Relative multiplicative extended Kalman filter for observable GPS-denied navigation

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.