# Analysis, Design, and Experimental Research of a Novel Wheelchair-Stretcher Assistive Robot
## 2 Mechanical Structure Design of the Wheelchair-Stretcher Assistive Robot
### Kinematics Analysis and Structure Design of the Walking Mechanism
#### 2.1.1 Kinematics Analysis of the Walking Mechanism  
The mecanum wheel, a kind of omnidirectional wheel, is selected for the design of the walking mechanism. The reason is that a combination of three or more mecanum wheels can move in any direction on a flat surface [22]. The typical four-wheel longitudinal symmetrical layout structure is applied in the walking mechanism, as shown in Figure 3. The global coordinate frame \(\left\{{{{\alpha}_{xy}}}\right\}\) is assigned on the center of the walking platform and the velocity of the center point is described as \(\left[{{\begin{array}{*{20}{c}}{{v}_{x}}&{{v}_{y}}&{{w}_{z}}\\ \end{array}}}\right]^{\rm T}\).  
Figure 1: Designing scheme of the wheelchair-stretcher assistive robot.  
Figure 2: The wheelchair-stretcher assistive robot; (**a**) prototype of the robot; (**b**) remote control of the robot.  
Assuming that the roller does not skid against the ground and the platform moves on the horizontal plane, the velocity \(v_{4}\) of the fourth wheel can be established by kinematics analysis:  
\[v_{4}=\left[\begin{array}{cc}0&-\sin\alpha\\ R&\cos\alpha\end{array}\right]\hskip-5.690551 pt\left\|\begin{array}{c}w_{4} \\ v_{g4}\end{array}\right\}, \tag{1}\]  
where \(R\) is the radius of the Mecanum wheel; \(\alpha\) is the bias angle of the roller and the diagonal line in Figure 3 indicates the layout of the rollers in contact with the ground; \(w_{4}\) is the angular velocity of the fourth wheel; \(v_{g4}\) is velocity to perpendicular to the roller.  
Meanwhile, in the global coordinate system, the following equation can be obtained,  
\[v_{4}=\left[\begin{array}{ccc}1&0&m_{2}\\ 0&1&m_{1}\end{array}\right]\left[\begin{array}{c}v_{x}\\ v_{y}\\ w_{z}\end{array}\right] \tag{2}\]  
where \(m_{1}\), \(m_{2}\) are horizontal and vertical distance from the center of the platform to the center of the wheel respectively.  
Combining Equations (1) and (2), we can obtain:  
\[\left[\begin{array}{cc}0&-\sin\alpha\\ R&\cos\alpha\end{array}\right]\hskip-5.690551 pt\left[\begin{array}{c}w_{4} \\ v_{g4}\end{array}\right]=\left[\begin{array}{ccc}1&0&m_{2}\\ 0&1&m_{1}\end{array}\right]\hskip-5.690551 pt\left[\begin{array}{c}v_{x}\\ v_{y}\\ w_{z}\end{array}\right]. \tag{3}\]  
Similarly, the kinematics expression from the first wheel to the three wheels can be computed; finally, the angular velocity vector \(\mathbf{w}\) for the four wheels can be described as,  
\[\mathbf{w}=\left[\begin{array}{c}w_{1}\\ w_{2}\\ w_{3}\\ w_{4}\end{array}\right]=J_{1}\left[\begin{array}{c}v_{x}\\ v_{y}\\ w_{z}\end{array}\right], \tag{4}\]  
where  
\[J_{1}=\left[\begin{array}{ccc}\frac{1}{R\tan\alpha}&\frac{1}{R}&\frac{-(m_{1 }\tan\alpha+m_{2})}{R\tan\alpha}\\ \frac{-1}{R\tan\alpha}&\frac{1}{R}&\frac{m_{1}\tan\alpha+m_{2}}{R\tan\alpha} \\ \frac{1}{R\tan\alpha}&\frac{1}{R}&\frac{m_{1}\tan\alpha+m_{2}}{R\tan\alpha} \\ \end{array}\right].\]  
Figure 3: Layout analysis of mecanum wheel.  
The angular velocity is obtained by the transmission of the motor and the gearing; therefore, the corresponding relationship between the rotational velocity of the motor and the velocity of the platform can be obtained as follows,  
\[n=\frac{\mathbf{w}}{2\pi\mathbf{m}}, \tag{5}\]  
where \(\mathbf{m}\) is the reduction ratio of the mecanum wheels.