Research Article | Open Access
Bing Hua, Guang Yang, Yunhua Wu, Zhiming Chen, "Angle-Only Target Tracking Method for Optical Imaging Micro-/Nanosatellite Based on APSO-SSUKF", Space: Science & Technology, vol. 2022, Article ID 9898147, 13 pages, 2022. https://doi.org/10.34133/2022/9898147
Angle-Only Target Tracking Method for Optical Imaging Micro-/Nanosatellite Based on APSO-SSUKF
To ensure the safety of the space station and improve the accuracy of the estimated trajectory tracking of noncooperative target, an optical imaging micro-/nanosatellite based on APSO-SSUKF (adaptive particle swarm optimization-spherical simplex unscented Kalman filter) is proposed to track low-orbit target using angle-only measurement. First, the algorithm considers the effect of J2 perturbation, uses the angle-only data as the observation vector, and uses spherical simplex unscented Kalman filter (SSUKF) to reduce the cost of calculation of the UKF in space noncooperative target tracking. Secondly, it is proposed to use the actual and theoretical covariance of the innovation sequence for real-time estimation of measurement noise, designing the adaptive particle swarm optimization (APSO) algorithm for real-time tracking of the process noise in the SSUKF that improves the accuracy of the filter in angle-only tracking. Finally, the tracking simulation of low-orbit satellite is carried out by using optical imaging micro-/nanosatellite, and the result shows that, compared with UKF, SSUKF, and PSO-SSUKF, APSO-SSUKF reduces the root mean square of the error in predicting the position in space target tracking by 45.44%, 35.26%, and 20.94%, and APSO-SSUKF reduces the root mean square of the error in velocity by 45.58%, 33.53%, and 16.33%, respectively; in the angle-tracking target, APSO-SSUKF improves the convergence and estimated accuracy of the algorithm in tracking.
Space Situation Awareness (SSA) [1–4] is currently a key research project for various spacefaring nations, which can achieve high-precision real-time tracking of noncooperative targets in space by establishing a combined space and sky satellite surveillance system. Since ground-based observation devices depend on meteorological, geographic, and geophysical conditions, the use of space-based observation systems  can circumvent these limitations and play an important role in space deployment, surveillance, and early warning [6–8]. Compared with traditional large satellites, micro-/nanosatellites  have a shorter development cycle, lower cost, and more flexible system applications, but micro-/nanosatellites suffer from limited power supply, small size, and insufficient computing power, which lead to great limitations in the observation equipment carried by micro-/nanosatellites; therefore, it is of great significance to study optical imaging micro-/nanosatellites to achieve high-precision tracking of space target.
Optical detection equipment mainly includes optical telescopes, microwave radar, and LIDAR. The large size and power consumption of space-based radar increase the load burden and reduce the service life of optical imaging micro-/nanosatellites, while space-based optical sensors (SBOS) to obtain measurement can achieve low-cost, long-range, and high-precision real-time tracking of noncooperative targets, which plays a key role in SSA and is highly valued and widely used, such as the U.S. Space-Based Space Surveillance System (SBSS) , Canadian Space Surveillance System (CSSS) , and Geosynchronous Space Situational Awareness Program (GSSAP) . GSSAP is a space surveillance satellite system operated by the U.S. Air Force that provides precise orbit tracking to avoid satellite collisions as the geostationary orbit becomes increasingly crowded. ESA  also verified that space debris can be effectively detected using SBOS, proposed the observation concept, sensor architecture, and system performance, and included SBOS for space surveillance and target tracking in the SSA program in 2007. In addition, many scholars have applied SBOS to orbit determination of noncooperative targets and autonomous navigation of spacecraft. For example, the short observation arcs of geosynchronous Earth orbit (GEO) objects are obtained by using only angular measurement information, and a multipoint optimal initial orbit determination method is proposed in . SBOS is used to track the target in , and a shifted Rayleigh filter (SRF) is designed to improve the tracking accuracy. Koenig et al.  completed autonomous, distributed, and scalable navigation for spacecraft swarms using optical sensors. Kruger et al.  used angle-only measurement for autonomous navigation of spacecraft swarms around planetary bodies.
