ABSTRACT Uncertainty propagation of dynamical systems is a common need across many domains and di... more ABSTRACT Uncertainty propagation of dynamical systems is a common need across many domains and disciplines. In nonlinear settings, the extended Kalman filter is the de facto standard propagation tool. Recently, a new class of propagation methods called sigma-point Kalman filters was introduced, which eliminated the need for explicit computation of tangent linear matrices. It has been shown in numerous cases that the actual uncertainty of a dynamical system cannot be accurately described by a Gaussian probability density function. This has motivated work in applying the Gaussian mixture model approach to better approximate the non-Gaussian probability density function. A limitation to existing approaches is that the number of Gaussian components of the Gaussian mixture model is fixed throughout the propagation of uncertainty. This limitation has made previous work ill-suited for nonstationary probability density functions either due to inaccurate representation of the probability density function or computational burden given a large number of Gaussian components that may not be needed. This work examines an improved method implementing a Gaussian mixture model that is adapted online via splitting of the Gaussian mixture model components triggered by an entropy-based detection of nonlinearity during the probability density function evolution. In doing so, the Gaussian mixture model approximation adaptively includes additional components as nonlinearity is encountered and can therefore be used to more accurately approximate the probability density function. This paper introduces this strategy, called adaptive entropy-based Gaussian-mixture information synthesis. The adaptive entropy-based Gaussian-mixture information synthesis method is demonstrated for its ability to accurately perform inference on two cases of uncertain orbital dynamical systems. The impact of this work for orbital dynamical systems is that the improved representation of the uncertainty of the space object can then be used more consistently for identification and tracking.
IEEE Transactions on Aerospace and Electronic Systems, 2012
The population of space objects (SOs) is tracked with sparse resources and thus tracking data are... more The population of space objects (SOs) is tracked with sparse resources and thus tracking data are only collected on these objects for a relatively small fraction of their orbit revolution (i.e., a short arc). This contributes to commonly mistagged or uncorrelated SOs and their associated trajectory uncertainties (covariances) to be less physically meaningful. The case of simply updating a catalogued SO is not treated here, but rather, the problem of reducing a set of collected short-arc data on an arbitrary deep space object without a priori information, and from the observations alone, determining its orbit to an acceptable level of accuracy. Fundamentally, this is a problem of data association and track correlation. The work presented here takes the concept of admissible regions and attributable vectors along with a multiple hypothesis filtering approach to determine how well these SO orbits can be recovered for short-arc data in near realtime and autonomously. While the methods presented here are explored with synthetic data, the basis for the simulations resides in actual data that has yet to be reduced, but whose characteristics are replicated as well as possible to yield results that can be expected using actual data.
Multi-sensor networks can alleviate the need for high-cost, high-accuracy, single-sensor tracking... more Multi-sensor networks can alleviate the need for high-cost, high-accuracy, single-sensor tracking in favor of an abundance of lower-cost and lower-accuracy sensors to perform multi-sensor tracking. The use of a multi-sensor network gives rise to the need for a fusion step that combines the outputs of all sensor nodes into a single probabilistic state description. When considering Gaussian uncertainties, the well-known covariance intersection technique may be used. In the more general, non-Gaussian case, covariance intersection is not sufficient. This paper examines a fusion method based on logarithmic opinion pools and develops algorithms for multi-sensor data fusion as well as investigates weight selection schemes for the opinion pool. The proposed fusion rules are applied to the tracking of a space object using multiple ground-based optical sensors. Non-Gaussian orbit determination methods are applied to each sensor individually, and the fusion rule is applied to the combined outputs of each sensor node. It is shown that the multi-sensor fusion rule leads to an increase of nearly two orders of magnitude in the position tracking accuracy as compared to the traditional single-sensor tracking method.
An approach for space object tracking utilizing particle filters is presented. New methods are de... more An approach for space object tracking utilizing particle filters is presented. New methods are developed and used to construct a robust constrained admissible region given a set of angles-only measurements, which is then approximated by a finite mixture distribution. This probabilistic initial orbit solution is refined using subsequent measurements through a particle filter approach. A proposal density is constructed based on an approximate Bayesian update and samples, or particles, are drawn from this proposed probability density to assign and correct weights, which form the basis for a more accurate Bayesian update. A finite mixture distribution is then fit to these weighted samples to reinitialize the cycle. This approach is compared to methods that approximate all probability densities as finite mixtures and process them as such. Both approaches utilize recursive estimation based on Bayesian statistics, but the benefits of densely sampling the support probability based on incoming measurements is weighed against remaining solely within the finite mixture approximation and performing measurement corrections there.
