Passive Electro-Optical Tracking of Resident Space Objects for Distributed Satellite Systems Autonomous Navigation

Abstract

Autonomous navigation (AN) and manoeuvring are increasingly important in distributed satellite systems (DSS) in order to avoid potential collisions with space debris and other resident space objects (RSO). In order to accomplish collision avoidance manoeuvres, tracking and characterization of RSO is crucial. At present, RSO are tracked and catalogued using ground-based observations, but space-based space surveillance (SBSS) represents a valid alternative (or complementary asset) due to its ability to offer enhanced performances in terms of sensor resolution, tracking accuracy, and weather independence. This paper proposes a particle swarm optimization (PSO) algorithm for DSS AN and manoeuvring, specifically addressing RSO tracking and collision avoidance requirements as an integral part of the overall system design. More specifically, a DSS architecture employing hyperspectral sensors for Earth observation is considered, and passive electro-optical sensors are used, in conjunction with suitable mathematical algorithms, to accomplish autonomous RSO tracking and classification. Simulation case studies are performed to investigate the tracking and system collision avoidance capabilities in both space-based and ground-based tracking scenarios. Results corroborate the effectiveness of the proposed AN technique and highlight its potential to supplement either conventional (ground-based) or SBSS tracking methods.

1. Introduction

Despite growing awareness of the orbital debris problem, recent developments such as launch ride-sharing, growth in the availability of small launch vehicles, and particularly large-scale satellite constellation deployments are dramatically increasing the orbital congestion. According to space environment statistics recently published by ESA [1] the current space environment situation is alarming. Figure 1a,b, illustrate the current space situation in terms of number of launches and debris population in space. Further exacerbating the space situation are an unfortunate series of events such as anti-satellite weapons (ASAT) tests, on orbit collisions, and satellite breakups. Further, several commercial entities have made plans to launch larger constellations (700–5000 spacecraft each) in the coming months. Currently, the space domain usage is unsustainable. If this continues, the population of space debris will increase multifold, ceasing space activities in the near future. An increase in space objects will increase the probability of collisions, which can, in- turn, lead to Kessler Syndrome [2].

Spacecraft operators must become more aware of the potential threats posed by the existing resident space objects (RSO) in order to prevent further collisions in Earth orbit. This includes not only tracking the total objects in space but also continuously estimating the probability of other accidental collisions. Unfortunately, neither of these tasks are trivial and require considerable tracking resources (optical telescopes and radar), computing power, and sophisticated software to calculate numerous satellite–satellite or satellite–debris conjunctions on a daily basis. The above tasks are referred to as ‘space situational awareness’ (SSA) or space domain awareness (SDA). Traditionally, these tasks are accomplished by a network of ground-based observation facilities known as the Space Surveillance Network (SSN), owned and operated by the US Department of Defence (DoD) [3]. Besides this, various other ground-based telescopes and space surveillance systems contribute to SSA. Different SSA systems in current use are briefly described in [4].

Table 1. Main ground-based SSA systems.

USA		Russia		Japan		Europe
System	Year	System	Year	System	Year	System	Year
Thule radar	1943	Dnepr radar	1963	BSGC	2002	GRAVES (France)	2005
Eglin radar	1969	Dunay-3U radar	1968	KSGC	2004	TIRA (Germany)	2009
GEODSS	1980	Daryal radar	1984
SST	2011	Don-2N radar	1996
Space fence	2020	Okno optical complex	1997
		Krona system	2008

provides a summary of major ground-based systems and the time frames during which they were introduced. Although these ground-based radars, laser, and telescope systems continue to play a pivotal role in providing situational awareness in the space environment, whether or not these systems can effectively achieve the goal in the future, adapting to the evolving space domain, is still a question. This is due to the following reasons.

• Most of the ground-based systems are able to perform regional surveillance and then randomly look at other areas.
• They lack persistency in surveillance. In order to achieve true surveillance, it is necessary to monitor objects or regions for extended periods of time.
• Due to space perturbations, there is an on-orbit change in the RSO position. This will decrease the revisit frequency of the RSO within the field of view of the sensors on ground.
• Weather conditions are still a significant concern for ground-based systems. In typical ground-based observation sites, weather restricts visibility more than half the time, with some sites having a visibility of no more than 25% [4].
• For optical sensors on the ground, daylight observations represent a significant challenge because the passage of objects between the Earth and the Sun is almost always difficult to monitor.

1.1. Space-Based Space Surveillance (SBSS)

The gaps associated with ground-based measurements pave a path for tracking RSO by exploiting space borne measurements [5,6]. This type of approach is termed space-based space surveillance (SBSS). SBSS is an effective approach due to its capability to offer enhanced performances in terms of tracking accuracy and weather independence, allowing space-borne measurements to provide a broader range of useful observations [7]. In addition, space-based observation systems are not prone to atmospheric scattering, diffraction, turbulence, and aberrations [8]. Historically, radar sensors have been explored for space-borne measurements; however, the challenges associated with size and power consumption have shifted the focus towards optical sensors. Technological developments in optical sensor principles (e.g., coupled charged device (CCD)) [9], complementary metal–oxide–semiconductor (CMOS), and photon counting sensors [7,10]) have significantly enhanced optical detection performance, demonstrating the ability to track a 3 cm diameter object at a 3000 km range [7,9].

The precision in state information of the RSO will continue to be pivotal for the future space traffic management (STM) applications. These estimations can be provided using two approaches. Cooperative surveillance relies on state estimates from on-board time and space position information (TSPI)/navigation systems (e.g., GNSS, INS) and on the collaborative exchange of information among all other vehicles in the course of a potential collision. Whereas non-cooperative surveillance is typically carried out by ground or space-based radar or electro-optical sensors that do not require communication with the observed object. These systems are prone to errors caused by physical phenomena or by the mathematical extrapolation itself. In other words, a non-cooperative scenario is described as an encounter between a host platform and a RSO or possibly a non-cooperative spacecraft, in which only the host spacecraft is capable of preventing a potential collision. On the other hand, a cooperative scenario is described as a scenario in which all the potentially colliding RSO are capable of communicating position data and, if necessary, they can perform manoeuvres to avoid a collision [11].

SBSS has already been attempted in the past; [12] describes various SBSS missions and Figure 2 summarizes different attempts made by different countries for debris tracking. The current space-based SSA systems face issues such as latency and limited coverage, making them unreliable for the future to come. Hence, there is a need for a reliable and sustainable SSA system to fill the gaps in the present.

In 1996, the space-based visible instrument (SBV) onboard the mid-course space experiment led to the first SBSS mission [13,14,15]. This was followed by the United States Air Force’s SBSS mission, also known as the SBSS block 10 satellite. The SBSS constellation aimed to detect and track all space resident objects in orbit around the Earth in real-time [16]. A similar space surveillance mission was conducted by Sapphire, a Canadian satellite for US SSN. It was able to provide accurate measurements in rate track or sidereal track modes [17]. Furthermore, the ORS-5 demonstrator, also known as Sensor-Sat, was launched by the United States with an objective to demonstrate technologies for geosynchronous SSA. The success of the above-mentioned technological demonstrations paved a path to create new evolutionary frameworks for SBSS and to create new profitable pathways for accurate RSO tracking for various commercial actors and orbital startup companies [18]. The overall business of SDA faces formidable challenges. Most objects are extremely difficult to observe when they pass between the Earth and the Sun. Although space-based sensors can provide observations much closer to the Sun, they have limitations when the target positioned between Sun and the sensor [19].

Recent technological advances have led to the concept of multiple spacecraft operating in optimal coordination to accomplish desired mission goals [20]. Considering this, distributed satellite systems (DSS) is a promising concept for the future of SSA and STM. DSS mission architectures move away from the monolith system concept to adopt multiple elements that interact, cooperate, and communicate with each other [21,22,23], resulting in new systematic properties and/or emerging functions. Different types of DSS have been proposed to date, including constellations [24], clusters, swarms, trains, fractionated spacecraft, and federated satellites [25,26,27,28]. From the system architecture point of view, these concepts vary across different verticals, such as the number of assets, homogeneity of the assets, physical separation between the assets, and so forth [29]. For better understanding of various DSS configurations, the reader is referred to references [30,31,32,33,34,35,36].

In contrast to the conventional ground-based systems whose observations are conducted from accurately surveyed locations, SBSS platforms are subjected to positional errors and tracking errors caused by onboard TSPI/Navigation systems and tracking sensors, respectively. These errors can be represented geometrically in the form of ellipsoids which can be combined to form the total uncertainty volume that determines the total uncertainty in the tracking estimates [37] to support separation assurance (SA) and collision avoidance (CA) between the RSO, which is followed by the application of relevant collision avoidance manoeuvres. There exists a duality to the approach mentioned previously. On one hand, for the existing DSS constellation, optimized manoeuvres can be performed to avoid collisions with RSO. Alternatively, the design of the astrionics onboard could be optimized to mitigate errors and achieve mission objectives. In this paper, a non-cooperative SBSS scenario is analysed, in which DSS spacecraft collaboratively track the RSO subject to specific errors in tracking and navigation systems for positioning that will ultimately determine the uncertainty volume or a confidence region around the detected RSO.

