# NURSE-2 DoF Device for Arm Motion Guidance: Kinematic, Dynamic, and FEM Analysis
## 3 Kinematic Analysis and Design Parameters
The variable \(r_{2}\) can be isolated from Equation (7) as:  
Figure 1:1  
Then, the angle \(\beta_{2}\) in Equation (8) can be substituted by \(\beta_{1}\) from Equation (6). The angle \(\beta_{1}\) is a known variable since it can be calculated using Equation (5). Therefore, Equation (8) can be rewritten as:  
Figure 1:2  
Since the angle \(\alpha\) and the variable \(r_{2}\) can be respectively calculated using Equations (1) and (9), the Cartesian coordinates of the point \(E\) with respect to the Cartesian reference frame can be defined for both solutions in the form:  
Figure 1:3  
Figure 1:4  
Using \(x_{E}\) and \(y_{E}\) values, the angular positions \(\theta_{1}\) and \(\theta_{2}\) of the active joints \(M_{1}\) and \(M_{2}\) can be computed for both solutions as follow:  
Figure 1:5  
where:  
Figure 1:6  
Figure 1:7  
Figure 1:8  
and:  
Figure 1:9  
where:  
Figure 2:0Figure 2:1  
Figure 2:2  
The inverse kinematics is formulated by Equations (12) and (16). Using the inverse kinematics equations, the mechanism workspace can be determined. However, first the link lengths must be defined so that the end-effector can cover most of the workspace of the human arm motion in Figure 2:3 Therefore, the limits of the mechanism workspace have been approximated by indicating a maximum height (in \(X\)-axis) of 720 mm and a maximum width of 1440 mm (in \(Y\)-axis) when a safety factor of approximately 5 cm is considered [24]. According to the limit of movement of \(M_{1}\) and \(M_{2}\) (0\({}^{\circ}\) to 180\({}^{\circ}\)), and the relationship between the bars (\(L_{1}=L_{2}=L_{3}=L_{4}\)), five-bar can reach a maximum distance equivalent to the length of each symmetrical link \(L_{1}\), \(L_{2}\), \(L_{3}\), and \(L_{4}\). Thus, the link lengths of the five-bar should be equal to 720 mm to cover the needed workspace without taking in consideration the pantograph. Therefore, since the pantograph can amplify the workspace of the five-bar, a decreases scale of 1/4 has been implemented to the link lengths of five-bar, obtaining that \(L_{1}=L_{2}\)= \(L_{3}\)= \(L_{4}\) = (720 mm/4) = 180 mm. Consequently, the pantograph needs to increase the five-bar workspace by a factor of 4 to cover the required workspace. Therefore, the amplification scale of the pantograph should be \(|AC|\)\(\langle AB|\) = 4. Since the angles \(\theta_{1}\) and \(\theta_{2}\) have been delimited from 0\({}^{\circ}\) to 180\({}^{\circ}\), the distances \(|AB|\)+\(|BE|\) should be equal to 180 mm so that the tracing point \(E\) reaches the usable workspace of the five-bar. According to the dimensional ratios of the pantograph, since \(|AB|\)=\(|BE|\) and \(|AB|\)+\(|BE|\) = 180 mm, then \(|AB|\)=\(|BE|\) = 180 mm/2 = 90 mm; since \(|BE|\)=\(|CD|\), then \(|CD|\) = 90 mm; since the amplification scales is defined by \(|AC|\)\(\langle AB|\) = 4, then \(|AC|\) = 4 x 90 mm = 360 mm; since \(|AC|\)=\(|CF|\), then \(|CF|\) = 360 mm; since \(|BC|\)=\(|AC|\) - \(|AB|\), then \(|BC|\) = 360 mm - 90 mm = 270 mm; and since \(|BC|\) = \(|ED|\), then \(|ED|\)= 270 mm. Subsequently, \(L_{6}\) = 360 mm, \(L_{7}\) = 270 mm, \(L_{8}\) = 90 mm, and \(L_{9}\) = 360 mm. Since the link sizes were defined, the workspace can be calculated using Equations (12) and (16). Figure 4 shows the mechanism workspace (in black) versus the workspace of the upper limb motion on a horizontal plane (in grey). The circles in Figure 4 indicate the uncovered areas of the workspace of upper limb motion. It is important to notice that the patient should be located in a convenient and comfortable place in front to the end-effector (point \(F\) in Figure 3) at the opposite side of the active joints. The uncovered areas of the workspace of the upper limb motion are not necessary to perform the proposed exercise and the exercises reported in [24; 27]. Figure 2:4 Workspace of the upper limb motion on a horizontal plane (in grey) and mechanism workspace (in black).