# Transition Analysis and Its Application to Global Path Determination for a Biped Climbing Robot
## 4 Fast Determination of Feasible Global Paths
### Principle and Flowchart
From the table, if the accessibility is taken into account, all the operational regions in four cases are shortened, with a percentage from 7.3% to 100%. To further investigate the importance of considering accessibility, we conducted 1000 rounds of comparative transition analysis with randomly-generated members. In total 94.6% of operational regions were narrowed down, with an average shortening percentage of 59.9% at a time cost of 32.7 ms (over 19.6 ms if we do not account for accessibility). In fact, configurations corresponding to the boundaries always suffer from singularity or joint rotation limitation. When planning specific grips subsequently, optimal grips should keep a distance from the boundaries of operational regions. Therefore, it is necessary to consider accessibility in addition to reachability for each grip. ### Result of Global Path Determination

The second part of the simulation is to verify the proposed algorithms used to determine global paths. Two virtual truss scenes, simple and complicated, consisting of 7 and 25 members, respectively, were deployed for validation. Figures 9 and 10 show us the results. In the simulation, the starting \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Characteristics**} & \multicolumn{4}{c}{**Operational Regions \({}^{\text{a}}\)**} & \multirow{2}{*}{**Shortening Percentage \({}^{\text{b}}\)**} \\ \cline{2-3} \cline{5-5}  & & **On** _Member_ **1** & & **On** _Member_ **2** \\ \hline \multirow{3}{*}{Without} & \multirow{3}{*}{\([408,\ 653.7]\)} & Case I & \multirow{3}{*}{\([881.0,\ 229.35]\)} & \multirow{3}{*}{7.3\%} \\ With & & \([426.7,\ 653.7]\) & & \([831.4,\ 229.3]\) & \\ \hline \multirow{3}{*}{Without} & \multirow{3}{*}{\([628.6,\ 801.0]\)} & Case II & \multirow{3}{*}{\([600.0,\ 535.0]\)} & \multirow{3}{*}{100.0\%} \\ With & & \(\backslash\) & & \(\backslash\) \\ \hline \multirow{3}{*}{Without} & \multirow{3}{*}{\([694.0,\ 742.6]\)} & Case III & \multirow{3}{*}{\([760.9,\ 770.1]\)} & Case III & \multirow{3}{*}{\([230.5,\ 212.2]\)} & \multirow{3}{*}{100.0\%} \\ With & & \(\backslash\) & & \(\backslash\) \\ \hline \multirow{3}{*}{Without} & \multirow{3}{*}{\([0,\ 470.6]\)} & Case IV & \multirow{3}{*}{\([837.0,\ 660.0]\)} & \multirow{3}{*}{84.5\%} \\ With & & \([252.0,\ 325.0]\) & & \([742.0,\ 714.5]\) & \\ \hline \hline \end{tabular}

* \({}^{\text{a}}\) Represented with parameters of \(t_{1}\) and \(t_{2}\), respectively. \({}^{\text{b}}\) Computed as \(1-\Delta t_{1}^{\prime}/\Delta t_{1}\), where \(\Delta t_{1}^{\prime}\) and \(\Delta t_{1}\) refer to operational regions with and without the consideration of accessibility.

\end{table} 
Table 2: Comparison of transition analysis results with and without the consideration of accessibility. Figure 8: Results of member-to-member transition analysis: (**a**) without the consideration of grip accessibility; (**b**) considering grip accessibility (no operational region for the cases II and III shown in Sub-figure (**a**)). point and the destination are specified manually but arbitrarily, highlighted with a green and a red sphere, respectively. The feasible global routes are illustrated with member-to-member transitions. Corresponding operational regions for transition are also highlighted with green. Nearby numbers indicate the sequence of transitions to be performed along the way. Sub-figures on the right side are simplified diagrams of routes for easy understanding. Arrows indicate the directions Climbot goes forwards from the starting point to the destination. Figure 9: Results of global path determination in Scene I consisting of 7 members. (**a**) Route I; (**b**) Route II; (**c**) Route III. From Figure 9, there are in total three possible routes globally, passing through 3, 2, and 1 in-between members, respectively. Correspondingly, the robot needs to perform 4, 3, and 2 periods of transition. Among the possible routes, Route III shown in Figure 9c is the best solution, regardless of whether the point of transition numbers or the total path length are evaluated.