1.2. Aim and Structure of the Article

This article proposes a unified approach that combines RSO tracking and AN/manoeuvring algorithms with an aim to realize trusted autonomy in heterogenous DSS platforms for SA and CA. The remainder of the article is structured as follows. Section 2 presents an overview of various RSO tracking approaches and the equations corresponding to the proposed tracking algorithm. Section 3 summarises autonomous navigation (AN) and various trajectory optimization approaches for spaceflight applications and the equations for PSO algorithm. Verification case studies carried out in various scenarios are defined in Section 4. The results obtained from the case studies are discussed in Section 5. Section 6 comprises the conclusions and scope for future research.

2. Tracking Algorithms Overview

A single angles-only sensor is not sufficient for accurate RSO tracking. In contrast, using two angles-only sensors allows one to determine the range, and thus the 3D location, of an object via simple triangulation [38,39,40,41]. Typically, an error in the sensor measurement always prevails because the sensors do not exactly point towards the RSO, making it necessary to find the most probable RSO position. Without the error in measurement, the triangulation becomes trivial. In [42,43] a ground-based optical system was proposed, consisting of two ground-based optical sensors located at two different sites to track the debris using simultaneous optical measurements.

Several algorithms are proposed in the literature to solve this problem. For instance, in [44], the proposed algorithm estimates the most probable position of the RSO as the midpoint of the shortest line joining the two lines of sight. In [45], the most probable RSO position is estimated as a point on the shortest line between the two lines of sight that subtends an equal angle with respect to the observation sites. Although numerical solutions are proposed to calculate position errors in the triangulation solution, relating the measurement errors with the experimental parameters is a huge challenge. In [46], the analytical expressions for the position error for tracking an RSO in space for several observation-station configurations is presented. The current work deals with AN and tracking through simultaneous optical measurements performed by multiple heterogenous platforms that constitute a DSS architecture. In order to justify the aim of the current work, a suitable tracking algorithm with high position accuracy is tested in different operating scenarios (ground-based tracking, space-based tracking).

2.1. Triangulation Problem

This section introduces a suitable tracking algorithm and the corresponding equations required to estimate the position of the RSO, and the errors associated with the measurements. Attention is then turned towards the uncertainty quantification of navigation and tracking errors in the form of a covariance matrix to generate an overall uncertainty volume. Figure 3 and Figure 4 illustrate the tracking problem in both ground-based and space-based scenarios respectively.

2.2. Tracking Algorithm

In order to compute the RSO position estimate, it is necessary to know the location of each sensor $\left({\mathrm{x}}_{\mathrm{i}},{\mathrm{y}}_{\mathrm{i}},{\mathrm{z}}_{\mathrm{i}}\right)$, as well as the line of sight (LOS) azimuth and elevation pointing angles $\left(\mathsf{\theta },\mathsf{\varphi }\right)$ from each sensor to the target. An inaccuracy in determining these 10 parameters will result in an error in the target position estimate. Furthermore, the relationship between the measurement error and the errors in the estimated target location is a function of the sensor–target–sensor geometry, where a sensor–target–sensor separation of 90° results in the lowest error sensitivity, while a sensor separation of $0°$ or $180°$ results in impossible solutions (as seen from the target). To define the triangulation problem, we need to define the positions of the sensors and the RSO in a right-handed coordinate system as illustrated in Figure 4.

The $\mathrm{x}$ and $\mathrm{y}$ axes form the horizontal plane, and the $\mathrm{z}$ axis points out of plane vertically. ${\mathsf{\theta }}_1$ and ${\mathsf{\theta }}_2$are the azimuth angles of the corresponding sensors measured clockwise from positive y axis towards positive $\mathrm{x}$ axis. The corresponding elevation angles are denoted by ${\mathsf{\varphi }}_1$ and ${\mathsf{\varphi }}_2$ which increase from $0^{°}$ in the $\mathrm{xy}$ plane to ${90}^{°}$ pointing vertically. The separation angle ${\mathsf{\theta }}_{\mathrm{sep}}$ is measured from sensor $1$ through the RSO to sensor $2$. The sensor positions and the LOS from sensor to target allows the 3D target position computation.

The equations to calculate the position estimates are defined below [47]:

$${\mathrm{x}}_{\mathrm{t}}=\frac{{\mathrm{x}}_2\tan \left({\mathsf{\theta }}_1\right)-{\mathrm{x}}_1\tan \left({\mathsf{\theta }}_2\right)+\left({\mathrm{y}}_1-{\mathrm{y}}_2\right)\tan \left({\mathsf{\theta }}_1\right)\tan \left({\mathsf{\theta }}_2\right)}{\tan \left({\mathsf{\theta }}_1\right)-\tan \left({\mathsf{\theta }}_2\right) }$$

(1)

$${\mathrm{y}}_{\mathrm{t}}=\frac{{\mathrm{y}}_1\tan \left({\mathsf{\theta }}_1\right)-{\mathrm{y}}_2 \tan \left({\mathsf{\theta }}_2\right)+\left({\mathrm{x}}_2-{\mathrm{x}}_1\right)}{\tan \left({\mathsf{\theta }}_1\right)- \tan \left({\mathsf{\theta }}_2\right) }$$

(2)

$${ \mathrm{r}}_{\mathrm{i}}=\sqrt{{\left({\mathrm{x}}_{\mathrm{i}}-{\mathrm{x}}_{\mathrm{t}}\right)}^2+{\left({\mathrm{y}}_{\mathrm{i}}-{\mathrm{y}}_{\mathrm{t}}\right)}^2}$$

(3)

$${\mathrm{z}}_{\mathrm{t}}=\frac{{\mathrm{r}}_1\tan \left({\mathsf{\phi }}_1\right)+{\mathrm{z}}_1+{\mathrm{r}}_2\tan \left({\mathsf{\phi }}_2\right)+{\mathrm{z}}_2}{2}$$

(4)

where:

• (x_t,y_t,z_t) = target RSO position coordinates,
• (x_i,y_i,z_i) = corresponding sensor position coordinates
• r_i = horizontal ranges from the x − y components of the corresponding sensor to the x and y components of the RSO,
• φ_i = corresponding sensor to RSO elevation angle,
• θ_i = corresponding sensor to RSO azimuth angle.
• i = 1, 2 (number of sensors used to perform triangulation)

These equations relate the target location to measurements of the sensor position and LOS from the sensors to the target, aiding the target position estimation. The error propagation equations corresponding to the respective position coordinates are derived in [48], with a key assumption that the error generated by each sensor follows a Gaussian distribution. The sigmas for each ${\mathrm{x}}_{\mathrm{t}},{ \mathrm{y}}_{\mathrm{t}},{ \mathrm{z}}_{\mathrm{t}}$are the sums of various partial derivatives that are simplified as [49]:

$${ \mathsf{\sigma }}_{\mathrm{xt}}=\sqrt{{\mathrm{c}}_{\mathrm{x},\mathrm{p}}^2{\mathsf{\sigma }}_{\mathrm{p}}^2+{\left({\mathrm{rc}}_{\mathrm{x},\mathsf{\theta }}\right)}^2{\mathsf{\sigma }}_{\mathsf{\theta }}^2}$$

(5)

$${\mathsf{\sigma }}_{\mathrm{yt}}=\sqrt{{\mathrm{c}}_{\mathrm{y},\mathrm{p}}^2{\mathsf{\sigma }}_{\mathrm{p}}^2+{\left({\mathrm{rc}}_{\mathrm{y},\mathsf{\theta }}\right)}^2{\mathsf{\sigma }}_{\mathsf{\theta }}^2}$$

(6)

$${\mathsf{\sigma }}_{\mathrm{ri}}=\sqrt{{\mathrm{c}}_{\mathrm{r},\mathrm{p}}^2{\mathsf{\sigma }}_{\mathrm{p}}^2+{\left({\mathrm{rc}}_{\mathrm{r},\mathsf{\theta }}\right)}^2{\mathsf{\sigma }}_{\mathsf{\theta }}^2}$$

(7)

$${\mathsf{\sigma }}_{\mathrm{zt}}=\sqrt{{\mathrm{c}}_{\mathrm{z},\mathrm{r}}^2{\mathsf{\sigma }}_{\mathrm{r}}^2+{\mathsf{\sigma }}_{\mathrm{p}}^2+{\left({\mathrm{rc}}_{\mathrm{z},\varnothing }\right)}^2{\mathsf{\sigma }}_{\mathsf{\phi }}^2}$$

(8)

The various c’s mentioned in the equations are the error coefficients that are functions of the sensor-to-sensor separation angle [48,49]. The error coefficients describe the dependence of target position estimates $({\mathrm{x}}_{\mathrm{t}},{ \mathrm{y}}_{\mathrm{t}},{ \mathrm{z}}_{\mathrm{t}}$) on errors in each of the sensor position coordinates $\left({\mathrm{x}}_{\mathrm{i}},{\mathrm{y}}_{\mathrm{i}},{\mathrm{z}}_{\mathrm{i}}\right)$. For instance, the ${\mathrm{c}}_{\mathrm{x},\mathrm{p}}$ indicates the error coefficient for the $\mathrm{x}$coordinate of the target position. ${\mathrm{c}}_{\mathrm{x},\mathsf{\theta }}$ corresponds to error coefficient for azimuth error in ${\mathrm{x}}_{\mathrm{t}}$. Equations (5)–(8) relate the target position uncertainties to the standard deviations of measurement errors: ${\mathsf{\sigma }}_{\mathrm{p}}$ for position; ${\mathsf{\sigma }}_{\mathsf{\theta }}$ for the azimuth; and ${\mathsf{\sigma }}_{\varnothing }$ for elevation measurements.

