Adaptive Robust Cubature Filtering-Based Autonomous Navigation for Cislunar Spacecraft Using Inter-Satellite Ranging and Angle Data

Abstract

The Linked Autonomous Interplanetary Satellite Orbit Navigation (LiAISON) technique enables cislunar spacecraft to obtain accurate position and velocity information, allowing full state estimation of two vehicles using only inter-satellite range (ISR) measurements when both their dynamical states are unknown. However, its stand-alone use leads to significantly increased orbit determination errors when the orbital planes of the two spacecraft are nearly coplanar, and is characterized by long initial convergence times and slow recovery following dynamical disturbances. To mitigate these issues, this study introduces an integrated navigation method that augments inter-satellite range measurements with line-of-sight vector angles relative to background stars. Additionally, an enhanced Adaptive Robust Cubature Kalman Filter (ARCKF) incorporating a chi-square test-based adaptive forgetting factor (AFF-ARCKF) is developed. This algorithm performs adaptive estimation of both process and measurement noise covariance matrices, improving convergence speed and accuracy while effectively suppressing the influence of measurement outliers. Numerical simulations involving spacecraft in Earth–Moon L4 planar orbits and distant retrograde orbits (DRO) confirm that the proposed method significantly enhances system observability under near-coplanar conditions. Comparative evaluations demonstrate that AFF-ARCKF achieves faster convergence compared to the standard ARCKF. Further analysis examining the effects of initial state errors and varying initial forgetting factors clarifies the operational boundaries and practical applicability of the proposed algorithm.

1. Introduction

Since the beginning of the 21st century, leading spacefaring nations have launched ambitious lunar exploration programs, such as the Artemis Program and Moon Village, with the long-term goal of establishing a sustained human presence and utilizing lunar resources. Cislunar operations are poised for significant expansion in the coming years. Currently, orbit determination for these spacecraft relies primarily on measurements from the Deep Space Network, incorporating range and angle data with dynamic orbit models. However, this ground-based approach requires long observation arcs to achieve high accuracy and struggles to support simultaneous orbit determination for multiple spacecraft.

In order to provide accurate position and velocity information for cislunar spacecraft, LiAISON method was proposed in ref. [1], which utilizes range measurements between spacecraft in distinct three-body gravitational regimes, combined with dynamic filtering, to achieve autonomous state estimation. The United States and China have launched missions such as CAPSTONE and a DRO constellation, respectively, to validate this technology [2].

Reference [3] used covariance analysis to examine the observability performance of ISR-based libration point orbiter navigation. It was found that orbits with shorter periods and greater deviations from the lunar orbital plane yield higher accuracy, whereas accuracy degrades significantly when the orbital planes of the two spacecraft are nearly aligned. Further research in ref. [4] studied autonomous orbit determination using range measurements between spacecraft in Earth–Moon libration point orbits, DRO, and lunar orbits, concluding that Halo orbits with greater out-of-plane amplitudes can provide higher navigation accuracy and broader lunar coverage. An analysis of orbit determination accuracy in a hybrid constellation scenario involving libration point orbiters, DRO spacecraft, and medium Earth orbit satellites was also conducted in ref. [5].

In the area of filtering robustness, the integration of Kalman filtering with reinforcement learning has emerged as a promising approach to address noise and model uncertainties [6,7]. For instance, a reinforcement learning-based adaptive Kalman filter was developed in ref. [8], though its application remains limited by the substantial data requirements of reinforcement learning—a challenge in deep space autonomous navigation. Another line of research focuses on adaptive filtering based on innovation or residual sequences. A residual-based fuzzy adaptive extended Kalman filter was proposed in ref. [9], while adaptive factors were introduced to adjust the covariance matrix of dynamic model information in ref. [10], thereby balancing the contributions of model predictions and real-time GNSS observations while mitigating dynamic model errors.

Existing studies generally address the issue of degraded accuracy and slow convergence in near-coplanar configurations by selecting spacecraft in distinctly different orbital planes or by adding inter-satellite links [11]. However, many common cislunar trajectories—such as Earth–Moon transfer orbits, DROs, and L4 orbits—exhibit only small inclinations relative to the lunar plane. Moreover, establishing additional inter-satellite links across vast cislunar distances entails significant engineering challenges and costs. Therefore, developing high-precision autonomous orbit determination methods applicable to near-coplanar orbital configurations is both a practical necessity and valuable for future missions.

Building on the LiAISON framework, this paper introduces inter-satellite angle measurements and proposes an improved Adaptive Robust Cubature Kalman Filter (AFF-ARCKF) that utilizes a chi-square test to adaptively adjust the forgetting factor. This multi-source data fusion approach enhances system observability, ensures filtering robustness, and improves both convergence speed and accuracy, thereby enabling high-precision autonomous orbit determination for cislunar spacecraft in near-coplanar orbits.

In summary, the main contributions of this work are as follows:

• An autonomous navigation method for two near-coplanar Earth–Moon spacecraft: Based on the LiAISON framework, the proposed method combines inter-satellite ranging with inter-satellite stellar background angle measurement. By incorporating angular observations, it addresses the challenge of achieving high-precision state estimation for two spacecraft in near-coplanar orbits.
• Improved convergence speed of the navigation filter: Compared to the conventional LiAISON method and the standard ARCKF, the proposed navigation scheme and algorithm significantly reduce the convergence time of state estimation errors. This is achieved by augmenting the measurement set with angular data and adaptively tuning the forgetting factor for process noise covariance estimation via a chi-square test.
• Enhanced robustness of the navigation algorithm: Relative to the standard ARCKF, the improved method further strengthens the filter’s ability to suppress measurement anomalies. By adjusting the forgetting factor adaptively based on the chi-square test, it mitigates filter fluctuations caused by abnormal measurements and increases overall operational stability.

2. System Model for Orbit Determination

This study focuses on the performance of spacecraft navigation in cislunar space, utilizing the Circular Restricted Three-Body Problem (CRTBP) model as the core dynamical framework. Through reasonable simplification, this model effectively characterizes the complex motion patterns of the spacecraft under the combined gravitational effects of the Earth and the Moon, while retaining the main characteristics of their gravitational forces. It provides a reliable theoretical foundation for in-depth analysis of navigation performance.

