Uncertainty analysis of initial orbit determination in the minimum radar admissible region

Abstract

The poor accuracy of the single solution of the initial orbit determination (IOD) affects the precise orbit determination, target matching and arc-segment correlation. This paper proposes a method for analyzing the uncertainty of the IOD results based on the Minimum Radar Admissible Region (MRAR), which describes the IOD results in the form of probability in the region bounded by the minimum admissible region. The MRAR can contain the real orbit state with a probability greater than 99.7%; the weight coefficients of each sampling point in the MRAR are calculated by using the Bayesian formula, and the probability density function of the IOD is obtained by using the weighted kernel density estimation method. The simulation experiment proves that the proposed method can well describe the possibility of each point in the MRAR to be the real orbit state, and can provide a better initial value for the subsequent space mission analysis.

Introduction

The observation and surveillance of space objects using ground-based radars is an important part of space situational awareness activities, and the determination of reliable orbit for space objects using radar observation is a challenge^1,2. For initial state estimation of dynamic systems, there are usually multiple solutions that all satisfy the state embodied in the observation data. This problem also exists in the mentioned challenge. Existing initial orbit determination (IOD) algorithms take the unique minimum root mean square of the residuals as the solution objective, but in the real situation (with measurement errors), the orbit state corresponding to this solution may not be close to the real orbit^3,4. Therefore, the results of IOD should be described probabilistically within a reliable range so as to reflect the real situation to the greatest extent possible⁵. The results of orbit determination described with probability will be beneficial to the correlation between different radar observation arc segments and the calculation of collision probability, and can also provide more reasonable initial values for subsequent orbit improvement⁶. In this process, how to determine a reliable range of parameter estimation and conduct uncertainty analysis within this range becomes the key to solve the problem⁷.

A popular approach for IOD and arc-segment correlation in situational awareness missions is the Admissible Region (AR) method^8,9,10,11. This method limits the parameters that cannot be observed by ground-based equipment by applying astrodynamics constraints on the space target, and then combines them with the parameters that can be observed to estimate the possible orbital states. The dimension of the IOD problem can be reduced from six to two dimensions by finding the orbit that best matches the actual observations in the AR, which can greatly reduce the difficulty of orbit determination¹².

The original AR only considers energy constraints, which contains a large number of orbits with different semi-major axes and eccentricities, so the scope of orbital states included in the original AR is very big¹³. Previous scholars have narrowed the scope of the original AR by adding additional constraints to the AR based on some prior information (e.g., semi-major axis, eccentricity, perigee, apogee altitude, etc.) to form a constrained admissible region (CAR)¹⁴. DeMars et al.¹⁵ used a mixture of Gaussian distributions to approximate the probability description of the AR using a uniform distribution to obtain a probabilistic description of the AR and therefore of the initial orbital state. Psiaki et al.¹⁶ further developed the method by using a new approach to implement the estimation of Gaussian mixing components and using position-velocity space instead of the range-range rate space to better describe the uncertainty of the orbital states. Hussein et al.¹⁷ formally introduced the concept of probabilistic Admissible Region (PAR) and proposed a method for constructing the PAR based on the error characteristics of the observed data, and pointed out that the PAR is not uniformly distributed. Han Cai et al.¹⁸ used the theory of external probability measures to obtain the uncertainty of the AR under the condition of unknown systematic errors, and named it as Possibilistic Admissible Region (PAR+).

Existing theoretical research on ARs proceeds in two directions: one is to continuously reduce the original admissible region according to different constraints or using other information. The most significant effect is that the patent¹⁹ provides a method for constructing a minimum optical admissible region (MOAR) based on optical observation data, which is able to greatly reduce the range of the optical AR in the range-range rate plane, and to improve the solving efficiency and accuracy for the subsequent arc correlation. The other is: using uncertainty analyses based on different ARs to provide a probabilistic description of the corresponding regions. The most typical is the PAR + method constructed by Han Cai¹⁸ mentioned above, characterized by the fact that no a priori information is required and that it is applicable to both optical and radar observation data.

The radar admissible region (RAR) was firstly proposed by Tommei²⁰, who modified the concept of arc-segment attributes based on the optical admissible region and represented the RAR on a plane, which is the fundamental difference between the RAR and the optical admissible region. Jones and Zucchelli^21,22,23 proposed PAR for radar observations and combined it with Cardinalized Probability Hypothesis Density (CPHD) filter to realize the orbit determination and tracking of space objects based on radar observations; after that, Jones introduced Outer Probability Measures (OPMs) into the RAR theory²³, which quantitatively represents the random error and systematic error in space target tracking. Cai¹⁸ used the theory of OPMs to describe more realistically the uncertainty in the original RAR and defined it as PAR+, that can be appropriately interpreted as initializing our inference of a dynamical system with a so-called uninformative prior.This concept provides ideas for our study. TAO²⁴ proposed describing the probability distribution of IOD solutions on the semi-major axis-eccentricity plane and utilizing prior knowledge to enhance the accuracy of orbit determination schemes. Luo²⁵ conducted a systematic review of uncertainties in orbital mechanics, which provides valuable references for this study.

We propose a probabilistic description method for initial orbit determination (IOD) results based on the MRAR. This approach significantly improves the representation of the probability distribution of IOD results within this region compared to directly describing them in the original RAR. By employing the MRAR, our method reduces the area by more than two orders of magnitude while maintaining a 99.7% probability of including the true orbital state. This enhancement offers several key advantages:

• Enhanced precision: The reduced uncertainty range allows for more accurate predictions and estimations of an object’s trajectory.

