Overview on Space-Based Optical Orbit Determination Method Employed for Space Situational Awareness: From Theory to Application

Abstract

Leveraging space-based optical platforms for space debris and defunct spacecraft detection presents several advantages, including a wide detection range, immunity to cloud cover, and the ability to maintain continuous surveillance on space targets. As a result, it has become an essential approach for accomplishing tasks related to space situational awareness. However, the prediction of the orbits of space objects is crucial for the success of such missions, and current technologies face challenges related to accuracy, reliability, and practical efficiency. These challenges limit the performance of space-based optical space situational awareness systems. To drive progress in this field and establish a more effective and reliable space situational awareness system based on space optical platforms, this paper conducts a retrospective overview of research advancements in this area. It explores the research landscape of orbit determination methods, encompassing orbit association methods, initial orbit determination methods, and precise orbit determination methods, providing insights from international perspectives. The article concludes by highlighting key research areas, challenges, and future trends in current space situational awareness systems and orbit determination methods.

1. Introduction

Since the launch of the first satellite on 4 October 1957, approximately 15,302 artificial spacecraft have been deployed through 6346 launch missions worldwide by the end of 2022 (as shown in Figure 1 and Figure 2) [1]. The fragments generated from those rocket launches and disabled spacecraft have resulted in millions of pieces of space debris, posing a significant threat to operational satellites in space [2,3]. In 1983, the space shuttle Challenger experienced the first ever significant collision with space debris in human history. This incident resulted in damage to Challenger’s window and the premature termination of the mission. Subsequently, several severe instances of space debris colliding with operational satellites in orbit have been observed. In each case, the impact caused substantial damage or even the complete disintegration of the affected spacecraft [4]. With the space environment becoming increasingly crowded, it is crucial to ensure the safety of space assets in orbit and prevent potential damage from space debris collisions. This urgency highlights the need for timely and continuous sensing and measurement of the operational orbits of objects in space.

Recognizing the imperative of protecting on-orbit spacecraft from collisions with space debris, major spacefaring nations worldwide have universally developed their own space situational awareness systems. In the mid-20th century, the United States and the Soviet Union pioneered the development of ground-based space situational awareness systems. As we entered the 21st century, other spacefaring nations, including Europe, Japan, India, and China, significantly enhanced their capabilities in space situational awareness, aiming to address the challenges posed by the increasingly congested space environment and ensure the security of their respective space assets.

In recent years, significant attention has been directed towards space-based optical systems for space situational awareness. Unlike ground-based systems, those dependent on space-based optical platforms provide unique advantages, such as a wide observational field, extended daily monitoring time, and immunity to atmospheric or cloud cover. These features facilitate the timely and continuous surveillance of both space regions and targets. Furthermore, through orbital networking or orbital maneuvers, space-based optical space situational awareness systems can attain persistent monitoring and conduct close-range reconnaissance of space targets.

In the technical framework of space situational awareness utilizing optical monitoring systems in space, the determination of the orbits of space targets constitutes its core and foundation. After acquiring detection data, space-based situational awareness platforms employ orbit determination methods to convert sensor data into precise parameters defining the orbital state of the targets. This process serves as the essential bridge for transforming sensor detection information into assessments of the orbits of space targets and alert information. This foundational procedure is integral for estimating the on-orbit operational status of space targets, cataloging space objects, and conducting analyses related to space contact and collisions.

In this paper, a comprehensive overview of the development history of space situational awareness systems and the present state of research on orbit determination methods employing space-based optical platforms is provided. First, it delineates the status of space situational awareness policies and equipment in major spacefaring nations, analyzing the practical engineering demands in this domain. Subsequently, the existing state of relevant theoretical research in the three stages of orbit determination—short-arc association, initial orbit determination, and precise orbit determination—is introduced. Finally, the paper summarizes key issues in the current theoretical research and underscores research directions that merit particular attention in the future development of this field.

2. The Evolution of Space Situational Awareness Policies

Research on space situational awareness originated in the 1980s and gained significant attention in the 1990s [5]. In August 1998, the U.S. government underscored the critical significance of space situational awareness in its official document [6]. In 2018, space situational awareness was officially recognized as one of the four primary U.S. space missions. The related document explicitly emphasized that a comprehensive understanding of the space environment is foundational to the execution of U.S. space commands and operations in the space domain [7].

In 2019, the United States formally introduced the term Space Domain Awareness (SDA) to supplant space situational awareness. This signifies the official acknowledgment of space as an operational domain, alongside land, sea, and air [8]. The documents titled “Space Capstone Publication, Spacepower” published in 2020 and “Space Doctrine Notes, Operations” published in 2022 by the U.S. government underscored that the Space Domain Awareness mission entails understanding all factors, intentions, capabilities, and current conditions that may impact U.S. space activities. These documents particularly highlighted the importance of space situational awareness in anticipating potential future scenarios in the space domain [9,10]. Important milestones in the development of the U.S. space situational awareness policies are shown in Figure 3.

In addition to the United States, European countries have embarked on establishing an independent system to detect, assess, and forecast space threats since the last century. The development of a European space situational awareness system was initially led by European spacefaring countries, notably France [11]. In 2008, the Council of the European Union adopted resolutions to advance European space policies, emphasizing the need for Europe to enhance its monitoring capabilities for space facilities and debris to strengthen its global standing in space endeavors [12]. Since 2009, the European Space Agency (ESA) has spearheaded the Space Situational Awareness Program, categorizing it among the six European major space programs alongside the Galileo Global Satellite Navigation System [13].

The primary objectives of the European space situational awareness system include monitoring and tracking space objects, observing and forecasting space environments, and detecting space targets in low Earth orbits (LEOs). These objectives aim to achieve space collision avoidance, re-entry trajectory analysis, and orbit evolution of space targets to ensure the safety of European space facilities [14,15,16]. In 2010, the European Council formally acknowledged the importance of space situational awareness in both military and civilian areas in its seventh Space Council resolution [17]. Subsequently, the European Council reiterated the significance of space situational awareness in a series of related meetings in 2011 [18,19].

