This work presents a multiplicative extended Kalman filter (MEKF) for estimating the relative state of a multirotor vehicle operating in a GPS-denied environment. The filter fuses data from an inertial measurement unit and altimeter with relative-pose updates from a keyframe-based visual odometry or laser scan-matching algorithm. Because the global position and heading states of the vehicle are unobservable in the absence of global measurements such as GPS, the filter in this article estimates the state with respect to a local frame that is colocated with the odometry keyframe. As a result, the odometry update provides nearly direct measurements of the relative vehicle pose, making those states observable. Recent publications have rigorously documented the theoretical advantages of such an observable parameterization, including improved consistency, accuracy, and system robustness, and have demonstrated the effectiveness of such an approach during prolonged multirotor flight tests. This article complements this prior work by providing a complete, self-contained, tutorial derivation of the relative MEKF, which has been thoroughly motivated but only briefly described to date. This article presents several improvements and extensions to the filter while clearly defining all quaternion conventions and properties used, including several new useful properties relating to error quaternions and their Euler-angle decomposition. Finally, this article derives the filter both for traditional dynamics defined with respect to an inertial frame, and for robocentric dynamics defined with respect to the vehicle’s body frame, and provides insights into the subtle differences that arise between the two formulations.

中文翻译：

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用于可观测 GPS 拒绝导航的相对乘法扩展卡尔曼滤波器

这项工作提出了一种乘法扩展卡尔曼滤波器 (MEKF)，用于估计在 GPS 拒绝环境中运行的多旋翼飞行器的相对状态。该过滤器将来自惯性测量单元和高度计的数据与来自基于关键帧的视觉里程计或激光扫描匹配算法的相对位姿更新相融合。由于在没有 GPS 等全球测量的情况下，车辆的全球位置和航向状态是不可观察的，因此本文中的滤波器根据与里程计关键帧共处一地的局部帧来估计状态。因此，里程计更新提供了相对车辆姿态的几乎直接测量，使这些状态可观察。最近的出版物严格记录了这种可观察参数化的理论优势，包括改进的一致性、准确性和系统鲁棒性，并在长时间的多旋翼飞行测试中证明了这种方法的有效性。本文通过提供相关 MEKF 的完整、自包含的教程派生来补充这项先前的工作，该派生已被彻底激发，但迄今为止仅对其进行了简要描述。本文介绍了滤波器的一些改进和扩展，同时清楚地定义了所有使用的四元数约定和属性，包括与误差四元数及其欧拉角分解相关的几个新的有用属性。最后，本文为相对于​​惯性框架定义的传统动力学和相对于车辆车身框架定义的以机器人为中心的动力学导出了过滤器，