# Practical Adaptive Sliding-Mode Control Approach for Precise Tracking of Robot Manipulators
## 3 Simulation
### Simulation Result  
Figure 4 shows the update parameters that are time-varying parameters inherent in the parent adaptive law (Equation (13)). Given that the switching gains (Figure 5) can be sufficiently increased to suppress the undesirable side effects through fast adaptation rate, the update parameters move towards its low bound when the rotational direction of motor of the robot manipulator is changed at the moment. On the other hand, the update parameters are increased in the vicinity of the sliding manifold and therefore help decreasing the switching gains (Figure 5) while generating fast and stable adaptation rate. It results in producing proper switching gains without any negative effects including the chattering, as shown in Figure 5.  
Figure 5 shows a comparison between the switching gains of the proposed SS-ASMC approach and the existing ASMC approach. As shown in Figure 5a,b, the switching gains of the proposed SS-ASMC approach are smoothly increased or decreased depending on whether the sliding variables (Figure 6) are close to the sliding manifold or not. By contrast, as shown in Figure 5c,d, the existing ASMC approach can suppress the disturbances temporarily through powerful switching gains. It seems to be superior to the tracking performance at the moment large disturbances occur because their switching gains adopt larger values than those of the proposed SS-ASMC approach. In other words, it  
Figure 4: Update parameters generated by the proposed SS-ASMC approach: (**a**) Joint 1 (\(\times 10^{2}\)). (**b**) Joint 2 (\(\times 10^{2}\)). Figure 5: A comparison between the switching gains of the proposed SS-ASMC approach and the existing ASMC approach: (**a**) Switching gain of the proposed SS-ASMC approach for the Joint 1 (\(\times 10^{2}\)). (**b**) Switching gain of the proposed SS-ASMC approach for the Joint 2 (\(\times 10^{2}\)). (**c**) Switching gain of the ASMC approach for the Joint 1 (\(\times 10^{2}\)). (**d**) Switching gain of the ASMC approach for the Joint 2 (\(\times 10^{2}\)). Figure 3: Command joint angle trajectories with sinusoidal signal: (**a**) Joint 1. (**b**) Joint 2.  
is observed that the switching gains of the proposed SS-ASMC approach are slower when compared to those of the existing ASMC approach. However, there are residual negative effects because of the effect of inertia in the system, as shown in Figure 7. Moreover, the existing ASMC approach only depends on the convergence rate of the TDC approach in the transient sections because of the switching gains in the excessive adaptation rate. It leads to a result of the insufficient behavior in the tracking performance, as shown in Figure 7. Fortunately, the residual switching gains of the proposed SS-ASMC approach can solve the problems in the existing ASMC approach. The switching gains remaining during the decrease do not cause undesired side effects, which result in the chattering-free switching gains. Moreover, given that the switching gains of the proposed SS-ASMC approach are appropriately adjusted through fast and stable adaptation rate, it can be observed that they help to improve the nominal tracking performance in transient sections where the disturbances does not occur directly, as shown in Figure 7.  
Figure 6 shows the sliding variables of the proposed SS-ASMC approach which increase temporarily when the disturbances occur. When the sliding variables leave near sliding manifold, it is strongly influenced by the switching gains of the SS-ASMC approach. Since then, the sliding variables are close to the sliding manifold. It is observed that the magnitude on the sliding variables tend to be similar to that on the tracking errors, as shown in Figure 7.  
Figure 7 shows the tracking errors generated from the TDC approach, the existing ASMC approach, and the proposed SS-ASMC approach. As shown in Figure 7, the TDC approach is sufficiently affected by both unmatched/matched disturbances and the changes in the payload on the end-effector of the robot manipulator.