In 2014, the European Parliament adopted Resolution No. 541 concerning the establishment of the Space Surveillance and Tracking (SST) System, systematically outlining concepts, necessity, mission goals, and series of resolutions related to SST [20]. In 2015, the EU established the SST Alliance to coordinate SST-related affairs. In 2016, the European Space Strategy was promulgated, emphasizing Europe’s commitment to strengthening existing space target monitoring and tracking capabilities and evolving them into a more robust space situational awareness system in the future [21]. In 2018, the EU Commission submitted the European SST system development report for the years 2014–2017 to the European Parliament. The report systematically summarized the development of the European space monitoring and space target tracking system during that period, providing prospects for the future development of the system in both technical and managerial aspects, underscoring European high regard for constructing a space target monitoring system [22].

In 2022, European leaders proposed considering space as a strategic zone for Europe and called for the formulation of a European Space Security and Defense Strategy. In response, the European Commission put forth the first European Space Security and Defense Strategy, emphasizing the prominent position of space situational awareness [23,24]. Presently, aligning with the European development plan for its space situational awareness system, Europe has established ground radar systems, such as the Grand Réseau Adapté à la Veille Spatiale (GRAVES), for space situational awareness. Concurrently, the EU is planning to respectively launch the Hera satellite in 2024 and the ClearSpace-1 satellite in 2026 to carry out tasks related to space debris cleanup. Plans are also underway to launch the Vigil satellite in the 2020s to monitor the space environment, gradually advancing toward the goal of constructing the European space situational awareness and space defense system. Figure 4 shows the important milestones in the development of European space situational awareness policies.

In addition to the United States and Europe, Russia has developed its space situational awareness system known as the Russian Space Surveillance System (RSSS) [25]. The origins of the Russian space situational awareness system can be traced back to the Soviet era in the 1960s when a world-class network of ground-based radar stations and optical telescopes was established, ensuring the efficient monitoring of space targets [26]. Moreover, owing to historical factors, Russia has maintained a top-tier level of theoretical research in the field of space situational awareness [27].

In recent years, India has demonstrated a growing interest in space situational awareness. India established the Directorate of Space Situational Awareness and Management and, in 2019, initiated the construction of the Space Situational Awareness Control Centre to specifically oversee India’s activities and research programs in space situational awareness [28,29]. In 2022, India entered into a cooperation agreement with the United States, covering aspects such as the sharing of space situational awareness information [30]. Additionally, starting from 2022, the Indian Space Research Organisation (ISRO) has annually released a Space Situational Assessment Report on its official website. This report provides information on the global launch scenario and typical space collision risks of the previous year, highlights future trends in space situation, and offers insights into the development in the field of space situational awareness [31,32]. Japan is also in the planning stages for constructing a space-based situational awareness system composed of radars, optical telescopes, and analysis systems. The goal is to monitor space debris and analyze its potential threats to on-orbit satellites and astronauts [33].

In general, the concept of space situational awareness emerged in the 1990s and has garnered widespread attention in the 21st century. With the ongoing advancement of space technology, the concept of space situational awareness has evolved, expanding from its initial focus on the discovery and perception of space objects to encompass the measurement and estimation of the motion states of these objects, continuous monitoring of key space targets and regions, orbit prediction of space objects, and warning of collision threats.

Looking ahead, as space technology continues to advance, more complex space missions will demand higher requirements for the scope and precision of space situational awareness. Existing space situational awareness systems primarily detect and perceive unknown targets one by one within their detection range. The development of large-scale spacecraft constellations urgently calls for perception systems capable of addressing clustered targets.

Furthermore, to ensure the long-term operation of increasingly sophisticated space payloads on orbit, it is crucial to minimize the probability of collisions between these payloads and space debris. This places higher demands on the sensitivity of space situational awareness systems, necessitating the research and development of equipment and systems capable of detecting extremely small space debris.

In order to reduce the detection time of space collision threats, providing more time for spacecraft to generate and execute collision avoidance strategies, space situational awareness systems need to have a longer detection range and higher recognition speed. This enables the early and swift identification of potential close encounters and collision threats.

3. Typical Space-Based Situational Awareness Facilities

The current global systems for detecting and alerting against threats to space assets are broadly categorized into ground-based and space-based surveillance systems based on their spatial distribution [34,35,36,37]. Ground-based systems, being the initial development, formed the early space surveillance infrastructure for major spacefaring nations, leveraging their high power and capacity for large detection payloads. However, the Earth’s curvature and susceptibility to atmospheric and cloud cover may limit ground-based systems, imposing significant constraints on their detection range and effective operational time. Additionally, their distance from space targets results in lower resolution for detecting targets on HEO.

In contrast, space-based surveillance systems utilize detection payloads on satellites, mitigating issues related to atmosphere and clouds on Earth. They offer a broader field of view compared to ground-based systems through maneuverability. Moreover, these systems, equipped with satellite maneuvering and multi-satellite networking capabilities, can ensure continuous monitoring and close observation of specific space targets or regions. To achieve comprehensive, continuous, and high-precision space situational awareness on a global scale, nations worldwide have increasingly emphasized the development of space-based platforms in recent years [34]. In this section, typical detection payloads for space situational awareness systems shown in

Table 1. Key performance parameters of space situational awareness satellite payloads.

Satellite	Optical Aperture	Pixel Size	Pixel Number	Field of View
MSX	15 cm	Unknown	Unknown	$1.4^{\circ }\times 5.6^{\circ }$
SBSS	30 cm	Unknown	2.4 million	$3^{\circ }\times 3^{\circ }$
STARE	8.5 cm	6.7 $\mathsf{\mu }$m	1280 × 1024	$2.{08}^{\circ }\times 1.{67}^{\circ }$
MOST	15 cm	Unknown	1024 × 1024	$2^{\circ }\times 2^{\circ }$
Sapphire	15 cm	Unknown	Unknown	$1.4^{\circ }$
SBO	20 cm	18 $\mathsf{\mu }$m	Unknown	$6^{\circ }$
Asteroid	25 cm	13 $\mathsf{\mu }$m	2k × 2k	$2^{\circ }\times 2^{\circ }$

are presented to provide a more intuitive and comprehensive understanding of current development status of this field.

