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.
