On the Space Observation of Resident Space Objects (RSOs) in Low Earth Orbits (LEOs)

Abstract

Space debris is an increasingly severe problem in the space industry. According to projections, the number of satellites will increase from the current 10,000 to 100,000 by 2030, specially in LEO orbits. This significant rise in the number of satellites threatens space sustainability, forcing satellites to perform more maneuvers to avoid impacts or leading to the production of more and more space debris due to collisions (Kessler Syndrome). Consequently, substantial efforts have been made to detect and track space debris, leading to the development of the current catalogs. However, with existing technology, detecting and tracking small debris remains challenging. In order to improve the current system, several proposals of Space-Based Situational Awareness (SBSA) have been made. These proposals involve satellites equipped with telescopes to detect space debris and determine their orbits. Unlike prior works, focused primarily on detection rates, this research aims to quantify their accuracy in orbit determination as a function of observation duration, the number of observers, and sensor precision. The Unscented Kalman Filter (UKF) is employed as the core estimation algorithm, leveraging both simulated single-case analyses and Monte Carlo simulations to evaluate system performance under various configurations and uncertainties. The results indicate that a constellation of at least three observers with high-precision instruments and sub-kilometer positioning accuracy can reliably estimate debris orbits within an observation period of 4–7 min, with the mean error in position and velocity obtained being 2.2–3 km and 3–4 m/s, respectively. These findings offer critical insights for designing future SBSA constellations and optimizing their operational parameters to address the growing challenge of orbital debris.

1. Introduction

At present, space debris is a severe problem in the space sector, imposing a significant cost, estimated at around USD 58 M annually, in relation to evasive manoeuvres and satellite losses [1]. The current estimations establish that there are around 36,500 objects larger than 10 cm, 1 million objects between 1 and 10 cm, and 130 million objects between 1 mm and 1 cm, encompassing a wide variety of materials and shapes [2]. With the advent of megaconstellations and the increasing accessibility of space, it is projected that the number of satellites will increase from the current 10,000 to 100,000 by 2030 [2]. This increase in the number of satellites will radically increase this cost.

Space debris is not only a financial burden, it is also a threat to space sustainability (Kessler Syndrome [3]). Due to its relevance, a lot of effort has been made to detect and track space debris, resulting in the North American Aerospace Defense Command (NORAD) catalogs. However, with the existing technology, the detection and tracking of small debris remain a challenge. To enhance the current system, several proposals for Space-Based Situational Awareness (SBSA) have been made. These proposals primarily involve satellites equipped with telescopes to detect both man-made objects such as space debris and natural objects such as asteroids, and to determine their orbits [4].

The main actors in the development of Optical Space-Based (OSB) Space Situational Awareness (SSA) systems are the U.S., Canada, and Europe [4]. The technological maturity presented by each block differs. The US provided the first demonstration of space-based space surveillance, with the Midcourse Space Experiment (MSX) satellite, part of the Space-Based Visible (SBV) program [5]. This mission achieved the detection of deep-space objects; its success led to the development of state-of-the-art space-based systems, including close-proximity observation programs (i.e., XSS-10 [6] and MiTEx [7] missions), as well as Geosynchronous Earth Orbit (GEO) surveillance missions (OSR-5 mission [8]). Canada has launched two space-based SSA missions, NEOSSat and Sapphire, on a shared launch. Sapphire was designed to detect objects at altitudes ranging from 6000 to 40,000 km [9]. NEOSSat was developed to perform high and Near-Earth orbit surveillance, and also to detect GEO objects and asteroids [10]. The results from NEOSSat demonstrated remarkable success, including the detection of some objects in low Earth Orbit (LEO) [11]. In 2014, Europe completed a feasibility study of a Space-Based Space Surveillance (SBSS) demonstration system to observe objects in GEO, middle Earth Orbit (MEO), Molniya, and high elliptical orbits [12]. A secondary objective was to detect objects in LEO (700 to 2000 km altitude). This study revealed that while it was possible to detect some objects at these altitudes, many objects were too small to generate strong enough signals or moved too quickly for effective tracking.

The missions proposed or launched by the main global actors in OSB SSA show that most systems focus on the detection of GEO or MEO objects, with only one mission being designed to observe LEO objects above 700 km in altitude, and only as a secondary objective. There is a remarkable gap in space-based systems specifically designed for LEO surveillance. Even if the SBSS system proves successful, it does not detect most LEO objects above 700 km [12]. Furthermore, there is a surveillance gap in the 400–700 km altitude range, where the majority of conjunction events occur [2]. Therefore, it is clear that a space-based observation system focused on LEO objects can benefit SSA systems and improve conjunction predictions. This is the main idea behind the Skylark constellation [13], which will consist of an initial constellation of 12 to 16 U-CubeSat satellites for tracking other satellites across multiple orbits and 40 followed-up satellites with infrared and hyperspectral sensors for Earth. However, this proposal has not yet been materialized and its fundamental aspects remain unclear. This scarcity of missions or information on proposed missions may be attributed to the immature state of the technologies required for a functional SSA LEO system, including the need for both high autonomous onboard processing capabilities and reliable Artificial Intelligence (AI) algorithms.

Space-based systems for SSA in LEO are currently under investigation. The status of key technologies for space-based SSA and their future prospects have been reviewed in [14]. Concerning the detection of space debris and the type of hardware needed, different sensor technologies can be utilized, including binocular-vision sensors [15,16,17], laser radars [18], infrared sensors [19,20], and optical telescopes [5,21,22]. Each sensor technology has its own advantages and disadvantages, but the consensus is that a space-based SSA system would primarily rely on visible-light telescopes. However, it is suggested that a complete characterization of the space debris (its orbit, size, shape, etc.) would require fusing the information of other types of sensors with the measurements taken with optical telescopes [14].

Another critical issue for optical systems is the recognition of debris. It is necessary to identify whether an observed object is a star, an asteroid, or a piece of space debris. Most of the methods for debris recognition used on the ground can be applied ti space-based observation. For instance, a Convolutional Neural Network (CNN) has been proposed to identify space objects and distinguish them from background stars [23]. Notably, this research suggested using commercial-grade star trackers or cameras with similar characteristics to observe LEO objects, a concept independently proposed by other authors, such as well [24]. The use of commercial-grade star trackers to observe LEO objects was recently validated in a balloon flight [25]. However, until in-orbit validation is achieved, algorithms developed to identify space debris will continue to primarily rely on simulated images [26].

Regarding the algorithms used to extract the orbital parameters of space debris, most of the methods used for ground observation can be employed, such as the Unscented Kalman Filter (UKF) [27], Kalman-consensus filter [28], asynchronous measurement schemes [29], or non-linear least squares methods [30]. Most of the algorithms and applications used to determine the orbit of space debris correspond to ground radar systems. For example, [31] studies the orbit determination of space debris using radar signals, reporting an uncertainty of 100 m in the along-track component and 0.01 m/s in its velocity. In fact, precise knowledge of the uncertainty is as important as the accuracy of the estimate. For example, in the work of Armellin et al. [32], orbit determination and its associated uncertainty are studied.

