Popular trajectory planning techniques for systems with many state variables and complicated constraints are the Rapidly Exploring Random Trees (RRT) framework and its extension RRT∗. However, both algorithms heavily depend on the availability of a Local Planner that can cope with the system's dynamics and constraints. We describe a local planning technique for a wide range of orbitally flat systems, i.e. differentially flat systems with time scaling. Although differential equations appear in their descriptions, these systems can be handled by the RRT∗ with only very little overhead compared to differentially flat ones. We show that solving the differential equations can be avoided during most invocations of the local planner without resorting to approximations. The approach is applied to the task of 3D-trajectory planning for a fixed-wing aircraft and planning results are compared to a previous implementation that did use approximations.