# Angle-Only Cooperative Orbit Determination Considering Attitude Uncertainty
## 1 Introduction  
Autonomous orbit determination (OD), due to its considerable value in space systems engineering, has been of widespread interest in the last several decades [1; 2; 3; 4]. The ability of spacecraft to determine their own states, without the help of ground-based tracking equipment, can improve their 'intelligence' and survivability and may also reduce operational management costs [5; 6; 7; 8]. With the current plans and development of future multispacecraft constellations, the latter issue, i.e., efficient constellation management, becomes particularly important, and autonomous OD for a spacecraft constellation is highly desirable [9; 10; 11]. Filter algorithms have been widely used in many OD applications since [PERSON] proposed his famous recursive method Kalman filter (KF) to solve discrete linear filtering problems in 1960 [12; 13; 14]. Those filter algorithms could utilize the provided measurements coming from different sensors to obtain the required estimates of the orbit states [15; 16; 17; 18]. Thus, current filter algorithms rely on the accurate measurement information measured by different sensors equipped on the spacecraft. In this situation, many filter-based methods have been proposed to solve the autonomous OD problem for a single spacecraft using GPS, radar, guidepost and magnetic field vector measurements [19; 20; 21; 22]. Although the above methods have been proven to be efficient in providing considerable estimation, these measurements rely on complicated measurement equipment, which is impossible for constellation spacecraft. Recently, a variety of methods have been proposed to solve the autonomous OD problem for a satellite constellation by using only interspacecraft relative measurements [23; 24; 25; 26]. However, they all face the rank deficiency, meaning that the multispacecraft system withinterspacecraft measurements alone is unobservable [27]. Interspacecraft ranging systems are widely used to obtain range-only measurements between two spacecraft, but they lack observability in some specific situations [23]. It has been concluded that in the multibody system, due to the nonsymmetric gravity from the noncentral gravitational body, all the orbit elements of two spacecraft can be obtained using only intersatellite range measurements [28; 29]. However, intersatellite range measurements are not adequate to obtain all orbit elements in two-body dynamics because the rotation of the orbits with respect to the inertial reference system cannot be determined, and only the shape, size, and relative orientation of two orbits can be obtained using only intersatellite range measurements. Another well-known method employs onboard cameras to obtain the relative angle measurements, which only include the relative direction information [24; 25]. The angle-only navigation is simple, robust, and well proven in many applications [30]. However, the inertial states of the spacecraft cannot be observed without additional measurements. At least one beacon (usually noted as attitude-reference spacecraft) must be added to the system to ensure that there is a reference in inertial space or that other sources of measurements should be added [31]. The basic requirement for orbit determination using angle-only measurements is that the line-of-sight vector must be measured with respect to the inertial frame, which requires several types of sensors, such as a laser sensor to measure the relative range, an optical camera to measure the relative direction, and a star sensor to measure the inertial attitude [24; 25; 32]. However, due to the size, mass, and power consumption problem of active sensors such as star sensors, they might be unable to be on-boarded for small and microspacecraft to estimate the inertial attitude. Therefore, the inertial attitude of observing spacecraft should also be added into the OD problem when using intersatellite line-of-sight angle measurements. Fortunately, the conclusion of OD using angle-only measurements in two-body dynamics does not hold in other dynamical cases.