• Efficiency improvement: A smaller search space leads to significant reductions in computational costs associated with subsequent arc association and precise orbit determination processes.

• Reliability increase: By focusing on a more confined region that is highly likely to contain the actual orbit, the reliability of the IOD process is greatly improved.

In summary, the proposed method not only refines the solution space but also enhances the overall accuracy and efficiency of the IOD procedure, making it a valuable contribution to the field of space situational awareness and debris tracking. In this paper, Section “Introduction” introduces the current state of admissible domain theory and analyzes how existing literature describes IOD results. Section “Initial orbit determination in the admissible region” explains how to achieve IOD within the admissible domain. Section “Minimum radar admissible region (MRAR)” constructs the MRAR. Section “Kernel density estimation” provides a probabilistic description of sampling points in the MRAR and solves for the probability density function within this region. Section “Instance verification” designs simulation experiments to validate the proposed method with practical examples. Finally, Section “Conclusion” summarizes the findings.

Initial orbit determination in the admissible region

Admissible region orbit elements

In general, we are more accustomed to using classical orbit elements and Cartesian orbit elements to describe the orbital state of space objects in near-Earth space, and these two descriptions provide a comprehensive and accurate framework for analyzing the orbital state of space objects¹⁴.

$$\sigma ={(a,e,i,\Omega ,\omega ,f)^T}$$

(1)

$${\varvec{x}}={(x,y,z,\dot {x},\dot {y},\dot {z})^{\text{T}}}$$

(2)

As we all know, classical orbit elements and cartesian orbit elements are the two most commonly used ways to describe orbital states. However, for space situational awareness, the theory of AR is able to represent classical orbit elements more intuitively in the observation space, i.e., the admissible region orbit elements (AROE). According to the actual observation, the AROE are divided into observable elements and estimation elements. For radar observation data, observable elements include range, range rate, right ascension and declination; estimation elements include right ascension rate and declination rate.

$$\mathfrak{X}=\left[ {\begin{array}{*{20}{c}} {{\mathfrak{X}_{OB}}} \\ {{\mathfrak{X}_{ES}}} \end{array}} \right]=\left[ {\begin{array}{*{20}{c}} {(\rho ,\dot {\rho },\alpha ,\delta )} \\ {(\dot {\alpha },\dot {\delta })} \end{array}} \right]$$

(3)

All types of RARs are constructed on a two-dimensional plane consisting of the right ascension rate and the declination rate, which is defined as the RAR plane. By constructing the RAR, the theoretical support for the subsequent arc-segment correlation is provided.

Conversion of observations

In order to facilitate the description and calculation of the orbital state of space objects in the observation space, the RAR is usually constructed on the right ascension rate - declination rate plane²⁶. In order to convenience the subsequent calculation, it is necessary to map the actual observation data to the RAR plane, that is, to convert the radar observation data (including: range, range rate, azimuth, elevation) to the attributes of the RAR arc segment (including: range, range rate, right ascension, declination), i.e., ${{\mathbf{B}}_{rad}}=\left( {\rho ,A,E} \right)$ is converted to ${\mathbf{\tilde {B}}}=\left( {\rho ,\alpha ,\delta } \right)$.This step converts the radar observation data from the station coordinate system to the geocentric inertial coordinate system with the following transformation matrix.

$${\mathbf{\tilde {B}}}={\varvec{M}_3}\left[ { - {\lambda _S} - {\Omega _G}} \right]{\varvec{M}_2}\left[ {{\varphi _S}} \right]{{\mathbf{B}}_{rad}}$$

(4)

where M₂[θ ] is the second primitive transformation matrix, and M₃[θ ] is the third primitive transformation matrix, ${\lambda _S}$ is the station longitude, ${\varphi _S}$ is the station latitude, and ${\Omega _G}$ is the angle between the geocentric inertial coordinate system and the X-axis of the geocentric coordinate system as stated, which can be obtained by looking up the astronomical almanac. The above process does not use the first primitive transformation matrix M₁.

Original radar admissible region

In the original RAR construction, only the energy constraints of the orbit being operated need to be considered. For near-Earth space objects, the spacecraft operates in an elliptical orbit with orbital energy less than 0.

$$\varepsilon =\frac{1}{2}{\left\| {\dot {{\varvec{r}}}} \right\|^2} - \frac{\mu }{{\left\| {\varvec{r}} \right\|}}<0$$

(5)

Observing the above equation, the key to solving the problem is to find the relationship between the AROEs and the Cartesian orbit elements. Analyze the positional relationship between the observing radar equipment, the space target and the center of the earth²⁰.

$$\left\{ \begin{gathered} {\varvec{r}}={\varvec{R}}+\rho {\varvec{L}} \hfill \\ \dot {{\varvec{r}}}=\dot {{\varvec{R}}}+\dot {\rho }{\varvec{L}}+\rho \dot {\alpha }{{\varvec{L}}_\alpha }+\rho \dot {\delta }{{\varvec{L}}_\delta } \hfill \\ \end{gathered} \right.$$

(6)

where ${\varvec{R}}$ and $\dot {{\varvec{R}}}$ are the position and velocity vectors of the observation platform, respectively.

$$\left\{ \begin{array}{l}L = {(cos\delta cos\alpha ,cos\delta sin\alpha ,sin\delta )^T}\\{L_\alpha } = {( - cos\delta \sin \alpha ,cos\delta \cos \alpha ,0)^T}\\{L_\delta } = {( - \sin \delta \cos \alpha , - \sin \delta \sin \alpha ,\cos \delta )^T}\end{array} \right.$$

(7)

