# Concept, Design, Initial Tests and Prototype of Customized Upper Limb Prosthesis
## 2 Materials and Methods
In order to simplify the transformation matrix elements' notation, the symbols are: \(\theta_{1}=\theta_{10}\), \(\theta_{2}=\theta_{20}\), and \(\theta_{3}=\theta_{30}\). Coordinates transformation of a point, from \(T_{i}\) trihedral to \(T_{i+1}\) trihedral, is:  
\[T_{i+1}=A_{i}\cdot T_{i} \tag{2}\]  
where:  
\[A_{i}=\left[\begin{array}{ccc}\cos\theta_{i}&\sin\theta_{i}&0\\ -\sin\theta_{i}\cdot\cos\alpha_{i}&\cos\theta_{i}\cdot\cos\alpha_{i}&\sin \alpha_{i}\\ \sin\theta_{i}\cdot\sin\alpha_{i}&-\cos\alpha_{i}\cdot  
and  
\[AS^{T}=\left[ \begin{array}{cccccccccccc}a_{1}&a_{2}&a_{3}&a_{4}&a_{5}&s_{1}&s_{2 }&s_{3}&s_{4}&s_{5}&s_{6}\end{array} \right] \tag{11}\]  
For the designed index finger mechanism (see Figure 2), there are some customized values that equal zero, as follows:  
* the axial displacements: \(s_{3}=0\); \(s_{4}=0\); \(s_{5}=0\)
* the distances between axes versors: \(a_{1}=0\); \(a_{2}=0\); \(a_{5}=0\). Results of the kinematic analysis are further presented in Section 3. The mechatronic system for the upper limb has been designed so that it enables finger and forearm motion like that of a real limb as much as possible. Each finger is driven by three micromotors, and its mechanism allows for complex motion, in both the horizontal and vertical planes. The hand and forearm are to be driven by two motors each (and, if required, by assigned gearboxes), and their motion will be possible in two planes (horizontal and vertical). From the point of view of the system's appearance and its components' dimensions, it has been envisaged to look similar to a real hand. The reverse engineering technique has been used to scan a real hand and to obtain its surface dimensions. The technique used was laser scanning, the specific equipment was a MetraSCAN 3D scanner, and the scanning process did not last more than 3 min. Still, if any slight movement of the real hand occurs while scanning, the acquired data (point cloud) generate a contour surface with errors. This is why the process had to be repeated few times till a correct and complete surface had been generated. The scanning process was performed using a 0.1 mm mesh resolution at a measurement rate of 800,000 measurements/s. The finished scanned body was achieved by aligning 3 scans, with each scan containing overlapping areas. The resulting body can be seen in Figure 2(b),c.  
After each scan, the concerned body and also the table body on which the hand is resting (see Figure 2(a)) were cleaned by different surplus meshes by using the proprietary  
Figure 3: Reverse engineering technique for the real upper limb: (**a**) scanning process, (**b**) 3D scan surface—with errors, (**c**) 3D scan surface—accurate. software (VXmodel) and different types of commands and functions within the software. The images taken while scanning are presented in Figure 3.  
Once the surface of the real upper limb was obtained (by combining and aligning 3 different scans of the same hand), further processing to determine the hand and fingers' dimensions was performed. Information about the dimensions and contour characteristics (shape and size) of the hand were obtained by using reverse engineering software (Geo-magic DesignX and Solidworks) and plane generation, as can be seen in Figure 4. For this process, we used 43 planes from the tips of the fingers to the back of the hand.. The planes were generated using the offset function, and the distance between two consecutive planes was set to 10 mm. If the interest on dimensions referred to the horizontal plane, the scanning process, using the proprietary software (VXmodel), generated a triangle mesh in real time through the alignment or registration method. This process can be seen in Figure 5a. After this is done, the dimensions and dimension parameters were obtained by using reverse engineering software (Geomagic DesignX) and specific tools, such as in Figure 5b. Then, a sketch was generated and parametrized, as in Figure 5c.