## 1 Introduction  
Delta parallel robot is the most successful commercial parallel robot to date. The parallel robot has been employed for high-precision tracking and high-speed motions to boost output [1, 2]. The robot has three kinematics branches consisting of two arms, and the branches connect a base and a moving plate. Due to the complexity of the closed-loop structure and kinematic constraints of the robot, the robot model is a multi-variable, multi-parameter, coupling, and multi-degree-of-freedom nonlinear system. Many studies have shown the significance of kinematic [3, 4, 5] and dynamic models [6, 7, 8, 9] on path planning and tracking precision. Accordingly, numerous model-based control techniques are employed to improve the performance of parallel robots. For instance, [PERSON] proposed a proportional derivative feedforward (PDFF) controller based on a dynamic model of the Delta robot in [10, 11] to give more precise trajectory tracking than a kinematic controller. In a recent work [12], a dynamic feedforward decoupling control scheme was proposed for a compensated feedforward gravity and an independently adjustable three PID control loop corresponding to the position, velocity, and current controls of a four-degrees-of-freedom parallel robot. Although the simulation results reveal that the system can reduce tracking errors, it is difficult to be applied due to the decoupling factors in serial control loops, which depend not only on the correctness of the dynamic modal transformations but also on the noises of the feedback signal. In addition, the two model-based methods require not only the accuracy of parameters but also lack influences of uncertainty parameters and unknown disturbances in the systems. To solve robust tracking problems and reduce the influence of external disturbance, a sliding mode control (SM) technique based on the Delta robot dynamics with uncertain parameters was proposed in [13]. This approach guarantees stability by keeping system states on a sliding plane. Another SM improvement by a terminal attractor technique is the so-called terminal sliding mode (TSM) control, proposed in [14] for finite-time stabilization of robotic manipulators. However, both SM and TSM techniques have a drawback, which produces high-frequency oscillations of the controller outputs, known as chattering noises. The noises can impose vibration on the mechanical actuators. To eliminate the chattering phenomenon, another control methodology for modern and complex systems led to the introduction of the synergetic control (SC) theory [15] by [PERSON]. Similar to SM properties, the advantages of synergetic control are well-suited for order reductions of the controlled system, robustness to external disturbance, and better control of the off-manifold dynamics. In synergetic control papers [16; 17; 18], synergetic-based controllers generate chattering-free dynamics of nonlinear systems. Inspired by the TSM, finite-time synergetic control (FS) schemes to control serial robot manipulators were proposed in [19; 20]. The finite-time synergetic control (FS) scheme employed the synergetic theory and a terminal attractor technique to control serial robot manipulators. The new synergetic approach is allowed to perform well in multiple connected nonlinear systems with continuous, discontinuous, discrete-time, terminal, and adaptive control, such as robots [20; 21], electric drives [22], and flying apparatus [23]. Based on those studies, the benefits of a synergetic approach, such as robustness, quick dynamic reactions, and chattering-free phenomena, are well suited for the Delta robot. In addition, the Delta parallel robot is a highly nonlinear complexity system. Therefore, developing a suitable control scheme for the robot with parametric and non-parametric uncertainties and external disturbance represents a significant challenge. [PERSON] and co-workers proposed a nonlinear controller to solve robust and precise trajectory problems using SM or a nonlinear PD controller based on iterative learning [24; 25; 26].