# Design, Fabrication, and Control Algorithm of Self-Reconfigurable Modular Intelligent Vehicles
## 3 Kinematic Analysis  
In the kinematic model analysis, the motion of the modular vehicle in the horizontal plane can be divided into three independent DOFs: X-axis translation, Y-axis translation, and rotation around the Z-axis (yaw angle). Based on the inverse kinematics model, the corresponding relationship between the four Mecanum wheels and the motion state of the modular car is established, and the forward kinematics model should also satisfy this relationship [12]. First, the coordinate system is established at the geometric center of the modular vehicle. The forward movement direction of the modular car is the positive direction of the X-axis, the left translation direction of the modular car is the positive direction of the Y-axis, and the vertical modular car is the positive direction of the Z-axis, as illustrated in Figure 9. Assume that the linear velocity of the modular vehicle is \(v\); \(V_{x}\) and \(V_{y}\) are components of \(v\) in the X- and Y-axis, the rotational angular velocity of the modular vehicle is \(w\); \(V_{m}\) (\(n\) = 0,1,2,3) is the rotational angular velocity; \(V_{wn}\) (\(n\) = 0,1,2,3) is the vector sum of \(v\) and \(V_{m}\) in the world coordinate, which represents the velocity vector of each Mecanum wheel; a and b are 1/2 of the distance between Mecanum wheels in longitudinal and lateral directions. Assume that (i) the Mecanum wheel is closely attached to the ground and there is no idle, and (ii) the geometry center is identical to the gravity center [13]. The resultant speed of each Mecanum wheel \(V_{wn}\) is:  
\[\vec{V_{wn}}=\vec{V_{m}}+\vec{v} \tag{1}\]  
Figure 8: Magnetic suction structure. Therefore, the resultant speed on each wheel is:  
\[ \begin{cases}\overrightarrow{V_{w0}}=\left[\begin{array}{c}V_{wx0}\\ V_{wy0}\\ \overrightarrow{v_{w1}}=\left[\begin{array}{c}V_{wx1}\\ V_{wy1}\\ \overrightarrow{v_{w2}}=\left[\begin{array}{c}V_{wx2}\\ V_{wy2}\\ \overrightarrow{v_{w3}}\end{array}\right]=\overrightarrow{V_{r2}}+\overrightarrow {v}=\left[\begin{array}{c}V_{x}-aw\\ V_{y}+bw\\ \overrightarrow{v_{w1}}+\overrightarrow{v}=\left[\begin{array}{c}V_{x}-aw\\ V_{x}-aw\\ V_{y}-bw\\ \overrightarrow{v_{w2}}\end{array}\right]=\overrightarrow{V_{r2}}+\overrightarrow {v}=\left[\begin{array}{c}V_{x}+aw\\ V_{y}+bw\\ V_{x}+aw\\ V_{y}-bw\end{array}\right]\end{array}\right.\end{cases}  \tag{2}\]  
Assume that the unit vector alongside the roller is:  
\[\overrightarrow{V}_{wn}\parallel=\overrightarrow{V}_{wn}\cdot\vec{u_{n}} \tag{3}\]  
Each \(u_{n}\) is related to the orientation of the roller. The corresponding \(u_{n}\) of each Mecanum wheel is:  
\[\overrightarrow{u_{0}}=\left[ \begin{array}{c}\frac{\sqrt{2}}{2}\\ -\frac{\sqrt{2}}{2}\end{array} \right]\overrightarrow{u_{1}}=\left[ \begin{array} []{c}\frac{\sqrt{2}}{2}\\ \frac{\sqrt{2}}{2}\end{array} \right]\overrightarrow{u_{2}}=\left[ \begin{array} []{c}\frac{\sqrt{2}}{2}\\ \frac{\sqrt{2}}{2}\end{array} \right]\overrightarrow{u_{3}}=\left[ \begin{array} []{c}\frac{\sqrt{2}}{2}\\ -\frac{\sqrt{2}}{2}\end{array} \right] \tag{4}\]  
The velocity component of the four Mecanum wheels parallel to the roller direction is:  
\[ \begin{cases}\overrightarrow{V}_{w0}\parallel=\overrightarrow{V}_{w0}\cdot \overrightarrow{u_{0}}=\frac{\sqrt{2}}{2}V_{wx0}-\frac{\sqrt{2}}{2}V_{wy0}\\ \overrightarrow{V}_{w1}\parallel=\overrightarrow{V}_{w1}\cdot \overrightarrow{u_{1}}=\frac{\sqrt{2}}{2}V_{wx1}+\frac{\sqrt{2}}{2}V_{wy1}\\ \overrightarrow{V}_{w2}\parallel=\overrightarrow{V}_{w2}\cdot \overrightarrow{u_{2}}=\frac{\sqrt{2}}{2}V_{wx2}+\frac{\sqrt{2}}{2}V_{wy2}\\ \overrightarrow{V}_{w3}\parallel=\overrightarrow{V}_{w3}\cdot \overrightarrow{u_{3}}=\frac{\sqrt{2}}{2}V_{wx3}-\frac{\sqrt{2}}{2}V_{wy3} \end{cases}  \tag{5}\]  
The relationship of each wheel between the velocity component and the rotation speed \(w_{n}\) is:  
\[w_{n}=\frac{1}{r}\frac{\overrightarrow{V}_{w0}\parallel}{\cos\frac{\pi}{4}} \tag{6}\]  
Figure 9: Kinematic analysis of a modular vehicle. The speed of each Mecanum wheel is:  
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