## 1 Introduction
Initially, the H4 machine, based on the Delta robot concept, a 3T-1R parallel kinematic machine (PKM), was proposed, and its position and velocity relationships were analyzed. It was demonstrated that the H4 principle could serve as the foundation for a complete family of machines [14; 15]. Subsequently, scholars in the field have designed and studied various types of 4-DOF parallel manipulators. [PERSON] [16] designed two 4-DOF parallel mechanisms. [PERSON] et al. [17] presented a partially decoupled, four-degrees-of-freedom 3T1R parallel manipulator with high load-carrying capacity. [PERSON] et al. [18] proposed an over-constrained RRPU + 2 UPU parallel manipulator (PM) with three legs. [PERSON] et al. [19] introduced a novel 4-DOF decoupled parallel manipulator with Schonflies motions [20] called the Pantopteron-4. [PERSON] et al. [21] presented a hierarchical approach for the type-synthesis of limb structures of the PKMs, with 3T1R motion with a variable rotational axis. In machining processes, particularly milling, it is crucial that the tool remains perpendicular to the workpiece surface throughout the machining process to avoid errors and ensure high-quality machining. However, existing 4-DOF parallel mechanisms either have a rotational DOF around the vertical axis or two rotational DOFs suitable for pick-and-place and positioning tasks. Studies have shown that using parallel kinematic structures with different layouts may shorten the time it takes to produce parts by increasing the number of degrees of freedom to four, mostly through turning around the z-axis [22]. A further investigation included the creation of a delta-type parallel kinematic framework that had the capability to rotate along the x-axis. This advancement enhanced the system's maneuverability [23]. Another study presented a high-speed parallel robot that exhibited Schonflies motion. In this study, the robot had great potential for achieving efficient and rapid pick-and-place manipulation in packaging manufacturing lines. The robot was equipped with four identical appendages and a solitary platform. The robot's small shape and single-platform idea provided it with a strong potential for dynamic responsiveness. The movement properties of the proposed robot are examined using a line graph approach that relies on Grassmann line geometry. This approach also incorporates a generalized Blanding rule to facilitate the translation between line graphs representing movements and restrictions. Afterward, the inverse kinematics is calculated, and the problem of singularity in the robot is examined using both qualitative and quantitative methods [24]. A novel parallel robot, Heli4, with four degrees of freedom, was provided in [25]. It drew inspiration from Delta architecture but was specifically engineered to surpass its constraints by including an articulated traveling plate. Contrary to the majority of articulated traveling plates, Heli4's moving plate was very small in size. Another research study introduced a novel parallel mechanism with four degrees of freedom that was specifically designed for manipulating heavy objects in a wide operating area [26]. The four degrees of freedom consisted of three translational movements and one rotational movement around a specified axis. This kind of mechanism is uncommon; the majority of parallel mechanisms typically contain either three or six degrees of freedom. Another study presented a specialized parallel manipulator with four degrees of freedom, which was designed for the purpose of pick-and-place tasks. Its development was aimed at achieving very high velocities. This research demonstrated that its design was well-suited for handling rapid changes and fluctuations. It was a development of the Delta, H4, and I4 robot designs, including the strengths of these robots while addressing their limitations [27]. Another example is an adaptive slicing method that can create ideal slices for five-axis hybrid layered manufacturing, allowing for deposition without the need for support structures.