The prototype testing method is still in the research stage, and three problems exist. First, the prototype design lacks a uniform standard, such as the prototypes designed by Erkan [9] and Liebrich et al. [10], and the spatial error of the five-axis machine can be obtained by measuring the main ball spacing shown in Fig. 1(b). Keaveney [11] and Kato [12] designed a circular table prototype, and by performing simulated cutting, the accuracy of machine motion can be checked. However, there is no optimal prototype structure that can characterize the accuracy of a 5-axis machine. Second, existing tests are still mainly simulating the tool trajectory and are still performed in the cold and no-cut condition of the machine. Although Ibaraki [13] and Hong [14] performed actual machining of sample parts, they did not consider the effect of the cutting state on the dynamic error pattern of the 5-axis machine. Third, lack of sensitivity of the sample testing process to the machine error sources, which makes it difficult to separate and identify the machine errors. Ibaraki et al. [15] and Mchichi et al. [16] have explored this problem, but have not obtained an ideal identification model.
To test the prototype machining of a 5-axis machine, the stepped-axis prototype was designed as shown in Fig. 2 based on the machine error measurement method in circular trajectory motion.
In order to optimize the selection of the sample machining parameters, the author conducted an orthogonal experimental design. This experiment is a four-factor experiment. According to the main factors affecting the machining error (spindle speed, feed, radial depth of cut and axial depth of cut), L9 (34) table is selected for the orthogonal design, and the material is 45 steel as an example. During the machining of the sample part, the radial depth of cut Ae was selected as 20 mm, 40 mm and 60 mm, and the axial depth of cut Ap was set as 0.5 mm, 1.0 mm and 1.5 mm, and the cutting speed and feed were set with reference to the tool description, and the preliminary experimental scheme is shown in Table 1. After completing a set of experiments with 45 steel samples, experiments were conducted in the same way with Inconel 600 and 6066 aluminum alloys to obtain the optimal processing parameters for different materials.
The integrated error model is the difference between the actual and ideal positions of the tool forming point in the workpiece coordinate system Pw- Pwideal. The integrated error model of a five-axis machine along the X-axis was expressed in Eq. (1).
Table 1
Sample processing experiment plan
serial number
|
Factor A Feed quantity(mm/min)
|
Factor B Spindle speed(r/min)
|
Factor C Radial cutting depth(mm)
|
Factor D axial cutting depth(mm)
|
Error of experimental results(mm)
|
1
|
100
|
1000
|
20
|
0.5
|
0.0095
|
2
|
100
|
1500
|
40
|
1.0
|
0.0087
|
3
|
100
|
2000
|
60
|
1.5
|
0.0093
|
4
|
150
|
1000
|
40
|
1.5
|
0.0085
|
5
|
150
|
1500
|
60
|
0.5
|
0.0098
|
6
|
150
|
2000
|
20
|
1.0
|
0.0086
|
7
|
200
|
1000
|
60
|
1.0
|
0.0092
|
8
|
200
|
1500
|
20
|
1.5
|
0.0082
|
9
|
200
|
2000
|
40
|
0.5
|
0.0081
|
K1
|
0.092
|
0.091
|
0.088
|
0.091
|
|
K2
|
0.090
|
0.089
|
0.084
|
0.088
|
|
K3
|
0.085
|
0.087
|
0.094
|
0.087
|
|
R
|
0.007
|
0.004
|
0.010
|
0.004
|
|
where Δx represents the deviation of the tool forming point along the X-axis in the workpiece coordinate system, x, y, and z represent the position offset of X, Y, and Z axes, respectively, α, γ denote the directional shift of the A and C axes, respectively, the remaining parameters related to C, S, and ε, such as Cae, Sce, and εx(y), all represent different error terms of the five-axis machine.
As can be seen from Eq. (1), after establishing a comprehensive error model, the mathematical relationship between the machining error of the sample and the error term of the machine tool is determined. The x, y, z and α and γ in Eq. (1) are regarded as known, and Δx can be obtained by sample machining error. By inverting Eq. (1), relevant error terms of the five-axis machine tool can be indirectly identified.
In order to obtain the x, y, z and α, γ values required to solve Eq. (1), the moving and rotating axis offsets of the five-axis machine are investigated in this paper. During machining, the X, Y and Z axis offsets are determined according to the spatial position of the forming point, and the number of offset pairs corresponds to the number of measurement points, which is determined according to the unknown quantity of the equation. Different from moving axes, there is no theoretical basis for the biasing of two rotating axes A and C, so the biasing of two rotating axes A and C is mainly designed, and the specific scheme is shown in Fig. 3.
According to the analytical function between the machining error of the sample and the error of the 5-axis machine, different machining trajectories of the sample are determined, and the bias combinations of A and C axes are initially designed according to the different machining trajectories. On this basis, the Jacobi matrix of the error identification model is solved by optimizing the design and simulation processing of the A and C axis biases, and determining whether its condition number is reduced. If the number of conditions decreases, it means that the bias combination is good for machine error identification, otherwise, the bias combination should be deleted and the simulation process of the next bias combination should be optimized. The above process is repeated until the optimal A and C axis bias combination is finally determined.
Since the actual coordinates are measured by the coordinate measuring machine (CMM), the obtained coordinate values are relative to the CMM coordinate system and the Δx in Eq. (1) is not the actual machining error of the sample. In order to obtain the actual coordinate values of the sample in the workpiece coordinate system, the spatial posture model between the coordinate system of the measuring machine and the workpiece coordinate system is established. The actual coordinates of the sample in the workpiece coordinate system are obtained and compared with its theoretical coordinates to obtain the required Δx.