This paper considers a position control system of a 3-link robot manipulator. The authors reviewed publications on nonlinear compensation of dynamic errors with the use of neural networks in robot manipulator control systems. The paper presents mathematical description of the control system with the compensation of nonlinear dynamics of the robot mechanism. We carried out training of multivariable neural network compensators of dynamic errors occurring because of the influence of inertia, Coriolis and gravity load torques. We developed computer models of the control system with different types of neural network compensators which are included in feedforward and feedback of the system and carried out a computer simulation of control systems with prototype and different kinds of neural network compensators. We also conducted a comparative analysis of dynamic errors in the system with different combinations of neural network compensators and gave recommendations on program realization of neural network compensators for real robot manipulator position control systems.