###### Abstract
As a result, many optical systems are being optimized at the same time as the requirement to reduce the tolerance sensitivity of the optical system, which provides a reference for the design of optical system that realizes image motion compensation through the movement of the optical element. At present, there are two main types of methods to reduce tolerance sensitivity. The first type is to find a system with low sensitivity from the optimization of the initial structure which can be considered as a positive thinking method. The second type is to find a system with low sensitivity from analyzing the variation of the system when the tolerance is introduced. It can be considered as a reverse thinking method. The first type of method mainly includes the global optimization method [7], the multiple structure method [8] and the use of aspherical or free-form surfaces. The second type of method mainly includes the small aberration complementary method [9; 10; 11], the desensitization method based on the optical path difference and the desensitization method based on the Nodal aberration theory (NAT). The desensitization method of positive thinking usually focuses on the adjustment of the initial structure. For example, [PERSON] pointed out, a desensitization lens usually requires more degrees of freedom and change of the lens form [12]. Therefore, the desensitization method of positive thinking usually requires multiple iterations or sensitivity analysis on a sufficient number of samples. In the desensitization method of reverse thinking, the small aberration complementary method points out that the change of the primary aberration caused by the tolerance is a function of the primary aberration coefficient and the incident angle and the refraction angle. Therefore, [PERSON] [13] employed the operands of various primary aberrations to optimize the initial structure in the zemax software. [PERSON] [14] optimized the curvature of each optical surface and balanced the incident angle of each optical surface to achieve small aberration complementation. [PERSON] et al. [15] proposed a suitable adjustment of the incident angle on the sensitive surface. However, the above designs consider a single angle, and the desensitization effect is limited. [PERSON] [16; 17] proposed using optical path difference induced by the tolerance as an evaluation criterion for tolerance sensitivity of optical system. This method derives the optical path difference from the changes of the intersection of the light and the surfaces and the image plane. When there are many components, more optical path changes need to be considered. Therefore, the desensitization method based on the optical path difference is suitable for optical systems with a simple optical structure and few optical components,and the optimization effect of the complex optical system such as aerial mapping camera is poor. In the desensitization research based on the NAT, [PERSON] and [PERSON] [18] established a linear relationship between tolerance, compensator, and wave aberration. [PERSON] [19] derived the estimated wavefront aberration for a given surface eccentricity and tilt tolerance based on NAT and established an optical system performance evaluation model. The above methods are all based on improving the performance evaluation of the optical system, which has strong versatility and a small calculation amount. However, it has some drawbacks. The analytical derivation of the methods mentioned above is insufficient. In order to realize the matrix operation, [PERSON] and [PERSON] ignore the quadratic term of the aberration field decenter vector, and [PERSON]'s optimization does not analyze the dependence of the sensitivity of the optical component on the field of view. Based on the NAT of the misaligned system, the analytical expression of the tilt sensitivity of the secondary mirror in different tilt directions is given in the form of zernike polynomial coefficients.