Online state-time trajectory planning in highly dynamic environments remains an unsolved problem due to the unpredictable motions of moving obstacles and the curse of dimensionality from the state-time space. Existing state-time planners are typically implemented based on randomized sampling approaches or path searching on discretized state graph. The smoothness, path clearance, and planning efficiency of these planners are usually not satisfying. In this work, we propose a gradient-based planner over the state-time space for online trajectory generation in highly dynamic environments. To enable the gradient-based optimization, we propose a Timed-ESDT that supports distance and gradient queries with state-time keys. Based on the Timed-ESDT, we also define a smooth prior and an obstacle likelihood function that is compatible with the state-time space. The trajectory planning is then formulated to a MAP problem and solved by an efficient numerical optimizer. Moreover, to improve the optimality of the planner, we also define a state-time graph and then conduct path searching on it to find a better initialization for the optimizer. By integrating the graph searching, the planning quality is significantly improved. Experiment results on simulated and benchmark datasets show that our planner can outperform the state-of-the-art methods, demonstrating its significant advantages over the traditional ones.
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