## 2 Related Work  
PMs known as parallel wrists (PWs) only move spherically around a fixed point when there is a relative motion between the base and moving platforms [7]. Three-DOF spherical parallel manipulators (SPMs) have received a careful examination in the literature cited in this paper. For a considerable amount of time, only two manipulator architectures have been employed to obtain PWs. These are known as not-over-constrained and over-constrained PWs. The base and moving platforms of the over-constrained PWs are joined by three legs of type RRR (R stands for revolute pair), each of which has its axes intersecting at a fixed point. Geometric errors cause mechanism jams or excessive internal loads in the links of an over-constrained architecture [8]. Two categories of manipulators exist in PWs that are not over-constrained. The first group consists of 3 UPU PMs (U and P stand for universal joint and prismatic pair), and the second set contains 3-SPS-1-S PMs, where S stands for spherical pair. Three legs of type SPS control the moving platform in 3-SPS-1-S PWs. To carry out the necessary orientation, the prismatic pairs are active ones. Only rotational motion is ensured by the fixed central leg with the spherical pair connecting the base platform and moving platform, which also makes the robot more rigid. This PW has three DOFs and a smaller workspace because of the passive spherical pair [9]. Systems for defence and surveillance employ the 3 SPS-1S PM. A test scenario showcasing the robot's use as a sentry gun was presented in [9]. The robot works well in the test scenario; however, to put this robot into use, a system that recognizes the intruder and plans the trajectory is required. [PERSON] et al. [6] demonstrated that it is possible to achieve infinitesimal spherical motions using 3-UPU translational manipulators under specific manufacturing and mounting scenarios. It was therefore referred to as a 3-UPU wrist. Because the 3-UPU wrists overcome the limitations of two PWs (3-SPS-1-S and RRR) and do not require a passive spherical pair, they are of particular interest to researchers. While it is desirable to be able to perform an infinitesimal spherical motion, this requirement alone does not guarantee that the platform performs a finite spherical motion, i.e., a PW manipulator. Their research was therefore only useful in identifying potential PW architecture. The latter decade of the 20 th century saw the discovery of these robots. Although these architectures are not used in commercial robots, researchers find them to be quite interesting. These designs could be reconfigured as PWs, translational PMs, or metamorphic PMs. A 3 UPU PW was created to follow the path of the sun. The fact that the fixed point on the heliostat is not precisely at the correct distance from the base, suggesting that the manufactured 3-UPU is not a perfect PW, is one of the main problems with sun tracking experiments. The ideal 3-UPU wrist has only three rotational DOFs. However, because of manufacturing flaws that cause the fixed point to become misaligned, the 3 UPU moves in both position and orientation, making control more challenging with its existing controller. A greater manufacturing tolerance level needs to be maintained in order to remove pointing errors [4]. Commercial robots do not exist because, as the literature shows, all PW manipulators are unable to achieve the appropriate platform orientations. Researchers have also used a combination of 3 UPU PW and 3 SPS/1S PW like the 3 UPS/S, 3-UPS/RPU, and 3-PUS/S PMs in specific applications. For better design and control of a robot, solving kinematics is the foremost task [3]. [PERSON] [10] established a recursive relation between dynamics and kinematics for a 3-UPS/S manipulator. Inverse kinematics was solved using the recursive matrix method, and then using the principle of virtual work, inverse dynamics was solved. [PERSON] et al. [11] solved the inverse kinematics using the matrix variation method. Then, forward kinematics was solved using the numerical analysis method.