An investigation on mobility and stiffness of a 3-DOF translational parallel manipulator via screw theory

Abstract

This paper analyzes the mobility and stiffness of a three-prismatic-revolute-cylindrical (3-PRC) translational parallel manipulator (TPM). Firstly, the original 3-PRC TPM is converted into a non-overconstrained manipulator since there exist some practical problems for the overconstrained mechanism. By resorting to the screw theory, it is demonstrated that the conversion brings no influences to the mobility and kinematics of the manipulator. Secondly, the stiffness matrix is derived intuitively via an alternative approach based upon screw theory with the consideration of actuations and constraints, and the compliances subject to both actuators and legs are taken into account to establish the stiffness model. Furthermore, the stiffness performance of the manipulator is evaluated by utilizing the extremum stiffness values over the usable workspace, and the influences of design parameters on stiffness properties are presented, which will be helpful for the architecture design of the TPM.

Introduction

A parallel manipulator typically consists of a mobile platform that is connected to a fixed base by several limbs or legs in parallel. Generally, if carefully designed, parallel manipulators can provide several attractive advantages over their serial counterparts in terms of high stiffness, high accuracy, and low inertia [1], which enable them become challenging alternatives for wide applications such as in assembly lines, flight simulators, machine tools, ultra-precision instruments, medical devices, and so on. Recently, the progress on the development of parallel manipulators with less than six degree-of-freedom (DOF) has been accelerated because these limited-DOF parallel manipulators own several other advantages including the total cost reduction in manufacturing and operations in addition to the inherent merits of parallel mechanisms. As a result, limited-DOF parallel manipulators have been investigated and applied more and more extensively.

Among the limited-DOF manipulators, those translational parallel manipulators (TPM) possessing three spatial pure translational DOF have drawn particular interests from numerous researchers, since they satisfy the requirements of many specific applications. Various TPM architectures have been proposed in the literatures [2], [3], and the type syntheses of a 3-DOF TPM have been carried out by investigations based on the screw theory [4], [5], group theoretic approach [6], [7], and several other methods [8], [9], [10], [11].

In our previous research [12], a 3-PRC (three-prismatic-revolute-cylindrical) TPM with three P joints intersecting at a common point was presented and designed so as to eliminate all the singularities from the workspace. It should be noted that this 3-PRC TPM possessed the same mechanism with the one that was proposed in [13] with three P joints parallel to one another, and the used PRC linkage was equivalent to the PŘŘP (letters Ř denote the revolute joints with parallel axes) translational parallel kinematic chain that was enumerated by the type synthesis in [5].

As an overconstrained mechanism, the 3-PRC TPM has a very simple structure. However, in practice, the problems of variable frictions in passive joints and large reaction moment have to be considered so as to assure the mobility of the mobile platform for a 3-PRC TPM. Otherwise, the mobile platform may not move or the manipulator cannot work if there are some kinematic errors. Unfortunately, due to the unavoidable errors arising from the manufacturing tolerance and imperfect assembly, there will always exist kinematic errors. Considering this point, the original 3-PRC TPM will be converted into a non-overconstrained TPM in this paper in order to solve the movability problem by resorting to the screw theory [14], [15], [16], [17].

Additionally, since high stiffness is one of the advantages of parallel manipulators, it is necessary to investigate the stiffness issue of the 3-PRC TPM in more detail. Actually, the stiffness of a manipulator has direct impact on its position accuracy. Hence, in the early design stage, it is desired to perform the stiffness modeling and evaluation of a parallel manipulator for the precise manipulation purpose. The second objective of this paper is to accomplish the stiffness analysis of the 3-PRC TPM since there are no efforts made toward the stiffness characterization of this type of manipulator yet.

In the remainder of this paper, after a brief review of the screw and reciprocal screw systems in Section 2, and a short description of the 3-PRC TPM in Section 3, the mobility of the manipulator is analyzed in Section 4, where the overconstrained conditions of the mechanism are eliminated without any influences on its mobility and kinematics. Then in Section 5, the stiffness model is identified with the consideration of compliances subject to both actuators and legs, and the overall stiffness matrix is established. Afterwards, the stiffness assessment is carried out in Section 6 along with the derivation for the influences of design parameters on stiffness characteristics. Finally, some concluding remarks are summarized in Section 7.