# Practical Adaptive Sliding-Mode Control Approach for Precise Tracking of Robot Manipulators
## 2 Proposed SS-ASMC Approach  
### The Dynamics of Robot Manipulator  
The \(n\)-axes robot manipulator model [20; 21; 22] can be described as follows,  
\[\mathbf{M}(\mathbf{q}_{t})\ddot{\mathbf{q}}_{t}+\mathbf{C}(\mathbf{q}_{t}, \dot{\mathbf{q}}_{t})\dot{\mathbf{q}}_{t}+\mathbf{g}(\mathbf{q}_{t})+\mathbf{ f}(\mathbf{q}_{t},\dot{\mathbf{q}}_{t})=\boldsymbol{\tau}_{t}+\boldsymbol{\tau}_{d,t} \tag{1}\]  
where \(\mathbf{q}_{t}=(q_{1,t},q_{2,t},\cdots,q_{n,t})\in\Re^{n}\), \(\dot{\mathbf{q}}_{t}=(q_{1,t},q_{2,t},\cdots,\dot{q}_{n,t})\in\Re^{n}\), and \(\ddot{\mathbf{q}}_{t}=(\ddot{q}_{1,t},\ddot{q}_{2,t},\cdots,\ddot{q}_{n,t}) \in\Re^{n}\) are the joint angle, the joint angular velocity, and the joint angular acceleration, respectively. \(\mathbf{M}(\mathbf{q}_{t})\in\Re^{n\times n}\) is the symmetric positive definite that is called the moment of inertia. \(\mathbf{C}(\mathbf{q}_{t},\dot{\mathbf{q}}_{t})\in\Re^{n\times n}\) is the Coriolis matrix. \(\mathbf{g}(\mathbf{q}_{t})\) is the gravity matrix. \(\mathbf{f}(\mathbf{q}_{t},\dot{\mathbf{q}}_{t})\in\Re^{n}\) is the friction matrix. Multiplying \(\mathbf{M}^{-1}(\mathbf{q}_{t})\) by both sides in Equation (1), we have  
\[\ddot{\mathbf{q}}_{t}=-\mathbf{M}^{-1}(\mathbf{q}_{t})\big{[}\mathbf{C}(  
manipulator. To obtain the desired performance in tracking the trajectory for each joint of the robot manipulator, we shall employ  
\[\mathbf{s}_{t}=\dot{\mathbf{e}}_{t}+\tilde{\mathbf{K}}_{s}\mathbf{e}_{t} \tag{6}\]  
where \(\mathbf{s}_{t}=(s_{1},s_{2},\cdots,s_{n})\in\Re^{n}\) is the sliding variable. \(\tilde{\mathbf{K}}_{s}=\text{diag}(\tilde{K}_{s,1},\tilde{K}_{s,2},\cdots, \tilde{K}_{s,n})\in\Re^{n\times n}\) are the diagonal positive design parameter that is called \"sliding gain\". It is noted that the sliding gain \(\tilde{\mathbf{K}}_{s}\) is directly related to the rate at which the tracking error \(\mathbf{e}_{t}\) converges to zero, assuming the sliding variable \(\mathbf{s}_{t}\) is zero. From Equation (6), we construct the following control approach as  
\[\mathbf{\tau}_{t} =-\tilde{\mathbf{M}}\big{[}\hat{\mathbf{\mathcal{L}}}_{t}+\bar{ \mathbf{q}}_{d,t}+\tilde{\mathbf{K}}_{s}\mathbf{e}_{t}+\tilde{\mathbf{K}}_{s} \mathbf{s}_{t}\big{]}\] \[=-\tilde{\mathbf{M}}\big{[}\hat{\mathbf{\mathcal{L}}}_{t}+\bar{ \mathbf{q}}_{d,t}+\big{(}\tilde{\mathbf{K}}_{s}+\tilde{\mathbf{K}}_{g}\big{)} \dot{\mathbf{e}}_{t}+\tilde{\mathbf{K}}_{s}\tilde{\mathbf{K}}_{g}\mathbf{e}_ {t}\big{]} \tag{7}\]  
where \(\tilde{\mathbf{K}}_{g}=\text{diag}(\tilde{K}_{g,1},\tilde{K}_{g,2},\cdots, \tilde{K}_{g,n})\in\Re^{n\times n}\) is the diagonal positive design parameter. The estimation value \(\hat{\mathbf{\mathcal{L}}}_{t}\in\Re^{n}\) can be obtained from one-sample delayed measurement of \(\mathbf{\mathcal{L}}_{t}\in\Re^{n}\), which is called time-delayed estimation (TDE), and then we have  
\[\hat{\mathbf{\mathcal{L}}}_{t}\cong\mathbf{\mathcal{L}}_{t-L}=\bar{\mathbf{q}}_{t-L}- \tilde{\mathbf{M}}^{-1}\mathbf{\tau}_{t} \tag{8}\]  
where \(L\) is the sampling time. In other words, the smaller the sampling time \(L\) is, the higher the accuracy of estimation value in \(\hat{\mathbf{\mathcal{L}}}_{t}\) is. As known in some papers [26, 27, 28], the sampling time \(L\) yields the positive effects above 100 Hz, and therefore will also be set above 100 Hz in the proposed control approach.