## 1 Introduction  
The term \"6-DOF\" refers to the six degrees of freedom of motion that a mechanism or virtual object is capable of exhibiting. These degrees of freedom include movement in the vertical (up-down) plane, the horizontal (left-right) plane, the longitudinal (forward-back) plane, as well as rotation about the x (roll), y (pitch), and z (yaw) axes. The ability to move in these six distinct ways is crucial for a variety of applications, particularly in simulating the behavior of aircraft, robotics systems, and augmented/virtual reality (AR/VR) systems. The utilization of 6-DOF simulations is prevalent within the aerospace industry, serving as a valuable tool for both research and development, as well as for the training and evaluation of pilots and the assessment of aircraft designs [1]. It is also crucial for robotic applications, as it enables the capability of the robotic arm to effectively access and manipulate objects in various positions and orientations. In industries such as manufacturing, transportation, and surgical procedures, the implementation of 6-DOF robotic arms allows for the successful completion of intricate tasks that would be difficult or impossible for humans to perform. In recent years, the application of 6-DOF robotic systems has been an area of significant research interest, as it aims to develop robots that can perform complex tasks in various environments. For example, [2] presents new designs of 6-DOF robot manipulators for the transportation of building materials and 3D printing with concrete mixtures. In [3], [PERSON] et al. proposed a system that uses augmented reality to assist in the autonomous view adjustment of a 6-DOF robotic stereo flexible endoscope. [PERSON] et al. [4] proposed the use of a 6-DOF sensor-embedded soft robot for robotic-assisted minimally invasive surgery. The precise manipulation of objects is essential for tasks such as sorting, and 6-DOF robotic arms have been proven to be effective tools for such tasks. One example of this is the method presented in [5], which utilizes a 6-DOF robotic arm for cutlery sorting through object detection and grasping techniques. In AR and VR as well as Mixed Reality (MR), 6-DOF allows for a greater level of realism, as the user is able to move and interact with virtual objects in a similar way to how they would in the real world [6]. For example, a 6-DOF controller allows the user to move their virtual hand in a natural, lifelike way, making it possible to grasp and manipulate virtual objects [7; 8; 9]. AR and MR technologies can enhance minimally invasive surgery by providing improved visualization and precision during procedures. AR overlays virtual images and data onto the patient's body, while MR combines AR and VR to create an immersive experience for the surgeon [10; 11; 12]. These technologies can also be used to train surgeons and provide remote assistance [13]. They can improve patient outcomes by reducing the invasiveness of procedures and increasing the precision and skill of surgeons [12]. Specially, the utilization of AR and MR technology in cardiac catheterization has been demonstrated to potentially be a valuable tool in improving diagnostic and therapeutic capabilities [14]. The overlay of virtual images of the heart and its vasculature onto a real-time 3D representation of the patient's anatomy allows for enhanced visualization of the heart's structure, thereby facilitating the precise navigation and guidance of the catheter during the procedure [13; 14]. This can result in an increase in the safety and efficiency of the procedure. Furthermore, the use of AR and MR technology can aid in the planning and execution of complex procedures such as transcatheter aortic valve implantation (TAVI) [15; 16] and transcatheter mitral valve replacement (TMVR) [17; 18]. Although many cardiac catheters are tube-like objects that require only 5-DOF, there are many that are rotationally asymmetric, and thus 6-DOF tracking is critical.