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A translational oscillator with a rotational actuator (TORA) is an underactuated nonlinear mechanical system with two degrees of freedom (DOF). This paper concerns the robust stabilization control problem for the system with multiple external disturbances. First, a disturbance observer is constructed based on the internal nonlinear dynamic behavior of the system. Second, a robust stabilization controller is designed by the estimated disturbances and the fixed-time sliding mode control method. The controller realizes the global robust stabilization control objective of the TORA system, and the stability of both disturbance observer and robust closed-loop control system are analyzed using the Lyapunov theorem. Finally, the effectiveness of the theoretical results are verified by numerical experiments.

In this paper, a gradient dynamics-based control method is proposed to directly tackle the singularity problem in the backstepping control design of the TORA system. This method is founded upon the construction of an energy-like positive function, which includes an auxiliary variable in terms of the intermediate virtual control law. On this basis, a gradient dynamics is created to obtain a new virtual control command, which is capable of making the auxiliary variable gradually approach zero, thereby mitigating the issue of division by zero. The core innovation is the integration of the gradient dynamics into the recursive backstepping design to overcome the singularity problem and stabilize the system at the equilibrium quickly. In addition, it rigorously proves that all the signals in the closed-loop control system are uniformly ultimately bounded, and the tracking errors converge to a small neighborhood around zero through a Lyapunov-based stability analysis. Comparative simulations demonstrate that the proposed approach not only avoids the singularity issue, but also achieves a better transient performance over other methods.

A control for the rotation of an eccentric wheel that asymptotically stabilizes the equilibrium state of the Translational Oscillator with Rotating Actuator (TORA) is proposed. The nonlinear dependence of the force arising from the deformation of the elastic elements of the model on the displacement is taken into account since neglecting this nonlinearity can lead to the loss of model accuracy. It is shown that such a control is robust, and a method for assessing the domain of robustness in the parameter space of the mechanical system is proposed. The obtained results are illustrated using a real model as an example.

The 2-Dimensional Translational Oscillators with Rotating Actuator (2DTORA) is a novel underactuated system which has one actuated rotor and two unactuated translational carts. This paper focuses on dynamical modelling and simulation analysis of the underactuated 2DTORA on a slope. Based on Lagrange equations, the dynamics of the 2DTORA is achieved by selecting a transverse position of a cart, a travelling position of a cart, and the rotor angle as the general coordinates and torque acting on the rotor as the general force. When the slope angle is set to zero, the dynamics of 2DTORA on a slope is reduced to that of 2DTORA on the horizontal plane. Moreover, by eliminating one degree of translational cart motion, the dynamics of 2DTORA is reduced to that of TORA which is a benchmark of underactuated systems. In addition, the equilibrium and controllability of the 2DTORA system on a slop are discussed. Finally, numerical simulations are performed to verify the feasibility of the developed dynamic models.

The horizontal translational oscillator with a rotational actuator (TORA) is a typical underactuated mechanical system, whose control problem is still open and theoretically challenging. At present, the existing control methods are structurally complicated and require an exact knowledge of the system parameters. Moreover, few works have considered the output feedback stabilization of the TORA system subject to practical constraints of input saturation and angular velocity unmeasurement. To address these problems, this paper proposes a novel model-free amplitude-limited control approach to stabilize the TORA system at the origin using only angle feedback. Firstly, the passivity of the horizontal TORA system is analyzed, based on which a novel Lyapunov function augmented with an auxiliary signal is constructed by taking the input saturation into account. Then, an amplitude-limited control law is derived in a straightforward manner. In order to make the control law independent of velocity feedback, the auxiliary signal is designed in terms of the ball rotational angle and an output of a dynamic system. The asymptotic stability of the entire control system is rigorously guaranteed by utilizing Lyapunov theory and LaSalle’s invariance principle. Finally, simulation results with comparisons to existing methods demonstrate the effectiveness and superiority of the proposed control approach.

In this paper, we consider the robust stabilization control problem of underactuated translational oscillator with a rotating actuator (TORA) system in the presence of unknown matched disturbances by employing continuous control inputs. A nonlinear continuous robust control approach is proposed by integrating the techniques of backstepping and linear extended state observer (LESO). Specifically, based on the backstepping design methodology, a hyperbolic tangent virtual control law is designed for the first subsystem of the cascaded TORA model, via which an integral chain error subsystem is subsequently constructed and the well-known LESO technique is easy to implement. Then, an LEO is designed to estimate the lumped matched disturbances in real-time, and the influence of the disturbances is compensated by augmenting the feedback controller with the disturbance estimation. The convergence and stability of the entire control system are rigorously proved by utilizing Lyapunov theory and LaSalle’s invariance principle. Unlike some existing methods, the proposed controller is capable of generating robust and continuous control inputs, which guarantee that both the rotation and translation of TORA systems are stabilized at the origin simultaneously and smoothly, attenuating the influence of disturbances. Comparative simulation results are presented to demonstrate the effectiveness and superior control performance of the proposed method.