## 1 Introduction
should be used, how must the different controls be organized hierarchically, and how many degrees of freedom must the controller have to achieve the desired performance?\". In [4], as in [26], another architecture is proposed for the control structure. An NN-based control structure is used in an outer loop (i.e., for position control) as feedforward compensation on either a quadrotor with a deep NN or a 7-DOF robot arm with an RNN. These papers highlighted three main advantages of combining an NN system as feedforward compensation with a position PD controller. First, NN architectures can be applied to various systems with complex dynamics while ensuring their stability [26]. Secondly, this approach is easy to use because it does not need any prior information on the system and can be applied to unseen trajectories without any adaptation process [5; 11]. Thirdly, with a good training process, this approach demonstrates good accuracy in trajectory tracking while being computationally efficient with a small model [4]. In [4], the learning of the direct dynamic model of a flexible Baxter robot was conducted at the joint position control level by combining a proportional-integral-derivative (PID) controller with either a bidirectional RNN or an RNN associated with an iterative learning control (ILC) algorithm, both with around \(10^{3}\) parameters, directly as one integrated model. The results showed a reduction in tracking errors between 12% and 54% depending on the joints, compared to the reference for random trajectories. In [5], the inverse dynamic learning process of a 7-DOF industrial robot was first conducted in a simulation by implementing the control for many trajectories to collect data for the NN on the real system. However, this technique required a good dynamic simulation of the system, which is rarely available for non-industrial robots because a large amount of data is needed to train the model. In this paper, a trained NN with around \(2\cdot 10^{4}\) parameters--2 layers of 100 neurons--is employed as a feedforward component to compensate for the dynamic errors of the position controller of an ABB IRB6640-180 robot with 6 DOF. This approach resulted in a diminution of tracking errors by 80% to 90%compared to tracking errors without compensation, achieving a global mean squared error of 2.145\({}^{\circ}\) in their tests. Unlike their previous work [4], which used a single NN for the whole system, the authors mentioned that on the studied ABB robot (Zurich, Switzerland), the joint dynamics were decoupled--i.e., the impact of the joint movements was negligible on the dynamics of the other joints--allowing the possibility to train six separate NN's, each approximating the dynamic inversion for one joint. However, only a global neural network for all joints was implemented. In [11], we proposed a model-free control structure for high-speed trajectory tracking that combined the advantages of both proportional-derivative (PD) controllers, which are stable and easy to use, and neural networks, which are efficient in estimating the system dynamics. This method differed from [4] and [5] by using one NN per joint (the INN method) instead of one global NN for the entire robot arm (the GNN method). The architecture--illustrated in Figure 1--combines an NN with a PD controller. It consists of learning the dynamic response of the PD controller with an NN and using it in an outer loop as a feedforward compensation to predict and correct the errors of the PD controller, which is effective for robot trajectory tracking [27]. A summary of existing methods for trajectory tracking is presented in Figure 2.  
Figure 1: Model-free control structure for high-speed trajectory tracking. Figure 2: Summary of existing control structures for trajectory tracking [4; 5; 7; 8; 12; 13; 14; 15; 16; 25; 26].