Angle-only Measurements Research Articles

Distributed orbit determination and observability analysis for satellite constellations with angles-only measurements

Autonomous absolute orbit determination (OD) for a satellite constellation using only inter-spacecraft measurements has considerable value in space systems engineering. This paper proposes a novel solution for cooperative autonomous OD for a system comprised of an arbitrary number of spacecraft, with inertial angles-only measurements. We establish a link between observability and a graph representation of measurements among spacecraft. For a system with a beacon or without beacons, it is shown that if a measurement graph topology has graph theoretic equivalence properties with a two-level tree form, the system is second-order and third-order locally weakly observable, respectively. To solve the distributed OD problem, the new interleaved distributed extended Kalman filter (IDEKF) is proposed. Several evaluation cases are designed to demonstrate the influence of measurement topology, number of spacecraft, and orbital configuration on the degree of system observability. Finally, the performance of the IDEKF estimation is illustrated for a ten-spacecraft constellation.

Toward space situational awareness: A space-based autonomous optical tracking method for space objects

<p indent=0mm>Space-based optical (SBO) surveillance is one of the important means of space situational awareness. Traditional space-based observational platforms cannot operate autonomously because accurate ephemeris needs to be obtained before determining the states of the observed objects. In this article, the concept of autonomous space-based surveillance is proposed. A space-based optical observation platform can carry out space situational awareness (SSA) autonomously using the obtained angle-only measurements, even if an accureate ephemeris is unknown. In contrast to traditional space object tracking, the states of an observed space object can be determinted by only using the obtained angle-only measurements. In combination with long-arc observational data, an autonomous orbit improvement algorithm based on historical data is presented. The orbit of the original object can be improved by using historical observational data obtained by observing other space objects. Thus, a real-time autonomous tracking strategy with the assistance of additional measurements is presented, which can be performed by tracking multiple space objects using an observational satellite that has a payload of multiple SBO sensors. Finally, the proposed method was demonstrated by simulation cases. Compared with traditional object tracking, the estimation accuracy of autonomous tracking is worse but it will be similar when an observational satellite tracks two objects at the same time, especially if one of them operates in low earth orbit. This paper provides a feasible method for the development of an autonomous space-based SSA platform.

Three-spacecraft autonomous orbit determination and observability analysis with inertial angles-only measurements

Autonomous orbit determination (OD) for a group of spacecraft using only inter-spacecraft measurements has considerable value in space systems engineering. This paper proposes a novel solution for cooperative autonomous OD for a system comprised of three spacecraft, by employing inertial angles-only measurements. Based on the two-body dynamics of each spacecraft, the observability problem for the two-spacecraft and three-spacecraft systems is analyzed with the Lie-derivative criterion. It is deduced that the two-spacecraft system is third-order locally weakly unobservable, while the three-spacecraft system with three sets of inter-spacecraft observations is third-order locally weakly observable under certain sufficient conditions. To complement the theoretical analysis, we use a numerical method to compute the observability measure to illustrate the degree of observability. Several scenarios are designed to demonstrate the influence of orbital configurations, of J2 dynamics, and measurement noise on the degree of observability. In addition, the observability of the three-spacecraft system is also compared with the two-spacecraft system. Finally, practical cases of three-spacecraft autonomous OD are simulated which demonstrate the application and feasibility of the proposed method.

A novel estimation algorithm for torpedo tracking in undersea environment

A novel estimation algorithm is introduced to handle the popular undersea problem called torpedo tracking with angle-only measurements with a better approach compared to the existing filters. The new algorithm produces a better estimate from the outputs produced by the traditional nonlinear approaches with the assistance of simple noise minimizers like maximum likelihood filter or any other algorithm which belongs to their family. The introduced method is extended to the higher version in two ways. The first approach extracts a better estimate and covariance by enhancing the count of the intermediate filters, while the second approach accepts more inputs so as to attain improved performance without enhancement of the intermediate filter count. The ideal choice of the placement of towed array sensors to improve the performance of the proposed method further is suggested as the one where the line of sight and the towed array are perpendicular. The results could get even better by moving the ownship in the direction of reducing range. All the results are verified in the MATLAB environment.

