# Kinematics and Performance Analysis of the 2T1R/1T2R Reconfigurable Decoupled Parallel Mechanism
## 1 Introduction
[PERSON] and [PERSON] [19] conducted a detailed study on the topological representation and characteristics of variable kinematic pairs, proving that the topological features of variable motion joints have properties such as reversibility, continuity, degree of freedom variability, joint homomorphism, contractility, and scalability. Finally, a series of variable topological structures were synthesized. [PERSON] et al. [20; 21; 22] proposed the structural formula and composition method of a planar metamorphic mechanism based on Assur groups, and studied the basic issues of migration rate and synthesis of constrained metamorphic work and migration rate configuration, providing a reference theory for the kinematic research of metamorphic mechanisms. [PERSON] et al. [23] designed a class of reconfigurable modular parallel robots based on 3-CPS underactuated topology, which switches between the three-dimensional revolute mode and the three-dimensional translational mode of the parallel mechanism by locking the revolute pair. The advantages of this mechanism when applied to parallel machine tools are very obvious. [PERSON] et al. [24] proposed a new reconfigurable mechanical gripper based on the screw theory. The fingers of the gripper are composed of deformable mechanism modules, which can realize expansion and grasping actions through deformation. [PERSON] et al. [25] analyzed a 2R1T parallel mechanism with two continuous rotational axes (CRAs) branches through screw theory and spatial geometry theory, obtained the spatial arrangement relationship of the constraint force of the two branches on the moving platform, and obtained a series of 2R1T parallel mechanisms with two CARs. [PERSON] al. [26; 27] analyzed the degrees of freedom and relationships between the branches of parallel mechanisms based on screw theory, derived the rotation conditions of parallel mechanisms, integrated the decoupling rules of parallel mechanisms, and summarized the type of synthesis method of parallel mechanisms with rotational characteristics. [PERSON] et al. [28; 29] proposed a type of synthesis method for decoupling parallel mechanisms based on Lie group and screw theory, and derived a series of new 2T1R parallel mechanisms that are fully decoupled in space and plane. [PERSON] and [PERSON] [30] developed a 3R ankle rehabilitation parallel mechanism with decoupling characteristics. The inverse kinematics, velocity Jacobian matrix, and singularity of the mechanism were analyzed and compared with the 3-SPS/S ankle rehabilitation mechanism. [PERSON] [31] studied the decoupling of parallel mechanisms based on screw theory, replacing translational motion with a planar four-bar parallelogram. The singularity of the mechanism was analyzed using the Plucker coordinates of the parallel mechanism's kinematic chain, and finally a parallel mechanism with three degrees of freedom, four degrees of freedom, and six degrees of freedom was synthesized. This article proposes a reconfigurable decoupled parallel mechanism based on the rR reconfigurable kinematic pair, and analyzes the kinematic characteristics and performance index of the parallel mechanism. The advantage of this reconfigurable parallel mechanism is that decoupling can be achieved in both motion modes, and the motion of the parallel mechanism can be achieved through simple control. This provides a certain research idea for the study of decoupled parallel mechanisms and reconfigurable parallel mechanisms. The organizational structure of the article is as follows. In Section 2, the configuration and degrees of freedom of the reconfigurable decoupled parallel mechanism are analyzed. The parallel mechanism has two motion modes, 2R1T and 1R2T, and its mobility is full-cycle. In Section 3, based on the prepositive DH coordinates and the decoupling characteristics of parallel mechanisms, the inverse kinematics, forward kinematics, and operating speed of parallel mechanisms are analyzed.