## 1 Introduction
However, the schemes must require a long time to follow continuous repetitive or non-repetitive trajectories before improving the tracking errors. To online estimate uncertain plant parameters, different kinds of adaptive control strategies have been performed to deal with the uncertain parameter or disturbance. The work in [27] proposed an adaptation law for dynamics and friction compensation of the dynamic model of the 2-DOF parallel mechanism based on a Lyapunov method. In ref. [28], an online estimation was proposed for simulating trajectory tracking control with nonlinear joint friction models, which were compensated to the nominal Delta model. Another adaptive control inside an active disturbance rejection framework used the least mean square technique in [29] to produce a reliable tracking control for the low-speed Delta platform, where the researchers did not consider the joint velocity data of the system. As can be seen, uncertainty parameters in the adaptive controllers must be analyzed explicitly in a specific model, which represents the effects of the system tracking errors. In recent years, there have been numerous adaptive intelligent control techniques. On the one hand, a systematic analysis with sensory noise and uncertainty robustness for a type 2 fuzzy logic was used to position the control of the end-effector of the parallel Delta robot [30]. Or a utilization of fuzzy logic systems can estimate the unknown nonlinear behavior of simple examples and adapt successfully via synergetic control theory [31]. However, the linguistic rules and optimal dataset for stability guarantee remain challenging issues. On the other hand, uncertainty parameters in a model are estimated by approximations using neural networks (NNs). The study in [32] is an example of using NNs to create compensators for the dynamic model. In particular, the robust integral of the sign error (RISE) controller with a B-spline NN feedforward compensation was applied to a Delta robot to regulate the trajectory tracking for a pick-place application. An intelligent compensation term added to the system control can reduce the tracking error considerably and cancel the steady-state error. However, only simulating validations of these NNs are taken into account. Based on a linear motor model with unknown frictions [33], a processing structure of two-layer NNs was proposed successfully to compensate for the effects of the friction model in both simulations and experiments. Ref. [34] gave an output feedback controller with a feedforward term based on a radial basis function (RBF) neural network that dealt with uncertainty compensation in a robotic exoskeleton. It is shown that an RBF network may approximate a continuously nonlinear function with benefits such as low-cost computation and quick convergence. In this work, RBF networks are utilized in addition to a robust synergetic controller to estimate unknown parameters for control quality enhancement. Based on the benefits and limitations of the above literature review, a novel combination of the advantages of a proportional derivative, a robust synergetic, and an RBF network controller based on the object-oriented technique [7] in a Delta robot system with the generalized disturbance is proposed. Therefore, the main contributions of this paper lie in the following aspects:  
1. A systematic controller using a PD controller with a finite-time synergetic technique is developed based on the Delta robot dynamics for robust trajectory tracking control. In the proposed controller structure, unknown parameters and external disturbance of the DC motors and the robot manipulator are analyzed in a generalized disturbance model, then compensated torques are estimated online through RBF 2-layer neural networks. 2. The stability of the proposed control framework is analyzed and proven through the Lyapunov stability theory. 3. The features of the proposed controller are successfully applied to the parallel Delta robot in simulation and practical experiments with different trajectory examples.