# Parasitic Motions of 3-PRS Parallel Mechanisms with Two Different Branch Chain Arrangements
## 1 Introduction  
In recent decades, PMs have attracted much research focus. Since most practical applications do not require six degrees of freedom (DOFs), research on lower-mobility PMs with a high speed, stiffness, and load-carrying capacity is gradually expanding into many fields [1]. Lower-mobility PMs, whose DOFs are fewer than six, are usually implemented in industrial applications, such as machining [2; 3]. Two rotational DOFs and one translation DOF (2R1T) PMs are simple to design and have low fabrication, actuation, control, and maintenance costs. In industrial fields, 2R1T PMs, as the spindle head of five-DOF computer numerical control (CNC) machine tools, require a high speed and high precision for machining. The sprint Z3 head [4], Tricept hybrid robot [5], and Exechon parallel robot [6; 7] are typical examples of successful 2R1T PMs. As a key component that greatly determines the function and performance of robots or CNC machine tools, such as spindle heads and high-performance reducers (such as RV reducers), 2R1T PMs are an effective solution to related problems. The most famous 2R1T PM may be the 3-RPS PM proposed by [PERSON] [8], where R denotes a revolute pair, P a prismatic pair, and S a spherical joint. The 3-RPS PM or its variant, 3-PRS PM, has attracted much attention, and has broad application prospects and potential in industrial fields [9; 10]. Research was performed regarding kinematics analysis [11; 12], dimensional synthesis [13; 14; 15], singularity [16], and dynamics [17; 18]. [PERSON] et al. [19] proposed the concept of the parasitic motion of the 3-PRS PM, which is unwanted in the mechanism but has a great effect on the application of the mechanism. [PERSON] et al. [20] discussed whether the accompanying motion would affect the dynamic characteristics of the mechanism and suggested that the parasitic motion depends on the used transformation matrix combination. Mathematical methods, such as fractional differential equations, have risen to prominence in scientific and engineering fields [21; 22; 23; 24]. [PERSON] [25] used the Newton homotopy continuation method to analyze the kinematics of a 3-RRS PM, facilitating calculating the accessible workspace. [PERSON] et al. [26] analyzed lightweight 3-DOF flexural prismatic joints based on 2R1T 3-PRS PM. [PERSON] et al. [27] proposed a type of PPS PM to simplify the description of the parasitic motion of 3-PRS PM, satisfying the specific geometric condition that three planes of branch chains intersect at a common line. [PERSON] et al. [28] systematically described the relations between the parasitic motion and the chain arrangements of a 3--PRS PM and synthesized a series of asymmetric PMs without parasitic motion. [PERSON] et al. [29] proposed a developed overconstrained 2 UPR-RPU-RPR PM that improved the bearing capacity of the mechanism. [PERSON] et al. [30] developed an optimal selection that considered both transmission and constraint performance on the basis of the equivalent mechanism of the asymmetric 3-PRS mechanism. [PERSON] et al. [31] changed the design parameters to minimize the parasitic motion through the size ratio of the moving and fixed platform of the reconfigurable 3-RPS parallel robot. [PERSON] [32] used interval analysis to simplify the complex constrained optimization problem for the optimal design of 3--PRS parallel manipulators in order to minimize parasitic motions. [PERSON] et al. [33] deduced the synthesis criteria and geometric conditions of a 1T2R PM without parasitic motion, which were obtained by analyzing the condition of the rotational axes of a PM without parasitic motion on the basis of the spatial geometry theory and exploring the spatial position relationship between the plane of the moving platform and the constraint wrenches using the screw theory. The positions of the constraint wrenches and rotational axes directly caused parasitic motions that were verified by kinematics analysis. On the basis of the inverse kinematics of the 3-PRS PM, [PERSON] et al.