## 1 Introduction  
Undoubtedly, manipulator robots have many applications in many fields. Researchers are looking for hybrid manipulators to improve and develop these manipulators [1]. The combination of hybrid serial-parallel (HSP) manipulators has many advantages. Serial manipulators are distinguished in that they can provide a large workspace and simple control. In contrast, the parallel manipulator can provide high payload capacity and precision [2]. Furthermore, the hybrid manipulator is for the mechanism and can also be in the control and algorithm [3]. In hybrid manipulators, there are many categories, such as hybrid serial-parallel (HSP) [4; 5; 6; 7], hybrid parallel-serial (HPS), and hybrid parallel-parallel (HPP). Each combination has its features and properties. A mechanism with a seven-degree-of-freedom hybrid serial-parallel (HSP) arm of a humanoid robot manipulator has been proposed. The model comprises a 4-DOF serial robotic representing the shoulder and elbow joints connected with a 3-DOF parallel manipulator as a wrist joint [8]. Fusing the equilibrium and deformation coordination equations has also created a fully dynamic model. Moreover, three different configuration designs for the hybrid robot have been implemented using SCARA and SCARA-Tau mechanisms to show their advantages and disadvantages [9]. To eliminate oscillations for the HSP manipulator, they presented a dynamic stabilizing strategy to create an equilibrium between the swaying force and the swaying moment consisting of five interconnected links and having 3-DOF in the joint space [10]. Furthermore, the cable-linkage serial-parallel pelletizing robot (CSPR) has been presented as a study and a design focused on a series-parallel hybrid device powered by flexible wires to reduce inertia and improve dynamical responsiveness [11]. They also obtained the kinematic and dynamic models of the system. A novel proposed HSP design with 6-DOF [12]. The mechanism combines a 3-P parallel with 3-DOF and a 3-PSP with 3-DOF in serial form. This new layout will help regulate the attitude of the soaking cover in an immersion lithography system. A 6-DOF hybrid manipulator was composed of the 3-SPR (spherical prismatic revolute joints) connected with the 3-RPS (revolute prismatic spherical joints). It presents a solution for the end-effector terminal constraint and mobility problems with another configuration composed of the 3-RPS connected with the 3-SPR serial-parallel manipulators [13]. The workspace analysis and actuation distribution of a novel \(n\)(3-RRS) mechanism were developed. This hybrid manipulator is designed to compose a serial-parallel mechanism for gripping non-cooperative objects in space [14]. A new ideal hierarchical methodology is proposed for a serial-parallel hybrid kinematic machine with five degrees of freedom [15]. In addition, the kinematics and dynamics model for the new system was obtained. The suggested method simultaneously empowers the machine to execute significant kinematic, dynamic, and challenging operations. Furthermore, a 9-DOF worm robot with a hybrid serial-parallel mechanism is designed with two improved segments of a 3-RPS mechanism. The findings demonstrate that the robot can move in any direction for any given distance, and the theoretical analysis and the actual locomotion performance agree [16]. A new finger with a serial-parallel mechanism using twisted and coiled polymers (TCP) is implemented and designed for excellent performance [17]. Moreover, the kinematic and dynamic model was shown for the mechanical hand's increased power-to-weight ratio. The results show that comparing experimental and theoretical data verified the finger's mechanistic kinematic design. Moreover, to climb the transmission tower, a unique series-parallel hybrid robot with three degrees of freedom translation, parallel legs, and a body linkage is planned and constructed [18]. Furthermore, the design is equipped with an electromagnet system to climb onto the transmission tower.