Integrity Assessment and Verification Procedure of Angle-only Data for Low Earth Orbit Space Objects with Optical Wide-field PatroLNetwork (OWL-Net)

The Optical Wide-field patroL-Network (OWL-Net) is a global optical network for Space Situational Awareness in Korea. The primary operational goal of the OWL-Net is to track Low Earth Orbit (LEO) satellites operated by Korea and to monitor the Geostationary Earth Orbit (GEO) region near the Korean peninsula. To obtain dense measurements on LEO tracking, the chopper system was adopted in the OWL-Net’s back-end system. Dozens of angle-only measurements can be obtained for a single shot with the observation mode for LEO tracking. In previous work, the reduction process of the LEO tracking data was presented, along with the mechanical specification of the back-end system of the OWL-Net. In this research, we describe an integrity assessment method of time-position matching and verification of results from real observations of LEO satellites. The change rate of the angle of each streak in the shot was checked to assess the results of the matching process. The time error due to the chopper rotation motion was corrected after re-matching of time and position. The corrected measurements were compared with the simulated observation data, which were taken from the Consolidated Prediction File from the International Laser Ranging Service. The comparison results are presented in the In-track and Cross-track frame.

A Novel Space-Based Orbit Determination Method Based on Distribution Regression and Its Sparse Solution

There is an inherent need to track and catalog space debris (objects) in geosynchronous earth orbits (GEO) based on space-based surveillance networks and a large amount of observation data. However, for objects in GEO, angle-only measurements containing noises have been regarded as difficult for short-arc orbit determination (OD) when pursuing high accuracy. In this paper, from a data-driven perspective, we propose a novel method for space-based OD based on distribution regression (DR), which is called the weighting distribution-regression OD (WDR-OD) method. The OD is treated as a regression process, which is learned from abundant observation data and the corresponding orbits of known objects. First, we propose the structure of space-based OD samples, wherein the feature variables with a weighting matrix are introduced to enhance prediction accuracy. Second, a two-stage sampled learning theory is employed to learn the mapping from measurements to objects' orbit through kernel mean embedding. The proposed method is experimentally compared with the improved Laplace method and shows greater robustness in measurements with white Gaussian noise (WGN) and colored noise. The positional RMSE reaches 0.8793 km with WGN and 1.6972 km with colored noise, which are significantly smaller than the corresponding Laplace method's 5.0804 km and 14.8132 km. Furthermore, we propose a RIP-based ROMP algorithm to provide the theoretical bound of sparsity and then to pursue a sparse solution. Although the positional RMSE increases to 1.6554 km in the sparse method, it shrinks the 90% to 93% nonzero elements of the coefficients matrix to zero, which is helpful in reducing the computing load, and it meaningfully extends the application domain of the WDR-OD method.

Space-Based Relative Orbit Estimation Using Information Sharing and the Consensus Kalman Filter

The consensus extended Kalman filter is employed for the problem of multiple chief spacecraft cooperatively estimating the orbit of one deputy spacecraft in a connected communication network. The improvements on system observability by using consensus estimation are shown by the analytical observability criteria using Lie derivatives for the case of angles-only measurements. The conditions for the ambiguous orbits to persist in the presence of consensus feedback are obtained for range-only measurements. Through numerical results, it is shown that use of the consensus estimation strategy can improve the convergence performance of the filter and increase the system observability by excluding ambiguous orbits associated with angles-only and range-only measurements. The benefits of faster convergence and consensus speed by using communication networks with more connections are also illustrated. The effects of periodic link failures are studied through multiple numerical examples. Finally, the numerical observability measures, observability Gramian, and associated observability index and condition number are used to verify the advantages of the proposed consensus estimation strategy.

