Orbit determination and thrust estimation for noncooperative target using angle-only measurement

First, the author introduced the non-maneuvering prediction model, the target maneuvering prediction model, and the angle-only measurement model used by the filter. In the non-maneuvering prediction model, the filter’s state variables include the three-axis position and velocity in the J2000 coordinate system, represented as \( X_1 = [r_x, r_y, r_z, v_x, v_y, v_z]^T \). The nonlinear differential equation of the state variables \( X_1 \) uses 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 complexity. The state variables of the filter in the target maneuvering prediction model consist of the three-axis position and velocity in the J2000 coordinate system and the acceleration in the local vertical local horizontal (LVLH) coordinate system, represented as \( X_2 = [r_x, r_y, r_z, v_x, v_y, v_z, a_x^{LVLH}, a_y^{LVLH}, a_z^{LVLH}]^T \), with the assumption that the target has 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 measurement model, as shown in Figure 1, involves two cooperative satellites (i.e., tracking spacecraft) forming a dual-satellite system that uses detection cameras to photograph and track the target’s center of mass, 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 that the camera coordinate system coincides with the LVLH coordinate system of the tracking spacecraft. The observation equation is \( Z = [\alpha_1, \beta_1, \alpha_2, \beta_2]^T = h(X) \), which includes measurement noise and the components of the relative LOS unit vector between the camera and the target; the corresponding observation matrix is \( H = [H_{\alpha1}, H_{\beta1}, H_{\alpha2}, H_{\beta2}; 01×3, 01×3, 01×3, 01×3]^T \), containing \( \alpha_i \) and \( \beta_i \).

Subsequently, the author briefly introduced the classic IMM algorithm and elaborated on the proposed VAIMM-STEKF algorithm. The iterative process of the classic IMM algorithm consists of four steps: model interaction, parallel filtering, model probability update, and fusion estimation. Based on the framework of the classic IMM algorithm, the proposed VAIMM-STEKF algorithm is composed 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 selected: 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 for unbiased expansion of Model 1. The adaptive TPM guides model switching during the model interaction step, where changes in model probability are mainly related to the likelihood functions of the two filters and the predicted model probabilities. Noise can weaken the likelihood function of the matching model, reducing its probability, which may lead to inaccurate model selection; however, by reasonably defining a transition probability correction function, noise can be suppressed by modifying the transition probability. Meanwhile, Model 2 is designed as a strong tracking filter to ensure sufficient sensitivity to changes in measurement information and higher tracking accuracy; to suppress the problem of bandwidth amplification caused by noise in the strong tracking filter, a smoothing factor γ is also introduced.

Finally, the author presented the simulation results and designed two sets of simulations to verify the VAIMM-STEKF algorithm’s (1) advantages in tracking accuracy and convergence speed compared to four other algorithms and (2) applicability in high-dynamic scenarios. In the orbit determination and acceleration estimation simulations for maneuvering spacecraft, the spacecraft’s acceleration was assumed to be unknown. In Simulation 1, the tracking accuracy and acceleration estimation errors of the VAIMM-STEKF, IMM, AIMM-STEKF, VIMM-STEKF, and VAIMM-EKF algorithms were compared. All five algorithms included Model 1 and Model 2 and were tested under 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 improved steady-state tracking accuracy during non-maneuvering phases; under different maneuvering accelerations, the position accuracy of VAIMM-STEKF improved by at least 27%, and the velocity accuracy improved by at least 17%; (2) The VAIMM-STEKF algorithm converged faster than the IMM algorithm during spacecraft maneuvers; (3) The VAIMM-STEKF algorithm provided the most accurate acceleration estimates and could be applied to a broader range of acceleration magnitudes. Simulation 2 also simulated the smoothing factor γ and attenuation factor b, showing that: (1) The VAIMM-STEKF algorithm is well-suited for high-dynamic scenarios; (2) When γ is large, the convergence speed is slightly slower, and the steady-state tracking accuracy is slightly better; when γ is small, the opposite is true; the value of γ indicates the sensitivity of the strong tracking filter’s adaptive bandwidth adjustment; (3) The smaller the value of b, the larger the value of ρ, and the more sensitive the filter is to changes in model probability; if b is particularly small, the filter’s estimation results may exhibit significant jumps.