Research on Modeling and Load Simulation Algorithm of Tracked Vehicle Braking System on Mu-Split Road Surfaces

Abstract

The efficient work of the braking system ensures the safety of the driver’s life and property during the driving process. For tracked vehicles, a reliable braking system is the guarantee of completing the job or the task. This paper studies a high-speed electric-driven tracked vehicle’s driving state on different mu-split road surfaces and establishes the braking system model and the longitudinal dynamics model of the vehicle and driving wheels. Steering stability analysis was carried out and a relative motion between the two-sided braking systems resulting in a transfer torque was found. A load simulation test system has been built and the improved forward speed tracking algorithm has been used to adjust the inertia by combining the flywheel and the loading motor to simulate the load of the driving wheel. The results of the motor load simulation test show that the load simulation algorithm in this paper can effectively simulate the load on the driving wheel and restore working conditions of the real tracked vehicle. This paper also conducts the combined braking test of the two-sided system which gives the torque situation on both sides of the vehicle and obtains the transfer torque by applying the load simulation algorithm.

1. Introduction

Ensuring the safety of vehicles is the most basic and intrinsic issue in vehicle design and manufacture. Active safety [1,2], as an important part of vehicle safety, requiring the driver to operate and control the car as freely as possible. Vehicle braking performance [3,4,5] is one of the main factors of active safety, and improved braking performance can enhance vehicle active safety.

At present, the analysis of the brake mainly adopts computer-aided engineering (CAE) simulations, numerical simulation and experimental testing. Vdovin, A et al. [6] summarized the advantages and disadvantages between CAE simulations and experimental testing and proposed that these two approaches can be used together to complement each other. Cantoni et al. [7] analyzed the existing literature, summarized the mathematical models of brake vibration and noise, and pointed out that the application of time domain models and nonlinear systems can improve the model accuracy when establishing a brake dynamics model. Popescu, F. D. et al. [8] proposed a new numerical calculation model for the temperature of the brake discs of a mine hoist under emergency braking and applied realistic material properties to brake discs, cable drive wheels and brake pads. Yevtushenko et al. [9,10] studied a numerical method and a three-dimensional finite element model to calculate the temperature of a disc brake when the friction coefficient changes with temperature. Based on the traditional disc brake system, Mahmoud et al. [11] proposed a mathematical model of the dynamic behavior of the wedge disc brake and formulated the mathematical equations of friction coefficient, normal force, and sliding speed through experiments.

Compared to wheeled vehicles [12], tracked vehicles [13,14] have much more complicated working conditions and can operate in various harsh and extreme environments. They are widely used in military warfare, agricultural life, engineering operations, and polar exploration.

Weather conditions, different paving materials, and other factors make the road conditions unpredictable. Complex road conditions can be simplified and roughly divided into two types of road surfaces, namely butt-jointed [15,16] and mu-split road surfaces [17,18,19,20,21]. The butt-jointed road means that the vehicle travels from pavement with one adhesion coefficient to pavement with another adhesion coefficient, and the mu-split road means that the wheels/tracks on both sides of the vehicle travel on the road surface with different adhesion coefficients. Due to the different road adhesion coefficients, the vehicle is prone to drift or even rollover when braking. Figure 1 shows the butt-jointed road surface on the left and the mu-split road surface on the right. The dots and triangles in the figure represent pavements with different adhesion coefficients.

The research on the braking of tracked vehicles on split roads is in its infancy, and the relevant literature and experimental research is relatively lacking. At present, road brake research with different adhesion coefficients is mainly aimed at ABS, which has a certain reference value for track vehicle brake research. Starting from the cooperative control of the steering system and the braking system, Ahn et al. [22] designed the steering angle controller which solved the model error of the existing design and effectively improved the braking stability of the vehicle on the mu-split road. Ping et al. [21] constructed a strong tracking unscented Kalman filter, which can quickly converge on time-varying road surfaces to estimate the friction coefficient, and achieved good results on both the joint road and the bisectional road. Chen et al. [23] proposed a novel anti-lock braking strategy with a sliding mode controller and a rule based braking torque allocating method was developed to enhance the braking stability. Simulations were conducted under a typical low adhesion condition. They compared the control performances of the proposed strategy with a normal full braking strategy and a traditional anti-lock braking strategy transplanted from wheeled vehicles. Chen et al. [24] further develped a novel emergency braking control strategy using a sliding mode slip ratio controller and a rule-based braking torque allocating method. The simulation results of the proposed strategy were much better than three other existing strategies including the full braking strategy, traditional antilock braking strategy, and the sliding mode slip ratio strategy without the use of motors.

