Inverse Dynamics of Flexible Manipulators

High performance robot manipulators, in terms of cycle time and accuracy, require well designed control methods, based on accurate dynamic models. Robot manipulators are traditionally described by the flexible joint model or the flexible link model. These models only consider elasticity in the rotational direction. When these models are used for control or simulation, the accuracy can be limited due to the model simplifications, since a real manipulator has a distributed flexibility in all directions. This work investigates different methods for the inverse dynamics of a more general manipulator model, called the extended flexible joint model. The inverse dynamics solution is needed for feedforward control, which is often used for high-precision robot manipulator control. The inverse dynamics of the extended flexible joint model can be computed as the solution of a high-index differential algebraic equation (DAE). One method is to solve the discretized DAE using a constantstepsize constant-order backwards differentiation formula (BDF). This work shows that there is only a small difference between solving the original high-index DAE and the index-reduced DAE. It is also concluded that scaling of the algebraic equations and their derivatives is important. The inverse dynamics can be solved as an initial-value problem if the zero dynamics of the system is stable, i.e., minimum phase. For unstable zero dynamics, an optimization approach based on the discretized DAE is suggested. An optimization method, using a continuous DAE formulation, is also suggested and evaluated. The solvers are illustrated by simulation, using a manipulator with two actuators and five degrees-of-freedom.