To derive the equations of motion for a multibody system using the Gibbs-Appell calculus – the partial derivatives of the Gibbs function G = 1/2 ∫ a2 dm with respect to the generalized accelerations equal the generalized forces-shows special advantages. Describing the kinematics with Jacobi matrices and local terms, these equations can be written in such a way that the partial derivations need not be performed explicitly. Kinetic effects of fast rotating driving devices attached to the moving links can be included in a similar way. Though an analytical formulation of the equations of motion is especially desirable with respect to its application for industrial robots, such a formulation becomes too extended and susceptible to errors for systems with more than 3 or 4 bodies. Therefore an approach is developed for tree structured robots with rotational or translational joints for calculating the Jacobi matrices and the local terms without employing any differentiation process. So it is possible to use the Gibbs-Appell method numerically in a recursive way e.g. for calculating the torques of the actuators of a robot with 6 or more degrees of freedom for a given motion.

The stability analysis of active spatial mechanisms comprising both powered and unpowered joints is carried out for the first time using aggregation-decomposition method via Lyapunov vector functions. This method has already been used for analysis of mechanisms with all powered joints. To extend the application of the method to the stability analysis of mechanisms containing unpowered joints we developed modelling of special subsystem consisting of one powered and one unpowered joint. Then, we consider the stability of the complete system without neglecting any dynamic effect. The stability analysis is demonstrated by a numerical example of a particular biped system.