The Hamel Representation: A Diagonalized Poincaré Form

The Poincaré equations, also known as Lagrange’s equations in quasicoordinates, are revisited with special attention focused on a diagonal form. The diagonal form stems from a special choice of generalized speeds that were first introduced by Hamel (Hamel, G., 1967, Theorctische Mechanik, Springer-Verlag, Berlin, Secs. 235 and 236) nearly a century ago. The form has been largely ignored because the generalized speeds create so-called Hamel coefficients that appear in the governing equations and are based on the partial derivative of a mass-matrix factorization. Consequently, closed-form expressions for the Hamel coefficients can be difficult to obtain. In this paper, a newly developed operator overloading technique is used within a simulation code to automatically generate the Hamel coefficients through an exact partial differentiation together with a numerical evaluation. This allows the diagonal form of Poincaré’s equations to be numerically integrated for system simulation. The diagonal form and the techniques used to generate the Hamel coefficients are applicable to general systems, including systems with closed kinematic chains. Because of Hamel’s original influence, these special Poincaré equations are called the Hamel representations and their usefulness in dynamic simulation and control is investigated.