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Abstract
The diversity of the shape functions used in the spatial discretization of the dynamics of flexible robots complicates the task of selecting a particular shape function for a particular robot type. For a desired precision, an appropriate choice of the shape functions may result in a smaller order model, leading to a simpler simulation and an easier control design. This thesis develops the model of a one-link flexible system considering gravity effects. Several assumed modes are used in the spatial discretization process. Namely, the eigenfunctions of a rotating beam, the eigenfunctions of a clamped-free beam, the eigenfunctions of a clamped-payload beam, the polynomial functions, the cubic splines, and the cubic B-splines. A detailed comparison of the eigenvalues, the eigenmodes and their derivatives, and the static deformations and their derivatives is performed on a slewing beam in the vertical plane. Load parameters are changed from their nominal values to test the sensitivity of the shape functions. The comparisons show that the clamped-free eigenfunctions are mostly inadequate while the clamped-payload eigenfunctions are good candidates even when the payload parameters are changed. Overall, the cubic spline shape functions using curvatures as generalized coordinates offer the best compromise between good precision and low calculation complexity.
In this thesis, we apply the concept of passivity to control the position of the system with one flexible link using the input torque at the base. Since the choice of the tip position as an output leads to a non-passive system, we select a new output, namely, the noncollocated output measurement consisting of the angle of rotation at the hub augmented with a weighted value of the angle of rotation of the link's extremity. A technique for determining the closest output to the tip resulting in a passive system is given. We analyse the passivity of the input-output relation for the system in the horizontal plane with and without damping. We also analyse system's passivity in the vertical plane. In the horizontal case, analytical methods based on reactivity and (real) positivity of the transfer function are used to select the output. In the vertical one, we also apply the concept of passivity to prove asymptotic convergence of the dynamics of the error between the selected output and the reference.
