## 2 Upper Limb Modeling
### Range of Motion for the Upper Limb Model  
In [36], the shoulder's joint motion is explained in detail, while defining the corresponding ranges of motion. From an anatomic point of view the shoulder girdle consists of the clavicle, the scapula and the humerus. The shoulder joint motion is determined by a combination of these body parts distinct motions, making the analysis more complex. A ball-and-socket joint is adopted to model the shoulder girdle motion ability. Hence, \begin{table}
\begin{tabular}{c c} \hline \hline
**Location** & **Description** \\ \hline Chest [C] & Chest Position—Origin Frame \\ Shoulder [S1] & Flexion and Extension \\ Shoulder [S2] & Ab- and Adduction \\ Shoulder [S3] & In- and External Rotation \\ Elbow [E1] & Flexion and Extension \\ Elbow [E2] & Pronation and Supination \\ Wrist [W1] & Flexion and Extension \\ Wrist [W2] & Deviation \\ Hand [End-effector] & End-effector \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Frames of the upper limb model. Figure 1: Upper limb representation as a robotic arm. the shoulder is described by 3 DoFs, permitting shoulder abduction-adduction motion, flexion-extension and internal-external rotation. The elevation plain of upper limb defines the kind of exercise that the subject executes. The elevation plane of \(0^{\circ}\) denotes the upper limb elevation over the frontal plane. In this case, the abduction-adduction of the upper limb occurs with the corresponding range of motion being \([0^{\circ},180^{\circ}]\). The elevation plane of \(90^{\circ}\) indicates that the shoulder flexion-extension takes place. The corresponding range of motion is \([0^{\circ},180^{\circ}]\) during elevation through flexion and \([0^{\circ},-60^{\circ}]\) for the shoulder's elevation through extension (hyperextension). In both cases, the rotation of \(0^{\circ}\) is defined for an upper limb posture with its shaft parallel to the thorax vertical axis. The upper arm internal-external rotation is performed when the shoulder's elevation angle is \(0^{\circ}\) and the elbow is flexed at \(90^{\circ}\), leading the forearm to be lying in the sagittal plane. The range of motion for this shoulder's DoF that accomplishes the upper arm internal-external rotation is \([-90^{\circ},20^{\circ}]\), with the negative value being when the upper arm rotates internally towards the human body. The elbow joint main rotation is performed around a fixed axis that is located among the center point of the trochlear sulcus and the center point of the capitulum of the humerus. This DoF permits the elbow flexion-extension with a range of \(0^{\circ}\) for the case of the full extension for the elbow to \(130^{\circ}\) depicting its flexion. Furthermore, a second rotation occurs a bit distal from the elbow joint, which is responsible for the forearm pronation-supination motion. In this case, the forearm rotates around an axis that is defined among the center points of the radial head and the distal ulna. The range of this motion is \(90^{\circ}\) for the pronation case to \(-90^{\circ}\) for the supination one. The forearm neutral position, where its rotation is \(0^{\circ}\), occurs when having the shoulder and also the wrist at their neutral positions, the hand is lying in the sagittal plane. At last, the motion of the wrist joint around the axes that are defined in [37], allow for the wrist flexion-extension and wrist deviation. The first ranges from \(-70^{\circ}\) for the case of extension to \(70^{\circ}\) for the wrist flexion. The wrist deviation is defined as radial and ulnar. In the radial, the wrist deviates internally towards the body and till \(-10^{\circ}\), while during the ulnar deviation the wrist deviates outwards of the body and to an upper limit of \(25^{\circ}\). The neutral position of \(0^{\circ}\) for both flexion and deviation DoFs of the wrist occurs when the third metacarpal and the longitudinal axis of the forearm are aligned.