# Optimal Collision-Free Grip Planning for Biped Climbing Robots in Complex Truss Environment
## 3 Computation of Operational Regions for Negotiating Obstacles
### Constraints  
#### 4.2.1 Moving Distance and the Gaits  
Benefiting from the bipedal climbing pattern inspired by arboreal primates, BiCRs normally have several climbing gaits. Climbot, for example, has at least three basic gaits, namely the inchworm-like gait, the swinging-around gait, and the flipping-over gait [8]. Each gait has its own characteristics, among which the step length plays a key role in the determination of the required number of climbing steps given a moving distance. Table 1 summarizes the minimum and maximum step lengths of each gait. In the table, the so-called hybrid gait refers to a three-step climbing pattern derived from the basic gaits, as shown in Figure 7.  
To go through a given distance, i.e., from a landing segment to a takeoff segment on a certain member, Climbot always can climb with a combination of different gaits. Based on Table 1, we analyze the relationship between the moving distance and the minimum number of required climbing steps as follows.  
* DiMov (direct movement) gait: the moving distance is equal to \(S_{0}\). In this case, the landing segment partially (or completely) overlaps with the takeoff segment. Climbot can pass through  
\begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Gaits**} & \multicolumn{2}{c}{**Step Lengths**} \\ \cline{2-3}  & **Minimum** & **Maximum** \\ \hline Inchworm-like gait & \(S_{0}=0\) & \(S_{1}=S_{3}-S_{2}\) \\ Hybrid gait & \(S_{1}\) & \(S_{2}=2l_{2}(1-cos(\theta_{max}/2))\) \\ Swinging-around or & \(S_{2}\) & \(S_{3}=\sqrt{(2l_{2})^{2}-l_{safe}^{2}}\) \\ Flipping-over gait & \(S_{3}\) & \(S_{4}=2l_{2}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Minimum and maximum step lengths with different gaits.  
Figure 7: The three-step hybrid gait. The solid lines represent the current configuration, while the dashed ones stand for the following configuration one step forward.  
just with one grip performing two climbing cycles continuously. As a result, the required number of climbing steps in this case is 0.
* Inch gait: the moving distance is within \((S_{0},S_{1})\). Only the inchworm-like gait is applied to this condition. At least two steps (stretching and shrinking) are required when climbing with the Inch gait.
* SaFo gait: the moving distance is within \((S_{2},S_{3})\). One step climbing with the swinging-around gait or flipping-over gait meets the distance well.
* Hyb gait: the moving distance is within \((S_{1},S_{2})\) or \((S_{3},S_{4})\). Under such circumstances, a mixture of the inchworm-like gait and swinging-around or flipping-over gait should be applied (hybrid gait). Figure 7 illustrates the climbing patterns with Climbot. It moves a step with the swinging-around gait or the flipping-over gait, and then climbs two steps with the inchworm-like gait. Thus, at least three steps are required.
* SaFo gait: the moving distance is larger than \(S_{4}\). The minimum number of climbing steps can be calculated as \([S_{k}/S_{3}]\), where the symbol \([\ ]\) represents the ceiling operation. Accordingly, the robot moves with the swinging-around gait or the flipping-over gait.  
In summary, the mathematical relationship between the minimum number of climbing steps \(N_{k}\) and the corresponding moving distance \(S_{k}\) on a member can be expressed as a piecewise function,  
\[g(^{j}x_{k},\ ^{j}y_{k})=N_{k}=\left\{\begin{array}{ll}0,&S_{k}=0,\\ 2,&S_{k}\in(S_{0}\ S_{1}],\\ 3,&S_{k}\in(S_{1}\ S_{2}),\\ 1,&S_{k}\in[S_{2}\ S_{3}],\\ 3,&S_{k}\in(S_{3}\ S_{4}),\\ \left\lceil S_{k}/S_{3}\right\rceil,&S_{k}\in[S_{4}\ \infty),\end{array}\right. \tag{11}\]  
where \(S_{k}=|^{j}x_{k}-^{j}y_{k}|\). This piecewise function is plotted in Figure 8. Given the moving distance, the required minimum climbing steps can be computed with Equation (11).