# Study on the Control of Torque Distribution of 4 WD Corn Harvester Operation Drive
## 2 Design and Modeling
### Establishment of the Mathematical Model
#### 2.2.5 The Mathematical Model of Brakes  
The harvester has a high degree of inertia, to ensure good braking control performance, reduce the braking torque load of the brake, miniaturize the brake structure, combine the speed-torque with other mechanical relationships, this study designed a harvester brake with the mounting structure shown in Figure 4, which was not mounted directly on the drive wheels but was fastened as a component to the differential part of the harvester drive axle.  
Each of the four drive wheels adopted independent floating electronically controlled disc brakes, which makes the drive torque active distribution control between the drive wheels more precise and also makes the harvester have a better safety performance. The principle of active inter-wheel torque distribution control is shown in Figure 4. If one of the two drive wheels on the same drive axle takes a certain degree of braking, the torque input to the drive axle will be transferred from the braking wheel to the other drive wheel through the differential, thus realizing the inter-wheel torque distribution control.  
In this study, a logic gate algorithm was used to design the brake control system with three control states: pressurization, pressure-holding, and decompression, and the corresponding output control signals were digital in and out (DIO) = 1, 2, 3; the input signals were the angular acceleration \(\dot{\omega}\), and the relative slip rate \(S_{Ri}\) of the corresponding drive wheels.  
Figure 4: Brake mounting structure.  
The corresponding slip rate of each tire is  
\[\left\{\begin{array}{l}S_{R1}=\frac{(\omega_{1}-\omega_{2})}{\omega_{1}}\\ S_{R2}=\frac{(\omega_{2}-\omega_{1})}{\omega_{2}}\\ S_{R3}=\frac{(\omega_{3}-\omega_{4})}{\omega_{3}}\\ S_{R4}=\frac{(\omega_{4}-\omega_{3})}{\omega_{4}}\end{array}\right. \tag{20}\]  
where \(S_{R1}\), \(S_{R2}\), \(S_{R3}\), and \(S_{R4}\) in \(S_{Ri}\)\(i=1,2,3,4\) are the left front, right front, left rear, and right rear drive wheel slip rates, respectively; \(S_{ob}\) is the target control setting slip rate. The specific control strategy is shown in Table 2.  
On this basis, the torque model and pressure model of the brakes were established. The pressure model inputs the control signal \(in\) and outputs the pressure signal \(P\). The output pressure signal \(P\) is used as the input signal of the torque model to make it output the braking torque \(M_{b}\).  
Mathematical model of the brakes:  
\[P=\left\{\begin{array}{l}P_{o}+P_{max}\Big{(}1-e^{-iK_{t}}\Big{)}\quad,in=0\\ P_{o}e^{-iK_{t}}\quad\quad\quad\quad\quad\quad,in=1\\,in=3\end{array}\right. \tag{21}\]  
\[M_{b}=i_{b}K_{b}P \tag{22}\]  
the solenoid brake command in Equations (21) and (22) is entered as 0 for brake pressurization, 1 for brake pressure-holding, and 3 for brake decompression.  
Where \(P_{o}\) is the initial brake pressure; \(t\) is the duration; \(K_{t}\) is the brake braking time coefficient; \(i_{b}\) is the ratio of the brake transmission mechanism; \(K_{b}\) is the braking coefficient.