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Collision Probability Prediction and Orbit Maneuvering Probability Determination of Non-Cooperative Space Object Orbit

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Collision Probability Prediction and Orbit Maneuvering Probability Determination of Non-Cooperative Space Object Orbit remote sensing Article Collision Probability Prediction and Orbit Maneuvering Probability Determination of Non-Cooperative Space Object Orbit 1,2 3 1, 1 1, , Yuanlan Wen , Zhuo Yu , Lina He y, Qian Wang and Xiufeng He * y 1 School of Earth Science and Engineering, Hohai University, Nanjing 210098, China; [email protected] (Y.W.); [email protected] (L.H.); [email protected] (Q.W.) 2 Research School of High Technology, Hunan Institute of Traffic Engineering, Hengyang 421099, China 3 Xi’an Satellite Control Center, Xi’an 710043, China; [email protected] * Correspondence: [email protected] These authors contributed equally to the work. y Received: 5 September 2020; Accepted: 1 October 2020; Published: 12 October 2020 Abstract: Probability of collision between non-cooperative space object (NCSO) and the reference spacecraft (RS) has been increased drastically over the past few decades. The traditional method is difficult to identify the maneuvering of non-cooperative space object. In the present paper, not only positions and velocities, but also accelerations of non-cooperative space object are estimated as parameters by the extended Kalman filtering based on setting up the state linear equation and measurement model of the non-cooperative space object. The algorithm for predicting collision probability is derived from position error ellipsoid, and the algorithm for determining maneuvering probability is derived from maneuvering acceleration and its error ellipsoid, which can be employed to identify whether the upcoming space object is being maneuvered. An epoch Earth-centered inertial (EECI) coordinate system is suggested to replace Earth-centered inertial (ECI) to simplify coordinate transformation. Finally, a set of simulations were conducted to validate the proposed algorithms with the simulated measurement data of the reference spacecraft space-borne millimeter-wave (MMW) radar. Keywords: non-cooperative space object; acceleration estimation; kalman filtering; collision probability; maneuvering identification 1. Introduction The non-cooperative space object (NCSO) refers to the space object which cannot provide effective information, such as the disabled or discarded satellites, space debris, or the rival spacecraft in space which may surveil or harm the reference spacecraft (RS) that is performing a mission. With the development of astronautics technology over the past decades, the number of space activities surges rapidly. On one end, collision probability between RS and NCSOs has therefore increased. On the other end, some NCSOs might approach RS intentionally for some purposes. These can be serious threats to RS mission. It is crucial to monitor the status of the NCSOs nearby in order to avoid collision [1]. Technically, estimating orbit parameters, identifying orbit maneuvering, and predicting collision probability of NCSOs are the three main tasks of NCSOs monitoring. A variety of research on orbital mechanics modeling can be found in the literature. A non-Gaussian error propagation model was set up by Junkins et al. [2]; the model was evaluated in different cases and proved to be successful. A modeling error index was proposed by Alfriend et al. [3] for grading the accuracy of different satellite relative motion models. The modeling error index provides a foundation for differentiating various relative orbital variables and models. By using the measurement data from Remote Sens. 2020, 12, 3310; doi:10.3390/rs12203310 www.mdpi.com/journal/remotesensing Remote Sens. 2020, 12, 3310 2 of 20 on-board sensors on the spacecraft, there are three main methods for space object orbit estimation: geometric method [4], dynamic method [5], and geometric dynamic method [6,7]. A method to determine the orbital elements and to predict the orbit of a space object using measurements acquired by space-borne millimeter-wave (MMW) radar is studied, and the encounter of the space station and a space object within a week is predicted by the means of simulation by Liu [8]. The orbit determination methods for primary spacecraft and other space objects were presented by Qui et al., and the criteria for collision prediction algorithm and to filter the cataloged objects were proposed [9]. When it comes to the NCSOs risk evaluation, minimum distance forecast and collision probability predictions are commonly conducted. Under the assumption that the position error covariance matrices of the two objects are not correlated [10], and a three-dimensional probability density function to demonstrate the relative position relationship between our spacecraft and a space object is defined by Chan [10]. By