Semi-analytical design of PDE endpoint controller for flexible manipulator with non-homogenous boundary conditions

This paper proposes a new semi-analytical design and implementation method for nonlinear partial differential equation (PDE) control of flexible manipulator. The proposed scheme considers the effects of boundary input force and gravity on the payload, which results in non-homogenous boundary conditions. This objective is achieved based on an appropriate model transformation scheme for homogenizing boundary conditions, which enables obtaining semi-analytical solutions for the corresponding PDE model. Model transformation is assigned as a hybrid exponential–polynomial function whose coefficients are conveniently calculable without the need for any additional boundary condition measurements. This results in elimination of the need for using intensive numerical solvers,  e.g., those based on finite element analysis, and allows for implementation of sophisticated PDE control methods considering fully nonlinear PDE models with high computation speed. Precision and efficiency of calculating distributed states using proposed model transformation is demonstrated based on experimental data for the manipulator with respect to ground truth camera-based motion capture system. The model transformation is also numerically implemented for the proposed nonlinear endpoint control method based on original PDE model. Note to practitioners—This paper investigates difficulty of obtaining data describing flexible manipulator pose required for precise control and analysis, and proposes a computationally efficient method to overcome this issue.