## 2 Upper Limb Exoskeleton Robot
### Workspace and Singularity Analysis  
The two most important properties that influence the geometrical design of a robotic exoskeleton are workspace and singularity analysis [28]. The kinematic model is used to analyze the workspace of the human upper limb and exoskeleton robot. Given the position of any point in the workspace, it is important to determine whether it belongs to the actual workspace or not, and helps to verify if at least one solution for the joint angles exists [2]. Therefore, a direct search method is employed to essentially evaluate the existence of an inverse kinematics solution for the human and robotic exoskeleton, shown in Figure 2b,c. The kinematic properties selected for this study are given in Appendix C.  
Apart from analyzing the reachable workspace, implementation of safe and stable operation is also required due to kinematic singularities within the workspace. Hence, it is necessary to identify all singular configurations while planning trajectories for the robotic exoskeleton. The manipulability ellipsoid and determinant of the Jacobian matrix are the two important indices that characterize the degree of singularity [29]. Our study determines the kinematic performance of the exoskeleton system by analyzing the manipulability  
Figure 2: Workspace analysis of the 4-DOF upper limb exoskeleton (cyan) within the human arm workspace (red) (measured in meters): (**a**) human-exoskeleton system, (**b**) isotropic view of upper limb exoskeleton and human arm workspace, (**c**) sagittal plane view of upper limb exoskeleton and human arm workspace, (**d**) different configurations of human–exoskeleton system in high manipulability region, (**e**) configurations of human–exoskeleton system in low manipulability region.  
index, which gives us information about the low and high manipulability regions, shown in Figure 2d,e.  
In the manipulability analysis, we look at the position of the wrist only. Thus, we take Jacobian in the form of  
\[\mathbf{J}=\frac{\partial\boldsymbol{n}}{\partial\boldsymbol{\theta}_{i}}= \begin{bmatrix}-n_{24}+L_{1}c\theta_{1}&-n_{34}c\theta_{1}&-L_{2}(s\theta_{1} s\theta_{3}+c\theta_{1}c\theta_{2}c\theta_{3})\\ n_{14}+L_{1}s\theta_{1}&-n_{34}s\theta_{1}&L_{2}(c\theta_{1}s\theta_{3}-s \theta_{1}c\theta_{2}c\theta_{3})\\ 0&-L_{2}c\theta_{2}s\theta_{3}&-L_{2}s\theta_{2}c\theta_{3}\end{bmatrix} \tag{4}\]  
where \(\boldsymbol{n}=[n_{14}\ n_{24}\ n_{34}]^{T}\), \(\boldsymbol{\theta}_{i}=[\theta_{1}\ \theta_{2}\ \theta_{3}]^{T}\). The manipulability index can be determined after computing the Jacobian as follows:  
\[\mu(\mathbf{J})=\sqrt{|\mathbf{J}|^{T}|} \tag{5}\]  
where \(\mu\) is the manipulability index. Figure 2d,e display different configurations of the human upper limb and exoskeleton system and their corresponding manipulability ellipses in high and low region of manipulability. Moreover, the manipulability analysis gives us information about the uniform distribution of the forces and torques applied by the exoskeleton system to the human upper limb [30]. Another important aspect of analyzing the manipulability ellipse is to identify the singular configuration of the exoskeleton system in the workspace. If the determinant of the Jacobian matrix is zero, the robot encounters singularity or exhibit zero manipulability. Hence, this analysis can be used for the robot path planning where it will try to avoid the low region of manipulability.