A Unified Framework for Dynamics and Lyapunov Stability of Holonomically Constrained Rigid Bodies
Khoder Melhem*, Zhaoheng Liu**, and Antonio Loría***
*Département de Génie Mécanique, École Polytechnique de Montréal, Montréal, H3C 3A7, Canada
**Département de Génie Mécanique, École de Technologie Supérieure, Montréal, H3C 1K3, Canada
***CNRS-LSS, Supélec, Plateau de Moulon, 91192, Gif sur Yvette, France
A new dynamic model for interconnected rigid bodies is proposed here. The model formulation makes it possible to treat any physical system with finite number of degrees of freedom in a unified framework. This new model is a nonminimal realization of the system dynamics since it contains more state variables than is needed. A useful discussion shows how the dimension of the state of this model can be reduced by eliminating the redundancy in the equations of motion, thus obtaining the minimal realization of the system dynamics. With this formulation, we can for the first time explicitly determine the equations of the constraints between the elements of the mechanical system corresponding to the interconnected rigid bodies in question. One of the advantages coming with this model is that we can use it to demonstrate that Lyapunov stability and control structure for the constrained system can be deducted by projection in the submanifold of movement from appropriate Lyapunov stability and stabilizing control of the corresponding unconstrained system. This procedure is tested by some simulations using the model of two-link planar robot.
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