In the specific modeling process, the Earth–Moon centroid rotating coordinate system is selected as the reference frame. This coordinate system rotates synchronously with the Earth–Moon system. Describing spacecraft motion within this framework can visually present its relative spatial geometric configuration and naturally separate effects such as centrifugal force and Coriolis force, facilitating theoretical analysis and numerical calculations. To further achieve the universality and numerical stability of the equations, the study adopted normalization processing based on characteristic quantities: using the average distance between the Earth and the Moon as the length unit, the reciprocal of the system’s orbital angular velocity as the time unit, and scaling parameters such as mass accordingly. This processing not only eliminates dimensions and simplifies the form of the equations, but also enables the research results to be easily scaled and applied to different specific task scenarios.

In the Earth–Moon barycentric rotating frame, the origin O is located at the barycenter of the Earth–Moon system, while points E and M represent the centers of the Earth and the Moon, respectively. The O-XY plane coincides with the Moon’s orbital plane, with the X-axis pointing from the origin toward the Moon’s center. The Z-axis is aligned with the normal direction of the orbital plane, and the Y-axis forms a right-handed coordinate system together with the X- and Z-axes. The configuration of the autonomous orbit determination method, which utilizes both inter-satellite ranging and lunar vector observations, is illustrated in Figure 1. In the figure, ${\mathit{r}}_1$ and ${\mathit{r}}_2$ denote the position vectors of the two spacecraft, ${\mathit{l}}_{12}$ and ${\mathit{l}}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{r}}$ represent the line-of-sight vectors from Spacecraft 2 to Spacecraft 1 and to a reference star, respectively, and $\theta$ denotes the angular separation between ${\mathit{l}}_{12}$ and ${\mathit{l}}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{r}}$.

2.1. State Equation

The motion equation of the Spacecraft in the Earth Moon centroid convergence coordinate system is [12]:

$${\dot{\mathit{X}}}_{\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{e}\mathrm{c}\mathrm{r}\mathrm{a}\mathrm{f}\mathrm{t}}=\left\{\begin{array}{c}\dot{x}=v_{\mathrm{x}}\hfill \\ \dot{y}=v_{\mathrm{y}}\hfill \\ \dot{z}=v_{\mathrm{z}}\hfill \\ {\dot{v}}_{\mathrm{x}}=x-{\displaystyle \frac{\left(1-\mu \right)\left(x+\mu \right)}{{\left({\left(x+\mu \right)}^2+y^2+z^2\right)}^{3/2}}}-{\displaystyle \frac{\mu \left(x+\mu -1\right)}{{\left({\left(x+\mu -1\right)}^2+y^2+z^2\right)}^{3/2}}}+2v_y\hfill \\ {\dot{v}}_{\mathrm{y}}=y-{\displaystyle \frac{\left(1-\mu \right)y}{{\left({\left(x+\mu \right)}^2+y^2+z^2\right)}^{3/2}}}-{\displaystyle \frac{\mu y}{{\left({\left(x+\mu -1\right)}^2+y^2+z^2\right)}^{3/2}}}-2v_x\hfill \\ {\dot{v}}_{\mathrm{z}}=-{\displaystyle \frac{\left(1-\mu \right)z}{{\left({\left(x+\mu \right)}^2+y^2+z^2\right)}^{3/2}}}-{\displaystyle \frac{\mu z}{{\left({\left(x+\mu -1\right)}^2+y^2+z^2\right)}^{3/2}}}\hfill \end{array}\right.$$

(1)

where ${\mathit{X}}_{\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{e}\mathrm{c}\mathrm{r}\mathrm{a}\mathrm{f}\mathrm{t}}={\left[x,y,z,v_{\mathrm{x}},v_{\mathrm{y}},v_{\mathrm{z}}\right]}^{\mathrm{T}}$ represents the position and velocity of the spacecraft, and $\mu =m_2/\left(m_1+m_2\right)$, $m_1=5.965\times {10}^{24}\mathrm{kg}$, $m_2=7.342\times {10}^{22}\mathrm{k}\mathrm{g}$ denote the masses of the Earth and Moon, respectively. In the normalization process, the average Earth–Moon distance is taken as 384,401 km.

The proposed integrated orbit determination method is investigated using two spacecraft operating in distinct cislunar orbits as an example. The methodology can be readily extended to scenarios involving multiple spacecraft for autonomous orbit determination. For the two-spacecraft system, we define the augmented state vector as:

$$\mathit{X}={\left[{\mathit{X}}_{\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{e}\mathrm{c}\mathrm{r}\mathrm{a}\mathrm{f}\mathrm{t}\_1}^{\mathrm{T}},{\mathit{X}}_{\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{e}\mathrm{c}\mathrm{r}\mathrm{a}\mathrm{f}\mathrm{t}\_2}^{\mathrm{T}}\right]}^{\mathrm{T}}={\left[x_1,y_1,z_1,v_{\mathrm{x}1},v_{\mathrm{y}1},v_{\mathrm{z}1},x_2,y_2,z_2,v_{\mathrm{x}2},v_{\mathrm{y}2},v_{\mathrm{z}2}\right]}^{\mathrm{T}}$$

where ${\mathit{X}}_{\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{e}\mathrm{c}\mathrm{r}\mathrm{a}\mathrm{f}\mathrm{t}\_i}^{\mathrm{T}}$ represents the state of the ith spacecraft $\left(i=1,2\right)$. The state equation of the orbit determination system can be expressed as

$$\dot{\mathit{X}}=\mathit{F}\left(\mathit{X}\right)$$

(2)

2.2. Measurement Equation

2.2.1. Dual One-Way Ranging

The spacecraft can perform dual one-way ranging via inter-satellite links [13]. The geometric distance between the two spacecraft is given by:

$${\rho }_{\mathrm{true}}(t)=\sqrt{{(x_1-x_2)}^2+{(y_1-y_2)}^2+{(z_1-z_2)}^2}$$

(3)

After time synchronization, the actual range measurement obtained from the dual one-way ranging process is

$${\rho }_{\mathrm{m}}(t)={\rho }_{\mathrm{true}}(t)+v_{\mathrm{d}}(t)=h_{\mathrm{d}}(\mathit{X}(t))+v_{\mathrm{d}}(t)$$

(4)

where $v_{\mathrm{d}}\left(t\right)$ represents the ranging noise.

