Sensor Network Localization on the Group of Three-Dimensional Displacements
Jeffrey R. Peters, Domenica Borra, Bradley E. Paden, Francesco Bullo · SIAM Journal on Control and Optimization · 2015
We consider the problem of estimating relative configurations of nodes in a sensor network based on noisy measurements. By exploiting the cyclic constraints induced by the sensing topology to the network, we derive a constrained optimization on $SE(3)^n$. For the case of a network with a single cyclic constraint, we present a closed-form solution. We show that, in certain cases, namely restriction to pure rotation and pure translation, this solution is independent of the particular representation of the constraint function and is the unique, constrained minimizer of an appropriate cost. For sensing topologies represented by a general, weakly connected digraph, we present a solution method which is based on the limits of the solution curves of a continuous-time ordinary differential equation. We show that solutions obtained by our method satisfy all semicycle (generalized cycle) constraints induced by the sensing topology of the network. Further, we show through numerical simulation that for “Gaussian-like” noise models with small variance, our solution achieves noise reduction comparable to that of the least squares estimator for an analogous linear problem.