To describe the configuration of a multi-body system, Cartesian coordinate systems are attached to all bodies comprising the system. Their connections through joints and force elements are efficiently expressed by using 4×4 matrices of the homogeneous transformation, presented by Denavit and Hartenberg in 1955. However, at this time, there is no systematic method to compute velocities and angular velocities using the matrices of such homogeneous transformations.
In this paper, homogeneous transformation matrices are identified as a subset of a Lie group, called the special Euclidean group denoted by SE(3). This observation enables the usage of the Lie group theory in multibody kinematics. The effective use of the theory is built upon a platform of a moving frame method as presented in this paper. In this method, for each body-attached Cartesian coordinate system, the coordinate vector basis is written explicitly following Élie Cartan. This moving frame notation enables us to use the Lie algebra of SE(3), denoted by se(3), to compute velocities and angular velocities by minimizing the complexities of the Lie group theory.
For kinetics, a variational method is established in se(3) by deriving a relationship between a virtual angular velocities and the corresponding virtual rotational displacements. This constrained variation of virtual angular velocities allows the derivation of the d’Alembert principle of virtual work from Hamilton’s principle for multibody systems. Utilizing this variational tool, we present a systematic computation of equations of motion from Hamilton’s principle.
Finally, we reduce the spatial dynamics to planar dynamics and list the simplifications achieved in the two-dimensional problems using SE(2). Then, for a two-degree-of-freedom manipulator the analytical equations of motion are obtained to demonstrate the power of the moving frame method.