Optimization-based methods are commonly applied in autonomous driving trajectory planners, which transform the continuous-time trajectory planning problem into a finite nonlinear program with constraints imposed at finite collocation points. However, potential violations between adjacent collocation points can occur. To address this issue thoroughly, we propose a safety-guaranteed collision-avoidance model to mitigate collision risks within optimization-based trajectory planners. This model introduces an embodied footprint, an enlarged representation of the vehicle's nominal footprint. If the embodied footprints do not collide with obstacles at finite collocation points, then the ego vehicle's nominal footprint is guaranteed to be collision-free at any of the infinite moments between adjacent collocation points. According to our theoretical analysis, we define the geometric size of an embodied footprint as a simple function of vehicle velocity and curvature. Particularly, we propose a trajectory optimizer with the embodied footprints that can theoretically set an appropriate number of collocation points prior to the optimization process. We conduct this research to enhance the foundation of optimization-based planners in robotics. Comparative simulations and field tests validate the completeness, solution speed, and solution quality of our proposal.

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