## 2 Problem Statement
### Case Study  
The wrist joint performs its movements along two axes, each corresponding to a DOF: Flexion/Extension (FL/EX) and Radial Deviation (RD)/Ulnar Deviation (UD). It is oriented by a third DOF, the Pronation (P)/Supination (S) movement of the elbow joint. Figure 0(a) shows the rehabilitation device proposed by [PERSON] et al. [40] to reproduce these 3-DOF. It is part of a join project between the Instituto Politecnico Nacional (Mexico) and the Universidad Militar Nueva Granada (Colombia), to develop low-cost rehabilitation systems expressly designed for the population of Latin American countries. The specifications of the device components were determined from the motion ranges proposed in [1,44] and the hand anthropometry for the Latin American population [45]. The torques required to drive each articulation were considered for the structural analysis, taking into account both the requirements of the DLA and the values proposed in literature [1,3,46]. The RD/UD and FL/EX movements are actuated directly by servomotors connected to the structural components. For this reason, an adequate design is critical for a safe operation.  
The development in this paper (Figure 0(b)) is a continuation of the aforementioned project, and consists in a modification to the extensor arm of the mechanism with two links instead of one to facilitate the wrist movements. The size of each link is determined by its length \(l\), width \(W\), thickness \(t\), and diameter \(d\) (Figure 1(a)), and both of them form an open kinematic chain. The links should be as light as possible, to take advantage of the highest torque in the hand movement during the therapy. The solution method tackles the simultaneous design of more than one structural element, considering the interaction and joint effect while designing. Although this work only analyzes two links, all the mobile elements are designed with the same methodology. A complex mixed-optimization problem is obtained, with both the objective function and the constraints of non-linear type. The free-body diagrams (Figure 1(b)) consider the bending and torsional external loads for the combined stress. The bending stress is calculated for each link by Equation (2), with Equations (3) and (4) for the bending moments and the second moments of area, respectively. The torque is \(T_{xy}=3.571\) Nm, and the forces are \(F_{21}=F_{22}=25.3N\), and \(F_{y1}=F_{y2}+\)weight of link 2, and \(F_{y2}=30.4N+EF\)[40], with an extra load \(EF=49N\) for this case. The shear stress produced by torsion is calculated by Equation (5), with \(W>t\).  
\[\sigma_{f}=\frac{M_{y}}{I_{yy}}+\frac{M_{z}}{I_{zz}} \tag{2}\]  
\[M_{y}=\frac{l\times W\times F_{y}}{2},\qquad\qquad M_{z}=\frac{l \times t\times F_{z}}{2} \tag{3}\] \[I_{yy}=\bigg{(}\frac{1}{12}t\bigg{)}(W^{3}-d^{3}),\qquad I_{zz} =\bigg{(}\frac{1}{12}W-d\bigg{)}(t^{3}) \tag{4}\]  
Figure 1: Wrist rehabilitation mechanism.  
\[\tau_{s}=\frac{3T_{yy}}{8 ab^{2}}\bigg{(}1+0.6095\bigg{(}\frac{t}{W}\bigg{)}+0.88665 \bigg{(}\frac{t}{W}\bigg{)}^{3}+0.91\bigg{(}\frac{t}{W}\bigg{)}^{4}\bigg{)} \tag{5}\]  
There is no load generating an axial force, so \(\sigma_{a}=0\). The maximum total deflection is obtained using Equations (6) and (7), where \(E\) is the elastic modulus of the selected material:  
\[\delta_{max}=\sqrt{(y_{xy})^{2}+(y_{xz})^{2}} \tag{6}\] \[y_{xy}=\frac{F_{y}l^{3}}{3 EI_{yy}},\hskip 28.452756 pty_{xz}= \frac{F_{2}l^{3}}{3 EI_{zz}} \tag{7}\]