# Nonlinear Robust Adaptive Control of Universal Manipulators Based on Desired Trajectory
## 1 Introduction  
In recent years, robots have been widely used in industrial automation [1; 2], the service industry [3; 4], and the medical field [5; 6]. In many applications, trajectory tracking control of robots is key to ensuring their accurate execution of tasks. In the early days, the PID control method, which is widely used in linear time-invariant systems, was also used in robot control [7]. However, due to the strong nonlinearity and strong coupling of the robot dynamics, the PID method has great limitations and disadvantages in the high-precision tracking of the desired trajectory. In later studies, nonlinear control was shown to be effective by introducing a dynamic model into the robot control design [8; 9; 10]. The realization of this method depends on the accurate robot dynamic model. However, the robot system is very complex with many uncertainties, including the error of dynamic parameters of links, unmodeled dynamics, joint friction errors, and other uncertainties in the system itself [11; 12; 13], as well as being affected by external uncertainties such as environmental disturbances [14] and variable loads on the end-effector [15]. Therefore, it is difficult to obtain a complete and accurate dynamic model of the robot, which restricts this model-based control method. Typical control methods used to address robot uncertainties in existing research are robust control and adaptive control. Robust control is a deterministic control strategy, i.e., its control parameters are not adjusted online during the control process. The main principle is to assume that the system has a priori known boundaries and use the feedback signalto compensate for the uncertainties, which has the advantages of disturbance rejection and compensation for unmodeled dynamics and non-structural uncertainties [16; 17]. Sliding mode control, which introduces the concept of sliding surfaces, is widely used in robot control as a typical robust control method [18; 19; 20]. However, in the actual control, it is inevitable that the chattering of the control signal will exist, which will damage the actuator and reduce the performance of the system [21]. Using the boundary layer design to replace the switching function with a continuous interpolation function can balance the range of uncertainties and the control effect to a certain extent [22]. [PERSON] [23] proposed an alternative simple linear feedback control which can also avoid the control chattering. Adaptive control is based on the parameterization of the structural uncertainties in the robot. The unknown parameters are estimated online by the adaptive law, and the control strategy is modified at any time by using the estimated values of the parameters to eliminate the parametric uncertainties of the system [24]. On this basis, the introduction of neural networks and fuzzy logic can better estimate and compensate for uncertainties [25]. Nevertheless, the combined algorithm also involves complex high-order calculations, which increase the convergence time and reduce the response speed [26]. [PERSON] et al. presented the incorporation of an adaptive term into robust control to improve the control performance, where the adaptive term is considered as a bounded disturbance and does not affect the stability of the system [27]. This work inspired subsequent researchers to combine robust control and adaptive control. In addition, a control method called model reference adaptive control [28; 29] achieves tracking by introducing a reference model, giving inputs to both the real system and the reference model, recording the output errors of the two systems, and utilizing the errors for adaptive adjustment of the controller's parameters so as to make the errors converge to zero [30; 31; 32]. Many of the methods proposed above ultimately use model-based control [33], i.e., there exists a nominal robot dynamic model, and the uncertainties in the remaining part are compensated for using robust or adaptive control.