Distributed synchronization of Euclidean transformations with guaranteed convergence

Abstract

Abstract-This paper addresses synchronization of Euclidean transformations over graphs. Synchronization in this context, unlike rendezvous or consensus, means that composite transformations over loops in the graph are equal to the identity. Given a set of non-synchronized transformations, the problem at hand is to find a set of synchronized transformations approximating well the non-synchronized transformations. This is formulated as a nonlinear least-squares optimization problem. We present a distributed synchronization algorithm that converges to the optimal solution to an approximation of the optimization problem. This approximation stems from a spectral relaxation of the rotational part on the one hand and from a separation between the rotations and the translations on the other. The method can be used to distributively improve the measurements obtained in sensor networks such as networks of cameras where pairwise relative transformations are measured. The convergence of the method is verified in numerical simulations.

Consensus-based Distributed 3D Pose Estimation with Noisy Relative Measurements

“…1) in an undirected measurement graph. The characterization of these conditions yields a distributed initialization method for the estimation and also helps to explain update laws in existing distributed estimation algorithms [13], [14]. Simulation results corroborate the theoretical findings of the paper.…”

“…This type of pairwise averaging also appears in [14] (Algorithm 1) despite the different form of rotation. We also relate our findings to formation control literature such as [15], where they require global consistency on the desired poses for the proof of convergence of the coupled translation and rotation formation.…”

“…Most notably, Tron et al proposed a distributed gradient descent method that sequentially considers three separate objectives: first a convex relaxation of the rotation-only objective, then a translation-only objective, and finally the full 3D pose objective [13]. Recently, Thunberg et al also studied distributed synchronization over all of SE(3) and guarantee convergence with an iterative algorithm that projects the rotation estimates at each step onto SO(3) [14]. These works consider distributed estimation for 3D pose but do not characterize the consistency conditions on the noisy measurements for consensus-based estimation systems as we do in this paper.…”

Pose Localization of Leader-Follower Networks with Direction Measurements

“…For example, some necessary and sufficient conditions are provided for orientation localizability of triangular sensing networks of relative orientation measurements in [6], without providing a distributed orientation localization law. Network localization schemes using relative poses (relative orientations and relative positions), which are measured by a vision-based technique, are investigated in [7,12]. The estimation of relative poses, however, requires the agents to have views of a common scene and complicated estimation algorithms.…”

“…Under the proposed position localization law, positions of all followers are also globally and asymptotically determined. The proposed network pose localization laws can work exclusively with inter-agent directional vectors and does not require a common scene and extra algorithms to compute relative poses; unlike [7,12]. The rest of this paper is organized as follows.…”