The influence factors of the test piece in the bench test and the real driving process of the vehicle are different. In order to ensure the authenticity and accuracy of the bench load simulation, it is necessary to perform compensation loading on the test piece running on the bench, and the core technology is the load simulation technology [25].

As early as 1999, Akpolat et al. [26] proposed a dynamometer control strategy and compared the experimental results of the closed-loop control of some nonlinear loads with the computer simulation results to verify the performance of the proposed load simulation algorithm. Kyslan et al. [27,28,29] proposed a torque closed-loop control method and speed algorithm based on feedforward compensation of inertia and friction torque and applied it to practical engineering.

The purpose of this paper is to investigate the braking load of a tracked vehicle when it travels in a straight line on different mu-split road surfaces. According to the driving posture of the vehicle, this paper divides the split pavement into three types: horizontal split pavement, longitudinal split pavement and lateral split pavement, and the tracked vehicles are modeled respectively. In this paper, the hybrid inertia method is used to simulate the braking load, where the combined flywheel and the motor are used to jointly adjust the inertia. The speed tracking algorithm based on a forward dynamics model is adopted as the motor load simulation control algorithm.

To explore the braking law of tracked vehicles on the mu-split road surfaces, we construct bench tests based on the major horizontal project “High-efficiency Braking System Performance Test Bench”. The layout of the test bench for this project is shown Figure 2 and the arrangement of mechanical brake is shown in Figure 3.

Relying on the “High-efficiency Braking System Performance Test Bench”, this paper verified the effectiveness of the algorithm and simulated the braking conditions of tracked vehicles on three opposite roads.

2. Longitudinal Dynamics Modeling of Tracked Vehicles

2.1. Braking System

The schematic diagram of the brake system of the electric drive tracked vehicle is shown in Figure 4. The tracked vehicle’s braking system is mainly composed of a mechanical brake, a hydraulic retarder and permanent magnet synchronous motors (PMSMs), which worked together when braking the vehicle during actual driving as an electro-hydraulic combined braking system, of which is symmetrical about the coupling mechanism.

To avoid interference from the coupling mechanism, we used a hard axle to connect the brakes on both sides of the vehicle when experimenting. We also installed a torque meter in the test chamber to measure the torque between the two brakes. The arrangement is shown in Figure 3.

Taking the left side as an example, the force and motion of the brake subsystem are shown in the following figure:

In Figure 5, ${\omega }_{em}$ is the speed of the driving wheel, $T_{bem}$ is the required braking torque, and $T_{bh}$, $T_{bmec}$ and $T_m$ are the torque equivalent to the hydraulic retarder, mechanical brake and motor, respectively.

$$T_{bem}=i_r\cdot \left(T_{bh}+T_{bmec}+T_{bmotor}\right)$$

(1)

In the formula, $i_r$ is the speed ratio of the wheel reducer.

2.2. Longitudinal Dynamics

Dynamic analysis of the electric drive tracked vehicle and establishment of an appropriate dynamic model are the basis for the load simulation study. Figure 6 shows the force diagram of the tracked vehicle when braking on an uneven road. The instantaneous position of the tracked vehicle in the figure, its longitudinal slope angle is $\alpha$, the transverse slope angle is $\beta$, the approach angle and the departure angle are ${\phi }_A$ and ${\phi }_D$, respectively.

Assuming that the mass of the tracked vehicle is $m$ and that the speed of the tracked vehicle is $v$ at a certain time, then the driving resistance of the tracked vehicle is

$${\displaystyle \sum F}=F_f+F_w+F_i+F_j$$

(2)

In this formula, $F_f$ is the rolling resistance, $F_f=f\cdot mg\cos \alpha$, in which $f$ is the rolling resistance coefficient. For tracked vehicles, the rolling resistance can be summarized as internal rolling resistance (related to the structure of the track walking system) and external rolling resistance (related to the properties of the road surface). Tracked vehicles have similar coefficients of rolling resistance when traveling on roads of similar nature.

$F_w$ is the wind resistance, $F_w=\frac{C_{D}A}{21.15}{v}^2$, in which $C_D$ is the air drag coefficient and $A$ is the windward area of the tracked vehicle.

$F_i$ is the slope resistance, $F_i=mg\sin \alpha$.

$F_j$ is the inertial resistance, $F_j=m\dot{v}$.

If the vehicle is in the braking condition, it is also constrained by the braking force from the ground. Figure 7 shows the relationship between the ground braking force, the braking force of the brake and the ground adhesion during the braking process.

