### A Finite Element Scheme for Calculating Inverse Dynamics of Link Mechanisms

Abstract

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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