# Optimal Collision-Free Grip Planning for Biped Climbing Robots in Complex Truss Environment
## 3 Computation of Operational Regions for Negotiating Obstacles
### The Key Point for Negotiating Obstacles  
As mentioned earlier, for the \(j\)-th member, the applicable gripping orientation \({}^{W}\mathbf{R}_{j}\), the transition landing segment \([\![y,\,\,y]\!]\) and next takeoff segment \([\![\underline{x},\,\,/\overline{x}]\!]\) are determined by the global path planner. BiCRs with planar configuration can only move within a plane, i.e., the robot plane \(\eta\) as shown in Figure 5, along the via member. Therefore, the possible range of collision can be segmented by a cuboid \(\Delta\). Denote \(l_{cub}\), \(w_{cub}\) and \(h_{cub}\) as the length, width and height of this cuboid, respectively. The dimension of the cuboid can be set as,  
\[\begin{cases}l_{cub}=max(||^{j}x-\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\  
where \(r_{link}\) and \(r_{mem}\) are the radii of the robot links and the truss members; \(l_{1}\) and \(l_{2}\) refer to the robot link lengths indicated in Figure 1, while \(\theta_{max}\) is the maximum rotation angle of the T-type joint of the robot; \(l_{safe}\) is a pre-defined safe distance to facilitate the gripping and grip-releasing operations. In this paper, only the truss members are considered as potential obstacles during climbing. Thus after defining this cuboid, the robot can only collide with those members that intersect the cuboid during the movement from the transition landing segment to the next takeoff segment. Conversely, the potential obstacle members can be extracted out by an intersection check between each member and the cuboid.  
Next, we define a _Key Point_ which plays a key role to determine the reasonable obstacle-negotiating configuration. As shown in Figure 5, the obstacle-negotiating configuration is a state that the robot grips on the \(j\)-th member with both grippers, surrounding the obstacle with its body (Figure 5(a)) or avoiding collision with the obstacle by lowering its body (Figure 5(b)). The _Key Point_ is actually the point that most likely to collide with the robot in the obstacle-negotiating configuration.  
Suppose the angle between the obstacle member axis and the robot plane is \(\lambda\). If \(\lambda<\lambda_{threshold}\), where \(\lambda_{threshold}\) is a small angle, the obstacle member is approximately parallel to the robot plane. In this case, the _Key Point_ is one of the end points of the obstacle member axis, according to the striding-over mode or the creeping mode selected for negotiating obstacles. If \(\lambda>\lambda_{threshold}\), where the obstacle member intersects the robot plane, the _Key Point_ is selected as,  
Figure 5: The definition of collision avoidance space and _Key Point_.  
Figure 6: Negotiating obstacles with the striding-over mode and the creeping mode.  
* the intersection point between the obstacle member axis and \(\eta\), if this intersection point locates within the cuboid, or
* the intersection point between the obstacle member axis and the cuboid, if the intersection point between the obstacle member axis and \(\eta\) is outside the cuboid. The _Key Point_, in this case, is closest to \(\eta\).