In this paper, several estimation algorithms called the variational unscented Kalman filters (UKF-Vs) are proposed for matrix Lie groups. The proposed filters are inspired by the unscented Kalman filter in Euclidean space and they exhibit advantages over conventional methods, as the prediction step and the measurement update step are established on the Lie algebra and its dual space, which therefore avoids direct operations on highly nonlinear Lie group configuration spaces. Correspondingly, the proposed UKF-Vs exhibit significant improvements in terms of the estimation error and mean square error. This also makes it possible to construct a computationally efficient geometric integrator for the filtering dynamics. The obtained formulation is independent of the nonlinear Lie group state-space, and it sheds light on fully predicting and updating on the Lie algebra and its dual which are endowed with vector space structures. In particular, these formulations can avoid singularities or the well-known gimbal lock in the attitude estimation problem. Furthermore, the stochastic stability of the proposed filters is studied. The performances of the proposed filters are demonstrated for the satellite attitude estimation problem, which is an important benchmark from a control perspective. Numerical results show that the proposed UKF-Vs keep less computational complexity and perform significantly high accuracy compared with two unscented Kalman filters on Lie groups.

中文翻译：

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矩阵 Lie 群上的变分无迹卡尔曼滤波


在本文中，为矩阵李群提出了几种称为变分无迹卡尔曼滤波器 （UKF-Vs） 的估计算法。所提出的滤波器受到欧几里得空间中无迹卡尔曼滤波的启发，它们表现出优于传统方法的优势，因为预测步骤和测量更新步骤建立在 Lie 代数及其对偶空间上，因此避免了对高度非线性 Lie 群配置空间的直接操作。相应地，所提出的 UKF-Vs 在估计误差和均方误差方面表现出显着改进。这也使得为滤波动力学构建计算高效的几何积分器成为可能。所得公式独立于非线性 Lie 群状态空间，它阐明了对 Lie 代数及其具有向量空间结构的对偶的全面预测和更新。特别是，这些公式可以避免姿态估计问题中的奇点或众所周知的万向节锁。此外，还研究了所提出的滤波器的随机稳定性。该文针对卫星姿态估计问题演示了所提出的滤波器的性能，从控制角度来看，这是一个重要的基准。数值结果表明，与两个无迹卡尔曼滤波器相比，所提出的 UKF-Vs 在 Lie 群上的计算复杂度较低，并且具有显著的高精度。