Equation (8) refers to the error in measurement of ${\mathrm{z}}_{\mathrm{t}}$ from a single sensor. However, since the measurements performed during triangulation involve two sensors, we can compute the error estimates from:

$${\mathsf{\sigma }}_{{\mathrm{z}}_{\mathrm{t}}}=\frac{1}{2}\sqrt{{\mathsf{\sigma }}_{{\mathrm{z}}_{\mathrm{t}}}^2\left(1\right)+{\mathsf{\sigma }}_{{\mathrm{z}}_{\mathrm{t}}}^2\left(2\right)+2\frac{\partial {\mathrm{z}}_{\mathrm{t}}}{\partial {\mathrm{r}}_1}\frac{\partial {\mathrm{z}}_{\mathrm{t}}}{\partial {\mathrm{r}}_2}\mathrm{Cov}\left({\mathsf{\delta }\mathrm{r}}_1,{\mathsf{\delta }\mathrm{r}}_2\right)}$$

(8a)

$$\mathrm{where}: \mathrm{Cov}\left({\mathsf{\delta }\mathrm{r}}_1,{\mathsf{\delta }\mathrm{r}}_2\right)=\frac{\partial {\mathrm{r}}_1}{\partial {\mathrm{x}}_{\mathrm{t}}}\frac{\partial {\mathrm{r}}_2}{\partial {\mathrm{x}}_{\mathrm{t}}}{\mathsf{\sigma }}_{{\mathrm{x}}_{\mathrm{t}}}^2+\frac{\partial {\mathrm{r}}_1}{\partial {\mathrm{y}}_{\mathrm{t}}}\frac{\partial {\mathrm{r}}_2}{\partial {\mathrm{y}}_{\mathrm{t}}}{\mathsf{\sigma }}_{{\mathrm{y}}_{\mathrm{t}}}^2$$

(8b)

2.3. Uncertainty Quantification

In multi-sensor platforms, the uncertainty position of the RSO is influenced by the accuracy of the sensor positions and the sensor to target LOS. It is crucial to know each sensor location (${\mathrm{x}}_1,{ \mathrm{y}}_1,{\mathrm{z}}_1$), (${\mathrm{x}}_2,{ \mathrm{y}}_2,{\mathrm{z}}_2$) and the corresponding LOS vector’s azimuth and elevation ($\mathsf{\theta }$and$\mathsf{\phi }$), respectively. The tracked RSO position can be analysed by combining the tracking and the navigation uncertainty using the proposed mathematical framework. To obtain the covariance matrix corresponding to the navigation error, the reader is directed to [37] in which the steps are described in detail. Therefore, navigation measurements are assumed to be provided by an on-board GNSS system, and the corresponding uncertainty values are derived from a literature LEO GPS accuracy experiment [50]. The uncertainty in the tracking measurements can be expressed in terms of the covariances as follows:

$${\mathrm{Q}}_{\mathrm{TRK}}=\left[\begin{array}{ccc}{\mathsf{\sigma }}_{\mathrm{x}}^2& 0& 0\\ 0& {\mathsf{\sigma }}_{\mathrm{y}}^2& 0\\ 0& 0& {\mathsf{\sigma }}_{\mathrm{z}}^2\end{array}\right]$$

(9)

where ${\mathsf{\sigma }}_{\mathrm{x}}^2$, ${\mathsf{\sigma }}_{\mathrm{y}}^2$ and ${\mathsf{\sigma }}_{\mathrm{z}}^2$ are obtained using Equations (5), (6) and (8a).

To determine the total covariance matrix Gauss–Helmert formulation, [51,52] are used in order to relate the sensor measurement errors $\left({\mathsf{\sigma }}_{\mathrm{p}1},{\mathsf{\sigma }}_{\mathrm{p}2},{\mathsf{\sigma }}_{\mathsf{\theta }1},{\mathsf{\sigma }}_{\mathsf{\theta }2},{\mathsf{\sigma }}_{\mathsf{\phi }1},{\mathsf{\sigma }}_{\mathsf{\phi }2}\right)$ to the final RSO position (${\mathrm{x}}_{\mathrm{t}},{ \mathrm{y}}_{\mathrm{t}},{\mathrm{z}}_{\mathrm{t}}$). We define the $\mathrm{L},\mathrm{X}$ vectors of estimated observations and estimated parameters, respectively:

$$\mathrm{L}={\left[{\mathrm{x}}_1,{\mathrm{y}}_1,{\mathrm{z}}_1,{\mathsf{\theta }}_1,{\mathsf{\phi }}_1,{\mathrm{x}}_2,{\mathrm{y}}_2,{\mathrm{z}}_2,{\mathsf{\theta }}_2,{\mathsf{\phi }}_2\right]}^{\mathrm{T}}$$

(10)

$$\mathrm{X}={\left[{\mathrm{x}}_{\mathrm{t}},{ \mathrm{y}}_{\mathrm{t}},{\mathrm{z}}_{\mathrm{t}}\right]}^{\mathrm{T}}$$

(11)

The total covariance matrix can then be expressed as:

$${\mathrm{Q}}_{\mathrm{TOT}\left(3\ast 3\right)}={ \mathrm{BC}}_{\mathrm{r}}{\mathrm{B}}^{\mathrm{T}}$$

(12)

where ${\mathrm{C}}_{\mathrm{r}}$ is the covariance matrix of the observations which can be written as a 10·10 diagonal matrix with elements [${\mathsf{\sigma }}_{\mathrm{x}1},{\mathsf{\sigma }}_{\mathrm{y}1},{\mathsf{\sigma }}_{\mathrm{z}1},{\mathsf{\sigma }}_{\mathsf{\theta }1},{\mathsf{\sigma }}_{\mathsf{\phi }1},{\mathsf{\sigma }}_{\mathrm{x}2},{\mathsf{\sigma }}_{\mathrm{y}2},{\mathsf{\sigma }}_{\mathrm{z}2},{\mathsf{\sigma }}_{\mathsf{\theta }2},{\mathsf{\sigma }}_{\mathsf{\phi }2}]$. For matrix B, we first define the function $\mathrm{F}\left(\mathrm{X},\mathrm{L}\right)=0$ as:

$${\mathrm{x}}_{\mathrm{t}}-\frac{{\mathrm{x}}_2\tan \left({\mathsf{\theta }}_1\right)-{\mathrm{x}}_1\tan \left({\mathsf{\theta }}_2\right)+\left({\mathrm{y}}_1-{\mathrm{y}}_2\right) \tan \left({\mathsf{\theta }}_1\right) \tan \left({\mathsf{\theta }}_2\right)}{\tan \left({\mathsf{\theta }}_1\right)-\tan \left({\mathsf{\theta }}_2\right) }=0$$

$${\mathrm{y}}_{\mathrm{t}}-\frac{{\mathrm{y}}_1\tan \left({\mathsf{\theta }}_1\right)-{\mathrm{y}}_2\tan \left({\mathsf{\theta }}_2\right)+\left({\mathrm{x}}_2-{\mathrm{x}}_1\right)}{\tan \left({\mathsf{\theta }}_1\right)-\tan \left({\mathsf{\theta }}_2\right) }=0$$

(13)

$${\mathrm{z}}_{\mathrm{t}}-\frac{{\mathrm{r}}_1\tan \left({\mathsf{\phi }}_1\right)+{\mathrm{z}}_1+{\mathrm{r}}_2\tan \left({\mathsf{\phi }}_2\right)+{\mathrm{z}}_2}{2}=0$$

Then, matrix $\mathrm{B}$ is defined as $\frac{\partial \mathrm{F}}{\partial \mathrm{L}}$ and, therefore, has a 3 × 10 dimension.

