# NURSE-2 DoF Device for Arm Motion Guidance: Kinematic, Dynamic, and FEM Analysis
## 3 Kinematic Analysis and Design Parameters  
To control an end-effector position consisting of cartesian compounds \(X\) and \(Y\), two DoFs are needed. On the other hand, a planar mechanism is enough to perform movements on the planar workspace in Figure 1. Therefore, a planar mechanism of two DoFs has been proposed (See Figure 3). The suggested mechanism consists of by a five-bar and a pantograph mechanism. The pantograph was designed to enlarge the five-bar workspace and therefore to use tiny connections. The five-bar mechanism mobile bars are \(L_{1}\), \(L_{2}\), \(L_{3}\), and \(L_{4}\) and the pantograph mobile bars are \(L_{6}\), \(L_{7}\), \(L_{8}\), and \(L_{9}\). The bar \(L_{5}\) is the attachment structure set frame. Point \(E\) is the pantograph tracing point which is associated to the five-bar mechanism end-effector. Point \(F\) is the amplified tracing point to follow the arm exercise path. \(M_{1}\) and \(M_{2}\) are the active joints that will be actuated by motors and they are  
Figure 1: Workspace of the upper limb on a horizontal plane. Figure 2: \(X\) and \(Y\) cartesian coordinates to trace the number eight. related with \(\theta_{1}\) and \(\theta_{2}\) angles, respectively. The distance \(H\) is defined as \(H=L_{1}=L_{2}=L_{3}=L_{4}\). It is important to notice that variable \(H\) is included in the kinematics formulation. The \(\beta_{1}\), \(\beta_{2}\), and \(\alpha\) angles are required for the kinematic analysis. The point \(A\), \(E\), and \(F\) are always aligned for any position of the pantograph so that a straight line connects the three points. The proposed mechanism has been protected by the patent in [28]. The rotational movements of \(M_{1}\) and \(M_{2}\) were reduced from \(0^{\circ}\) to \(180^{\circ}\) to prevent physical collisions. In addition, in order to obtain the maximum possible workspace using the five-bar mechanism [29; 30; 31] the bars \(L_{1}\), \(L_{2}\), \(L_{3}\), and \(L_{4}\) have equal lengths so that \(L_{1}\)= \(L_{2}\)= \(L_{3}\)= \(L_{4}\), and the joint \(M_{1}\) is aligned with joint \(M_{2}\). Distance \(|\)AC\(|\)/\(|\)AB\(|\) give the amplification scale of the pantograph. The dimensional ratios \(|\)\(BE\)\(|\)=\(|\)\(CD\)\(|\); \(|\)\(BC\)\(|\)=\(|\)\(ED\)\(|\); \(|\)\(AB\)\(|\)=\(|\)\(BE\)\(|\); and \(|\)\(AC\)\(|\)=\(|\)\(CF\)\(|\) define the pantograph structure [32]. The mechanism end-effector should, according with the workspace in Figure 1, cover a maximum height of 720 mm and a maximum width of 1440 mm based on an about 5 cm safety factor. From Figure 3, the angle \(\alpha\) can be defined as:  
\[\alpha=tan^{-1}\frac{(y_{F}-H)}{x_{F}} \tag{1}\]  
where \(Y_{F}\) and \(X_{F}\) are the coordinates of the point \(F\) (end-effector positions) with respect to the Cartesian reference frame. \(Y_{F}\) and \(X_{F}\) are known inputs in the inverse kinematics. In Figure 3, the triangle formed by points \(A\), \(C\), and \(F\) can be defined by using law of cosines in the form:  
\[L_{6}^{2}+L_{9}^{2}-2L_{6}L_{9}\ cos\ (\beta_{1})=r_{1}^{2}. \tag{2}\]  
Figure 3: Proposed mechanism for arm motion assistance. On the other hand, by using the Pythagorean theorem, \({r_{1}}^{2}\) can be defined as:  
\[{r_{1}}^{2}={x_{F}}^{2}+{(y_{F}-H)}^{2}, \tag{3}\]  
then, replacing Equation (2) in Equation (1):  
\[{L_{6}}^{2}+{L_{9}}^{2}-2{L_{6}}{L_{9}}\;cos(\;{\beta_{1}})={x_{F}}^{2}+{(y_{F} -H)}^{2}. \tag{4}\]  
So that the angle \(\beta_{1}\) can be computed from Equation (4) in the form:  
\[\beta_{1}=cos^{-1}\frac{{L_{6}}^{2}+{L_{9}}^{2}-{x_{F}}^{2}-{(y_{F}-H)}^{2}}{2{ L_{6}}{L_{9}}}\;, \tag{5}\]  
where \(\beta_{1}\) is the angle between links \(L_{6}\) and \(L_{9}\). In a pantograph mechanism \(\beta_{2}\) is defined as:  
\[\beta_{2}=\beta_{1}. \tag{6}\]  
In addition, the triangle formed by points \(A\), \(B\), and \(E\) can be identify by using law of cosines in the form:  
Figure 1:0  
where \(|AB|\) is the distance between the point \(A\) and the point \(B\).