2.1 Optimization objective of OMM efficiency

The efficient measurement process of complex curved surfaces is given below (see Figure 2):

Figure 2全屏查看

The efficient measurement process of complex curved surfaces

Step 1: The probe ruby ball center is moved to the safe point located on the safe Z-plane. The position of the workpiece within the machine is also determined by the rotary axes’ motion.

Step 2: The probe moves backward along the Z-axis of the probe coordinate system, causing the ruby ball center to move toward the anchor point .

Step 3: The probe tries to touch the to-be-measured point along the opposite direction of the normal vector .

Step 4: The probe returns to the anchor point after contacting the workpiece.

Step 5: The probe is moved within the impeller channel to the next anchor point . Subsequently, Steps 3 and 4 are repeated.

Step 6: After measuring all to-be-measured point , the probe moves along the stylus orientation to the safe point .

All the to-be-measured points on the complex curved surfaces in the workpiece coordinate system are denoted as . The directions of normal vector and stylus orientation are respectively represented by and . The terms and are used to indicate the anchor points and safe points, respectively.

Unlike the traditional measurement path, this path does not require the probe to lift out of the impeller channel during the measurement process, leading to a significantly shorter measurement path. To prevent interference between the probe and the impeller during movement between anchor points, it is necessary to add path points [28], denoted by .

In the measurement process, the position and attitude of the workpiece coordinate system in space are determined once the rotary axes angle combinations and for are given. When all the above rotary axes angle combinations are determined, the measurement path is determined in the machine tool coordinate system . Based on the kinematic chain of the machine tool, the transformation matrix from to is established as , and the transformation matrix from the stylus coordinate system to is established as .

The position and the direction in can be respectively calculated by

(1)

(2)

Similarly, the position in can be calculated by

(3)

The position in is

(4)

where the represents the detection distance and the represents the radius of the ruby ball.

The direction of stylus orientation in is calculated by

(5)

The position in is

(6)

According to the OMM process, the distance from the anchor point to the to-be-measured point is specified. Therefore, for a more efficient measurement, other motion paths should be made the shortest. This path is defined as a plannable path, and its length is

(7)

2.2 Constraints of optimization model

Constraint 1: The touch position must be chosen inside the common calibration position region.

First of all, it should be ensured that there is no interference between the workpiece and the probe when measuring the to-be-measured points. Many angle combinations are discretized over the entire angular ranges of the rotary axes. Collision detection based on the bounding box method is used to determine the range formed by the angle combinations that do not produce interference [29]. This range is defined as the region of interference-free angle combinations (see Figure 3).

Figure 3全屏查看

Regions of interference-free angle combinations

The touch position on the ruby sphere can be represented by in the spherical coordinate system at the center of the ruby sphere. The direction in the stylus coordinate system can be calculated by

(8)

Then the touch position can be obtained by

(9)

(10)

where the variables , , are the components of in the X, Y, and Z directions, respectively.

Combining Eqs. (8) to (10), the region of interference-free angle combinations can be transformed into the interference-free region of the touch position (see Figure 4). If there are several overlapping regions , the is defined as the set of serial numbers of in relation to . The overlapping regions are given a higher serial number when there are more relevant interference-free regions. If the numbers are equal, the larger overlapping region receives a higher serial number. Figure 4 shows the two overlapping regions with the greatest number of relevant interference-free regions. If the touch position is selected within the overlapping region, the same pre-travel error compensation value can be applied to all to-be-measured points connected to the overlapping region.

Figure 4全屏查看

Interference-free regions of the touch positions and their overlapping regions

To keep the number of calibration positions to a minimum, it is necessary to use the same touch position for as many to-be-measured points as possible. This is a typical set covering problem (SCP). First, the set containing all the serial numbers of to-be-measured points and the set consisting of several sets of elements within are created. By the greedy algorithm [30], the subset of that can cover and have the least number of elements is obtained. The touch positions of the to-be-measured points associated with in the set W need to be selected within the corresponding overlapping region .

Constraint 2: The change in stylus orientation should be minimized over the entire measurement path.

The stylus orientation for each to-be-measured point can be mapped to a point on the unit sphere centered on the to-be-measured point (see Figure 5(a)). The touch positions in the overlapping areas do not have a one-to-one correspondence with the stylus orientations. Without accounting for interference, the stylus orientations correspond to a spherical circle on the unit sphere as described above. To determine the feasible stylus orientations corresponding to the given touch position, it is necessary to establish a range of stylus orientations on the unit sphere where no interference occurs at each to-be-measured point. The designated range is termed the feasible direction cone (FDC) of the to-be-measured point.

Figure 5全屏查看

(a) The stylus orientation on the unit sphere centered on the to-be-measured point; (b) The feasible stylus orientations corresponding to the touch positions

The direction of stylus orientation in the workpiece coordinate system is calculated by

(11)

According to Eqs. (5) and (11), the interference-free angle combinations can be converted into FDC on the unit sphere. The feasible stylus orientations corresponding to the touch positions are represented in Figure 5(b) by the intersection of the accessible cones and the spherical circles. In the figure, various to-be-measured points are differentiated by distinct colors. In the five-axis OMM, the swing of the rotary axes will alter dramatically due to the unequal variety of stylus orientations, which makes it simple to create potential interference issues and introduce additional effects of machine tool kinematics error [31, 32]. To prevent the fluctuation in stylus orientations, the minimizing of the tool’s total amount of axis rotation is proposed as a constraint. Here, the accumulated angle minimization algorithm [33] commonly used in tool axis optimization is used to determine the unique probe axis direction, which minimizes the amount of machine tool rotary axes angle variation in the entire path.

