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This paper concerns the swing-up control of an underactuated two-link robot with a linear torsional spring-attached first joint and an actuated second joint (called the FA robot below) moving in a vertical plane. First, we present a necessary and sufficient condition such the FA robot is linearly controllable at the upright equilibrium point (UEP, where two links are both upright). Second, we prove without any assumption that the FA robot is at an equilibrium point provided that its actuated joint angle is constant under a constant torque. Third, for the FA robot with its torsional stiffness of the spring being no greater than a value determined by the coefficients of its gravitational terms, we propose an energy-based controller without singular points to swing it up. We conduct a global motion analysis for the FA robot under the proposed controller. For the case that the total mechanical energy of the FA robot converges to its desired value, we present its phase portrait. For the case that the convergence is not achieved, we show that the FA robot approaches an equilibrium point belonging to a set of equilibrium points, and give a sufficient condition to check its stability. From the motion analysis, we present a sufficient condition such that the FA robot can be swung-up close to the UEP under the proposed swing-up controller. Finally, we verify our theoretical results through a numerical simulation.

In this paper, for the six-degree-of-freedom (six-DOF) model of hypersonic flight vehicle (HFV) subject to actuator faults, state constraints, parametric uncertainties, and external disturbances, an adaptive fault tolerant control (FTC) scheme is proposed based on barrier Lyapunov functions (BLFs). The study is begun with a series of control-oriented manipulations: at first, due to the high complexity of the six-DOF model, the corresponding simplified model is proposed under reasonable assumptions; then, through the stability analysis of the internal dynamics, we can conclude that the vehicle model is a non-minimum phase system, namely, having unacceptable zero-dynamics. In order to solve the non-minimum phase problem, the elevator-to-lift coupling term is regarded as uncertainty of the model. Subsequently, in consideration of the insufficient control torque caused by the fault of the rudder or elevators, an adaptive fault-tolerant controller is designed based on BLFs, backstepping method, and Nussbaum gains. In the control law, the uncertain parameters are replaced by their estimates updated by adaptive laws. And the angle of attack and the roll angle of the aircraft are constrained in the preset range. Additionally, the convergence of the proposed FTC algorithm and the boundedness of all the signals of the closed system is proved by Lyapunov stability theory. At last, the numerical simulation results of the six-DOF model are carried out to manifest the effective tracking performance of the proposed FTC scheme.

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