Observability of a Thrusting/Ballistic Trajectory in 3-D From a Single Fixed Passive Sensor

In previous works, it has been shown that the motion of a thrusting/ballistic object in a three-dimensional (3-D) space is observable with 2-D measurements from a stationary passive sensor. The measurements can either start from the launch point or can be delayed. The observability was investigated by testing the invertibility of the Fisher Information Matrix (FIM). This paper discusses the observability via the uniqueness of the target state vector for a certain sequence of 2-d angle-only measurements from a single fixed passive sensor. The discussion starts with polynomial motion from which the results are extended to nonlinear thrusting/ballistic motion. To illustrate the observability of a thrusting/ballistic target, the estimation problem of such a target with delayed acquisition after burn-out time is solved with a 7-d parameter vector (velocity vector azimuth angle and elevation angle, drag coefficient, 3-d acquisition position, and target speed at the acquisition time). A maximum likelihood (ML) estimator is used for the motion parameter estimation at acquisition time. The impact point prediction is then carried out with the ML estimates. The FIM is investigated via simulations to prove the observability numerically and the maximum likelihood estimator is shown to be efficient.

Flight demonstration of spaceborne real-time angles-only navigation to a noncooperative target in low earth orbit

The paper describes the onboard vision-based navigation software employed during the DLR's AVANTI experiment and presents its flight performance. The navigation task relies on a dedicated target detection module in charge of extracting the line-of-sight observations from the images taken by a monocular camera. The recognition of the target is based on the kinematic identification of its trajectory over a sequence of images. Subsequently, the resulting angles-only measurements are validated dynamically and processed by an Extended Kalman Filter to derive in real-time the relative state estimate. A computationally light implementation of the filter is achieved by the use of an analytical relative motion model. Two autonomous rendezvous to a noncooperative object have been performed in orbit, first from 13 km to 1 km separation, then from 3 km to 50 m. Despite the poor visibility conditions and the strong orbit perturbations encountered at low altitude, the filter was able to supply the onboard guidance and control algorithms with accurate and reliable relative state estimation, enabling thus a safe and smooth approach.

Optical measurement aided autonomous navigation for pinpoint Mars landing

The pinpoint planetary landing is an essential technology for future planetary exploration missions. An optical measurement aided autonomous navigation (OMAAN) scheme is proposed to improve autonomous navigation accuracy during the Mars landing. Based on Inertial Measurement Unit (IMU), states of the lander can be determined precisely using the angles-only measurements to one landmark obtained by optical equipment. Unlike other autonomous navigation based on optical measurements, this navigation scheme only needs one landmark and takes full advantage of historical measurements. On-board computation can be reduced largely and accuracy of state estimation also can be improved. To confirm the validity and the performance of proposed navigation scheme, simulations using batch least squares with differential correction are performed. As a result, the proposed method is feasible, and observation duration and measurement accuracy are vital important for pinpoint landing. Compared with other autonomous navigation methods, this method has the same navigation accuracy as the other methods based on multiple landmarks whose landing position on the order of 20m.

On Kalman Filtering and Observability in Nonlinear Sequential Relative Orbit Estimation

To study the effects of incorporating higher-order nonlinearities with different measurement types on observability and filter performance in sequential relative orbit estimation, an extended Kalman filter is implemented with four dynamic models of different orders of nonlinearity and either angles- or range-only measurements. The Kalman filtering studies compare the filter performance for these estimation scenarios and illustrate the lack of observability when using the first-order (Hill–Clohessy–Wiltshire) dynamic model, as well as the benefits of using higher-order nonlinear dynamic models on increased observability and faster filter convergence. Observability properties are then studied analytically using Lie derivatives, including the new concept of an “observability angle” for the case of angles-only measurements, as well as numerically with two observability measures obtained from the observability gramian for both angles- and range-only measurements and for a variety of different relative orbit scenarios, including for both circular and elliptic chief orbits. The analytical and numerical observability results are found to agree and to qualitatively confirm and verify the Kalman filtering studies.