Constraint 3: The local measurement paths between neighboring anchor points are the shortest and interference-free.

To improve the measurement efficiency, the movement between the neighboring anchor points is always in the impeller channel, imposing stringent demands on the positioning of path points. Two key requirements govern the addition of path points.

First, it should be ensured that there is no interference between the probe and the workpiece during the whole movement process. Second, the path point should be as close as possible to the line connecting the neighboring anchor points to ensure the shortest measurement path. The local measurement path planning method based on dynamic searching volume (DSV) is used to add the path point [34]. The path point is at a distance from the midpoint of and along the summation direction of and , which can be calculated by

(12)

takes the minimum value which satisfies that the line does not interfere with the workpiece. Afterward, based on the DSV, the combination of angles can be determined to ensure that the local measurement path between neighboring anchor points is interference-free.

2.3 Optimal efficiency measurement solution

The optimization objective and constraints for the optimization of OMM efficiency are given in the previous sections. Therefore, a single-objective optimization problem with constraints can be constructed. Based on the above constraints, it is possible to identify the stylus orientation while determining the touch point and the combination of measurement angles synchronously. This allows for the identification of the measurement path length. The touch positions are therefore employed as optimization variables. The best set of touch positions is determined by using Particle Swarm Optimization (PSO) to solve this issue [35]. Additionally, the optimal efficiency angle combinations that satisfy the constraints and have the shortest plannable path can also be found.

3.1 Compensation method through workpiece coordinate system adjustment

Due to the influence of positioning error, the position and attitude of the workpiece in space will deviate from the theoretical position under the given conditions of rotary axes’ angle combination . When the rotary axes move to the angle combination , the theoretical workpiece coordinate system will be shifted to the practical workpiece coordinate system due to the influence of the rotary axes’ positioning errors. In order to minimize errors in the final measurement position of the to-be-measured point and direction of the normal vector, the workpiece coordinate system in the machine tool needs to be compensated to make it consistent with .

The homogeneous transformation matrix from to can be represented by (, are the position errors of the B and C axes, respectively). Then the homogeneous transformation matrix from to can be computed by

(13)

The compensation of the workpiece coordinate system is divided into two steps.

Firstly, the origin position of the workpiece coordinate system is compensated (see Figure 6(a)). The compensation parameters of the origin position in are defined as , which can be computed by

(14)

Figure 6全屏查看

(a) Workpiece coordinate system origin compensation method; (b) The Tait-Bryan angles for workpiece coordinate system attitude compensation method

Secondly, the attitude of the workpiece coordinate system is compensated, that is, the Tait-Bryan angles from to are calculated. The rotation order can be specified artificially. For uniformity, the rotation order is specified as X-Y-Z, and the parameters are defined as (see Figure 6(b)). The directions of the X-axis and Z-axis of in are defined as and , which can be calculated as follows:

(15)

(16)

Since the rotation of the coordinate system is completed in the CNC of machine tools, the rotation angle parameters are not constrained, so it is only necessary to give any set of Tait-Bryan angles solutions. The calculations are as follows.

According to the specified rotation order, the rotation angle about the X-axis is calculated first by

(17)

After this rotation, the direction of the Y-axis and Z-axis are respectively defined as and , which can be computed by

(18)

(19)

where represents the rotation transformation matrix around the X-axis.

Then the rotation angle about the Y-axis is calculated by

(20)

where .

After the first two rotations, the direction of the X-axis is defined as , which can be computed by

(21)

where represents the rotation transformation matrix around the Y-axis.

The final rotation angle about the Z-axis is calculated by

(22)

where .

3.2 Compensation method of measurement result

Through the compensation method of workpiece coordinate system position and attitude, the influence of positioning error on the workpiece position and attitude can be eliminated. After the position and attitude of the workpiece are determined, the relative motion between the probe and the workpiece is only controlled by three linear axes. In this process, linear axes positioning errors will be introduced.

After the compensation method of coordinate system position and attitude, the practical position in is

(23)

The practical direction of the normal vector in is

(24)

When the probe touches the to-be-measured point, due to the positioning error, the position recorded by the CNC system will be shifted in the practical position in the direction of the normal vector. The offset can be calculated by

(25)

Considering that the OMM results recorded by the CNC system are coordinates in , the OMM result compensation vector is obtained by

(26)

3.3 Compensation package within machine tools

To ensure the promptness of the compensation method, the concept of compensation package within the machine tools is proposed. This refers to the inclusion of CNC subroutines within the machine tool that compensate for the attitude, position, and measurement result of the workpiece coordinate system.

According to the optimization method of OMM efficiency proposed in Section 2, the optimal set of angle combinations for planning the measurement path can be obtained. For each angle combination, according to the aforementioned compensation method in this section, we can calculate the compensation parameters of the origin position , the Tait-Bryan angles for attitude compensation and the OMM result compensation vector under each angle combination in the optimal set. When measuring each to-be-measured point, the compensation package is called in the CNC program to integrate the compensation operation with the measurement process.