One-Shot Initial Orbit Determination in Low-Earth Orbit

 

Comparison of uncertainty ellipsoid between the proposed method and trilateration method for the position estimation.

One-Shot Initial Orbit Determination in Low-Earth Orbit

Ricardo Ferreira, Marta Guimarães, Filipa Valdeira, Cláudia Soares

Due to the importance of satellites for society and the exponential increase in the number of objects in orbit, it is important to accurately determine the state (e.g., position and velocity) of these Resident Space Objects (RSOs) at any time and in a timely manner. State-of-the-art methodologies for initial orbit determination consist of Kalman-type filters that process sequential data over time and return the state and associated uncertainty of the object, as is the case of the Extended Kalman Filter (EKF). However, these methodologies are dependent on a good initial guess for the state vector and usually simplify the physical dynamical model, due to the difficulty of precisely modeling perturbative forces, such as atmospheric drag and solar radiation pressure.

Other approaches do not require assumptions about the dynamical system, such as the trilateration method, and require simultaneous measurements, such as three measurements of range and range-rate for the particular case of trilateration. We consider the same setting of simultaneous measurements (one-shot), resorting to time delay and Doppler shift measurements. Based on recent advancements in the problem of moving target localization for sonar multistatic systems, we are able to formulate the problem of initial orbit determination as a Weighted Least Squares.

With this approach, we are able to directly obtain the state of the object (position and velocity) and the associated covariance matrix from the Fisher's Information Matrix (FIM). We demonstrate that, for small noise, our estimator is able to attain the Cramér-Rao Lower Bound accuracy, i.e., the accuracy attained by the unbiased estimator with minimum variance. We also numerically demonstrate that our estimator is able to attain better accuracy on the state estimation than the trilateration method and returns a smaller uncertainty associated with the estimation.

Subjects: Systems and Control (eess.SY); Instrumentation and Methods for Astrophysics (astro-ph.IM); Machine Learning (cs.LG); Optimization and Control (math.OC) Cite as: arXiv:2312.13318 [eess.SY]   (or arXiv:2312.13318v1 [eess.SY] for this version)   https://doi.org/10.48550/arXiv.2312.13318 

Authors

Authors:

• Ricardo Ferreira - Ph.D. student at NOVA School of Science and Technology, Portugal. Background in computer science.
• Marta Guimaraes - Ph.D. student at NOVA School of Science and Technology, AI researcher at Neuraspace. Background in aerospace engineering.
• Filipa Valdeira - Postdoctoral researcher at NOVA School of Science and Technology. Background in aerospace engineering and mathematical sciences.
• Claudia Soares - Assistant professor at NOVA School of Science and Technology. Background in engineering.

Institutions:

• The authors are affiliated with NOVA School of Science and Technology in Portugal. This institution focuses on science and technology research and education.
• Marta Guimaraes is also affiliated with Neuraspace, a company focused on AI solutions.

Related Works:

• The authors build on prior work on formulating moving target localization for sonar systems as a weighted least squares problem, adapting it to the orbit determination problem.
• They compare their method to standard approaches like extended Kalman filters and trilateration methods for orbit determination.
• References are made to foundational works on Lambert's orbital boundary value problem, gibbs methods for orbit determination, and statistical methods like the Cramer-Rao lower bound.

In summary, the authors have backgrounds spanning computer science, aerospace engineering, and mathematical sciences, with affiliations in academia and industry, and build on a base of statistical estimation theory and classic orbital mechanics works.

Summary

This approach would seem to be applicable to tracking missiles and FOB satellites for a missile defense system.

The documents include a research paper and its source PDF file on developing a new method for initial orbit determination of objects in low Earth orbit (LEO). The key points are:

• Accurately determining the orbit (position and velocity) of resident space objects like satellites and debris is important for space situational awareness and collision avoidance.
• Standard methods like extended Kalman filters (EKF) have drawbacks like needing good initial guesses for the orbit and simplifying complex orbital dynamics.
• The proposed method formulates the problem as a weighted least squares optimization to directly estimate the object's state and uncertainty from time delay and Doppler shift measurements.
• It doesn't require propagating a reference trajectory or linear approximations of dynamics.
• Experiments show it can attain the Cramér-Rao lower bound accuracy for small noise levels.
• It provides better accuracy than trilateration methods and lower uncertainty estimates for the orbit.
• This method could complement decision-making for collision avoidance and trajectory tracking by providing efficient and accurate orbit determination from simultaneous measurements.
• Further work could evaluate performance for medium and geostationary orbits and incorporate satellite-to-satellite tracking data.

