We consider the problem of jointly estimating the attitude and spin-rate of a spinning spacecraft. Psiaki (J. Astronautical Sci., 57(1-2):73--92, 2009) has formulated a family of optimization problems that generalize the classical least-squares attit
ude estimation problem, known as Wahbas problem, to the case of a spinning spacecraft. If the rotation axis is fixed and known, but the spin-rate is unknown (such as for nutation-damped spin-stabilized spacecraft) we show that Psiakis problem can be reformulated exactly as a type of tractable convex optimization problem called a semidefinite optimization problem. This reformulation allows us to globally solve the problem using standard numerical routines for semidefinite optimization. It also provides a natural semidefinite relaxation-based approach to more complicated variations on the problem.
We develop a general framework for MAP estimation in discrete and Gaussian graphical models using Lagrangian relaxation techniques. The key idea is to reformulate an intractable estimation problem as one defined on a more tractable graph, but subject
to additional constraints. Relaxing these constraints gives a tractable dual problem, one defined by a thin graph, which is then optimized by an iterative procedure. When this iterative optimization leads to a consistent estimate, one which also satisfies the constraints, then it corresponds to an optimal MAP estimate of the original model. Otherwise there is a ``duality gap, and we obtain a bound on the optimal solution. Thus, our approach combines convex optimization with dynamic programming techniques applicable for thin graphs. The popular tree-reweighted max-product (TRMP) method may be seen as solving a particular class of such relaxations, where the intractable graph is relaxed to a set of spanning trees. We also consider relaxations to a set of small induced subgraphs, thin subgraphs (e.g. loops), and a connected tree obtained by ``unwinding cycles. In addition, we propose a new class of multiscale relaxations that introduce ``summary variables. The potential benefits of such generalizations include: reducing or eliminating the ``duality gap in hard problems, reducing the number or Lagrange multipliers in the dual problem, and accelerating convergence of the iterative optimization procedure.
