Performance enhancement of a three-degree-of-freedom parallel tool head via actuation redundancy
Introduction
Parallel kinematic machines (PKMs), if properly designed, can provide considerable advantages over their serial counterparts in terms of high stiffness, high accuracy and low inertia. Due to these virtues, PKMs are suitable candidates for high-speed and high-precision machine tools, especially for those designed for high-hardness materials' machining. The early stage PKM-based machine tools were mainly designed in fully parallel forms [1] and could hardly be used for 5-axis complex surface machining because of problems such as small workspace, small tilting capacity and bundles of configuration singularities in the workspace [2], [3], [4].
To enlarge the tilting capacity and workspace volume of the PKM, many researches have been focused on the hybrid machine tools (HMTs). Many practical 5-DOF hybrid solutions had been proposed, which were mainly realized either by adding a 2-DOF serial spherical wrist to a 3-DOF PKMs such as the Tricept robot [5], [6], Exechon machine tool [7] and Trivariant robot [8], [9], [10], or by combining a 3-DOF PTH to a 2-DOF serial translational platform [11], [12], [13], e.g. the DS Ecospeed machine tool. However, since every coin has two sides, there still exist some drawbacks to the HMTs, e.g. the DS Ecospeed machine tool with Sprint Z3 head only has a maximum tilting angle of ± 40° [1], which is hardly possible to meet the demands in practical machining of aerospace parts such as impeller and blisk. Furthermore, manipulator singularities are still inherent limitations to the manufacturing capabilities of the HMTs. Nevertheless, actuation redundancy is a potential to diminish these problems. In literatures, actuation redundancy had been proved to be an effective way to eliminate singularities and to improve the stiffness of a PKM [14], [15], [16], [17]. In addition, the usable workspace of a HMT can also be increased by the elimination of configuration singularities.
In this paper, based on a 3-DOF non-redundant PTH, a 3-DOF PTH with actuation redundancy is developed by introducing an additional leg with active actuator. The two types of PTHs, i.e. the non-redundant PTH and the redundant one, are fully compared with each other in terms of singularity distribution, workspace size, kinematic performance and stiffness behavior.
The paper is organized as follows: in Section 2, the two types of PTH are introduced and their geometric models are shown. In Section 3, kinematic analysis of the PTH is performed and Jacobian matrix is generated. In Section 4, singularity distributions of the two types of PTH are compared in the entire orientation workspace on the basis of Jacobian analysis. In Section 5, reachable workspaces are generated. The dexterity behavior and the stiffness performance are then analyzed in 6 Dexterity analysis, 7 Stiffness analysis, respectively. Finally, the conclusion to this paper is given.
Section snippets
Architectures of the 3-DOF PTH
As shown in Fig. 1(a), the basic form of the proposed PTH is developed on basis of a 3-PRS parallel mechanism. The basic form of the PTH consists of a swivel table, a fixed frame, three identical prismatic–revolute–spherical (PRS) kinematic legs and a machine tool spindle. The three PRS legs are arranged in 120° intervals on the guide ways and each leg connects the swivel table to the fixed base with a modified spherical (S) joint followed by a revolute (R) joint and a prismatic (P) joint in
Parasitic motions analysis
The proposed PTH, which is geometrically developed from a 3-PRS PKM, always has three parasitic motions: one rotation about z-axis of the fixed frame and two translations along x- and y-axes of the fixed frame O [19]. In fact, those parasitic motions arise synchronously with the other three independent motions, under the mechanical constraints imposed by the R joints.
In Fig. 2, the position vectors of ai (i = 1, 2, 3) and bi (i = 1, 2, 3) with respect to reference frames O and P can be respectively
Singularity types and criterion
In the view of PKM design, singularity is important for singularity-free path planning and for the geometric design of a desired workspace free from singularities. Many researchers have investigated singularity problems of PKM for various purposes [21], [22], [23], [24]. Concerning the basic form of the proposed PTH, singularity analysis is generally based on the instantaneous kinematics, which can be described aswhere denotes the velocities of the three actuations.
For
Workspace analysis
Compared with serial ones, parallel manipulators generally have relatively small workspace, which largely limits the applications of the parallel manipulators. Thus, the workspace of a PKM is one of the most important indexes to evaluate its work capability. In previous studies, the workspace is mainly generated under physical constraints imposed by the cone angle limits of the S joints, motion ranges of the linear actuators and interferences between the kinematic legs, whereas manipulator
Dexterity analysis
Dexterity is a measure of kinematic performance of a parallel manipulator. It depicts the ability to arbitrarily change its position and orientation, or apply force and torques in arbitrary directions while working. Until now, the most commonly used performance indices are all based on Jacobian matrix, because the Jacobian matrix describes the internal relationship of both velocities and forces between the swivel table and the actuators.
In previous literatures, two performance indices,
Stiffness matrix generation
During practical machining, the external forces applied on the swivel table of the PTH will induce a slight deviation from the desired tool path, while the deviation is usually related with the stiffness of the proposed manipulator. For the sake of accuracy, the stiffness is a critical performance for the proposed PTH.
With reference to Eqs. (21), (26), the inverse velocity solution of the PTH with actuation redundancy can be written aswhere indicates the velocity vectors
Conclusions and future works
In this paper, the advantages of actuation redundancy have been comprehensively investigated. A 3-DOF PTH for hybrid machine tool is constructed from a basic form 3-PRS PKM by adding a UPS leg between the fixed frame and the swivel table. Generally, the volume of the workspace of a PKM is largely reduced by physical constraints imposed by S joints. To overcome this problem, a modified spherical joint with a maximum tilting angle of 120° is adopted. Manipulator singularities within the reachable
Acknowledgment
This research is support by National Science and Technology Major Project of the Ministry of Science and Technology of China (2012ZX04010-021).
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