  To be applicable to real world scenarios trajectory planning schemes for
mobile autonomous systems must be able to efficiently deal with obstacles in
the area of operation. In the context of optimization based trajectory planning
and control a number of different approaches to formulate collision avoidance
constraints can be found in the literature. Here the contribution of the
present work is twofold. First, the most popular methods to represent obstacles
are summarized, namely the simple ellipsoidal representation, the constructive
solid geometry (CSG) method as well as a direct and an indirect implementation
of a signed distance based approach. The formulations are characterized with
respect to the impact on the complexity of the optimization problem, as well as
the ability to meet different problem requirements. Second, this work presents
a novel variant of the CSG method to describe collision avoidance constraints.
It is highly efficient due to a very low number of nonlinear inequality
constraints required for a given number of obstacles and sample points and in
contrast to the original CSG formulation allows to consider the controlled
system's shape. The good performance of the proposed formulation is
demonstrated by a comparison to the previously mentioned alternatives. To this
end optimal trajectory planning for marine surface vessels formulated as a
nonlinear programming problem is used as a benchmark example where the scenario
is designed based on the maritime test field in Kiel, Germany.