In the 1990s, the United States launched one of the earliest and most representative space-based situational awareness satellites, the Midcourse Space Experiment (MSX). It carried a suite of optical detection instruments, including an infrared imaging telescope, a space-based visible (SBV) camera, and the ultraviolet and visible imagers and spectrographic imagers (UVISI) [38]. The SBV camera is considered a groundbreaking space-based visible light detection instrument. The optical aperture of SBV is 15 cm, operating in the spectral range of 0.3–0.9 $\mathsf{\mu }$m. Its focal plane is composed of four low-noise visible light CCD chips with a field of view of $1.4^{\circ }\times 1.4^{\circ }$ each, stitched together to form a total field of view of $1.4^{\circ }\times 5.6^{\circ }$ [39]. This camera signifies the validation of space-based optical detection technology and provides crucial insights for the development of subsequent space detection systems.

Building upon the success of the MSX satellite, the United States further strengthened its space situational awareness capabilities by establishing the space-based surveillance system (SBSS). The first satellite of the SBSS mission, launched in 2010, carried a TMA visible light camera with 30 cm aperture, $3^{\circ }\times 3^{\circ }$ field of view, and 2.4 million pixels. Mounted on a gimbal, TMA demonstrated the capability to flexibly adjust its field of view under a fixed satellite posture. The detection range of TMA covered LEO, MEO, and HEO, with a positioning accuracy of 10 m for low-orbit targets and 500 m for high-orbit targets. According to the plan of the SBSS project, the system was originally intended to establish a constellation of space-based monitoring satellites operating in LEO, thereby enhancing positioning accuracy for targets in HEO to approximately 250 m [40].

To address the issue of too many false collision alerts caused by insufficient ground-based detection resolution, the Space-based Telescopes for the Actionable Refinement of Ephemeris (STARE) satellite was launched into a polar orbit to validate the capability of space-based surveillance systems to update the orbital parameters of on-orbit targets. According to relevant research, this satellite carries a coaxial reflection system payload with an 85 mm aperture, a field-of-view angle of $2.{08}^{\circ }\times 1.{67}^{\circ }$, and a pixel size of 6.7 $\mathsf{\mu }$m, comprising 1280 × 1024 pixels [39,40].

To validate whether satellite maneuvers can enhance target detection capability, Canada launched the smallest space-based telescope to date, the Microvariability and Oscillations of Stars (MOST) satellite [41]. This satellite carries a 15 cm aperture optical telescope with a field-of-view angle of $2^{\circ }\times 2^{\circ }$ and features 1024 × 1024 pixels [42].

After the launch of the MOST satellite in Canada, the United States launched the Wide Area Space Surveillance System (WASS), utilizing a network of multiple LEO satellites to monitor stationary orbit targets on Earth. Each satellite in this system carries multiple wide-angle cameras, with each camera having a field-of-view angle of up to ${60}^{\circ }\times 4^{\circ }$ and an aperture of 28 mm. The cameras are equipped with a long sunshade to achieve a $4^{\circ }$ sun avoidance angle [41].

The most well-known space situational awareness satellite of Canada is the Sapphire satellite. This satellite employs a three-mirror anastigmat design which is similar to the SBV, with a 15 cm aperture and a field of view of $1.4^{\circ }$. The detector of this satellite utilizes a trajectory speed mode to achieve a higher signal-to-noise ratio. It is capable of monitoring and tracking objects with a magnitude as faint as 15 [40,42].

The space-based optical telescope (SBO), developed in Europe for space debris monitoring, features a CMOS sensor with an optical aperture of 20 cm, pixel size of 18 $\mathsf{\mu }$m, and a field-of-view angle of 6°. It possesses robust capabilities for monitoring space debris targets, enabling observation of 2 cm-sized objects in HEO [39].

In addition, the AsteroidFinder satellite developed by Germany carries an optical payload with a 25 cm aperture. It employs an off-axis three-reflection dispersive imaging catadioptric optical system with a field-of-view angle of $2^{\circ }\times 2^{\circ }$, pixel size of 13 $\mathsf{\mu }$m, and a resolution of 2k × 2k pixels. By incorporating a field stop in the optical system, the camera of this satellite achieves effective stray light suppression, enabling the detection of spacecraft and space debris in LEO. This satellite validates the capability of space-based optical detection systems for detecting centimeter-sized space targets, marking a significant milestone in the development of space situational awareness systems [39,40].

In general, space-based situational awareness facilities are predominantly equipped with an optical detection payload to monitor and assess space objects and the space environment. Compared to radar detection, optical detection is envisioned as the primary means for future space-based target monitoring. Optical detection consumes less energy for long-range detection and can be combined with real-time image processing to enhance resolution. In terms of monitoring capabilities, space-based situational awareness systems predominantly rely on visible light cameras as their effective payloads. However, due to the diversity of space targets, there is a growing trend towards multi-spectral joint detection, such as combining visible light with infrared detection or utilizing multi-band infrared detection.

Regarding the selection of monitoring constellations and the number of satellites, with the advancement of small satellite technology and space satellite formation techniques, space situational awareness payloads are increasingly integrated with small or multiple satellites. Additionally, a single monitoring satellite may carry various sensors to cover a broader monitoring range and multiple spectral bands.

In the choice of space camera, off-axis three-mirror optical systems and even more complex optical systems are becoming more widely used to address the growing demand for wide-field space-based surveillance.

4. Orbit Association Methods

Space-based optical space situational awareness systems offer several advantages, including a short detection wavelength, a large frame information capacity, and the ability to monitor multiple space targets. Currently, this kind of system has emerged as the mainstream solution for monitoring space target and has entered the experimental and application phases.

In the development of space-based space situational awareness systems, one of the crucial technologies is predicting the orbits of identified space targets based on optical imaging results. This process, often referred to as orbit determination, serves as the core and foundation of space-based space situational awareness tasks [35]. Orbit determination can be categorized into batch processing and sequential processing, depending on specific processing methods [36]. It can also be classified into real-time tracking and post-event tracking based on processing timeliness.