An expression for the orbital energy about $(\rho ,\dot {\rho })$ can be obtained:

$$\varepsilon =\frac{1}{2}\left( {{{\dot {\rho }}^2}+{\omega _1}\dot {\rho }+{\omega _2}{\rho ^2}+{\omega _3}\rho +{\omega _4} - \frac{{2\mu }}{{\sqrt {{\rho ^2}+{\omega _5}\rho +{\omega _0}} }}} \right)$$

(8)

where,

$$\left\{ \begin{gathered} {\omega _0}={\left\| {\varvec{R}} \right\|^2} \hfill \\ {\omega _1}=2\dot {{\varvec{R}}} \cdot {\varvec{L}} \hfill \\ {\omega _2}={{\dot {\alpha }}^2}co{s^2}\delta +{{\dot {\delta }}^2} \hfill \\ {\omega _3}=\dot {\alpha }{\omega _{3,\alpha }}+\dot {\delta }{\omega _{3,\delta }} \hfill \\ {\omega _4}={\left\| {\dot {{\varvec{R}}}} \right\|^2} \hfill \\ {\omega _5}=2{\varvec{R}} \cdot {\varvec{L}} \hfill \\ {\omega _{3,\alpha }}=2\dot {{\varvec{R}}} \cdot {{\varvec{L}}_\alpha } \hfill \\ {\omega _{3,\delta }}=2\dot {{\varvec{R}}} \cdot {{\varvec{L}}_\delta } \hfill \\ \end{gathered} \right.$$

(9)

The orbital energy of a space target can be expressed as

$$\varepsilon (\dot {\alpha },\dot {\delta })={z_{11}}{\dot {\alpha }^2}+2{z_{12}}\dot {\alpha }\dot {\delta }+{z_{22}}{\dot {\delta }^2}+2{z_{13}}\dot {\alpha }+2{z_{23}}\dot {\delta }+{z_{33}}$$

(10)

Therefore, the original RAR is

$$\varepsilon (\dot {\alpha },\dot {\delta })<0$$

(11)

Loss function for candidate orbit

An orbit can be found in the admissible region that best fits the observations. For any combination of $(\dot {\alpha },\dot {\delta })$, the theoretical observations $\rho (\dot {\alpha },\dot {\delta })$,$\alpha (\dot {\alpha },\dot {\delta })$,$\delta (\dot {\alpha },\dot {\delta })$ can be calculated from their corresponding orbits. The weighted sum of squares of the theoretical and real observation series can characterize the estimation accuracy of the orbit state. Adopting the idea of least squares, uniform sampling in the admissible region is performed, and the loss function ${J_{AR}}(\dot {\alpha },\dot {\delta })$ of each sampling point is calculated.

$${J_{AR}}(\dot \alpha ,\dot \delta ) \equiv \frac{1}{{\sqrt {3m} }}\sum\limits_{i = 1}^m {\left[ {\frac{{{\kern 1pt} {{\left( {{{\tilde \rho }_i} - {\rho _i}(\dot \alpha ,\dot \delta )} \right)}^2}}}{{\sigma _\rho ^2}} + \frac{{{{\left( {{{\tilde \alpha }_i} - {\alpha _i}(\dot \alpha ,\dot \delta )} \right)}^2}}}{{{\sigma ^2}/{{\cos }^2}{{\tilde \delta }_i}}} + {\kern 1pt} {\kern 1pt} \frac{{{{\left( {{{\tilde \delta }_i} - {\delta _i}(\dot \alpha ,\dot \delta )} \right)}^2}}}{{{\sigma ^2}}}} \right]}$$

(12)

m is the number of observation points, $\tilde {\rho }$ is the actual observed distance with noise, $\tilde {\alpha }$ is the actual observed right ascension with noise, $\tilde {\delta }$ is the actual observed declination with noise, ${\sigma _\rho }$ is the radar’s range observation error, and $\sigma$ is the radar’s angle observation error. A smaller value of ${J_{AR}}(\dot {\alpha },\dot {\delta })$ means a better fit to the observation. The Levenberg-Marquardt algorithm is utilized to find the sampling point with the best fit, which is the least squares solution in the original RAR.

Minimum radar admissible region (MRAR)

Uncertainty propagation in arc segment attributes

When analyzing the RAR, various possibilities and uncertainties need to be considered, and these lead to two sources of uncertainty in the RAR²⁷. The first is the randomness inherent in the observing system, which can be characterized by probability distributions, such as the observation errors of the observing equipment in terms of range and angle, while the second refers to the uncertainty propagated by the observation errors, which is not a random phenomenon. The latter can be analyzed in the RAR by deriving the propagation of the uncertainty in the arc attributes into the observation space. Therefore, the errors in the theoretical observations of range, right ascension, and declination are jointly affected by two parts: the first part is the observation error of the sensor $\sigma _{\rho }^{2}$, $\sigma _{{}}^{2}/cos\tilde {\delta }_{i}^{2}$,$\sigma _{{}}^{2}$; and the second part is the propagation error of the uncertainty in the tolerance domain ${\varPhi _{{\rho _i}}}{\Sigma _{\mathbf{B}}}\varPhi _{{{\rho _i}}}^{T}$, ${\varPhi _{{\alpha _i}}}{\Sigma _{\mathbf{B}}}\varPhi _{{{\alpha _i}}}^{T}$, ${\varPhi _{{\delta _i}}}{\Sigma _{\mathbf{B}}}\varPhi _{{{\delta _i}}}^{T}$. Among the two errors, the second part of the propagation error is more complicated and is analyzed below.

