## 1 Introduction  
Nowadays, small bodies are interesting targets of in-suite explorations. Up to now, many missions have visited these objects. Determining the spacecraft's orbit and the asteroid's gravity field is one of the goals for these missions. To this end, the current technology heavily depends on observations from ground stations. There are several types of observation data often used in asteroid missions. In the earliest missions, Doppler and range-measurement are main types of data used in missions, such as the NEAR-Shoemaker and the DAWN mission [1; 2; 3]. VLBI data is also widely used. For example, during the Chang'e-2 flyby of the asteroid Toutatis, its orbit was determined by the VLBI data together with the USB (Unified S-Band) data [4]. Delta Differential One-way Ranging (DDOR) is a highly accurate measurement based on the VLBI technology [5] and was used in the OSIRIS-Rex mission [6; 7]. Most asteroids are irregularly shaped. Many studies have been carried out on asteroid shapes and gravity-field models. For example, the polyhedron model is one of the most widely used asteroid shape models [8], but its associated form of gravity field is inconvenient to use in the gravity field recovery from observations. For elongated asteroids, a simple rotating mass dipole model is used to rapidly model the gravity field [9]. Despite of these gravity models, the most commonly used one in the asteroid's gravity field recovery is still the well-known spherical harmonics. By recovering the gravity harmonic coefficients, density and internal structures of an asteroid can be inferred [10; 11; 12]. The gravity coefficients are usually measured simultaneously with the spacecraft's orbit. For the first asteroid exploration mission, NEAR-Shoemaker, the gravity field up to the fourth order and degree was determined [1]. Using Doppler and range data of the Dawn spacecraft, [PERSON] and [PERSON] et al. obtained a gravity field up to 20 th order and degree for Vesta and 18 thorder and degree for Ceres, the two most massive asteroids in the asteroid belt [2; 3; 10; 13]. For asteroids of hundreds of meters in diameter or even smaller, it is usually difficult to obtain a high order and degree gravity field using traditional methods. One exception is the asteroid Bennu which is only about 450 m in diameter. In the OSIRIS-Rex mission, angle-measurement data between the spacecraft and particles around the asteroid made it possible to determine the gravity field to a high precision of 10 th order and degree [14; 15]. Experience from the OSIRIS-Rex mission indicates that accurate relative measurement around the asteroid may help improve the accuracy of the gravity-field recovery. We do not always have the chance to obtain relative angle measurement data as we did in the OSIRIS-Rex mission because not all targets have particles ejected from their surface, but we can use artificial relative-measurement data such as the inter-satellite range measurement, which is already extensively used in many applications around the Earth. For example, in the global navigation satellite system (GNSS), the addition of inter-satellite range data to the ground-observation data can significantly improve the accuracy of orbit determination [16; 17; 18]. In addition, in the GRACE-FO mission, the precise inter-satellite range data are used to obtain the high-order gravity field of the Earth [19]. An intuitive idea is to borrow this technology for asteroid missions. That is to say, we build accurate inter-satellite links between the spacecraft formation with one chief spacecraft and some cheap deputies which can move closer to the asteroid's surface. For spacecraft formations in the proximity of asteroids, autonomous navigation using multiple sensors was already studied [20]. For Earth satellites, it is impossible to autonomously determine the satellites' orbit using only the relative inter-satellite range data, due to the well-known rank-deficiency problem. This is no longer the case for missions around irregular asteroids.