# Modular Robotic Design and Reconfiguring Path Planning
## 2 Robotic Design and Modeling
### Kinematic Analysis  
Before a module is assembled, the robot's configuration needs to be described in the form of a mathematical description. It is used as the module's assembly target to guide the module through the reconfiguration tasks. The configuration description of a modular robot needs contains essential information about the module parameters and the inter-module connection relationships [28]. We select the five-DOF reconfigured robot for kinematic analysis. In comparison, it can meet most of the requirements without excessive corresponding loads on the module joints. It consists of three rotating modules, two swing modules, two clamping modules, and six interstitial modules. The coordinate system is established at the geometric center point of each functional module according to the robot configuration characteristics, as shown in Figure 2.  
Set the clamping function module at the left end of the reconstructed robot as the fixed end, and the right end as the free gripping end. The D-H parameter table of the five-DOF configuration robot is analyzed, as shown in Table 2.  
The spatial positional information of the reconfigured robot is expressed by resolving the positional transformations between functional modules by the alignment transformation matrix. The alignment transformation matrix of the reconfigured robot is:  
\[A_{n+1}={}_{n+1}^{\#}T=\left[\begin{array}{cccc}cos\theta_{n+1}&-sin\theta_ {n+1}cos\alpha_{n+1}&sin\theta_{n+1}sin\alpha_{n+1}&a_{n+1}cos\theta_{n+1}\\ sin\theta_{n+1}&cos\theta_{n+1}cos\alpha_{n+1}&-cos\theta_{n+1}sin\alpha_{n+1} &a_{n+1}sin\theta_{n+1}\\ 0&sin\alpha_{n+1}&cos\alpha_{n+1}&d_{n+1}\\ 0&0&0&1\end{array}\right] \tag{1}\]  
\begin{table}
\begin{tabular}{c c c c c c} \hline
**Joint** & **Turn a** & **Length a** & **Offset d** & **Angle \(\theta\)** & **Range** \\ \hline Module 1 & \(0^{\circ}\) & \(0\) & \(d_{1}+d_{2}\) & \(\theta_{1}\) & \(-180\)–\(180^{\circ}\) \\ Module 2 & \(90^{\circ}\) & \(0\) & \(0\) & \(\theta_{2}\) & \(-90^{\circ}\) \\ Module 3 & \(0^{\circ}\) & \(a_{3}\) & \(0\) & \(\theta_{3}\) & \(-90\)–\(90^{\circ}\) \\ Module 4 & \(0^{\circ}\) & \(a_{4}\) & \(0\) & \(\theta_{4}\) & \(-90\)–\(90^{\circ}\) \\ Module 5 & \(-90^{\circ}\) & \(0\) & \(0\) & \(\theta_{5}\) & \(-180\)–\(180^{\circ}\) \\ Module 6 & \(0^{\circ}\) & \(0\) & \(d_{5}+d_{6}\) & \(\theta_{6}\) & \(0^{\circ}\) \\ \hline \end{tabular}
\end{table}
Table 2: Modular robotic D-H parameter table.  
Figure 2: Schematic diagram of modular robotic D-H coordinate.  
When the clamping module of the end of the reconfigured robot is clamped and fixed, the velocity of the end module of the reconfigured robot in the reference coordinate system can be known by the reference [29] and is expressed:  
\[\dot{p}=[\begin{array}{c}v_{e}\\ \omega_{e}\end{array}]=J\left[\begin{array}{c}\dot{\theta}_{1}\\ \vdots\\ \dot{\theta}_{5}\end{array}\right]=J[\dot{\theta}] \tag{2}\]  
where: \"\(p\)\" is the velocity of the reconfigured robotic functional module; \"\(v_{e}\)\" is the linear velocity of the robotic end clamping module; \"\(\omega_{e}\)\" is the angular velocity of the robotic end clamping module; \"\(J\)\" is the Jacobi matrix; \"\([\theta]\)\" is the angular velocity vector of the reconfigured robotic joint.  
The Jacobi matrix \"\(J\)\" can be expressed by Equation (3).  
\[J(\theta)=\left[\begin{array}{ccc}J_{v1}&J_{v2}&\cdots&J_{v5}\\ Q_{1}&Q_{2}&\cdots&Q_{5}\end{array}\right] \tag{3}\]  
in the formula:  
\[Q_{1}=\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right]\text{; }Q_{2}=\left[\begin{array}{c}s_{1}\\ c_{1}\\ 0\end{array}\right]\text{; }Q_{3}=\left[\begin{array}{c}s_{1}\\ -c_{1}\\ 0\end{array}\right]\text{; }Q_{4}=\left[\begin{array}{c}s_{1}\\ c_{1}\\ 0\end{array}\right]\text{; }Q_{5}=\left[\begin{array}{c}c_{1}c_{234}\\ s_{1}c_{234}\\ -s_{234}\end{array}\right]\]  
\[s_{i}{=}\;sin\;\dot{o}_{i}\;c_{i}=cos\dot{o}_{i}\;\text{;}\]