2.4. Covariance Matrix to Ellipsoid

It is crucial to understand the influence of a single term of the covariance matrix on the size and orientation of the ellipsoid. The principle component analysis (PCA) framework [11,53] describes the relationship between the covariance matrix $\mathrm{Q}$ and the rotated ellipsoid as follows:

$$\mathrm{Q}=\left[\begin{array}{ccc}{\mathsf{\sigma }}_{\mathrm{x}}^2& {\mathsf{\rho }}_{\mathrm{xy}}{\mathsf{\sigma }}_{\mathrm{x}}{\mathsf{\sigma }}_{\mathrm{y}}& {\mathsf{\rho }}_{\mathrm{xz}}{\mathsf{\sigma }}_{\mathrm{x}}{\mathsf{\sigma }}_{\mathrm{z}}\\ {\mathsf{\rho }}_{\mathrm{xy}}{\mathsf{\sigma }}_{\mathrm{x}}{\mathsf{\sigma }}_{\mathrm{y}}& {\mathsf{\sigma }}_{\mathrm{y}}^2& {\mathsf{\rho }}_{\mathrm{yz}}{\mathsf{\sigma }}_{\mathrm{y}}{\mathsf{\sigma }}_{\mathrm{z}}\\ {\mathsf{\rho }}_{\mathrm{xz}}{\mathsf{\sigma }}_{\mathrm{x}}{\mathsf{\sigma }}_{\mathrm{z}}& {\mathsf{\rho }}_{\mathrm{yz}}{\mathsf{\sigma }}_{\mathrm{y}}{\mathsf{\sigma }}_{\mathrm{z}}& {\mathsf{\sigma }}_{\mathrm{z}}^2\end{array}\right]$$

(14)

where the correlation coefficient $\mathsf{\rho }$ is expressed as:

$${\mathsf{\rho }}_{\mathrm{xy}}=\frac{\mathrm{cov}\left(\mathrm{x},\mathrm{y}\right)}{{\mathsf{\sigma }}_{\mathrm{x}}{\mathsf{\sigma }}_{\mathrm{y}}}$$

$${\mathsf{\rho }}_{\mathrm{xz}}=\frac{\mathrm{cov}\left(\mathrm{x},\mathrm{z}\right)}{{\mathsf{\sigma }}_{\mathrm{x}}{\mathsf{\sigma }}_{\mathrm{z}}}$$

(15)

$${\mathsf{\rho }}_{\mathrm{yz}}=\frac{\mathrm{cov}\left(\mathrm{y},\mathrm{z}\right)}{{\mathsf{\sigma }}_{\mathrm{y}}{\mathsf{\sigma }}_{\mathrm{z}}}$$

A convenient way to express the ellipsoid equation is to use the covariance matrix $\mathrm{Q}$:

$${\overrightarrow{\mathrm{x}}}^{\mathrm{T}}{\mathrm{Q}}^{-1}\overrightarrow{\mathrm{x}}=1$$

(16)

where $\overrightarrow{\mathrm{x}}={\left\{\mathrm{xyz}\right\}}^{\mathrm{T}}$ is a vector expressed in Cartesian coordinates. ${\mathrm{Q}}^{-1}$ can be written as:

$${\mathrm{Q}}^{-1}=\frac{1}{\mathsf{\rho }}\left[\begin{array}{ccc}\frac{1-{\mathsf{\rho }}_{\mathrm{yz}}^2}{{\mathsf{\sigma }}_{\mathrm{x}}^2}& \frac{-{\mathsf{\rho }}_{\mathrm{xy}}+{\mathsf{\rho }}_{\mathrm{yz}}{\mathsf{\rho }}_{\mathrm{xz}}}{{\mathsf{\sigma }}_{\mathrm{x}}{\mathsf{\sigma }}_{\mathrm{y}}}& \frac{-{\mathsf{\rho }}_{\mathrm{xz}}+{\mathsf{\rho }}_{\mathrm{yz}}{\mathsf{\rho }}_{\mathrm{xy}}}{{\mathsf{\sigma }}_{\mathrm{x}}{\mathsf{\sigma }}_{\mathrm{z}}}\\ \frac{-{\mathsf{\rho }}_{\mathrm{xy}}+{\mathsf{\rho }}_{\mathrm{yz}}{\mathsf{\rho }}_{\mathrm{xz}}}{{\mathsf{\sigma }}_{\mathrm{x}}{\mathsf{\sigma }}_{\mathrm{y}}}& \frac{1-{\mathsf{\rho }}_{\mathrm{xz}}^2}{{\mathsf{\sigma }}_{\mathrm{y}}^2}& \frac{-{\mathsf{\rho }}_{\mathrm{yz}}+{\mathsf{\rho }}_{\mathrm{xy}}{\mathsf{\rho }}_{\mathrm{xz}}}{{\mathsf{\sigma }}_{\mathrm{y}}{\mathsf{\sigma }}_{\mathrm{z}}}\\ \frac{-{\mathsf{\rho }}_{\mathrm{xz}}+{\mathsf{\rho }}_{\mathrm{yz}}{\mathsf{\rho }}_{\mathrm{xy}}}{{\mathsf{\sigma }}_{\mathrm{x}}{\mathsf{\sigma }}_{\mathrm{z}}}& \frac{-{\mathsf{\rho }}_{\mathrm{yz}}+{\mathsf{\rho }}_{\mathrm{xy}}{\mathsf{\rho }}_{\mathrm{xz}}}{{\mathsf{\sigma }}_{\mathrm{y}}{\mathsf{\sigma }}_{\mathrm{z}}}& \frac{1-{\mathsf{\rho }}_{\mathrm{xy}}^2}{{\mathsf{\sigma }}_{\mathrm{z}}^2}\end{array}\right]$$

(17)

where:

$$\mathsf{\rho }=\left(1+2{\mathsf{\rho }}_{\mathrm{xy}}{\mathsf{\rho }}_{\mathrm{xz}}{\mathsf{\rho }}_{\mathrm{yz}}-{\mathsf{\rho }}_{\mathrm{yz}}{}^2-{\mathsf{\rho }}_{\mathrm{xz}}{}^2-{\mathsf{\rho }}_{\mathrm{xy}}{}^2\right)$$

(18)

Hence, the rotated ellipsoid equation can be written as:

$$\begin{array}{c}\frac{\left(1-{\mathsf{\rho }}_{\mathrm{yz}}^2\right)}{{\mathsf{\sigma }}_{\mathrm{x}}^2}{\mathrm{x}}^2+\frac{\left(1-{\mathsf{\rho }}_{\mathrm{xz}}^2\right)}{{\mathsf{\sigma }}_{\mathrm{y}}^2}{\mathrm{y}}^2+\frac{\left(1-{\mathsf{\rho }}_{\mathrm{xy}}^2\right)}{{\mathsf{\sigma }}_{\mathrm{z}}^2}{\mathrm{z}}^2-\frac{{\mathsf{\rho }}_{\mathrm{xy}}-{\mathsf{\rho }}_{\mathrm{yz}}{\mathsf{\rho }}_{\mathrm{xz}}}{{\mathsf{\sigma }}_{\mathrm{x}}{\mathsf{\sigma }}_{\mathrm{y}}}2\mathrm{xy}\\ -\frac{{\mathsf{\rho }}_{\mathrm{xz}}-{\mathsf{\rho }}_{\mathrm{yz}}{\mathsf{\rho }}_{\mathrm{xy}}}{{\mathsf{\sigma }}_{\mathrm{x}}{\mathsf{\sigma }}_{\mathrm{z}}}2\mathrm{xz}-\frac{{\mathsf{\rho }}_{\mathrm{yz}}-{\mathsf{\rho }}_{\mathrm{xy}}{\mathsf{\rho }}_{\mathrm{xz}}}{{\mathsf{\sigma }}_{\mathrm{y}}{\mathsf{\sigma }}_{\mathrm{z}}}2\mathrm{yz}=\mathsf{\rho }\end{array}$$

(19)

Equation (20) expresses the contribution of the individual terms of the covariance matrix to the overall ellipsoid shape and size:

$$\mathrm{Q}={\mathrm{U}\mathsf{\Lambda }\mathrm{U}}^{\mathrm{T}}$$

(20)

where:

$$\mathsf{\Lambda }=\left[\begin{array}{ccc}{\mathsf{\lambda }}_1& \cdots & 0\\ ⋮& \ddots & ⋮\\ 0& \cdots & {\mathsf{\lambda }}_{\mathrm{n}}\end{array}\right]$$

(21)

Hence, the ellipsoid equation in (17) can be modified as:

$${(\overrightarrow{\mathrm{x}}-\overrightarrow{\mathrm{c}})}^{\mathrm{T}}{\mathrm{RA}}^{-1}{\mathrm{R}}^{\mathrm{T}}\left(\overrightarrow{\mathrm{x}}-\overrightarrow{\mathrm{c}}\right)=1$$

(22)

where:

• $\overrightarrow{\mathrm{x}}$ is a vector in the Cartesian coordinates about the nominal position (origin), $\overrightarrow{\mathrm{c}}$ of the ellipsoid.
• R = rotation matrix,
• A = diagonal eigenvalues matrix respectively, which are derived from Q.

$$\overrightarrow{\mathrm{x}}=\left[\begin{array}{c}{\mathrm{x}}_{\mathrm{ECI}}\\ {\mathrm{y}}_{\mathrm{ECI}}\\ {\mathrm{z}}_{\mathrm{ECl}}\end{array}\right]$$

(23a)

$$\overrightarrow{\mathrm{c}}=\left[\begin{array}{c}{\mathrm{x}}_{{\mathrm{ECI}}_{\mathrm{N}}}\\ {\mathrm{y}}_{{\mathrm{ECI}}_{\mathrm{N}}}\\ {\mathrm{Z}}_{{\mathrm{ECI}}_{\mathrm{N}}}\end{array}\right]$$

(23b)