In recent years, with the rapid development of optical integration and technology of CMOS, volume and power consumption have been reduced significantly, and power consumption is only watt level, to make up for the shortcomings of the limited load of optical imaging micro-/nanosatellites, which have been widely used in SSA, such as the United States observation of space debris telescope, weighing 4 kg and the British SNAP-1 satellite, weighing 6.5 kg. Swedish “Prism” satellite includes two small satellites, with masses of 140 kg and 10 kg and spatial resolution of 30 m. For the micro-/nanosatellites of the United States staring into space, a single satellite weighs less than 5 kg and runs on an orbit of 490~760 km. There are also many applications for reconnaissance missions with micro-/nanosatellites, as shown in Figure 1, the U.S. Army’s satellite “HawkEye” weighs 50 kg and has an image resolution of 1.5 m, and a constellation of 30 micro-/nanosatellites can achieve global all-weather reconnaissance . In Figure 2, the satellite “NanoEye” developed by Microcosm weighs 20 kg and operates at an orbital altitude of 200~300 km with an image resolution of 0.5~0.7 m . The small sensitive tactical satellite weighs about 32 kg and has an image resolution of 1.5~2 m and has three modes of instantaneous scanning, real-time video, and fast imaging. The use of optical imaging micro-/nanosatellite can realize the role of low-orbit antimissile early warning, which is of great significance to protect the space station. The Chinese space station had a dangerous approach event with SpaceX Starlink-1095 satellite on July 1, 2020, and SpaceX Starlink-2305 satellite approached the Chinese space station again on October 21, 2021; the safety of the Chinese space station is greatly affected, and the Chinese space station implemented emergency avoidance for both dangerous events.
How to achieve precise orbit determination of noncooperative targets using optical imaging micro-/nanosatellite has been a hot topic of research. The constraints of SBOS such as Earth occlusion, daylight conditions, and shadow conditions of the Earth lead to short observation arcs for space targets, and the uncertainty of noncooperative target orbits also increases the difficulty of orbit estimation. For the problem of estimating the orbit of space targets with short observation arcs, Ansalone and Curti  used genetic algorithm (GA) to determine the initial orbit, but the algorithm takes long time as well as low tracking accuracy. Li et al.  fused differential evolution and estimation of distribution algorithm for short-arc segment initial orbit determination, which improves the performance of algorithm and accuracy compared with GA. Sciré et al.  used the Levenberg-Marquardt algorithm and Powell dog-leg algorithm for low-orbit target trajectory tracking and the algorithm can converge quickly, but for long-range target, the algorithm suffers from singularity of the Jacobi matrix and convergence speed reduction. Lei et al.  use the geometric method for space-based very short-arc LEO orbit determination, which improves the computational efficiency of orbit determination. Giannitrapani et al., Ceccarelli et al., and Xiong et al. proposed the unscented Kalman filter (UKF), which has since been widely used to track space targets [23–25]. Since the noise statistical properties in the UKF are all a priori and fixed, the process noise and measurement noise will vary with the environment in the problem of tracking space target, but the Kalman filter itself cannot perceive this characteristic; therefore, considering the impact of the practical application environment, it is necessary to study the use of optical imaging micro-/nanosatellite to achieve real-time estimation of noise during the tracking of space noncooperative target with angle-only information. Karamat et al.  proposed a sliding window-based extended Kalman filter (EKF) robust method and used innovative statistics to adaptively adjust system noise and measurement noise. However, EKF is a first-order Taylor formula expansion approximation, and the filtering accuracy is low. Yang et al.  proposed a generalized maximum likelihood-based robust filtering method; due to changes in the environment, the estimation of measurement noise is difficult to attain. Gaussian process regression (GPR) uses observational information to make real-time predictions for sliding windows , and variable Bayesian (VB) method is used to statistically approximate the time-varying noise. However, this method involves a machine learning process, which has a high time cost. In this paper, an intelligent optimization algorithm is proposed to realize real-time estimation of process noise in the process of angle measurement tracking target. The PSO has the advantages of simple structure and fast convergence speed. However, the PSO is easy to fall into the local optimal solution. Dash and Mallick  use the modified PSO (MPSO) to optimize the noise in UKF, but the MPSO does not have high precision for parameter optimization. In this paper, we propose an APSO-SSUKF algorithm to track targets using only the measured angular information, and the main contributions of this paper, compared with previous research, are as follows: (1)SSUKF is used in angle-only tracking of spatial targets, and compared to UKF, SSUKF can reduce the computational effort with guaranteed filtering accuracy(2)The adaptive particle swarm optimization (APSO) has been proposed, in which adaptive learning factors and the adaptive inertia weight are set to increase the global search capability of the particle swarm. Compared with [29, 30], APSO has improved accuracy in parameter optimization(3)In angle-only tracking of targets, APSO-SSUKF can solve the problem of noise variation. Compared with , APSO-SSUKF tracking accuracy is improved, and for noise estimation, APSO has better stability than PSO
The full paper is organized as follows: in Section 2, the model of space-based angle tracking space noncooperative target and the model of space motion system are given. Section 3 proposes the target tracking algorithm based on APSO-SSUKF, uses SSUKF to reduce the cost of calculation of the UKF, adopts the innovation sequence for real-time estimation of measurement noise, and designs the APSO algorithm to combine the estimated measurement noise for real-time tracking of process noise. Section 4 simulates and compares the convergence and estimated accuracy of UKF, SSUKF, PSO-SSUKF, and APSO-SSUKF algorithms for angle-only tracking of space noncooperative target. Section 5 concludes the full paper.
