This work focuses on the problem of estimating the locations of mobile agents by agent displacement measurements with agent relative position measurements. The problem of distributed Kalman smoothing for this application is reformulated as a parameter estimation problem. The graph structure underlying the reformulated problem makes it computable in a distributed manner using iterative methods of solving linear equations.

The first part of this dissertation presents a smoothing algorithm that computes an approximation of the centralized optimal estimates. The algorithm is distributed in the sense that each agent can estimate its own position by communication only with nearby agents. With finite memory and limited number of iterations before new measurements are obtained, the algorithm produces an approximation of the Kalman smoother estimates. As the memory of each agent and the number of iterations between each time step are increased, the approximation improves. The error covariances of the location estimates produced by the proposed algorithm are significantly lower than what is possible if inter-agent relative position measurements are not available.

The second part of this dissertation presents an algorithm that reduces communication for a class of distributed estimation problems, which includes the aforementioned smoothing algorithm. Herein, the agents are viewed as nodes in a graph and the relative measurements between agents are viewed as the graph's edges. The algorithm exploits the existence of cycles in the graph to compute the best linear state estimates. For large graphs, the algorithm significantly reduces the total number of message exchanges that are needed to obtain an optimal estimate. The algorithm is guaranteed to converge for planar graphs and provide explicit formulas for its convergence rate for regular lattices.