With the development of satellite miniaturization, intelligence, and clustering, a single satellite platform can no longer support high-power detection equipment. Monocular angle-only cameras have become standard detection devices for satellites due to their low power consumption, high reliability, and long detection range. In the context of space confrontation, the requirement for target maneuvering recognition necessitates that the filter can accurately and quickly track the target. However, due to the limitations of observability, a single satellite can only measure angles and cannot fully achieve precise tracking and estimation of the maneuvering spacecraft’s acceleration. A dual-satellite angle-only observation system and the Interactive Multiple Model (IMM) algorithm are effective methods to improve the overall system observability to meet the requirements of long-distance tracking and acceleration estimation of maneuvering spacecraft. However, traditional IMM algorithms have shortcomings such as poor steady-state tracking accuracy, strong dependence on sub-filter parameter selection, model mismatches, and the inability to determine the motion of the target spacecraft.

In a newly published research article in *Space: Science & Technology*, scholars from the University of Chinese Academy of Sciences propose a variable-dimensional adaptive IMM strong tracking filter algorithm (VAIMM-STEKF) to address these issues. This algorithm features faster convergence speed, higher steady-state tracking accuracy, simpler filter parameter selection, and fewer models.

Firstly, the authors introduce the non-maneuvering prediction model, the maneuvering prediction model, and the angle-only observation model used by the filter. In the non-maneuvering prediction model, the state variables of the filter include the three-axis positions and velocities in the J2000 coordinate system, i.e., \( X_1 = [r_x, r_y, r_z, v_x, v_y, v_z]^T \). The nonlinear differential equations of state variables \( X_1 \) adopt the GGM03S gravitational field model expressed by a 21st-degree spherical harmonic function; the Jacobian matrix \( A_1 \) of the recursive model in the EKF is calculated using a simplified two-body model to reduce computational load. The state variables of the maneuvering prediction model filter consist of the three-axis positions and velocities in the J2000 coordinate system and the accelerations in the local vertical local horizontal (LVLH) coordinate system of the target, i.e., \( X_2 = [r_x, r_y, r_z, v_x, v_y, v_z, a_x^{LVLH}, a_y^{LVLH}, a_z^{LVLH}]^T \), assuming constant acceleration \( da/dt = 0 \); the Jacobian matrix \( A_2 \) of the recursive model is also calculated using a simplified two-body model. The angle-only observation model, as shown in Figure 1, involves two cooperating satellites (i.e., tracking spacecraft) forming a dual-satellite system, using detection cameras to photograph and track the target centroid, obtaining line-of-sight (LOS) information (azimuth angle \( \beta \) and elevation angle \( \alpha \)). It is assumed that the dual-satellite system simultaneously outputs angle measurements of the target, and the camera coordinate system coincides with the LVLH coordinate system of the tracking spacecraft. The observation equation is \( Z = [\alpha_1, \beta_1, \alpha_2, \beta_2]^T = h(X) \), containing measurement noise and the components of the relative LOS unit vector between the camera and the target; the corresponding observation matrix is \( H = [H_{\alpha_1}, H_{\beta_1}, H_{\alpha_2}, H_{\beta_2}; 0_{1×3}, 0_{1×3}, 0_{1×3}, 0_{1×3}]^T \), containing \( \alpha_i \) and \( \beta_i \).

Fig. 1. Double satellite observation model.

Subsequently, the authors briefly introduce the classical IMM algorithm and detail the proposed VAIMM-STEKF algorithm. The iterative process of the classical IMM algorithm includes four steps: model interaction, parallel filtering, probability model update, and fusion estimation. Based on the framework of the classical IMM algorithm, the proposed VAIMM-STEKF algorithm consists of three parts: variable-dimension model interaction and fusion, adaptive transition probability matrix (TPM), and normalized residual strong tracking filter. For model interaction, an appropriate model mixing estimation strategy must be chosen: in the state interaction of the low-dimensional model (Model 1), the redundant components of the high-dimensional model can be directly discarded; in the state interaction of the high-dimensional model (Model 2), the dimensions of the low-dimensional model must be appropriately expanded and then combined. Specifically, if the true state of the target acceleration is zero, Model 1 can be expanded with zero mean and zero covariance; in the case of target maneuvering, the acceleration and covariance matrix of the state variables in Model 2 can be used to unbiasedly expand Model 1. The adaptive TPM guides model transitions in the model interaction step, with model probability changes mainly related to the likelihood functions of the two filters and the model prediction probabilities. Noise can weaken the likelihood function of the matching model, reducing the probability of the matching model and leading to inaccurate model selection; defining a reasonable transition probability correction function allows the transition probability to be modified to suppress noise. Additionally, Model 2 is designed as a strong tracking filter to ensure sufficient sensitivity to changes in measurement information and higher tracking accuracy; a smoothing factor \( \gamma \) is introduced to suppress the bandwidth amplification problem of the strong tracking filter caused by noise.

Fig. 2. VAIMM-STEKF algorithm flow chart.

Finally, the authors present simulation results, designing two sets of simulations to verify the advantages of VAIMM-STEKF in tracking accuracy and convergence speed compared to four other algorithms, and its applicability in highly dynamic scenarios. In the orbit determination and acceleration estimation simulation of a maneuvering spacecraft, the spacecraft’s acceleration is assumed to be unknown. In Simulation 1, the tracking accuracy and acceleration estimation errors of VAIMM-STEKF, IMM, AIMM-STEKF, VIMM-STEKF, and VAIMM-EKF algorithms are compared, with all five algorithms including Model 1 and Model 2, and testing three acceleration values: 0.1 m/s², 0.01 m/s², and 0.001 m/s². The main simulation results are shown in Figures 6-8, leading to the following conclusions:

1. Compared to the IMM algorithm, the VAIMM-STEKF algorithm improves the steady-state tracking accuracy in non-maneuvering phases; the position accuracy of VAIMM-STEKF can be improved by at least 27% and the velocity accuracy by at least 17% under different maneuvering accelerations.

2. The VAIMM-STEKF algorithm has a faster convergence speed during spacecraft maneuvers than the IMM algorithm.

3. The VAIMM-STEKF algorithm provides the most accurate acceleration estimates and can be applied to a wider range of acceleration magnitudes.

Simulation 2 also simulates the smoothing factor \( \gamma \) and decay factor \( b \), showing:

1. The VAIMM-STEKF algorithm is well-suited to highly dynamic scenarios.

2. Larger \( \gamma \) values result in slightly slower convergence speed but better steady-state tracking accuracy; smaller \( \gamma \) values result in the opposite; the size of \( \gamma \) indicates the sensitivity of the strong tracking filter’s adaptive bandwidth adjustment.

3. Smaller \( b \) values and larger \( \rho \) values make the filter more sensitive to model probability changes; if \( b \) is too small, the filter’s estimation results exhibit significant jumps.

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