2.2.2. Astronometric Angle Measurement

Traditional astronometric angle measurement methods typically use the angular separation between a nearby celestial body and a background star as the observation [14]. Since the states of both spacecraft are unknown, independent astronometric angle measurements for each are required to enhance the observability of the state components orthogonal to the orbital plane.

This paper employs the spacecraft 1–spacecraft 2–star angular separation as the angle measurement. This observable contains state information from both spacecraft simultaneously, effectively reducing the complexity of the orbit determination system.

The cosine of the angle between the inter-satellite vector and the star direction is given by:

$$\cos {\theta }_{\mathrm{ture}}={\displaystyle \frac{{\mathit{l}}_{12}\cdot {\mathit{l}}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{r}}}{\left|{\mathit{l}}_{12}\right|\cdot \left|{\mathit{l}}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{r}}\right|}}$$

(5)

The actual astrometric measurement is [15,16]:

$$\cos {\theta }_{\mathrm{m}}(t)=\cos {\theta }_{\mathrm{true}}(t)+v_{\mathrm{a}}(t)=h_{\mathrm{a}}(\mathit{X}(t))+v_{\mathrm{a}}(t)$$

(6)

where $v_{\mathrm{a}}(t)$ represents the ranging noise.

The measurement equation for the integrated orbit determination system is:

$$\mathit{Z}(t)=\mathit{h}(X(t))+\mathit{V}(t)=\left[\begin{array}{c}{h}_{\mathrm{d}}(\mathit{X}(t))\\ h_{\mathrm{a}}(\mathit{X}(t))\end{array}\right]+\left[\begin{array}{c}{v}_{\mathrm{d}}(t)\\ v_{\mathrm{a}}(t)\end{array}\right]$$

(7)

where $\mathit{h}(\mathit{X}(t))$ is the deterministic measurement model vector, and $\mathit{V}(t)$ is the measurement noise vector at time $t$.

3. Filtering Algorithm

3.1. Cubature Kalman Filter

The inter-satellite angle-and-range orbit determination system is nonlinear. The Cubature Kalman Filter (CKF) is employed to ensure orbit determination accuracy within this nonlinear framework.

The computational procedure of the CKF algorithm is as follows [17]:

(1)

Set initial parameters ${\widehat{\mathit{X}}}_0$, ${\mathit{P}}_0$, ${\mathit{R}}_0$, and ${\mathit{Q}}_0$.

(2)

Calculate the volume point at k − 1 based on volume transformation.

$${\mathit{P}}_{k-1}={\mathit{S}}_{k-1}{\mathit{S}}_{k-1}^{\mathrm{T}}$$

(8)

Calculate Volume Point:

$$\begin{array}{c}{\mathit{\chi }}_{k-1}^i={\widehat{\mathit{X}}}_{k-1}+{\mathit{S}}_{k-1}{\xi }_i\left(i=1,2,\dots ,2n\right)\hfill \\ {\xi }_i=\sqrt{n}{\left[1\right]}_i\hfill \end{array}$$

(9)

where ${\left[1\right]}_i$ Represents the i-th column in point set $\left[1\right]$,$\left[1\right]=\left[{\mathit{I}}_{n\times n},-{\mathit{I}}_{n\times n}\right]$ represents a complete set of completely symmetric points.

(3)

Time update:

$$\begin{array}{ccc}\hfill {\mathit{\chi }}_{k-1,k-1}^i& =& \mathit{f}\left({\mathit{\chi }}_{k-1}^i,k-1\right)\hfill \\ \hfill {\widehat{\mathit{X}}}_{k,k-1}& =& {\displaystyle \frac{1}{2n}}{\displaystyle \sum _{i=1}^{2n}{\mathit{\chi }}_{k-1,k-1}^i}\hfill \\ \hfill {\mathit{P}}_{k,k-1}& =& {\displaystyle \frac{1}{2n}}{{\displaystyle \sum _{i=1}^{2n}\left[{\mathit{\chi }}_{k-1,k-1}^i-{\widehat{\mathit{X}}}_{k,k-1}\right]\left[{\mathit{\chi }}_{k-1,k-1}^i-{\widehat{\mathit{X}}}_{k,k-1}\right]}}^{\mathrm{T}}+{\mathit{Q}}_{k-1}\hfill \\ \hfill {\mathit{P}}_{k,k-1}& =& {\mathit{S}}_{k,k-1}{\mathit{S}}_{k,k-1}^{\mathrm{T}}\hfill \\ \hfill {\mathit{\chi }}_{k,k-1}^i& =& {\widehat{\mathit{X}}}_{k,k-1}+{\mathit{S}}_{k,k-1}{\xi }_i\hfill \\ {\overline{\mathit{h}}}_{k,k-1}^i\hfill & =& \mathit{h}\left({\mathit{\chi }}_{k,k-1}^i,k-1\right)\hfill \\ \hfill {\widehat{\mathit{Z}}}_{k,k-1}& =& {\displaystyle \frac{1}{2n}}{\displaystyle \sum _{i=1}^{2n}{\overline{\mathit{h}}}_{k,k-1}^i}\hfill \end{array}$$

(10)

(4)

Filter gain update:

$$\begin{array}{ccc}\hfill {\mathit{P}}_{{\left(\mathrm{h}\mathrm{h}\right)}_{k,k-1}}& =& {\displaystyle \frac{1}{2n}}{{\displaystyle \sum _{i=1}^{2n}\left[{\overline{\mathit{h}}}_{k,k-1}^i-{\widehat{\mathit{Z}}}_{k,k-1}\right]\left[{\overline{\mathit{h}}}_{k,k-1}^i-{\widehat{\mathit{Z}}}_{k,k-1}\right]}}^{\mathrm{T}}\hfill \\ \hfill {\mathit{P}}_{{\left(\mathrm{Z}\mathrm{Z}\right)}_{k,k-1}}& =& {\mathit{P}}_{{\left(\mathrm{h}\mathrm{h}\right)}_{k,k-1}}+{\mathit{R}}_{k-1}\hfill \\ \hfill {\mathit{P}}_{{\left(\mathrm{X}\mathrm{Z}\right)}_{k,k-1}}& =& {\displaystyle \frac{1}{2n}}{{\displaystyle \sum _{i=1}^{2n}\left[{\mathit{\chi }}_{k,k-1}^i-{\widehat{\mathit{X}}}_{k,k-1}\right]\left[{\overline{\mathit{h}}}_{k,k-1}^i-{\widehat{\mathit{Z}}}_{k,k-1}\right]}}^{\mathrm{T}}\hfill \\ \hfill {\mathit{K}}_k& =& {\mathit{P}}_{{\left(\mathrm{X}\mathrm{Z}\right)}_{k,k-1}}{\mathit{P}}_{{\left(\mathrm{Z}\mathrm{Z}\right)}_{k,k-1}}^{-1}\hfill \end{array}$$

