###### Abstract  
While curved surface microstructures have wide applications in optical components and devices, how to achieve high machining accuracy of the microstructures is crucial for their applications. In the present work, we fabricate sinusoidal modulation microstructures on a curved copper surface by ultra-precision diamond cutting, with the combination of a rotary \(B\)-axis and a fast tool servo system. Specifically, tool path planning, together with the consideration of a curved, sinusoidal surface meshing and tool tip arc segmentation compensation, is carried out. Preliminary cutting experiments are firstly carried out on a homemade four-axis ultra-precision lathe, which demonstrates the advantages of additionally applying the rotary \(B  
conducted experimental research on the fabrication of aspheric arrays microstructures on aluminum cylinders. [PERSON] et al. (2018) designed a four-axis linkage tool-workpiece motion system and considered the lens design, tool geometry and roller parameters in the tool path generation, which led to the realization of an average surface roughness of 8.25 nm of lens microstructures fabricated on a brass roller by using the FTS system. [PERSON] et al. (2018) proposed a new data processing algorithm to achieve an overall contour error of less than 2 \(\upmu\)m for microstructures fabricated on a brass surface by using the FTS system. [PERSON] et al. proposed a high-precision two-degrees-of-freedom compliant mechanism. However, achieving the fabrication of high-precision surface textures on a curved surface with a large curvature radius by diamond cutting is still challenging. Specifically, in the fabrication of a sinusoidal modulation surface on a curved surface with a large curvature radius by solely using the FTS system, the height difference exceeds the stroke of the FTS from a certain position, which thus requires the extension of the stroke of the FTS. However, the stroke of the FTS system is severely limited by the high-frequency characteristics of the system. [PERSON] et al. presented a novel FTS diamond turning method with layered tool trajectories, which enables the extending of the limited stroke length to fivefold of its maximum possible stroke length ([PERSON] et al., 2018). [PERSON] et al. presented a long-stroke FTS to obtain a large-amplitude tool motion, which led to the fabrication of a sinusoidal surface with five sinusoidal waves and an amplitude of 20 \(\upmu\)m on 6061 aluminum ([PERSON] et al., 2018). More recently, [PERSON] et al. fabricated a multiscale optical surface by diamond turning using the high-dynamic performance of FTS and the long stroke of a slow tool servo ([PERSON] et al., 2018). [PERSON] et al. proposed a dual-axial FTS diamond turning method for the one-step fabrication of hierarchical micro-nano-structured surfaces, in which the additional \(x\)-axial motion is effective for deterministic generation of a variety of secondary structures ([PERSON] et al., 2018). In addition, the currently utilized method of tool path planning for the fabrication of modulation microstructures on curved surfaces generally has the disadvantages of complex processing procedures and low processing efficiency, which subsequently lowers the machining efficiency. [PERSON] et al. demonstrated that the tool path has a strong influence on the surface form error, such as roughness, waviness and profile errors ([PERSON] et al., 2018). [PERSON] et al. found that the machining errors have a strong influence on the optical performance of optical aspheric components machined by FTS ([PERSON] et al., 2018). More recently, [PERSON] et al. developed a new tool path generation method and discussed the tool geometry parameter selection for diamond turning utilizing non-zero rake angle tools, based on which a typical freeform surface is fabricated ([PERSON] et al., 2018). [PERSON] et al. proposed an optimization method of tool path generation based on the interpolation of the lens edge, which can effectively reduce the overcut of the lens edge ([PERSON] et al., 2018).