During the braking process, depending on the state of the driving wheel, only two situations of locking and rolling are considered. When the brake pedal force $F_P$ or the hydraulic pressure $p$ of the brake system is small, the ground braking force $F_b$ is equal to the brake braking force $F_{\mu }$, which means $F_b=F_{\mu }=\frac{T_{\mu }}{r}$, of which $r$ is the drive wheel radius. When $F_P/p$ rises to a certain value $F_{Pa}/p_a$, the ground braking force $F_b$ reaches the value of the adhesion force $F_{\phi }$, which means $F_b=F_{\phi }=\phi \cdot mg$, of which $\phi$ is the adhesion coefficient, the wheels are locked and slippage occurs, and the ground braking force does not increase any more. Point $a$ in Figure 7 is called the lock-up point. Therefore, the expression for the ground braking force is

$$F_b=\left\{\begin{array}{c}{F}_{\mu }\left(rolling\right)\\ F_{\phi }\left(locking\right)\end{array}\right.$$

(3)

In summary, the longitudinal dynamics equation of the tracked vehicle is

$$F_f+F_w+F_i+F_j+F_b=0$$

(4)

Arrange the above formula to get

$$\left\{\begin{array}{c}{F}_{vehicle}=F_f+F_w+F_i+F_b\\ F_{vehicle}=m\dot{v}\end{array}\right.$$

(5)

For a single-sided driving wheel,

$$\left\{\begin{array}{c}{T}_{vehicle}=F_{f}r+F_{i}r+T_{bem}\\ T_{vehicle}=J_{eq}\dot{\omega }\end{array}\right.$$

(6)

In the formula, $T_{bem}$ is the total braking torque output by the braking system to the driving wheels, and its composition has been introduced in the braking system model. $J_{eq}$ is the sum of the rotational inertia of the driving wheel and its equivalent transmission system. $\omega$ is the rotational speed of the driving wheel.

In this paper, the mu-split road is divided into three categories according to the attitude of the tracked vehicle when driving. For different road types, the ground forces on the tracks on both sides of the vehicle are analyzed, respectively.

3. Analysis of Steering Stability under Different Split Road Surfaces

The tracked vehicle runs on the split road with the adhesion coefficients ${\phi }_1$ and ${\phi }_2$ (assuming that ${\phi }_1<{\phi }_2$), respectively. During the braking process, if the braking system applies the same size and direction braking torque to the driving wheels on both sides, the track on the road with the low adhesion coefficient first reaches the maximum ground braking force ($F_{b_1}=F_{{\phi }_1}={\phi }_1\cdot \frac{1}{2}mg$), which is smaller than that ($F_{b_2}=F_{{\phi }_2}={\phi }_2\cdot \frac{1}{2}mg$) on the road with the high adhesion coefficient, so that the speed difference between the two sides of the track is generated, prompting the vehicle to turn.

3.1. Horizontal Mu-Split Pavement

3.1.1. Track-Ground Mechanical Characteristics Analysis

When a tracked vehicle is running on a horizontal road, the normal and lateral forces on its ground surface are rectangular and evenly distributed. As shown in Figure 8, $C$, $C_1$ and $C_2$ are the geometric centers of the complete vehicle, the track on the side with the low adhesion coefficient and the track on the side with the high adhesion coefficient, respectively.

The normal load per unit length of the track is

$$q=\frac{G}{2L}$$

(7)

Due to the different surfaces of the road on both sides, the lateral loads per unit length of the tracks on both sides are different.

$$\left\{\begin{array}{c}{q}_1={\mu }_{1}q\\ q_2={\mu }_{2}q\end{array}\right.$$

(8)

In the formula, ${\mu }_1$ and ${\mu }_2$ are the lateral drag coefficients of the road surfaces on both sides.

The lateral load forms a steering resistance torque, the magnitude of which is

$$\begin{array}{ll}{M}_{\mu }& =2\cdot \left({\mu }_{1}q\frac{L}{2}\right)\cdot \frac{L}{2}+2\cdot \left({\mu }_{2}q\frac{L}{2}\right)\cdot \frac{L}{2}\\ & =\frac{{\mu }_1+{\mu }_2}{2}\cdot q{L}^2\\ & & =\frac{{\mu }_1+{\mu }_2}{4}\cdot GL\end{array}$$

(9)

