# Transition Analysis and Its Application to Global Path Determination for a Biped Climbing Robot
## 2 Problem Statement
### Transition from One Member to Another  
Operational regions for transition are the junction of a route. Therefore, transition analysis plays a fundamental role in global path planning on trusses. However, transition analysis for BiCRs is challenging due to many constraints, such as robot kinematics, geometries of members, and gripper features. Basically, a BiPCR performing a transition from _Member 1_ to _Member 2_ is illustrated in Figure 4 and is described below, supposing the robot originally grips on _Member 1_ at the beginning. 1. The robot starts to move its end effector, i.e., the swinging gripper, towards _Member 2_, as indicated in Figure 4a. 2. After aligning well with _Member 2_, the robot shifts its swinging gripper to the desired gripping point, as shown in Figure 4b. Figure 3: An illustration of global path for BiCRs traveling on a truss. 3. The robot grips on _Member 2_ with its swinging gripper and holds the two members at the same time, as shown in Figure 3(c). 4. The robot releases the base gripper from _Member 1_. Afterwards the two grippers alternate their roles. 5. The new swinging gripper moves away from _Member 1_, then towards its new target gripping point. Procedures (d) and (e) correspond to Figure 3(d). Accordingly, a valid transition has the following requirements. * Safety. The robot must be able to support itself reliably with only one grip, as required in phases (a)-(e), respectively. * Reachability. The robot must be able to simultaneously grip both members with its two grippers, satisfying kinematic constraints, as illustrated in phase (c). * Accessibility. Grips on both members must be accessible by corresponding grippers, as required in phases (b) and (e). Each grip's accessibility must be considered in two aspects: when the gripper moves towards the grip, and when the gripper moves away from the grip after alternating its role. Considering the 3/4 envelope pattern used, possibilities for the gripper moving forwards and backwards with respect to a grip in the gripping direction for a specified safe distance, are accounted the corresponding grip's accessibility. These three criteria form the basis for transition analysis. Regarding safety checks, strict computation considering dynamics is neither necessary nor practical at the global path planning stage, which will be discussed later on. For reachability and accessibility inspection we propose a mathematical model for further analysis. Without loss of generality, the transition problem can be depicted as a geometric model, as shown in Figure 5. Denote \(\{W\}\) as the world coordination frame, and arrange \(\{U\}\), \(\{V\}\), \(\{B\}\) and \(\{E\}\) to the reference points of two members, the current and target gripping points, respectively. We use parametric equations to express the gripping positions and transformation matrices to represent grips for BiCRs. Considering reachability and accessibility, we have the following problem statement:  
Given: \(\begin{array}{l} \begin{array}{l}W\,\mathbf{P}=\,W\,\mathbf{P}+t_{1}{}^{W}\mathbf{d}_{1} \end{array} \) Points on _Member 1_  
\(\begin{array}{l}\begin{array}{l}\begin{array}{l}\begin{array}{l}\begin{array}{l} \begin{array}{l}\begin{array}{l}\begin{array}{  
\[\exists\;\mathbf{q}=IK(\;\prescript{W}{B}{\mathbf{T}}^{-1W}_{\;\;\;E}{\mathbf{T}}\;), \tag{2}\]  
\[\exists\;\mathbf{q}=IK(\;\prescript{W}{B}{\mathbf{T}}^{-1W}_{\;\;\;E}{\mathbf{T}}^{E}_{\;\; \;E^{\prime}}{\mathbf{T}}\;), \tag{3}\]  
\[\exists\;\mathbf{q}=IK(\;\prescript{TR}{\mathbf{T}}{\mathbf{T}}^{-1W}_{\;\;\;B}{\mathbf{T}}^{B} _{\;\;\;B^{\prime}}{\mathbf{T}}{\mathbf{T}}_{Rx}\;), \tag{4}\]  
where \(\frac{E}{E^{\prime}}{\mathbf{T}}\) and \(\frac{B}{B^{\prime}}{\mathbf{T}}\) refer to the safe distances for grippers approaching _Member 2_ and moving away from _Member 1_ linearly. \(\mathbf{T}_{RxB}^{\;\;\;E}{\mathbf{T}}{\mathbf{T}}_{Rx}\) is the operation in order to use the same inverse kinematics as in the case of \(\frac{B}{E}{\mathbf{T}}\), while the robot interchanges grippers for attaching.