In classical orbital mechanics, the Lambert problem consists of computing the full state of an or... more In classical orbital mechanics, the Lambert problem consists of computing the full state of an orbiting body given two positions and the time elapsed between them. Similarly, a relative Lambert problem may cast to determine the relative velocity given two relative positions and the time elapsed between them. When considering linear relative dynamics, the relative Lambert problem may be solved analytically using matrix inversion. In this work, an analytic solution to the relative Lambert problem is derived using second-order Clohessy-Wiltshire equations and is applied directly in the relative frame. The problem is represented geometrically by the intersection of three quadric surfaces. The associated multivariate algebraic set is solved non-iteratively using Macaulay resultant expressions to high accuracy. The method is compared to the linear relative Lambert method as well as the classic Lambert routine.
Vol. 152 of Advances in the Astronautical Sciences, 2014
Finding zeros of algebraic sets is a fundamental problem in aerospace engineering computation. Ge... more Finding zeros of algebraic sets is a fundamental problem in aerospace engineering computation. Geometrically, this problem can often be represented by the intersection of multiple conic or quadric surfaces. A common example is GPS trilateration, which is geometrically given by the intersection of three spheres. In this work, Macaulay resultant expressions are used to compute the solutions of a set of multivariate polynomial expressions. Both two- and three-dimensional algebraic sets are considered, and examples of two geometric systems and their solutions are provided.
Vol. 152 of Advances in the Astronautical Sciences, 2014
Linear and unscented covariance analyses are implemented to compare the nominal performance of an... more Linear and unscented covariance analyses are implemented to compare the nominal performance of an extended and unscented Kalman filter for estimating the relative position and velocity of a deputy spacecraft using two camera line-of-sight measurements from a chief spacecraft, as well as the position, velocity, and attitude of the chief spacecraft. Dead reckoning of IMU data is used with external aiding from GPS and magnetometer measurements aboard the chief spacecraft. The ratio of nongravitational acceleration acting on the deputy spacecraft with respect to the chief spacecraft is included in order to stochastically include the effects of perturbations to relative motion. Camera specifications, such as resolution and baseline, are investigated to determine their effects on the nominal performance.
Vol. 152 of Advances in the Astronautical Sciences, 2014
The unobservability of space-based angles-only orbit determination can be mitigated by the inclus... more The unobservability of space-based angles-only orbit determination can be mitigated by the inclusion of angle measurements from a second optical sensor fixed at a known baseline on the observing spacecraft. Previous approaches to the problem have used these stereoscopic angles to triangulate the position of a second satellite at a given time step. However, due to the nonlinearity of stereo triangulation, zero-mean Gaussian noise of these measurements cannot be assumed. This work investigates a modified approach in which the uncertainty of both angle measurements is used to bound a region for all possible positions of the second satellite. A Gaussian mixture that represents uniform uncertainty across the bounded region for the position of the second object is constructed at two initial time steps. Linkage of the Gaussian mixtures is performed using a new second-order relative Lambert solver in order to formulate a full state probability density function that can be further refined through processing subsequent measurement data in a Bayesian framework.
Vol. 152 of Advances in the Astronautical Sciences, 2014
A comparison between common coordinate systems used for state representation in orbital mechanics... more A comparison between common coordinate systems used for state representation in orbital mechanics is presented for track initialization in orbit determination and follow-on tracking utilizing optical angles-only measurements. A Gaussian mixture parameterized probability density function representing uniform uncertainty across all possible Earth-bound constrained orbits is constructed. This distribution is mapped into each coordinate system, propagated forward in time, and refined via a Bayesian filter. Performance measures related to uncertainty characterization are applied to judge the efficacy of the method in each coordinate system.
Vol. 152 of Advances in the Astronautical Sciences, 2014
In classical orbital mechanics, the Lambert problem consists of computing the full state of an or... more In classical orbital mechanics, the Lambert problem consists of computing the full state of an orbiting body given two positions and the time elapsed between them. Similar analytic methods exist for the relative motion problem, but rely on linearized equations of motion and matrix inversion. In this work, an analytic solution to the “relative” Lambert problem is derived using second-order Clohessy-Wiltshire equations and is applied directly in the relative frame. The problem is represented geometrically by the intersection of three quadric surfaces. The associated multivariate algebraic set is solved non-iteratively using Macaulay resultant expressions to high accuracy.
