The equivalence of Boltzmann-Hamel and Gibbs-Appell equations in modeling constrained systems

The model-based control design process for mechanical systems such as robotic vehicles and manipulators requires a clear mathematical model that describes the dynamics of the system to be available. Formulation of such mathematical models can be challenging especially if constraints are involved. There exist several methods of developing dynamic models for mechanical systems. The popular methods can be carried out in the natural inertial space, configuration space or in the quasi-coordinate space. Inertial space models such as those based on Newton-Euler formulation are only suitable for small systems because of the need to determine all unknown constraint reaction forces. Configuration space models such as Euler-Lagrange models are preferred to inertial space models because they reduce the number of unknown reaction forces into Lagrange multipliers where each constraint is associated with one Lagrange multiplier. However, these Lagrange multipliers also increase the dimension of the system and can be prohibitive if the system has many constraints. The most suitable alternative approach for multi-constrained systems is the use of quasi-coordinate space models such as the Maggi, Boltzmann-Hamel and the Gibbs-Appell formulations that eliminate the constraints altogether from the process. By doing so, they reduce the size of the working space compared to both the configuration space and the inertial space. Despite these advantages, quasi-coordinate space methods are not yet sufficiently popular in control design applications. The objective of this paper is to not only try to popularize the Boltzmann-Hamel and the Gibbs-Appell formulations but also to show that the two approaches are equivalent. Two examples are provided at the end to show this equivalence by yielding the same equations for the tested systems.