Geometrical distribution of rotational axes of 3-[P][S] parallel mechanisms
Introduction
A lower-mobility PM (parallel mechanism) has less than six DOF (degrees of freedom). The 3-DOF PM with one translational DOF and two rotational DOF is an important category of the lower-mobility PM. Such a kind of PM can be constructed by using three legs generating a planar-spherical kinematic bond [1]. Hence, such a kind of PM can be called 3-[P][S] PM. The 3-[P][S] family includes four typical types of architectures, namely, 3-RPS, 3-PRS, 3-RRS and 3-PPS, where R denotes a revolute pair, P a prismatic pair, and S a spherical joint.
Since Hunt [2] proposed the 3-RPS PM in 1983, the 3-[P][S] PM has been used in many fields. For example, Z3 head in machine tool [3], telescope application [4], motion simulator [5], micro-manipulator [6], and coordinate measuring machine [7].
The kinematics and dynamics of the 3-[P][S] PM have been extensively studied. Lee and Shah [8] presented the inverse and forward kinematic analysis of a 3-RPS PM and studied the effects of the physical constraints of ball joints on the range of motion. Tsai and his colleagues [9] presented the direct kinematic solutions for a 3-PRS PM. Li and Xu [10] analyzed the kinematics, workspace and dexterity of a 3-PRS PM with adjustable layout angle of actuators. Joshi and Tsai [11] found two singular configurations of the 3-RPS PM by screw theory. Liu and Cheng [12] obtained the direct singular positions of a 3-RPS PM. Lee and Shah [13] analyzed the dynamics of a 3-RPS PM using the Lagrangian approach. Farhat and his colleagues [14] addressed the identification of dynamic parameters of a 3-RPS PM.
Another topic receiving much interest is the dimensional synthesis of the 3-[P][S] PM. Tsai and Kim [15] discussed how to determine the design parameters of a 3-RPS PM to satisfy six prescribed poses of the moving platform. Rao and Rao [16] modified Kim and Tsai's method to synthesize the 3-RPS PM for any number of prescribed poses of the moving platform. They [17] further studied the dimensional synthesis of a 3-RPS PM for a prescribed range of motion of spherical joints. Liu and Bonev [18] presented the orientation capability, error analysis and dimensional optimizations of a 3-PPS and a 3-PRS PM. Pond and Carretero [19] optimized the dexterous workspace volume of three variants of 3-PRS PM by using a homogeneous Jacobian matrix. Carretero and his colleagues [20] optimized a 3-PRS PM to minimize its parasitic motion.
There are two kinds of interpretations of the two rotational DOFs of the 3-[P][S] PM, more specifically, two rotational axes. One widely-accepted statement is that the moving platform can rotate about the X and Y axes of the fixed coordinate frame [10], [20]. Although such an interpretation is correct, it introduces a third rotation and two translations that are dependent on the two rotations about the X and Y axes. These dependent motions are also called parasitic motions [20]. The parasitic motion in a 3-[P][S] PM may cause problems in analysis and control and needs to be reduced [20].
The other kind of interpretation is based on the analysis of instantaneous rotation axis. This interpretation directly illustrates the rotational axes of the 3-[P][S] PM. As Huang, Tao and Fang pointed out in 1996 [21], any lines in the plane determined by the three spherical joint centers in a 3-RPS PM can be an instantaneous rotational axis. Further, the line connecting any two spherical joint centers can be a finite rotational axis. Later, Huang, Wang and Fang [22] identified the principal screws of the 3-RPS PM by means of the conic section degeneration theory and presented the spatial distribution of axes of the twists of the moving platform.
Nevertheless, the research on the rotational axes of the 3-[P][S] PM is far from being adequate. First, the identification of instantaneous rotational axes of the 3-[P][S] PM with different limb arrangements has not been investigated systematically. Such an identification is necessary to know the rotation capabilities of the 3-[P][S] PM with different limb arrangements. Second, relations between the geometrical distribution of the rotational axes and kinematic performance are not disclosed. This paper attempts to solve the first problem by identifying the rotational axes of the 3-[P][S] PM with different limb arrangements.
The organization of this paper is as follows. Section 2 presents a brief introduction of mobility analysis based on screw theory. Section 3 shows that the limb constraint generated by a PRS, RPS, RRS, and PPS limb is the same. Section 4 investigates the geometrical distribution of the rotational axes of 3-PRS PMs. Finally, conclusions are given in Section 5.
Section snippets
Modified Grübler–Kutzbach criterion
Consider a PM with M DOF (M < 6). The PM comprises p limbs, each exerting qi structural constraints on the moving platform. The constraints form the mechanism constraint system, which must be a 6-M system in the general configuration.
The modified Grübler–Kutzbach criterion [23] is given bywhere M denotes the mobility of the mechanism, d the order of the mechanism, n the number of links, g the number of kinematic pairs, fi the freedom of the ith pair and v the number of
Limb constraint of a 3-[P][S] PM
Using reciprocal screw theory, Huang, Tao and Fang [21] found that each RPS limb exerted a constraint force on the moving platform. The mobility of the 3-RPS PM was determined by the combined effect of the three limb constraint forces. The constraining effect on the rotation caused by forces increases the complexity of mobility analysis of the 3-[P][S] PM.
Fig. 2 shows PRS, RPS, RRS and PPS limbs, all of which generate a planar–spherical kinematic bond. Note that the center of the spherical
Rotational axes of 3-[P][S] PMs with different limb arrangements
Without losing generality, we focus on 3-[P][S] PMs with three LPs being parallel to a line. Then, 3-[P][S] PMs can be classified into the following categories based on the relative positions of the three LPs and three spherical centers. Note that a more comprehensive classification of PMs with 5-DOF limbs including 3-[P][S] PMs can be found in [1].
Because the limb constraint generated by PRS, RPS, RRS, and PPS limbs are the same, we investigate the rotational axes of the 3-PRS PM while
Conclusions
This paper systematically investigates the rotational axes of 3-[P][S] PMs with different limb arrangements. Based on the geometrical condition of LPs and spherical joint centers, the 3-PRS PMs are classified into four categories and seven subcategories. Reciprocal screw theory is applied to identify of the rotational axes of the 3-PRS PM. It is indicated that the existence of a finite rotational axis relies on the geometrical condition of LPs and spherical joint centers. The results also help
Acknowledgments
This work was supported by the National Natural Science Foundation of China (NSFC) under grant 51075369, 51135008 and by Natural Science Foundation of Zhejiang Province under grant R1090134. The authors also would like to thank the reviewers for their pertinent and helpful comments.
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