The most complete description of the state of a system at any time is given by knowledge of the p... more The most complete description of the state of a system at any time is given by knowledge of the probability density function, which describes the locus of possible states conditioned on any available measurement information. When employing optical data, the concept of the admissible region provides a physics-based region of the range/range-rate space that produces Earth-bound orbit solutions. This work develops a method that employs a probabilistic interpretation of the admissible region and approximates the admissible region by a Gaussian mixture to formulate an initial orbit determination solution. The Gaussian mixture representation of the probability density function is then forecast and updated with subsequent data to iteratively refine the region of uncertainty. Simulation results are presented using synthetic data over a range of orbits, in which it is shown that the new method is consistently able to initialize a probabilistic orbit solution and provide iterative refinement via follow-on tracking.
Uncertainty propagation of dynamical systems is a common need across many domains and disciplines... more Uncertainty propagation of dynamical systems is a common need across many domains and disciplines. In nonlinear settings, the extended Kalman filter is the de facto standard propagation tool. Recently, a new class of propagation methods called sigma-point Kalman filters was introduced, which eliminated the need for explicit computation of tangent linear matrices. It has been shown in numerous cases that the actual uncertainty of a dynamical system cannot be accurately described by a Gaussian probability density function. This has motivated work in applying the Gaussian mixture model approach to better approximate the non-Gaussian probability density function. A limitation to existing approaches is that the number of Gaussian components of the Gaussian mixture model is fixed throughout the propagation of uncertainty. This limitation has made previous work ill-suited for nonstationary probability density functions either due to inaccurate representation of the probability density function or computational burden given a large number of Gaussian components that may not be needed. This work examines an improved method implementing a Gaussian mixture model that is adapted online via splitting of the Gaussian mixture model components triggered by an entropy-based detection of nonlinearity during the probability density function evolution. In doing so, the Gaussian mixture model approximation adaptively includes additional components as nonlinearity is encountered and can therefore be used to more accurately approximate the probability density function. This paper introduces this strategy, called adaptive entropy-based Gaussian-mixture information synthesis. The adaptive entropy-based Gaussian-mixture information synthesis method is demonstrated for its ability to accurately perform inference on two cases of uncertain orbital dynamical systems. The impact of this work for orbital dynamical systems is that the improved representation of the uncertainty of the space object can then be used more consistently for identification and tracking.
ABSTRACT Uncertainty propagation of dynamical systems is a common need across many domains and di... more ABSTRACT Uncertainty propagation of dynamical systems is a common need across many domains and disciplines. In nonlinear settings, the extended Kalman filter is the de facto standard propagation tool. Recently, a new class of propagation methods called sigma-point Kalman filters was introduced, which eliminated the need for explicit computation of tangent linear matrices. It has been shown in numerous cases that the actual uncertainty of a dynamical system cannot be accurately described by a Gaussian probability density function. This has motivated work in applying the Gaussian mixture model approach to better approximate the non-Gaussian probability density function. A limitation to existing approaches is that the number of Gaussian components of the Gaussian mixture model is fixed throughout the propagation of uncertainty. This limitation has made previous work ill-suited for nonstationary probability density functions either due to inaccurate representation of the probability density function or computational burden given a large number of Gaussian components that may not be needed. This work examines an improved method implementing a Gaussian mixture model that is adapted online via splitting of the Gaussian mixture model components triggered by an entropy-based detection of nonlinearity during the probability density function evolution. In doing so, the Gaussian mixture model approximation adaptively includes additional components as nonlinearity is encountered and can therefore be used to more accurately approximate the probability density function. This paper introduces this strategy, called adaptive entropy-based Gaussian-mixture information synthesis. The adaptive entropy-based Gaussian-mixture information synthesis method is demonstrated for its ability to accurately perform inference on two cases of uncertain orbital dynamical systems. The impact of this work for orbital dynamical systems is that the improved representation of the uncertainty of the space object can then be used more consistently for identification and tracking.
IEEE Transactions on Aerospace and Electronic Systems, 2012
The population of space objects (SOs) is tracked with sparse resources and thus tracking data are... more The population of space objects (SOs) is tracked with sparse resources and thus tracking data are only collected on these objects for a relatively small fraction of their orbit revolution (i.e., a short arc). This contributes to commonly mistagged or uncorrelated SOs and their associated trajectory uncertainties (covariances) to be less physically meaningful. The case of simply updating a catalogued SO is not treated here, but rather, the problem of reducing a set of collected short-arc data on an arbitrary deep space object without a priori information, and from the observations alone, determining its orbit to an acceptable level of accuracy. Fundamentally, this is a problem of data association and track correlation. The work presented here takes the concept of admissible regions and attributable vectors along with a multiple hypothesis filtering approach to determine how well these SO orbits can be recovered for short-arc data in near realtime and autonomously. While the methods presented here are explored with synthetic data, the basis for the simulations resides in actual data that has yet to be reduced, but whose characteristics are replicated as well as possible to yield results that can be expected using actual data.
