  In real-world applications for automatic guided vehicle (AGV) navigation, the
positioning system based on the time-of-flight (TOF) measurements between
anchors and tags is confronted with the problem of insufficient measurements
caused by blockages to radio signals or lasers, etc. Mounting multiple tags at
different positions of the AGV to collect more TOFs is a feasible solution to
tackle this difficulty. Vehicle localization by exploiting the measurements
between multiple tags and anchors is a rigid body localization (RBL) problem,
which estimates both the position and attitude of the vehicle. However, the
state-of-the-art solutions to the RBL problem do not deal with missing
measurements, and thus will result in degraded localization availability and
accuracy in harsh environments. In this paper, different from these existing
solutions for RBL, we model this problem as a sensor network localization
problem with missing TOFs. To solve this problem, we propose a new efficient
RBL solution based on Euclidean distance matrix (EDM) completion, abbreviated
as ERBL-EDMC. Firstly, we develop a method to determine the upper and lower
bounds of the missing measurements to complete the EDM reliably, using the
known relative positions between tags and the statistics of the TOF
measurements. Then, based on the completed EDM, the global tag positions are
obtained from a coarse estimation followed by a refinement step assisted with
inter-tag distances. Finally, the optimal vehicle position and attitude are
obtained iteratively based on the estimated tag positions from the previous
step. Theoretical analysis and simulation results show that the proposed
ERBL-EDMC method effectively solves the RBL problem with incomplete
measurements. It obtains the optimal positioning results while maintaining low
computational complexity compared with the existing RBL methods based on
semi-definite relaxation.