According to the orbit model used in the orbit determination process, orbit determination tasks can be classified into dynamic orbit determination, kinematic orbit determination, and reduced-dynamic orbit determination. In terms of processing steps, the orbit determination process involves orbit association, initial orbit determination (IOD), and orbit refinement (or precise orbit determination) [37].

In scenarios where optical observations are the sole source of information, obtaining direct measurements of the distance and velocity of space targets in relation to the observing platform is unattainable. Only information about the relative angles and angular velocities of the target’s motion can be gathered. Before commencing the initial orbit determination, it is crucial to confirm, through orbit association, whether multiple observation results pertain to the same target [43].

The admissible region method is a classical approach for correlating diverse observation outcomes [44]. This method utilizes the mechanical principles governing the motion of space targets as constraint conditions, restricting the feasible range of distances and its change rate (equivalent to the orbital elements) from observation platforms to the target during angle-only measurement process. Subsequently, the orbital element of the target will be determined within the feasible range.

Expanding upon the admissible region method, as shown in Table 2, Tommei et al. introduced an approach that integrates the gravity as a constraint on the the motion of the targets. Simultaneously they utilized the orbital altitudes of the targets as a constraint to construct the feasible domain of their motion. This method, grounded in the admissible region framework, successfully achieves the association of the observed short arcs of the space targets [45].

Following the principles laid out by Tommei et al., Maruskin et al. proposed a recursive intersection method to address the problem of orbit association. This approach employs orbital mechanical energy, distance from Earth, and permissible ranges for perigee and apogee as constraints for feasible target orbital solutions. It calculates intersections of different orbital segments corresponding to multiple observations, and associates different observed orbit segments according to the existence of intersections associated with various observation results [46]. The recursive intersection method simplifies the orbital association process but requires manual screening and processing of results during implementation, making it unsuitable for associating large-scale orbital observation data.

Fujimoto et al. mapped the feasible domain where the targets might exist to a 6D Poincaré space, which is divided into multiple hypercubes. In the Poincaré space, the density of virtual particles, representing the probability of the existence of the observed target in space, is distributed among different hypercubes. Bayesian theorem is then employed to achieve the association between different observation segments [47,48,49]. The approach, characterized by linearizing the mapping process and discretizing the feasible domain, effectively mitigates computational costs and maintains robustness even in the presence of observation errors. Nevertheless, it does not apply to the GEO association problems [50].

Siminski et al. reframed the orbit association problem into an optimization process by employing a loss function and corresponding optimization methods. Depending on whether the optimization problem used in the trajectory association process is an initial value problem (IVP) or a boundary value problem (BVP), the associated methods are categorized into two classes [51,52]. Among them, the BVP-based approach utilizes distance assumptions and Lambert problem solvers to determine feasible orbits for observed targets, exhibiting better completeness and efficiency. On the other hand, the IVP-based methods determine the orbits of the targets through the admissible region method, offering advantages in terms of clarity.

Table 2. Comparison of the orbit association methods mentioned in this section.

Developers	Characteristics	Inputs	Outputs	Main Equations
Tommei et al. [45]	Regard the gravity as a constraint on the motion of targets and utilized the orbital altitudes as a constraint to construct the feasible domain	Arc of observations and the corresponding optical attributable	Admissible region of the orbit of target	$\mathcal{C}={\mathcal{C}}_1\cap {\mathcal{C}}_2$
Maruskin et al. [46]	Simplified the association process and further limited the size of admissible region in the range range-rate plane	Arc of observations and the corresponding optical attribute	Admissible region of the orbit of target	$\mathcal{C}=\underset{i=1}{\overset{4}{\cap }}{\mathcal{C}}_i$
Fujimoto et al. [47,48,49]	Mapped the feasible domain where the target might exist to a 6D Poincaré space	Multiple observations and the sensor position	Correction of observations	$\mathit{X}\approx {\mathit{X}}^{\ast }+\Phi {[\delta \rho ,\delta \dot{\rho }]}^T$
Siminski et al. [51,52]	Reframed the association problem into an optimization process	Range and range-rate hypothesis	Improved range and range-rate hypothesis	$L\left(\mathit{p}\right)={({\mathit{a}}_2-{\widehat{\mathit{a}}}_2)}^T({\mathit{C}}_{{\widehat{a}}_2}+{\mathit{C}}_{a_2})({\mathit{a}}_2-{\widehat{\mathit{a}}}_2)$; $L(\mathit{p},k)={(\dot{\mathit{z}}-\widehat{\dot{\mathit{z}}})}^T({\mathit{C}}_{\widehat{\dot{z}}}+{\mathit{C}}_{\dot{z}})(\dot{\mathit{z}}-\widehat{\dot{\mathit{z}}})$

Table 2. Comparison of the orbit association methods mentioned in this section.

Developers	Characteristics	Inputs	Outputs	Main Equations
Tommei et al. [45]	Regard the gravity as a constraint on the motion of targets and utilized the orbital altitudes as a constraint to construct the feasible domain	Arc of observations and the corresponding optical attributable	Admissible region of the orbit of target	$\mathcal{C}={\mathcal{C}}_1\cap {\mathcal{C}}_2$
Maruskin et al. [46]	Simplified the association process and further limited the size of admissible region in the range range-rate plane	Arc of observations and the corresponding optical attribute	Admissible region of the orbit of target	$\mathcal{C}=\underset{i=1}{\overset{4}{\cap }}{\mathcal{C}}_i$
Fujimoto et al. [47,48,49]	Mapped the feasible domain where the target might exist to a 6D Poincaré space	Multiple observations and the sensor position	Correction of observations	$\mathit{X}\approx {\mathit{X}}^{\ast }+\Phi {[\delta \rho ,\delta \dot{\rho }]}^T$
Siminski et al. [51,52]	Reframed the association problem into an optimization process	Range and range-rate hypothesis	Improved range and range-rate hypothesis	$L\left(\mathit{p}\right)={({\mathit{a}}_2-{\widehat{\mathit{a}}}_2)}^T({\mathit{C}}_{{\widehat{a}}_2}+{\mathit{C}}_{a_2})({\mathit{a}}_2-{\widehat{\mathit{a}}}_2)$; $L(\mathit{p},k)={(\dot{\mathit{z}}-\widehat{\dot{\mathit{z}}})}^T({\mathit{C}}_{\widehat{\dot{z}}}+{\mathit{C}}_{\dot{z}})(\dot{\mathit{z}}-\widehat{\dot{\mathit{z}}})$