Given a pair of angles the corresponding radial distance $\left(\mathrm{R}\right)$ from the centre of the ellipse, this can be calculated using:

$$\begin{array}{l}{\mathrm{r}}^2\left[{(\overrightarrow{\mathrm{x}}-\overrightarrow{\mathrm{c}})}^{\mathrm{T}}{\mathrm{RA}}^{-1}{\mathrm{R}}^{\mathrm{T}}\left(\left(\overrightarrow{\mathrm{x}}-\overrightarrow{\mathrm{c}}\right)\right)\right]=1\\ \mathrm{r}={\left[{(\left(\overrightarrow{\mathrm{x}}-\overrightarrow{\mathrm{c}}\right))}^{\mathrm{T}}{\mathrm{RA}}^{-1}{\mathrm{R}}^{\mathrm{T}}\left(\left(\overrightarrow{\mathrm{x}}-\overrightarrow{\mathrm{c}}\right)\right)\right]}^{-\frac{1}{2}}\end{array}$$

(24)

where $\mathsf{\alpha }\in \left\{-{180}^{\circ }:{180}^{\circ }\right\},\mathsf{\epsilon }\in \left\{-{90}^{\circ }:{90}^{\circ }\right\}$

3. Trajectory Optimization Techniques Overview

Spacecraft trajectory optimization is of paramount importance due to its direct influence on the system performance [54]. It is one of the highly studied problems in the field of aerospace engineering [55,56]. An optimal trajectory for a spacecraft can be defined as the one that meets some criteria, including initial and final conditions. The first serious attempt to classify these approaches was conducted back in 1998 by Betts [57], known as the direct and indirect methods. Both branches aim to minimize cost functions and constraint through discrete approximations. The direct methods transcribe the continuous optimal control problem into an optimization problem [58] adopting the state-space representation. The system equations are satisfied by integrating them implicitly or explicitly within a finite interval, thereby converting it into a non-linear programming problem (NLP) [59]. Although the solution obtained from direct methods is not necessarily optimal, its simple implementation, reduced problem size, and larger domain of convergence makes this method a widely used one [60]. One of the direct methods is referred to as the shape based method [61,62]. In the method, the state variables are interpolated, and the control variables are taken in the objective function. This is followed by a gradient-based technique or metaheuristic to minimize the cost by varying the state variables. The most popular technique among the direct methods is the Fourier series, specifically for low-thrust trajectory optimization applications [63,64,65]. The solution obtained by the shape-based method satisfies the equation of motion and the boundary conditions. Moreover, the solution obtained from the shape-based method can serve as a very good guess for other optimization approaches.

Gradient-based methods such as NLP appear to be the most popular computational techniques for addressing trajectory optimization problems. The NLP uses the gradient information and is often capable of quick convergence and accurate results, leading to its popularity. However, the most noticeable limitation of gradient-based methods is that the analyst must have some a priori knowledge of the optimal trajectory. In other words, the initial guess should be near to the global optimal solution. Failing to do so will lead to a non-global optimal solution or the optimization routine will become extremely slow.

An entirely different class of optimization approaches are referred to as metaheuristics, an alternative approach that addresses the issues faced by gradient-based methods. Essentially, a metaheuristic is an iterative approach to exploring and exploiting the search space that intelligently combines different concepts that are inspired by phenomena occurring in nature for guiding a subordinate heuristic [66]. These algorithms belong to a special class of artificial intelligence (AI) techniques, which also include evolutionary algorithms such as genetic algorithms (GA), PSO, and other insect/swarm inspired algorithms [67]. The metaheuristic approaches are classified into two types, single solution algorithms and population based algorithms, the latter are often used for spacecraft trajectory optimization problems [68,69]. The second branch of optimization approaches is that of indirect methods. Indirect methods consider the dualized form of the equations, including the states and co-states within the discretized time frame [65]. Although the indirect methods are better than direct methods in terms of accuracy, they have certain limitations, such as:

• The necessary conditions, including the co-state differential equations, the Hamiltonian, and the optimality conditions, must be expressed analytically.
• Due to discretization of co-states, the problem size becomes large.
• The analyst must guess certain aspects of the solution, such as portions of the time domain containing constrained or unconstrained control arcs.
• The domain of convergence decreases due to the requirement of the initial guess being close to optimal solution.

The feasibility of performing the on-board optimization routine depends on computation time and cost. Hence, the PSO technique is chosen as a primary optimization routine due to its capability for global convergence and its robustness to solve highly non-linear problems with greater computational efficiency. Moreover, this technique is used widely to solve diverse spacecraft trajectory optimization problems, and its on-board implementation was verified, through various case studies, for adequate convergence time and low computation cost [70].

Autonomous Navigation Models

As discussed, PSO algorithms were used for trajectory optimization. PSO was first introduced in 1995 by J. Kennedy and R.C. Hebarhart [71]. Among several metaheuristic algorithms available, PSO is the most frequently used technique, as demonstrated by several works that exploit this approach [72,73,74]. PSO is a metaheuristic algorithm with an enhanced ability to perform global optimization. The PSO comprises a fixed-size population of particles, N, that are candidate solutions to the problem which move inside the search domain, modifying their position through appropriate perturbation called velocity, which allows the particle to perform a displacement. The particles move iteratively until they converge onto a global optimal solution considering the goal of optimality and minimizing the cost function, which describes the quality of the solution and the imposed constraints. The position of the particles iterates according to:

$${\mathrm{X}}_1^{\mathrm{k}+1}={\mathrm{X}}_{\mathrm{i}}^{\mathrm{k}}+{\mathrm{V}}_{\mathrm{i}}^{\mathrm{k}+1}$$

(25)

$${\mathrm{V}}_{\mathrm{i}}^{\mathrm{k}+1}={\mathrm{V}}_{\mathrm{i}}^{\mathrm{k}}+{\mathrm{c}}_1\cdot {\mathrm{r}}_1\cdot \left({\mathrm{p}}_{\mathrm{i}}^{\mathrm{k}}-{\mathrm{x}}_{\mathrm{i}}^{\mathrm{k}}\right)+{\mathrm{c}}_2\cdot {\mathrm{r}}_2\cdot \left({\mathrm{p}}_{\mathrm{g}}^{\mathrm{k}}-{\mathrm{x}}_{\mathrm{i}}^{\mathrm{k}}\right)$$

(26)

where ${\mathrm{V}}_{\mathrm{i}}^{\left(\mathrm{k}+1\right)}$ is the velocity required to move from k^th iteration to ${\left(\mathrm{k}+1\right)}^{\mathrm{th}}$ iteration.

• ${\mathrm{p}}_{\mathrm{i}}^{\mathrm{k}}$ = best position of particle i at time k,
• ${\mathrm{p}}_{\mathrm{g}}^{\mathrm{k}}$ = global best solution for all particles at time k,
• r₁ and r₂ = random numbers between 0 and 1,
• c₁ = cognitive parameter assigned with a value 2. This is a hyperparameter that enables defining the ability of the swarm to be influenced by the best local solutions found over the iterations.
• c₂ = scaling parameter assigned with a value 2. This is a hyperparameter that enables defining the ability of the swarm to be influenced by the best global solutions during the iterations.

In our case, the PSO implementation requires an appropriate spacecraft dynamics model, optimality criteria, and constraints that fulfil the collision avoidance requirements. Typically, satellite motion in an orbit can be modelled using the classical orbital elements based on Gaussian variational equations. However, using these equations will result in ambiguity, especially for orbits with low eccentricities or inclinations [75]. In order to avoid this ambiguity, a new model that employs a set of modified equinoctial elements (MEE) developed in [76,77] is used to solve the low thrust transfer problem. The MEE parameters, in terms of the classical orbit parameters, are used as inputs for the PSO algorithm and can be written as follows:

$$\mathrm{p}= \mathrm{a}\left(1-{ \mathrm{e}}^2\right)$$

(27)

$$\mathrm{f}= \mathrm{ecos}\left(\mathsf{\omega }+\mathsf{\Omega }\right)$$

(28)

$$\mathrm{g}= \mathrm{esin}\left(\mathsf{\omega }+\mathsf{\Omega }\right)$$

(29)

$$\mathrm{h}=\tan \left(\frac{\mathrm{i}}{2}\right)\cos \left(\mathsf{\Omega }\right)$$

(30)

$$\mathrm{k}=\tan \left(\frac{\mathrm{i}}{2}\right)\sin \left(\mathsf{\Omega }\right)$$

(31)

$$\mathrm{L} = \mathsf{\omega } + \mathsf{\Omega } + \mathsf{\theta }$$

(32)

where:

• p = semi-latus rectum,
• a = semimajor axis,
• e = orbital eccentricity,
• i = orbital inclination,
• ω = argument of perigee,
• Ω = RAAN,
• L = true longitude,
• f,g = x,y components of the eccentricity vector in the orbital frame,
• h,k = x,y components of the node vector in the orbital frame.

