## 1 Introduction
[14] developed a 3-PRPiR parallel robot, which can operate in five operation modes without crossing any constraint singularity, including a 3T operation mode and four 2T1R operation modes, by using a lockable prismatic pair Pi and a revolute pair R. [PERSON] et al. [15] proposed a reconfigurable parallel mechanism based on a 3-RRR spherical parallel mechanism, which can switch between the 1T3R motion mode and 3R motion mode, and analyzed its workspace, dexterity and singularity. [PERSON] et al. [16] proposed a single-loop and multi-loop reconfigurable platform, and they analyzed and optimized the workspace, dexterity and payload index of a 3-PUU reconfigurable parallel mechanism based on this reconfigurable platform. [PERSON] et al. [17] proposed a 3-rRPS reconfigurable parallel mechanism that can be converted between the 3R motion mode and 1T2R motion mode. During the course of rehabilitation training, the rotation center of the rehabilitation mechanism being used is consistent with the rotation center of the ankle. The traditional 1T2R parallel mechanism generally has a parasitic motion, which can easily cause secondary damage to patients. [PERSON] et al. [18] proposed a 3-RRPRR variable spherical symmetric parallel mechanism with circular arc-shaped kinematic pairs and proved that there is no parasitic motion of the mechanism by analyzing its geometric characteristics. [PERSON] et al. [19] proposed a 2 RPU-RPS-UPS parallel mechanism without parasitic motion, verified its non-parasitic motion characteristics by using the screw theory, and analyzed its motion/force transmission performance and dynamic performance. [PERSON] et al. [20] used the maximum parasitic motion in the maximum inscribed circle of the workspace as the parasitic motion index of a 3-[PP]S parallel mechanism, and they compared the directional motion ability and parasitic motion of the 3-PPS parallel mechanism in different modes. [PERSON] et al. [21] formulated a parasitic motion equation in terms of velocity and used the quasi-Newtonmethod to optimize the parasitic motion in the workspace. [PERSON] et al. [22] proposed a kind of 2T1R parallel mechanism without parasitic motion, and analyzed and compared the performance indexes, such as the workspace and rotation ability of the mechanism, under different configurations. A branched chain with actuation redundancy can improve the accuracy of a mechanism, enhance the load capacity and improve the stiffness of the mechanism [23; 24]. [PERSON] et al. [25] proposed a redundantly actuated 1T2R parallel lower-limb rehabilitation mechanism and optimized the performance index of the mechanism by means of a genetic algorithm. [PERSON] et al. [26] proposed a planar 2T1R redundant parallel mechanism and analyzed its workspace and stiffness. [PERSON] et al. [27; 28] analyzed the motion/force transmission performance of a 1T2R redundantly actuated parallel mechanism and optimized the mechanisms based on this performance index. [PERSON] et al. [29] concluded that the reachable workspace of a 2 UPR&1 RPS&1 RPU redundantly actuated parallel mechanism is the intersection of its non-redundantly actuated forms. [PERSON] et al. [30] obtained the dimensionless Jacobian matrix of a 2 PUR-2 PRU redundantly actuated parallel mechanism by using the characteristic length method and analyzed the dexterity of the mechanism based on the dimensionless Jacobian matrix. According to the structure characteristics of the ankle joint, this paper proposes a reconfigurable parallel mechanism without parasitic motion, which can be converted to the 1T2R motion mode and 2T1R motion mode to meet the degree of freedom required for ankle rehabilitation training. The rest of the paper is organized as follows. In Section 2, the mechanism configuration, vA kinematic pair, degree of freedom and non-parasitic motion characteristics are analyzed. In Section 3, an inverse kinematics model of the mechanism in the two motion modes is established, and the singularity of the mechanism is analyzed.