# 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  
Figure 6 depicts the geometrical structure of a mobile robot with four Swedish wheels for the proposed networked mobile robot system. Compared to the conventional differential-drive (non-holonomic) mobile robots, this kind of mobile robot with omnidirectional capability has superior mobility. The kinematic model of the four-wheeled Swedish mobile robot is expressed by:  
\[\left[ \begin{array}{c}v_{1}(t)\\ v_{2}(t)\\ v_{3}(t)\\ v_{4}(t)\end{array} \right]=\left[ \begin{array}{c}r\omega_{1}(t)\\ r\omega_{2}(t)\\ r\omega_{3}(t)\\ r\omega_{4}(t)\end{array} \right]=T(\theta(t))\left[ \begin{array}{c}\dot{x}( t)\\ \dot{y}(t)\\ \dot{\theta}(t)\end{array} \right], \tag{12}\]  
where:  
\[T(\theta(t))=\left[ \begin{array}{ccc}-\sin(\delta+\theta)&\cos(\delta+ \theta)&L\\ -\cos(\delta+\theta)&-\sin(\delta+\theta)&L\\ \sin(\delta+\theta)&-\cos(\delta+\theta)&L\\ \cos(\delta+\theta)&\sin(\delta+\theta)&L\end{array} \right],\]  
\(\delta\) is \(\pi/4;r\) represents the radius of the Swedish wheel; \(L\) denotes the distance from the Swedish wheel's center to the geometric center of the mobile robot; \(v_{i}(t)\) and \(\omega_{i}(t),i=1,2,3,4\) respectively denote the linear and angular velocities of each omnidirectional wheel. \([x(t)\ y(t)\ \theta(t)]^{T}\) is the pose vector that includes the position and orientation of the mobile robot measured at time \(t\). In mobile robotic research, robots with over three degrees-of-freedom (DOFs) are classified as redundant robots because they provide redundancy. Note that \(T(\theta(t))\) in Equation (12) is singular for any \(\theta\) in this redundant mobile robot system. This study adopts the pseudo inverse matrix approach to address the redundant control problem of mobile robots. Considering the left pseudo-inverse matrix \(T^{\#}(\theta(t))\) of \(T(\theta(t))\) by using \(T^{\#}(\theta(t))P(\theta(t))=I\), the matrix \(T^{\#}(\theta(t))\) is expressed by:  
\[T^{\#}(\theta(t))=\left[ \begin{array}{ccc}\frac{-\sin(\delta+\theta)}{2}& \frac{-\cos(\delta+\theta)}{2}&\frac{\sin(\delta+\theta)}{2}&\frac{\cos( \delta+\theta)}{2}\\ \frac{\cos(\delta+\theta)}{2}&\frac{-\sin(\delta+\theta)}{2}&\frac{-\cos( \delta+\theta)}{2}&\frac{\sin(\delta+\theta)}{2}\\ \frac{1}{4L}&\frac{1}{4L}&\frac{1}{4L}&\frac{1}{4L}\end{array} \right]. \tag{13}\]  
Combining Equations (12) and (13), the kinematics of the four-wheeled omnidirectional mobile robot is derived as follows:  
\[\left[ \begin{array}{c}\dot{x}(t)\\ \dot{y}(t)\\ \dot{\theta}(t)\end{array} \right]=T^{\#}(\theta(t))\left[ \begin{array}{c}v_ {1}(t)\\ v_{2}(t)\\ v_{3}(t)\\ v_{4}(t)\end{array} \right]. \tag{14}\]  
Figure 6: Geometry of the omnidirectional mobile robot with four Swedish wheels. After the kinematics analysis of the Swedish mobile robots, the next step is to design a motion control law and prove its stability using Lyapunov theory. The current pose of the omnidirectional Swedish mobile robot at time \(t\) is defined as \(S=\left[ \begin{array}{cc}x(t)&y(t)&\theta(t)\end{array} \right]^{T}\), and the desired reference trajectory of the Swedish mobile robot is expressed as \(S_{r}=\left[ \begin{array}{cc}x_{r}(t)&y_{r}(t)&\theta_{r}(t)\end{array}  \right]^{T}\). With the two pre-defined vectors, the tracking error of the mobile robot is given by:  
\[S_{e}=\left[ \begin{array}{c}x_{e}(t)\\ y_{e}(t)\\ \theta_{e}(t)\end{array} \right]=\left[ \begin{array}{c}x(t)\\ y(t)\\ \theta(t)\end{array} \right]-\left[ \begin{array}{c}x_{r}(t)\\ y_{r}(t)\\ \theta_{r}(t)\end{array} \right]=S-S_{r}, \tag{15}\]  
which gives:  
\[\dot{S}_{e}=\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}(t)\\ \dot{y}(t)\\ \dot{\theta}(t)\end{array} \right]-\left[ \begin{array}{c}\dot{x}_{r}(t)\\ \dot{y}_{r}(t)\\ \theta_{r}(t)\end{array} \right]=T^{\#}(\theta(t))\left[ \begin{array}{c}r \omega_{1}(t)\\ r\omega_{2}(t)\\ r\omega_{3}(t)\\ r\omega_{4}(t)\end{array} \right]-\left[