3.1.2. Kinematics and Dynamics

Figure 9 shows the kinematic relationship of the tracked vehicle and the force on the tracks on both sides when the tracked vehicle drives on the horizontally split road. In this figure, $O$ is the center for vehicle steering, $B$ is the center distance of the tracks on both sides, $R$ is the instantaneous radius of curvature; $v$, $v_1$ and $v_2$ are the speeds of the vehicle and the geometric center of the tracks on both sides, respectively; $a$,$a_1$ and $a_2$ are the corresponding acceleration; $\theta$ is the steering angle; $\omega$ is the angular velocity of the steering. Suppose the acceleration of the steering is $\alpha$, then

$$a=R\cdot \alpha$$

(10)

From the related properties of similar triangles, we can get:

$$\left\{\begin{array}{c}{a}_1=\left(1-B/2R\right)\cdot a\\ a_2=\left(1+B/2R\right)\cdot a\end{array}\right.$$

(11)

As shown in Figure 9, the tracked vehicle is affected by wind resistance $F_w$, rolling resistance $F_f$ and ground braking force $F_b$. If the force of the tracked vehicle is equivalent to the track on both sides, it can be considered that the rolling resistance and wind resistance of the track on both sides are the same.

$$\left\{\begin{array}{c}{F}_{f1}=F_{f2}=\frac{F_f}{2}\\ F_{w1}=F_{w2}=\frac{F_w}{2}\end{array}\right.$$

(12)

Figure 10 shows the schematic diagram of the ground braking force on both sides of the track under the split road. When the braking force of the brakes on both sides is the same, the ground braking force on the two sides of the track is

$$\begin{array}{l}{F}_{b1}=\left\{\begin{array}{c}{F}_{\mu }\left(\mathit{rolling}\right)\\ F_{\phi 1}\left(\mathit{sliding}\right)\end{array}\right.\\ F_{b2}=\left\{\begin{array}{c}{F}_{\mu }\left(\mathit{rolling}\right)\\ F_{\phi 2}\left(\mathit{sliding}\right)\end{array}\right.\end{array}$$

(13)

The resultant external forces on both sides of the track are

$$\left\{\begin{array}{c}{F}_1=F_{b1}+F_{f1}+F_{w1}\\ F_2=F_{b2}+F_{f2}+F_{w2}\end{array}\right.$$

(14)

The dynamic model of the track on both sides is

$$\left\{\begin{array}{c}{F}_1+F_d=\frac{1}{2}m{a}_1\\ \begin{array}{l}{F}_2-F_d=\frac{1}{2}m{a}_2\\ T_d=F_d\cdot r\end{array}\end{array}\right.$$

(15)

In the formula, $T_d$ is a pair of equally large and directional torques generated by the motor coupling mechanism, which is called the transfer torque, and $r$ is the radius of the driving wheel.

Simultaneous Equations (11) and (15) can be obtained

$$T_d=\frac{F_2-F_1}{2}\cdot r+\frac{B}{2R}\left(F_1+F_2\right)\cdot r$$

(16)

For the whole, according to the force and moment balance, there are

$$\left\{\begin{array}{c}{F}_1+F_2=ma\\ \left(F_2-F_1\right)\cdot \frac{B}{2}-M_{\mu }=J\alpha \end{array}\right.$$

(17)

In the formula, $J$ is the equivalent moment of inertia of the tracked vehicle, and $M_{\mu }$ is the steering resistance torque.

From the Equations (10) and (17), the instantaneous radius of curvature can be expressed as

$$R=\frac{J\left(F_1+F_2\right)}{m\left[\left(F_2-F_1\right)\cdot \frac{B}{2}-M_{\mu }\right]}$$

(18)

In the formula, moment of inertia $J$, mass $m$ and track center distance $B$ are determined by the tracked vehicle itself, which are known quantities. After determining the ground braking force and the steering resistance torque $M_{\mu }$, the instantaneous radius of curvature is determined accordingly.

3.2. Longitudinal Ramp Mu-Split Road

3.2.1. Track-Ground Mechanical Characteristics Analysis

When the tracked vehicle travels at speed $v$ on the longitudinal slope with the slope angle $\alpha$, the force analysis of its braking is shown in Figure 11, in which the tracked vehicle is affected by its own gravity, wind resistance, ground support and braking force. The height of the center of mass of the tracked vehicle from the slope is $h_g$; due to the longitudinal slope, the center of mass of the tracked vehicle is shifted backward, and the longitudinal offset is $e$; the track is subjected to ground pressure, and the offset of the center of the ground pressure is $x_0$; according to the moment balance

$$\left\{\begin{array}{c}{F}_N\cdot \left(x_0-e\right)-G\sin \alpha \cdot h_g=0\\ F_N=G\cos \alpha \end{array}\right.$$

(19)

