## 1 Introduction
The same technique was used in [16; 17; 18] to reject the additive disturbance and the friction model that contains uncertain nonlinear parameterizable terms for Euler-Lagrange systems. The RISE feedback was used in [19; 20] for trajectory tracking control and active vibration control for quadrotor unmanned aerial vehicles (UAVs) and flexible refueling hose, respectively. In the same manner, the RISE feedback was used in [21] for suppressing limit cycle oscillations in UAVs with dynamic model uncertainty and parametric actuator uncertainty. Furthermore, the authors of [22] used RISE feedback to compensate for uncertain, nonautonomous disturbances for a class of coupled, fully-actuated underwater vehicles. These controllers are continuous and yield asymptotic tracking errors. Some other studies focus on disturbance torques that are caused by vibration and payload variation. In [23], the authors use finite impulse response FIR digital filters to design a controller that is robust to vibration and payload variation. However, digital filters work better for linear, time-invariant systems and they have some limitations with nonlinear systems. Additionally, this controller requires knowledge of the natural frequencies of the payload. In [24], a sliding mode control was applied to solve the position tracking problem of a single-link flexible robot arm. The controller is robust with regards to payload and actuator friction changes; however, it requires a high bandwidth actuator, which increases the control effort. [PERSON], [PERSON]. et al. [25] designed a control algorithm to cancel vibrations that are originated by the structural flexibility of the manipulator during movement. The control algorithm works efficiently for trajectory tracking without saturating the actuators, however, it requires a perfect kinematic model and a perfect knowledge of the compliance matrix. This paper studies the regulation problem in joint space for a manipulator subjected to uncertain torque due to vibration and payload variation. For this objective, we consider that there is no uncertainty in the kinematics of the manipulator. Therefore, the effect of the vibration and the payload variation are modeled as bounded disturbance torques [26] to the joints, which are driving the links of the robot manipulator. Two approaches are used to design the control input depending on the characteristics of the disturbance torques based on Lyapunov analysis. The first approach considers the vibration and payload variation torques are bounded. The controller derived from this approach introduces UUB output for the tracking error with an acceptable control effort. The second approach exploits the differentiable property plus the boundness of the disturbance torques. Therefore, the controller derived from the second approach shows that the error goes to zero as time goes to infinity depending on the initial condition of the state (i.e., semiglobal stable result of the tracking error). The performance of the controllers is evaluated as follows:  
1. The controller derived from the first approach is compared with a PD controller in the presence of bounded disturbance torques that are caused by vibration and payload variation. The simulation results show a UUB tracking error for this controller with a good control effort, while the PD controller performs poorly in term of accuracy. 2. The controller derived from the second approach is also compared with the PD controller in the presence of a differentiable and bounded vibration and payload variation torques and with a specific initial condition. The simulation results show an asymptotic tracking error for this controller with low control effort. The PD controller behaves almost the same as previously. In comparison with the mentioned existing advanced techniques, the proposed approaches are conducted based on the best compact dynamic model of a rigid-link robot arm in the presence of disturbance torques due to vibration and payload variation.