Multi-sensor networks can alleviate the need for high-cost, high-accuracy, single-sensor tracking... more Multi-sensor networks can alleviate the need for high-cost, high-accuracy, single-sensor tracking in favor of an abundance of lower-cost and lower-accuracy sensors to perform multi-sensor tracking. The use of a multi-sensor network gives rise to the need for a fusion step that combines the outputs of all sensor nodes into a single probabilistic state description. When considering Gaussian uncertainties, the well-known covariance intersection technique may be used. In the more general, non-Gaussian case, covariance intersection is not sufficient. This paper examines a fusion method based on logarithmic opinion pools and develops algorithms for multi-sensor data fusion as well as investigates weight selection schemes for the opinion pool. The proposed fusion rules are applied to the tracking of a space object using multiple ground-based optical sensors. Non-Gaussian orbit determination methods are applied to each sensor individually, and the fusion rule is applied to the combined outputs of each sensor node. It is shown that the multi-sensor fusion rule leads to an increase of nearly two orders of magnitude in the position tracking accuracy as compared to the traditional single-sensor tracking method.
An approach for space object tracking utilizing particle filters is presented. New methods are de... more An approach for space object tracking utilizing particle filters is presented. New methods are developed and used to construct a robust constrained admissible region given a set of angles-only measurements, which is then approximated by a finite mixture distribution. This probabilistic initial orbit solution is refined using subsequent measurements through a particle filter approach. A proposal density is constructed based on an approximate Bayesian update and samples, or particles, are drawn from this proposed probability density to assign and correct weights, which form the basis for a more accurate Bayesian update. A finite mixture distribution is then fit to these weighted samples to reinitialize the cycle. This approach is compared to methods that approximate all probability densities as finite mixtures and process them as such. Both approaches utilize recursive estimation based on Bayesian statistics, but the benefits of densely sampling the support probability based on incoming measurements is weighed against remaining solely within the finite mixture approximation and performing measurement corrections there.
In classical orbital mechanics, the Lambert problem consists of computing the full state of an or... more In classical orbital mechanics, the Lambert problem consists of computing the full state of an orbiting body given two positions and the time elapsed between them. Similarly, a relative Lambert problem may cast to determine the relative velocity given two relative positions and the time elapsed between them. When considering linear relative dynamics, the relative Lambert problem may be solved analytically using matrix inversion. In this work, an analytic solution to the relative Lambert problem is derived using second-order Clohessy-Wiltshire equations and is applied directly in the relative frame. The problem is represented geometrically by the intersection of three quadric surfaces. The associated multivariate algebraic set is solved non-iteratively using Macaulay resultant expressions to high accuracy. The method is compared to the linear relative Lambert method as well as the classic Lambert routine.
Vol. 152 of Advances in the Astronautical Sciences, 2014
Finding zeros of algebraic sets is a fundamental problem in aerospace engineering computation. Ge... more Finding zeros of algebraic sets is a fundamental problem in aerospace engineering computation. Geometrically, this problem can often be represented by the intersection of multiple conic or quadric surfaces. A common example is GPS trilateration, which is geometrically given by the intersection of three spheres. In this work, Macaulay resultant expressions are used to compute the solutions of a set of multivariate polynomial expressions. Both two- and three-dimensional algebraic sets are considered, and examples of two geometric systems and their solutions are provided.
Vol. 152 of Advances in the Astronautical Sciences, 2014
Linear and unscented covariance analyses are implemented to compare the nominal performance of an... more Linear and unscented covariance analyses are implemented to compare the nominal performance of an extended and unscented Kalman filter for estimating the relative position and velocity of a deputy spacecraft using two camera line-of-sight measurements from a chief spacecraft, as well as the position, velocity, and attitude of the chief spacecraft. Dead reckoning of IMU data is used with external aiding from GPS and magnetometer measurements aboard the chief spacecraft. The ratio of nongravitational acceleration acting on the deputy spacecraft with respect to the chief spacecraft is included in order to stochastically include the effects of perturbations to relative motion. Camera specifications, such as resolution and baseline, are investigated to determine their effects on the nominal performance.