An interesting approach to orbit determination and the characterization of space debris using only optical data seems to be the study of the light curve of the objects [33]. Light curve analysis shows promise for systems employing only optical sensors. One significant limitation noted by the authors is the lack of real images for validation, as most analyses rely on simulated images. The potential of light curve analysis can be further enhanced by combining information from different observers (multi-agent observation) and the use of AI techniques.

Beyond detection technologies, a crucial aspect of SBSS is defining the mission in terms of the chosen orbit and the observation strategy. Tentative orbits for an SSA constellation in LEO are proposed by [34], emphasizing detection and the possibility of obtaining orbital parameters with a minimal observation time. However, it does not establish the accuracy of these parameters, which is essential for a realistic SSA system. Concerning observation strategies, there are two main options: passive observation and active observation [4]. In passive observation, debris passes through the field of view (FOV) of the sensors, while in active observation, the satellite or sensor actively follows the debris. Active observation is more accurate but also more complex, as it requires an Attitude Determination and Control System (ADCS) capable of changing the satellite’s attitude at the speed needed to follow the space debris. Additionally, an active system necessitates selecting objects to track, presenting a sensor tasking problem, where the satellite or a ground station will have to decide which objects to track to optimize the observations to be made [35,36]. In fact, the use of the word optimization is already a problem in itself, since it would be necessary to know which type of observation is more interesting: the one that provides more observations, the one that provides observations with a longer duration, or an intermediate strategy that seeks a trade-off between time and the number of observations.

As discussed in [14], the future projections of an SBSS suggest that a multi-agent observing strategy has the potential to solve most of the problems that were detected for LEO SSA systems. Multi-agent systems could allow for an increase in observation time and the gathering of measurements of the same object from different angles, thus improving the accuracy and facilitating the characterization of space debris.

This brief summary of the state of the art of SSA systems using space-based observations highlights the growing interest in developing SSA constellations to detect and track space debris. Optical instrumentation-based systems, in particular, show significant promise, especially when AI techniques and multi-agent observing systems are introduced. Although these solutions seem promising in terms of their ability to solve some of the aforementioned problems, the literature lacks discussions on the optimal configuration of constellations and the specific observation requirements for such systems. While [34] proposes a series of orbits for a constellation, it focuses primarily on the detectability of objects. It is equally important to establish the accuracy in orbit estimation that is achievable using these systems. To the authors’ knowledge, only one paper was found regarding this topic [37], which analyzes the performance of an algorithm of orbit determination for very short observation arcs (10–20 s) using a passive observation strategy, which requires at least two uncorrelated tracks.

The aim of the present paper is to analyze the accuracy of a multi-agent, active, optical SSA system as a function of several parameters, including the observation period, the number of observers, and the accuracy of the measurement instruments. The study described in this paper specifically addresses observing debris in LEOs at altitudes of 400–900 km, which is the main gap left by the major constellations proposed to date.

The algorithm described in this paper is primarily based on [38], which utilizes an Unscented Kalman Filter (UKF) algorithm. In comparison to the algorithm presented in [37], the UKF has the potential to estimate more characteristics of the observed Resident Space Object (RSO), such as shape and attitude (which can enhance the prediction and reduce the uncertainty of the future position of the observed object), as well as the possibility of incorporating simultaneous measurements made by different observers. In Linares’ paper [38], the UKF is used to estimate the position, velocity, shape, and attitude of RSOs in geosynchronous orbit from ground-based observations. According to this work, the estimation algorithm typically takes a minimum of 10 min of observation to converge, with some variables requiring over 20 min to reduce uncertainty around the estimated values. While a 10–20 min observational window is possible for RSOs in geosynchronous orbit, LEO objects, whether observed from ground-based or space-based SSA systems in active tracking observation mode, will have, at most, an observation window of 10 min, while passive observers in space would have an average observation time of 20 s [37]. Therefore, it is crucial to reduce the convergence time to make this type of algorithm feasible for use with LEO objects.

In this paper, the feasibility and conditions that must be met by space-based SSA systems that use these algorithms to obtain a good estimation of the observed RSOs are analyzed. To this end, the UKF algorithm and the models used in this work are described in Section 2. Section 3 is devoted to explaining the case study scenario (hypotheses and assumptions), along with the main analyses to be performed, and the results obtained. Finally, the main conclusions derived from the research carried out are summarized in Section 4.

2. Orbit Determination Algorithm

To estimate the orbit accurately and predict the future state of an RSO, it is essential to determine a set of at least six parameters. There are several sets of parameters that express the orbit and position of an object, with the Keplerian and Cartesian parameters being the most well-known and widely used, especially for objects orbiting Earth. Keplerian parameters provide an intuitive understanding of an object’s orbit, while Cartesian parameters, which represent position and velocity, are preferred for predicting future states due to their simplicity in state propagation:

$$\mathbf{x}={\left[\begin{array}{c}xyz\dot{x}\dot{y}\dot{z}\end{array}\right]}^T={\left[\begin{array}{c}\mathit{r}\mathit{v}\end{array}\right]}^T.$$

(1)

This simplicity is crucial for any SSA system which aims for precise predictions of RSO’s orbits and future position. More parameters can be estimated, such as the shape, size, or attitude, which provides valuable information for accurately predicting the future state of the observed object. However, the estimation of these additional parameters is not possible for the limited observation windows of an LEO object for a space- or ground-based observer. Therefore, for this first iteration, the minimum set of variables to estimate the orbit was considered in order to explore the characteristics that a space-based SSA system should meet. The estimation of the selected parameters was achieved using an Unscented Kalman Filter (UKF) algorithm very similar to the one described in [38]. Later, based on these results, the possibility of extending this initial set of parameters could be discussed.

2.1. Unscented Kalman Filter

In order to better understand the estimation process of RSO orbit parameters, and the implementation of the algorithm to solve the determination problem, it seems necessary to somehow describe the UKF algorithm [39]. Let us consider a non-linear system described by the following equations:

$${\mathit{x}}_{k+1}^{-}=f({\mathit{x}}_k,{\mathbf{\nu }}_k),$$

(2)

$${\mathit{y}}_{k+1}=h({\mathit{x}}_{k+1},{\mathit{u}}_k).$$

(3)

The first equation corresponds to the prediction of the system state in the future, $\mathit{x}$, which is calculated using the non-linear function $f(\mathit{x},\mathit{\nu })$ which depends on the system state and the process noise, $\mathbf{\nu }$, which is related to unmodelled perturbations. The second expression represents the measurements $\mathit{y}$, which can be taken by or from the system. These measurements can also be modeled or predicted using the measurement function, $h(\mathit{x},\mathit{u})$, which depends on the state of the system and the overall measurement noise, $\mathit{u}$. The subscript k is introduced to differentiate the state and measurement of the system at different instants.

Now, let us also consider that we know the most probable state of the system, $\mathit{x}$, at an arbitrary instant, k. Also, assume that the covariance matrix of the system, $\mathbf{P}$, is known, or at least reasonably well estimated. With this information, it is possible to propagate and predict the future state of the system ${\mathit{x}}_{k+1}^{-}$, and also its predicted covariance matrix, ${\mathbf{P}}_{k+1}^{-}$, using the information provided by the system function $f(\mathit{x},\mathit{\nu })$. The calculation of these predicted (or a priori) values is known as the prediction step.

