# New Methods of Series Expansions between Three Anomalies
## 1 Introduction  
With the development of the Chinese BeiDou Navigation Satellite System (BDS) globalization process, the construction of the third-generation BeiDou Navigation Satellite System with global coverage (BDS-3) was basically completed by the end of 2018 [1, 2]. In 2020, China completed the global networking of the Beidou Navigation System and provided conventional PNT services as well as short message communication and international search and rescue services to global users [3]. The nominal space constellation of the Beidou-3 satellite consists of three geostationary orbit (GEO) satellites, three inclined geostationary orbit (IGSO) satellites, and twenty-four medium Earth orbit (MEO) satellites. Geostationary orbit (GEO) satellites have an orbit altitude of 35,786 km, located at 80 degrees east longitude, 110.5 degrees east longitude, and 140 degrees east longitude, respectively; the inclined geostationary orbit (IGSO) satellites have an orbit altitude of35,786 km, and the inclination angle of the track surface is 55 degrees. The medium Earth orbit (MEO) satellites have an orbit altitude of 21,528 km, and the inclination angle of the track surface is 55 degrees. The Beidou Navigation Satellite System deploys in orbit backup satellites according to specific circumstances [4]. The Chinese space industry, represented by the BeiDou Navigation Satellite System (BDS), has developed vigorously and made great achievements [5; 6]. With the needs of scientific research, business, and other aspects, more and more spacecrafts have been launched; most of these spacecrafts orbit the Earth. Some satellites, represented by the Beidou Navigation System, require the precise determination of their orbits [7; 8]. And they may need to perform precise calculations on their orbital formulas, such as calculating satellite ephemeris. In these precise calculation processes, three types of anomalies of satellite orbits are often involved [9]. For example, in the process of converting the orbital elements to the vector of the position and velocity, the time element in the orbital elements is often used as the mean anomaly. But in the calculation, it needs to be converted to the true anomaly and the elliptic anomaly before calculation, which involves the transformation between the three types of anomalies. In addition, in many problems, it is necessary to express some parameters as explicit functions of time through the mean anomaly. However, due to the fact that [PERSON]'s equation is a transcendental equation [10], it is difficult to obtain a rigorous analytical solution. We cannot directly convert the three types of anomalies. Many scholars have studied the [PERSON] equation. For example, [PERSON] et al. proposed a high-precision numerical solution for the [PERSON] equation [11]. [PERSON] et al. solved [PERSON]'s equation using the symbolic iteration method of computer algebra analysis [12]. [PERSON] et al. proposed a new analytical approach for constructing approximate solutions to the elliptic [PERSON]'s equation [13]. In 1968, [PERSON] applied the Lie-Series to [PERSON]'s problem of an undisturbed planet around the sun (two-body problem) [14]. [PERSON] and [PERSON] proposed a method that solves the hyperbolic [PERSON] equation in a very efficient manner, and to an accuracy that proves to be always better than \(10^{-20}\)[15]. Taking into account the monotony and convexity properties of the sine hyperbolic Kepler equation (SHK) and by making a detailed analysis of the discretization error, [PERSON] et al. provided a monotonic piecewise starter for this equation which has the q-convergence property for the Newton iteration [16]. [PERSON] developed an idea to solve [PERSON]'s equation with a CORDIC-like algorithm, which solves [PERSON]'s equation using only bitshifts, additions, and one initial multiplication [17]. [PERSON] and [PERSON] proposed a numerical technique, based on [PERSON]'s fixed-point theorem, to obtain an approximate solution of the elliptical Kepler equation [18]. [PERSON] et al.