## 1 Introduction  
The tooth profile is the basic factor determining gear transmission performance and a fundamental factor affecting the sensitivity of the gear to error. At present, most of the research regarding gears with a few teeth are based on an involute tooth profile. However, inherent geometrical properties affect the load carrying capacity, transmission quality, and improvement of lubrication performance. Theoretical analysis [1] and experimental research [2] has shown that the involute profile has a large curvature owing to the meshing of two convex surfaces, which makes the tooth prone to fatigue damage and the load capacity of gears with a few teeth (gear pair with 2-6 pinion teeth) more sufficient. Therefore, certain experts and scholars have proposed new types of tooth profile, new theories, and new methods to meet the application needs of gears in special or extreme occasions [3; 4; 5; 6; 7]. By using a variable involute forming principle [2], a novel S-shaped profile, machined by a rack cutter with a sine curve and suitable for a gear pair with a few teeth to greatly improve the bearing capacity, have been proposed in several patents [8; 9]. At present, other scholars have also made some progress in the research of S-profile gears. Although the generation mechanism of an S-shape is different, its practical application can be realized. The literature [10; 11; 12] verifies the correctness of the S-shape gear theory and the feasibility of practical application from the aspects of S-curve formation theory, actual manufacturing, and gear failure forms, etc. It has a greater advantages than an involute tooth shape. At the same time, new problems were introduced; the involute tooth has a good contact performance and separability of the center distance. What are the characteristics of the meshing performance and the sensitivity to shaft misalignments of an S-shaped tooth profile gear pair? And its separability of the center distance? These issues will be analyzed in this article. To explore the above questions, mathematical models of an S-shaped tooth surface are established and the tooth contact analysis and transmission error analysis of an S-shaped tooth profile gear pair with shaft misalignments are carried out. [PERSON] et al. [13] proposed a complex geometric modeling method of a helical face gear pair with an arc-tooth generated by an arc-profile cutting (APC) disc. Its tooth contact characteristics were analyzed; through the contact analysis of the tooth surface, the sensitivity of the APC face gear to different installation errors was obtained. The results showed that the APC face gear pair is most sensitive to the installation error in the direction of tooth thickness. [PERSON] [14] took the cylindrical gear pair as the research object, considering the influence of the flexibility of the shaft and bearing and the installation form of the gear pair on the contact characteristics of the gear system. The corresponding modification methods have been put forward for the meshing extension of the gear pair and the meshing misalignment of the gear system. [PERSON] and [PERSON] [15] put forward a computer simulation method to carry out loaded contact analysis of a large contact ratio spur gear pair of a modified tooth profile and an arc-involute tooth profile helical gear pair, respectively, considering tip corner contact and shaft misalignments, which obtained the contact path and contact stress distribution on the tooth surface under the condition of different modification parameters and errors. Based on the penetration gear contact model, [PERSON] [16] performed an accurate and efficient numerical analysis of the tooth surface contact of spiral level gears and simulated the alignment error of gears. The developed gear contact model with surfaces of roll angles, computed for the gear pair's tooth flanks in the absence of misalignments, is shown to be well capable of predicting a misaligned gear pair's contact performance.