(11)

(5)

Calculate filter value:

$$\begin{array}{ccc}\hfill {\widehat{\mathit{X}}}_k& =& {\widehat{\mathit{X}}}_{k,k-1}+{\mathit{K}}_k\left[{\mathit{Z}}_k-{\widehat{\mathit{Z}}}_{k,k-1}\right]\hfill \\ \hfill {\mathit{P}}_k& =& {\mathit{P}}_{k,k-1}-{\mathit{K}}_k{\mathit{P}}_{{\left(\mathrm{Z}\mathrm{Z}\right)}_{k,k-1}}{\mathit{K}}_k^{\mathrm{T}}\hfill \end{array}$$

(12)

3.2. Enhanced Adaptive Robust Cubature Kalman Filter

The standard CKF does not dynamically update the ${\mathit{R}}_k$ and ${\mathit{Q}}_k$ This limitation becomes apparent when outlier errors are present in the measurement data, as the ${\mathit{R}}_k$ fails to accurately represent the reliability of such measurements during filtering, leading to increased estimation errors. Furthermore, dynamic model inaccuracies, notably those arising from uncertainties in solar radiation pressure, significantly impact spacecraft in cislunar three-body orbits. An inaccurate ${\mathit{Q}}_k$ also adversely affects the filter’s final accuracy and convergence speed. To comprehensively address the aforementioned issues, this paper proposes an improved adaptive robust cubature Kalman filter (AFF-ARCKF), which incorporates a chi-square test to adjust the forgetting factor, thereby enhancing the robustness, convergence rate, and precision of the orbit determination filter.

Defining the innovation vector of the system as ${\mathit{V}}_{{\mathrm{z}}_k}$, its expression is given by

$${\mathit{V}}_{z_k}={\widehat{\mathit{Z}}}_k-{\widehat{\mathit{Z}}}_{k,k-1}$$

(13)

The adaptive ${\overline{\mathit{R}}}_k$ is defined as

$${\overline{\mathit{R}}}_k={\displaystyle \frac{1}{{\mathit{A}}_k}}{\mathit{R}}_k$$

(14)

where ${\mathit{A}}_k=diag\left(\left[\begin{array}{cccc}{\alpha }_k^1& {\alpha }_k^2& \dots & {\alpha }_k^m\end{array}\right]\right)$ is the robust factor matrix, and m is the measurement dimension.

The robust factor ${\alpha }_k^i$ can be constructed as follows [18]:

$${\alpha }_k^i=\left\{\begin{array}{cc}1& {\overline{v}}_{z_k}^i\le k_0\\ {\displaystyle \frac{k_0}{{\overline{v}}_{z_k}^i}}\left({\displaystyle \frac{k_1-{\overline{v}}_{z_k}^i}{k_1-k_0}}\right)& k_0<{\overline{v}}_{z_k}^i\le k_1\\ {10}^{-20}& {\overline{v}}_{z_k}^i>k_1\end{array}\right.$$

(15)

Here, $k_0$ and $k_1$ are empirical parameters whose values can be determined through trial-and-error analysis of the innovation characteristics of the orbit determination method. ${\overline{v}}_{z_k}^i$ represents the theoretical innovation covariance, satisfying

$$\begin{array}{c}{\overline{\mathit{V}}}_{z_k}={\left[\begin{array}{cccc}{\overline{v}}_{z_k}^1& {\overline{v}}_{z_k}^2& \dots & {\overline{v}}_{z_k}^m\end{array}\right]}^{\mathrm{T}}\hfill \\ =\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left(\left|\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left({\mathit{V}}_{z_k}\right)\cdot \mathrm{s}\mathrm{q}\mathrm{r}\mathrm{t}\left(\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left({\mathit{P}}_{{\left(ZZ\right)}_{k,k-1}}^{-1}\right)\right)\right|\right)\hfill \end{array}$$

(16)

To mitigate the impact of dynamic model errors on the accuracy of position and velocity estimation, ${\mathit{Q}}_k$ is adaptively estimated:

$$\begin{array}{c}{\mathit{Q}}_k^{\ast }=\left({\widehat{\mathit{X}}}_k-{\widehat{\mathit{X}}}_{k,k-1}\right){\left({\widehat{\mathit{X}}}_k-{\widehat{\mathit{X}}}_{k,k-1}\right)}^{\mathrm{T}}-\left[{\mathit{P}}_k-\left({\mathit{P}}_{k,k-1}-{\mathit{Q}}_{k-1}\right)\right]\hfill \\ {\mathit{Q}}_k={\mathit{Q}}_{k-1}+{\beta }_k\left({\mathit{Q}}_k^{\ast }-{\mathit{Q}}_{k-1}\right)\hfill \end{array}$$

(17)

Here, $\beta \in \left(0,1\right)$ is a weighting factor. A larger $\beta$ value means the estimation of ${\mathit{Q}}_k$ relies more heavily on the observed process noise covariance at time k, whereas a smaller $\beta$ makes the estimation favor the previous value ${\mathit{Q}}_{k-1}$. Regarding the selection of the weighting factor, Deok-Jin Lee et al. employed a numerical optimization method to solve for ${\beta }_k$, but this approach is computationally intensive. Alternatively, Lichtfuss et al. introduced a forgetting factor $d_k$ that adjusts the magnitude of the weighting factor dynamically during the filtering process, thereby enabling the updating of ${\mathit{Q}}_k$ to progressively stabilize.

$${\beta }_k=\left(1-d_k\right)/\left(1-d_k^{\left(k+1\right)}\right)$$

(18)

where $0<d_k<1$ is the forgetting factor.