In calculations of orbital dynamics, scholars are accustomed to computing the propagation of orbits in a Cartesian coordinate system. Denoting the mapping from the AROEs $\mathfrak{X}$ to the Cartesian orbit elements ${{\varvec{x}}_0}$ as I₁.

$${I_1}({\kern 1pt} {\kern 1pt} \cdot {\kern 1pt} {\kern 1pt} ):{{\varvec{x}}_0}={I_1}(\mathfrak{X})$$

(13)

The orbital state x_i at any moment t_i can be obtained from the orbital state x₀ at the moment t₀ and the orbital dynamics model, denoted as the mapping I₂.

$${I_2}({\kern 1pt} {\kern 1pt} \cdot {\kern 1pt} {\kern 1pt} ):{{\varvec{x}}_i}={I_2}({{\varvec{x}}_0};{t_i})$$

(14)

The theoretical observation ${\mathfrak{X}_{OB}}$ can be solved for from the orbital state ${\varvec{x}}=({\varvec{r}};\dot {{\varvec{r}}})$, denoted as the mapping I₃.

$${I_3}({\kern 1pt} {\kern 1pt} \cdot {\kern 1pt} {\kern 1pt} ):{\mathfrak{X}_{OB}}={I_3}({\varvec{x}})$$

(15)

Take the right ascension observation α_i at time t_i as an example to illustrate the analysis method. Fix the rate of change of right ascension and declination $(\bar {\dot {\alpha }},\bar {\dot {\delta }})$, and write the theoretical observation value of the corresponding orbit under the actual arc segment attributes ${\mathbf{\tilde {B}}}$ as

$${\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha } _i}({\mathbf{\tilde {B}}};\bar {\dot {\alpha }},\bar {\dot {\delta }})={I_{3\alpha }}({I_2}({I_1}({\mathbf{\tilde {B}}};\bar {\dot {\alpha }},\bar {\dot {\delta }});{t_i}))$$

(16)

For ease of derivation, the mapping of the above given $(\bar {\dot {\alpha }},\bar {\dot {\delta }})$ and the observation moment ${t_i}$ is denoted as:

$${f_{{\alpha _i}}}({\mathbf{B}}) \triangleq {I_{3\alpha }}({I_2}({I_1}(({\mathbf{B}};\bar {\dot {\alpha }},\bar {\dot {\delta }}));{t_i}))$$

(17)

Then the above equation simplifies to ${\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha } _i}={f_{{\alpha _i}}}({\mathbf{\tilde {B}}})$. The real orbital state can be reconstructed using the real arc segment attributes ${\mathbf{\bar {B}}}$ and $(\bar {\dot {\alpha }},\bar {\dot {\delta }})$. Therefore, the theoretical observed value of the real orbital state can be expressed as ${\bar {\alpha }_i}={f_{{\alpha _i}}}({\mathbf{\bar {B}}})$.

The true value ${\bar {\alpha }_i}$ is expanded by a first order Taylor expansion at ${\mathbf{\tilde {B}}}$.

$${\bar {\alpha }_i}={f_{{\alpha _i}}}({\mathbf{\bar {B}}}) \approx {f_{{\alpha _i}}}({\mathbf{\tilde {B}}})+{\left. {\frac{{\partial {f_{{\alpha _i}}}}}{{\partial {\mathbf{B}}}}} \right|_{{\mathbf{\tilde {B}}}}}({\mathbf{\bar {B}}} - {\mathbf{\tilde {B}}})={\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha } _i}+{\left. {\frac{{\partial {f_{{\alpha _i}}}}}{{\partial {\mathbf{B}}}}} \right|_{{\mathbf{\tilde {B}}}}}({\mathbf{\bar {B}}} - {\mathbf{\tilde {B}}})$$

(18)

Noting that the Jacobian matrix ${\varPhi _{{\alpha _i}}}={\left. {\frac{{\partial {f_{{\alpha _i}}}}}{{\partial {\mathbf{B}}}}} \right|_{{\mathbf{\tilde {B}}}}}$ (the derivation of which will be given in appendix), the error between the theoretical observation and the true value can be expressed as

$${\bar {\alpha }_i} - {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha } _i}={\varPhi _{{\alpha _i}}}({\mathbf{\bar {B}}} - {\mathbf{\tilde {B}}})$$

(19)

According to the observation error characteristics, ${\mathbf{\bar {B}}} - {\mathbf{\tilde {B}}}$ obeys the following normal distribution.

$${\mathbf{\bar {B}}} - {\mathbf{\tilde {B}}}\sim \mathcal{N}(0,\Sigma )$$

(20)

where,

$$\Sigma =\left( {\begin{array}{*{20}{c}} {\sigma _{{{\rho _0}}}^{2}}&0&0&0 \\ 0&{\sigma _{{{{\dot {\rho }}_0}}}^{2}}&0&0 \\ 0&0&{\sigma _{{{\alpha _0}}}^{2}}&0 \\ 0&0&0&{\sigma _{{{\delta _0}}}^{2}} \end{array}} \right)$$

(21)

$${\sigma _{{\rho _0}}}{\text{=}}\frac{{3{\sigma _\rho }}}{{2\sqrt N }}, {\sigma _{{{\dot {\rho }}_0}}}{\text{=}}\frac{{2\sqrt 3 {\sigma _\rho }}}{{T\sqrt N }}, {\sigma _{{\alpha _0}}}{\text{=}}\frac{{3\sigma /\cos (\tilde {\delta })}}{{2\sqrt N }},{\sigma _{{\delta _0}}}{\text{=}}\frac{{3\sigma }}{{2\sqrt N }}$$

(22)

where ${\sigma _{{\rho _0}}},\, {\sigma _{{{\dot {\rho }}_0}}},\, {\sigma _{{\alpha _0}}},\, {\sigma _{{\delta _0}}}$ is the accuracy information of range, range rate, right ascension and declination²⁸, ${\sigma _\rho }$ is the accuracy of range, $\sigma$ is the accuracy of angle, T is the time span of the observation arc, and N is the number of observation points.

