## 2 Adaptive Time-Delayed Control Schemes  
### Conventional Adaptive Time-Delayed Control Scheme  
The mathematical model of single-input single-output (SISO) robot manipulator can be described as  
\[M(q_{t})\dot{q}_{t}+C(q_{t},\dot{q}_{t})\dot{q}_{t}+g(q_{t})+f(\dot{q}_{t})= \tau_{t}+\tau_{d,t} \tag{1}\]  
where \(q_{t},\dot{q}_{t},\dot{q}_{t}\) are the joint angle, joint angular velocity, joint angular acceleration, respectively. \(M(q_{t})\) is the moment of inertia, \(C(q_{t},\dot{q}_{t})\dot{q}_{t}\) is the Coriolis force, \(g(q_{t})\) is the gravity force, and \(f(\dot{q}_{t})\) is the friction force. Finally, \(\tau_{t}\) and \(\tau_{d,t}\) are the control torque input and the disturbance torque, respectively. A constant \(\bar{M}\) is introduced in Equation (1), resulting in the following equation:  
\[\tau_{t}=C(q_{t},\dot{q}_{t})\dot{q}_{t}+g(q_{t})+f(\dot{q}_{t})-\tau_{d,t}+ \big{(}M(q_{t})-\bar{M}\big{)}\dot{q}_{t}+\bar{M}\dot{q}_{t} \tag{2}\]  
where \(\bar{M}\) will be introduced shortly in this section. In this paper, we applied the following well-known assumptions [23]:  
* \(\mu_{m}\leq|M(q_{t})|\leq\mu_{M}\) for certain positive values \(\mu_{m}\) and \(\mu_{M}\)
* \(|\circ_{t}-\circ_{t-L}|\leq\circ_{M}\) for a small sampling time \(L\) and the upper-bounded value \(\circ_{M}\). Representing Equation (2) in a compact and simple form yields  
\[\dot{q}_{t}=N_{t}+\bar{M}^{-1}\tau_{t} \tag{3}\]  
where \(N_{t}=-\bar{M}^{-1}\{C(q_{t},\dot{q}_{t})\dot{q}_{t}+g(q_{t})+f(\dot{q}_{t})- \tau_{d,t}-\big{(}M(q_{t})-\bar{M}\big{)}\dot{q}_{t}\}\). As it is difficult to obtain \(N_{t}\) in industrial fields, we use its estimate \(\hat{N}_{t}\)[24; 25] as follows:  
\[\hat{N}_{t}\approx N_{t-L}=\ddot{q}_{t-L}-\bar{M}^{-1}\tau_{t-L} \tag{4}\]  
which is one-sampled delayed estimation based on the sampling time \(L\). It is called time-delayed estimation (TDE). \(\ddot{q}_{t-L}\) is the one-sample delayed joint angular acceleration of \(\ddot{q}_{t}\). As a simple estimate of \(N_{t}\) in Equation (4), the conventional adaptive time-delayed control (ATDC) scheme [21] can be constructed as  
\[\tau_{t}^{A}=-\hat{M}_{t}\tilde{q}_{t-L}+\tau_{t-L}^{A}+\hat{M}_{t}\big{(}\tilde {q}_{d,t}+K_{s}\dot{e}_{t}+\beta s_{t}+\bar{K}\text{sgn}(s_{t})\big{)} \tag{5}\]  
where \(\tau_{t}^{A}\) is the control input of the conventional ATDC and \(s_{t}\) is the sliding variable to be determined in the next section. The adaptive control gain \(\hat{M}_{t}\) is defined as  
\[\hat{M}_{t}=\left\{ \begin{array}{lcl}\bar{M}(1+a^{A}\omega_{t})&\text{if}& \
u_{t}\geq 0\\ \bar{M}&\text{if}&\
u_{t}<0\end{array} \right. \tag{6}\]  
where \(\
u_{t}=s_{t}\big{[}-\tilde{q}_{t-L}+\tilde{q}_{d,t}+K_{s}\dot{e}_{t}+\beta s _{t}+\bar{K}\text{sgn}(s_{t})\big{]}\) and \(a^{A}\) is a positive value. Subsequently, the adaptive law \(\dot{\omega}_{t}\) is  
\[\dot{\omega}_{t}=\left\{ \begin{array}{lcl}-\bar{a}^{A}|s_{t}|&\text{if}& \omega_{t}\geq\omega^{*}\\ \bar{a}^{A}|s_{t}|\text{sgn}(|s_{t}|-\bar{\epsilon})&\text{if}&0<\omega_{t}< \omega^{*}\\ \bar{a}^{A}|s_{t}|&\text{if}&\omega_{t}=0\end{array} \right. \tag{7}\]  
where \(\bar{a}^{A}\) and \(\bar{\epsilon}\) are positive design values. \(\omega_{t}\) is a positive update parameter, which is upper-bounded by \(\omega^{*}\). The stability of the conventional ATDC of Equation (5) is guaranteed under the following Lemma 1. Lemma 1: _[_21_]_ _If the control gain \(\hat{M}_{t}\) inEquation (6) is chosen to satisfy the following criteria:_  
\[\big{|}\big{[}1-M^{-1}(q_{t})\bar{M}\big{]}\cdot\hat{M}_{t}\hat{M}_{t-L}^{-1} \big{|}<1\]  
_for all time \(t>0\), then the \(|\tilde{q}_{t}-\tilde{q}_{t-L}|\to 0\) as \(L\to\infty\) and the TDE errors \(E_{t}\) are bounded by constant \(E^{*}\), i.e., \(|E_{t}|=|N_{t}-N_{t-L}|\leq E^{*}\)._  
It implies that the control gain \(\hat{M}_{t}\) should be chosen to guarantee the boundedness of the TDE error \(E_{t}\).