# Nonlinear Robust Adaptive Control of Universal Manipulators Based on Desired Trajectory
## 1 Introduction
Therefore, they are very dependent on the real-time calculation of the robot dynamics, i.e., the position and velocity signals collected in real time are substituted into the dynamic terms for calculation. Yet, the extraordinary complexity of the robot dynamics equations makes it difficult to apply these methods in real industrial systems. In this article, a robust adaptive control method based on the desired trajectory is designed, considering the real-time computational complexity of the control process and the objective uncertainties of universal manipulator. Specifically, a parameter identification of the robot body is carried out prior to the actual control design to obtain the inertia parameters of the links and the friction coefficients of the joints to obtain a nominal dynamic model of the robot. Then, this article classifies the objective uncertainties of universal robot into two categories according to the time span: one is long-term uncertainty, i.e., the inertia parameters of links will be slightly changed after long-term operation, deviating from the inertia parameter during identification; and the other is short-term uncertainty, i.e., the joint friction coefficients, which vary with different working conditions, environments, and even different trajectories. According to the above two types of uncertainties, a robust control term is designed to compensate for the inertia parameters' perturbation of the links, and an adaptive control term is designed to learn the change of the friction coefficients of the joints online during the control process in order to correct them. On this basis, the proposed method innovatively employs the desired trajectory instead of the real-time feedback signal in the traditional method for the computation of the nominal dynamic term and the regression matrix (the core part of the robust adaptive control term), and additionally designs the compensation term in order to mathematically satisfy the Lyapunov stability condition of the system in a rigorous manner. The contributions of this article are summarized as follows. 1. Compared with the traditional robust adaptive control method, which requires real-time computation of the robot's complex dynamic model and regression matrix during the control process, the proposed method computes most of the complex control terms using the a priori known desired trajectory, so that the computation results can be stored before the start of the control and only the stored results need to be queriedduring the real-time control, thus greatly reducing the need for real-time computation of the feedback signals. Therefore, it is guaranteed that the controller can be realized even on universal manipulators with poor computational power. 2. The objective uncertainties of the universal robot are classified according to the time span, and robust adaptive control is used to compensate the error caused by the uncertainties. Combined with the above use of the desired trajectory to calculate most of the control items, an additional feedback control item is designed to strictly prove the Lyapunov stability of the proposed method. 3. Instead of a simulation, experiments are carried out on a real universal manipulator to compare the results with traditional PID, dynamics feedforward control, real-time robust adaptive control, and one state-of-the-art control method. The feasibility and rationality of the control method proposed in this article are analyzed in terms of control performance and real-time computational efficiency. The remainder of the article is structured as follows: Section 2 describes the preliminaries needed for modeling robot dynamics and controller design. In Section 3, the structure of the robust adaptive controller based on desired trajectory is presented and the Lyapunov stability of the control system is proved theoretically. Comparative experiments on an universal manipulator are conducted in Section 4, and the effects are analyzed and discussed as an illustration of the feasibility and superiority of the proposed control method.