Convexity Analysis of Optimization Framework of Attitude Determination from Vector Observations

TL;DR

This paper analytically proves that the attitude determination optimization problem becomes convex under quaternion normalization, eliminating local optima and improving the reliability of derivative-based algorithms.

Contribution

It provides the first analytic proof that quaternion normalization renders the attitude determination problem convex, addressing a key challenge in existing methods.

Findings

Loss function is convex with quaternion normalization

Eliminates local optima in attitude determination

Supports more reliable optimization algorithms

Abstract

In the past several years, there have been several representative attitude determination methods developed using derivative-based optimization algorithms. Optimization techniques e.g. gradient-descent algorithm (GDA), Gauss-Newton algorithm (GNA), Levenberg-Marquadt algorithm (LMA) suffer from local optimum in real engineering practices. A brief discussion on the convexity of this problem is presented recently \cite{Ahmed2012} stating that the problem is neither convex nor concave. In this paper, we give analytic proofs on this problem. The results reveal that the target loss function is convex in the common practice of quaternion normalization, which leads to non-existence of local optimum.