This paper considers the problem of planning reference trajectories for cooperating autonomous vehicles. Our previous work [1] relied on mixed-integer quadratic programming (MIQP) for planning, while controllable sets of hybrid automata were used to assess feasibility. In this paper, we supersede MIQP in the online part and obtain plans by directly making use of controllable sets and adapting ideas from approximate multi-parametric programming. The run-time of the algorithm is not only lower, but also more predictable than that of MIQP solvers, which is crucial for real-time operation. Even though it returns suboptimal solutions, the algorithm also enables the use of smaller sampling times or longer planning horizons.