A Finite Element Scheme for Calculating Inverse Dynamics of Link Mechanisms


There is a difficulty in calculating the inverse dynamics for the closed-loop mechanism using conventional methods such as the Newton-Euler method or the Lagrangian method. This is due to the interdependence variables between the constituting links, which become impossible to derive when a chain is closed in the system using the former method. The latter method is also difficult to apply, since the derivation process of an equation considering the binding condition is very complicated. Generally, robotic tasks include motions that generate open and closed loops alternatively, and the dynamic equations of the system (or the numerical algorithm) require an instant revision during the motion. Therefore, a unified numerical scheme for calculating the inverse dynamics is strongly desired, particularly for those cases of massive, quick-motion robots controlled by force.
Isobe and Nakagawa proposed to apply the Finite Element Method (FEM), a widely used computational tool for analyzing structures, fluids, and so forth, to a control system of connected piezoelectric actuators, and achieved good control not only of the actuator itself but also of the entire system. Then, Isobe et al. implemented the FEM to a calculation scheme of inverse dynamics for hyper-redundant link mechanisms. Using the characteristic of the FEM, which is the capability of expressing the behavior of each discrete element as well as that of the entire continuous system, local information such as nodal forces or displacements can be calculated in parallel. The FEM does not require re-implementation of dynamic equations in the software, and revision can be achieved simply by changing the input data in the case of a physical change in the hardware system.
This study describes a finite element scheme for calculating inverse dynamics of link mechanisms. Link mechanisms are modeled using linear Timoshenko beam elements based on the Shifted Integration (SI) technique, which was originally used in finite element analyses of framed structures. Nodal forces for obtaining target trajectories are calculated using the FEM, and the joint torque of each link is calculated based on a matrix-formed conversion equation between nodal forces and the joint torque. Some numerical tests are carried out for several types of link mechanisms, to verify the validity of the proposed scheme as a unified numerical scheme independent of the system configuration.

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