The following equations describe the inverse relationship amongst classical and modified equinoctial elements:

$$\mathrm{a}=\frac{\mathrm{p}}{1-{\mathrm{f}}^2-{\mathrm{g}}^2}$$

(33)

$$\mathrm{e}=\sqrt{{\mathrm{f}}^2+{\mathrm{g}}^2}$$

(34)

$$\mathrm{i}={\tan }^{-1}\left(2\sqrt{\left({\mathrm{h}}^2+{\mathrm{k}}^2\right)},1-{\mathrm{h}}^2-{\mathrm{k}}^2\right)$$

(35)

$$\mathsf{\omega }={\tan }^{-1}\left(\mathrm{gh}-\mathrm{fk},\mathrm{fh}+\mathrm{gk}\right)$$

(36)

$$\mathsf{\Omega }={\tan }^{-1}\left(\mathrm{k},\mathrm{h}\right)$$

(37)

$$\mathsf{\theta }= \mathrm{L}- \mathsf{\omega }- \mathsf{\Omega }$$

(38)

The ECI state vector is expressed in terms of MEE using:

$$\overrightarrow{\mathrm{r}}=\left\{\begin{array}{c}\frac{\mathrm{r}}{{\mathrm{s}}^2}\left[\cos \left(\mathrm{L}\right)+{\mathsf{\alpha }}^2\cos \left(\mathrm{L}\right)+2\mathrm{hksin}\left(\mathrm{L}\right)\right]\\ \frac{\mathrm{r}}{{\mathrm{s}}^2}\left[(\sin \left(\mathrm{L}\right)-{\mathsf{\alpha }}^2\cos \left(\mathrm{L}\right)+2\mathrm{hkcos}\left(\mathrm{L}\right)\right]\\ \frac{2\mathrm{r}}{{\mathrm{s}}^2}\left[\mathrm{hsin}\left(\mathrm{L}\right)-\mathrm{kcos}\left(\mathrm{L}\right)\right]\end{array}\right\}$$

(39)

$$\overrightarrow{\mathrm{v}}=\left\{\begin{array}{c}-\frac{1}{{\mathrm{s}}^2}\sqrt{\frac{\mathsf{\mu }}{\mathrm{p}}}\left[\left(\sin \left(\mathrm{L}\right)+{\mathsf{\alpha }}^2\sin \left(\mathrm{L}\right)-2\mathrm{hkcos}\left(\mathrm{L}\right)+\mathrm{g}-2\mathrm{fhk}+{\mathsf{\alpha }}^2\mathrm{g}\right)\right]\\ -\frac{1}{{\mathrm{s}}^2}\sqrt{\frac{\mathsf{\mu }}{\mathrm{p}}}\left[\left(-\cos \left(\mathrm{L}\right)+{\mathsf{\alpha }}^2\sin \left(\mathrm{L}\right)+2\mathrm{hksin}\left(\mathrm{L}\right)-\mathrm{f}+2\mathrm{ghk}+{\mathsf{\alpha }}^2\mathrm{f}\right)\right]\\ \frac{2}{{\mathrm{s}}^2}\sqrt{\frac{\mathsf{\mu }}{\mathrm{p}}}\left[\left(\mathrm{hcosL}+\mathrm{ksin}\left(\mathrm{L}\right)+\mathrm{fh}+\mathrm{gk}\right)\right]\end{array}\right\}$$

(40)

where α, s, r, q can be expressed in terms of MEE as:

$${\mathsf{\alpha }}^2={ \mathrm{h}}^2-{\mathrm{k}}^2$$

(41)

$${\mathrm{s}}^2=1+{\mathrm{h}}^2+{\mathrm{k}}^2$$

(42)

$$\mathrm{r} =\frac{\mathrm{p}}{\mathrm{q}}$$

(43)

$$\mathrm{q}=1+\mathrm{fcos}\left(\mathrm{L}\right)+\mathrm{gsin}\left(\mathrm{L}\right)$$

(44)

The motion of the spacecraft is modelled by the following second order differential equation:

$$\frac{{\mathrm{d}}^2\overrightarrow{\mathrm{r}}}{{\mathrm{dt}}^2}+\frac{\mathsf{\mu }\overrightarrow{\mathrm{r}}}{{\mathrm{r}}^3}={\overrightarrow{\mathrm{a}}}_{\mathrm{d}}$$

(45)

where:

• r = magnitude of the inertial position,
• a_d = disturbing acceleration.

The disturbing acceleration is replaced by$\Delta$ when MEE are used. The perturbations will therefore be composed of the Earth effects and the applied thrust. The disturbing acceleration is expressed in the rotating RSW frame whose principal axes are defined by:

$${\mathrm{Q}}_{\mathrm{T}}=\left[\frac{\overrightarrow{\mathrm{r}}}{\left|\overrightarrow{\mathrm{r}}\right|}\frac{\left(\overrightarrow{\mathrm{r}}\times \overrightarrow{\mathrm{v}}\right)\times \overrightarrow{\mathrm{r}}}{\left|\left(\overrightarrow{\mathrm{r}}\times \overrightarrow{\mathrm{v}}\right)\times \overrightarrow{\mathrm{r}}\right|}\frac{\overrightarrow{\mathrm{r}}\times \overrightarrow{\mathrm{v}}}{\left|\overrightarrow{\mathrm{r}}\times \overrightarrow{\mathrm{v}}\right|}\right]$$

(46)

The ${\mathrm{J}}_2$ perturbations are modelled using the following equations:

$${\mathrm{f}}_{\mathrm{r}}=-\frac{3{\mathsf{\mu }\mathrm{J}}_2{\mathrm{R}}^2}{2{\mathrm{r}}^4}\left[1-12\frac{{\left(\mathrm{h}\sin \mathrm{L}-\mathrm{k}\cos \mathrm{L}\right)}^2}{{\mathrm{s}}^4}\right]$$

(47)

$${\mathrm{f}}_{\mathrm{s}}=-\frac{12{\mathsf{\mu }\mathrm{J}}_2{\mathrm{R}}^2}{2{\mathrm{r}}^4}\left[\frac{\left(\mathrm{h}\sin \mathrm{L}-\mathrm{k}\cos \mathrm{L}\right)\left(\mathrm{h}\cos \mathrm{L}-\mathrm{k}\sin \mathrm{L}\right)}{{\mathrm{s}}^4}\right]$$

(48)

$${\mathrm{f}}_{\mathrm{w}}=-\frac{6{\mathsf{\mu }\mathrm{J}}_2{\mathrm{R}}^2}{2{\mathrm{r}}^4}\left[\frac{\left(\mathrm{h}\sin \mathrm{L}-\mathrm{k}\cos \mathrm{L}\right)\left(1-{\mathrm{k}}^2-{\mathrm{h}}^2\right)}{{\mathrm{s}}^4}\right]$$

(49)

The total perturbation, if the thrust applied in the low thrust manoeuvre is very small, can be expressed as:

$$\overrightarrow{\Delta }={\overrightarrow{\Delta }}_{\mathrm{pert}}+\overrightarrow{{\Delta }_{\mathrm{T}}}$$

(50)

where ${\Delta }_{\mathrm{pert}}$ corresponds to ${\mathrm{J}}_2$ and ${\Delta }_{\mathrm{T}}$is thrust. The thrust acceleration $\overrightarrow{{\mathrm{a}}_{\mathrm{T}}}$ can be expressed as:

$$\overrightarrow{{\mathrm{a}}_{\mathrm{T}}}=\frac{\mathrm{T}}{\mathrm{m}}\overrightarrow{{\mathrm{u}}_{\mathrm{T}}}=\frac{{\mathrm{cn}}_0}{\mathrm{c}-{\mathrm{n}}_0\mathrm{t}}\mathrm{u}\left[\begin{array}{c}\sin \mathsf{\alpha }\cos \mathsf{\beta }\\ \cos \mathsf{\beta }\cos \mathsf{\alpha }\\ \sin \mathsf{\beta }\end{array}\right]$$

(51)

where:

• $\overrightarrow{{\mathrm{u}}_{\mathrm{T}}}$ = thrust control vector,
• T = thrust,
• c = effective exhaust gas velocity,
• n₀ = thrust to mass ratio at the initial time.

The thrust control magnitude is expressed in the form of a square wave with frequency ${\mathrm{f}}_{\mathrm{T}}$and phase ${\mathrm{k}}_{\mathrm{T}}$to reduce the infinitely dimensional optimal control problem into a finite non-linear programming problem.

$$\mathrm{u}= \mathrm{square} \left({\mathsf{\pi }\mathrm{f}}_{\mathrm{T}}\frac{\mathrm{t}}{{\mathrm{t}}_{\mathrm{f}}}+{\mathrm{k}}_{\mathrm{T}}\right)$$

(52)

where:

• t_f = time duration of the total manoeuvre, which is determined by the optimization algorithm.

