A Principal Bundle Perspective on Differential Flatness in Complex Robotic and Biological Systems

In recent decades, research has achieved deep insight into the locomotion of those robotic and biological systems whose evolution can be described in terms of a principal connection thanks to their natural symmetries. Meanwhile, the property of differential flatness has afforded another powerful method for the control of underactuated systems, but a tractable means of finding the requisite flat output for a given mechanical system has remained elusive. Recently, we have shown that the principal bundle structure induced by symmetry also furnishes a powerful tool for flat output discovery. Specifically, we give a sufficient condition for the direct construction of flat outputs from any section of the system's principal bundle that is orthogonal to a computable distribution. These flat outputs are the group variables of a trivialization of the bundle (a "choice of gauge" in the physics parlance), thus they are equivariant and typically global or almost-global. This perspective yields insight into long-standing open questions on the role of symmetry in differential flatness, while also facilitating flatness-based modeling of complex systems such as free-flying multibody robots and airborne insects, encompassing modes of locomotion that transcend the classical principal connection model.