# Practical Adaptive Sliding-Mode Control Approach for Precise Tracking of Robot Manipulators
## 2 Proposed SS-ASMC Approach
### The Dynamics of Robot Manipulator
\(\hat{\hat{K}}_{i,t}^{s}\) is the parent adaptive law of the switching gain \(\hat{\hat{K}}_{i,t}^{s}\). \(\hat{\eta}_{i,t}\) is called \"update parameter\" whose the derivative is defined as \(\hat{\eta}_{i,t}\), and then it has a upper value \(\bar{\eta}_{i}^{*}\) and a lower value \(\underline{\eta}_{i}^{*}\) for all \(i=1,2,\cdots,n\). \(\bar{\epsilon}\) can determine that the switching gain \(\hat{\mathbf{K}}_{i}^{s}\) increases or decreases. \(\bar{\delta}_{i}\) and \(\bar{\gamma}_{i}\) can adjust the adaptation rate of the switching gain \(\hat{\mathbf{K}}_{t}^{s}\) and the update parameter \(\hat{\eta}_{i,t}\), respectively. \(\bar{\mu}\) is set to be restricted in the adaptation rate of the switching gain \(\hat{\mathbf{K}}_{t}^{s}\) because the excessive adaptation rate may cause the control input fluctuation making the system instability. \(a\) is also set to the positive value for the sake of changes in the update parameter \(\hat{\eta}_{i,t}\). The signum function \(\text{sgn}(s_{i,t})\) is defined by  
\[\text{sgn}(s_{i,t})= \begin{cases}1&\text{if}\quad s_{i,t}\geq 0\\ -1&\text{if}\quad s_{i,t}<0.\end{cases}  \tag{15}\]  
As shown in Equations (13) and (14), both the parent adaptive law \(\hat{\hat{K}}_{i,t}^{s}\) and the child adaptive law \(\hat{\eta}_{i,t}\) do not require the information on the uncertain terms, which can be adjusted in accordance with the magnitude of the sliding variable. The parent adaptive law \(\hat{\hat{K}}_{i,t}^{s}\) is set to be proportional to the magnitude of the sliding variable. If the sliding variables \(\mathbf{s}_{t}\) are far from the sliding manifold, i.e., \(\|\mathbf{s}_{t}\|_{\infty}\geq\bar{\epsilon}\), the adaptive law \(\hat{\hat{K}}_{i,t}^{s}\) is sufficiently large for all \(i=1,2,\cdots,n\), and therefore the switching gain \(\hat{\mathbf{K}}_{t}^{s}\) will be increased quickly through fast adaptation rate. It provides a result that the sliding variables \(\mathbf{s}_{t}\) go to the sliding manifold more quickly owing to the large switching gain. As another point, it means that the sliding variables \(\mathbf{s}_{t}\) enter the vicinity of the sliding manifold. However, the large switching gain may cause the undesirable side effect, e.g., chattering, when the sliding variable \(\mathbf{s}_{t}\) stays the vicinity of the sliding manifold. To alleviate this negative effect, the boundedness \(\bar{\epsilon}\) is set to decrease the switching gain \(\hat{\mathbf{K}}_{t}^{s}\) when \(\|\mathbf{s}_{t}\|_{\infty}<\bar{\epsilon}\). However, the adaptation rate of the switching gains \(\hat{\mathbf{K}}_{t}^{s}\) may be slow because it is directly proportional to the magnitude of the sliding variables. In other words, when the sliding variables \(\mathbf{s}_{t}\) stay in the boundedness \(\bar{\epsilon}\), inappropriately large switching gain may still cause the chattering that generates serious problems such as wear in the motor of the robot manipulator. To handle all aspects of this matter, the proposed control approach proposes a new update parameter \(\hat{\eta}_{i,t}\) whose derivative, i.e., \(\hat{\hat{\eta}}_{i,t}\), can be generated when the sliding variable \(\mathbf{s}_{t}\) stays in the vicinity of the sliding manifold. Therefore, the update parameter \(\hat{\eta}_{i,t}\) is asymptotically increased for decreasing the switching gain \(\hat{\mathbf{K}}_{t}^{s}\) quickly. Then, given that the update parameter \(\hat{\eta}_{i,t}\) can be increased to be upper-bounded, it generates a finite value which helps to avoid the excessive adaptation rate of the switching gain \(\hat{\mathbf{K}}_{t}^{s}\). As a result, the proposed SS-ASMC approach provides the strong and stable switching gains without the undesirable side effects, including chattering, owing to the parent adaptive law \(\hat{\hat{K}}_{i,t}^{s}\) and the child adaptive law \(\hat{\eta}_{i,t}\).