Kinematic Analysis of a Serial - Parallel Machine Tool: the VERNE machine

Abstract
The paper derives the inverse and the forward kinematic equations of a serial – parallel 5-axis machine tool: the VERNE machine. This machine is composed of a three-degree-of-freedom (DOF) parallel module and a two-DOF serial tilting table. The parallel module consists of a moving platform that is connected to a fixed base by three non-identical legs. These legs 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. In this paper we propose symbolical methods that able to calculate all kinematic solutions and identify the acceptable one by adding analytical constraint on the disposition of legs of the parallel module. Keywords: Parallel kinematic machines; Machine tool; Complex motion; Inverse kinematics; Forward kinematics.

1. Introduction Parallel kinematic machines (PKM) are well known for their high structural rigidity, better payload-to-weight ratio, high dynamic performances and high accuracy [1, 2, 3]. Thus, they are prudently considered as attractive alternatives designs for demanding tasks such as high-speed machining [4]. Most of the existing PKM can be classified into two main families. The PKM of the first family have fixed foot points and variable–length struts, while the PKM of the second family have fixed length struts with moveable foot points gliding on fixed linear joints [5, 6]. In the first family, we distinguish between PKM with six degrees of freedom generally called Hexapods and PKM with three degrees of freedom called Tripods [7, 8]. Hexapods have a Stewart–Gough parallel kinematic architecture. Many prototypes and commercial hexapod PKM already exist, including the VARIAX (Gidding and Lewis), the TORNADO 2000 (Hexel). We can also find hybrid architectures such as the TRICEPT machine (SMT Tricept) [9], which is composed of a two-axis wrist mounted in series to a 3-DOF “tripod” positioning structure. In the second family, we find the HEXAGLIDE (ETH Zürich) that features six parallel and coplanar linear joints. The HexaM (Toyoda) is another example with three pairs of adjacent linear joints lying on a vertical cone [10]. A hybrid parallel/kinematic PKM with three inclined linear joints and a two-axis wrist is the GEORGE V (IFW Uni Hanover). Many three-axis translational PKMs belong to this second family and use architecture close to the linear Delta robot originally designed by Clavel for pick-and-place operations [11]. The Urane SX (Renault Automation) and the QUICKSTEP (Krause and Mauser) have three non-coplanar horizontal linear joints [12]. Because many industrial tasks require less than six degrees of freedom, several lower-DOF PKMs have been developed [13-15]. For some of these PKMs, the reduction of the number of DOFs can result in coupled motions of the mobile platform. This is the case, for example, in the RPS manipulator [13] and in the parallel module of the Verne machine. The kinematic modeling of these PKMs must be done case by case according to their structure. Many researchers have contributed to the study of the kinematics of lower-DOF PKMs. Many of them have focused on

2 the discussion of both analytical and numerical methods [16, 17]. This paper investigates the inverse and direct kinematics of the VERNE machine and derives closed form solutions. The VERNE machine is a 5-axis machine-tool that was designed by Fatronik for IRCCyN [18, 19]. 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. The purpose of this paper is to formulate analytic expressions in order to find all possible solutions for the inverse and forward kinematics problem of the VERNE machine. Then we identify and sort these solutions in order to find the one that satisfies the end-user.

Figure 1: Overall view of the VERNE machine The following section describes the VERNE machine. In section 3, we study the kinematics of the parallel module of the VERNE machine. In section 4 the methods presented in section 3 are extended to study the kinematic of the full VERNE machine. Finally Section 5 concludes this paper. 2. Description of the VERNE machine The VERNE machine consists of a parallel module and a tilting table as shown in Fig. 2. The vertices of the moving platform of the parallel module are connected to a fixed-base plate through three legs Ι, ΙΙ and ΙΙΙ. Each leg uses a pair of rods linking a prism

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