It is notable that a significant challenge in existing orbit association methods lies in the high rate of erroneous orbit associations for distinct targets within the same satellite constellation. To enhance the accuracy of orbit associations for different targets within the same constellation, Cai et al. introduced a novel approach known as the “Common Ellipse Method”. This method addresses the orbit association problem by examining the proximity of the hypothetical ellipses representing the target trajectories in the observational data to a common ellipse. Simulation results demonstrate a notable improvement in the accuracy of orbit associations using this method [43].

5. Initial Orbit Determination Methods

In the initial stages of orbit determination, challenges arise due to the limited field of view of monitoring camera and the high relative velocity between space targets and monitoring platforms. A single monitoring spacecraft often struggles to maintain continuous tracking of the target motion. Consequently, the estimation of a space target orbit is usually based on limited observational data, a technique referred to as short-arc orbit determination [53]. Representing the orbit of a space target uniquely requires six orbital elements. However, a single optical observation can only provide information about two angles of the target and their corresponding times. Hence, achieving short-arc orbit determination during the initial phase typically requires obtaining six sets of observations, including angles and times from three separate instances as depicted in Figure 5.

The most classic algorithms for initial orbit determination are the Gauss method and the Laplace method shown in Table 3 [54,55]. Among them, the Gauss method initiates by formulating a system of equations for target positioning, involving 18 unknowns. This process leverages the geometric relationship between the observation platform and the target, along with the spatial target position vector expression based on Lagrange coefficients and the conservation theorem of angular momentum. Finally, the Gauss method utilizes multiple observations, incorporating target angle information and the position of the observation platform, to solve these equations [56]. However, the accuracy of the Gauss method diminishes when the curvature of the observed arc is low since the curvature of the observed trajectory is used as a divisor during the computation, and it may encounter singularity and instability problems during its usage [50]. Considering the impact of observational errors, results obtained by the Gauss method may face challenges in subsequent differential corrections for least squares fitting, potentially resulting in divergence in the orbit determination process [57].

On the other hand, the Laplace method solves the geocentric distance of the target iteratively by utilizing the azimuth of the observed vector, the azimuthal velocity, and the angular acceleration derived differentially from the observed data. Building upon this, the method utilizes observational geometry to ascertain the position and acceleration of the target, enabling the determination of the target’s six orbital elements. Theoretical research and application instances show that the Laplace method proves to be a more straightforward and practical method for orbit determination missions, particularly when dealing with orbit determination problems with very limited observational data [58]. However, it is crucial to acknowledge the inherent difficulty in accurately determining the geocentric distance of space targets due to errors in determining the target’s geocentric distance relative to angle information from the observation platform. This challenge becomes particularly pronounced in scenarios with sparse observational data during short-arc orbit determination. The inherent limitations of the Laplace method make it difficult to precisely determine the geocentric distance of observed targets, potentially leading to an ill-condition during the matrix inversion process when calculating the position vector of the target, sometimes even resulting in orbit determination failure [43,59,60].

Distinct from the Gauss method and the Laplace method, the Gooding method and the double r method are other two widely recognized techniques for initial orbit determination [61,62]. Schaeperkoetter systematically compared the Gauss method, the Laplace method, the double r method and the Gooding method in terms of accuracy, convergence, robustness and suitability for space-based platforms. The results consistently showed that the Gooding method exhibited optimal performance in initial orbit determination problems in the majority of cases [63]. When estimating the direction of the target, the accuracy of the Gooding method typically surpassed other methods by several orders of magnitude. In terms of the shape estimation of orbits, the precision of the Gooding method and double r method was generally comparable. In scenarios involving near-polar orbits and issues where the initial distance estimation for the target orbit is unknown, the Laplace method often performs the best.

To further enhance the precision and speed for solving the initial orbit determination problems, Schmidt et al. investigated the observability of the target relative to the monitoring platform. They proposed an algorithm for determining the shape of the target orbit under the condition of angle-only measurements [64]. Newman and others addressed observability issues in the angle-only initial orbit determination problems through nonlinear second-order Volterra sequences [65,66]. Subsequently, Shubham et al. optimized the algorithms proposed by Newman and achieved higher computational efficiency [67]. Geller et al. developed another orbit determination method by setting the camera offset from the vehicle [68,69]. Gong et al. employed neural networks to map the observation vectors to space orbits, effectively addressing the nonlinear challenges introduced by complex dynamic models [70]. While machine learning methods exhibit commendable performance in orbit determination, they concurrently escalate the computational demands for space situational awareness tasks, significantly hampering the operational efficiency of associated algorithms. Within the highly constrained computational hardware of onboard computers, these substantial increases in computational workload may even lead to a considerable escalation in overall computation. According to the research of Moniruzzaman et al. [71], GPUs can extend the computing speed of CPU-based algorithms, providing the possibility of real-time computing. Dai et al. [72] pointed out that GPU can improve the computational efficiency by concurrently executing the matrix operation process that decouples each other in the calculation of orbit dynamics problems, thereby greatly reducing the computational time complexity. Based on this principle and in order to reduce the on-board computation workload, Lim et al. utilized GPU to handle the increased computational demands caused by artificial intelligence algorithms in space situational awareness, and they achieved significant acceleration effects in addressing these concerns [42].

