Kinematics for an actuated flexible n-manifold

Recent years show an increasing interest in flexible robots due to their adaptability merits. This paper introduces a novel set of hyper-redundant flexible robots which we call actuated flexible manifold (AFM). The AFM is a two-dimensional hyper-redundant grid surface embedded in R² or R³. Theoretically, such a mechanism can be manipulated into any continuous smooth function. We introduce the mathematical framework for the kinematics of an AFM. We prove that for a nonsingular configuration, the number of degrees of freedom (DOF) of an AFM is simply the number of its grid segments. We also show that for a planar rectangular grid, every nonsingular configuration that is also energetically stable is isolated. We show how to calculate the forward and inverse kinematics for such a mechanism. Our analysis is also applicable for three-dimensional hyper-redundant structures as well. Finally, we demonstrate our solution on some actuated flexible grid-shaped surfaces.