2. The Model of Optical Imaging Micro-/Nanosatellite Angle-Only Target Tracking
2.1. The Model of Space Target Motion
As shown in Figure 3, in the process of space-based target tracking, the target satellite is mainly subjected to J2 perturbation in Earth orbit caused by the nonspherical Earth. The state variable x of the target satellite in the Earth-centered inertial (ECI) coordinate frame, and x = (XT, YT, ZT, vx, vy, vz) T. The equation of motion of the target satellite when considering the J2 perturbation is as follows: where is the standard gravitational constant, is the position vector of the target satellite under ECI coordinate frame, is the distance from the target satellite to the center of the earth, is the Earth radius, and .
Equation (1) can be expressed as . If the state variable at time is known, then the state update at the is as follows:
Since the state equation of the system contains high-order terms, the linearized state prediction equation will bring a large truncation error. Runge-Kutta is a high-precision single-step algorithm. In Equation (3), the local truncation error of the fourth-order Runge-Kutta is . For the discretization processing of Equation (2), the accuracy of the fourth-order Runge-Kutta can meet the requirements, and the formulation is as follows:
2.2. The Model of Observation Based on SBOS
Compared with traditional infrared sensors, SBOS is lighter in weight and lower in power consumption and has higher accuracy of measurement. As shown in Figure 3, this paper divides the space-based noncooperative target tracking into the observation phase and the tracking phase. In the observation phase, the rough orbit of the noncooperative target with errors is obtained through SBOS, and the observation satellite uses SBOS tracking to provide camera pointing through attitude adjustment, reducing the camera pointing deviation and improving the accuracy of detection. In Figure 3, the relative position of the observation satellite and the target satellite in the observation satellite center-of-mass coordinate frame at time is shown, and . The azimuth angle and the elevation angle are used as the equation of measurement as follows: where is the measurement noise.
Assuming that the center of the sensor and the centroid of the observation satellite are coincident, since the measurement is in the observation satellite center-of-mass coordinate frame, and the state variables are in the ECI coordinate frame, the measurement information needs to be converted in real time. As shown in Figure 3, ( and represent the observation satellite and the target satellite, respectively) is the state in the ECI coordinate frame. The formulation for the coordinate transformation of the position vector is as follows:
Equation (5) takes the first derivative with respect to time to get where
When using SBOS to obtain measurement information, in addition to the constraint of earth occlusion, the influence of the solar angle on SBOS imaging is also considered. The geometric relationship between the sun, SBOS, and target satellite is shown in Figure 4.
According to the cosine theorem, the solar angle satisfies the following: where is the distance between the target satellite and the sun and is the distance between the optical device and the sun.
From , the solar angle is 0°~20°, 20°~40°, 40°~60°, 60°~75°, and 75°~90°, and the measurement noises at time are 0.8vĸ, 0.9vĸ, vĸ, 1.1vĸ, and 1.2vĸ, respectively.
3. Target Tracking Based on APSO-SSUKF
3.1. Spherical Simplex Unscented Kalman Filter (SSUKF)
UKF needs to determine the sampling strategy of Sigma points before unscented transformation; the cost of calculation is proportional to the number of Sigma points. To improve the real-time tracking of noncooperative target, this paper uses the SSUKF algorithm, and Sigma points are reduced from to , which on the surface of a sphere with the mean of state as the center of the sphere and a radius of 1. The weights of each Sigma point are required to be equal, except for the weights at the mean of state, and the specific process of SSUKF is as follows: (1)State parameter initialization(2)Calculate Sigma points using spherical simplex unscented transformation (SSUT)
Calculate the weights corresponding to Sigma points.