In an Extended Kalman Filter (EKF) scheme, the a priori state is directly calculated by propagating the system function, and the predicted covariance matrix is calculated using the Jacobian matrix of the system function. This methodology is suitable for systems which are linear or have low-order nonlinearities. However, for highly non-linear systems, this approximation lacks accuracy, leading to significant errors, especially in the prediction of the a priori covariance matrix. In order to improve the estimation of the predicted states, the UKF algorithm was developed [40,41]. The UKF uses the Unscented Transformation (UT) to predict the state and covariance matrix of a system given a nonlinear model and a probability distribution characterized by a finite set of points within the distribution. The process of calculating the a priori state using UT is as follows:

First, it is necessary to compute what are called sigma points $\mathit{\chi }$. These sigma points are a representative sample of different states of the system within the statistical distribution given by the covariance matrix. There are a total of $2n+1$ sigma points, where n is the dimension of the state vector. The first point corresponds to the mean (or most probable) state of the system:

$${\mathit{\chi }}_k^0={\mathit{x}}_k.$$

(4)

The other sigma points are calculated as a deviation of the mean value using the following expression:

$$\begin{array}{cc}\hfill {\mathit{\chi }}_k^i={\mathit{x}}_k+{\left(\sqrt{\left(n+\lambda \right){\mathbf{P}}_k}\right)}_i;& i=1\dots n,\hfill \\ \hfill {\mathit{\chi }}_k^i={\mathit{x}}_k-{\left(\sqrt{\left(n+\lambda \right){\mathbf{P}}_k}\right)}_{i-n};& i=n+1\dots 2n,\hfill \end{array}$$

(5)

here $\lambda$ represents a scaling parameter known as the spread parameter, which is calculated as follows:

$$\lambda ={\alpha }^2\left(n+\kappa \right)-n,$$

(6)

where $\alpha$ and $\kappa$ are the fine-tuning parameters of the filter. $\alpha$ is the most important parameter and controls the spread of the sigma points. The value of this parameter is usually set as a small positive number (e.g., $1\times {10}^{-3}$). $\kappa$ is a secondary scaling parameter and, in most cases, can be set to 0. Returning to Equation (5), it is possible to observe that it is necessary to calculate the square root of the covariance matrix, which is typically solved using the Cholesky decomposition. Note that there is a subscript, i, denoting the i-th column of the square root of the covariance matrix. Each sigma point is always associated with a specific column of that matrix.

The next step in the UT is the calculation of the weights, W, associated with each sigma point:

$$\begin{array}{cc}\hfill W^{0,m}& ={\displaystyle \frac{\lambda }{\lambda +n}},\hfill \\ \hfill W^{0,c}& ={\displaystyle \frac{\lambda }{\lambda +n}}+1-{\alpha }^2+\beta ,\hfill \\ \hfill W^{i,m}& =W^{i,c}={\displaystyle \frac{\lambda }{2\left(\lambda +n\right)}}i=1\dots 2n,\hfill \end{array}$$

(7)

where $\beta$ is another tuning parameter, whose value is set based on the knowledge of the distribution. For a Gaussian distribution, $\beta =2$ is optimal. The propagated sigma points, ${\mathit{x}}_{k+1}^{-}$ (denoted with a minus sign to identify that it is the a priori state), can be calculated using the non-linear function for each sigma point:

$${\mathit{\chi }}_{k+1}^{-,i}=f\left({\mathit{\chi }}_k^i\right)$$

(8)

Once the propagated sigma points and their respective weights have been calculated, the predicted state and covariance matrix are computed as follows:

$${\mathit{x}}_{k+1}^{-}=\sum _{i=0}^{2n}{W}^{i,m}{\mathit{\chi }}_{k+1}^{-,i},$$

(9)

$${\mathbf{P}}_{k+1}^{-}=\sum _{i=0}^{2n}{W}^{i,c}\left({\mathit{\chi }}_{k+1}^{-,i}-{\mathit{x}}_{k+1}^{-}\right){\left({\mathit{\chi }}_{k+1}^{-,i}-{\mathit{x}}_{k+1}^{-}\right)}^T+\mathbf{Q},$$

(10)

where $\mathbf{Q}$ is a matrix that models the system noise. This propagation step is repeated consecutively until a new measurement is available, at which point the correction step begins.

In the correction step, the first task is to calculate the sigma points of the expected measurement, ${\mathit{Y}}^{-}$. This is achieved using the measurement function h and the predicted value of the propagated sigma points ${\mathit{\chi }}^{-}$, as is expressed in the following equation:

$${\mathit{Y}}_{k+1}^{-,i}=h\left({\mathit{\chi }}_{k+1}^{-,i}\right).$$

(11)

With the sigma points of the expected measurement calculated, the mean value can be obtained with an expression similar to that of Equation (9).

$${\mathit{y}}_{k+1}^{-}=\sum _{i=0}^{2n}{W}^{i,m}{\mathit{Y}}_{k+1}^{-,i}.$$

(12)

Once the expected measurement and its sigma points are calculated, it is possible to obtain the Kalman gain $\mathbf{K}$ as follows:

$${\mathbf{P}}_{yy}=\sum _{i=0}^{2n}{W}^{i,c}\left({\mathit{Y}}_{k+1}^{-,i}-{\mathit{y}}_{k+1}^{-}\right){\left({\mathit{Y}}_{k+1}^{-,i}-{\mathit{y}}_{k+1}^{-}\right)}^T+\mathbf{R},$$

(13)

$${\mathbf{P}}_{xy}=\sum _{i=0}^{2n}{W}^{i,c}\left({\mathit{\chi }}_{k+1}^{-,i}-{\mathit{x}}_{k+1}^{-}\right){\left({\mathit{Y}}_{k+1}^{-,i}-{\mathit{y}}_{k+1}^{-}\right)}^T,$$

(14)

$$\mathbf{K}={\mathbf{P}}_{xy}{\mathbf{P}}_{yy}^{-1}.$$

(15)

Here, $\mathbf{R}$ is the matrix that contains the noise information of each measurement. Finally, once the Kalman gain $\mathbf{K}$ has been computed, the predicted state vector and covariance matrix of the system can be corrected:

$${\mathit{x}}_{k+1}={\mathit{x}}_{k+1}^{-}+\mathbf{K}\left({\mathit{y}}_{k+1}-{\mathit{y}}_{k+1}^{-}\right),$$

(16)

$${\mathbf{P}}_{k+1}={\mathbf{P}}_{k+1}^{-}-\mathbf{K}{\mathbf{P}}_{yy}{\mathbf{K}}^T.$$

(17)

2.2. Propagation Model

To implement the UKF algorithm to estimate the orbital parameters of RSOs, a propagation model and a measurement model should be defined. The system being modeled primarily corresponds to the case of an RSO orbiting Earth. Therefore, the simplest model that can be implemented for the propagation step is that of a particle that is affected by the Earth’s gravitational field:

$$\left\{\begin{array}{c}\dot{x}\\ \dot{y}\\ \dot{z}\\ \ddot{x}\\ \ddot{y}\\ \ddot{z}\end{array}\right\}=\left\{\begin{array}{c}\dot{\mathit{r}}\\ -{\displaystyle \frac{\mu }{r^3}}\mathit{r}+{\mathit{a}}_p\end{array}\right\},$$

(18)

where $\mathit{r}$ corresponds to the RSO’s position in the Earth-Centered Inertial (ECI) frame, $\dot{\mathit{r}}$ refers to the velocity of the RSO’s, $\mu$ is the Earth’s gravitational parameter ($\mu \approx 3.986\times {10}^{14}$${\mathrm{m}}^3·{\mathrm{s}}^{-2}$) and ${\mathit{a}}_p$ denotes the modeled perturbation terms. It is important to note that the model described corresponds to a function of the time derivative of the state variables, ${\displaystyle \frac{\mathrm{d}\mathit{x}}{\mathrm{d}t}}=f\left(\mathit{x}\right)$. Therefore, in order to calculate the future state of these variables, numerical integration of this function is needed.

