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In this article, a distributed control strategy using an adaptive sliding mode controller (ASMC) is proposed for a 13-degree-of-freedom (13-DOF) cooperative robotic system in the field of automated fiber placement (AFP). A distributed control structure with event-triggered mechanism is developed to guarantee the desired cooperation performance and reduce the communication burden. To address dynamic uncertainties and external disturbances, an adaptive sliding mode control approach is designed for the robots. A deep recurrent neural network (DRNN) is incorporated into the ASMC to estimate lumped system uncertainties. The DRNN features a feedforward structure through three hidden layers and a feedback loop connecting the output layer to the input layer. This architecture demonstrates superior online learning capability and dynamic adaptability compared to shallow feedforward neural networks. To ensure the stability of the controller, the adaptation laws of the neural network parameters are formulated through Lyapunov theorem. The feasibility and advantages of the distributed DRNN-based adaptive sliding mode control strategy have been validated by simulation and experimental results.