  In this paper, we propose a novel approach to the problem of augmenting
distance-based formation controllers with a secondary constraint for the
purpose of preventing 3D formation ambiguities. Specifically, we introduce
three controlled variables that form an orthogonal space and uniquely
characterize a tetrahedron formation in 3D. This orthogonal space incorporates
constraints on the inter-agent distances and the signed volume of tetrahedron
substructures. The formation is modeled using a directed graph with a
leader-follower type configuration and single-integrator dynamics. We show that
the proposed decentralized formation controller ensures the \textit{global}
asymptotic stability and the local exponential stability of the desired
formation for an \textit{n}-agent system with no ambiguities. Unlike previous
work, this result is achieved without conditions on the tetrahedrons that form
the desired formation or on the control gains.
