## 1 Introduction
The results are compared with the approach utilizing direct evaluations of the objective function, and it is shown that the newly developed adaptive mechanism is faster in convergence and effectiveness in error minimization, and in controlling multiple-link flexible robots. The trajectory tracking of a robotic arm utilizing a feed-forward ANN, which maps between a dataset of numeric inputs and a set of numeric targets, is given in [10]. Training data for the ANN are calculated using forward kinematics equations of the given robot. The ANN is trained to solve an inversion problem. The efficacy of this approach is confirmed using Matlab. In [11], the dynamics identification and multi-link industrial manipulator control are given using Runge-Kutta-Gill Neural Networks (RKGNNs). The neural network method can accurately grasp the changing rates of the states; his method can be effectively used for long-term prediction of the states of the multi-link robot manipulator dynamics. Once identified, the gathered dynamics data can also be used to control the robot for a new trajectory. In [12], researchers designed an industrial robot manipulator controller using two parallel subsystems. First, the PD controller is designed to minimize trajectory tracking errors. Then, the ANN controller is used to generate the required torque by a given dynamic trajectory. Data used for ANN training and online weights update are calculated based on the simplified dynamic model of the robot. The proposed method is tested in simulation and experimentally, and both methods confirm its usefulness. The ANN is implemented in a 2-DOF industrial-grade manipulator [13], in which position controllers utilize neural networks and provide satisfactory performance of trajectory tracking for sinusoidal references. The ANN can be effectively used to adjust PID parameters [14]. Thanks to this, the accuracy of the tracking control is improved over the standard PID algorithm. More advanced adaptive ANN solutions are utilized in complex systems. For example, in [15], an adaptive neural network controller is proposed for robot manipulators with multiple fingers and uncertain kinematics, dynamics, and Jacobian matrix. Utilizing ANNs allows one to limit the position error despite the presence of uncertainties. Simulations are carried out to show the performance of the proposed controller. In [16], researchers propose global adaptive neural control with finite-time convergence learning performance for a general class of nonlinear robot manipulators. What is more, an adaptive learning algorithm driven by estimated weights error is used to guarantee the neural weights to converge to optimal values in a finite amount of time. The method performance is proven in a simulation environment against conventional methods. In [17], a robust adaptive control method based on dynamic structure Fuzzy Wavelet Neural Networks (FWNNs) is proposed for trajectory tracking control of industrial robot manipulators with uncertainties and disturbances via adaptive sliding mode control. Good control performance is achieved by utilizing an adaptive learning algorithm that allows FWNNs' weights to be modified online, and the Lyapunov stability theorem. The researchers present a mathematical proof to confirm the functionality and parameters of the given controller. In [18], the authors present a neural network tracking controller for a SCARA manipulator. A three-layer neural network is trained based on reinforcement learning methodology. In [19], a feed-forward NN controller with robust integral of the sign of the error (RISE) feedback is proposed to control a multiaxial hydraulic manipulator. The performance of the control system is verified using a simulation. An interesting approach using the auto-encoder method is used in [20] for kinematic control of a manipulator. The model is built directly from the kinematic parameters of the robot model rather than from a training dataset.