# Practical Adaptive Sliding-Mode Control Approach for Precise Tracking of Robot Manipulators
###### Abstract  
We present a practical adaptive sliding-mode control approach, namely, the strong and stable adaptive sliding-mode control (SS-ASMC), in this paper. There is a significant effort towards addressing the technical challenges associated with the switching gains with two adaptive laws, which are called parent and child adaptive laws. A parent adaptive law helps achieve strong switching gains through fast adaptation rate when sliding variable moves away from the sliding manifold. A child adaptive law updates the parameter of the parent adaptive law, which helps to achieve the switching gains with fast and stable adaptation rate in the vicinity of the sliding manifold. Such switching gains with two adaptive laws provide remarkably precise tracking performance while enhancing the robustness. Besides, to yield desirable closed-loop poles and simplicity of control approach structure, the proposed SS-ASMC approach employs a combination of time-delayed estimation and pole-placement method, which makes it unnecessary to have a rather complete system dynamics. It is shown by the bounded-input-bounded-output stability through the Lyapunov approach, and thus the tracking errors are also proved to be uniformly ultimately bounded. The effectiveness of the proposed SS-ASMC approach is illustrated in simulations with robot manipulators, which is compared with that of the existing control approaches. adaptive sliding-mode control; adaptive law; adaptation rate; time-delayed estimation; robot manipulator +
Footnote †: journal: Applied Soft  
system instability. Linear quadratic regulator (LQR) control approaches [3; 4] have been studied to be designed in the optimal parameters. These control approaches have a simple structure and therefore offer ease-of-use in the robot manipulators. However, the LQR control approaches may deteriorate the tracking performance because they may degrade the robustness when the undesirable side effects, e.g., internal disturbances and unstructured/structured uncertainties, occur in the robot manipulators. H\({}_{\infty}\) control approaches [5; 6] have been developed to overcome the above-mentioned problems. They have provided a result in increasing the robustness against the internal disturbances. However, given that H\({}_{\infty}\) control approaches depend on feasible sets of solutions, i.e., optimal parameters, to the strict groups of linear matrix inequality and Riccati equation, they construct a class of complex control structure. Besides, all control approaches described above should be required in establishing an exact mathematical system model for the robot manipulators. Unfortunately, it is impossible to obtain the exact system model in real robot manipulators. Furthermore, they require the linearization method so that the requirement of prior system information is indisputable, and they are only governed by all of the linear subsystems at each operating point. To summarize, as a practical problem under the occurrence of unpredictable system uncertainties, they may still face a problem in achieving the enhanced robustness against undesired side effects, including parameter variations and external disturbances. Conventional sliding-mode control (SMC) approaches have been designed to remedy the above-mentioned problems. They have mainly obtained much attention for their enhanced robustness even in the presence of a significant class of uncertainties and disturbances. However, the conventional SMC approaches require switching gains with large positive constants, yielding the chattering in the sliding surface which may excite high-frequency oscillation [7; 8]. They produce a result in limiting the life cycle of the robot manipulators due to wear of actuators. This is why the conventional SMC approaches are not used well in real robot manipulators [9]. To avoid these problems, the boundary-layer SMC approaches [10; 11] have been employed as a way to replace the signum function which is typically discontinuous function in the property of the conventional SMC.