Dynamic Modelling of Lower-Mobility Parallel Manipulators Using the Boltzmann-Hamel Equations

Abstract

Explicit Dynamic equations are needed for simulation and control in the field of Mechatronics. Several classical methods are available to get Dynamic equations. Regarding those methods that aim exclusively at the equations relating applied torques and motion generated, i.e. avoiding any calculus of joint wrenches, Analytical mechanics offers several approaches. For serial mechanisms, Lagrange equations are very convenient and systematic. However, finding such mathematical expressions can be cumbersome when facing closed-loop mechanisms even with the help of Lagrange multipliers. Moreover, this is quite complex if spatial rotations are considered, and hence, generalized coordinates are very much coupled in the expressions of Lagrange functions to be differentiated. Boltzmann-Hamel equations come to help in this regard. In this paper, authors show the finding of the dynamic equations of the $3\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{P} RS$ lower-mobility parallel manipulator using the Boltzmann-Hamel equations and exploring the effect of coupled freedoms in the rotation of the end-effector.