Variational Principles for the Trajectory Tracking Control of Underactuated Mechanical Systems

Robotics is undergoing dynamic progression with the spread of soft robots and compliant mechanisms. The mechanical models describing these systems are underactuated, having more degrees-of-freedom than inputs. The trajectory tracking control of underactuated systems is not straightforward. The solution of the inverse dynamics is not stable in all cases, as it only considers the actual state of the system. Therefore employing the advances of optimal control theory is a reasonable choice. However, the real-time application of these is challenging as the solution to the discretized optimization problems is numerically expensive. This paper presents a novel iterative approach to solving nonlinear optimal control problems. The authors first define the iteration formula after which the obtained equations are discretized to prepare the numerical solution, contrarily to the accessible works in the literature having reverse order. The main idea is to approximate the cost functional with a second-order expansion in each iteration step, which is then extremized to get the subsequent approximation of the optimum. In the case of nonlinear optimal control problems, the process leads to a sequence of time-variant linear–quadratic regulator (LQR) problems. The proposed technique was effectively applied to the trajectory tracking control of a flexible rotational-rotational joint (RR) manipulator. The case study showed that the initialization of the iteration is simple, and the convergence is rapid.