## 1 Introduction  
Modern gear transmissions are being developed for higher speeds and heavier loads, which pose higher requirements for the static accuracy and dynamic performance of gears [1; 2; 3]. As an important transmission component, herringbone gears are widely used in high-speed, heavy-duty, and high-power transmission systems such as aviation and vessel transmission equipment and other mechanical transmission areas because of their advantages in transmission stability and strong bearing capacity [4; 5; 6]. Therefore, high accuracy in the manufacture of herringbone gears is increasingly required [7; 8; 9; 10]. Even the spatial symmetry of herringbone gears directly affects transmission errors, transmission efficiency and service life, which are significant for transmission systems [11; 12]. At present, tooth profile and tooth direction modifications are research priorities that can improve the stability of and reduce vibrations in gear transmission systems [13; 14; 15; 16; 17]. Thus far, manufacturers have lacked a high-precision and high-efficiency control processing method for on-line detection and compensation of spatial symmetry errors in herringbone gears [18]. In recent years, many scholars at home and abroad have conducted numerous related studies on herringbone gear machining methods and the principles of processing error influence on transmission characteristics. Meanwhile, advanced machining methods based on online detection and compensation machining technology have great reference value. [PERSON] et al. [19] proposed the development of kinematic error models accounting for geometric and thermal errors in the vertical machining center, and used the error model to calculate and predict the resultant error vector at the tool-workpiece interface for error compensation. [PERSON] et al. [20] developed a feature-based inspection and control system to realize completion of whole machining and detection and compensation processes on machine tools. [PERSON] et al. [14] developed a new double-helical test setup for operating a double-helical gear pair under realistic torque and speed ranges. [PERSON] et al. [21] proposed a dynamic model that includes friction and tooth profile error excitation for herringbone gears and used the proposed model in the dynamic analysis of the variable speed process of a herringbone gear transmission system. [PERSON] et al. [22] proposed a cutting compliance coefficient model to estimate corrections for the tool path at the finish cut based on a finite number of measured errors at discrete locations for previous cuts, and presented an integrated methodology for compensation errors detected with an on-machine touch probe. [PERSON] et al. [23] studied the influence mechanism of manufacturing error and assembly error on the load sharing characteristics of a transmission system. [PERSON] et al. [24] investigated the effects of centering error and angular misalignment on crack initiation life in herringbone gears. [PERSON] et al. [25] compared gear surface characteristics generated by grinding, honing and superfinishing of case-hardened steel. [PERSON] et al. [26] proposed an error compensation machining method that improves complex surface components based on analyzing the error factors influencing the inspection accuracy of on-machine detection systems. [PERSON] et al. [27] established a mathematical model of comprehensive error compensation for complex thin-wall parts and performed machining tests of error compensation. In a word, the above research findings developed on-line detection and compensation machining technology to some degree. However, they had some limitations that could not be effectively applied to real-time spatial symmetry error detection and compensation machining processes for herringbone gears. Further, investigations reporting on the measurement and evaluation of spatial symmetry errors in herringbone gears and machining methods for their compensation have not been reported so far.