This study proposes a unified optimization-based planning framework that addresses the precise and efficient navigation of a controlled object within a constrained region, while contending with obstacles. 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.

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