The accuracy of the propagation step greatly depends on how well these perturbation terms are known. However, in order to obtain a good estimation of all the perturbation terms, more information about the system is needed. For example, the drag from the residual atmosphere or the solar radiation pressure needs the RSO’s attitude, shape, area, and surface properties to be accurately modeled. Since this filter does not estimate either the shape, size, or attitude, the only applicable perturbation terms are those related to the Earth’s gravity field or the gravitational influence of other massive bodies, such as the Sun or the Moon. At least the first correction terms of the gravitational field (J2, J3...) should be considered, especially for LEO objects [42]. The other unknown perturbations can be introduced into the problem as system noise using the $\mathbf{Q}$ matrix. To assemble the $\mathbf{Q}$ matrix, an approximation of the magnitude of the most significant unmodeled perturbation term is needed. This approximation depends on many factors, with the best option being a worst-case estimation considering the most common size, shape, and frontal area of RSOs at that altitude. Once the magnitude of the perturbation terms has been calculated, the $\mathbf{Q}$ matrix can be assembled as follows:

$${\mathbf{\sigma }}_r={\displaystyle \frac{1}{2}}{\mathit{a}}_{pert}·\Delta t^2,$$

(19)

$${\mathbf{\sigma }}_v={\mathit{a}}_{pert}·\Delta t,$$

(20)

$$\mathbf{Q}=\left[\begin{array}{cccccc}{\sigma }_{r,x}& 0& 0& 0& 0& 0\\ 0& {\sigma }_{r,y}& 0& 0& 0& 0\\ 0& 0& {\sigma }_{r,z}& 0& 0& 0\\ 0& 0& 0& {\sigma }_{v,x}& 0& 0\\ 0& 0& 0& 0& {\sigma }_{v,y}& 0\\ 0& 0& 0& 0& 0& {\sigma }_{v,z}\end{array}\right],$$

(21)

where ${\mathbf{\sigma }}_r$ and ${\mathbf{\sigma }}_v$ represent the uncertainties associated with the position and velocity, respectively.

2.3. Measurement Model

Regarding the measurement model, cameras or star trackers mounted on a satellite capture images that include several bright objects, such as RSOs and stars. The first step is to identify which objects are RSOs and which are stars. Once the RSOs are identified, their pixel positions and pixel values are used as the raw measurements of the system. However, using raw measurements as input to the filter depends on the specific characteristics of the equipment. This means that to obtain the expected pixel position and value, the characteristics of every piece of equipment that introduces measurements into the filter shall be modeled. To make the filter as general as possible, it is better to preprocess the data from each observer to provide more meaningful data for the problem that does not require knowing the characteristics of the camera itself, aside from the noise or covariance matrix associated with the measurement.

In this filter, the measurement obtained from each observer is the unit vector pointing from the observer to the RSO, $\widehat{\mathit{\rho }}$. Since the measurement vector depends on the attitude of the host satellite, or where it is pointing the ground station, in a situation with multiple observers, it is better to express all measurements in the same reference frame. For the implemented algorithm, the measurements are expressed in the ECI reference system as the evolution of the state vector of the RSO. The transformation from one reference system to another can be computed using the following equation:

$${\widehat{\mathit{\rho }}}_{\mathrm{ECI}}={\mathbf{C}}_{\mathrm{bi}}^T{\widehat{\mathit{\rho }}}_{\mathrm{body}},$$

(22)

where ${\mathbf{C}}_{\mathrm{bi}}$ is the transformation matrix from the inertial frame to the body or camera reference frame.

A sketch of the geometric configuration of the problem is shown in Figure 1. The RSO’s position can be expressed as follows:

$${\mathit{r}}_{RSO}={\mathit{r}}_O+\rho \widehat{\mathit{\rho }},$$

(23)

where ${\mathit{r}}_{RSO}$ is the position of the observed object, ${\mathit{r}}_O$ denotes the position of the observer (host satellite or ground station), and $\rho$ is the range between the observer and the RSO. As the only geometrical measurement available is the unit vector that points from the observer to the RSO, the measurement model can be expressed as follows:

$$\mathit{y}=\widehat{\mathit{\rho }}={\displaystyle \frac{{\mathit{r}}_{RSO}\left(\mathit{x}\right)-{\mathit{r}}_O}{\parallel {\mathit{r}}_{RSO}\left(\mathit{x}\right)-{\mathit{r}}_O\parallel }}.$$

(24)

2.4. Initialization of the Filter

At the beginning of the algorithm, it is necessary to provide initial values for the state vector and its covariance matrix. The values given to these variables at this moment have a great impact on the convergence to a solution and the number of measurements needed to achieve a desired accuracy. The accuracy of the initial guess depends greatly on prior knowledge of the object and the number of simultaneous observations. Although it is possible to initialize the algorithm with a random state of the RSO, this approach carries the risk that the initial guess may be too far from the actual state, leading to non-convergence or a solution with a higher level of error than if a more accurate initial guess were used. Using the first measurements to derive an initial guess can help the algorithm converge more reliably and with fewer errors. In the following paragraphs, various initialization methods based on the number of observers are presented, providing strategies to improve the initial guess and enhance the overall performance of the algorithm. The better the initial estimate of the state, the better the results that can be expected.

2.4.1. One Observer

When there is only one observer, initializing the RSO’s position and velocity is challenging. Using the information from a single image, it is not possible to obtain the relative range and velocity of the object from the observer. However, an initial guess can be obtained using consecutive measurements, such as with the Gauss method for preliminary orbital estimation [43]. This method requires three different measurements with small time intervals between them. One problem with this method is that it relies on orbital properties, such as the assumption that orbits lie in a plane and the conservation of angular momentum. These equations are applicable to perfect or nearly perfect measurements. When measurements contain errors, the resulting error can be significant, particularly for velocity estimates.

2.4.2. Two Observers

With two observers targeting the same object, it is possible to estimate the relative range of the object to the observers. Each observer’s position (a point in space) and the unit vector (direction) from the observer to the RSO are known. This setup defines a line for each observer. As the two observers are looking at the same RSO, the intersection of these lines will provide the RSO’s position (see Figure 2).

