Study of invariance properties in robotic systems: a Lagrangian formulation approach

We present here some interesting properties of regularity regarding firstly the flexural deformation spectra and secondly the flexural and torsional spectra of mechanical loaded bodies which can be useful for the finite control of these systems. In these two cases, two main cases are studied: the non compliant and non dissipative case, and the compliant and non dissipative case. By studying flexural spectra of general deformable bodies versus the load, we found a noteworthy invariance property of the eigenfrequencies versus the system load. Roughly we found that under a preload condition the eigenfrequencies of deformation of the system are invariant versus loads, i.e. on a diagram eigenfrequencies versus loads, the curves are horizontal lines. As shown by our curves, these behaviors are not very distant from the asymptotic behaviors but occur for very small values of loads and frequencies, allowing the implementation of this property to real technological devices. For loaded flexural and torsional deformable bodies, we found that under a preload condition, each eigenfrequency is invariant versus the load m_N. For the mode 7 and beyond, the preload conditions are less than m_N=1, and the seven first modes (from 0 to 6) can be easily modelled by polynomial functions of m_N in the neighborhood of useful m_N