## 1 Introduction  
Space optical remote sensing camera is one of the important means to obtain terrestrial remote sensing images. It is widely used in the fields of meteorology, geological exploration, agriculture and forestry, ocean, environment, military, disaster relief, and so on [1]. The focusing mechanism is mainly to solve the effect of defocus on the image quality of space [2]. The traditional focusing mechanism is a combined transmission mechanism, which is mainly composed of two-stage worm gear pair reduction mechanism and eccentric cam or leadscrew nut mechanism [3]. Generally, with the increase of the complexity of motion law and trajectory requirements, the number of parts of the combined mechanism increases, the structure becomes complex, and the reliability of the mechanism often decreases. Therefore, the combined mechanism is not suitable for occasions where the structural size is limited and the motion law is too complex. According to the design requirements of the focusing mechanism of space remote sensing camera, the curve-face gear pair, a new type of gear pair with variable transmission ratio was used for the focusing mechanism. With the dual advantages of gear pair and cam pair [4], the curve-face gear compound transmission can adapt to heavy-duty and high-precision conditions such as aerospace [5]. Therefore, researches on the curve-face gear compound transmission is necessary, which can enrich the theory and expand the application of non-circular gears. So far, the fundamental theory system of gear transmission has been established by using space vector and differential geometry as theoretical basis and mathematical tools [6; 7; 8]. Remarkably, screw theory was introduced into mechanism kinematics by many researchers, which can provide a deeper understanding of motion characteristics and promoted the engineering applications of space gears [9; 10]. [PERSON] [11] used the theory of screw to study gear transmission. [PERSON] [12; 13] proposed the three laws of spatial gearing based on screw theory, and optimized the design concept of spatial and planar gear. [PERSON] [14] studied spatial involute gear transmission with skew axis by using the theory of screw as a mathematical tool. Therefore, screw theory and the principle of spatial gearing are expected to apply in characteristics analysis of curve-face gear compound transmission pair, which is of great significance to the design and practical applications of the curve-face gear pair. As a new type of gear, the curved-face gear has been studied preliminarily. For common variable contact ratio transmission combined with non-cylindrical gear and curve-face gear, the mathematical equations for the tooth surface of non-circular gear and curve-face gear based on the external generating method with the same shaper cutter were established [15]; the equations of instantaneous contact ratio of helical curve-face gear pair are proposed by transforming helical curve-face gear pair into helical non-circular rack and pinion [16]; a calculating method of the tooth root bending strength of the eccentric straight and helical curve face gear was proposed [17]; the loaded tooth contact analysis of straight and helical curve-face gear pair were taken into consideration [18; 19]; to study the nonlinear dynamic characteristics of curve-face gear drive, a generalized nonlinear dynamic model based on Lagrange Bond graphs was presented [20]. For compound transmission combined with cylindrical gear and curve-face gear, the movement characteristics of the compound curve-face gear transmission were analyzed, including transmission ratio, rotation, movement, and quick-return [21]; the calculation of meshing efficiency of compound motion has been proposed [22]. It can be seen that, compared with common variable contact ratio transmission, the research on curve-face gear compound transmission is not perfect.