With the development of spacecraft control and cluster techniques, observing space objects through multiple observation stations, as shown in Figure 6, has become practical and attracted widespread attention [61,73]. In the late 20th century, Jia et al. proposed the reference vector method which utilizes statistical characteristics of the observation data to achieve an approximate optimal estimation of the target state [74]. However, this method has low accuracy and cannot be used in practice [75]. Subsequently, Lu et al. introduced a unit vector method to iteratively approximate the initial state of space targets by constructing two coordinate systems. This method has a wide application range and converges fast, but it may be inaccurate or even diverge when the observed arcs are too long [76,77]. Building upon the unit vector method, Yang et al. proposed a space target initial orbit determination method for multi-camera arraies. This method first establishes equations for solving the initial state of the targets, then incorporates measurements from different time into the equation to form a system of equations. By solving the equations using the least squares method, the dynamic state of the target will be ultimately obtained. By comprehensively utilizing the computational results from multiple measurement devices, this approach leverages the advantages of camera arrays, achieving higher computational accuracy compared to traditional methods such as the Laplace method and the Gauss method [78].

Table 3. Comparison of the mentioned initial orbit determination methods in this section.

Methods	Characteristics	Inputs	Outputs	Main Equations
Gauss method	The most classic method for initial orbit determination	The direction of three observations	Geocentric distance of the target	${\mathbf{r}}_2=c_1{\mathbf{r}}_1+c_3{\mathbf{r}}_3$
Laplace method	Straightforward and practical	The direction of three observations	Geocentric distance of the target	$\mathbf{r}+\mathbf{R}=\rho \widehat{\mathit{\rho }}$
Proposed by Schmidt et al. [64]	Proposed method for determining the shape of orbit	relative angle measurements	values of several of the relative orbit elements	${[i_x,i_y,i_z]}^T=([x_d-a_{e}cos\left({\beta }_0\right)/2]\widehat{i}+[y_{d0}+a_{e}sin\left({\beta }_0\right)]\widehat{j}+\left[z_{max}sin\left({\psi }_0\right)\right]\widehat{k})$ /$sqrt(a_e^2(1-3co{s}^2\left({\beta }_0\right)/4)+a_e(2y_{d0}sin\left({\beta }_0\right)-x_{d}cos\left({\beta }_0\right))+x_d^2+y_{d0}^2+z_{max}^{2}si{n}^2\left({\psi }_0\right))$
Proposed by Geller et al. [68,69]	Developed novel orbit determination method by setting the camera offset from the vehicle	three line-of-sight observations for relative motion coasting trajectory to the center of mass of an object	initial position and velocity of target	$k_i{\mathit{i}}_{los}\left(i\right)={\varphi }_{rr}\left(i\right)\{k_0{i}_{los}\left(0\right)-\mathit{d}\left(0\right)\}+{\varphi }_{rv}\left(i\right)$$\left\{{\varphi }_{rv}^{-1}\left(1\right)[k_1{i}_{los}\left(1\right)-{\varphi }_{rr}\left(1\right)\{k_0{i}_{los}\left(0\right)-\mathit{d}\left(0\right)\}-\mathit{d}\left(1\right)]\right\}+\mathit{d}\left(i\right)$
Proposed by Gong et al. [70]	Effectively addressed the nonlinear challenges in orbit determination by using machine learning methods	three sets of bearing angle and the absolute orbit state of the observer	the initial relative orbit state of the target	$L={\displaystyle \frac{1}{m}}{\sum }_{i=1}^m{(y_i-{\widehat{y}}_i)}^2+{\displaystyle \frac{\lambda }{2n}}{\sum }_{\omega }{\omega }^2$ (Loss Function)

Table 3. Comparison of the mentioned initial orbit determination methods in this section.

Methods	Characteristics	Inputs	Outputs	Main Equations
Gauss method	The most classic method for initial orbit determination	The direction of three observations	Geocentric distance of the target	${\mathbf{r}}_2=c_1{\mathbf{r}}_1+c_3{\mathbf{r}}_3$
Laplace method	Straightforward and practical	The direction of three observations	Geocentric distance of the target	$\mathbf{r}+\mathbf{R}=\rho \widehat{\mathit{\rho }}$
Proposed by Schmidt et al. [64]	Proposed method for determining the shape of orbit	relative angle measurements	values of several of the relative orbit elements	${[i_x,i_y,i_z]}^T=([x_d-a_{e}cos\left({\beta }_0\right)/2]\widehat{i}+[y_{d0}+a_{e}sin\left({\beta }_0\right)]\widehat{j}+\left[z_{max}sin\left({\psi }_0\right)\right]\widehat{k})$ /$sqrt(a_e^2(1-3co{s}^2\left({\beta }_0\right)/4)+a_e(2y_{d0}sin\left({\beta }_0\right)-x_{d}cos\left({\beta }_0\right))+x_d^2+y_{d0}^2+z_{max}^{2}si{n}^2\left({\psi }_0\right))$
Proposed by Geller et al. [68,69]	Developed novel orbit determination method by setting the camera offset from the vehicle	three line-of-sight observations for relative motion coasting trajectory to the center of mass of an object	initial position and velocity of target	$k_i{\mathit{i}}_{los}\left(i\right)={\varphi }_{rr}\left(i\right)\{k_0{i}_{los}\left(0\right)-\mathit{d}\left(0\right)\}+{\varphi }_{rv}\left(i\right)$$\left\{{\varphi }_{rv}^{-1}\left(1\right)[k_1{i}_{los}\left(1\right)-{\varphi }_{rr}\left(1\right)\{k_0{i}_{los}\left(0\right)-\mathit{d}\left(0\right)\}-\mathit{d}\left(1\right)]\right\}+\mathit{d}\left(i\right)$
Proposed by Gong et al. [70]	Effectively addressed the nonlinear challenges in orbit determination by using machine learning methods	three sets of bearing angle and the absolute orbit state of the observer	the initial relative orbit state of the target	$L={\displaystyle \frac{1}{m}}{\sum }_{i=1}^m{(y_i-{\widehat{y}}_i)}^2+{\displaystyle \frac{\lambda }{2n}}{\sum }_{\omega }{\omega }^2$ (Loss Function)

6. Precise Orbit Determination Methods

In the initial orbit determination phase, the perturbation factors are usually not taken into consideration [58]. To augment the accuracy of the predicted orbit of the target, it is imperative to iteratively refine the initial orbit based on subsequent observational data and in conjunction with a precise orbit dynamic model.

