  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.