Mathematically, this problem can be solved using the following procedure:

The first step is to find the equation that defines each line in the space. There are several formulations to express the equation of a line. For this case, the line is formulated as the intersection of two planes since it facilitates the programming and solving of the intersection of two lines. Consequently, given a point $P(x,y,z)$ and a direction vector $\widehat{\mathit{\rho }}({\widehat{\rho }}_1,{\widehat{\rho }}_2,{\widehat{\rho }}_3)$, find the normal direction of two planes that contain the line. The normal direction of the first plane can be computed as follows:

$${\mathit{n}}_1={\displaystyle \frac{{\widehat{\mathit{\rho }}}_1\times \mathit{u}}{\parallel {\widehat{\mathit{\rho }}}_1\times \mathit{u}\parallel }},$$

(25)

where $\mathit{u}$ is any random vector non-parallel to the observation vector. Then, the normal direction of the second plane is obtained using the following equation:

$${\mathit{n}}_{\mathbf{2}}={\widehat{\mathit{\rho }}}_1\times {\mathit{n}}_1.$$

(26)

With the normal direction of the plane and one point (the observer’s position), each plane can be mathematically defined. The general equation of a plane is given by the following:

$$x{n}_{i,x}+y{n}_{i,y}+z{n}_{i,z}+A_i=0,$$

(27)

where $\left[n_x,n_y,n_z\right]$ are the components of the normal direction of the plane, and A is a constant that ensures the equation equals zero when the coordinates of the point are substituted into the equation of the plane. The subscript i is used to identify each plane.

Once the equations of the planes are obtained, the line can be described as the intersection of these planes. Consequently, the line is defined as a system of two equations, with the following general form:

$$\left\{\begin{array}{c}x{n}_{1,x}+y{n}_{1,y}+z{n}_{1,z}+A_1=0\hfill \\ x{n}_{2,x}+y{n}_{2,y}+z{n}_{2,z}+A_2=0\hfill \end{array}\right..$$

(28)

After obtaining the equations of the two lines, the intersection point can be calculated. The intersection of two lines involves three unknown quantities. Each line is represented by two equations, resulting in a total of four equations for two lines. Since one of the equations can be derived as a linear combination of the other three, there are only three independent equations. Therefore, the intersection point and the solution to the problem are found by solving the system of three independent equations from the four available equations:

$$\left\{\begin{array}{c}x{n}_{1,x}+y{n}_{1,y}+z{n}_{1,z}+A_1=0\hfill \\ x{n}_{2,x}+y{n}_{2,y}+z{n}_{2,z}+A_2=0\hfill \\ x{n}_{3,x}+y{n}_{3,y}+z{n}_{3,z}+A_3=0\hfill \end{array}\right..$$

(29)

This method provides the position of the RSO, and the velocity can be derived using the Gibbs method [43]. However, it is important to remark that if the measurement error is considered, it is possible that the system of equations does not have a solution, and it is necessary to obtain the solution that approaches the system closest to 0 (in this case, ideally, the fourth equation can be introduced to minimize the error). As this method could present problems, it is better to use the generalized method for multiple observers.

2.4.3. Multiple Observers

As previously discussed, the main problem with earlier methods is their reliance on very accurate measurements to obtain a possible solution. Therefore, it is preferable to use a method that can account for measurement errors and be generalized for multiple observers. Following the same idea as the one proposed with two observers, there will be several lines that go from the observers to the RSO. With perfect measurements, all these lines would intersect at a common point. However, when measurement errors are introduced, the lines may intersect at different points or may not intersect at all. As is shown in Figure 3, the true solution is closely surrounded by the various intersection points, so a possible solution can be achieved by computing the average value of these intersections. However, if two observers are close to each other, the intersection point can be far from the object, resulting in a significant error in the initial guess.

Another possible approach to this problem is to find the point in the space that minimizes the distance to the different observation lines. This transforms the geometrical problem into an optimization one:

$$J={\displaystyle \frac{1}{n}}\sum _{i=1}^n{\displaystyle \frac{1}{{\sigma }_i^2}}{\displaystyle \frac{\parallel (x-x_o^i,y-y_o^i,z-z_o^i)\times {\widehat{\mathit{\rho }}}_i\parallel }{\parallel {\widehat{\mathit{\rho }}}_i\parallel }},$$

(30)

where $\sigma$ represents the standard deviation of the measurement error of the observer i and $(x_o,y_o,z_o)$ denotes its position.

Once again, this method only provides range measurements; in order to derive the velocity, it is necessary to use other methods, such as the one described by Gibbs, or to try and find the mean velocity between two consecutive points if the measurement rate is sufficiently high. It is important to note that while it is possible to achieve a reasonably accurate position determination, the velocity determination is highly susceptible to measurement errors, often leading to poor-quality results.

2.5. Back- and Forward-Propagation Process

One of the main challenges faced by the algorithm is the short observation period typically available for LEO objects passing through the observer’s FOV. This limited observation time can lead to two potential scenarios.

In the first scenario, the algorithm does not converge to a solution within the observation time. In this case, the algorithm will require a longer observation time (not to be confused with the number of measurements) to accurately capture the motion of the RSO.

In the second scenario, the algorithm does converge to a solution within the observation time, but the obtained solution may not reach the maximum accuracy that the filter is able to achieve based on the given data. In these cases, an iterative process can be employed to improve the final estimation of the orbital parameters. The idea behind this iterative process is to obtain a new initialization point based on the last calculated orbital parameters (see Figure 4).

Following this idea, the algorithm back-propagates the last estimated state of the RSO. In the second iteration, it uses the back-propagated state as the new initialization point. Since the algorithm is based on an Unscented Kalman Filter, it is also important to propagate the covariance matrix of the state variables during the back-propagation phase.

2.6. Summary of the Determination Process

Now all the parts of which the algorithm is composed have been explained, it is helpful to explain how all of them are integrated and what the complete process of determination is.

The orbit determination of an RSO has two fundamental steps. The first step is where the instrument observes an object and these images must be processed to identify the object as an RSO. Once identified, measurements of the actual direction between the observing satellite and the target must be obtained. The second step corresponds to the determination of the orbit.

When an RSO is identified as such, it is first decided whether to calculate the initial value of the observed RSO position and velocity. This decision is made depending on the number of available measurements, the number of observers, etc. For example, if three observers are available, a state initialization should be performed, but if only one observer is available, with few initial measurements, a random initialization with an initial covariance matrix with high values should be performed. Once the state values and the initial covariance matrix are available, the UKF is fed by the preprocessed measurements from the instrument. Once the state has been propagated and all the measurements of the observed arc have been processed, whether the algorithm performed the maximum iterations is checked. If the algorithm has not performed all the iterations, the estimated state is back-propagated to obtain a new initial state and re-run the UKF process. When all iterations are completed, the last state obtained is considered the estimated state of the RSO. A summary of this process is graphically presented in Figure 5.