The thrust vector angles are expressed using the following equations:

$$\mathsf{\alpha }\left(\mathrm{t}\right)={\mathsf{\pi }\mathsf{\alpha }}_{\mathrm{M}}\sin \frac{\mathsf{\pi }}{2}\left[{\left(\frac{\mathrm{t}}{{\mathrm{t}}_{\mathrm{f}}}\right)}^{{\mathrm{p}}_1}+{\left(\frac{\mathrm{t}}{{\mathrm{t}}_{\mathrm{f}}}\right)}^{{\mathrm{p}}_2}\right]$$

(53)

$$\mathsf{\beta }\left(\mathrm{t}\right)={\mathsf{\pi }\mathsf{\beta }}_{\mathrm{M}}\sin \frac{\mathsf{\pi }}{2}\left[{\left(\frac{\mathrm{t}}{{\mathrm{t}}_{\mathrm{f}}}\right)}^{\mathrm{p}1}+{\left(\frac{\mathrm{t}}{{\mathrm{t}}_{\mathrm{f}}}\right)}^{{\mathrm{p}}_2}\right]$$

(54)

where ${\mathsf{\alpha }}_{\mathrm{M}}$, ${\mathsf{\beta }}_{\mathrm{M}}$, ${\mathrm{p}}_1$, ${\mathrm{p}}_2$are the control parameters for the thrust orientation control. The thrust acceleration vector in the RSW frame is expressed as:

$${\overrightarrow{\mathrm{a}}}_{\mathrm{T}}^{\mathrm{RSW}}=\frac{\mathrm{T}}{\mathrm{m}}\left[\begin{array}{c}\sin \mathsf{\alpha }\cos \mathsf{\beta }\\ \cos \mathsf{\beta }\cos \mathsf{\alpha }\\ \sin \mathsf{\beta }\end{array}\right]$$

(55)

The thrust control vector can be transformed to the ECI frame as follows:

$${\overrightarrow{\mathrm{u}}}_{\mathrm{T}}^{\mathrm{ECI}}={ \mathrm{Q}}_{\mathrm{T}}{\overrightarrow{\mathrm{u}}}_{\mathrm{T}}^{\mathrm{RSW}}$$

(56)

Since the satellite operates in low earth orbit, the effects of aerodynamic drag must be considered. MSIS90 atmospheric model was employed in the PSO algorithm [78].The drag in RSW frame is expressed as:

$$\left[\begin{array}{c}{\Delta }_{{\mathrm{D}}_{\mathrm{R}}}\\ {\Delta }_{{\mathrm{D}}_{\mathrm{S}}}\\ {\Delta }_{\mathrm{DW}}\end{array}\right]=\frac{1}{2}{\mathsf{\rho }\mathrm{SC}}_{\mathrm{D}}{\mathrm{v}}_{\mathrm{R}}\left[\begin{array}{c}{\mathrm{v}}_{\mathrm{r}}\\ {\mathrm{v}}_{\mathrm{s}}\\ 0\end{array}\right]$$

(57)

where:

• ρ = atmospheric density,
• S = aerodynamic reference area,
• C_D = drag coefficient,
• V_R = velocity magnitude.

The velocity components ${\mathrm{v}}_{\mathrm{r}}$, ${\mathrm{v}}_{\mathrm{s}}$ are expressed as:

$${\mathrm{v}}_{\mathrm{r}}=\sqrt{\frac{\mathsf{\mu }}{\mathrm{p}}}\left(\mathrm{f}\sin \mathrm{L}-\mathrm{g}\cos \mathrm{L}\right)$$

(58)

$${\mathrm{v}}_{\mathrm{s}}=\sqrt{\frac{\mathsf{\mu }}{\mathrm{p}}}\left(1+\mathrm{f}\cos \mathrm{L}-\mathrm{g}\sin \mathrm{L}\right)$$

(59)

4. Case Studies

4.1. SBSS Scenario

Globally, the UAE is classified as a country with a high vulnerability to climate change, adversely affecting the infrastructure, human health, and natural habitat, which in turn impacts various sectors and policies including socioeconomic, health, and environment [79]. Adding on to the problem of climate change are intermittent events of oil spills that have a long term effect on eco-systems and economies. An Earth observation constellation comprising 40 satellites evenly spaced across four orbital planes with RAAN distributed between 0 degrees and 360 degrees (i.e., 0, 90, 180, 360) as illustrated in Figure 5 is hypothesized to detect oil spills, monitor climate change, and track RSOs simultaneously for SA and CA. The spacecraft in the DSS constellation are assumed to be placed in nearly circular arrangement (eccentricity = 0.001) at an altitude of 500 km at an inclination of 99 degrees, and are equipped with a hyperspectral payload with artificial intelligence (AI) to process the data on-board of the satellite for oil spill detection and climate change analysis [80,81,82]. The characteristics of the on-board payload are listed in

Table 2. Hyperspectral payload specifications.

Payload	HyperScout-2
Field Of View (FOV)	channel 1: 31° × 16° channel 2: 31° × 16°
Ground Sample Distance (GSD)	channel 1: 75 m channel 2: 490 m
Swath	310 × 150 km
Active Pixels	channel 1: 4000 × 1850 px channel 2: 1024 × 768 px
Spectral Range	channel 1: 400–1000 nm channel 2: 8000–14,000 nm
Spectral Bands	channel 1: 45 channel 2: 3
Spectral Resolution	channel 1: 16 nm channel 2: 1100 nm (B1, B2) and 6000 nm (B3)
Signal to Noise Ratio (SNR)	channel 1: 50–100 channel 2: 0.5–3000
Power	12 W

. To ensure the safety of DSS assets, the DSS constellation performs RSO tracking and establishes a collaborative exchange of tracking information through inter-satellite links to accomplish AN (AN) for CA [83,84]. The DSS constellation also acts as a service provider for other operational satellites or sub-orbital vehicles by establishing a data exchange using inter-satellite links so that they can perform AN for CA, as illustrated in Figure 6.

In recent studies, star trackers have been investigated as an alternative sensor for debris surveillance [85]. The participating spacecraft in the constellation are equipped with star trackers for attitude determination and RSO tracking. The spacecraft are assumed to simultaneously perform Earth observation operations along with AN for CA. RSO tracking and detection are performed using optical sensors and star trackers due to [86]:

• The passage of the RSO across the sensor’s field of view [10].
• The exploitation of brightness, angular velocity, and other threshold parameters for RSO travelling across the FOV of the sensor [87].

The major focus is to come up with a suitable mission architecture that accomplishes Earth observation and SSA mission objectives expeditiously. The requirements of the mission architecture are as follows:

• The system configuration must be feasible for multiple applications. For instance, AN for CA (current interest), SBSS for STM, multi-domain traffic management (MDTM) [88], and point-to-point sub orbital space transport, which are envisioned in the longer term.
• The satellites in the DSS architecture should complement each other and form ad-hoc or optional teams to make autonomous decisions and maximize mission objectives without involving the ground control segment, making them distinct compared to conventional satellite systems.

As mentioned earlier, the on-board payload for Earth observation is HyperScout-2, as specified in Table 2, whereas, for RSO tracking, the DSS assets are equipped with an MAI-SS star tracker, which has the specifications tabulated in

Table 3. Star tracker payload specifications.

Performance Parameter	Specification
Accuracy (Cross Axis/Boresight)	5.7 arcsec/27 arcsec
Acquisition Time	130 ms Acq, 105 ms Track (typical)
Max Tracking Rate	>2.0°/s
Update Rate	4 Hz
Lens	0.9in f1.2 BK7 Glass

. The two payloads are illustrated in Figure 7.

For the proposed SBSS mission architecture, the following assumptions are critical:

• The star trackers on board track the RSOs with the stars in the background.
• The participating spacecraft are equipped with state-of-the-art GPS for positioning and navigation that provide a full set of navigation data.
• The RSO position is estimated by simultaneous optical measurements obtained from two different spacecraft.
• The participating spacecraft share their position information and the estimated RSO position through a network.
• Mutual separation between the spacecraft belonging to the DSS constellation is guaranteed using intersatellite links and continuous monitoring from the ground stations.

The proposed AN system comprises the following components. Figure 8 illustrates the system architecture and its individual components:

• Navigation hardware comprises the state-of-the-art GPS to obtain a full set of navigation data comprising the DSS satellite positions, velocities, and attitude rates.
• Tracking hardware comprises star trackers that track the RSO.
• The obtained data from the hardware is used as inputs by the on-board Tracking System to obtain the RSO position estimates, error measurement budget, and to generate the uncertainty ellipsoids.
• The navigation and guidance system exploits the data generated by the tracking system for trajectory optimization and AN/manoeuvring to generate the steering commands.
• Actuators use the steering commands to perform the collision avoidance manoeuvres in order to avoid a collision with the RSO.

4.2. Case Study 2—Ground-Based Surveillance

The ground-based surveillance system comprises two electro-optical (EO) sensors that are located at two different observation sites, as illustrated in Figure 9. As illustrated in Figure 4b, two ground-based EO sensors track the RSO simultaneously and share the estimated RSO position data with a distributed satellite system (DSS) constellation through a network [24,81,82,91]. For the ground-based tracking scenario, the following assumptions are adopted [92]:

• The DSS assets are placed in a nearly circular low earth orbit (LEO) at an altitude of 500 km to carry out Earth observation activities.
• The participating spacecraft are equipped with sophisticated GPS for positioning and navigation that provide a full set of navigation data.
• The estimated RSO position from ground-based sensors is uplinked to the DSS assets to ensure their safety.
• The participating spacecraft share their position information and the RSO position estimates with other satellites using inter-satellite links (ISL).
• The ground-based EO sensors assumed in this scenario are similar to the sensors used in [93] and operate in the infra-red (IR) region between the wavelengths of 3–12 µm.

The proposed AN system comprises the following components:

• Navigation hardware comprises the state of the art GPS to obtain a full set of navigation data comprising the DSS satellite positions, velocities, and attitude rates.
• Tracking hardware comprises ground-based EO sensors that track the RSO by simultaneous optical measurements.
• The obtained data from the hardware are used as inputs by the On-Board System (OBS) to obtain the RSO position estimates, error measurement budget, and to generate the uncertainty ellipsoids.
• The navigation and guidance system exploits the data generated by the OBS for trajectory planning and optimization to generate the steering commands.
• Actuators use the steering commands to perform the collision avoidance manoeuvres in order to avoid a collision with the RSO.

