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A natural modal expansion for the flexible robot arm problem via a self-adjoint formulation

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4 Author(s)
Y. Chait ; Dept. of Mech. Eng., Massachusetts Univ., Amherst, MA, USA ; M. Miklavcic ; C. R. Maccluer ; C. J. Radcliffe

The equations of motion of a flexible robot arm consist of a coupled partial differential equation describing the arm's transverse vibrations and an ordinary differential equation describing the hub's rigid motion. Many researchers obtained a solution using a modal expansion based on the arm's equation alone, which has erroneous eigenfunctions and eigenvalues. A novel method is presented for obtaining an equivalent but self-adjoint form for the problem. This self-adjoint form leads to a natural modal expansion, where the equations decouple. This method is used to show that the effect of the hub-arm model coupling depends exclusively on the hub-inertia-to-arm-mass ratio. The need for a self-adjoint form arises in many control applications. This is because, typically, the control design is based on approximate models, and in order to guarantee robust performance, a prior estimate of the approximation error is required. When a self-adjoint form is available, obtaining approximate modes and the associated error bounds becomes an easy task

Published in:

IEEE Transactions on Robotics and Automation  (Volume:6 ,  Issue: 5 )