Vol. 152 of Advances in the Astronautical Sciences, 2014
The unobservability of space-based angles-only orbit determination can be mitigated by the inclus... more The unobservability of space-based angles-only orbit determination can be mitigated by the inclusion of angle measurements from a second optical sensor fixed at a known baseline on the observing spacecraft. Previous approaches to the problem have used these stereoscopic angles to triangulate the position of a second satellite at a given time step. However, due to the nonlinearity of stereo triangulation, zero-mean Gaussian noise of these measurements cannot be assumed. This work investigates a modified approach in which the uncertainty of both angle measurements is used to bound a region for all possible positions of the second satellite. A Gaussian mixture that represents uniform uncertainty across the bounded region for the position of the second object is constructed at two initial time steps. Linkage of the Gaussian mixtures is performed using a new second-order relative Lambert solver in order to formulate a full state probability density function that can be further refined through processing subsequent measurement data in a Bayesian framework.
Vol. 152 of Advances in the Astronautical Sciences, 2014
A comparison between common coordinate systems used for state representation in orbital mechanics... more A comparison between common coordinate systems used for state representation in orbital mechanics is presented for track initialization in orbit determination and follow-on tracking utilizing optical angles-only measurements. A Gaussian mixture parameterized probability density function representing uniform uncertainty across all possible Earth-bound constrained orbits is constructed. This distribution is mapped into each coordinate system, propagated forward in time, and refined via a Bayesian filter. Performance measures related to uncertainty characterization are applied to judge the efficacy of the method in each coordinate system.
Vol. 152 of Advances in the Astronautical Sciences, 2014
In classical orbital mechanics, the Lambert problem consists of computing the full state of an or... more In classical orbital mechanics, the Lambert problem consists of computing the full state of an orbiting body given two positions and the time elapsed between them. Similar analytic methods exist for the relative motion problem, but rely on linearized equations of motion and matrix inversion. In this work, an analytic solution to the “relative” Lambert problem is derived using second-order Clohessy-Wiltshire equations and is applied directly in the relative frame. The problem is represented geometrically by the intersection of three quadric surfaces. The associated multivariate algebraic set is solved non-iteratively using Macaulay resultant expressions to high accuracy.
The most complete description of the state of a system at any time is given by knowledge of the p... more The most complete description of the state of a system at any time is given by knowledge of the probability density function, which describes the locus of possible states conditioned on any available measurement information. When employing optical data, the concept of the admissible region provides a physics-based region of the range/range-rate space that produces Earth-bound orbit solutions. This work develops a method that employs a probabilistic interpretation of the admissible region and approximates the admissible region by a Gaussian mixture to formulate an initial orbit determination solution. The Gaussian mixture representation of the probability density function is then forecast and updated with subsequent data to iteratively refine the region of uncertainty. Simulation results are presented using synthetic data over a range of orbits, in which it is shown that the new method is consistently able to initialize a probabilistic orbit solution and provide iterative refinement via follow-on tracking.
Uncertainty propagation of dynamical systems is a common need across many domains and disciplines... more Uncertainty propagation of dynamical systems is a common need across many domains and disciplines. In nonlinear settings, the extended Kalman filter is the de facto standard propagation tool. Recently, a new class of propagation methods called sigma-point Kalman filters was introduced, which eliminated the need for explicit computation of tangent linear matrices. It has been shown in numerous cases that the actual uncertainty of a dynamical system cannot be accurately described by a Gaussian probability density function. This has motivated work in applying the Gaussian mixture model approach to better approximate the non-Gaussian probability density function. A limitation to existing approaches is that the number of Gaussian components of the Gaussian mixture model is fixed throughout the propagation of uncertainty. This limitation has made previous work ill-suited for nonstationary probability density functions either due to inaccurate representation of the probability density function or computational burden given a large number of Gaussian components that may not be needed. This work examines an improved method implementing a Gaussian mixture model that is adapted online via splitting of the Gaussian mixture model components triggered by an entropy-based detection of nonlinearity during the probability density function evolution. In doing so, the Gaussian mixture model approximation adaptively includes additional components as nonlinearity is encountered and can therefore be used to more accurately approximate the probability density function. This paper introduces this strategy, called adaptive entropy-based Gaussian-mixture information synthesis. The adaptive entropy-based Gaussian-mixture information synthesis method is demonstrated for its ability to accurately perform inference on two cases of uncertain orbital dynamical systems. The impact of this work for orbital dynamical systems is that the improved representation of the uncertainty of the space object can then be used more consistently for identification and tracking.
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Papers by Kyle DeMars