# Analysis, Design, and Experimental Research of a Novel Wheelchair-Stretcher Assistive Robot
## 2 Mechanical Structure Design of the Wheelchair-Stretcher Assistive Robot
### Kinematics, Dynamics Analysis and Optimization of the Folding Mechanism
#### 2.2.2 Force Analysis of the Folding Mechanism  
The force research for the folding mechanism can provide a basis for the selection of the motor, as well as the setting, and optimization of the control parameters. In this section, force analysis is conducted in detail. The stress analysis diagram is shown in Figure 10.  
According to the stress analysis diagram, the force equation in \(x\)-axis direction can be established as,  
\[F_{Ax}+F_{Bx}=0 \tag{15}\]  
where \(F_{Ax}\) is the force of the horizontal direction on fulcrum \(A\), and \(F_{Bx}\) is the force of the horizontal direction on fulcrum \(B\).  
Figure 10: Force analysis of the folding mechanism.  
Figure 9: Parameter optimization of the folding mechanism.  
In the meanwhile, the force equation in \(y\)-axis direction also can be expressed as,  
\[F_{Ay}+F_{By}=F_{1}+F_{2} \tag{16}\]  
where \(F_{Ay}\) is the force of the vertical direction on fulcrum \(A\), \(F_{By}\) is the force of the vertical direction on fulcrum \(B\), \(F_{1}\) is the force on the calf support plate, and \(F_{2}\) is the force on the back plate.  
So resultant moment on fulcrum \(A\) can be obtained as,  
\[F_{1}\cdot\frac{1}{2}\cdot l_{5}\cdot\sin\theta_{2}+F_{By}\cdot l_{1}\cdot\cos( e)-F_{2}\cdot\left[\left(l_{2}+\frac{1}{2}l_{6}\right)\cdot\cos\theta_{1}+l_{1} \cdot\cos(e)\right]+M=0 \tag{17}\]  
where \(l_{5}\) is the length of the calf support plate, \(l_{6}\) is the length of the back plate, \(\theta_{1}\) is the angle between the back plate and direction of \(x\)-axis, \(\theta_{2}\) is the angle between the side link 2 and direction of \(x\)-axis, \(M\) is the torque of the motor, \(\gamma\) is the angle between the seat plate and the direction of \(x\) axis.  
Correspondingly, the resultant moment on the fulcrum \(B\) can be described as:  
\[\begin{array}{l}F_{1}\cdot\left(\frac{1}{2}\cdot l_{5}\cdot\cos\theta_{2}+l _{1}\cdot\cos(e)\right)-F_{Ay}\cdot l_{1}\cdot\cos(e)\\ -F_{2}\cdot\left(l_{2}+\frac{1}{2}l_{6}\right)\cdot\sin\theta_{1}+F_{Ax}\cdot l _{1}\cdot\sin(e)+M=0\end{array} \tag{18}\]  
In the folding mechanism process, the force applied at the center of the calf support plate can be equaled by the weight of human lower limbs and mechanism weight of this part; and the force applied at the center of the back plate can be equaled by the weight of the upper body and the mechanism weight in this part. According to the standards of ergonomics and adult body size in China [30; 31], the values of parameters \(l_{5}\), \(l_{6}\), \(F_{1}\), \(F_{2}\) are set as 490 mm, 510 mm, 300 N, and 600 N, respectively.  
Through the force analysis of \(l_{4}\) and \(l_{5}\), the resultant moment on point \(D\) can be obtained as:  
\[F_{1}\cdot\left(\frac{1}{2}l_{5}+l_{4}\right)\cdot\cos\theta_{2}+F_{Ax}\cdot l _{4}\cdot\sin\theta_{2}-F_{Ay}\cdot l_{4}\cdot\cos\theta_{2}=0. \tag{19}\]  
Through the force analysis of \(l_{2}\) and \(l_{6}\), the resultant moment on point \(C\) can be obtained as:  
\[-F_{2}\cdot\left(\frac{1}{2}\cdot l_{6}+l_{2}\right)\cdot\cos\theta_{1}+M-F_{ By}\cdot l_{2}\cdot\cos\theta_{1}+F_{Bx}\cdot l_{2}\cdot\sin\theta_{1}=0. \tag{20}\]  
The following geometric relationships between \(\theta_{1}\) and \(\theta_{2}\) can be described as:  
\[\left\{\begin{array}{l}l_{4}\cdot\cos(\theta_{2})+l_{3}\cdot\cos\alpha-l_{2 }\cdot\cos\theta_{1}=l_{1}\cdot\cos(e),\\ l_{1}\cdot\sin(e)=l_{4}\cdot\sin(\theta_{2})-l_{3}\cdot\sin\alpha-l_{2}\cdot \sin\theta_{1}.\end{array}\right. \tag{21}\]  
Merging Equations (15)-(21) by using Matlab software to program, the torque \(M\) required by the folding mechanism can be solved as:  
\[M=162\cos\theta_{1}. \tag{22}\]  
It is found that the maximum torque is 162 N\(\cdot\)m. If such a high output torque is chosen, the size and power of the motor are too big for the folding mechanism. It is necessary to add a booster device to the mechanical structure.