods are among the prevalent trajectory
planners for autonomous driving. In a numerical optimization-based planner, the
nominal continuous-time trajectory planning problem is discretized into a
nonlinear program (NLP) problem with finite constraints imposed on finite
collocation points. However, constraint violations between adjacent collocation
points may still occur. This study proposes a safety-guaranteed
collision-avoidance modeling method to eliminate the collision risks between
adjacent collocation points in using numerical optimization-based trajectory
planners. A new concept called embodied box is proposed, which is formed by
enlarging the rectangular footprint of the ego vehicle. If one can ensure that
the embodied boxes at finite collocation points are collide-free, then the ego
vehicle's footprint is collide-free at any a moment between adjacent
collocation points. We find that the geometric size of an embodied box is a
simple function of vehicle velocity and curvature. The proposed theory lays a
foundation for numerical optimization-based trajectory planners in autonomous
driving.Comment: 12 pages, 13 figure