# Configuration Design of an Upper Limb Rehabilitation Robot with a Generalized Shoulder Joint
## 6 Comprehensive Kinematic Performance Analysis of the Shoulder Joint Configuration
### Operational Ellipsoid Analysis  
The reciprocal of the condition number of the Jacobian matrix can be used to judge whether the current posture of the mechanical arm is singular. If the condition number is large, then the current Jacobian matrix is ill conditioned. However, for some actions, the reciprocal of the condition number is too large and does not affect the normal operation of the mechanism, so the concept of operability is proposed. [PERSON] [38] defines operability:  
\[m=\sqrt{\det(JJ^{T})}=\sqrt{\lambda_{1}\lambda_{2}\cdots\lambda_{n}}=\sigma_{1 }\sigma_{2}\cdots\sigma_{n} \tag{19}\]  
where \(\lambda_{i}\) (\(i\) = 1,2,\(\cdots\)-\(n\)) is the eigenvalue of matrix \(JJ^{T}\), and \(\sigma_{i}\) (\(i\) = 1,2,\(\cdots\)-\(n\)) is the singular value of Jacobian matrix \(J\). To compare the isotropic motion capabilities of the two mechanisms in different poses, the operability ellipsoids of the two mechanisms are compared at different positions on the coronal, sagittal and horizontal planes of the human body. Figures 16 and 17 show the operability ellipsoids with \(2P_{a}1P3R_{a}\) and \(5R_{a}1P\) in the lift plane \(\alpha\) = 30\({}^{\circ}\) of the human body at positions \(\beta\) = 30\({}^{\circ}\), 60\({}^{\circ}\), 90\({}^{\circ}\), 120\({}^{\circ}\). Through comparison, when \(\beta\) = 90\({}^{\circ}\), the operable ellipsoids of the two configurations are closest to the sphere. At this time, \(5R_{a}1P\) has better isotropic movement ability in all directions than \(2P_{a}1P3R_{a}\). In the lifting plane \(\alpha\) = 30\({}^{\circ}\), when the angles change as \(\beta\) = 30\({}^{\circ}\), 60\({}^{\circ}\), 90\({}^{\circ}\), 120\({}^{\circ}\), the operability ellipsoids of the two configurations have similar cha  
Figure 15: Comparison between the reciprocal of the condition number and the curve of the uplift angle. that the two have the same movement flexibility in \(\alpha=30^{\circ}\), and the same-sex athletic ability is the same in the angular position. Figures 18 and 19 show the operability ellipsoids with \(2P_{a}1P3R_{a}\) and \(5R_{a}1P\), respectively, in the lift plane \(\alpha=90^{\circ}\) (sagittal plane) of the human body at position \(\beta=30^{\circ}\), \(60^{\circ}\), \(90^{\circ}\), \(120^{\circ}\). Through comparison, it can be seen that when \(\beta=90^{\circ}\), the operable ellipsoid of the two configurations is closest to the sphere, and the operability ellipsoid of \(5R_{a}1P\) is very close to the sphere compared to \(2P_{a}1P3R_{a}\). In this lift plane, the isotropic movement ability in all directions is better. In the sagittal plane, when the lift angle changes at \(\beta=30^{\circ}\), \(90^{\circ}\), \(120^{\circ}\), the change in the operability ellipsoid of \(5R_{a}1P\) is smaller than that of \(2P_{a}1P3R_{a}\). When \(\beta=60^{\circ}\), the change in the operability ellipsoid of \(5R_{a}1P\) changes more than \(2P_{a}1P3R_{a}\). However, the overall performance shows that \(5R_{a}1P\) has slightly better movement flexibility in the \(\alpha=30^{\circ}\) lifting plane and has slightly better single-sex movement ability under the corresponding lifting angle posture. It can be seen from Figures 20 and 21 that \(2P_{a}1P3R_{a}\) has the same operability ellipsoid in the horizontal plane, indicating that the motion performance of the mechanism configuration corresponding to different lift angles on the horizontal plane remains unchanged. In the horizontal plane, when the lifting surface angle changes at \(\alpha=30^{\circ}\), \(90^{\circ}\), \(120^{\circ}\), the overall operability of the\(5R_{a}1P\) ellipsoid is closer to a spherical shape than \(2P_{a}1P3R_{a}\). The overall performance shows that \(5R_{a}1P\) has better movement flexibility in the horizontal plane, and the same-sex sports ability is stronger in the corresponding lift angle pose. The changes in the operability ellipsoids of the \(2P_{a}1P3R_{a}\) and \(5R_{a}1P\) in four different forms on three planes are summarized in Table 3. A comparison index \(S\) is proposed.