By substituting the updated measurement noise covariance matrix ${\overline{\mathit{R}}}_k$ from Equation (23) back into Equation (17) and completing the update of the process noise covariance matrix ${\mathit{Q}}_k$ via Equation (27) following Equation (21), the Adaptive Robust Cubature Kalman Filter (ARCKF) is obtained. It should be noted that the updated ${\overline{\mathit{R}}}_k$ is only utilized in the filtering process at time step k, and reverts to its original form at time k + 1.

As shown in Equation (21), after multiple iterations, ${\beta }_k$ converges to $\left(1-d_k\right)$. Once $d_k$ is determined, the adjustment rate of ${\mathit{Q}}_k$ becomes fixed, resulting in limited dynamic response capability to changes in the dynamical environment. To address this limitation, this paper employs a two-sided chi-square test based on innovation covariance to adaptively adjust the forgetting factor β in Equation (28), thereby enhancing the algorithm’s convergence speed while maintaining filtering performance:

According to the orthogonality principle of the Kalman filter, the normalized innovation squared (NIS), denoted as $\gamma$, follows a chi-square distribution:

$$\gamma ={\mathit{V}}_{z_k}^{\mathbf{T}}{\mathit{P}}_{{\left(ZZ\right)}_{k,k-1}}^{-1}{\mathit{V}}_{z_k}\sim {\chi }^2\left(m\right)$$

(19)

where ${\mathit{V}}_{z_k}$ is the innovation vector and ${\mathit{P}}_{{\left(ZZ\right)}_{k,k-1}}$ is its theoretical covariance matrix. Consequently, the system status can be evaluated by calculating the squared Mahalanobis distance of the residual sequence and applying hypothesis testing principles. Abnormal system states or observations will cause ${\mathit{V}}_{z_k}$ to deviate from the chi-square distribution. The degree of this deviation is used to adaptively adjust the forgetting factor:

$$d_k=\left\{\begin{array}{c}\begin{array}{cc}{d}_k\cdot 1.05& \gamma >{\chi }_{\mathrm{u}\mathrm{p}\mathrm{p}\mathrm{e}\mathrm{r}}^2\end{array}\\ \begin{array}{cc}{d}_k\cdot 0.95& \gamma <{\chi }_{\mathrm{l}\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{r}}^2\end{array}\end{array}\right.$$

(20)

Here, ${\chi }_{\mathrm{u}\mathrm{p}\mathrm{p}\mathrm{e}\mathrm{r}}^2$ and ${\chi }_{\mathrm{l}\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{r}}^2$ represent the upper and lower chi-square test thresholds, determined by [19]:

$$\begin{array}{c}{\chi }_{\mathrm{upper}}^2={\mathit{F}}_{{\chi }^2(m)}^{-1}(1-\alpha /2)\hfill \\ {\chi }_{\mathrm{lower}}^2={\mathit{F}}_{{\chi }^2(m)}^{-1}(\alpha /2)\hfill \end{array}$$

(21)

where ${\mathit{F}}_{{\chi }^2(m)}^{-1}(\cdot )$ is the inverse cumulative distribution function of the chi-square distribution with m degrees of freedom, and α is the significance level, typically set to 0.05. This defines the maximum and minimum reasonable values for the $\gamma$ at a 95% confidence level.

The adjusted forgetting factor undergoes exponential smoothing with boundary constraints to ensure algorithm stability:

$$\begin{array}{c}{d}_{k+1}=\left(1-\eta \right)d_k+\eta d_k\hfill \\ d_{k+1}=\max \left(d_{\min },\min \left(d_{\max },d_{k+1}\right)\right)\hfill \end{array}$$

(22)

where $\eta$ is a smoothing factor between 0 and 1, and $d_{\min }$, $d_{\max }$ represent the permissible range for the forgetting factor.

The algorithm flow is shown in Figure 2.

4. Observability Theory

The performance of the orbit determination system can be partially analyzed through its observability. Since the inter-satellite ranging/astronometric angle measurement orbit determination system is nonlinear and time-varying with high dimensionality, the piecewise constant method is adopted to analyze its observability degree over small time intervals. The observability degree can be calculated using the condition number $\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}\left({\mathit{Q}}_{\mathrm{o}\mathrm{b}}\right)$ of the system observability matrix ${\mathit{Q}}_{\mathrm{o}\mathrm{b}}$[20]:

$${\gamma }_j={\displaystyle \frac{1}{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}\left({\mathit{Q}}_{\mathrm{o}\mathrm{b}}\right)}}={\displaystyle \frac{1}{‖{\mathit{Q}}_{\mathrm{o}\mathrm{b}}‖\cdot ‖{\mathit{Q}}_{\mathrm{o}\mathrm{b}}^{-1}‖}}={\displaystyle \frac{{\sigma }_{\min }}{{\sigma }_{\max }}}$$

(23)

where ${\sigma }_{\max }$ and ${\sigma }_{\min }$ represent the largest and smallest singular values of ${\mathit{Q}}_{\mathrm{o}\mathrm{b}}$, respectively. The condition number $\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}\left({\mathit{Q}}_{\mathrm{o}\mathrm{b}}\right)$ reflects the upper bound of the relative error in the solution caused by disturbances in the orbit determination system. A larger value indicates greater error susceptibility to disturbances, corresponding to a lower observability degree.

