Range-based relative localization using a fixed number of measurements

Abstract

Storms block the global positioning system and reduce visibility. The key to success in rescuing a wrecked robot is to find it first; however, in hostile territory, broadcasting distress signals is not an option. In this paper, we design an iterative algorithm for a rescue robot that identifies the relative pose of a wrecked robot using five range measurements and communications at different time points. Just as architects build real frameworks to hold up a building, we use the above information as building blocks to construct a hypothetical geometric framework, which consists of vertices bounded by straight, stiff edges. Then, the relative pose can be calculated from the coordinates of vertices in the framework. Theoretical analysis shows that measuring and communicating four or more times can form only one framework; thus, this framework provides one relative pose. Therefore, we use only one more measurement than the theoretical lower bound. To restrain noise, we suppress the fact that there are many possible frameworks. The algorithm finds a framework whose edge lengths are close to the expectations of the edge lengths of these frameworks. The algorithm is guaranteed to give an error-bounded estimate with an adjustable possibility rather than gradually stabilizing estimates or numerous possible estimates. The entire positioning scheme contains no matrix operations and does not require a polynomial toolbox. Additionally, this approach requires no prior information of the relative pose, no coordinated motion of the two robots, and no parameter must be adjusted through experience. In summary, a robot in distress can be located if five range measurements and communications can be performed between the two robots.

Appendix

See Figs. 6, 7 and 8.

Fig. 6

The histogram and kernel density estimation of error along x direction under different noise levels (full data). The true relative position and orientation of the target varied randomly within $\left[ -9,9 \right]$ and $\left[ 0,2\pi \right)$ for each experiment. In Algorithm 1, $\iota$ and $\varUpsilon$ were 1e-6 and 100, respectively. The stop condition $\tau$ was calculated from (42) with $m=1.93$. We completed 105 trials for each noise level

Fig. 7

The histogram and kernel density estimation of error along y direction under different noise levels (full data). The true relative position and orientation of the target varied randomly within $\left[ -9,9 \right]$ and $\left[ 0,2\pi \right)$ for each experiment. In Algorithm 1, $\iota$ and $\varUpsilon$ were 1e-6 and 100, respectively. The stop condition $\tau$ was calculated from (42) with $m=1.93$. We completed 105 trials for each noise level

Fig. 8

The histogram and kernel density estimation of error along $\phi$ direction under different noise levels (full data). The true relative position and orientation of the target varied randomly within $\left[ -9,9 \right]$ and $\left[ 0,2\pi \right)$ for each experiment. In Algorithm 1, $\iota$ and $\varUpsilon$ were 1e-6 and 100, respectively. The stop condition $\tau$ was calculated from (42) with $m=1.93$. We completed 105 trials for each noise level