The ability to predict the intentions and driving trajectories of other vehicles is a key problem for autonomous driving. We propose an integrated planning and prediction system which leverages the computational benefit of using a finite space of maneuvers, and extend the approach to planning and prediction of sequences (plans) of maneuvers via rational inverse planning to recognise the goals of other vehicles. Goal recognition informs a Monte Carlo Tree Search (MCTS) algorithm to plan optimal maneuvers for the ego vehicle. Inverse planning and MCTS utilise a shared set of defined maneuvers to construct plans which are explainable by means of rationality, i.e. plans are optimal in given metrics. Evaluation in simulations of four urban driving scenarios demonstrate the system's ability to robustly recognise the goals of other vehicles while generating near-optimal plans. In each scenario we extract intuitive explanations for the recognised goals and maneuver predictions which justify the system's decisions.
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