# Biologically-Inspired Learning and Adaptation of Self-Evolving Control for Networked Mobile Robots
## 3 Application to Self-Evolving Control of Networked Mobile Robots
### Modeling and Lyapunov-Based Control
\begin{array}{c}\dot{x}_{r}(t)\\ \dot{y}_{r}(t)\\ \dot{\theta}_{r}(t)\end{array} \right]. \tag{16}\]  
The goal of control law design is to derive the angular velocity vector \(\left[ \begin{array}{cc}\omega_{1}(t)&\omega_{2}(t)&\omega_{3}(t)&\omega_{4}( t)\end{array} \right]^{T}\) for tracking the desired differentiable trajectory \(\left[ \begin{array}{cc}x_{r}(t)&y_{r}(t)&\theta_{r}(t)\end{array} \right]^{T}\) with asymptotical stability. Based on the PID (Proportional-Integral-Derivative) control strategy, the following redundant control law is proposed:  
\[\left[ \begin{array}{c}v_{1}(t)\\ v_{2}(t)\\ v_{3}(t)\\ v_{4}(t)\end{array} \right]=T(\theta(t))\left(-K_{P}\left[ \begin{array}{c}x_{ e}(t)\\ y_{e}(t)\\ \theta_{e}(t)\end{array} \right]-K_{I}\left[ \begin{array}{c}\int_{0}^{t}x_{e}( \tau)d\tau\\ \int_{0}^{t}y_{e}(\tau)d\tau\\ \int_{0}^{t}\theta_{e}(\tau)d\tau\end{array} \right]-K_{D}\left[ \begin{array}[] {c}\dot{x}_{e}(t)\\ \dot{y}_{e}(t)\\ \dot{\theta}_{e}(t)\end{array} \right]+\left[ \begin{array}{c}\dot{x}_{r}(t) \\ \dot{y}_{r}(t)\\ \dot{\theta}_{r}(t)\end{array} \right]\right), \tag{17}\]  
where \(K_{P},K_{I}\) and \(K_{D}\) are the control matrices. They are diagonal and positive, thus \(K_{P}=diag[k_{xp}\;k_{yp}\;k_{\theta p}]\), \(K_{I}=diag[k_{xi}\;k_{yi}\;k_{\theta i}]\), and \(K_{D}=diag[k_{xd}\;k_{yd}\;k_{\theta d}]\). By substituting Equations (17) into (16), the closed-loop error system is obtained:  
\[\dot{S}_{e}=\left[ \begin{array}{c}\dot{x}_{e}(t)\\ \dot{y}_{e}(t)\\ \dot{\theta}_{e}(t)\end{array} \right]=\left(-K_{P}\left[ \begin{array}{c}x_{ e}(t)\\ y_{e}(t)\\ \theta_{e}(t)\end{array} \right]-K_{I}\left[ \begin{array}{c}\int_{0}^{t}x_{e}( \tau)d\tau\\ \int_{0}^{t}y_{e}(\tau)d\tau\\ \int_{0}^{t}\theta_{e}(\tau)d\tau\end{array} \right]-K_{D}\left[ \begin{array}[] {c}\dot{x}_{e}(t)\\ \dot{y}_{e}(t)\\ \dot{\theta}_{e}(t)\end{array} \right]\right). \tag{18}\]  
To prove the asymptotical stability of the closed-loop error system in (18) via Lyapunov theory, the following Lyapunov function is selected:  
\[V(t)=\mathbb{1}\left[ \begin{array}{cc}x_{e}(t)&y_{e}(t)&\theta_{e}(t)\end{array}  \right]\left[ \begin{array}{c}\dot{x}_{e}(t)\\ \dot{y}_{e}(t)\\ \theta_{e}(t)\end{array} \right]+\mathbb{1}\left[ \begin{array}{cc}\int_{0}^{t}x _{e}(\tau)d\tau&\int_{0}^{t}y_{e}(\tau)d\tau&\int_{0}^{t}\theta_{e}(\tau)d\tau \end{array} \right]K_{I}\left[ \begin{array}{c}\int_{0}^{t}y_{e}(\tau)d\tau\\ \int_{0}^{t}y_{e}(\tau)d\tau\end{array} \right]+\mathbb{1}\left[ \begin{array}{ cc}x_{e}(t)&y_{e}(t)&\theta_{e}(t)\end{array} \right]K_{D}\left[ \begin{array}{c} \dot{x}_{e}(t)\\ \dot{y}_{e}(t)\\ \theta_{e}(t)\end{array} \right]>0\]  
one obtains:  
\[\dot{V}(t)=-\left[ \begin{array}{cc}x_{e}(t)&y_{e}(t)&\theta_{e}(t)\end{array}  \right]K_{P}\left[ \begin{array}{c}x_{e}(t)\\ y_{e}(t)\\ \theta_{e}(t)\end{array} \right]<0.\]  
Since \(\dot{V}\) is negative definite, the asymptotical stability is therefore proven. The proposed motion control law can steer the mobile robot to achieve \(S\to S_{r}\) as \(t\rightarrow\infty\).