Service robots are one of the relevant areas of modern robotics. Many service robots are equipped with a pair of anthropomorphic manipulators, so that they are able to perform complex operations. However, this approach leads to new challenges in development of the robot control systems. In this paper we propose an algorithm for training the control system of two anthropomorphic manipulators with 7 degrees of mobility having intersecting work areas. The algorithm is based on deep reinforcement learning approach applied to the artificial neural network (ANN). The paper also describes the practical implementation of the ANN-based manipulator control system that avoids collisions and achieves an average accuracy of reproducing target positions of manipulator end effector of 98.3%. The ANN training was carried out using Keras framework. The obtained results indicate the promise of applying the proposed method for the development of control systems for anthropomorphic manipulators based on deep reinforcement learning.