The tracking control of underactuated systems is a challenging problem due to the structural differences compared to fully actuated systems. Contrarily to fully actuated systems, resolving the inverse kinematics problem of underactuated systems is not possible independently from the dynamic equations. Instead, the inverse dynamics must be addressed. It is common to extend the computed torque control (CTC) technique with servoconstraints. Besides the CTC's clearness, the stability of the system cannot be always guaranteed. A novel predictive controller (PC) is presented in this paper. Our PC applies the variational principle to design the motion of the system in order to achieve a stable motion with the lowest possible tracking error. To demonstrate the applicability and the performance of the PC method, a numerical study is presented for a planar manipulator resulting in about 20% RMS error compared to the CTC method from the literature.

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.

Contradictory demands are present in the dynamic modeling and analysis of legged locomotion: on the one hand, the high degrees-of-freedom (DoF) descriptive models are geometrically accurate, but the analysis of self-stability and motion pattern generation is extremely challenging; on the other hand, low DoF models of locomotion are thoroughly analyzed in the literature; however, these models do not describe the geometry accurately. We contribute by narrowing the gap between the two modeling approaches. Our goal is to develop a dynamic analysis methodology for the study of self-stable controlled multibody models of legged locomotion. An efficient way of modeling multibody systems is to use geometric constraints among the rigid bodies. It is especially effective when closed kinematic loops are present, such as in the case of walking models, when both legs are in contact with the ground. The mathematical representation of such constrained systems is the differential algebraic equation (DAE). We focus on the mathematical analysis methods of piecewise-smooth dynamic systems and we present their application for constrained multibody models of self-stable locomotion represented by DAE. Our numerical approach is demonstrated on a linear model of hopping and compared with analytically obtained reference results.