According to the law of covariance propagation, ${\bar {\alpha }_i} - {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha } _i}$ obeys a normal distribution with a mean of 0 and a variance of ${\varPhi _{{\alpha _i}}}\Sigma \varPhi _{{{\alpha _i}}}^{T}$:

$${\bar {\alpha }_i} - {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha } _i}\sim \mathcal{N}(0,{\varPhi _{{\alpha _i}}}\Sigma \varPhi _{{{\alpha _i}}}^{T})$$

(23)

According to the observation error characteristics, the actual observation error obeys the following normal distribution:

$${\tilde {\alpha }_i} - {\bar {\alpha }_i}\sim \mathcal{N}(0,\sigma _{{}}^{2}/cos\tilde {\delta }_{i}^{2})$$

(24)

Then,

$${\tilde {\alpha }_i} - {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha } _i}\sim \mathcal{N}(0,\sigma _{{}}^{2}/cos\tilde {\delta }_{i}^{2}+{\varPhi _{{\alpha _i}}}\Sigma \varPhi _{{{\alpha _i}}}^{T})$$

(25)

Similarly, noting that ${f_{{\delta _i}}}({\mathbf{B}}) \triangleq {I_{3\delta }}({I_2}({I_1}({\mathbf{B}};\bar {\dot {\alpha }},\bar {\dot {\delta }});{t_i}))$, the observed value of declination corresponding to $(\bar {\dot {\alpha }},\bar {\dot {\delta }})$ is ${\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\delta } _i}={f_{{\delta _i}}}({\mathbf{\tilde {B}}})$. Set ${\varPhi _{{\delta _i}}}={\left. {\frac{{\partial {f_{{\delta _i}}}}}{{\partial {\mathbf{B}}}}} \right|_{{\mathbf{\tilde {B}}}}}$, then

$${\tilde {\delta }_i} - {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\delta } _i}\sim \mathcal{N}(0,\sigma _{{}}^{2}+{\varPhi _{{\delta _i}}}\Sigma \varPhi _{{{\delta _i}}}^{T})$$

(26)

Similarly, noting that ${f_{{\rho _i}}}({\mathbf{B}}) \triangleq {I_{3\rho }}({I_2}({I_1}({\mathbf{B}};\bar {\dot {\alpha }},\bar {\dot {\delta }});{t_i}))$, the observed value of declination corresponding to $(\bar {\dot {\alpha }},\bar {\dot {\delta }})$ is ${\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\delta } _i}={f_{{\rho _i}}}({\mathbf{\tilde {B}}})$. Set ${\varPhi _{{\rho _i}}}={\left. {\frac{{\partial {f_{{\rho _i}}}}}{{\partial {\mathbf{B}}}}} \right|_{{\mathbf{\tilde {B}}}}}$, then

$${\tilde {\rho }_i} - {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\rho } _i}\sim \mathcal{N}(0,\sigma _{\rho }^{2}+{\varPhi _{{\rho _i}}}\Sigma \varPhi _{{{\rho _i}}}^{T})$$

(27)

In summary, there is

$$\left\{ \begin{gathered} \frac{{{{\tilde {\rho }}_i} - {{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\rho } }_i}}}{{\sqrt {\sigma _{\rho }^{2}+{\varPhi _{{\rho _i}}}\Sigma \varPhi _{{{\rho _i}}}^{T}} }}\sim \mathcal{N}(0,1) \hfill \\ \frac{{{{\tilde {\alpha }}_i} - {{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha } }_i}}}{{\sqrt {\sigma _{{}}^{2}/cos\tilde {\delta }_{i}^{2}+{\varPhi _{{\alpha _i}}}\Sigma \varPhi _{{{\alpha _i}}}^{T}} }}\sim \mathcal{N}(0,1) \hfill \\ \frac{{{{\tilde {\delta }}_i} - {{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\delta } }_i}}}{{\sqrt {\sigma _{{}}^{2}+{\varPhi _{{\delta _i}}}\Sigma \varPhi _{{{\delta _i}}}^{T}} }}\sim \mathcal{N}(0,1) \hfill \\ \end{gathered} \right.$$

(28)

According to the 3σ property of the standard normal distribution, there is

$$\left\{ \begin{gathered} p\left( {\frac{{|{{\tilde {\rho }}_i} - {{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\rho } }_i}|}}{{\sqrt {\sigma _{\rho }^{2}+{\varPhi _{{\rho _i}}}\Sigma \varPhi _{{{\rho _i}}}^{T}} }}<3} \right)=99.7\% \hfill \\ p\left( {\frac{{|{{\tilde {\alpha }}_i} - {{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha } }_i}|}}{{\sqrt {\sigma _{{}}^{2}/cos\tilde {\delta }_{i}^{2}+{\varPhi _{{\alpha _i}}}\Sigma \varPhi _{{{\alpha _i}}}^{T}} }}<3} \right)=99.7\% \hfill \\ p\left( {\frac{{|{{\tilde {\delta }}_i} - {{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\delta } }_i}|}}{{\sqrt {\sigma _{{}}^{2}+{\varPhi _{{\delta _i}}}\Sigma \varPhi _{{{\delta _i}}}^{T}} }}<3} \right)=99.7\% \hfill \\ \end{gathered} \right.$$

