# Transition Analysis and Its Application to Global Path Determination for a Biped Climbing Robot
## 3 Transition Analysis
### Strict Constraints
#### Orientation Constraints  
Considering orientation constraints, we propose and prove two propositions which are important for analyzing the member-to-member transition problem with BiCRs, such as Climbot. Note that variable symbols without superscript are used with respect to frame \(\{B\}\) hereafter.  
**Proposition 1**.: _Given the current base at \(\{B\}\) and the goal gripping orientation \(\begin{smallmatrix}\frac{B}{E}\mathbf{R}=\begin{bmatrix}\mathbf{n}&\mathbf{o}&\mathbf{a}\end{bmatrix}\end{smallmatrix}\), there exists only one feasible configuration for BiCRs, such as Climbot, to transit between two staggered members._  
Proof.: Due to the structural configuration of Climbot, its links are always in a plane (\"robot plane\" for short, highlighted in gray in Figure 5). Therefore, the following orientation constraints must be satisfied,  
\[\left\{\begin{array}{l}\mathbf{k}\cdot\mathbf{a}=0\\ \tan\alpha=p_{y}^{E}/p_{x}^{E}\end{array},\right. \tag{12}\]  
where \(\mathbf{k}=\begin{bmatrix}-\sin\alpha&\cos\alpha&0\end{bmatrix}^{T}\) represents the norm vector of the robot plane. Substituting Equation (11) and \(\mathbf{n}=\mathbf{d}_{2}\) into Equation (12) yields  
\[t_{2}=\frac{p_{x}^{V}a_{y}-p_{y}^{V}a_{x}}{n_{y}a_{x}-n_{x}a_{y}}. \tag{13}\]  
On the right side of Equation (13), the values are all constants. Hence, we obtain a unique gripping position for feasible transition by substituting \(t_{2}\) into Equation (11), when \(n_{y}a_{x}-n_{x}a_{y}\
eq 0\).  
It should be underlined that _Member 2_ is on the robot plane when \(n_{y}a_{x}-n_{x}a_{y}=0\). Therefore, all points on _Member 2_ satisfy the orientation constraints. In this case, a gripping point on _Member 1_ may correspond to unlimited gripping points on _Member 2_ for feasible transition. The graspable region determination algorithm presented in [31] could be applied to this case by sampling attaching positions on both members in order to quickly obtain some operational regions for transition.  
**Proposition 2**.: _For BiCRs, such as Climbot, to transit between two staggered members, given the base and the goal gripping orientations as \(\begin{smallmatrix}\frac{W}{U}\mathbf{R}\end{smallmatrix}\) and \(\begin{smallmatrix}\frac{W}{V}\mathbf{R}\end{smallmatrix}\), respectively, the operational region for transition on Member 2 is linear with that on Member 1._  
Proof.: Recall Equation (10), and set \(\mathbf{\lambda}=\begin{smallmatrix}\frac{W}{U}\mathbf{R}^{T}(\begin{matrix}\mathbf{W}\mathbf{ P}-\begin{matrix}\mathbf{W}\mathbf{P}\\ \mathbf{U}\end{matrix}\end{matrix})\cdot\frac{B}{V}\mathbf{P}\) can be written as  
\[\begin{smallmatrix}\frac{B}{V}\mathbf{P}=\begin{bmatrix}p_{x}^{V}&p_{y}^{V}&p_{z}^{V }\end{bmatrix}^{T}=\begin{bmatrix}-t_{1}+\lambda_{x}\\ \lambda_{y}\\ \lambda_{z}\end{bmatrix}. \tag{14}\]Note that since \(\prescript{W}{U}{R}\), \(\prescript{W}{V}{P}\), and \(\prescript{W}{U}{P}\) are all known invariants, \(\Lambda\) is actually a constant vector. Substituting Equation (14) into Equation (13) results in  
\[t_{2}=\frac{(-t_{1}+\lambda_{x})a_{y}-\lambda_{y}a_{x}}{n_{y}a_{x}-n_{x}a_{y}}= \sigma t_{1}+\delta, \tag{15}\]  
where \(\sigma=-a_{y}/(n_{y}a_{x}-n_{x}a_{y})\), and \(\delta=(\lambda_{x}a_{y}-\lambda_{y}a_{x})/(n_{y}a_{x}-n_{x}a_{y})\). Both \(\sigma\) and \(\delta\) are constants. Therefore, \(t_{2}\) and \(t_{1}\) are linear, and thereby have one-to-one mapping with each other. Once \(t_{1}\) is determined, \(t_{2}\) can be calculated by Equation (15), and vice versa.