To deal with the task assignment problem of multi-AUV systems under kinematic constraints, which means steering capability constraints for underactuated AUVs or other vehicles likely, an improved task assignment algorithm is proposed combining the Dubins Path algorithm with improved SOM neural network algorithm. At first, the aimed tasks are assigned to the AUVs by the improved SOM neural network method based on workload balance and neighborhood function. When there exists kinematic constraints or obstacles which may cause failure of trajectory planning, task re-assignment will be implemented by changing the weights of SOM neurals, until the AUVs can have paths to reach all the targets. Then, the Dubins paths are generated in several limited cases. The AUV’s yaw angle is limited, which results in new assignments to the targets. Computation flow is designed so that the algorithm in MATLAB and Python can realize the path planning to multiple targets. Finally, simulation results prove that the proposed algorithm can effectively accomplish the task assignment task for a multi-AUV system.