The paper addresses geometric aspects of a spatial three-degree-of-freedom parallel module, which is the parallel module of a hybrid serial-parallel 5-axis machine tool. This parallel module consists of a moving platform that is connected to a fixed base by three non-identical legs. Each leg is made up of one prismatic and two pairs of spherical joint, which are connected in a way that the combined effects of the three legs lead to an over-constrained mechanism with complex motion. This motion is defined as a simultaneous combination of rotation and translation. A method for computing the complete workspace of the VERNE parallel module for various tool lengths is presented. An algorithm describing this method is also introduced.
The workspace calculation of a parallel manipulator is very important for the designer and for the end-user. If we consider a serial robot, the representation of the workspace is generally based on the illustration in 3 dimensions of the space reachable by the center of its wrist (characterizing translations) and by the space reachable by the extremity of the terminal link (characterizing orientations), these two zones being uncoupled. Unfortunately, it is not the case for parallel robots: the zone reachable by the center of the moving platform is dependent on the orientation of its platform. Thus a graphical representation of the workspace of parallel manipulators with more than three degrees of freedom is only possible if we fix parameters representing the exceeded degrees of freedom. As consequence, different types of workspace were used in the literature, according to the choice of the presented parameters [1].
Several methods may be used to calculate the workspace of a parallel manipulator. One can mostly distinguish between discretization methods, geometrical methods, and analytical methods. A simple way for determining the workspace of a parallel manipulator is to use a discretization method. In this method, a grid of nodes with position and orientation is defined.
Then each node is tested to see whether it belongs to the workspace or not [2,3]. The discretization algorithm takes into account all constraints and it is simple to implement but is has some serious drawbacks. It is expensive in computational time and the accuracy depends on the sampling step that is used to create the grid [4]. Geometrical methods are mostly used to determine the boundary of the workspace. The principle is to define geometrical models for the constraints that limit the workspace of the parallel manipulator [5]. These models are obtained Workspace Analysis of the Parallel Module of the VERNE Machine, draft paper proposed to the Journal IFToMM Problems of Applied Mechanics, D. Kanaan, P. Wenger and D. Chablat, November 2006. p2 for each leg separately and the workspace is the intersection between these models [1].
Analytical methods are more difficult to apply because they increase the dimension of the problem by introducing supplementary variables. They consist in solving an optimization problem with penalties at the borders [6]. Parallel kinematic machines (PKM) are commonly claimed to offer several advantages over their serial counterparts [7], such as high structural rigidity, better payload-to-weight ratio, high dynamic capacities and high accuracy [1,8]. Thus, they are prudently considered as promising alternatives for high-speed machining and have gained essential attention of a number of companies and researchers. Since the first prototype presented in 1994 during the IMTS in Chicago by Gidding and Lewis (the VARIAX) [9], many other parallel manipulators have appeared. However, most of the existing PKM still suffer from a limited range of motion [10]. This drawback can be diminished by designing a hybrid manipulator as for the VERNE machine, which is a 5-axis machine-tool built by Fatronik for IRCCyN [11]. This machine-tool consists of a parallel module and a tilting table as shown in Fig. 1. The parallel module moves the spindle mostly in translation while the tilting table is used to rotate the workpiece about two orthogonal axes.
A simplified workspace model of the Verne machine is used currently, but this model is reduced with respect to the real one. The purpose of this paper is to calculate the real workspace to enhance the working capability and to improve the productivity of the VERNE machine. In the following section, we present the VERNE parallel module and we formulate its geometric equations. Section ΙΙΙ is devoted to the calculation of the complete workspace for various tool lengths of the VERNE parallel module. In this section we define geometric models for constraints limiting the workspace. Then we apply a combination of geometric and discretization methods in order to calculate the complete workspace. Finally a conclusion is given in section IV.
Figure 2 shows a scheme of the parallel module of the VERNE machine. The vertices of the moving platform are connected to a fixed-base plate through three legs Ι, ΙΙ and ΙΙΙ. Each leg uses pairs of rods linking a prismatic joint to the moving platform through two pairs of spherical joints. Legs ΙΙ and ΙΙΙ are two identical parallelograms. Leg Ι differs from the other two legs in that 11 12
, that is, it is not an articulated parallelogram. The movement of the moving platform is generated by three sliding actuators along three vertical guideways.
In order to analyze the kinematics of the parallel module, two relative coordinates are assigned as shown in Fig. 2. A static Cartesian frame xyz is fixed at the base of the machine tool, with the z-axis pointing downward along the vertical direction. The mobile Cartesian frame,
x y z , is attached to the mo
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