# Configuration Design of an Upper Limb Rehabilitation Robot with a Generalized Shoulder Joint
## 2 Human Upper Limb Kinematics  
### Human Upper Limb Anatomical Structure and Equivalent Mechanism Model  
Broadly speaking, the shoulder joint is complex and involves the glenohumeral joint, acromolavicular joint, sternoclavicular joint and scapular-chest wall joints [28] as shown in Figure 1. These joints coordinate and cooperate to complete a complex and flexible shoulder movement. The glenohumeral joint (GH) is the main joint in the shoulder joint. The GH is composed of the ball and socket joint between the humeral head and the scapular glenoid [29]. The acromolavicular joint (AC) is composed of the acromion end surface of the scapula and the end surface of the clavicle acromion joint [29]. The sternoclavicular joint (SC) is composed of the clavicle end face of the sternum and the clavicle notch of the sternum stem [30]. The scapulothoracic joint (ST) is a junction point between the front of the scapula and the back of the rib cage. The motion form of each subdivision joint of the scapula belt is shown in Figure 3.  
Figure 1: Physiological structure of the human shoulder joint complex. Figure 2: Cont. Each joint in the shoulder joint is equivalent to the kinematic pair in the mechanism, in which the AC has 3-DOF and is equivalent to the ball pair \(S_{4}\). The SC has 3-DOF; however, its backward rotation is small, and it is less involved in the overall joint movement. Therefore, the SC is simplified to form Hooke's hinge with 2 DOF, \(R_{5}\), \(R_{6}\). The ST has 3-DOF, which can be simplified to two translation pairs, \(P_{1}\), \(P_{2}\) and a rotating pair, \(R_{3}\). The GH is a typical ball and socket joint, which is equivalent to a ball joint, \(S_{7}\). Therefore, the equivalent mechanism model of the anatomical structure of the scapula belt is shown in Figure 4. According to the Grubler-Kutzbach formula, the DOF of the parallel mechanism can be calculated as shown below. \[F=\lambda(n-g-1)+\sum_{i=1}^{g}f_{i}=6(6-6-1)+8=2 \tag{1}\]  
where \(F\) is the DOF of the mechanism; \(\lambda\) is the number of DOFs the mechanism rods should have in space, generally 6; \(n\) is the number of mechanism rods; \(g\) is the number of motion pairs in the mechanism; \(f_{i}\) is the number of DOFs of the \(i\)-th motion pair. Since the translation trajectory of \(P_{1}\), \(P_{2}\) is an arc trajectory centred on the axis of rotation of \(R_{5}\) and \(R_{6}\), \(R_{5}\) and \(R_{6}\) can be regarded as a virtual constraint of the translation pair of \(P_{1}\), \(P_{2}\). Since the \(R_{3}\) rotation axis is very close to the centre of the GH ball pair \(S_{7}\), it is usually regarded as the same position, so the \(R_{3}\) rotation effect overlaps the \(S_{7}\) motion range. Biologically, because GH \(S_{7}\) has a certain range of motion, scapula uprotation/downrotation \(R_{3}\) can be used to compensate for the range of motion of the GH. Therefore, the equivalent mechanism model of the scapula belt can be simplified to a 5-DOF tandem mechanism, as shown in Figure 5.  
Figure 4: Equivalent mechanism model of the scapula belt. Figure 3: Motion form of the articulation in the scapular region. (**a**) Motion form of the acromiodavicular joint (AC); (**b**) Motion form of the stemolavicular joint (SC); (**c**) Motion form of the scapulothoracic joint (ST). In addition, upper limb exercises also include elbow joint flexion/extension, forearm pronation/supination, wrist joint dorsiflexion/palm flexion and radial flexion/ulnar flexion. Since the rotation axes of the 2-DOF of the wrist joint intersect, to reduce the design difficulty and production cost, the wrist joint of the rehabilitation robot adopts a single DOF, and the wrist joints are separately trained with different DOFs.