Modelling friction in the dynamics analysis of selected one-DOF spatial linkage mechanisms

Abstract

A method of dynamics analysis of selected one-DOF spatial linkage mechanisms containing one spherical joint in their structure is presented in the paper. These mechanisms include neither redundant DOF nor passive constraints. In the procedure assumed here the spherical joint was treated as ideal, whereas friction can be taken into account in the remaining kinds of connections of links, i.e. in a rotational joint and a prismatic joint. The advanced LuGre model was used to take into account the phenomena of friction in the joints. The mechanism, in the form of a closed-loop kinematic chain, was divided by the cut-joint technique into two open-loop kinematic chains in the place of the spherical joint. Joint coordinates and homogeneous transformation matrices were used to describe the geometry of the system. Equations of the chains’ motion were derived using the formalism of Lagrange equations. Cut-joint constraints and joint forces, acting in the cutting place, were introduced to complete the equations of motion, thus a set of DAEs was obtained. To solve these equations, a procedure was applied based on double differentiation of the formulated constraint equations in relation to time. In order to determine the values of friction torque in a rotational joint and friction force in a prismatic joint, in each integrating step of the equations of motion, the joint forces and torques acting in these joints were calculated using the recursive Newton–Euler algorithm, which was taken from robotics. For the requirements of the method, models were developed of both a rotational joint and a prismatic joint. As an example, the dynamics analysis of a one-DOF RSUP spatial linkage mechanism, including rotational joints R, spherical joint S, universal joint U and prismatic joint P, is presented in the article.