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In this paper, we develop a hybrid distance-angle rigidity theory thatinvolves heterogeneous distances (or unsigned angles) and signed constraintsfor a framework in the 2-D and 3-D space. The new rigidity theory determines a(locally) unique formation shape up to a translation and a rotation by a set ofdistance and signed constraints, or up to a translation, a rotation andadditionally a scaling factor by a set of unsigned angle and signedconstraints. Under this new rigidity theory, we have a clue to resolve the flip(or reflection) and flex ambiguity for a target formation with hybriddistance-angle constraints. In particular, we can completely eliminate theambiguity issues if formations are under a specific construction which iscalled \myemph{signed Henneberg construction} in this paper. We then apply therigidity theory to formation shape control and develop a gradient-based controlsystem that guarantees an exponential convergence close to a desired formationby inter-neighbor measurements. Several numerical simulations on formationshape control with hybrid distance-angle constraints are provided to validatethe theoretical results.

Hybrid distance-angle rigidity theory with signed constraints and its applications to formation shape control