disregarding the spacecraft operation attitude, two error ellipsoids can be defined based on the position error model of the two objects. The collision probability can therefore be obtained by integrating the probability density in the area where the two error ellipsoids intersect. This method was validated first by Bérend who utilized Monte Carlo simulations [11,12]. A more general formula for calculating collision probability between two space objects was carried out by Wang et al. [13]. Specifically, the relative velocity is assumed to be a directionally invariant vector. The dimension parallel to the relative velocity vector is eliminated and the three-dimensional probability calculation problem is reduced into a two-dimensional approximation along the encounter plane, perpendicular to the relative velocity. As a result, the maximum instantaneous collision probability is predicted in an interested area [14]. An explicit method to calculate the collision probability was performed by Bai et al. [15] who assumed that the orbit of space object is circular. Factors that increase or decrease the collision probability were also analyzed in their work. Collision probability models were also applied to predict collisions with space debris to avoid excessive false warning in BOX criterion [16]. An algorithm that is simpler and more physically motivated to calculate the non-tangential collisions in Keplerian orbits was presented by Jeongahn and Malhotra [17], and the singularity was also regularized by making use of a parabolic approximation for the motion in the proximity of a tangential encounter. Furthermore, the relative motion of two spacecraft was investigated based on the two-body problem and Hill’s equations, which can be found in the work done by Lee et al. [18] and Lane and Axelrad [19]. NCSOs maneuvering detection was accomplished based on wavelet transformation and sliding window [20–22]. In addition, a maneuvering strategy has been proposed by Lee at el. [18] to reduce the collision probability. The orbital parameters estimation of NCSO is usually complex and computationally expensive. A common process can be summarized as follows. Firstly, RS determines its own position and velocity in the Earth-centered Earth-fixed (ECEF) coordinate system by global navigation satellite system (GNSS) such as GPS or Beidou system. Next, RS uses on-board sensors to make the relative measurement for the approaching of NSCO. Finally, orbit parameters (such as the classic orbital elements, or positions and velocities) of NCSO are determined in the Earth-centered inertial (ECI) coordinate system. In this process, there are some problems to be solved. Firstly, 6-parameter orbital elements (such as the classic Keplerian orbital elements or position and velocity vectors, but no accelerations) are estimated by traditional orbit determination methods, but the maneuvering acceleration of NCSO cannot be directly deduced. Secondly, the conventional orbit determination method is based on the known classical orbital mechanics models which cannot include the orbital maneuver model of NCSO, so the known classical orbital mechanics models are not suitable for the conventional orbit determination method of NCSO that is maneuvering. Thirdly, transformation between ECEF coordinate system and ECI coordinate system requires the parameters of precession, nutation, polar shift and rotation of the Earth [23,24], which greatly increase the computational complexity. In the present paper, in addition to positions and velocities, accelerations of NCSO are also estimated as parameters based on the linear state equation of the NCSO, using measurement data provided by RS space-borne MMW radar [8]. The linear state equation of the NCSO avoids orbit Remote Sens. 2020, 12, 3310 3 of 20 integral to calculate the orbit, and thus the amount of on-orbit calculation is reduced. Furthermore, the algorithms for the collision probability prediction and maneuvering probability determination are derived from positions, velocities, and accelerations of NCSO and their covariance matrices. An epoch Earth-centered inertial (EECI) coordinate system is suggested to replace ECI, thus the amount of on-orbit calculation for the transformation between coordinate systems can be reduced. 2. Analysis of Tracking Models of NCSO 2.1. Coordinate System Since the measurement models and the motion models of a space object are established based on coordinate systems, defining an appropriate system could make the corresponding problem easier to be solved and the computation amount to be reduced. Coordinate systems employed in the current paper are defined as follows: (1) Earth-centered Earth-fixed (ECEF) coordinate system Oe xe yeze [23,24]: The origin, Oe, is located − at center of mass of the Earth; xe-axis points from Earth center to intersection of the prime meridian and the Equator; ze-axis is aligned with the mean rotational axis
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