In summary, the documents present a new optimization-based approach for accurately and robustly determining the orbit of objects in low Earth orbit that overcomes limitations of standard methods. Key benefits are better accuracy and uncertainty estimates to support space situational awareness.

The key points about the proposed new formulation for initial orbit determination are:

• It formulates the problem as a weighted least squares (WLS) optimization using time delay and Doppler shift measurements.
• Intermediate variables are introduced to avoid non-linearities with respect to position and velocity.
• The problem is solved in two steps - an estimation stage using WLS to get an initial estimate, and a correction stage to refine the estimate.
• In the estimation stage, the WLS cost function is minimized to get estimates of the position, velocity and intermediate variables.
• In the correction stage, the relations between intermediate variables and position/velocity are incorporated to refine the estimates from the first stage.
• This two-stage approach provides the complete state vector (position and velocity) and covariance matrix directly, without needing to linearize dynamics or propagate a reference trajectory.
  

the key measurement data used by the proposed method for initial orbit determination are:

• Time delay measurements: The time delay (τij) is the time taken for a signal to travel from the transmitter to the object and then to the receiver. It depends on the distances between the object and transmitter/receiver.
• Doppler shift measurements: The Doppler shift (fij) is the shift in frequency of the signal caused by the relative motion between object and transmitter/receiver. It depends on the object's velocity.

These measurements are gathered simultaneously from multiple ground transmitter and receiver stations to the object in low Earth orbit.

Specifically:

• Time delays τij are measured between each transmitter i and receiver j pair, giving a set of MN measurements for M transmitters and N receivers.
• Doppler shifts fij are also measured for each transmitter-receiver pair, giving another set of MN measurements.
• The time delay measures distance while Doppler shift measures velocity. Together they can characterize the full orbit state.
• Gaussian noise is added to simulate noisy measurements for testing.

So in summary, the proposed method relies on simultaneous time delay and Doppler shift measurements from a distributed network of transmitters and receivers to efficiently determine the orbit.

the key measurement data used by the proposed method for initial orbit determination are:

• Time delay measurements: The time delay (τij) is the time taken for a signal to travel from the transmitter to the object and then to the receiver. It depends on the distances between the object and transmitter/receiver.
• Doppler shift measurements: The Doppler shift (fij) is the shift in frequency of the signal caused by the relative motion between object and transmitter/receiver. It depends on the object's velocity.

These measurements are gathered simultaneously from multiple ground transmitter and receiver stations to the object in low Earth orbit.

Specifically:

• Time delays τij are measured between each transmitter i and receiver j pair, giving a set of MN measurements for M transmitters and N receivers.
• Doppler shifts fij are also measured for each transmitter-receiver pair, giving another set of MN measurements.
• The time delay measures distance while Doppler shift measures velocity. Together they can characterize the full orbit state.
• Gaussian noise is added to simulate noisy measurements for testing.

So in summary, the proposed method relies on simultaneous time delay and Doppler shift measurements from a distributed network of transmitters and receivers to efficiently determine the orbit.

Applicability for Space Based Missile Defense

It would seem that a distributed network of ground, maritime, and space based sensor nodes could be used. The proposed method could potentially be applicable to using measurements from moving sensors in low Earth orbit, as long as their position and velocity are known accurately. Some key points:

• The method as formulated uses fixed ground transmitter and receiver stations. But it relies only on the geometry between sensor, object and transmitter to calculate time delays and Doppler shifts.
• So if the sensors are in known orbits, their position and velocity vectors can simply replace the fixed ground station coordinates used in the current formulation.
• The time delays and Doppler shifts can be calculated in the same way, using the known sensor orbit data and measured signal values.
• More processing may be needed to account for sensor movement during the measurement window and oscillating Doppler shifts.
• Moving sensors in different orbits provide better geometric diversity compared to fixed ground sites for localization.
• But errors in knowledge of the sensor orbit may degrade localization accuracy. Robustness against these errors would need to be evaluated.

In summary, the fundamental concepts of the method could be extended to moving sensors in LEO provided their orbital states are well-characterized. Additional engineering would be needed to adapt the measurements and account for sensor movement. Performance analysis in different sensor orbit configurations could determine the feasibility and benefits of this approach.