Linearizing Equation (2) yields the state transition matrix ${\mathit{\Phi }}_{k,k-1}\approx \mathit{I}+\mathit{A}\left(k\right)\cdot T$, expressed as

$$\mathit{A}\left(k\right)={\left.{\displaystyle \frac{\partial \mathit{f}\left(\mathit{X}\right)}{\partial {\mathit{X}}^{\mathrm{T}}}}\right|}_{\mathit{X}=\widehat{\mathit{X}}}=\left[\begin{array}{cccc}{\mathsf{0}}_{3\times 3}& {\mathit{I}}_{3\times 3}& {\mathsf{0}}_{3\times 3}& {\mathsf{0}}_{3\times 3}\\ {\mathbf{B}}_1& \mathbf{\Omega }& {\mathsf{0}}_{3\times 3}& {\mathsf{0}}_{3\times 3}\\ {\mathsf{0}}_{3\times 3}& {\mathsf{0}}_{3\times 3}& {\mathsf{0}}_{3\times 3}& {\mathit{I}}_{3\times 3}\\ {\mathsf{0}}_{3\times 3}& {\mathsf{0}}_{3\times 3}& {\mathit{B}}_2& \mathbf{\Omega }\end{array}\right]$$

(24)

where

$$\mathbf{\Omega }=\left[\begin{array}{ccc}0& 2& 0\\ -2& 0& 0\\ 0& 0& 0\end{array}\right],{\mathit{B}}_i=\left[\begin{array}{ccc}{s}_{11}^i& s_{12}^i& s_{13}^i\\ s_{21}^i& s_{22}^i& s_{23}^i\\ s_{31}^i& s_{32}^i& s_{33}^i\end{array}\right]\left(i=1,2\right)$$

(25)

$$\left\{\begin{array}{c}{s}_{11}^i=1-{\displaystyle \frac{\left(1-\mu \right)r_{\mathrm{e}i}^2-3\left(1-\mu \right){\left(x_i+\mu \right)}^2}{{\left|r_{\mathrm{e}i}\right|}^5}}-{\displaystyle \frac{\mu r_{\mathrm{m}i}^2-3\mu {\left(x_i+\mu -1\right)}^2}{{\left|r_{\mathrm{m}i}\right|}^5}}\hfill \\ s_{22}^i=1-{\displaystyle \frac{\left(1-\mu \right)r_{\mathrm{e}i}^2-3\left(1-\mu \right)y_i^2}{{\left|r_{\mathrm{e}i}\right|}^5}}-{\displaystyle \frac{\mu r_{\mathrm{m}i}^2-3\mu y_i^2}{{\left|r_{\mathrm{m}i}\right|}^5}}\hfill \\ s_{33}^i=-{\displaystyle \frac{\left(1-\mu \right)r_{\mathrm{e}i}^2-3\left(1-\mu \right)z_i^2}{{\left|r_{\mathrm{e}i}\right|}^5}}-{\displaystyle \frac{\mu r_{\mathrm{m}i}^2-3\mu z_i^2}{{\left|r_{\mathrm{m}i}\right|}^5}}\hfill \\ s_{12}^i=s_{21}^i={\displaystyle \frac{3\left(1-\mu \right)\left(x_i+\mu \right)y_i}{{\left|r_{\mathrm{e}i}\right|}^5}}+{\displaystyle \frac{3\mu \left(x_i+\mu -1\right)y_i}{{\left|r_{\mathrm{m}i}\right|}^5}}\hfill \\ s_{13}^i=s_{31}^i={\displaystyle \frac{3\left(1-\mu \right)\left(x_i+\mu \right)z_i}{{\left|r_{\mathrm{e}i}\right|}^5}}+{\displaystyle \frac{3\mu \left(x_i+\mu -1\right){\mathit{z}}_i}{{\left|r_{mi}\right|}^5}}\hfill \\ s_{23}^i=s_{32}^i={\displaystyle \frac{3\left(1-\mu \right)y_i{z}_i}{{\left|r_{\mathrm{e}i}\right|}^5}}+{\displaystyle \frac{3\mu y_i{z}_i}{{\left|r_{\mathrm{m}i}\right|}^5}}\hfill \\ r_{\mathrm{m}i}=\sqrt{{\left(x_i+\mu -1\right)}^2+y_i^2+z_i^2}\hfill \\ r_{\mathrm{e}i}=\sqrt{{\left(x_i+\mu \right)}^2+y_i^2+z_i^2}\hfill \end{array}\right.$$

(26)

Linearizing the measurement Equation (4) gives

$${\left.{\displaystyle \frac{\partial {\mathit{h}}_{\mathrm{d}}}{\partial X^{\mathrm{T}}}}\right|}_{X=\widehat{X}\left(k\right)}=\left[{\displaystyle \frac{x_1-x_2}{\rho }},{\displaystyle \frac{y_1-y_2}{\rho }},{\displaystyle \frac{z_1-z_2}{\rho }},{\mathsf{0}}_{1\times 3},-{\displaystyle \frac{x_1-x_2}{\rho }},-{\displaystyle \frac{y_1-y_2}{\rho }},-{\displaystyle \frac{z_1-z_2}{\rho }},{\mathsf{0}}_{1\times 3}\right]$$

(27)

Similarly, linearizing Equation (6) yields

$${\left.{\displaystyle \frac{\partial {\mathit{h}}_{\mathrm{a}}}{\partial {\mathit{X}}^{\mathrm{T}}}}\right|}_{X=\widehat{X}\left(k\right)}=\left[{\displaystyle \frac{\partial \mathit{f}}{\partial x_1}},{\displaystyle \frac{\partial \mathit{f}}{\partial y_1}},{\displaystyle \frac{\partial \mathit{f}}{\partial z_1}},0,0,0,{\displaystyle \frac{\partial \mathit{f}}{\partial x_2}},{\displaystyle \frac{\partial \mathit{f}}{\partial y_2}},{\displaystyle \frac{\partial \mathit{f}}{\partial z_2}},0,0,0\right]$$

(28)

where:

