Nonlinear optimal control for two cable-driven 3-DOF robotic cranes

Abstract The article is concerned with the nonlinear optimal control problem of two cable-driven 3-DOF robotic cranes. Such cranes can be used in ship maintenance and repair. These are underactuated robotic systems comprising a 2-cable driven cart with a payload suspended from it. Such robotic mechanisms have three degrees of freedom and can move in the entire 2D vertical plane. Using Euler-Lagrange analysis the state-space model of the two cable-driven 3-DOF robotic crane is obtained. It is also proven that this dynamic model is differentially flat. Next, to solve the associated nonlinear optimal control problem, the dynamic model of the two cable-driven 3-DOF robotic crane undergoes approximate linearization around a temporary operating point that is recomputed at each time-step of the control method. The linearization relies on Taylor series expansion and on the associated Jacobian matrices. For the linearized state-space model of the crane an optimal (H-infinity) feedback controller is designed. This controller stands for the solution to the nonlinear optimal control problem under model uncertainty and external perturbations. To compute the controller’s feedback gains an algebraic Riccati equation is repetitively solved at each iteration of the control algorithm. The stability properties of the control method are proven through Lyapunov analysis. The proposed nonlinear optimal control approach achieves fast and accurate tracking of reference setpoints under moderate variations of the control inputs. Besides, the method avoids change of state variables and state-space model transformations and the control inputs it computes are applied directly on the initial nonlinear state-space model of the crane.