Distinguished from the initial orbit determination, precise orbit determination involves optimizing the estimation of the initial states of space targets and parameters of the orbit dynamic model using a series of observations with associated errors. Essentially, it is a optimal state estimation problem [58].

The methods employed for solving precise orbit determination problems generally revolve around the concept of least squares estimation [79]. Building upon this concept, a series of estimation methods have been developed, including unbiased least squares estimation, standard least squares estimation, weighted least squares estimation, and least squares estimation. Typically, the methods grounded in the least squares framework for precise orbit determination begin with the initial orbit determination process. Then, the estimates are incorporated into the precise orbit determination model to update the calculation results iteratively to improve the accuracy.

As shown in Table 4, Kozai proposed the first analytical prediction algorithm for precisely determining the orbital elements of LEO objects [80]. Building upon that, Hilton et al. simplified the gravity model as well as the drag model and introduced the Simplified General Perturbations method (SGP) [81]. Lane et al. further refined the SGP, leading to the development of the first practical orbit precise determination method, the SGP4, used by the North American Aerospace Defense Command (NORAD) [82,83]. Hujsak extended SGP4 and introduced the Simplified Deep-space Perturbation 4 method (SDP4) to precisely determine the orbits of deep-space targets [81]. Han et al. conducted an accuracy test on SDP4 and SGP4 using typical orbits. The results showed that SGP4/SDP4 models exhibit high computational speeds and can achieve the required accuracy for orbit prediction when dealing with near-circular orbits in MEO and GEO. The maximum deviation from the determined orbit to the nominal orbit was less than 3 km. However, for near-circular orbits with altitudes less than 50 km or highly eccentric orbits with eccentricity greater than 0.6 and perigee altitude less than 50 km, the computational results showed large discrepancies, and more precise computational models are needed to assist in updating the precise orbit determination of the targets [84,85].

The convergence speed and accuracy are core indicators for evaluating the performance of precise orbit determination methods and can be influenced by various factors such as the performance of the orbit prediction model, distribution of observation arcs, accuracy of the observation data, data density, and more [86]. A systematic study conducted by the U.S. National Research Council on the impact of different factors on orbit prediction models revealed that factors including the accuracy of atmospheric drag models, solar radiation pressure models, gravity field models, the performance of orbit correlation and orbit propagation methods during initial orbit determination, the algorithm for approximating the uncertainty of the orbit determination results, sensor measurement errors, and nonlinear estimation and filtering methods for handling uncertainties of measurement result may influence the orbit prediction results. Simultaneously, the research emphasized that with the rapid increase in the number of space objects, traditional orbit determination methods are no longer sufficient to meet current needs in determining the orbit of space objects. There is a need for research in orbit determination algorithms with superior performance to achieve higher demand in space situational awareness [87].

Against the backdrop of rapid development in space satellite networking technology, precise orbit determination techniques for multi-satellite platforms have emerged as a new research hotspot. Wang et al. proposed a dual-satellite optical orbit determination method for GEO satellites by deploying monitoring satellites on both sides of the targets. Experimental studies have shown that this method can improve the observability of targets, and the orbit determination accuracy of the target can be improved by three orders of magnitude compared to single-satellite orbit determination [88]. Liu et al. conducted research on the orbit determination effects for targets in near-Earth orbits under different conditions, deploying two, four and six observation satellites. The results revealed a positive correlation between orbit determination accuracy and the number of observation satellites. However, the enhancement effect diminishes gradually as the number of observing satellites increases. Therefore, adopting a dual-satellite evenly distributed space-based target orbit determination scheme is deemed to have the highest efficiency-to-cost ratio [89]. Shao et al. conducted a study on the monitoring effects of targets in GEO when deploying multiple surveillance satellites in a inclination LEO orbits. The findings suggest that increasing the number of observation satellites from two to six can significantly enhance the observation accuracy from the kilometer level to the decimeter level [90].

Due to the rapid increase in the number of space debris, precise orbit determination and prediction for space debris cluster have become another hot spot [91]. Building upon the investigation of factors influencing the growth of space debris, Toshiya et al. proposed a method to quantitatively estimate the density and quantity of space debris. However, it does not provide a method to estimate the orbits of space debris [92]. Hu et al. conducted a dedicated study on the selection of dynamic models in the orbit prediction of space debris. They systematically examined the influence of disturbance factors, including gravity field models, atmospheric drag, three-body gravity fields, and solar radiation, on the orbit prediction accuracy of space debris. The research offers quantitative insights into the selection of dynamic models in the precise orbit prediction processes for space debris in different orbital regimes. However, it primarily relied on established initial value problem integrators for recursive calculations and did not specifically delve into the investigation of the orbit estimation model [93]. In addressing the challenges of low computational efficiency with numerical methods and low accuracy with analytical methods in the precise orbit determination process, Li Bin proposed an analytical method for solving the orbit propagation problems with both rapidity and high precision. This method employs numerical integration method with large step size to compute the mean orbital elements of space targets, utilizes an analytical algorithm to reconstruct the short-period terms of the targets, and finally recombines the mean orbital elements and short-period terms to determine orbits fast and accurately [94]. Zhang et al. presented a method for orbit prediction in large-scale space debris clusters. They categorized the debris cluster into nominal and correlated debris. Numerical integration was employed for a small number of nominal debris to ensure accuracy in the prediction. For a larger quantity of correlated debris, a semi-analytical method based on Taylor expansion was utilized to enhance the computational speed while maintaining prediction accuracy. This approach achieved a balanced consideration between computational efficiency and accuracy [95].

Table 4. Comparison of the mentioned precise orbit determination methods in this section.

