For a team of robots to work collaboratively, it is crucial that each robot have the ability to determine the position of their neighbors, relative to themselves, in order to execute tasks autonomously. This letter presents an algorithm for determining the three-dimensional relative position between two mobile robots, each using nothing more than a single ultrawideband transceiver, an accelerometer, a rate gyro, and a magnetometer. A sliding window filter estimates the relative position at selected keypoints by combining the distance measurements with acceleration estimates, which each agent computes using an on-board attitude estimator. The algorithm is appropriate for real-time implementation, and has been tested in simulation and experiment, where it comfortably outperforms standard estimators. A positioning accuracy of less than 1 meter is achieved with inexpensive sensors. Index Terms-Localization; Range Sensing; Multi-Robot Systems I. INTRODUCTIONT HE ability to determine the relative position between two robots, or agents, is needed in many applications. For instance, multi-robot tasks such as collaborative exploration and mapping, search and rescue, and formation control, each require relative position information. Access to inter-robot distance measurements is becoming increasingly accessible due to technologies such as ultra-wideband radio (UWB). UWB in particular is attractive due to its low cost, low power, high accuracy, and ability to function in GPS-denied environments. Such advantages have even warranted the presence of UWB transceivers in smartphones and smartwatches, which could be used to provide position information of one user relative to another or, more generally, any other UWB transceiver. However, determining a full three-dimensional relative position estimate from a single distance measurement is impossible. There is an infinite set of relative positions that will produce the same distance measurement. Hence, to realize an observable relative positioning solution, more information is required.

The ability to accurately estimate the position of robotic agents relative to one another, in possibly GPS-denied environments, is crucial to execute collaborative tasks. Interagent range measurements are available at a low cost, due to technologies such as ultra-wideband radio. However, the task of three-dimensional relative position estimation using range measurements in multi-agent systems suffers from unobservabilities. This letter presents a sufficient condition for the observability of the relative positions, and satisfies the condition using a simple framework with only range measurements, an accelerometer, a rate gyro, and a magnetometer. The framework has been tested in simulation and in experiments, where 40-50 cm positioning accuracy is achieved using inexpensive off-the-shelf hardware.

The complex-step derivative approximation is a numerical differentiation technique that can achieve analytical accuracy, to machine precision, with a single function evaluation. In this paper, the complex-step derivative approximation is extended to be compatible with elements of matrix Lie groups. As with the standard complex-step derivative, the method is still able to achieve analytical accuracy, up to machine precision, with a single function evaluation. Compared to a central-difference scheme, the proposed complex-step approach is shown to have superior accuracy. The approach is applied to two different pose estimation problems, and is able to recover the same results as an analytical method when available.

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