We can get

$$x_0=h_g\tan \alpha +e$$

(20)

Figure 12a shows the normal load distribution of the track when the tracked vehicle is running on the longitudinal ramp. Due to the effect of gravity in the longitudinal direction, the load distribution will change from a rectangle to a trapezoid. In the figure, $q_l$ is the normal load per unit length at the front end of the track, $q_h$ is the normal load per unit length at the rear end of the track, and $q_m$ is the normal load per unit length in the middle of the track, that is, the average normal load per unit length. For the single-sided track, there is

$$q_m=\frac{q_l+q_h}{2}=\frac{G\cos \alpha }{2L}$$

(21)

Let $q^{\text{'}}$ be the difference between $q_l$ and $q_m$ (or $q_m$ and $q_h$), there is

$$\left\{\begin{array}{c}{q}_l=q_m-q^{\text{'}}\\ q_h=q_m+q^{\text{'}}\end{array}\right.$$

(22)

Combining Figure 11 and Figure 12, we get

$$F_N{x}_0=\left[\frac{1}{2}\left(2q^{\text{'}}\right)\frac{L}{2}\right]\cdot \left(\frac{2}{3}\cdot \frac{L}{2}\right)=\frac{q^{\text{'}}{L}^2}{6}$$

(23)

Simplify the above formula to get

$$q^{\text{'}}=\frac{6F_N{x}_0}{L^2}=\frac{6x_0}{L}\cdot q_m$$

(24)

Substitute the Equation (24) into the Equation (22) to get

$$\left\{\begin{array}{c}{q}_l=q_m\left(1-\frac{6x_0}{L}\right)\\ q_h=q_m\left(1+\frac{6x_0}{L}\right)\end{array}\right.$$

(25)

Figure 12b shows the lateral load distribution of the track when the tracked vehicle travels on the longitudinal slope, and the instantaneous steering center offset is $\lambda$.

For a single-sided track (taking the side road with the adhesion coefficient ${\phi }_1$ as an example), according to the force balance, there are

$$\frac{1}{2}{\mu }_1\left(q_h+q_{\lambda }\right)\left(\frac{L}{2}-\lambda \right)=\frac{1}{2}{\mu }_1\left(q_l+q_{\lambda }\right)\left(\frac{L}{2}+\lambda \right)$$

(26)

From the geometric relationship, we have

$$\frac{q_{\lambda }-q_m}{q_h-q_m}=\frac{\lambda }{\raisebox{1ex}{$L$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$$

(27)

Arranged from the above formula, we can get

$${\lambda }^2+\frac{L^2}{6x_0}\cdot \lambda -\frac{L^2}{4}=0$$

(28)

To solve the equation, we get the following

$$\lambda =\frac{-\frac{L^2}{6x_0}+\sqrt{{\left(\frac{L^2}{6x_0}\right)}^2+L^2}}{2}$$

(29)

Steering resistance torque is

$$\begin{array}{l}{M}_{\mu }=\left[\frac{1}{2}{\mu }_1\left(q_h-q_{\lambda }\right)\left(\frac{L}{2}-\lambda \right)\right]\cdot \left[\frac{2}{3}\left(\frac{L}{2}-\lambda \right)\right]+\left[{\mu }_1{q}_{\lambda }\left(\frac{L}{2}-\lambda \right)\right]\cdot \left[\frac{1}{2}\left(\frac{L}{2}-\lambda \right)\right]\\ \hspace{1em} +\left[\frac{1}{2}{\mu }_2\left(q_{\lambda }-q_l\right)\left(\frac{L}{2}+\lambda \right)\right]\cdot \left[\frac{1}{3}\left(\frac{L}{2}+\lambda \right)\right]+\left[{\mu }_2{q}_l\left(\frac{L}{2}+\lambda \right)\right]\cdot \left[\frac{1}{2}\left(\frac{L}{2}+\lambda \right)\right]\end{array}$$

(30)

Substituting Equations (21), (25) and (27) into the Equation (30), we get

$$M_{\mu }=\frac{{\mu }_{1}GL\cos \alpha }{4}{K}_1+\frac{{\mu }_{2}GL\cos \alpha }{4}{K}_2$$

(31)

where,

$$\left\{\begin{array}{c}{K}_1={\left(\frac{1}{2}-\frac{\lambda }{L}\right)}^2\left(1+\frac{4x_0}{L}+\frac{4x_0\lambda }{L^2}\right)\\ K_2={\left(\frac{1}{2}+\frac{\lambda }{L}\right)}^2\left(1-\frac{4x_0}{L}+\frac{4x_0\lambda }{L^2}\right)\end{array}\right.$$

