This paper provides a pair of summation formulas for a kind of combinatorial series involvingak＋b m as a factor of the summand. The construction of formulas is based on a certain series transformation formula [2, 7, ...This paper provides a pair of summation formulas for a kind of combinatorial series involvingak＋b m as a factor of the summand. The construction of formulas is based on a certain series transformation formula [2, 7, 9] and by making use of the C-numbers [3]. Various consequences and examples including several remarkable classic identities are presented to illustrate some applications of the formulas obtained.展开更多
This paper presents new methods for spacecraft relative pose estimation using the Unscented Kalman Filter(UKF),taking into account non-additive process and measurement noises.A twistor model is employed to represent t...This paper presents new methods for spacecraft relative pose estimation using the Unscented Kalman Filter(UKF),taking into account non-additive process and measurement noises.A twistor model is employed to represent the spacecraft's relative 6-DOF motion of the chaser with respect to the target,expressed in the chaser body frame.The twistor model utilizes Modified Rodrigues Parameters(MRPs)to represent attitude with a minimal number of parameters,eliminating the need for the normalization constraint that exists in the quaternion-based model.Additionally,it incorporates both relative position and attitude in a single model,addressing kinematic coupling of states and simplifying the estimator design.Despite numerous existing pose estimation algorithms,many rely on the simplification of additive noise assumptions.This work enhances the robustness and improves the convergence of non-additive noise algorithms by deriving two methods to accurately approximate process and measurement noise covariance matrices for systems with non-additive noises.The first method utilizes the Stirling Interpolation Formula(SIF)to obtain equivalent process and measurement noise covariance matrices.The second method employs State Noise Compensation(SNC)to derive the equivalent process noise covariance matrix and uses SIF to compute the equivalent measurement noise covariance matrix.These methods are integrated into the UKF framework for estimating the relative pose of spacecraft in proximity operations,demonstrated through two scenarios:one with a cooperative target using Position Sensing Diodes(PSDs)and another with an uncooperative target using LiDAR for 3-D imaging.The effectiveness of these methods is validated against others in the literature through Monte Carlo simulations,showcasing their faster convergence and robust performance.展开更多
Two-parameter gamma distributions are widely used in liability theory, lifetime data analysis, financial statistics, and other areas. Finite mixtures of gamma distributions are their natural extensions, and they are p...Two-parameter gamma distributions are widely used in liability theory, lifetime data analysis, financial statistics, and other areas. Finite mixtures of gamma distributions are their natural extensions, and they are particularly useful when the population is suspected of heterogeneity. These distributions are successfully employed in various applications, but many researchers falsely believe that the maximum likelihood estimator of the mixing distribution is consistent. Similarly to finite mixtures of normal distributions, the likelihood function under finite gamma mixtures is unbounded. Because of this, each observed value leads to a global maximum that is irrelevant to the true distribution. We apply a seemingly negligible penalty to the likelihood according to the shape parameters in the fitted model. We show that this penalty restores the consistency of the likelihoodbased estimator of the mixing distribution under finite gamma mixture models. We present simulation results to validate the consistency conclusion, and we give an example to illustrate the key points.展开更多
基金Supported by the National Natural Science Foundation of China (Grant No.11071183)
文摘This paper provides a pair of summation formulas for a kind of combinatorial series involvingak＋b m as a factor of the summand. The construction of formulas is based on a certain series transformation formula [2, 7, 9] and by making use of the C-numbers [3]. Various consequences and examples including several remarkable classic identities are presented to illustrate some applications of the formulas obtained.
基金the startup and UPAR grants funded by College of Engineering at United Arab Emirates University(UAEU).The grant codes are G00003527 and G00004562.
文摘This paper presents new methods for spacecraft relative pose estimation using the Unscented Kalman Filter(UKF),taking into account non-additive process and measurement noises.A twistor model is employed to represent the spacecraft's relative 6-DOF motion of the chaser with respect to the target,expressed in the chaser body frame.The twistor model utilizes Modified Rodrigues Parameters(MRPs)to represent attitude with a minimal number of parameters,eliminating the need for the normalization constraint that exists in the quaternion-based model.Additionally,it incorporates both relative position and attitude in a single model,addressing kinematic coupling of states and simplifying the estimator design.Despite numerous existing pose estimation algorithms,many rely on the simplification of additive noise assumptions.This work enhances the robustness and improves the convergence of non-additive noise algorithms by deriving two methods to accurately approximate process and measurement noise covariance matrices for systems with non-additive noises.The first method utilizes the Stirling Interpolation Formula(SIF)to obtain equivalent process and measurement noise covariance matrices.The second method employs State Noise Compensation(SNC)to derive the equivalent process noise covariance matrix and uses SIF to compute the equivalent measurement noise covariance matrix.These methods are integrated into the UKF framework for estimating the relative pose of spacecraft in proximity operations,demonstrated through two scenarios:one with a cooperative target using Position Sensing Diodes(PSDs)and another with an uncooperative target using LiDAR for 3-D imaging.The effectiveness of these methods is validated against others in the literature through Monte Carlo simulations,showcasing their faster convergence and robust performance.
基金supported by Grants from One Thousand Talents at Yunnan Universitya Discovery Grant from the Natural Sciences and Engineering Research Council of Canada (Grant No. RGPIN–2014–03743)
文摘Two-parameter gamma distributions are widely used in liability theory, lifetime data analysis, financial statistics, and other areas. Finite mixtures of gamma distributions are their natural extensions, and they are particularly useful when the population is suspected of heterogeneity. These distributions are successfully employed in various applications, but many researchers falsely believe that the maximum likelihood estimator of the mixing distribution is consistent. Similarly to finite mixtures of normal distributions, the likelihood function under finite gamma mixtures is unbounded. Because of this, each observed value leads to a global maximum that is irrelevant to the true distribution. We apply a seemingly negligible penalty to the likelihood according to the shape parameters in the fitted model. We show that this penalty restores the consistency of the likelihoodbased estimator of the mixing distribution under finite gamma mixture models. We present simulation results to validate the consistency conclusion, and we give an example to illustrate the key points.
