# Nonlinear Robust Adaptive Control of Universal Manipulators Based on Desired Trajectory
## 3 Controller Design
### Stability Analysis  
Based on the design given above for each control item of the control system, the stability of the control system is analyzed below.  
**Theorem 4**.: _(Asymptotic Tracking) When the robust adaptive control law in Equation (14) based on the desired trajectory is applied to the trajectory tracking control of the robot, the asymptotic tracking stability of the system can be guaranteed._  
Proof of Theorem 4.: Firstly, according to Equation (26), we agree that \(\left[\left\|r\right\|\left\|e\right\|\right]^{T}=\mu\); then:  
\[\dot{V}_{1}\leq-\mu^{T}Q\mu \tag{31}\]  
The \(\dot{V}_{2}\) and \(\dot{V}_{3}\) in Equation (23) are the derivative terms of the Lyapunov function corresponding to the robust and adaptive control terms, respectively. In addition, due to the introduction of the online estimated joint friction coefficients \(\dot{\theta}_{f}\), it is necessary to add a state variable \(\widetilde{\theta}=\dot{\theta}_{f}-\theta_{f}\) to the Lyapunov function:  
\[V_{4}=\frac{1}{2}\widetilde{\theta}^{T}\Gamma\widetilde{\theta} \tag{32}\]  
Finally, the derivative of the final Lyapunov function of the control system is written as:  
\[\dot{V}=\dot{V}_{1}+\dot{V}_{2}+\dot{V}_{3}+\dot{V}_{4} \tag{33}\]  
Combining Equations (23), (31) and (32), Equation (33) is further written:  
\[\dot{V}\leq-\mu^{T}Q\mu+r^{T}Y_{ld}\bigg{(}\widetilde{\theta}_{l}+\delta \theta_{l}\bigg{)}+r^{T}Y_{fd}\bigg{(}\dot{\theta}_{f}-\theta_{f}\bigg{)}+ \widetilde{\theta}^{T}\Gamma\widetilde{\theta} \tag{34}\]  
Notice that the joint friction coefficients \(\theta_{f}\) can be regarded as constant during a control process, so \(\dot{\widetilde{\theta}}=\dot{\theta}_{f}\). The parameter update law in Equation (30) is substituted into Equation (33) to obtain:  
\[\begin{split}&\dot{V}\leq-\mu^{T}Q\mu+r^{T}Y_{ld}\bigg{(} \widetilde{\theta}_{l}+\delta\theta_{l}\bigg{)}+r^{T}Y_{fd}\widetilde{\theta} -\widetilde{\theta}^{T}\Gamma\Gamma^{-1}Y_{fd}^{T}r\\ &=-\mu^{T}Q\mu+r^{T}Y_{ld}\bigg{(}\widetilde{\theta}_{l}+\delta \theta_{l}\bigg{)}\end{split} \tag{35}\]  
According to the Equation (29) for the robust term \(\delta\theta_{l}\), it is illustrated in two cases.  
* Case 1: When \(\left\|Y_{ld}r\right\|>\epsilon\), substitute it into Equation (35), and  
According to Equation (28), it can be seen that \(\left\|\widehat{\theta_{t}}\right\|-\rho\leq 0\), so the above equation can be written as: \[\dot{V}\leq-\mu^{T}Q\mu<0\] (37) At this time, according to Lyapunov stability theory and the Barbalat lemma, the system is stable, that is, \(e\) and \(r\) globally converge to 0.
* Case 2: When \(\|Y_{id}\mu^{T}\|\leq e\), substitute Equation (2) into Equation (35); then: \[\begin{array}{l}\dot{V}\leq-\mu^{T}Q\mu+w^{T}\left(\widehat{\theta_{t}}- \frac{\rho}{e}w\right)\\ \leq-\mu^{T}Q\mu+\|w\|\rho-\frac{\rho}{e}\|w\|^{2}\\ \leq-\mu^{T}Q\mu+e\frac{\rho}{4}\end{array}\] (38) In order to make the above equation less than 0, it is necessary to satisfy: \[\mu^{T}Q\mu>e\frac{\rho}{4}\] (39) And \(\mu^{T}Q\mu\geq\lambda_{min}(Q)\|\mu\|^{2}\); therefore: \[\lambda_{min}(Q)\|\mu\|^{2}>e\frac{\rho}{4}\] (40) Equivalently, \[\|\mu\|>\left(\frac{\epsilon\rho}{4\lambda_{min}(Q)}\right)^{\frac{1}{2}}\] (41) Since \(\mu=\left[\|r\|\left\|e\right\|\right]^{T}\), \(\dot{V}\) is constantly negative when the above equation is satisfied. Eventually, the state variable of the system will be retained in the final bounded set, i.e., the bounded set surrounded by the right side of the above equation, and thus, the uniform ultimate boundedness of the system is proven. \(\square\)