# Concept, Design, Initial Tests and Prototype of Customized Upper Limb Prosthesis
## 2 Materials and Methods  
One of the first steps in the conceptualization of the customized upper limb prosthesis is that of the kinematic analysis of its finger mechanism. In order to perform this, some basic aspects of the mechanism theory have been considered as follows [14; 15; 16; 17; 18; 19; 20; 21; 22]:  
* Link--is a rigid body. * Joint--is a permanent contact between two links. * Planar motion--is the motion \"in which all points belonging to a link move in a plane known as the plane of motion, while simultaneously the link is free to rotate about an axis perpendicular to the plane of motion\". * Degree of freedom of a rigid body--is the number of independent movements it has. Kinematic constraints are \"constraints between rigid bodies that result in the decrease of the degrees of freedom of the rigid body system\". An Assur group is a set of kinematic chains developed by [PERSON] whose degree of freedom is zero. A complex structure can be constructed by extending the Assur kinematic chain. The complex linkage mechanism can be regarded as originating from some Assur kinematic chains  
Kinematic analysis is the motion analysis aimed at determining the positions, speed, and acceleration distributions of each element, previously knowing their constructive characteristics and the relative movement of the active link elements. A dyad is an Assur group made of two parts, three joints, and two outputs (Figure 1a,b). A triad is an Assur group made of four parts, six joints, and three outputs (Figure 1c). Figure 1: _Cont._For the mechanism of the designed prosthesis, the kinematic analysis was done by the matrix-vector method [23; 24], which is simple, intuitive, and fits for open spatial contours. For its index finger mechanism, the kinematic scheme is shown in Figure 2. The final Hartenberg-Denavit trihedral, \(T_{1}\), \(T_{2}\),..., \(T_{i+1}\), is associated with the fingertips. The trihedral versos are \(\overline{n}_{i}\), \(\overline{\varepsilon}_{i+1}\)\(\times\overline{n}_{i}\). Figure 2: Schemes of the designed index finger mechanism: (**a**) kinematic scheme, (**b**) verso-rial scheme. Figure 1: Assur group types—dyad and triad: (**a**) kinematic scheme for the dyad (RRR) [21], (**b**) kinetostatic scheme for the dyad (RRR) [21], and (**c**) kinematic and kinetostatic schemes for the triad (RR-RR-RR) [22] where: \(R\) is rotation, \(\overline{F}\) is the force vector (components along axes), and \(\overline{R}\) is the joint reaction force vector (components along axes). The main steps of the calculus required for the kinematic analysis of the index finger mechanism are presented next. The simplified structure of the mechanism is a serial open kinematic chain made of dyads. The base trihedral, \(\overline{n}_{0}\), \(\overline{e}_{1}\times\overline{n}_{0}\), \(\overline{e}_{1}\), overlaps the orthogonal reference trihedral Oxyz with the \(\overline{i}\), \(\overline{j}\), \(\overline{k}\) versors. The position vector of the \(T_{1}\) customized point, \(\overline{r}_{T_{1}}\), is given by relation (1):  
\[\overline{r}_{T_{1}}=s_{1}\overline{e_{1}}+a_{1}\overline{n_{1}}+s_{2} \overline{e_{2}}+a_{2}\overline{n_{2}}+s_{3}\overline{e_{3}}+a_{3}\overline{n _{3}}+s_{4}\overline{e_{4}}+a_{4}\overline{n_{4}}+s_{5}\overline{e_{5}}+a_{5} \overline{n_{5}}+s_{6}\overline{e_{6}}+a_{6}\overline{n_{6}} \tag{1}\]  
where:  
* \(s_{1}\) represents the distance between normal versors, \(\overline{n}_{i-1}\) and \(\overline{n}_{i}\) (axial displacement), with a positive sign along the \(\overline{e}_{i}\) axis;
* \(a_{i}\) is the distance between the \(\overline{e}_{i}\) and \(\overline{e}_{i+1}\) axes, positive sign;
* \(\theta_{i,\ i-1}\) is the angle between the \(\overline{n}_{i-1}\overline{n}_{i-1}\) and \(\overline{n}_{i}\) versors, positive sign along \(\overline{e}_{i}\) axis;
* \(\alpha_{i}\) is the angle between the \(\overline{e}_{i}\) and \(\overline{e}_{i+1}\) versors (crossing angle), positive sign along \(\overline{n}_{i}\) axis.