Figure 10 illustrates the individual segments of the tracking system architecture.

5. Results and Discussions

Simulation case studies were performed in both space-based and ground-based tracking scenarios. The results obtained are presented and discussed in the following sections. However, it has to be noted that the current work focuses on estimating the RSO position and the associated errors. Assuming that the tracked RSO moves in a circular orbit, the equations used to estimate RSO velocity from position estimates are defined in [94]. Assuming that the RSO is moving freely in a circular orbit, then its motion can be governed by classic orbital dynamics, as:

$${\ddot{\mathrm{r}}}_{\mathrm{t}}=\frac{-\mathsf{\mu }}{{\left|{\mathrm{r}}_{\mathrm{t}}\right|}^3}+{\mathrm{f}}_{\mathrm{i}}$$

(60)

where:

• r_t = the position vector of the RSO in ECI frame,
• f_i = orbital perturbations that take into account J₂ perturbations and drag.
• The velocity of the RSO can be calculated by integrating Equation (60).

5.1. Space-Based Tracking Scenario

To estimate the target position using the algorithm defined in Section 2.2, we define the satellite orbital elements in the ECI frame and the Cartesian frame (

Table 4. Sensor orbital state elements.

Orbital Parameters	Sensor 1	Sensor 2	Cartesian Coordinates	Sensor 1	Sensor 2
a (km)	6878	6878	Xi (km)	6456.74	3843.01
e	0.001	0.001	Yi (km)	−367.63	−891.28
i (deg)	99	99	Zi (km)	2321.12	5627.35
ω (deg)	20	20	VX (km/s)	−2.6	−6.31
Ω (deg)	0	0	VY (km/s)	−1.12	−0.66
θ (deg)	0	36	VZ (km/s)	7.07	4.21

). The values of the corresponding error coefficients must be computed to calculate the errors in target position estimates. The error coefficients are sums of partial derivatives that are mentioned in [48]. The sigmas for each ${\mathrm{x}}_{\mathrm{t}},$ ${\mathrm{y}}_{\mathrm{t}},{ \mathrm{z}}_{\mathrm{t}}$ are calculated using Equations (5)–(8) with the corresponding azimuth angles ${\mathsf{\theta }}_1$ = 27 deg and ${\mathsf{\theta }}_2$ = 315 deg, and elevation angles ${\mathsf{\varphi }}_1$=58 deg and ${\mathsf{\varphi }}_2$ = 62 deg. Assuming that the state-of-the-art star trackers can provide the RSO sigma position ${\mathsf{\sigma }}_{\mathrm{p}}$= 0.5 m and sensor angular error ${\mathsf{\sigma }}_{\mathsf{\theta }}$ = 0.0022 degrees, running the algorithm allows us to obtain the sigmas listed in

Table 5. RSO position estimates and errors.

Cartesian Coordinate	xt (km)	yt (km)	zt (km)	R (km)
RSO-state parameters	5397.7	−2446	7908.3	2265.65
Total Error (σ’s)	4.7	10.46	20.36	9.12

.

Using the mathematical framework presented in Section 2.2 and Section 2.3, the covariance matrix corresponding to the navigation errors, tracking errors, and total error can be presented in the form of an uncertainty ellipsoid, illustrated in Figure 11.

5.2. Ground-Based Tracking Scenario

To track the RSO using the ground-based optical sensors, it is crucial to define the sensor positions in the Cartesian frame. For this particular scenario, two ground-based sensors are assumed to be located in Goldstone, California and Canberra, Australia. The geocentric coordinates of these two sites are converted to a Cartesian frame and the sensor positions are tabulated in

Table 6. Ground-based sensor positions.

Cartesian Co-Ordinates	Xi (km)	Yi (km)	Zi (km)
Sensor 1	1880.82	13.78	6221.92
Sensor 2	−1502.52	−5.24	6356.15

.

The sensor positions in Table 6 are used as inputs in the tracking algorithm to calculate the RSO position estimates and the corresponding errors listed in

Table 7. RSO position estimates and the corresponding errors.

Cartesian Co-Ordinates	${\mathbf{x}}_{\mathbf{t}} \left(\mathbf{km}\right)$	${\mathbf{y}}_{\mathbf{t}} \left(\mathbf{km}\right)$	${\mathbf{z}}_{\mathbf{t}} \left(\mathbf{km}\right)$	R (km)
RSO-state parameters	732.38	−2240.1	11,285	2818.1
Total Error (σ’s)	5.4	7.6	21.55	7.78

.

An uncertainty ellipsoid, as shown in Figure 12, can be generated using the mathematical framework presented in Section 2.2 and Section 2.3 to represent the navigation errors, tracking errors, and total error for the ground-based tracking scenario.

5.3. Trajectory Optimization for Collision Avoidance

It is assumed that a user segment satellite (Figure 5b) is present in an orbit that intercepts the orbit of the tracked RSO that was tracked in both the tracking scenarios, indicating a possible collision event. In order to avoid the total uncertainty volume, the spacecraft performs a collision avoidance manoeuvre based on the values obtained from Table 5 and Table 7 to reorient its initial orbit to a new orbit with a semimajor axis increased to 4.7 km in the SBSS scenario and 5.4 km in the ground-based tracking scenario (in-plane); failing to do so may result in a collision. It is assumed that the spacecraft that performs the orbit raising manoeuvre is equipped with RIT 10 EVO thruster (Ariane group) [95], which can generate a thrust of 5 mN with a specific impulse of 1900 s. An orbit raising manoeuvre (in-plane) is preferred over the out-of-plane manoeuvre (inclination change) due to its optimality in terms cost and manoeuvre time.

Table 8. Initial and final orbital parameters of the spacecraft after collision avoidance manoeuvres.

$\mathbf{O}\mathbf{r}\mathbf{b}\mathbf{i}\mathbf{t}\mathbf{a}\mathbf{l} \mathbf{P}\mathbf{a}\mathbf{r}\mathbf{a}\mathbf{m}\mathbf{e}\mathbf{t}\mathbf{e}\mathbf{r}$	a (km)	e	i (deg)	ω (deg)	Ω (deg)
$Initial state$	6878	0.001	99	20	360
$Final state \left(Space-based\right)$	6882.7	0.001	99	20	360
$Final state \left(Ground-based\right)$	6883.4	0.001	99	20	360

tabulates the orbital parameters of the spacecraft before and after the manoeuvre.

The control parameters for the constant thrust directions are tabulated in

Table 9. Generated control parameters for thrust directions.

Tracking Scenario	tf (min)	αM (deg)	βM (deg)	fT	kT (rad)	p1	p2
Space-based	28.97	3.02	−2.62	0.5	0.178	0.93	9.96
Ground-based	30.75	−1.44	−3.05	0.09	0.258	1	7.55

for both ground-based and space-based tracking scenarios.

The total manoeuvre time is higher for the ground-based scenario than the space-based scenario, as the variation in the semimajor axis is greater in comparison. The solution converged after 465,000 iterations with a total run time of 4255.9 s on an Intel Core $\mathrm{i}7 7\mathrm{th}$ generation processor in a MATLAB environment. The change in thrust vector angles with respect to time in both the tracking scenarios is illustrated in Figure 13 and Figure 14.

According to the results illustrated in Figure 13 and Figure 14, the total reorientation time in the SBSS and the ground-based scenarios is 100 min and 130 min, respectively. It can be observed that the thrust vector angles start from $0°$ at time t = 0 and return to $0°$ at the reorientation time, indicating the end of the orbit raising manoeuvre. The trajectory change from the initial trajectory to the final optimal trajectory is illustrated in Figure 15 and Figure 16.

The proposed approach results in orbit raising manoeuvres that require minimum $\Delta \mathrm{V}$ and manoeuvre time taking into account actual conditions, all modelled perturbations, and any chosen thrust profile, thereby granting the spacecraft with responsive trajectory planning and navigation capabilities.

6. Conclusions and Future Work

Over the past decade, space debris has caused an increased congestion of the orbital domain. Fragments of past space endeavours are trapped in orbit around Earth, posing a threat to the sustainability of the space sector. As space debris accumulates over time, it poses an increasingly higher risk to the satellites that are currently operational and to future space missions. This paper proposes an autonomous navigation (AN) and tracking framework for separation assurance (SA) in the low earth orbit (LEO) region. A suitable tracking algorithm was tested in both ground-based and space-based scenarios to accomplish AN/manoeuvring for avoidance of collisions with debris and other resident space objects (RSO). Particle swarm optimization (PSO) algorithms were developed to implement autonomous on-board trajectory generation and manoeuvre planning for RSO avoidance. Simulation case studies were performed in the presence of both ground-based surveillance and space-based space surveillance (SBSS), demonstrating the validity of the proposed techniques and the complementarity of ground-based and space-based tracking techniques to support DSS AN for RSO collision avoidance. Current research is addressing the optimal fusion of SBSS and ground-based sensors (using both conventional and AI-based techniques) to enhance the tracking accuracy, coverage, and timeliness of AN tasks in complex mission scenarios. By incorporating suitable AI techniques, trusted autonomous satellite operations (TASO) can be achieved using optimal combinations of SBSS and ground-based tracking data.