In this work, the author presents a method called Convex Model Predictive Control (CMPC) to control systems whose states are elements of the rotation matrices 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 model predictive control (MPC) scheme without the drawbacks associated with conventional linearization techniques such as slow computation time and local minima. Of particular emphasis is the application to aeronautical and vehicular systems, wherein the method removes many of the 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).
This paper proposes a new method for rigid body pose estimation based on spectrahedral representations of the tautological orbitopes of SE(2) and SE(3). The approach can use dense point cloud data from stereo vision or an RGB-D sensor (such as the Microsoft Kinect), as well as visual appearance data as input. The method is a convex relaxation of the classical pose estimation problem, and is based on explicit linear matrix inequality (LMI) representations for the convex hulls of SE(2) and SE(3). Given these representations, the relaxed pose estimation problem can be framed as a robust least squares problem with the optimization variable constrained to these convex sets. Although this formulation is a relaxation of the original problem, numerical experiments indicates that it is indeed exact - i.e. its solution is a member of SE(2) or SE(3) - in many interesting settings. We additionally show that this method is guaranteed to be exact for a large class of pose estimation problems. We study the convex hull of @math , the set of @math orthogonal matrices with unit determinant, from the point of view of semidefinite programming. We show that the convex hull of @math is doubly spectrahedral, i.e., both it and its polar have a description as the intersection of a cone of positive semidefinite matrices with an affine subspace. Our spectrahedral representations are explicit and are of minimum size, in the sense that there are no smaller spectrahedral representations of these convex bodies.  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 @math validate the approach.