An existence condition for the inverse dynamics solution of a slewing Euler–Bernoulli beam

Causal inverse dynamics problem of a slewing flexible beam has been identified as being ill-posed because it violates the stability of solution. Such problem has been studied intensively for the past three decades due to its application to the tip tracking control of flexible-link manipulators. A well known remedy has been to modify the location of the boundary point, or the output function in tracking control. This paper re-examines the problem by analyzing a closed form solution. Such a solution is made possible by truncating the assumed modes beam deflection model after the first mode. An existence condition for the solution is established, which links to the deflection mode shape, payload inertia, and the location of boundary point. This condition indicates that for certain system parameters, causal solutions may exist. Existing remedies as suggested by many published results can be explained by the condition. Extension to a multi-mode model is discussed and examined through numerical simulations.