This paper proposes a nonlinear stochastic complementary filter design for inertial navigation that takes advantage of a fusion of Ultra-wideband (UWB) and Inertial Measurement Unit (IMU) technology ensuring semi-global uniform ultimate boundedness (SGUUB) of the closed loop error signals in mean square. The proposed filter estimates the vehicle's orientation, position, linear velocity, and noise covariance. The filter is designed to mimic the nonlinear navigation motion kinematics and is posed on a matrix Lie Group, the extended form of the Special Euclidean Group $\mathbb{SE}_{2}\left(3\right)$. The Lie Group based structure of the proposed filter provides unique and global representation avoiding singularity (a common shortcoming of Euler angles) as well as non-uniqueness (a common limitation of unit-quaternion). Unlike Kalman-type filters, the proposed filter successfully addresses IMU measurement noise considering unknown upper-bounded covariance. Although the navigation estimator is proposed in a continuous form, the discrete version is also presented. Moreover, the unit-quaternion implementation has been provided in the Appendix. Experimental validation performed using a publicly available real-world six-degrees-of-freedom (6 DoF) flight dataset obtained from an unmanned Micro Aerial Vehicle (MAV) illustrating the robustness of the proposed navigation technique. Keywords: Sensor-fusion, Inertial navigation, Ultra-wideband ranging, Inertial measurement unit, Stochastic differential equation, Stability, Localization, Observer design.

翻译：本文提出了一种用于惯性导航的非线性随机互补滤波器设计，该设计利用了超宽带（UWB）与惯性测量单元（IMU）技术的融合，确保了闭环误差信号在均方意义上的半全局一致最终有界性（SGUUB）。所提出的滤波器能够估计飞行器的姿态、位置、线速度及噪声协方差。该滤波器旨在模拟非线性导航运动学模型，并建立在矩阵李群——扩展形式的特殊欧几里得群$\mathbb{SE}_{2}\left(3\right)$之上。基于李群的滤波器结构提供了唯一且全局的表示，避免了奇异性（欧拉角的常见缺陷）以及非唯一性（单位四元数的常见局限）。与卡尔曼型滤波器不同，本滤波器成功处理了具有未知上界协方差的IMU测量噪声。尽管导航估计器以连续形式提出，本文也给出了其离散版本。此外，附录中提供了单位四元数的实现。通过使用公开的真实世界六自由度（6 DoF）微型无人机（MAV）飞行数据集进行的实验验证，展示了所提出导航技术的鲁棒性。关键词：传感器融合、惯性导航、超宽带测距、惯性测量单元、随机微分方程、稳定性、定位、观测器设计。