$$\begin{array}{l}{\displaystyle \frac{\partial \mathit{f}}{\partial x_1}}={\displaystyle \frac{1}{L_{\mathrm{A}}{L}_{\mathrm{B}}}}\left(b_{\mathrm{x}}+a_{\mathrm{x}}-{\displaystyle \frac{\mathrm{D}{a}_{\mathrm{x}}}{L_{\mathrm{A}}^2}}-{\displaystyle \frac{\mathrm{D}{b}_{\mathrm{x}}}{L_{\mathrm{B}}^2}}\right)\\ {\displaystyle \frac{\partial \mathit{f}}{\partial y_1}}={\displaystyle \frac{1}{L_{\mathrm{A}}{L}_{\mathrm{B}}}}\left(b_{\mathrm{y}}+a_{\mathrm{y}}-{\displaystyle \frac{\mathrm{D}{a}_{\mathrm{y}}}{L_{\mathrm{A}}^2}}-{\displaystyle \frac{\mathrm{D}{b}_{\mathrm{y}}}{L_{\mathrm{B}}^2}}\right)\\ {\displaystyle \frac{\partial \mathit{f}}{\partial z_1}}={\displaystyle \frac{1}{L_{\mathrm{A}}{L}_{\mathrm{B}}}}\left(b_{\mathrm{z}}+a_{\mathrm{z}}-{\displaystyle \frac{\mathrm{D}{a}_{\mathrm{z}}}{L_{\mathrm{A}}^2}}-{\displaystyle \frac{\mathrm{D}{b}_{\mathrm{z}}}{L_{\mathrm{B}}^2}}\right)\\ {\displaystyle \frac{\partial \mathit{f}}{\partial x_2}}={\displaystyle \frac{1}{L_{\mathrm{A}}{L}_{\mathrm{B}}}}\left(-b_{\mathrm{x}}+{\displaystyle \frac{\mathrm{D}{a}_{\mathrm{x}}}{L_{\mathrm{A}}^2}}\right)\\ {\displaystyle \frac{\partial \mathit{f}}{\partial y_2}}={\displaystyle \frac{1}{L_{\mathrm{A}}{L}_{\mathrm{B}}}}\left(-b_{\mathrm{y}}+{\displaystyle \frac{\mathrm{D}{a}_{\mathrm{y}}}{L_{\mathrm{A}}^2}}\right)\\ {\displaystyle \frac{\partial \mathit{f}}{\partial z_2}}={\displaystyle \frac{1}{L_{\mathrm{A}}{L}_{\mathrm{B}}}}\left(-b_{\mathrm{z}}+{\displaystyle \frac{\mathrm{D}{a}_{\mathrm{z}}}{L_{\mathrm{A}}^2}}\right)\end{array}$$

(29)

$$\begin{array}{l}{a}_{\mathrm{x}}=x_1-x_2, a_{\mathrm{y}}=y_1-y_2, a_{\mathrm{z}}=z_1-z_2\\ b_{\mathrm{x}}=x_1-x_{\mathrm{s}}, b_{\mathrm{y}}=y_1-y_{\mathrm{s}}, b_{\mathrm{z}}=z_1-z_{\mathrm{s}}\\ L_{\mathrm{A}}=\sqrt{a_{\mathrm{x}}^2+a_{\mathrm{y}}^2+a_{\mathrm{z}}^2}\\ L_{\mathrm{B}}=\sqrt{b_{\mathrm{x}}^2+b_{\mathrm{y}}^2+b_{\mathrm{z}}^2}\\ \mathrm{D}=a_{\mathrm{x}}{b}_{\mathrm{x}}+a_{\mathrm{y}}{b}_{\mathrm{y}}+a_{\mathrm{z}}{b}_{\mathrm{z}}\end{array}$$

(30)

5. Simulation Results and Analysis

5.1. Simulation Conditions and Parameter Selection

To comparatively analyze the effectiveness of the proposed method in addressing the limitation posed by orbital plane inclination on the accuracy of LiAISON-based autonomous orbit determination for cislunar spacecraft, the method was applied to two scenarios: autonomous orbit determination for one spacecraft in an L4 planar short-period orbit and another in a DRO.

The initial nominal orbital states of the two spacecraft, comprising their position and velocity vectors in the Earth–Moon rotating frame, are provided in

Table 1. Orbital parameters.

No.	Orbit Type	Initial Orbital State
1	L4 Planar Short-Period Orbit	[0.591560618, 0.806886461, 0, −0.022946517, 0.057017081, 0]
2	DRO	[1.143936419, 0, 0, 0, −0.471735422, 0]

. Based on current ground-based tracking capabilities, the typical orbit determination and prediction accuracy for lunar spacecraft over a 6 h arc is on the order of approximately kilometer in position and several centimeters per second in velocity. at the kilometer level. Inter-satellite ranging accuracy depends on factors such as signal strength, ranging methodology, and instrument performance. BeiDou satellites can achieve centimeter-level ranging via inter-satellite microwave links within 70,000 km [21]. Considering the distance between triangular libration point spacecraft and lunar orbiters is about 380,000 km, achieving high-precision inter-satellite ranging is more challenging and was initially set at the meter level for this analysis. The accuracy of the optical angle measurement sensor is assumed to be approximately 1 arcsecond [22].

The parameters for the orbit determination filter are listed in

Table 2. Orbit determination filter parameters.

Parameter	Value
Simulation Period	1 January 2024 00:00:00 UTC to 29 February 2024 24:00:00 UTC
Step Size	15 min
Initial State Vector X	[0.591560618, 0.806886461, 0, −0.022946517, 0.057017081, 0, 1.143936419, 0, 0, 0, −0.471735422, 0]
Initial Position Error (per axis)	10 km
Initial Velocity Error (per axis)	1 × 10−3 km/s
Process Noise Covariance Matrix Q	$\begin{array}{l}{\mathrm{q}}_1={10}^{-6}\mathrm{k}\mathrm{m},{\mathrm{q}}_2={10}^{-9}\mathrm{k}\mathrm{m}/\mathrm{s}\\ \mathrm{Q}=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left[q_1^2,q_1^2,q_1^2,q_2^2,q_2^2,q_2^2,q_1^2,q_1^2,q_1^2,q_2^2,q_2^2,q_2^2\right]\end{array}$
Inter-satellite Ranging Error	1 m
Astrometric Angle Measurement Error	1″

.

5.2. Simulation Results

5.2.1. Accuracy Analysis of Method

For the autonomous navigation scenario involving two spacecraft (one located in a short-period orbit on the L4 plane and the other in a DRO, with both having unknown initial velocities), Figure 3 and Figure 4, respectively, present the orbit determination results obtained using the AFF-ARCKF algorithm, under two configurations: solely utilizing ISR and jointly utilizing ISR and angle measurements. In the figures, the black solid line represents the nominal reference orbit, the blue dashed-dotted line corresponds to the simulation results obtained solely using ISR, and the red dashed line represents the simulation results obtained by jointly utilizing ISR and angle measurements.

