Ultimate Boundedness and Output Convergence of Prioritized Output Tracking Control Under Nonsmooth and Imperfect Feedback Linearization

The concept of priority has been introduced to the robotic systems in 1980s as an effort to overcome problems caused by singularity and it has attracted considerable attention from both the robotics and control societies. However, none of the previous works have successfully addressed two fundamental degenerate properties of singularity: nonsmoothness and imperfect inversion. This technical note proposes a prioritized output tracking control method that guarantees the ultimate boundedness and the convergence of higher priority outputs when the input–output feedback linearization becomes nonsmooth and imperfect by singularity. For that purpose, we first prioritize the input–output feedback linearization and discuss how two degenerate properties can occur when the system becomes singular. Then, we introduce the differential inclusion to deal with the nonsmooth internal dynamics and establish a condition using the upper Dini derivative for asymptotically stable zero dynamics. Also, we propose a strictly positive realness-based feedback gain design and find a condition related to an M-matrix in order to handle nonlinearities that appear in the imperfect feedback linearization. Finally, we combine these two results and show the ultimate boundedness and the output convergence. In addition, we provide a motivational example with a planar four-link manipulator to illustrate the effectiveness and limitations of the proposed method.