The following is the sequence of spherical Sigma point vectors with extended dimensions : where . (3)Propagate the Sigma points by the system equation and predicting the state and the covariance (4)The Sigma point is propagated by the observation equation , and the latest observation value is added to update the measurement , and the measurement prediction variance and the cross-covariance are calculated(5)Calculate the filtering gain , and update the state estimate and covariance
3.2. SSUKF Process Noise Estimation Based on APSO
In the iterative process of the SSUKF filtering algorithm, the noise statistical properties are determined a priori and remain constant throughout the filtering process; in fact, when using SBOS for target tracking, the accuracy of measurement is affected by the solar angle on the one hand and related to the pointing of the sensor on the other hand , and the pointing deviation of the sensor will cause all the measurements to deviate from the actual value, causing the noise covariance matrix to change. This paper proposes to use the innovation sequence for real-time estimation of the measurement noise and combine the optimization algorithm for real-time tracking of the process noise to make the system more approximate to the actual value.
The actual covariance of the innovation sequence is as follows: where is the number of samples, usually taken 20~30.
In the EKF, the theoretical covariance matrix of the innovation is as follows:
The innovation theoretical covariance matrix in SSUKF is designed as follows:
The real-time update of measurement noise using innovative theoretical and practical covariance matrices is as follows:
From Equation (20), the theoretical innovation covariance contains both the process noise and the measurement noise . The premise of using the innovation sequence to estimate the noise is that one of the noises is known to estimate the other. During the tracking process of the space noncooperative target, when and change at the same time, SSUKF itself cannot estimate . To solve this problem, this paper proposes to use an optimization algorithm to construct a fitness function with the actual and theoretical covariance of innovation to track in real time. where trace is the trace operation on the matrix.
Considering the fast convergence speed and simple structure of the PSO, it is widely used in parameter optimization [29, 30, 34]. In this paper, the PSO is used to optimize the process noise of SSUKF. Since the inertia weight in PSO is a fixed constant, it is easy to cause the population to fall into a local optimal solution in the optimization process for nonlinear parameter optimization. An adaptive strategy is used to balance the global search ability and local search ability of the particle population, including the adaptive learning factors and the adaptive inertia weight. The number of particle populations in the D-dimensional space is defined as , and the velocity and position of APSO are updated as follows: where where and represent the velocity and position of the th particle in the th iteration (, ), respectively, , , and are random numbers between 0 and 1, is the local optimal solution of the particle at the th iteration, and represents the global optimal solution.
In Equation (24), the learning factors and affect the local search ability and global search ability of the particle population, and the adaptive learning factor is designed as follows: where
As is shown in Figure 5, in the early iteration, the value of is larger and the value of is smaller, which is favorable for the population to perform global search, and in the late iteration, the value of is smaller and the value of is larger, which is favorable for the algorithm to converge globally. is linearly increasing, which is conducive to obtaining information about other particles and improving the global search ability at a later stage to avoid falling into local optimal solutions.
The adaptive inertia weight is as follows: where , and are the fitness value of the th particle in the th iteration, the mean fitness of the th iteration particle, and the maximum fitness of the th iteration particle, respectively.
The flow chart of APSO-SSUKF algorithm is shown in Figure 6.
4. Simulation Analysis
4.1. Test of Convergence of APSO Algorithm
To verify the convergence of the APSO proposed in this paper to solve the optimization parameter, the comparison test of six functions was compared with standard PSO, MPSO in , and CSPSO in . The simulation parameters are set as follows: (i)PSO: , , , and (ii)MPSO: , , and (iii)CSPSO: , , , and (iv)APSO: , , , , , , , and
is the total number of iterations of the algorithm.
The test functions are as follows:
As is shown in Figure 7, compared with the PSO, the APSO has a faster convergence speed. While ensuring the convergence speed, it avoids falling into the local optimal solution and improves the accuracy of convergence.
