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@@ -99,15 +99,15 @@ <h1 class="title is-1 publication-title">Dynamically Feasible Trajectory Generat
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@@ -201,7 +201,7 @@ <h1 class="title is-1 publication-title">Dynamically Feasible Trajectory Generat
         <h2 class="title is-2">Abstract</h2>
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-            This paper aims to bridge perception and planning in navigation systems by learning optimal trajectories from depth information in an end-to-end fashion. However, using neural networks as black-box replacements for traditional mod- ules risks scalability and adaptability. Moreover, such methods often fall short in sufficiently incorporating the robot’s dynamic constraints, resulting in trajectories that are either inadequately executable or unexpectedly aggressive, diverging from user expectations. In this paper, we fuse the benefits of conventional methods and neural networks by introducing an optimization- embedded network based on a compact trajectory library. The network distills spatial constraints, which are then applied to model-based spatial-temporal trajectory optimization, yielding feasible and optimal solutions. By making the optimization differentiable, our model seamlessly approximates the optimal trajectory. Additionally, the introduced regularized trajectory library permits efficient capture of the spatial distribution of optimal trajectories with minimal storage cost, safeguarding multimodal planning features. Benchmarking demonstrates the outstanding performance of our method in trajectory smoothness, success rate, and constraint satisfaction. Real- world flight experiments with an onboard computer showcase the autonomous quadrotor’s ability to navigate swiftly through dense forests. 
+            This paper aims to bridge perception and planning in navigation systems by learning optimal trajectories from depth information in an end-to-end fashion. However, using neural networks as black-box replacements for traditional modules risks scalability and adaptability. Moreover, such methods often fall short in sufficiently incorporating the robot's dynamic constraints, resulting in trajectories that are either inadequately executable or unexpectedly aggressive, diverging from user expectations. In this paper, we fuse the benefits of conventional methods and neural networks by introducing an optimization-embedded network based on a compact trajectory library. The network distills spatial constraints,  which are then applied to model-based spatial-temporal trajectory optimization, yielding feasible and optimal solutions. By making the optimization differentiable, our model seamlessly approximates the optimal trajectory. Additionally, the introduced regularized trajectory library permits efficient capture of the spatial distribution of optimal trajectories with minimal storage cost, safeguarding multimodal planning features. Benchmarking demonstrates the outstanding performance of our method in trajectory smoothness, success rate, and constraint satisfaction. Real-world flight experiments with an onboard computer showcase the autonomous quadrotor’s ability to navigate swiftly through dense forests.
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