## 1 Introduction  
The multi-configuration changes in reconfigurable mechanisms enable them to transform their own motion manifold by switching between different configurations so as to adapt to different working conditions, different tasks, and different functions with greater flexibility and useability [1, 2]. They have an extremely high application value for realizing the engineering application goal of \"one machine for multiple purposes or one machine for multiple configurations\" [3, 4]. Reconfigurable mechanisms include metamorphic mechanisms and motion bifurcation mechanisms; the difference between them is mainly whether the topological structure of the mechanism has changed. There are six basic evolutionary rules of the metamorphic mechanism: (1) the mechanism's number of limbs changes [5]; (2) the relative position of each limb and platform of the mechanism changes [6]; (3) the parameters (number, relative position, and geometric parameters) in each limb of the mechanism change [7]; (4) the parameters of the kinematic pairs (type, quantity, direction, and relative position) in each limb of the mechanism change [8]; (5) the moving platform is transformed into a non-rigid series open-loop mechanism or a closed-loop mechanism with an integrated end-effector [9]; (6) the mechanism switches between the reference and moving plane [10]. The basic evolution rule of motion of bifurcation mechanisms is to change the instantaneous freedom of the mechanism by switching different motion modes through a singular configuration. Metamorphic parallel mechanisms (PMs for short) [11] have adaptive and reconfigurable characteristics. Among them, a kind of metamorphic mechanism changes from the parameters of kinematic pairs in the limb to the principle of switching motion modes [12], such as reconfigurable joints. The representative reconfigurable joints include vA joints [13; 14], which switch between two different motion modes by changing the angle between adjacent joint axes; rT joints [15; 16; 17; 18; 19; 20], in which the link motion can be changed from rotation in pitch and yaw directions to rotation in pitch and roll directions; and rR joints [21] for one-dimensional rotation in different directions and in which lockable joints [22; 23; 24; 25] of freedom can be changed by locking kinematic pairs. A conventional parallel robot is characterized by connecting two platforms using several series limbs. Parallel robots with multiple closed loops have the advantages of stable operation, high stiffness, and large load capacity. Different from traditional rigid parallel robots, the reconfigurable platform can be connected according to a series of open-loop linkage mechanisms in series or a closed-loop linkage mechanism integrated with the end actuator. By changing the shape of the non-rigid moving platform and kinematic pairs, the end actuator can be equipped with multiple motion modes [26]. For conventional rigid PMs, the number of driving parts should be equal to the degrees of freedom of the whole mechanism, and they should be evenly distributed on the kinematic pairs of each limb. For reconfigurable mechanisms, due to their unique topology, a multi-redundantly actuated system is often adopted to enable the switch between multiple motion modes; that is, the driver is greater than or equal to the instantaneous degrees of freedom of the mechanism [27], so as to realize the redundant actuation of the mechanism. By adding redundantly actuated limbs or joints, redundantly actuated PMs can achieve the optimal distribution of robot driving force, reduce the demand on the mechanical strength of the robot and the load capacity of the driving unit, effectively avoid or eliminate the singularity, reduce the influence of the PM's joint clearance on the robot motion accuracy, improve the robot motion accuracy, and increase the operability and reliability of the mechanism [28; 29]. Multi-redundantly actuated PMs have become a research hotspot in the field of parallel mechanisms.