Developers	Characteristics	Inputs	Outputs	Main Equations
Kozai [80]	Established the first analytical precise orbit determination method	Initial states of a close earth satellite	Prediction of satellite orbit	$R_1=GM{A}_2(I/3-Isi{n}^{2}i/2){(I-e^2)}^{-3/2}/a^3$
Wang et al. [88]	Proposed a dual-satellite optical orbit determination method for GEO satellites	Initial state of the target, priori of estimate and related covariance matrix	Optimal estimation of the orbits of space targets	$({\mathit{H}}^T{\mathit{R}}^{-1}\mathit{H}+{\overline{\mathit{P}}}_0^{-1}){\widehat{\mathit{x}}}_0={\mathit{H}}^T{\mathit{R}}^{-1}\mathit{y}+{\overline{\mathit{P}}}_0^{-1}{\mathit{X}}_0$
Li Bin [94]	Proposed an analytical method for solving the orbit propagation to determine orbit fast and accurately	Initial states of a close earth satellite	Prediction of satellite orbit	$\widehat{\mathbf{r}}=\mathit{f}{(\widehat{\mathbf{r}},t)}_{3\times 3}$
Zhang et al. [95]	Presented a method for orbit prediction in large-scale space debris clusters	Initial states of nominal space debris	Predicted orbits of space debris cluster	$\left[{\mathit{x}}^i\right]={\mathit{x}}_0+\Delta {\mathit{x}}_o^i$

Table 4. Comparison of the mentioned precise orbit determination methods in this section.

Developers	Characteristics	Inputs	Outputs	Main Equations
Kozai [80]	Established the first analytical precise orbit determination method	Initial states of a close earth satellite	Prediction of satellite orbit	$R_1=GM{A}_2(I/3-Isi{n}^{2}i/2){(I-e^2)}^{-3/2}/a^3$
Wang et al. [88]	Proposed a dual-satellite optical orbit determination method for GEO satellites	Initial state of the target, priori of estimate and related covariance matrix	Optimal estimation of the orbits of space targets	$({\mathit{H}}^T{\mathit{R}}^{-1}\mathit{H}+{\overline{\mathit{P}}}_0^{-1}){\widehat{\mathit{x}}}_0={\mathit{H}}^T{\mathit{R}}^{-1}\mathit{y}+{\overline{\mathit{P}}}_0^{-1}{\mathit{X}}_0$
Li Bin [94]	Proposed an analytical method for solving the orbit propagation to determine orbit fast and accurately	Initial states of a close earth satellite	Prediction of satellite orbit	$\widehat{\mathbf{r}}=\mathit{f}{(\widehat{\mathbf{r}},t)}_{3\times 3}$
Zhang et al. [95]	Presented a method for orbit prediction in large-scale space debris clusters	Initial states of nominal space debris	Predicted orbits of space debris cluster	$\left[{\mathit{x}}^i\right]={\mathit{x}}_0+\Delta {\mathit{x}}_o^i$

7. Challenges and Development Trends in Orbit Determination Methods for Space-Based Optical Platforms

The rapid increase in the quantity of space objects and space debris has heightened the need for improved speed and precision in orbit determination for space targets. Effectively addressing the challenge of enhancing the efficiency and accuracy of existing orbit determination methods within space situational awareness systems is of paramount importance. Furthermore, the development of strategies to swiftly and accurately predict the orbits of satellite constellations or extensive space debris resulting from space collisions has become a central concern for contemporary space situational awareness systems. Given the current state of space situational awareness systems and the anticipated future demands for orbit prediction, essential research in this field can be focused on the following key areas.

7.1. High-Precision Orbit Prediction Methods for Space Targets at Long-Distance

With the rapid escalation in the quantity of spacecrafts and space debris, the responsibilities of space situational awareness systems are undergoing a substantial surge in complexity. A critical focal point for advancing these systems lies in the development of algorithms for orbit determination that are not only more robust but also more efficient and accurate. This becomes especially challenging given the constraints posed by the limited number of space situational awareness platforms and their detection capabilities. Tackling these challenges is indispensable for realizing the precise and prompt tracking of distant space targets, thereby forming the linchpin for future endeavors in space collision prevention.

7.2. High-Performance Orbit Determination Methods for Large-Scale Satellite Cluster Targets

Due to the characteristics of space missions, large-scale constellations of satellites are commonly positioned in LEO, where they encounter specific perturbations from atmospheric drag. Compounding this, since these constellations tend to share similar orbital altitudes, the perturbations affecting their orbits may display similarities. The paramount challenge is to adeptly and precisely forecast the orbits of these extensive satellite constellations using their characteristics while operating within the constraints of limited on-orbit computational resources. Addressing this issue is imperative to fulfill the stringent requirements for avoiding space collisions arising from on-orbit activities of large constellations, elevating it to a pressing matter that demands attention.

7.3. Orbit Prediction Methods for Large-Scale Space Debris

Space collision will generate a large number of space fragments, posing a significant threat to orbiting space objects and creating substantial computational loads. Therefore, establishing a precise, low-computational-cost model for the evolution of fragment orbit after a space collision is of great significance for assessing the consequences of space collisions and defending against space debris.

7.4. Orbit Design for Space-Based Situational Awareness Systems

The operational orbits of a space-based surveillance systems have a crucial impact on the observability of space targets relative to the surveillance platforms and the precision of target trajectory measurements. Designing adaptive operational orbits tailored to specific space surveillance platforms and monitoring tasks to achieve the continuous and accurate monitoring of targets is a key research focus of space situational awareness.

8. Conclusions

This article provides a comprehensive overview of space situational awareness systems and orbit determination methods. The escalating presence of space debris has underscored the critical importance of space situational awareness systems, prompting major spacefaring nations to invest in diverse space target surveillance systems. To enhance space situational awareness capabilities, these nations are increasingly directing their efforts towards the development of space-based optical platforms.

Addressing the core challenge of orbit determination for space-based optical space situational awareness platforms, scholars have proposed theoretical frameworks, establishing a three-stage system comprising orbit association, initial orbit determination and precise orbit determination. However, the evolution of huge space constellations and the rapid proliferation of space debris impose heightened requirements on the construction of space situational awareness systems and the efficacy of associated methods. Numerous theoretical and technical challenges persist in achieving precise orbit determination for space targets.

Anticipating future demands for enhanced space situational awareness, further in-depth research into space target orbit determination methods is imperative and will serve as the theoretical bedrock for ensuring the secure operation of spacecraft.