Event-Triggered Model Predictive Control for target tracking with angles-only measurements

This study presents the definition of a new guidance law for a UAV tracking a mobile target using angles-only measurements. The objective of the guidance scheme is to follow the target so that it remains at a given range while enhancing its observability. It is based on a combination of two simple complementary behavior guidance laws: pure pursuit and lateral maneuver. Determination of suitable maneuvers is performed using Model Predictive Control approach using a function of the angles-rate as the optimality criterion. In order to reduce energy consumption, an additional event-triggered strategy, based on the evaluation of estimation uncertainty is used to determine the time when maneuvers are needed. Numerical simulations are presented to evaluate the resulting performances of systematic and event-triggered guidance. Event-triggered MPC strategy is shown to keep an equivalent estimation accuracy as systematic maneuvers while saving energy and computation budget.

Angles-Only Initial Relative Orbit Determination Performance Analysis using Cylindrical Coordinates

The solution of the initial relative orbit determination problem using angles-only measurements is important for orbital proximity operations, satellite inspection and servicing, and the identification of unknown space objects in similar orbits. In this paper, a preliminary relative orbit determination performance analysis is conducted utilizing the linearized relative orbital equations of motion in cylindrical coordinates. The relative orbital equations of motion in cylindrical coordinates are rigorously derived in several forms included the exact nonlinear two-body differential equations of motion, the linear-time-varying differential equations of motion for an elliptical orbit chief, and the linear-time-invariant differential equations of motion for a circular orbit chief. Using the nonlinear angles-only measurement equation in cylindrical coordinates, evidence of full-relative-state observability is found, contrary to the range observability problem exhibited in Cartesian coordinates. Based on these results, a geometric approach to assess initial relative orbit determination performance is formulated. To facilitate a better understanding of the problem, the focus is on the 2-dimensional initial orbit determination problem. The results clearly show the dependence of the relative orbit determination performance on the geometry of the relative motion and on the time-interval between observations. Analysis is conducted for leader-follower orbits and flyby orbits where the deputy passes directly above or below the chief.

Nonlinear Observability for Relative Orbit Determination with Angles-Only Measurements

This paper presents nonlinear observability criteria for the relative orbital dynamics represented by the solutions of the two-body problem. It is assumed that a chief is on a circular orbit with a prescribed orbital radius, and it measures lines-of-sight toward a deputy only. A differential geometric method, based on the Lie derivatives, is used to derive sufficient conditions for observability of the orbital properties of the deputy. It is shown that under certain geometric conditions on the relative configuration between the chief and the deputy, the nonlinear relative motion is observable from angles-only measurements. The second part of this paper presents a quantitative measure of observability for the relative orbits, and it is formulated by generalizing the observability Gramian of linear dynamic systems. An extended Kalman filter is also developed to numerically illustrate the observability of nonlinear relative orbits with angles-only measurements and to show correspondence between the proposed observability measure and filtered solution accuracy.

Observability analysis and enhancement of radome aberration estimation with line-of-sight angle-only measurement

This paper deals with the observability analysis and enhancement of target state variables and radome aberration estimation only with line-of-sight (LOS) angle measurement. The local observability conditions at an arbitrary time are derived, and a new maneuver algorithm, revised adaptive intermittent maneuver (RAIM), is introduced to enhance the estimation performance of the system. Simulations are conducted to verify the observability conditions and analyze the performance and characteristics of RAIM.

Observability Analysis on Radome Aberration Estimation

The observability of radome aberration estimation with line-of-sight (LOS) angle-only measurement is studied in this paper. When missile seeker can only measure LOS angle of target, target state variables are required to be estimated with this measurement for accurate missile guidance. Since LOS angle measurement contains error, caused by radome attached at missile nose and called radome aberration angle, the state estimation performance is degenerated. Thus, the radome aberration effects should be estimated and compensated for accurate target state variable estimation. The radome aberration angle is modeled as the product of radome slope and gimbal angle. The observability of state variables and radome slope estimation is analyzed for the relative dynamics with stationary target model in polar coordinate system. Simulations are conducted to verify the observability conditions obtained in this paper.