(29)

Preliminary simulation results

The above analyses will be used in the determination of the boundaries of the MRAR based on the radar observation data. The purpose of constructing the MRAR is to reduce the scope of the original RAR but ensure that the real orbital states can appear in the MRAR with the maximum possible probability²⁹. The loss function ${J_{{\text{MinAR}}}}(\dot {\alpha },\dot {\delta })$ is defined to describe the standardized deviation (absolute error divided by the standard deviation) of the theoretical observations relative to the actual observations.

$${J_{{\rm{MinAR}}}}(\dot \alpha ,\dot \delta ) = \frac{1}{{3m}}\sum\limits_{i = 1}^m {\left( {\frac{{|{{\tilde \rho }_i} - {{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over \rho } }_i}(\dot \alpha ,\dot \delta )|}}{{\sqrt {\sigma _\rho ^2 + {\Phi _{{\rho _i}{\rm{LS}}}}\Sigma \Phi _{{\rho _i}{\rm{LS}}}^T} }} + \frac{{|{{\tilde \alpha }_i} - {{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over \alpha } }_i}(\dot \alpha ,\dot \delta )|}}{{\sqrt {{\sigma ^2}/cos\tilde \delta _i^2 + {\Phi _{{\alpha _i}{\rm{LS}}}}\Sigma \Phi _{{\alpha _i}{\rm{LS}}}^T} }} + {\kern 1pt} {\kern 1pt} \frac{{|{{\tilde \delta }_i} - {{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over \delta } }_i}(\dot \alpha ,\dot \delta )|}}{{\sqrt {{\sigma ^2} + {\Phi _{{\delta _i}{\rm{LS}}}}\Sigma \Phi _{{\delta _i}{\rm{LS}}}^T} }}} \right)}$$

(30)

where J_MinAR represents the average angular deviation of the orbit. It can be seen that the loss function J_MinAR $(\dot {\alpha },\dot {\delta })$ strictly conforms to the normal distribution, and the threshold is set to β with reference to 3s criterion. For any sampling point $(\dot {\alpha },\dot {\delta })$ in the original RAR, judge whether the value of the loss function ${J_{{\text{MinAR}}}}(\dot {\alpha },\dot {\delta })$ is less than β. If ${J_{{\text{MinAR}}}}(\dot {\alpha },\dot {\delta }) \leqslant \beta$, the corresponding sampling point is considered to have a high probability of being the real orbit state, and will be included into the MRAR; if ${J_{{\text{MinAR}}}}(\dot {\alpha },\dot {\delta })>\beta$, it will not be included into the MRAR. In the actual setting of parameter β, let $\beta \geqslant 3$, so that the $(\bar {\dot {\alpha }},\bar {\dot {\delta }})$ corresponding to the real orbital state will be included in the MRAR with a probability greater than 99.7%. The way of setting the range of the MRAR excludes the sampling points with large mean angular deviation, which is also in accordance with the objective law.

Kernel density estimation

Sample orbit weight

By constructing the MRAR, the scope of the original RAR can be greatly reduced, and it is guaranteed that the real orbit is included with the maximum probability. However, because the AR is an irregular region constructed on the $(\dot {\alpha },\dot {\delta })$ plane, every point in it has the possibility of becoming a real orbit, and the probability of each point is different²⁰. To accurately describe the probability of each point becoming a real orbit, a probability density estimation of the MRAR is also required.

To perform the probability density estimation, the first step is to determine the weight of each sampling point in the MRAR. Each sampling point $({\dot {\alpha }_0},{\dot {\delta }_0})$ corresponds to a set of orbital elements ${{\varvec{x}}_0}$. Under the assumption of Gaussian noise, the probability density of ${{\varvec{x}}_0}$ can be calculated by Bayesian formula:

$$p({{\varvec{x}}_0}|{{\tilde {Y}}})=\frac{{p({{\tilde {Y}}}|{{\varvec{x}}_0})p({{\varvec{x}}_0})}}{{\int_{{{{\varvec{x}}_0}}} {p({{\tilde {Y}}}|{{\varvec{x}}_0})p{{(}}{{\varvec{x}}_0})d{{\varvec{x}}_0}} }}$$

(31)

In this paper, we address the problem of describing the probability of IOD in the absence of a priori information, which also implies that x₀ obeys a uniform distribution, and according to the expression of $p({{\tilde {Y}}}|{{\varvec{x}}_0})$, there is

$$p({{\varvec{x}}_0}|{{\tilde {Y}}}) \propto p({{\tilde {Y}}}|{{\varvec{x}}_0}) \propto \exp \left[ { - \sum\limits_{{i=1}}^{m} {\left( {\frac{{{{\left( {{{\tilde {\rho }}_i} - {{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\rho } }_i}({{\varvec{x}}_0},{t_i})} \right)}^2}}}{{\sigma _{\rho }^{2}+{\varPhi _{{\rho _i}{{LS}}}}\Sigma \varPhi _{{{\rho _i}{{LS}}}}^{T}}}+\frac{{{{\left( {{{\tilde {\alpha }}_i} - {{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha } }_i}({{\varvec{x}}_0},{t_i})} \right)}^2}}}{{\sigma _{{}}^{2}/cos\tilde {\delta }_{i}^{2}+{\varPhi _{{\alpha _i}{{LS}}}}\Sigma \varPhi _{{{\alpha _i}{{LS}}}}^{T}}}+{\kern 1pt} {\kern 1pt} \frac{{{{\left( {{{\tilde {\delta }}_i} - {{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\delta } }_i}({{\varvec{x}}_0},{t_i})} \right)}^2}}}{{\sigma _{{}}^{2}+{\varPhi _{{\delta _i}{{LS}}}}\Sigma \varPhi _{{{\delta _i}{{LS}}}}^{T}}}} \right)} } \right]{\kern 1pt} {\kern 1pt}$$

