In modern numerical control machining technology, five-axis machine tools are widely used in fields such as aerospace, mold manufacturing, and medical equipment due to their ability to efficiently and accurately process complex surfaces. Unlike traditional three-axis machine tools, five-axis machine tools add two rotating axes (usually A, B, or C axes) to the three linear axes of X, Y, and Z, allowing the tool to approach the workpiece at any angle, thereby significantly improving processing flexibility and accuracy. However, this also brings more complex problems of rotation angle calculation.



One, Basic Structure and Rotation Axis Definition of Five-axis Machine Tools



Five-axis machine tools usually consist of three linear motion axes (X, Y, Z) and two rotation axes. According to the different configurations of the rotation axes, common five-axis structures can be divided into the following types:

　　1. Worktable rotation + main shaft swing type (such as AC structure);



2. Double pendulum type (such as BC structure);



3. Double table type (such as AB structure).



In programming and calculation, the A-axis is usually defined as rotating around the X-axis, the B-axis as rotating around the Y-axis, and the C-axis as rotating around the Z-axis. The two rotation axes together determine the spatial posture of the tool relative to the workpiece.



Two, Calculation Principle of Rotation Angle



In five-axis simultaneous processing, in order to make the tool always contact the workpiece surface at the best angle, it is necessary to calculate the angle values of the two rotation axes in real time according to the geometric shape of the workpiece and the tool path. The calculation of rotation angles mainly includes the following steps:



# 1. Coordinate Transformation



Transform the coordinate of the tool's end point from the world coordinate system to the local coordinate system of the machine tool's rotating axis. This process usually involves the use of homogeneous coordinate transformation matrices.



# 2. Posture Calculation

　　Determine the angle at which the tool needs to tilt by the angle between the normal vector of the workpiece surface and the direction vector of the tool. Common methods include vector dot product and cross product methods, which are used to solve the deflection angle θ between the tool and the Z-axis.



# 3. Inverse Kinematics Solution



The core problem of five-axis machine tools lies in how to deduce the specific angle values of two rotation axes from the spatial posture of the tool. This involves the solution of inverse kinematics, which usually requires the establishment of a mathematical model based on the specific structure of the machine tool and the solution by numerical or analytical methods. For example, for AC structure machine tools, trigonometric equations can be established using geometric relationships to determine the angles of A and C axes.



Three, Challenges and Solutions in Practical Applications



Although theoretically the calculation of rotation angles can be completed through mathematical models, in practical applications, many challenges still exist:



- Tool interference and collision detection;



- Multi-solution problems (multiple rotation angle combinations may achieve the same tool posture);



- The travel limit of the machine tool causes certain angles to be unexecutable;



- High requirements for calculation efficiency and real-time performance.



To address these issues, modern CAM software (such as Mastercam, PowerMill, HyperMill, etc.) usually integrates automatic five-axis path planning and rotation angle optimization functions, which can automatically select the optimal rotation angle according to the characteristics of the machine tool while avoiding interference and singular points.



Four, Conclusion

　　The calculation of the rotation angle of a five-axis machine tool is the key to its efficient and high-precision processing. It not only involves complex mathematical modeling and coordinate transformation, but also requires optimization processing by combining the actual machine tool structure and processing requirements. With the continuous development of numerical control technology and CAM software, the threshold for using five-axis machine tools is gradually reducing, and their application prospects in the manufacturing industry will be more extensive in the future. For technical personnel engaged in numerical control programming and processing, it is of great significance to deeply understand the calculation principle of rotation angles.