(32)

3.2.2. Kinematics and Dynamics

Figure 13 shows the force of the track on both sides of the vehicle under the braking condition when traveling on the longitudinal split road surface. Compared with the horizontal road, the longitudinal offset of the track grounding steering center is $\lambda$ when running on the longitudinal ramp, the centroid position is offset in the longitudinal position by $e$ and there is a longitudinal component $G\sin \alpha$. Therefore, the resultant external forces on the tracks of both sides are

$$\left\{\begin{array}{c}{F}_1=F_{b1}+F_{f1}+F_{w1}+\frac{1}{2}G\sin \alpha \\ F_2=F_{b2}+F_{f2}+F_{w2}+\frac{1}{2}G\sin \alpha \end{array}\right.$$

(33)

Combined with the relevant content of the horizontal folio road, it can be seen that

$$T_d=\frac{F_2-F_1}{2}\cdot r+\frac{B}{2R}\left(F_1+F_2\right)\cdot r$$

(34)

where, the instantaneous radius of curvature is

$$R=\frac{J\left(F_1+F_2\right)}{m\left[\left(F_2-F_1\right)\cdot \frac{B}{2}-M_{\mu }\right]}$$

(35)

3.3. Lateral Mu-Split Pavement

3.3.1. Track–Ground Mechanical Characteristics Analysis

Figure 14 shows the force when the tracked vehicle travels on a lateral slope. The lateral offset of the center of mass is $e^{’}$, the normal support forces on the left and right tracks are $F_{N1}$ and $F_{N2}$, and the lateral resistance of the ground is $F_{S1}$ and $F_{S2}$, respectively. Due to the different ground normal forces in the grounding sections of the tracks on both sides, the pressure center is laterally shifted down the slope and the offset is $y_0$. According to the torque balance

$$y_0=h_g\tan \beta +e^{’}$$

(36)

Take the center of the tracks on both sides as the balance point and then

$$\left\{\begin{array}{c}{F}_{N1}B-G\sin \beta \cdot h_g-G\cos \beta \cdot \frac{B}{2}=0\\ F_{N2}B+G\sin \beta \cdot h_g-G\cos \beta \cdot \frac{B}{2}=0\end{array}\right.$$

(37)

Solve the equation to get

$$\left\{\begin{array}{c}{F}_{N1}=G\cos \beta \cdot \left(\frac{1}{2}+\frac{y_0}{B}\right)\\ F_{N2}=G\cos \beta \cdot \left(\frac{1}{2}-\frac{y_0}{B}\right)\end{array}\right.$$

(38)

The normal load distribution of the track on the upper and lower sides of the vehicle on the lateral slope is shown in the figure below.

As can be seen from Figure 15,

$$\left\{\begin{array}{c}{F}_{N1}=q_{1}L\\ F_{N2}=q_{2}L\end{array}\right.$$

(39)

Simultaneous Equations (25) and (26) get

$$\left\{\begin{array}{c}{q}_1=\frac{G\cos \beta }{L}\left(\frac{1}{2}+\frac{y_0}{B}\right)\\ q_2=\frac{G\cos \beta }{L}\left(\frac{1}{2}-\frac{y_0}{B}\right)\end{array}\right.$$

(40)

The size of the track adhesion on both sides is

$$\left\{\begin{array}{c}{F}_{\phi 1}={\phi }_1{F}_{N1}={\phi }_1{q}_{1}L\\ F_{\phi 2}={\phi }_2{F}_{N2}={\phi }_2{q}_{2}L\end{array}\right.$$

(41)

In the above formula,

$$\left\{\begin{array}{c}{\phi }_1<{\phi }_2\\ q_1>q_2\end{array}\right.$$

It is hard to directly judge the value size of the track adhesion $F_{\phi 1}$ and $F_{\phi 2}$ on both sides, so it can not judge which side locks first.

