13 May 2021

**Research background**

As a typical representative of ultra-precision machining technology, the ultra-precision turning technology of single point diamond has good control, which can not only obtain the shape accuracy of sub-micron level, but also obtain the surface roughness of nanometer level. Therefore, it has become one of the important methods to manufacture optical complex curved surface devices. Compared with the traditional multi-axis machining, a higher accuracy of the ultra-precision turning, so the cutting tool path calculation is also put forward higher standard, main requirements include adapt to the complex topology structure surface, satisfy different mathematical expressions, calculation of the amount of data efficiently knife loci, automatic calculation without interference in tool path, good cutting dynamics, etc. These requirements pose many challenges to the research and development of tool path algorithm. In this paper, through the analysis and research of three aspects of tool location calculation, tool path topology form and G code optimization, the following conclusions are drawn.

**Research conclusions**

1. Cutter location calculation

Based on the summary of the methods in the open literature, it can be seen that the general process of the existing spiral tool path calculation is as follows: A spiral of Archimedes is constructed as the driving line in a plane perpendicular to the axis of rotation axis C, which is discretized into a sequence of points as the driving points for tool position calculation. Then, a specific projection calculation is carried out along the fixed direction parallel to the axis of rotation of the workpiece, and the tool position corresponding to each driving point is obtained. Assuming that the rake Angle of circular turning tool is zero, each tool position should meet two basic conditions, as shown in the figure below :(a) the plane where the rake face of turning tool is always beyond the axis of rotation of axis C; (b) The tool tip circular arc is tangential to the design surface. In order to get the tool to meet the conditions, there are two basic ideas: (1) to calculate the spiral of Archimedes known on the driving point for a fixed point on the tool (generally for arc center or point point) and projection direction, the cutting tool rake face of (a) at the same time satisfy the above conditions, and then to design the cutting tool surface projection, makes the tool and the design surface tangent points to get the edge of the sword; (2) By projecting a point on the spiral of Archimedes onto a point on the design surface as a tool contact, the tool point is calculated according to the differential characteristics of the design surface at this point.

Fig 1 Two conditions are satisfied by the tool position of turning tool with zero rake Angle

2. Topological form of tool path

At present, most of the algorithms use Archimedes helix as the driving line to project along the direction of the workpiece rotation axis. One advantage of this projection algorithm is that it does not depend on the topological form of the machined surface. However, since the spiral of Archimedes is a plane curve, when the slope of the design surface relative to the projected direction is too large, the uniformity of the projected spiral tool path becomes worse. Where the slope is high, the trajectory becomes sparse, the error significantly increases, and even there is no tool path, as shown in Fig. 2. This problem can be improved by using the spatial Archimedean spiral. The plane on which the Archimedean spiral depends can be regarded as a special case of the rotary surface: the rotary surface when the bus is straight and perpendicular to the axis of rotation. If the straight generatrix on the plane is mapped to the generatrix on the general rotary surface by the way of preserving the length mapping, then the Archimedes spiral on the plane also has a spatial spiral mapped on the general rotary surface, which is defined as the spatial Archimedes spiral. One of its important characteristics is that the arc length between adjacent spiral points on the same bus is equal.

Fig 2 A problem driven by the spiral of Archimedes

3. Interpolation and G code optimization

The last link of tool path planning is to output the G code which can be recognized by the CNC system of the machine tool. Therefore, tool path planning must consider the interpolation form supported by the CNC system and combine it organically, which is of great significance to improve the processing efficiency and quality. Except linear interpolation, other interpolation methods have not been really applied in ultra-precision machining. Because spline interpolation has the characteristics of small amount of data and high smoothness, some advanced control systems of three-axis and five-axis milling centers support spline interpolation. However, it has not been widely used in the industrial CAM system for the following two reasons :(1) Linear interpolation (G01) is simply accepted by the industry and has formed a habit; (2) some manufacturers in the numerical control system to increase the G01 code fitting function, to improve the surface quality of processing. However, in order to ensure the nanoscale surface roughness of the optical surface, the ultra-precision turning trajectory must be very dense, which will lead to a huge amount of data based on linear interpolation. Therefore, the above reasons (2) for the function of G01 code fitting cannot solve the problem of large data volume. If the advantages of interpolation methods such as spline and PVT are fully utilized for ultra-precision tool path planning, it is expected to significantly reduce the amount of data input to the CNC system of the machine tool, and it is possible to further improve the surface quality of the optical surface and reduce the processing time by optimizing the dynamic performance of the tool path.