# Transition Analysis and Its Application to Global Path Determination for a Biped Climbing Robot
## 4 Fast Determination of Feasible Global Paths
### Principle and Flowchart
\)
2: calculate intermediate variables: \(\sigma\), \(\delta\), \(A\sim H\);
3:\(\Delta_{1}\Leftarrow F^{2}-4 EG\), \(\Delta_{2}\Leftarrow F^{2}-4 EH\);
4:if\(\Delta_{1}<0\) or (\(\Delta_{1}=0\) and \(\Delta_{2}>0\)) then // No operational region for transition
5:return\(t_{1}\Leftarrow\emptyset,t_{2}\Leftarrow\emptyset\);
6:elseif\(\Delta_{1}=0\)then // Unique configuration
7:\(t_{1}\Leftarrow-F/(2E)\);
8:elseif\(\Delta_{1}>0\) and \(\Delta_{2}\leq 0\)then // A segment of operational region
9:\(t_{1}\Leftarrow\left[\frac{-F-\sqrt{\Delta_{1}}}{2E},-\frac{F+\sqrt{\Delta_{1}} }{2E}\right]\);
10:elseif\(\Delta_{1}>0\) and \(\Delta_{2}>0\)then // Two segments of operational regions
11:\(t_{1}\Leftarrow\left[\frac{-F-\sqrt{\Delta_{1}}}{2E},\frac{-F-\sqrt{\Delta_{2}} }{2E}\right]\cup\left[\frac{-F+\sqrt{\Delta_{2}}}{2E},\frac{-F+\sqrt{\Delta_{1 }}}{2E}\right]\);
12:endif
13: check accessibility according to Equations (3) and (4);// Similar to Line 1 to 11
14: restrict \(t_{1}\) according to Equation (23);
15: check rotation limitations of other joints and modify \(t_{1}\);
16:if\(t_{1}\
eq\emptyset\)then // Transition feasible
17:\(t_{2}\Leftarrow\sigma t_{1}+\delta\); // Transition infeasible
18:else
19:\(t_{2}\Leftarrow\emptyset\);
20:endif
```

**Algorithm 4** Solving operational regions for transition

Algorithm 4 details the procedure to compute operational regions for each to-be-inspected transition, dispatched from Algorithm 3. The reachability is firstly taken into account to generate one or two segments of regions on _Member 1_ for further inspection, according to Table 1. After that, the resulting operational regions will be shortened if (a) points are not accessible; (b) points are not on _Member 2_; or (c) corresponding movement is beyond the robot joints' rotation limitation. Operational regions on _Member 2_ will be finally computed according to Equation (15). ## 5 Simulations

To verify the proposed analysis and algorithms, simulations are conducted with Climbot. A simulation environment is developed and algorithms are implemented on the platform of MATLAB R2015b. All the simulations are launched on a desktop with Intel Core i7-7700K CPU and 16 GB RAM, running with the 64-bit operating system Windows 10 Pro. ### Result of Transition Analysis

The first part of the simulations is to verify the effectiveness of transition analysis. In this simulation, given two arbitrary squared members in spatial environments, the aim is to solve the operational regions for transition. The two members used for illustration in this paper are

\[{}^{W}\boldsymbol{S}=\left\{ \begin{array}{l}1187\;372\;692\;428\;150\;878\;1 \;0.5\\ 225\;1182\;1331\;550\;807\;603\;1\;9\end{array} \right.^{,} \tag{24}\]

where each row represents a member, with the parameters \(\boldsymbol{P}_{1}\in\mathbb{R}^{3}\), and \(\boldsymbol{P}_{2}\in\mathbb{R}^{3}\) successively standing for positions of end points, and \(\zeta\in\mathbb{R}^{1}\) and \(\psi\in\mathbb{R}^{1}\) indicating the cross-section type and the rotating angle around the member's own axis, respectively. Figure 8 shows the simulation result of transition analysis. Without consideration of the accessibility of grips, there are four possible cases for transiting from _Member 1_ to _Member 2_, with different gripping orientations on the two members, respectively, as shown in Figure 8a. The operational regions are highlighted in green to distinguish them from the other parts of the member. Robots are displayed in specific configurations corresponding to the boundaries of operational regions. Taking the accessibility of grips into account, there remain only two possible cases for transition, as shown in Figure 8b. Moreover, the obtained operational regions in these two possible cases are narrowed down. Shortened parts, painted in red in Figure 8b, must be discarded, owing to corresponding grips being reachable but not accessible. Figure 8: _Cont._Table 2 quantitatively compares the operational regions for transition, computed with and without a consideration of the robot's accessibility.