Range-Based Rigid Body Localization With a Calibration Emitter for Mitigating Anchor Position Uncertainties

Rigid body localization (RBL) extends the traditional point positioning by determining not only the position but also the orientation of the body. This paper considers RBL using the range measurements between the sensors on the body and the outside anchors that have position uncertainties, where a calibration emitter at an inaccurate location is employed to mitigate the anchor position errors. This is a highly nonlinear constrained optimization problem as the rotation matrix defining the orientation must belong to the special orthogonal group. We first propose the use of rotation angles to parameterize the orientation, which enables the reduction of the problem to an unconstrained optimization. It is then sufficient to consider the unconstrained Cramér-Rao Lower Bound (CRLB) for analyzing the effects of anchor position errors and noisy calibration position on the localization performance, and such an analysis is prohibitive when using the constrained CRLB. We next propose and analyze three Maximum Likelihood estimators obtained from the unconstrained formulation that have different levels of approximation and complexity. We also enhance the divide and conquer and the semi-definite programming solution from the literature such that they can work with the presence of a calibration emitter and account for anchor position errors.