Geometrical distribution of rotational axes of 3-[P][S] parallel mechanisms

Highlights

• Classification of 3-[P][S] parallel mechanisms is presented.

• Rotational axes of 3-[P][S] parallel mechanisms are identified.

• Some new architectures of 3-[P][S] parallel mechanisms are disclosed.

Abstract

A 3-[P][S] parallel mechanism consists of three limbs and each limb can generate a planar-spherical ([P][S]) kinematic bond. Typically, the 3-[P][S] parallel mechanism family includes four 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. The 3-[P][S] parallel mechanism has received extensive attention due to its practical potential. But little is known about the geometrical distribution of the axes of the two rotational DOF (degrees of freedom) of the 3-[P][S] parallel mechanism. Consequently, although the kinematic derivations of the 3-[P][S] parallel mechanism are correct, the interpretation of the actual instantaneous rotation is not clear. This fact may hinder its application. This paper concentrates on the identification of the rotational axes of the 3-[P][S] parallel mechanism with different limb arrangements. First, the geometrical condition for the axis of a feasible rotation of a rigid body constrained by a force is discussed using screw theory. Then, the 3-[P][S] PMs are classified into four categories and seven subcategories based on the geometrical condition of their LPs (limb planes) and spherical joint centers, The instantaneous and finite rotational axes of the seven subcategories of 3-[P][S] parallel mechanism are identified using reciprocal screw theory. The results apply to all 3-[P][S] PMs.

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.