Semi-Analytical Design of PDE Endpoint Controller for Flexible Manipulator With Non-Homogenous Boundary Conditions

This study proposes a new semi-analytical design and implementation method for nonlinear partial differential equation (PDE) control of a flexible manipulator. The proposed scheme considers the effects of the boundary input force and gravity on the payload, which results in non-homogenous boundary conditions. This objective is achieved based on a model transformation scheme for homogenizing boundary conditions, obtaining semi-analytical solutions for the corresponding PDE model. Model transformation is assigned as a hybrid exponential&#x2013;polynomial function whose coefficients are conveniently calculable without the need for any additional boundary condition measurements. This eliminates the need to use intensive numerical solvers&#x2014;for example, methods based on finite element analysis&#x2014;and allows the implementation of sophisticated PDE control schemes considering fully nonlinear PDE models with high computation speed. The presented controller is robust to parametric model uncertainty due to its adaptive design. The precision and efficiency of calculating distributed states using the proposed model transformation are demonstrated based on experimental data for the flexible manipulator with respect to the ground truth camera-based motion capture system. Model transformation is also numerically implemented for the proposed nonlinear endpoint control method based on the original PDE model. <italic>Note to Practitioners</italic>&#x2014;This paper investigates the difficulty of obtaining data describing the flexible manipulator pose required for precise control and analysis and proposes a computationally efficient method to overcome this issue.