3. Numerical Simulations and Discussion

As previously mentioned, the primary objective of this work is to analyze the performance of a potential Optical-Based Space Situational Awareness Constellation. Various simulations were conducted to evaluate the achievable accuracy based on variables such as the number of observers, the observation time, and uncertainties related to the observers’ orbits, attitude, and instrument precision. To effectively interpret the simulation results, it is crucial to clarify the key assumptions as follows:

• Observers’ orbits are confined to low Earth circular orbits, within an altitude range of 400–700 km. The position and direction of velocity of each observer are randomly generated.
• The target (RSO) orbit is also randomly generated within the same low Earth circular orbit altitude range of 400–700 km.
• It is assumed that the RSOs are always visible to the observers, regardless of the relative positions of the Sun, the observer, and the RSO, as well as the observer’s distance from the instrument or whether the RSO is eclipsed by the Earth. This is the main simplification of the problem.
• All defined observers provide measurements of the RSO, and their measurements are perfectly synchronized.

Although these assumptions might initially seem unrealistic, it is essential to remember that the primary goal is not to assess the performance of a specific orbit or satellite constellation. Instead, the study aims to understand the characteristics that a constellation should possess to achieve the desired level of performance. With this perspective, the key assumptions can be justified as follows:

The rationale behind the random initialization of the observers’ orbits is to eliminate any bias toward specific orbits. Moreover, this random initialization simplifies the programming of Monte Carlo analyses, which are essential for studying the system’s performance beyond the results obtained from a single case.

The random generation of the RSOs’ orbits better reflects the real scenario, with the exception that the orbits are restricted to circular ones. This restriction simplifies orbit generation and ensures that the perigee does not enter the Earth’s atmosphere.

The assumption that the RSOs are always visible to the observers is justified for three reasons. First, it avoids the need to place both observers and RSOs in specific areas to ensure visibility. Second, it prevents introducing dependencies on the sensitivity of an undefined instrument. Third, it simplifies the coding and simulation of multiple cases in Monte Carlo analyses. Although detectability is crucial for the system’s real performance, it is advisable to compare the results of this study with those dedicated to constellation orbits based on the number and magnitude of observable objects, as discussed in [34].

The fourth assumption, that all observers detect the RSO simultaneously, simplifies the problem and provides a theoretical value of the system’s expected performance. In a real situation, not all observers can detect the RSO simultaneously, meaning the number of observers and their measurements vary over time. Without defining a specific constellation, predicting the number of observers viewing the same RSO at any given moment is impossible. Therefore, this analysis should be conducted at a later stage, once potential constellation options are narrowed down based on other variables.

Once the main assumptions of the simulation are considered, let us introduce the main variables considered in the simulation that have an impact on the system’s performance. The list of these variables is shown in

Table 1. The main considered variables and their default values.

Variable	Default Value
Observation time	5 min (300 s)
Elapsed time between measurements	0.2 s
Number of observers	3
Uncertainty of observers’ position error	1000 m (1$\sigma$ standard deviation)
Uncertainty of observer’s attitude error	0.05 deg (1$\sigma$ standard deviation)
Uncertainty of instrument error	50 arcsec (1$\sigma$ standard deviation)

, along with their default values, which are used for each case unless otherwise specified.

The default values given to the different parameters of the UKF are shown in

Table 2. Default values given to the UFK parameters.

Variable	Random Initialization	Initial Guest
Assumed uncertainty in position	5000 km	100 km
Assumed uncertainty in velocity	2600 m/s	10,000 m/s
Assumed uncertainty in measurements	$5\times {10}^{-4}$	$5\times {10}^{-4}$
Assumed uncertainty in propagation model	$1\times {10}^{-4}$$\mathrm{m}/{\mathrm{s}}^2$	$1\times {10}^{-4}$$\mathrm{m}/{\mathrm{s}}^2$
Maximum number of back-propagation steps	2	1

. In this table, different values are set for two different scenarios. In the first scenario (Random Initialization), the initial state corresponds to that of a randomly generated circular orbit in LEO. In the second scenario (Initial guest), one of the state initialization methods discussed above is used, preferably the multiple observers method. The choice of the chosen values is justified later in Section 3.4.1.

Regarding the convergence of the algorithm, it is assumed that the orbit determination has converged if the Root Mean Square Error (RMSE) for the position is less than 20 km and that of the velocity is 30 m/s. Although this RMSE is relatively large compared to the accuracy that would be sought in such a system, these values are sufficiently low considering the initial error in the estimation of position and velocity using both a random generation of the initial estimate and an initialization method.

To better understand the theoretical performance of an Optical-Based Space Situational Awareness Constellation, single-case analyses are first performed to extract relevant results regarding the number of observers and the time between measurements. Given the random generation of observers and space debris, a second set of analyses is performed using Monte Carlo methods. The first analysis focuses on the impact that the initialization method and the assumed initial uncertainty has on the convergence and accuracy of the filter. Secondly, the mean error of the orbit estimation for cases with a different number of observers and different observation times is determined. Thirdly, The impact of uncertainties on the observers’ position, attitude, and instrument precision on the system’s performance is analyzed using an additional Monte Carlo assessment. Finally, more realistic cases, where the number of simultaneous observers is not constant and changes throughout time, is considered.

3.1. Results with One Observer

The first analysis focuses on the case of a single observer, considering the evolution of the estimated state of the space debris over a 10 min observation period. The results are shown in Figure 6. The algorithm appears to slowly converge towards the solution. However, the error obtained at the end of the observation window remains unacceptably large, with the estimated orbit significantly differing from the actual one. The convergence of the RSO’s estimated state to the real one takes longer than the typical observation period available for any SSA space-based system in LEO. Further tests with a single observer revealed that this scenario is prone to failing to converge to a solution, particularly if the initial state provided to the UKF is too far from the actual state.

The main reason for the slow convergence of the estimated orbit is attributed to the observability problem of the observer–target distance. An observer alone cannot estimate the velocity and distance of the observed object. If a number of constraints are applied, such as the object orbiting a known object (Earth for example), then, through various measurements, the position and velocity can be derived if the time between measurements is known and the observed arc is sufficiently large. If the observation windows are not large enough, the information obtained from different observation arcs over time could be combined. The problem with this idea is that the revisit time between the arc and arc of observation will be high. As a compromise between having a single observer and several simultaneous observers, it is possible to have several observers that each see a piece of the observing arc at different times. Thus, by combining the information from the different observations over time, one can converge to a solution, as discussed in Section 3.5.

3.2. Multiple Observers Result

Considering the previous results, it is clear that a single observer is not suitable for LEO applications. When a similar simulation is conducted with two observers, a different behavior is observed, as shown in Figure 7. In this case, the algorithm successfully converges to the RSO state and its orbit within the observation window. The plotted results correspond to the first iteration of the UKF, without the backward propagation step, to better understand the algorithm’s behavior. It can be seen that the RSO’s position is estimated within the first 100 s of observation, while the velocity takes longer to converge, around 400–500 s.

To study the impact of multiple observers and observation time on the accuracy of the estimated RSO state, a series of simulations were carried out. In these simulations, while keeping the orbit of the target RSO constant, both the observation window and the number of observers were varied. The results of these analyses are shown in Figure 8. These graphs display the Root Mean Square Error (RMSE) in the estimated position and velocity of the RSO for different combinations of the number of simultaneous observers and as a function of the observation time. The RMSE calculation only takes into account the last 20% of the data of the entire observational window. This is to avoid incorporating the initial error and the filter error while converging to the solution. In this way, the RMSE value corresponds to the accuracy of the algorithm once it has converged to the solution.

