An investigation on mobility and stiffness of a 3-DOF translational parallel manipulator via screw theory
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.
Section snippets
Overview of screw and reciprocal screw systems
In screw theory, a unit (normalized) screw is defined by a pair of vectors:where s is a unit vector directing along the screw axis, r denotes the position vector pointing from an arbitrary point on the screw axis to the origin of the reference frame, the vector defines the moment of the screw axis with respect to the origin of the reference frame, and h represents the pitch of the screw. If the pitch equals to zero, the screw becomes:While in case of infinite pitch, the
Architecture description of the manipulator
The schematic diagram of a 3-PRC TPM is shown in Fig. 1. It consists of a mobile platform, a fixed base, and three limbs with identical kinematic structure. Each limb connects the fixed base to the mobile platform by a P (prismatic) joint, a R (revolute) joint, and a C (cylindrical) joint in sequence, where the P joint is driven by a linear actuator assembled on the fixed base. Thus, the mobile platform is attached to the base by three identical PRC linkages.
To facilitate the analysis, as shown
Mobility determination of a 3-PRC TPM
The mobility determination, i.e., the DOF identification, is the first and foremost issue in designing a parallel manipulator. The general Grübler–Kutzbach criterion is useful in mobility analysis for many parallel manipulators, however it is difficult to directly apply this criterion directly to mobility analysis of some kinds of limited-DOF parallel manipulators. For example, the number of DOF of a 3-PRC TPM given by the general Grübler–Kutzbach criterion is
Stiffness model identification
Concerning a rigid body elastically suspended by elastic devices, if only small displacements from its unpreloaded equilibrium position are considered, the overall spatial force–deflection relation of the mechanism is linear and described by a symmetric positive semidefinite matrix [21], i.e., the stiffness matrix. Generally, the stiffness characteristics of a parallel manipulator can be described by the stiffness matrix, which relates the vector of compliant deformations of the
Stiffness evaluation of the TPM
For a given design of a parallel manipulator, the stiffness varies with the variation of the manipulator configurations within its workspace as well as the direction of the applied wrenches. Once the stiffness model is obtained, it is desired to predict its stiffness characteristics over the workspace in order to assess whether the design is satisfied with the stiffness requirements or even further to perform an optimal design with the stiffness considered especially in the design stage.
As far
Conclusions
In this paper, the mobility determination and stiffness analysis for a 3-DOF TPM are carried out based upon the screw theory. A 3-CRC TPM is constructed from a previously presented overconstrained 3-PRC TPM by adding a revolute joint in each limb. It has been demonstrated that the conversion does not bring any impact on the mobility and kinematics of original manipulator since the added joints are idle indeed, but it can eliminate the overconstrained conditions of a 3-PRC TPM. The stiffness
Acknowledgments
The authors appreciate the fund support from the research committee of University of Macau under Grant No. RG068/05-06S/07R/LYM/FST and Macao Science and Technology Development Fund under Grant No. 069/2005/A.
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