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

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.