Cramér–Rao Lower Bounds on 3D Position and Orientation Estimation in Distributed Ranging Systems

Abstract

As radio frequency (RF) hardware continues to improve, many technologies that were traditionally impractical have suddenly become viable alternatives to legacy systems. Two-way ranging (TWR) is often considered a poor positioning solution for airborne and other vehicular navigation systems due to its low precision, poor angular resolution, and precise timing requirements. With the advent of modern RF hardware and advanced processing techniques, however, modern studies have experimentally demonstrated TWR systems with an unprecedented, sub-centimeter ranging precision with low size, weight, power, and cost (SWaP-C) consumer-grade hardware. This technique enables a new class of positioning, navigation, and timing (PNT) capabilities for urban and commercial aircraft but also instigates new system design challenges such as antenna placement, installation of new electronics, and design of supporting infrastructure. To inform these aircraft design decisions, we derive 2D and 3D Cramér–Rao lower bounds (CRLBs) on position and orientation estimation in a multi-antenna TWR system. We specifically formulate these bounds as a function of the number of antennas, platform geometry, and geometric dilution of precision (GDoP) to inform aircraft design decisions under different mission requirements. We simulate the performance of several classic position and orientation estimators in this context to validate these bounds and to graphically depict the expected performance with respect to these design considerations. To improve the accessibility of these highly theoretical results, we also present a simplified discussion of how these bounds may be applied to common airborne applications and suggest best practices for using them to inform aircraft design decisions.