# A General Euler Angle Error Model of Strapdown Inertial Navigation Systems
## 1 Introduction  
Inertial navigation systems (INS) can generally be classified into gimbaled inertial navigation systems (GINS) and strapdown inertial navigation systems (SINS) [1; 2; 3]. For SINS, gyro output is used to maintain a digital computational platform as its reference frame [4; 5]. Compared with GINS, SINS have many advantages, such as small size, light weight, low cost, easy maintenance, etc., gradually replacing GINS [6; 7]. Whether in GINS or in SINS, attitude error models play an important role in analyzing the characteristics of navigation error propagation, implementing the Kalman filter for aided navigation systems, and detecting failures in real-time [8; 9]. Attitude error formulations of GINS, including those of Psi-angle, Phi-angle, were developed for local level platform navigators [10]. With the development of SINS, [PERSON] et al. derived attitude error formulations for stationary SINS [11]. The static model had a better application in research for its simple structure, but it did not apply to a dynamic base. [PERSON] et al. established the dynamic error model based on the static model, where vehicle location changes would directly affect the error characteristics of SINS [12]. Assuming that the propagation characteristics of the navigation errors are linear, some linearized attitude error formulations are also modeled [13; 14]. [PERSON] and [PERSON] presented a SINS error model which contains all of the known models in the same framework [8]. [PERSON] et al. proved that these linearized error models are equivalent to each other [15], but linearized error models are not effective for nonlinear systems where large navigation errors, such as in-flight alignment,are generated [16]. [PERSON] introduced a rotation vector error and brought in white noise for error differential equations to analyze the large attitude errors of SINS [17]. Then, quaternion and dual quaternion errors were developed to describe attitude error propagation [18; 19]. As quaternions and Euler angles are not in one-to-one correspondence, both the converting relation and the corresponding relation have to be taken into consideration during an attitude control process [20]. The SINS attitude error formulations above mainly focus on misalignment for a specific computational frame rather than Euler angle errors. However, misalignment error formulations cannot describe attitude error propagation directly [21]. Euler angles are the most intuitive measurements of attitude movements, especially large-angle attitude movements. Real-time attitude information can be obtained by using Euler angles to express the attitude movement in controlled areas [22; 23], then appropriate optimization can be made. An adaptive Euler-angle-based unscented Kalman filter (UKF) is utilized to analyze attitude in real-time [24]. The manipulator control process is a large-angle attitude movement process which is very sensitive to Euler angles [25; 26]. The controllers for each degree of freedom are Euler angles. There are some specific applications, such as 6-DOF robot manipulators with Euler wrists [27; 28], and robotic-wheelchair controllers combined with multi-sensors to obtain accurate orientation in the form of Euler angles [29]. Additionally, [PERSON] proposed a quasi-Euler angle to control the attitude, where the terminal attitude of a satellite can be determined [20]. The control approaches mentioned above are based on Euler angles, and the Euler angle method turns out to be superior to the others in the applications because it shows better numerical accuracy, stability, and robustness [30; 31]. The misalignment angle is characterized by the calculation of the three-axis deviation angle between the computational frame and the navigation frame, which cannot directly reflect and optimize a body's attitude movement. The initial attitude of the navigation system is in Euler angle form [32].