(32)

The above equation expresses a positive proportionality relationship by which the weighting coefficients are defined. where η is the regulation parameter.

$$w({\varvec{x}_0}) = \eta \exp \left[ { - \frac{1}{{3m}}\sum\limits_{i = 1}^m {\left( {\frac{{{{\left( {{{\tilde \rho }_i} - {{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over \rho } }_i}({\varvec{x}_0},{t_i})} \right)}^2}}}{{\sigma _\rho ^2 + {\Phi _{{\rho _i}{\rm{LS}}}}\Sigma \Phi _{{\rho _i}{\rm{LS}}}^T}} + \frac{{{{\left( {{{\tilde \alpha }_i} - {{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over \alpha } }_i}({\varvec{x}_0},{t_i})} \right)}^2}}}{{\sigma _{}^2/cos\tilde \delta _i^2 + {\Phi _{{\alpha _i}{\rm{LS}}}}\Sigma \Phi _{{\alpha _i}{\rm{LS}}}^T}} + {\kern 1pt} {\kern 1pt} \frac{{{{\left( {{{\tilde \delta }_i} - {{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over \delta } }_i}({\varvec{x}_0},{t_i})} \right)}^2}}}{{\sigma _{}^2 + {\Phi _{{\delta _i}{\rm{LS}}}}\Sigma \Phi _{{\delta _i}{\rm{LS}}}^T}}} \right)} } \right]$$

(33)

Mapping onto the plane of the RAR, there is

$$w({\dot \alpha _0},{\dot \delta _0}) = \eta \exp \left[ { - \frac{1}{{3m}}\sum\limits_{i = 1}^m {\left( {\frac{{{{\left( {{{\tilde \rho }_i} - {{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over \rho } }_i}({{\dot \alpha }_0},{{\dot \delta }_0},{t_i})} \right)}^2}}}{{\sigma _\rho ^2 + {\Phi _{{\rho _i}{\rm{LS}}}}\Sigma \Phi _{{\rho _i}{\rm{LS}}}^T}} + \frac{{{{\left( {{{\tilde \alpha }_i} - {{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over \alpha } }_i}({{\dot \alpha }_0},{{\dot \delta }_0},{t_i})} \right)}^2}}}{{\sigma _{}^2/cos\tilde \delta _i^2 + {\Phi _{{\alpha _i}{\rm{LS}}}}\Sigma \Phi _{{\alpha _i}{\rm{LS}}}^T}} + {\kern 1pt} {\kern 1pt} \frac{{{{\left( {{{\tilde \delta }_i} - {{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over \delta } }_i}({{\dot \alpha }_0},{{\dot \delta }_0},{t_i})} \right)}^2}}}{{\sigma _{}^2 + {\Phi _{{\delta _i}{\rm{LS}}}}\Sigma \Phi _{{\delta _i}{\rm{LS}}}^T}}} \right)} } \right]$$

(34)

This coefficient is proportional to the probability density and can be used to characterize the magnitude of the chance that the orbit corresponding to the sampling point will be a real orbit. As can be seen from the expression for the weighting coefficient, the least squares orbit is also the orbit with the largest weight.

Probability density function (PDF)

There are many ways to estimate the overall probability density function through the sample, usually divided into parametric estimation methods, semi-parametric estimation methods and non-parametric estimation methods. Due to the strong nonlinear characteristics of the IOD problem, the orbit determination results do not necessarily obey a normal distribution or other specific forms of distribution. Therefore, both parametric and semiparametric estimation methods are no longer applicable. Kernel density estimation belongs to one of the nonparametric estimation methods, also known as Parzen window, which is currently the most effective and widely used one of the nonparametric probability density estimation algorithms³⁰. In this paper, the kernel density method is selected for the estimation of probability density function, in which the key lies in the selection of kernel function and the determination of window width³¹.

Suppose ${x_1},{x_2}, \cdots ,{x_n}$ are samples (independently and identically distributed) obtained by sampling a one-dimensional random variable x. The probability density function f(x) of x can be estimated as:

$${\hat {f}_h}\left( x \right)=\frac{1}{{nh}}\sum\limits_{{i=1}}^{n} {K\left( {\frac{{x - {x_i}}}{h}} \right)}$$

(35)

where K(·) is the kernel function, including Gaussian function, Epanechnikov function, Biweight function, etc.; h is the window width. When the weight of each variable is different, 1/n needs to be corrected to the weight value of each variable w_i, i.e.

$$f\left( x \right)=\frac{1}{h}\sum\limits_{{i=1}}^{n} {{w_i}K\left( {\frac{{x - {x_i}}}{h}} \right)}$$

(36)

