  This research addresses the increasing demand for advanced navigation systems
capable of operating within confined surroundings. A significant challenge in
this field is developing an efficient planning framework that can generalize
across various types of collision avoidance missions. Utilizing numerical
optimal control techniques, this study proposes a unified optimization-based
planning framework to meet these demands. We focus on handling two collision
avoidance problems, i.e., the object not colliding with obstacles and not
colliding with boundaries of the constrained region. The object or obstacle is
denoted as a union of convex polytopes and ellipsoids, and the constrained
region is denoted as an intersection of such convex sets. Using these
representations, collision avoidance can be approached by formulating explicit
constraints that separate two convex sets, or ensure that a convex set is
contained in another convex set, referred to as separating constraints and
containing constraints, respectively. We propose to use the hyperplane
separation theorem to formulate differentiable separating constraints, and
utilize the S-procedure and geometrical methods to formulate smooth containing
constraints. We state that compared to the state of the art, the proposed
formulations allow a considerable reduction in nonlinear program size and
geometry-based initialization in auxiliary variables used to formulate
collision avoidance constraints. Finally, the efficacy of the proposed unified
planning framework is evaluated in two contexts, autonomous parking in
tractor-trailer vehicles and overtaking on curved lanes. The results in both
cases exhibit an improved computational performance compared to existing
methods.