The position of the RSO is estimated within the first 30 s of observation in the presence of more than two observers. Increasing the observational window does not significantly enhance system performance. For this case, the accuracy in position estimation improves significantly when increasing from two to three observers. Further increases in the number of observers results in increasingly less significant improvements in accuracy, but these are still not negligible, especially in relation to the stability of the solution obtained, as in the case of going from three to five observers. A significant improvement is not clearly observed when changing from five to seven observers, showing that the system tends to reach a maximum precision after a certain number of observers. Focusing on the velocity error, it is evident that, unlike position, the velocity requires an observational window of more than 300 s for two observers and around 200–300 s for multiple observers to converge to the system’s maximum accuracy. The accuracy achieved with an increasing number of observers shows a similar trend to the position error. The improvement from two to three observers is significant and the increase from three to five observers makes the solution more stable, but additional observers beyond five do not seem to provide much better accuracy.

3.3. Impact of Time Interval Between Measurements

This analysis focuses on the impact that the time interval between measurements has on the system’s performance. The aim was to find the lowest measurement rate at which the accuracy starts to degrade. A simulation was conducted to measure this effect, and the results are shown in Figure 9. In this figure, the results of the velocity and position RMSE for different numbers of observers and the elapsed times between measurements over a 6 min observation window are displayed. It can be seen that the measurement rate should be smaller than one measurement per second. Interestingly, increasing the measurement rate beyond one measurement per second does not significantly impact the accuracy achieved. However, if the time between measurements exceeds one second, the error begins to increase. Regarding the number of observers, there is a significant performance improvement when increasing the number of observers from two to three. Better results are obtained when the total number of observers is five and seven, especially in the position error when the time between measurements is lower than 0.8 s.

3.4. Monte Carlo Analysis

The previous results illustrate the potential behavior of an SSA space-based system under one randomly generated scenario, providing insights into the impact of the number of observers, observation time, and measurement rate. However, it is important to remember that the random nature of RSO state generation and observer orbits means that results can vary slightly between scenarios. Furthermore, to understand the characteristics that would be desirable for the system and how they affect the accuracy obtained, it is essential to understand how the main errors or uncertainties that influence the measurement process affect system performance. In order to study the true impact of these variables, a series of Monte Carlo analyses were performed to determine the accuracy to which the system tends to converge. Each Monte Carlo analysis consisted of 100 randomly generated cases for both RSOs and observer satellites. Unless otherwise specified, all simulations are carried out using the default parameters listed in Table 1. Notably, these included an observation time of 5 min and a total of three observers.

3.4.1. Impact on the Initial Uncertainty and State Initialization

This analysis studies the impact that the assumed initial uncertainty in the UKF has on the convergence and accuracy of the algorithm. To perform this study, different values of the assumed position uncertainty, ${\sigma }_{pos}$, were considered, 10 km, 100 km and 1000 km, with the assumed uncertainty in velocity being swept in the range ${\sigma }_{vel}\in$ [0.1 m/s–10 km/s] and the velocity. Two different cases are shown in Figure 10 (both show the results for a 100-case Monte Carlo simulation). In the upper graph of the figure, the convergence rate when the initial state is randomly generated is shown, whereas in the lower graph, the results correspond to an initial state calculated with the first obtained measurements. In both cases, the algorithm starts to converge when the ${\sigma }_{vel}>5$ m/s. From this point, the convergence rate increases until ${\sigma }_{vel}=100$ m/s, where it reaches the vicinity of its maximum convergence rate (60–80% for the first case and almost 100% for the second one); then, the convergence rate remains stable if ${\sigma }_{vel}>100$ m/s.

Additionally, an extra analysis that compares the convergence of the filter based on the number of back-propagation processes was carried out. The results indicate that two iterations are enough to guarantee at least an 80% convergence rate for any initialization case. It should be noted that when an initial state is calculated using the first measurements, the convergence rate increases to 90% with just one back-propagation step.

3.4.2. Error Analysis

This analysis examines the impact of increasing the number of observers and the observation time with respect to an RSO. For each condition (one for each number of observers and each observation window selected), 100 simulations were performed (N = 100); each simulation corresponded to a completely randomly generated orbit for the RSO and the observers. The mean value of the Root Mean Square Error (RMSE) of the observed position, $e_{pos}$, and velocity, $e_{vel}$ of an RSO (these errors are referred to as Position RMSE and Velocity RMSE in Section 3.2—see also Figure 8), as a function of the number of observers during two different observational windows of 4 and 7 min, together with their corresponding standard deviations, ${\sigma }_{ep}$ and ${\sigma }_{ev}$, are shown in Figure 11.

The results indicate that the addition of observers improves accuracy. In the case of the position, a third observer reduces the mean error from 3 km to 2.5 km; the addition of more observers reduces the RMSE to 2 km if a 4 min observation is considered. In the case of a 7 min observation, this trend is similar, but the RMSE is lower when two observers are used (approximately 2.5 km). The standard deviation of the error in position is also improved when adding more observers, reaching a minimum standard deviation of 0.8 km. Regarding the velocity of the RSO, an improvement in the RMSE and the standard deviation is observed when increasing from two observers to three or more, obtaining a velocity error of 3.8 m/s and a standard deviation of 3.17 m/s if a 4 min observation is considered. In the case of a 7 min observation, the errors in velocity and its corresponding standard deviation are 3.4 m/s and 2.1 m/s, respectively. What seems to be significant is that a longer observation period leads to a much better estimate of the RSO velocity. This effect, however, is not appreciated in the results obtained for the position.

3.4.3. Impact of Observer Uncertainties in the Quality of Estimation

The main sources of errors and uncertainty considered in this work included the discrepancy between the true and estimated positions of the satellite, errors in the estimated attitude, and the uncertainty of the sensor or instrument that detects the light curves of various space objects. To isolate the effects of each error, an analysis focusing on one error at a time (and eliminating the influence of other errors) was carried out.

The mean value of the Root Mean Square Error (RMSE) of the observed position, $e_{pos}$, and velocity, $e_{vel}$, of an RSO obtained as a result of Monte Carlo Simulations (taking an observation period of 5 min), as a function of the standard deviation of the observer’s uncertainty about orbital position estimation, ${\sigma }_{SPU}$ (with ${\sigma }_{SPU}$ ∈ [0.1, 100] km), is shown in Figure 12. The results indicate that, logically, increasing this uncertainty significantly degrades the accuracy of the estimation. However, it also seems that this level of error is contained, or the effect of the observer’s orbital position estimation uncertainty is less significant, for values of this variable below 1 km. In this range, the errors in position and velocity increase from $e_{pos}$ = 1.52 km, (${\sigma }_{ep}$ = 0.11 km) and $e_{vel}$ = 3.58 m/s (${\sigma }_{ev}$ = 0.6 m/s) for ${\sigma }_{SPU}$ = 0.1 km, to $e_{pos}$ = 2.15 km (${\sigma }_{ep}$ = 0.76 km) and $e_{vel}$ = 3.66 m/s (${\sigma }_{ev}$ = 1.3 m/s) for ${\sigma }_{SPU}$ = 1 km. For larger values of uncertainty regarding the observer’s position estimation, the errors are significantly larger (e.g., $e_{pos}$ = 13.9 km (${\sigma }_{ep}$ = 6.75 km) and $e_{vel}$ = 17.4 m/s (${\sigma }_{ev}$ = 8.1 m/s) for ${\sigma }_{SPU}$ = 10 km).

