This study presents a fast generation method of time-optimal low-thrust trajectories based on deep neural networks (DNN) for the collision-free asteroid landings. An equivalent unconstrained differentiable problem is transformed via the introduction of smooth penalty function, so that it can be addressed with the optimal control framework. To efficiently solve the equivalent problem to generate sufficient dataset for training DNNs, a double homotopy method is developed. Specifically, the equivalent problem is connected to the time-optimal control problem without anti-collision through two consecutive simplifications, and a solving process with double backward continuation is also formulated. Besides, two DNN models are constructed to provide high-quality predictions for the minimum glide-slope constraint angle of an initial state, and the initial costates and landing time for time-optimal low-thrust trajectories with anti-collision constraints, respectively. As such, the efficiency of collision-free low-thrust landing trajectory generation is improved significantly. For the cases within the range of training dataset or within a certain extension, the low-thrust landing trajectories with a very small terminal landing error can be generated using the predicted initial costates. Meanwhile, with the high-quality initial costates, the high-precision landing trajectories can be obtained by the double homotopy method in only a few iterative steps. Finally, the promising results in a Bennu collision-free asteroid landing scenario validate the efficiency of the proposed method.