Noncooperative Rendezvous Using Angles-Only Optical Navigation: System Design and Flight Results

This paper presents system design and on-orbit results from the Advanced Rendezvous Demonstration using Global Positioning System and Optical Navigation (ARGON). ARGON has been conducted during the extended phase of the Prototype Research Instruments and Space Mission Technology Advancement (PRISMA) mission in April 2012. It represents one of the first rendezvous technology experiments using line of sight measurements from optical navigation and was motivated by the new generation of on-orbit-servicing and debris-removal missions which are discussed at national and international level. Its primary goal was to demonstrate the capability of an active servicer spacecraft to safely approach and rendezvous a non-cooperative passive client using angles-only optical navigation in a ground-in-the-loop fashion. To this end, a dedicated flight dynamics system has been developed for routine processing of the camera images collected on-board, for estimation of the relative orbit of the servicer with respect to the client vehicle, for maneuver planning and commanding. Despite the inherent difficulty to estimate the actual range to target through angles-only measurements and the constraints affecting the communication between ground-station and servicer, ARGON demonstrated an efficient and safe rendezvous from 30 km to the final hold point at 3 km mean separation selected before experiment start. This was possible due to the achieved relative navigation accuracy in combination with a guidance strategy based on the relative eccentricity/inclination vector separation method. As shown in the paper, the availability of independent and precise navigation information from carrier-phase differential Global Positioning System techniques gave the possibility to properly evaluate the achieved performance and cross-compare different relative navigation sensors after the conclusion of the technology demonstration.

Joint Registration and State Estimation for Two-Station Passive Tracking

Registration and nonlinearity are two crucial factors affecting the performance of the two-station passive locating system. In this paper, an online joint registration and data fusion algorithm is proposed to estimate the sensor bias and target state simultaneously using the angle-only measurements from the two ownship stations. The system model of the passive radar is firstly developed followed by the expectation-maximization (EM) approach dealing with the derivation of maximum likelihood (ML) function of the complete data. The unscented Kalman filter (UKF) is chosen to alleviate the influence caused by nonlinearity generated in the measurement function. Computer simulation shows that the proposed method is effective and reliable for this specific tracking scenario.

Passive Angle-Only Maneuvering Target Tracking Using Federated Filter in Hybrid Coordinates

An approach to federated filtering in hybrid coordinates is presented for a maneuvering target tracking through a multi-passive-sensor system. This new design accommodates a group of extended Kalman filters that the sensor-dedicated local processors process angle-only measurements extracted from a number of associated maneuverable aircrafts with onboard direction finders. Each filter utilizes the Reference Cartesian Coordinate (RCC) system for state and state covariance extrapolation and utilizes the modified spherical coordinate system for state and state covariance updating. In addition, a discrete-time state equation with the piecewise constant white target acceleration model is introduced into the RCC system for federated filtering. A weighted least squares estimator is used for the global processor aggregating the local estimates to generate a global estimate in the local inertial Cartesian coordinate system. Simulation results show that proposed approach drastically improves the tracking accuracy.

Integrated track initialization and maintenance in heavy clutter using probabilistic data association

Target tracking in high clutter or low signal-to-noise ratio (SNR) environments is an important topic and still a challenging task. Joint Maximum Likelihood Probabilistic Data Association (JML-PDA) is a well-known batch method for initializing the tracks of very low observable (VLO) targets in heavy clutter environments. On the other hand, the Joint Probabilistic Data Association (JPDA) algorithm, which is commonly used for recursive track maintenance, lacks track initialization capability. In this paper, we propose a Combined JML-PDA and JPDA (CJML-PDA) algorithm to automatically initialize and maintain the tracks. This combined approach seamlessly shares information between the initialization and maintenance stages of the tracker. In contrast, in other batch-recursive approaches the initialization and maintenance algorithms operate rather independent of each other. The effectiveness of the proposed algorithm is demonstrated on a heavy clutter scenario and its performance is tested on closely-spaced (but resolved) targets with association ambiguity using angle-only measurements.