Simultaneous Equations (40) and (41) get

$$\left\{\begin{array}{c}{F}_{\phi 1}={\phi }_{1}G\cos \beta \left(\frac{1}{2}+\frac{h_g\tan \beta }{B}\right)\\ F_{\phi 2}={\phi }_{2}G\cos \beta \left(\frac{1}{2}-\frac{h_g\tan \beta }{B}\right)\end{array}\right.$$

(42)

Let $F_{\phi 1}-F_{\phi 2}=0$, Solve these equations to get

$$\beta =\arctan \left(\frac{B}{2h_g}\cdot \frac{{\phi }_2-{\phi }_1}{{\phi }_2+{\phi }_1}\right)$$

(43)

Then,

$$\left\{\begin{array}{c}0\le \beta <\arctan \left(\frac{B}{2h_g}\cdot \frac{{\phi }_2-{\phi }_1}{{\phi }_2+{\phi }_1}\right),lowersidelockingfirst\\ \beta =\arctan \left(\frac{B}{2h_g}\cdot \frac{{\phi }_2-{\phi }_1}{{\phi }_2+{\phi }_1}\right),lockatthesametim\mathrm{e}\\ \beta >\arctan \left(\frac{B}{2h_g}\cdot \frac{{\phi }_2-{\phi }_1}{{\phi }_2+{\phi }_1}\right),uppersidelockingfirst\end{array}\right.$$

(44)

where it is the horizontal split road when $\beta =0$.

Figure 16 shows the lateral load distribution diagram of the vehicle’s inner and outer tracks when running on the lateral split road surfaces.

$$\left\{\begin{array}{c}{F}_{S1}=2{\mu }_1{q}_1\lambda \\ F_{S1}=2{\mu }_2{q}_2\lambda \end{array}\right.$$

(45)

According to the lateral force balance in Figure 13, it can be seen that

$$G\sin \beta =F_{S1}+F_{S2}$$

(46)

Simultaneous Equations (38), (45) and (46) are solved

$$\lambda =\frac{L\cdot \tan \beta }{2\left[{\mu }_1\left(\frac{1}{2}+\frac{y_0}{B}\right)+{\mu }_2\left(\frac{1}{2}-\frac{y_0}{B}\right)\right]}$$

(47)

As shown in Figure 16a, when the upper side locks up first, the steering resistance torque is:

$$\begin{array}{l}{M}_{\mu }={\mu }_1{q}_1\left(\frac{L}{2}+\lambda \right)\cdot \frac{1}{2}\left(\frac{L}{2}+\lambda \right)+{\mu }_1{q}_1\left(\frac{L}{2}-\lambda \right)\cdot \frac{1}{2}\left(\frac{L}{2}-\lambda \right)\\ +{\mu }_2{q}_2\left(\frac{L}{2}+\lambda \right)\cdot \frac{1}{2}\left(\frac{L}{2}+\lambda \right)+{\mu }_2{q}_2\left(\frac{L}{2}-\lambda \right)\cdot \frac{1}{2}\left(\frac{L}{2}-\lambda \right)\end{array}$$

(48)

Combined and organized to get

$$M_{\mu }=\frac{{\mu }_{1}GL\cos \beta }{4}{K}_1+\frac{{\mu }_{2}GL\cos \beta }{4}{K}_2$$

(49)

where,

$$\left\{\begin{array}{c}{K}_1=\left(\frac{1}{2}+\frac{y_0}{B}\right)\left[1+{\left(\frac{2\lambda }{L}\right)}^2\right]\\ K_2=\left(\frac{1}{2}-\frac{y_0}{B}\right)\left[1+{\left(\frac{2\lambda }{L}\right)}^2\right]\end{array}\right.$$

Due to the component force of gravity in the direction of the lateral slope, not only $M_{\mu }$, but also an external moment $G\sin \beta \cdot \lambda$, which helps steering, acts on the tracked vehicle, so the total external moment of lateral force is

$$M_{\mu \beta }=M_{\mu }+G\sin \beta \cdot \lambda$$

(50)

Simultaneously Equations (33) and (34) are solved

$$M_{\mu \beta }=\frac{{\mu }_{1}GL\cos \beta }{4}{K}_1^{\text{'}}+\frac{{\mu }_{2}GL\cos \beta }{4}{K}_2^{\text{'}}$$

(51)

where,

$$\left\{\begin{array}{c}{K}_1^{\text{'}}=\left(\frac{1}{2}+\frac{y_0}{B}\right)\left[1-{\left(\frac{2\lambda }{L}\right)}^2\right]\\ K_2^{\text{'}}=\left(\frac{1}{2}-\frac{y_0}{B}\right)\left[1-{\left(\frac{2\lambda }{L}\right)}^2\right]\end{array}\right.$$

3.3.2. Kinematics and Dynamics

Similarly, as shown in Figure 16b, when the lower side locks first, the instantaneous steering center shifts to the front of the vehicle body, but the lateral resistance graph on both sides is still a rectangle. Therefore, the steering resistance torque $M_{\mu }$ and the total external lateral torque $M_{\mu \beta }$ do not change.

