Generalization of Newton-Euler Formulation of Dynamic Equations to Nonrigid Manipulators

A recursive lumped mass/spring approximation model, based on the Newton-Euler formulation, is proposed to model the dynamics of manipulators with link flexibility. The model assumes that the displacement and rotation due to the link flexibility are measurable. For a small link deformation, a first-order lumped mass/spring approximation model is proposed, in which the parameters of each link are lumped to its joint and the link flexibility is modeled as a spring at each joint. For a larger deformation, the first-order lumped mass/spring approximation model is extended to model each nonrigid link by a series of small rigid segments connected by “pseudo-joints.” The link flexibility is modeled as a spring in each pseudo-joint. In both cases, the effects of torsion and extension are not included in the modeling. An anlaytical error analysis is performed to justify the approximation, and the mathematical relation between the maximum modeling error and the number of pseudo-joints in each link is derived. As the number of pseudo-joints approaches infinity, the joint torques computed by the extended lumped mass/spring approximation model approach the joint torques computed by other models obtained from the Lagrange’s equation.