For d-dimensional random variables ${\varvec{X}}={\left( {{x_1},{x_2}, \cdots ,{x_d}} \right)^T}$, set the sample to ${{\varvec{X}}_i}={({x_{i1}},{x_{i2}}, \cdots {x_{id}})^T}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (i=1,2, \cdots n)$ and Let ${\varvec{h}}=\left( {{h_1},{h_2}, \cdots ,{h_d}} \right)$ be a window width vector, the kernel density of X is estimated as:

$$\begin{array}{*{20}{c}} {{{\hat {f}}_{\varvec{h}}}\left( {\varvec{X}} \right)=\frac{1}{n}\sum\limits_{{i=1}}^{n} {{K_{\varvec{h}}}\left( {{\varvec{X}} - {{\varvec{X}}_i}} \right)} } \\ {{K_{\varvec{h}}}\left( {{\varvec{X}} - {{\varvec{X}}_i}} \right)=\frac{1}{{{h_1}{h_2} \cdots {h_d}}}\left\{ {K\left( {\frac{{{x_1} - {x_{i1}}}}{{{h_1}}}} \right)K\left( {\frac{{{x_2} - {x_{i2}}}}{{{h_2}}}} \right) \cdots K\left( {\frac{{{x_1} - {x_{in}}}}{{{h_n}}}} \right)} \right\}} \end{array}$$

(37)

In this paper, the Gaussian kernel function is chosen as the kernel density function with the expression

$$K\left( {\frac{{(\dot {\alpha },\dot {\delta }) - {{(\dot {\alpha },\dot {\delta })}_i}}}{{{h_1}{h_2}}}} \right)=\frac{1}{{\sqrt {2\pi } }}w{(\dot {\alpha },\dot {\delta })_i}{e^{ - \left[ {\frac{{{{\left( {\dot {\alpha } - {{\dot {\alpha }}_i}} \right)}^2}}}{{2h_{1}^{2}}}+\frac{{{{\left( {\dot {\delta } - {{\dot {\delta }}_i}} \right)}^2}}}{{2h_{2}^{2}}}} \right]}}$$

(38)

The above equation is the estimated probability density function of the MRAR. The results of the IOD in the region will be able to be described in probabilistic form.

Instance verification

The probabilistic description of the IOD result in the MRAR requires multiple steps. The following is a specific algorithm flow:

Algorithm Probabilistic description of IOD result in the MAR.

This study selected three satellites from GLONASS (NORAD ID: 32393), GPS (NORAD ID: 35752), and Starlink (NORAD ID: 44718) as research objects. The MRAR were constructed for each spacecraft, and their IOD results were expressed in the form of PDFs. Table 1 summarizes the orbit elements and observation error characteristics of the selected space objects.

Table 1 Parameter setting.

Fig. 1

Probability density of IOD results in the original RAR.

Figure 1 illustrates the probability density distribution of IOD results within the original RAR. The red solid circle represents the corresponding least-squares solution. It is evident that the probability density exhibits an elliptical ring pattern, with the highest probability concentrated around the ring containing the least-squares solution. This indicates that sample points closer to the least-squares solution have a higher likelihood of representing the true orbital state, while those further away have lower probabilities. The darker color of the central elliptical ring suggests that these sample points are more probable candidates for the true orbit state. However, this wide uncertainty range undermines the practical significance of the IOD. A smaller uncertainty range would significantly reduce computational costs for subsequent arc association and precise orbit determination processes.

In summary, Fig. 1 provides insights into the probabilistic nature of IOD but highlights the need for methods like the MRAR approach presented later in this paper to refine the solution space and improve accuracy.

Figure 2 illustrates the MRAR for Sat_32393 in the left panel, with the corresponding PDF within the MRAR shown in the right panel. As indicated in Table 1, the radar observation arc length for this dataset is 1 min, classifying it as an extremely short arc (ESA). The radar’s measurement accuracy—30 m in range and 0.05° in angular observation—aligns with operational specifications of modern space surveillance radars.

Fig. 2

MRAR (left) and PDF of IOD results(right) for Sat_32393.

Fig. 3

MRAR (left) and PDF of IOD results(right) for Sat_35752.

Fig. 4

MRAR (left) and PDF of IOD results(right) for Sat_44718.

Figure 2 demonstrates that the MRAR significantly reduces the spatial scope of the original RAR, enabling more focused uncertainty quantification in IOD. In the right panel, the color gradient from yellow to blue represents the decay of PDF values, where sampling points in yellow regions correspond to higher confidence in representing true orbital states. The weight coefficients of sampling points exhibit a negative correlation with loss function values as derived from Eqs. (26) and (30).

Figures 3 and 4 depict the MRAR and PDF for satellites Sat_35752 and Sat_44718, respectively. The PDF distributions reveal that the IOD results exhibit the highest probability density concentration around the least-squares solution, which aligns with statistical expectation in uncertainty quantification. Normally, the least-squares solution has the highest probability of becoming the true orbital state, which is in line with the normal logic of uncertainty analysis.

Conclusion

In this paper, for the inaccurate estimation of the orbit state in the IOD, radar observation data are used to determine the MRAR of the space target, and the probabilistic description method of the IOD results based on the MRAR is constructed. It solves the key problem of using radar observation data to determine the reliable orbit state of a space target, reduces the problem of IOD from six dimensions to two dimensions, greatly reduces the difficulty of IOD, and improves the accuracy of target orbit state positioning.

By calculating the loss function value of each sampling point in the original RAR, the MRAR is determined, and according to the theoretical derivation, it can be proved that the MRAR can contain the real track state with a probability greater than 99.7%, which means that the uncertainty analysis under the MRAR has the practical application value. On this basis, the Bayesian formula is used to calculate the weight coefficients of each sampling point in the MRAR, which objectively reflects the probability of the orbit corresponding to each sampling point to be the real orbit. Specifically, the higher the weight coefficient, the higher the possibility that the orbital state of the sampling point is the target orbital state. Based on the estimated weights of each sampling point, this paper uses the weighted kernel density estimation method to estimate the probability density, and obtains the probability density function of MRAR; compared with the more common parametric and semiparametric estimation methods, the use of the weighted kernel density estimation method is able to solve the strong nonlinear problem of IOD.