# Transition Analysis and Its Application to Global Path Determination for a Biped Climbing Robot
## 3 Transition Analysis
### Strict Constraints
#### 3.2.2 Position Constraints  
To confirm whether the transition is feasible, we need to further investigate if \(\frac{\theta}{E}\mathbf{P}\) is within the reachable workspace of the robot. Reflected in solving the inverse kinematics, the following inequality must be satisfied,  
\[c_{3}=\left|\frac{(p_{x}^{\prime}c_{1}+p_{y}^{\prime}s_{1})^{2}+(p_{z}^{\prime }-l_{01})^{2}-(l_{2}^{2}+l_{3}^{2})}{2l_{2}l_{3}}\right|\leq 1, \tag{16}\]  
where \(c_{1}\triangleq\cos\theta_{1}\), \(s_{1}\triangleq\sin\theta_{1}\), and \(l_{01}\triangleq l_{0}+l_{1}\) (similar expressions are used for shorthand hereafter); \(\tan\theta_{1}=a_{y}/a_{x}\), and \(\mathbf{P}^{\prime}=\mathbf{P}^{\prime}+t_{2}\mathbf{n}-t_{45}\mathbf{a}\) represents the position vector of the wrist joint of the robot, e.g., \(T_{2}\) in Figure 2 for Climbot. Let \(\omega_{1}=p_{x}^{\prime}c_{1}+p_{y}^{\prime}s_{1}\), and \(\omega_{2}=p_{z}^{\prime}-l_{01}\), and denote \(\theta_{lin}\) as the rotation limitation of joint \(T_{0}\) (see Figure 2). Since \(l_{2}=l_{3}\) for Climbot, Equation (16) can be simplified as  
\[2l_{2}\cos\frac{\theta_{lin}}{2}\leq\omega_{1}^{2}+\omega_{2}^{2}\leq 4l_{2} ^{2}. \tag{17}\]  
Utilizing \(t_{1}\) (Equation (15)) to express \(\omega_{1}^{2}+\omega_{2}^{2}\), we have  
\[\omega_{1}^{2}+\omega_{2}^{2}=(A^{2}+C^{2})t_{1}^{2}+2(AB+CD)t_{1}+B^{2}+D^{2}, \tag{18}\]  
where \(A=\sigma(n_{x}c_{1}+n_{y}s_{1})-c_{1}\), \(B=(\delta n_{x}-a_{x}l_{45}+\lambda_{x})c_{1}+(\delta n_{y}-a_{y}l_{45}+\lambda _{y})s_{1}\), \(C=\sigma n_{z}\), \(D=\delta n_{z}-a_{z}l_{45}+\lambda_{z}-l_{01}\).  
Let \(f(t_{1})=\omega_{1}^{2}+\omega_{2}^{2}-4l_{2}^{2}\), and \(g(t_{1})=\omega_{1}^{2}+\omega_{2}^{2}-2l_{2}\cos\frac{\theta_{lin}}{2}\). Equation (17) can be converted to two quadratic functions, as  
\[f(t_{1})=Et_{1}^{2}+Ft_{1}+G\leq 0, \tag{19}\]  
\[g(t_{1})=Et_{1}^{2}+Ft_{1}+H\geq 0, \tag{20}\]  
where \(E=(A^{2}+C^{2})\), \(F=2(AB+CD)\), \(G=B^{2}+D^{2}-4l_{2}^{2}\), and \(H=B^{2}+D^{2}-2l_{2}\cos\frac{\theta_{lin}}{2}\).  
In fact, Equation (19) accounts the reachability of the robot, while Equation (20) is the constraint from the rotation limitation of joint \(T_{0}\). Therefore, discriminants of \(f(t_{1})\) and \(g(t_{1})\) determine the feasibility and distribution of operational regions for transition, with the correspondences listed in Table 1.  
\begin{table}
\begin{tabular}{c c c} \hline
**Discriminants 1  & **Distribution 2  & **Operational Regions** \\ \hline \(\Delta_{1}<0\) & Out of the workspace & No \\ \(\Delta_{1}=0,\Delta_{2}>0\) & Out of the rotation limitation & No \\ \(\Delta_{1}=0,\Delta_{2}\leq 0\) & On the boundary of the workspace & Unique point \\ \(\Delta_{1}>0,\Delta_{2}\leq 0\) & Within the workspace and rotation limitation & One segment \\ \(\Delta_{1}>0,\Delta_{2}>0\) & Within the workspace but some points beyond rotation limitation of \(T_{0}\) & Two segments 3  & **Distribution 4  & **Operational Regions** \\ \hline \(\Delta_{1}\) & Out of the workspace & No \\ \(\Delta_{1}=0,\Delta_{2}>0\) & Out of the workspace & No \\ \(\Delta_{1}=0,\Delta_{2}\leq 0\) & Out of the workspace & No \\ \(\Delta_{1}>0,\Delta_{2}\geq 0\) & Within the workspace and rotation limitation & No \\ \(\Delta_{1}>0,\Delta_{2}>0\) & Within the workspace but some points beyond rotation limitation of \(T_{0}\) & Two segments 3  & **Operational Regions** \\ \hline \end{tabular}
\end{table}
Table 1: Discriminants corresponding to the distribution of operational regions for transition.  
Solving Equations (19) and (20), the operational regions on _Member 1_ and _Member 2_, respectively, satisfying Equation (2) could be obtained. Similar processes should be conducted for Equations (3) and (4), to discard those reachable but not accessible gripping points from the obtained operational regions.