Application of a new family of symplectic transformations in the adaptive control of mechanical systems

In the last years a special branch of soft computing aiming at the control of mechanical, electro-mechanical, and mechanical systems driven by hydraulic differential cylinders, was developed. Originally the internal symmetry of the classical mechanical systems described by the canonical transformations and the symplectic group was used as source of uniform structures also containing tuneable free parameters. Later on the special, so-called "minimum operation symplectic transformations", and, based on purely mathematical considerations, several Lie groups as the "generalized Lorentz group", the "stretched orthogonal group", and the "partially stretched orthogonal transformations" were successfully used for this purpose. Satisfactory conditions were proved for SISO, and later on for MIMO systems guaranteeing the complete stability of the control using the above groups in combination with the "modified renormalization transformation" and certain ancillary methods. The possibility for elaborating new versions of symplectic transformations of advantageous features hasn't yet been exhausted. In this paper a new family of symplectic matrices are introduced. As it is demonstrated via simulations made for 3 DOF SCARA arm these transformations have very advantageous properties in use in such control applications.