CNC lathes are capable of performing a wide range of cutting operations. However, the difference in tool positions between the previous and newly selected tools during tool changes can lead to inconsistencies. Additionally, errors may arise from improper tool installation, tool wear, or variations in the tool tip’s arc radius. Without using the tool compensation function, it would be impossible to produce parts that match the required shape. Moreover, utilizing tool compensation not only improves accuracy but also simplifies programming. Tool compensation on CNC lathes is generally categorized into two types: tool position compensation and tool radius compensation.
**1. Tool Position Compensation**
During machining, when multiple tools are used, the center of the tool holder is typically taken as the programming origin. This means that the program starts from the center of the tool holder, as illustrated in Figure 1. The actual movement path of the tool is controlled by the tool position compensation value. As shown in Figure 1(a), tool position compensation includes both geometric compensation values and wear compensation values.
Figure 1: Tool Position Compensation
There are two methods for implementing tool position compensation. One method involves setting the geometric and wear compensation values into a storage unit, following a specific format. Another method combines both geometric and wear offset values, as shown in Figure 1(b). The total compensation value storage unit serves two purposes: selecting the appropriate compensation value based on the tool number and canceling the position compensation when the memory unit number is 00, such as T0100, which clears the current tool's compensation value.
Figure 2 illustrates how position compensation works. The solid line represents the programmed trajectory of point A, while the dashed line shows the actual path of point A during compensation. The direction of the actual trajectory depends on the X and Z-axis compensation values. The corresponding program is as follows:
N010 G00 X10 Z-10 T0202;
N020 G01 Z-30;
N030 X20 Z-40 T0200;
Figure 2: Tool Position Compensation
The structure of the CNC lathe tool is shown in Figure 3. In this figure, P represents the imaginary tip, S is the arc center of the cutter head, r is the radius of the cutter head, and A is the reference point of the tool holder.
Figure 3: Turning Tool Structure
The control point of the lathe is the center of the tool holder, so tool position compensation is always necessary. It is used to convert between the tool nose circular arc center track and the tool holder reference point. This corresponds to the transition between points A and S in Figure 3. However, we cannot directly measure these two centers. Instead, we measure the distance between the imaginary tool tip P and the tool holder reference point A.
To simplify calculations, it is often assumed that the cutter head radius r = 0, and the coordinates of the imaginary tool tip P relative to the tool holder reference point A are measured and stored in the tool parameter table.
In the formula:
P —— Coordinates of the imaginary tip;
(X, Z) —— Coordinates of the tool holder reference point A.
It is straightforward to write the tool position compensation calculation formula:
After compensating the part contour trajectory using Formula (2), the tool holder reference point A can be controlled to achieve the desired result. For cases where r ≠0, the tool position compensation must also consider the tool installation method.
**2. Tool Radius Compensation**
When programming, the tool tip is usually treated as a point. However, in reality, the tool tip has a circular arc. While this does not affect the machining size or shape when cutting internal holes, external surfaces, or end faces, it becomes significant when cutting cones or arcs. The tool’s actual path may differ from the programmed path, leading to errors related to the arc radius r. Figure 4 shows the trajectory with and without radius compensation. When using the imaginary tool tip P for programming, the tool center arc trajectory appears as a two-dash line in Figure 4. The actual tool path and the workpiece contour require error correction, with the error size depending on the arc radius r. If the tool center is programmed with radius compensation, the arc center trajectory becomes a thin solid line, matching the required workpiece contour.
Figure 4: Trajectory with and without Radius Compensation
Because turning tools have complex geometry and installation, several aspects are further elaborated below.
**2.1 Tool Nose Orientation**
The orientation of the tool tip P determines its relationship with the arc center. This affects the calculation of tool compensation for circular tools. Figure 5 shows the imaginary nose orientation and code for an arc tool. As seen, there are eight possible azimuths for the tool tip P, represented by codes 1–8. Code 0 or 9 is used when the tool tip is at the arc center, indicating no arc compensation.
Figure 5: Arc Tool Imaginary Nose Orientation and Code
**2.2 Arc Radius Compensation and Position Compensation**
If the tool center A is used as the programming origin and arc radius compensation is not considered, the X and Z axis compensation values are determined as shown in Figure 1(b). In such cases, both position compensation and arc radius compensation must be considered. The X and Z axis position compensation values can be determined using the method shown in Figure 6, with the tool’s arc radius r added to the corresponding storage unit. During processing, the NC device automatically performs arc radius compensation. Each tool’s data, including X/Z length compensation, arc radius compensation, and imaginary tool nose orientation (0–9), is stored in the corresponding tool code T storage unit. These four data points are input into the tool compensation number storage unit, enabling automatic compensation (Table 1).
Figure 6: Arc Tool Position Compensation
Table 1: Tool Compensation Value
**2.3 Automatic Arc Radius Compensation**
Arc radius compensation is controlled using G40, G41, and G42 commands.
G40 – Cancels tool radius compensation.
G41 – Left-side tool radius compensation.
G42 – Right-side tool radius compensation.
Figure 7 shows the process of using arc radius compensation. The program format is as follows:
G40__; Cancel compensation
G41__; Start radius compensation
__;
Figure 7: Tool Compensation Process
As shown in Figure 7, during the initial block, the tool gradually adds the compensation value. At the end of the initial block, the tool’s arc center lies on the vertical line of the programmed coordinate point, with a distance equal to the radius compensation value.
**3. Tool Compensation Calculation When CNC Lathe Lacks Radius Compensation Function**
When a CNC lathe lacks a tool radius compensation function, special calculation methods are needed to determine the compensation amount when machining with round tools.
**3.1 Machining Conical Surfaces Using Imaginary Tool Tip Programming**
As shown in Figure 8, if the imaginary tool tip moves along the workpiece contour AB, programming according to AB’s dimensions will inevitably result in residual error ABCD. To avoid this, the cutting point of the turning tool should be moved along AB, assuming the imaginary cutting edge trajectory differs from the contour in the Z-direction by Δz.
Figure 8: Cone Cutter Compensation
**3.2 Programming Arcs Based on the Imaginary Tool Tip**
When turning an arc surface, as shown in Figure 9, the tool’s arc trajectory may not match the required shape due to the presence of P. The center of the arc is at “R + r,†and the programmer still programs based on the imaginary tool tip P, requiring compensation in the X and Z directions before machining. For concave arcs, the compensation is subtracted instead of added.
Figure 9: Sketch Map of Circular Arc Cutter Compensation
**3.3 Programming Using the Tool Nose Arc Center Trajectory**
For parts composed of multiple convex and concave arcs, such as the one in Figure 10, the three segmented equidistant lines can be used for programming. By calculating the coordinates of the arc endpoints based on the equidistant tangent point relationship, the program becomes more intuitive and widely used.
Figure 10: Programming by Center of Tool Nose Arc
**4. Conclusion**
The main purpose of the tool compensation function is to simplify programming, allowing the operator to program based on the part’s contour dimensions. Before machining, the operator measures the actual tool length and radius, determines the sign of the compensation, and inputs the parameters into the CNC system. Even if the tool size changes due to tool wear or replacement, the original program can still produce parts that meet the dimensional requirements. Furthermore, the tool compensation function can also satisfy special programming and machining needs.
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