# Cramer-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 Cramer-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.  
uncertainty quantification; reliability analysis; localization; geometric dilution of precision (GDoP); Cramer-Rao lower bound (CRLB); two-way ranging (TWR); distributed coherence; unmanned aerial vehicles (UAVs); urban air mobility (UAM) +
Footnote †: journal: Applied sciences  
and commercial aircraft but also introduce significant system design challenges such as antenna placement [6], installation of new electronics [7], and the design of supporting infrastructure [8].  
To inform these aircraft design decisions, we derive novel 2D and 3D Cramer-Rao lower bounds (CRLBs) on position and orientation estimation in a multi-antenna TWR system. These bounds are both a closed-form and tractable and are specifically formulated as a function of the number of antennas, platform geometry, and geometric dilution of precision (GDoP) to inform design decisions under different mission requirements. These bounds directly relate the position and orientation estimation precision to the geometric distribution of antennas on the platform. This can be used to infer the expected performance of existing installations or to inform the required distribution for new installations to achieve an arbitrary precision target. Since these geometries are often summarized by a single GDoP value, we also trace the performance manifold between ranging precision, positioning precision, and GDoP to graphically summarize the bounds, which can be explored visually without performing any calculations. To improve the accessibility of these highly theoretical results, we also present simplified commentary on how to navigate the trade space between performance and practical design considerations. These results are summarized into simple \"best practices\" to provide intuitive, geometric interpretations of the bounds. An example multi-antenna TWR configuration to which these bounds apply is depicted in Figure 1.