An analytical expression for the generalized forces in multibody Lagrange equations

This paper describes how the partial derivative of the kinetic energy, with respect to the generalized coordinates in the Lagrange equations, can be obtained in analytic form for structures consisting of rigid links connected by lower pair joints. We will prove that the expression for the derivative involves the time derivative of the line coordinates of the geometric lines, coinciding with the joint axes. As a consequence, the generalized force in the Lagrange equations can be written as a function of the inertia matrices and the line coordinates of the joint axes. The time derivative of the line coordinates can be expressed by using the adjoint matrix of the line vector.