Regarding the errors as a function of the standard deviation of the observer attitude estimation error, ${\sigma }_{SAU}$ (with ${\sigma }_{SAU}$ ∈ [${10}^{-2}$, 10] deg), and as a function of the standard deviation of the instrument uncertainty, ${\sigma }_{IMU}$ (with ${\sigma }_{IMU}$∈ [${10}^{-3}$, 1] deg), the results are shown, respectively, in Figure 13 and Figure 14. In the case of these two uncertainty sources, the errors in position and velocity seem to be less significant for values below ${\sigma }_{SAU}$ = 0.1 deg ($e_{pos}$ = 1.64 km, ${\sigma }_{ep}$ = 0.26 km and $e_{vel}$ = 5.1 m/s, ${\sigma }_{ev}$ = 1.5 m/s ) and ${\sigma }_{IMU}$ = 0.014 deg ($e_{pos}$ = 1.54 km, ${\sigma }_{ep}$ = 0.16 km and $e_{vel}$ = 4 m/s, ${\sigma }_{ev}$ = 1 m/s).

3.5. Performance of the Algorithm for a Non-Constant Number of Observers

Observing the same RSO using multiple observers, especially when the observers and the RSO are located in LEO, can be a challenge to achieve. Also, the greater the number of observers, the lower the probability that all of them will observe the same object simultaneously. Keeping this limitation in mind, a more realistic scenario, where the observation of the object is carried out by a non-constant number of observers, is considered. It should be noted that the back-propagation process was omitted for these analyses, since back-propagation smooths the results obtained and does not clearly show the effect of incorporating and losing different observers.

The scenario to be considered consists of a 5 min simulation where five different observers detect the RSO during different points of the whole observational period. The scenario in this analysis has the following structure:

• At the beginning of the simulation, observers 1, 2, and 3 will detect and observe the object until T + 60 s.
• From T + 60 s until T + 120 s, the object will be observed by satellites 2 and 3.
• From T + 120 s until T + 180 s, the object will be observed by satellites 2, 3, and 4.
• From T + 180 s until T + 210 s, the object will be observed by satellites 2, 3, 4, and 5.
• From T + 210 s until T + 270 s, the object will be observed by satellites 3, 4, and 5.
• From T + 270 s until T + 300 s, the object will be observed by satellites 4, and 5.

Since there are three observers at the beginning of the scenario, the initial state of the UFK is calculated using the first measurements.

The results obtained from this case are shown in Figure 15. As there are three observers and an initial state is calculated, the error in position is relatively low at the beginning of the estimation process, reaching a final error of around 2.5 km. In contrast, the velocity was estimated poorly, with an initial error larger than 10 km/s, but given the high value of the assumed uncertainty in velocity (${\sigma }_{vel}$ = 10 km/s), the initial estimation is rapidly corrected, later reaching a stable error value of around 5 m/s. From these results, it seems reasonable to conclude that a change in the number of simultaneous observers did not impact the estimations in the studied scenario.

Based on the above results and taking into account that observation by three simultaneous observers is still difficult to achieve, it was decided to create a much more restrictive but also more possible scenario in which the RSO is observed by a single observer for a period of time. Then, the RSO is no longer visible to that observer but is detected by a different one. The structure of the scenario is defined as follows:

• At the beginning, observer 1 will identify the RSO and it will be visible until T + 210 s.
• From T + 210 s, the object will be observed by a second satellite (observer 2) until the end of the simulation at time T + 300 s.

Since there is only one observer and the Gauss method does not provide good initialization results when the measurements are very close to each other, the value for the initial state was set randomly. The results obtained are shown in Figure 16. This figure provides a comparison between the true and estimated states for both the position and velocity. It is possible to observe how the state does not converge to the true solution while it is being observed by the first observer. However, when the second satellite starts to observe the RSO, the solution quickly converges to the true solution. This result shows that it is possible to converge to a solution without having simultaneous observers viewing the same object, with it being sufficient to merge the information from the different orbit arcs seen by different observers. This behavior may allow for the requirements of a hypothetical constellation of observers in terms of simultaneous observation to be relaxed to a certain extent. It is also fair to say that this scheme has some drawbacks, since, in order to merge the information from different observing arcs, it is necessary to make sure that the observed RSO is the same.

4. Conclusions

In this paper, the primary characteristics that an optical-only Space Situational Awareness constellation system must meet to achieve a certain level of performance using an Unscented Kalman Filter for orbit determination were studied. This study mainly focuses on determining the orbits of space debris or RSOs in LEOs (400–700 km circular orbits). Based on the results obtained from these analyses, the following conclusions were drawn:

• The orbit determination of RSOs in LEOs requires at least two observers to converge to a solution within the typical LEO observation time window, especially if the initial state of the RSO is unknown.
• With multiple observers, the RSO’s position can be obtained almost immediately (within the first 10 s), but velocity estimation requires a minimum observation time. According to the results, this time is approximately 400 s for two observers and between 200 and 300 s for three or more observers.
• The time interval between measurements should be less than one second.
• Increasing the number of observers improves the accuracy of the estimated orbit and reduces dispersion. However, it should be noted that more significant improvements are observed when increasing from two to three observers. When increasing the number of observers from three to five, the accuracy of the algorithm continues to improve, offering a more stable solution. From this point on, the increase in the number of observers produces less and less significant improvements, approaching the asymptotic error or maximum accuracy. For three observers, the mean error observed in position was 2.2 km.
• The errors in velocity that were obtained were much more dependent on the observation period that the position errors. For three observers and a 7 min observation period, the mean error obtained was 3.4 m/s.
• The correct initialization of the initial state increases the probability of converging to the solution. Regarding the initial uncertainty assumed in the initial state, it is concluded that the uncertainty in the velocity is the strongest determinant to ensure convergence, with ${\sigma }_{vel}$ values higher than 100 m/s being recommended to obtain the best results.
• The influence of different error sources on system accuracy was considered, including errors in the position estimation of observer satellites, attitude estimation error, and instrument/sensor uncertainty. Attitude estimation errors and instrument uncertainty are the most critical for achieving high accuracy. Position estimation uncertainty should be kept below 1 km, attitude uncertainty below 0.1 degrees, and instrument uncertainty below 30 arcseconds for a distance error below ten kilometers and a velocity error below 0.01 m/s.
• Based on the above conclusions, an SSA optical constellation should aim to have three observers viewing the same object for a period of 5 min, with a measurement rate of at least one per second, and maintain position estimation uncertainties below 1 km, attitude errors below 0.1 degrees, and sensor errors below 30 arcseconds.