A Convex Approach to Consensus on SO(n)

This paper introduces several new algorithms for consensus over the special orthogonal group. By relying on a convex relaxation of the space of rotation matrices, consensus over rotation elements is reduced to solving a convex problem with a unique global solution. The consensus protocol is then implemented as a distributed optimization using (i) dual decomposition, and (ii) both semi and fully distributed variants of the alternating direction method of multipliers technique -- all with strong convergence guarantees. The convex relaxation is shown to be exact at all iterations of the dual decomposition based method, and exact once consensus is reached in the case of the alternating direction method of multipliers. Further, analytic and/or efficient solutions are provided for each iteration of these distributed computation schemes, allowing consensus to be reached without any online optimization. Examples in satellite attitude alignment with up to 100 agents, an estimation problem from computer vision, and a rotation averaging problem on $SO(6)$ validate the approach.Comment: Accepted to 52nd Annual Allerton Conference on Communication, Control, and Computin

Convex Model Predictive Control for Vehicular Systems

In this work, we present a method to perform Model Predictive Control (MPC) over systems whose state is an element of $SO(n)$ for $n=2,3$. This is done without charts or any local linearization, and instead is performed by operating over the orbitope of rotation matrices. This results in a novel MPC scheme without the drawbacks associated with conventional linearization techniques. Instead, second order cone- or semidefinite-constraints on state variables are the only requirement beyond those of a QP-scheme typical for MPC of linear systems. Of particular emphasis is the application to aeronautical and vehicular systems, wherein the method removes many of the transcendental trigonometric terms associated with these systems' state space equations. Furthermore, the method is shown to be compatible with many existing variants of MPC, including obstacle avoidance via Mixed Integer Linear Programming (MILP)

Collaborative Perception From Data Association To Localization

During the last decade, visual sensors have become ubiquitous. One or more cameras can be found in devices ranging from smartphones to unmanned aerial vehicles and autonomous cars. During the same time, we have witnessed the emergence of large scale networks ranging from sensor networks to robotic swarms. Assume multiple visual sensors perceive the same scene from different viewpoints. In order to achieve consistent perception, the problem of correspondences between ob- served features must be first solved. Then, it is often necessary to perform distributed localization, i.e. to estimate the pose of each agent with respect to a global reference frame. Having everything set in the same coordinate system and everything having the same meaning for all agents, operation of the agents and interpretation of the jointly observed scene become possible. The questions we address in this thesis are the following: first, can a group of visual sensors agree on what they see, in a decentralized fashion? This is the problem of collaborative data association. Then, based on what they see, can the visual sensors agree on where they are, in a decentralized fashion as well? This is the problem of cooperative localization. The contributions of this work are five-fold. We are the first to address the problem of consistent multiway matching in a decentralized setting. Secondly, we propose an efficient decentralized dynamical systems approach for computing any number of smallest eigenvalues and the associated eigenvectors of a weighted graph with global convergence guarantees with direct applications in group synchronization problems, e.g. permutations or rotations synchronization. Thirdly, we propose a state-of-the art framework for decentralized collaborative localization for mobile agents under the presence of unknown cross-correlations by solving a minimax optimization prob- lem to account for the missing information. Fourthly, we are the first to present an approach to the 3-D rotation localization of a camera sensor network from relative bearing measurements. Lastly, we focus on the case of a group of three visual sensors. We propose a novel Riemannian geometric representation of the trifocal tensor which relates projections of points and lines in three overlapping views. The aforemen- tioned representation enables the use of the state-of-the-art optimization methods on Riemannian manifolds and the use of robust averaging techniques for estimating the trifocal tensor