A novel geometric error identification methodology for the tilting head of five-axis machine tools based on double ball bar

The paper proposes a novel geometric error identification methodology for the tilting head of five-axis machine tools using double ball bar(DBB).Firstly, based on the motion condition of the tilting head of five-axis machine tools, three measurement patterns in Y direction, X direction and Z direction are proposed respectively. Then, the relative displacement equations of two balls of DBB in three measurement patterns are established respectively on the basis of homogeneous transform matrix (HTM) and multi-body system (MBS) theory. Finally, the geometric error parameters of the tilting head are identified totally. The presented method is universal and provides a reference for the error identification for the tilting head of five-axis machine tools.


Introduction
With a rapid development of precision machining for the complex parts, five-axis machine tools (FAMTs) are preferred with the advantages of higher material removal rate, better the ability of positioning and orienting the tool with respect to the workpiece and lower production cost. However, the existing FAMTs cannot provide the same machining accuracy as their three-axis counterparts due to the additional rotary axes leading more geometric error sources, thus hindering the development and practical implementation of five-axis precision machining.
Over the past few decades, many researchers have investigated some efficient calibration approaches to identify the geometric error components of FAMTs based on double ball bar (DBB), which has been proved to be competent for rotary axes calibration. Zargarbashi et al. [1] presented a method to measure the position-dependent geometric errors of a trunnion-type A-axis using a DBB. Lee et al. [2] measured position-independent geometric errors of a rotary table axis(C-axis) and a trunnion axis (A-axis) of a five-axis machine tool using a DBB. Zhu et al. [3] proposed a new identification approach to recognize geometric errors of an A-tilting table by using a DBB. Zhang et al. [4] developed a novel measuring method to identify five geometric errors caused by the rotary table (C-axis) with a DBB. Lasemi et al. [5] provided a theoretical method to identify geometric errors of a rotary axis and C rotary axis utilizing a DBB. Chen et al. [6] proposed a comprehensive geometric error identification method for a tilt table using a DBB.
As can be observed in the above-mentioned researches, the geometric error identification methods for rotary axes of FAMTs are rarely aimed at the tilting head (B-axis). Therefore, an adequate geometric errors identification methodology for the tilting head (B-axis) of FAMTs should be developed.

The measuring principle based on the DBB
As shown in Figure 1, Every DBB measurement system has two sockets. The one clamped by the tool holder on the spindle is denoted as Socket 1, and the other set on the worktable is denoted as Socket 2. Two ball of the DBB, which is denoted as Ball 1 and Ball 2 respectively, are installed on the ball bowls of the sockets. The measurement of the rotary axes is usually realized by detecting position changes of two balls of the DBB. According to Ref. [3], kinematic chain and homogeneous transform matrix (HTM) have been widely used as a general modelling method. In addition, Fan et al. [7] proposed a generalized kinematics modelling method, regarding a machine tool as multi-body system (MBS) including many rigid bodies. These theories can describe the motion relationship between adjacent bodies simply and conveniently (see Yang et al. [8] and Wu et al. [9]). Meanwhile, the position coordinates of two balls of the DBB can be transformed to the same coordinate system through kinematic chain respectively. Therefore, the mapping relationship between the theoretical model and actual measurement values can be established by calculating the distance between Ball 1 and Ball 2.  Fig. 4 The motion diagram of measurement pattern of X direction for B axis As shown in Figure 2, the position array of P in the Bj body coordinate system is represented , , l e l e j j j j p p s s describe the relative position transformation matrix, the relative position error transformation matrix, the relative motion transformation matrix, the relative motion error transformation matrix between the Bi and the adjacent lower body Bj respectively.
The motion diagram of measurement pattern of Z direction for B axis

The Identification Principle of Geometric Error Parameters
Based on MBS theory, the tilting head (B-axis) has eight geometric errors, composed of six positiondependent geometric errors ( ( ), ( ), ( ), , and two position-independent geometric errors ( , xB Bz ).
In the measuring process of the DBB, B axis is moving by multi-axes synchronous linkage to drive the spindle rotating around the centre point of Ball 1, whose position is required to keep unchanged.  Fig.6 The position relation diagram of three measurement patterns for B-axis.

Measurement Pattern in Y Direction
As shown in Figure 2 and 6(a), by utilizing (1)   Then, Equation (4)

Measurement Pattern In X Direction
As shown in Figure 3 (10) Where x d indicates the original length of the ball bar in the measurement pattern of X direction.
By simplifying Eq. (10) and ignoring the higher order term, the distance between A' and O in coordinate system Where x d  indicates the length variation of the ball bar from the initial position to the j-th position in the measurement pattern of X direction.
Then, Equation (13) is obtained by simplifying (12) and ignoring the higher order term. cos ( ) cos ( ) sin ( ) Assuming that then Equation (14) is represented as follows, cos ( ) With two different values of L, the following equation is obtained. 1

Measurement Pattern in Z Direction
As shown in Figure 4 and 6(c), by utilizing (1)  So far, according to the above measured results in three different patterns in the X, Y, and Z direction, the five position-dependent geometric error parameters ( ()

Conclusions
Based on the motion condition of the tilting head of five-axis machine tools, three measurement patterns in Y direction, X direction and Z direction are proposed respectively using DBB . Then, the relative displacement equations between two balls of DBB in measurement patterns in Y direction, X direction and Z direction are established respectively on the basis of HTM and MBS theory. Finally, the geometric error parameters of the tilting head (B-axis) are identified totally. The presented method in this paper is universal and provides a reference for the error identification for the tilting head of five-axis machine tools.