The analysis indicates that when relying solely on inter-satellite ranging, the system’s estimation error in the Z direction (perpendicular to the orbital plane) cannot converge, with an average system observability measure of approximately 1.1266 × 10⁻⁵, confirming that the system lacks observability under this configuration. In contrast, after introducing inter-satellite angle measurements, the average system observability measure significantly increases to approximately 0.0021, indicating a fundamental improvement in the system’s observability. The corresponding orbit determination accuracy is summarized in the first row of

Table 3. Impact of the initial position and velocity errors on orbit determination performance.

Initial Position Error (km)	Initial Velocity Error (m/s)	Orbit	Position Error (m)	Velocity Error (m/s)
10	1	L4	631.26	0.0050
		DRO	229.27	0.0018
50	1	L4	684.37	0.0075
		DRO	256.29	0.0022
100	1	L4	665.44	0.0074
		DRO	265.43	0.0026
10	10	L4	655.77	0.0053
		DRO	255.63	0.0019
10	100	L4	805.97	0.0083
		DRO	376.32	0.0025

: the final position error for the L4 orbit spacecraft is approximately 631 m, and the position error for the DRO orbit spacecraft is approximately 229 m. This result visually demonstrates the crucial role of introducing angle observations in enhancing observability perpendicular to the orbital plane, thereby ensuring overall orbit determination accuracy.

In addition, it can be seen from the simulation that the position error is significantly greater than the velocity error, which may be related to the dynamic characteristics and filtering process. In the dynamic framework of the circular restricted three body problem, the system exhibits high sensitivity and numerical rigidity. The dynamic integration process in state prediction will transfer and amplify the uncertainty of velocity into an increasing uncertainty of position. The observation interval set in this simulation is 15 min. This design aims to reduce the frequency requirement for bidirectional ranging between binary star systems to alleviate the difficulty of engineering implementation. However, a longer integration period will also exacerbate the cumulative effect of velocity error on position error. In addition, the ranging and angle measurements used in this article are both geometric functions of the spacecraft’s position. The filter can directly correct the position in the update step, while the estimation of velocity needs to be indirectly obtained through the dynamic model associated with continuous position updates. This indirect estimation makes the speed more susceptible to model smoothing constraints, resulting in smaller apparent errors; However, any small deviation in velocity estimation is the main driving force for the growth of position errors in the prediction stage. In summary, larger position errors mainly reflect the combined effect of dynamic sensitivity and integrated accumulation of velocity uncertainty, while smaller velocity errors reflect the indirect estimation characteristics of the filter through the model. Although there is a visual difference in magnitude between the two, they are both within the error limits that enable autonomous navigation, and the convergence behavior proves the overall stability and effectiveness of the filter in the complex environment of CRTBP.

5.2.2. Comparative Analysis of Algorithm Performance

Employing the inter-satellite ranging and angle measurement method, a measurement outlier was introduced at a specific time during the filtering process to compare the performance of the CKF, RCKF, ARCKF, and the proposed AFF-ARCKF algorithms. The difference between RCKF, ARCKF and AFF-ARCKF algorithms lies in that RCKF does not perform adaptive estimation on ${\mathit{Q}}_k$. The comparative simulation results are shown in Figure 5. Through observation, it was found that RCKF, ARCKF, and AFF-ARCKF can effectively suppress measurement outliers, maintain the accuracy of orbit determination, and perform better than standard CKF. However, due to the lack of adaptive estimation of the ${\mathit{Q}}_k$, the final convergence accuracy of RCKF is not as good as the other two algorithms. In addition, compared with the ARCKF algorithm, the convergence speed of the AFF-ARCKF algorithm has been improved by more than 50%, and the fluctuations are smaller. In summary, AFF-ARCKF, through its adaptive mechanism, not only ensures robustness against abnormal disturbances, but also improves the convergence speed of state estimation errors, thereby comprehensively ensuring the accuracy and reliability of autonomous orbit determination for Cislunar spacecraft.

5.2.3. Analysis of Factors Influencing Algorithm Performance

To further analyze the performance of the proposed method and algorithm, simulation analyses were conducted with varying initial errors and initial forgetting factors.

(1)

Impact of Initial Errors

The influence of different initial position errors on orbit determination performance was investigated. As shown in Table 3, the accuracy of the proposed orbit determination method and algorithm exhibits only minor sensitivity to the magnitude of initial errors.

(2)

Impact of Initial Forgetting Factor

The influence of the initial forgetting factor on the final estimation results was examined. Simulation results in

Table 4. Impact of the initial forgetting factor on orbit determination performance.

dk	Orbit	Position Error (m)	Velocity Error (m/s)
0.3	L4	-	-
	DRO	-	-
0.6	L4	1070.41	0.0026
	DRO	352.5856	0.0047
0.9	L4	631.26	0.0050
	DRO	229.27	0.0018

demonstrate that the value of the initial forgetting factor $d_k$ significantly affects the robustness of the algorithm. When $d_k$ is too small, the state error covariance matrix may lose its positive definiteness during the filtering update process. To ensure the positive definiteness of ${\mathit{P}}_k$, omitting the second term in Equation (17) could be considered.

6. Conclusions

Addressing the challenges of degraded accuracy, slow convergence, and high re-convergence cost after disturbances in the Earth–Moon LiAISON autonomous orbit determination system based on inter-satellite ranging—particularly when the orbital planes of the two spacecraft are nearly coplanar—this paper introduces an integrated autonomous orbit determination method that combines inter-satellite angle and distance measurement, as well as the AFF-ARCKF algorithm, to enhance the observability of the system, improve the convergence speed of the filtering algorithm, and maintain robustness.

To verify the performance of the AFF-ARCKF algorithm, this study designed a comprehensive simulation testing strategy to systematically evaluate challenging scenarios such as measurement anomalies, model inaccuracies, and non-nominal initial conditions. The core robustness of the algorithm lies in its adaptive forgetting factor mechanism: The filter continuously monitors the new information sequence through chi square test. Once a statistically significant bias (usually caused by unmodeled disturbances or measurement outliers) is detected, the forgetting factor is automatically adjusted to increase or decrease ${\mathit{Q}}_k$, making it more in line with the actual dynamic environment and the speed of convergence of the speed filter.

It should be noted that the simulations in this study are based on the CRTBP model, which exhibits certain discrepancies compared to high-fidelity force models. In addition, the effects of orbit maintenance maneuvers on filter convergence were not considered. These aspects represent important directions for future research toward practical engineering application.