# Optimal Dimensional Synthesis of Ackermann Steering Mechanisms for Three-Axle, Six-Wheeled Vehicles
## 3 Steering Geometry of a Three-Axle, Six-Wheeled Vehicle
### Steering of a Three-Axle, Six-Wheeled Vehicle
#### 3.1.2 Reverse-Phase Steering Mode  
The six-wheeled reverse-phase steering system, as shown in Figures 5 and 6, has only the first (front) and third (rear) axles as steerable axles. The steering mechanisms on the front and rear axles steer the wheels, and the trapezoidal steering mechanism is arranged as shown and well-designed, which ensures the instantaneous turning center lies on the extension line of the middle axle. The two sets of steering mechanisms on the front and the rear shafts can be connected by gear or hydraulic system. Figure 5 illustrates the decomposition of the ideal Ackermann steering geometry for a left turn. In the figure, \(L\) is the distance between the front and rear axles, \(L_{1}\) is the distance between the front and intermediate axles, \(L_{3}\) is the distance between the intermediate and rear axles, \(W_{f}\) is the width between the front tires, and \(W_{r}\) is the width between the rear tires, \(\delta_{in\ f}\) and \(\delta_{out\ f}\) are the steering angles of the front inner and outer wheels. While \(\delta_{in\ r}\), and \(\delta_{out\ f}\), are the steering angles of the rear inner and outer wheels. Distance \(a_{1}\) and \(a_{3}\) are the perpendicular distance from the center of gravity of the front and rear parts of the vehicle, \(C_{1}\) and \(C_{3}\), to the instantaneous turning center O. The steering angles of the outer wheel and inner wheel of the front and rear four-bar steering mechanisms should satisfy the Ackermann geometrical relations, respectively. \[cot\delta_{outA\ f}-cot\delta_{in\ f}=\frac{W_{f}}{L_{1}} \tag{18}\]  
Figure 4: Front-axle steering system (left turn). \[cot\delta_{outA\ _{r}}-cot\delta_{in\ _{r}}=\frac{W_{r}}{L-L_{1}} \tag{19}\]  
Here, \(\delta_{outA\ _{f}}(i)\) and \(\delta_{outA\ _{r}}(i)\) mean the ideal Ackermann steering angles of the outer wheels of the front and rear axles. Figure 5: Ideal steering geometry of a three-axle vehicle (exploded view of left turning). Figure 6: Reverse-phase steering system of three-axle, six-wheeled (left turning). From the triangular relationship of Ackermann steering geometry in Figure 5, the steering angles of the inner wheels of rear and front axles, \(\delta_{in_{r}}\) and \(\delta_{in_{f}}\), should further satisfy the following relationship. \[tan\delta_{in_{r}}=-Ktan\delta_{in_{f}} \tag{20}\]  
where  
\[K=\frac{L-L_{1}}{L_{1}}=\frac{L_{3}}{L_{1}} \tag{21}\]  
Here, \(K\) is called the transmission ratio of front to rear axle. Equations (18) and (19) are the primary Ackermann geometry equations, and Equation (20) can be thought of as the dominant Ackermann geometry equation for a three-axle, six-wheeled vehicle with reverse-phase steering mode. Equation (20) can be substituted into Equation (19), and obtain the following equation:  
\[cot\delta_{outA~{}r}+\frac{cot\delta_{in_{f}}}{K}=\frac{W_{r}}{L_{3}} \tag{22}\]  
It indicates that the ideal Ackermann steering angle of the outer wheel of the rear steering mechanism, \(\delta_{outA~{}r}\), is mainly affected by the parameter \(K\). From Figure 5. The instantaneous turning centers of the front and rear parts of the vehicle, \(R_{c1}\) and \(R_{c3}\), can be expressed as follows:  
\[R_{c1}=\sqrt{a_{1}^{2}+L_{1}^{2}cot^{2}\delta_{f}} \tag{23}\]  
Figure 4:0  
Figure 4:1  
and  
Figure 4:2  
Figure 4:3  
Figure 4:4  
When the intermediate axle lies in the middle position of the front and rear axles, then \(L_{1}=L_{3}\), \(K=1\). If \(a_{1}=a_{3}\) and \(W_{f}=W_{r}\), then \(R_{c1}=R_{c3}\). If \(K\) is taken into account as a design parameter for optimization of the steering mechanism, then \(K=1\) can be considered as a special case. As shown in Figure 6 of the three-axle, six-wheeled, reverse-phase steering system, the four-bar steering mechanisms on axle 1 (front axle) and axle 3 (rear axle) are both Ackermann steering mechanisms. When the vehicle turns left, the movement of the front and rear Ackermann steering mechanisms, is shown by red dash line in Figures 7 and 8, respectively.