4.2. Optical Imaging Micro-/Nanosatellite for Tracking Noncooperative Target Based on APSO-SSUKF
This paper uses a 20 kg micro-/nanooptical imaging satellite carrying a high-resolution camera in  operating at an orbital altitude of 550 km, and the entire camera mass is 4.76 kg. For the Chinese space station risk avoidance event and SpaceX Starlink-1095 satellite as a tracking target, which has orbital data from the first batch of TLE data published by SpaceX, as shown in Table 2, the initial state of the observation satellite and the target satellite is described by six orbital elements of semi-major axis , eccentricity , orbital inclination , right ascension of ascending node , argument of perigee , true anomaly .
Due to the limited continuous working time of the optical camera, using the Access function in STK11.6 to generate the solar angle with a continuous visible arc of at the start time of May 5, 2022, at 4:00 (UTC), setting the duration of tracking phase to 2000 s, simulation step of filtering and Runge-Kutta are 1 s and 0.2 s, respectively, and the initial state error is [1, 1, 1, 0.001, 0.001, 0.001]ᵀ, in km and km/s. (10-12, 10-12, 10-12, 10-16, 10-16, and 10-16) and (4, 4). The number of Monte-Carlo ; comparing the errors in the velocity and position of UKF, SSUKF, PSO-SSUKF, and APSO-SSUKF, the performance indicators that define the trajectory tracking are the root mean square error of the position and the root mean square error of the velocity . where and are the estimated position and real position of the space target at time in the ECI coordinate frame in the th Monte-Carlo simulation, respectively.
Figure 8 shows the solar angle of the SBOS in the observation arc under the parameters in Table 2, and Figure 9 shows the orbits of the observation phase and the tracking phase. Since the order of magnitude of the process noise in the position and velocity directions is inconsistent, the optimization parameters are set to , , and dimension , and the rest of the initialization parameters are set with the test function simulation. It can be seen from Figure 6 that the SSUKF filter uses APSO to optimize once every updates. Setting and the update optimization result , the optimization parameters of by PSO-SSUKF and APSO-SSUKF are shown in Figure 10.
It is obvious from Figure 10 that the process noise can be tracked in real time by combining the optimization algorithm when measurement noise changes. When and are optimized, the APSO-optimized value fluctuates in a small range around the initial value. Compared with the PSO algorithm, the APSO algorithm has better stability.
Figure 11 shows the prediction errors of different algorithms for the position and velocity of space target; UKF and SSUKF diverge within the first 500 s; this is because the covariance of innovation changes in UKF and SSUKF with the change of measurement noise, and UKF and SSUKF cannot track the process noise. The optimization algorithm can track the process noise in real time, which can make the predicted trajectory closer to the real trajectory. The APSO-SSUKF has a faster convergence speed of the error curve in the angle-only target tracking, the fluctuation is smaller, and the algorithm is more stable.
From Table 3, compared with UKF, SSUKF, and PSO-SSUKF, the RMSE of position of APSO-SSUKF is reduced by 45.44%, 35.26%, and 20.94%, respectively, and the RMSE of velocity of APSO-SSUKF is reduced by 45.58% and 33.53% and 16.33%, respectively. SSUKF can reduce the cost of calculation on the premise of ensuring the accuracy of estimation, and the APSO-SSUKF algorithm improves the accuracy of estimation in angle-only noncooperative target tracking.
Optical imaging micro-/nanosatellite can achieve high-precision real-time tracking of space noncooperative target based on APSO-SSUKF considering J2 perturbation. The simulation results show that compared with the UKF, the SSUKF can reduce the cost of calculation under the premise of ensuring the accuracy of estimation. In the case of measurement noise changes, compared with PSO, the process noise estimated by APSO is more stable, which improves the stability of tracking, and APSO-SSUKF can avoid the problem of divergence caused by the filter not being able to perceive changes in process noise. From the simulation results, RMSE of the position and velocity of the APSO-SSUKF in the noncooperative target is significantly reduced, making the predicted trajectory closer to the real trajectory, and the algorithm improves the convergence speed of the error curve. APSO-SSUKF can use the angle data to achieve high-precision real-time tracking of noncooperative targets.
The simulation date and the program files in this paper cannot be shared with others as they are our basis for next research.
Conflicts of Interest
The authors declared that they have no conflicts of interest to this work.
B. Hua and G. Yang conceived the idea of this review. Y. Wu and Z. Chen supervised the study. B. Hua and G. Yang wrote the manuscript. B. Hua and G. Yang revised the manuscript. All authors discussed the results and contributed to the final version of the manuscript.
This work was supported by the National Natural Science Foundation of China under Grants 61973153 and 62073165.
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