Similarly, the longitudinal force analysis is carried out for the two cases under the lateral slope, as shown in Figure 17. What changes are only the position of the instantaneous steering center and the direction of the resultant external torque, and these have no effect on the expression of the transfer torque. Calculated and sorted as

$$T_d=\frac{F_2-F_1}{2}\cdot r+\frac{B}{2R}\left(F_1+F_2\right)\cdot r$$

(52)

where,

$$R=\frac{J\left(F_1+F_2\right)}{m\left[\left(F_2-F_1\right)\frac{B}{2}-M_{\mu \beta }\right]}$$

(53)

3.4. Dynamics Model of a Tracked Vehicle on Different Mu-Split Roads

In summary, when the tracked vehicle is running on the split road, the expression of the equivalent transfer moment $T_d$ between the tracks on both sides is as follows:

$$T_d=\frac{F_2-F_1}{2}\cdot r+\frac{B}{2R}\left(F_1+F_2\right)\cdot r$$

(54)

In the formula, $r$ is the drive wheel radius. $F_1$ and $F_2$ are the resultant external forces on the tracks on both sides, respectively. $B$ is the track center distance. $R$ is the instantaneous radius of curvature

$$R=\frac{J\left(F_1+F_2\right)}{m\left[\left(F_2-F_1\right)\frac{B}{2}-M_{\mu }\right]}$$

(55)

Steering resistance torque is

$$M_{\mu }=\frac{{\mu }_{1}GL\cos \theta }{4}{K}_1+\frac{{\mu }_{2}GL\cos \theta }{4}{K}_2$$

(56)

when the tracked vehicle is driving on the horizontal divided road,

$$\theta =0,K_1=K_2=1$$

when the tracked vehicle is driving on the longitudinal split road surfaces,

$$\theta =\alpha ,\left\{\begin{array}{c}{K}_1={\left(\frac{1}{2}-\frac{\lambda }{L}\right)}^2\left(1+\frac{4x_0}{L}+\frac{4x_0\lambda }{L^2}\right)\\ K_2={\left(\frac{1}{2}+\frac{\lambda }{L}\right)}^2\left(1-\frac{4x_0}{L}+\frac{4x_0\lambda }{L^2}\right)\end{array}\right.$$

when the tracked vehicle is driving on the lateral split road surfaces,

$$\theta =\beta ,\left\{\begin{array}{c}{K}_1=\left(\frac{1}{2}+\frac{y_0}{B}\right)\left[1-{\left(\frac{2\lambda }{L}\right)}^2\right]\\ K_2=\left(\frac{1}{2}-\frac{y_0}{B}\right)\left[1-{\left(\frac{2\lambda }{L}\right)}^2\right]\end{array}\right.$$

4. Load Simulation System and Algorithm

4.1. Load Simulation System

Taking the single-sided braking system of a tracked vehicle as an example, the construction of a load simulation system is shown in Figure 18. The load simulation system is mainly composed of the loading motor and its corresponding motor controller, flywheel box, braking system of the tested vehicle, corresponding sensors and data acquisition system. The electric drive tracked vehicle load simulation system can simulate various test conditions and carry out various forms of tests through the combination of different power sources (loading motor and flywheel group) and the test piece. This system can also be used for the double-side load simulation test of tracked vehicles, and is arranged symmetrically on both sides of the braking system.

4.2. Load Simulation Algorithm

The load with the vehicle driving wheel is the simulated target load. The speed of the single-side driving wheel is ${\omega }_{em}$, and the output torque is $T_{em}$ (for the brake condition in this paper, the output torque is the braking torque $T_{bem}$ on the side of the driving wheel). Therefore, the load transfer function of the vehicle unilateral driving wheel can be expressed as

$$G_{em}\left(s\right)=\frac{{\omega }_{em}\left(s\right)}{T_{em}\left(s\right)}$$

(57)

In the tracked vehicle brake bench test, the output speed of the tested piece is $\omega$, the output torque is $T$, and the torque provided by the bench loading motor is $T_m$. Then the load transfer function of the test piece can be expressed as

$$G\left(s\right)=\frac{\omega \left(s\right)}{T\left(s\right)-T_m\left(s\right)}$$

(58)

To make the actual load of the test piece the same as the target load and then

$$\frac{\omega \left(s\right)}{T\left(s\right)}=G_{em}\left(s\right)$$

(59)

4.2.1. Forward Speed Tracking Algorithm

The forward speed tracking algorithm is used to calculate the difference between the actual speed $\omega \left(s\right)$ and the target speed ${\omega }_{em}\left(s\right)$. The closed-loop controller $C\left(s\right)$ controls the output load torque $T_m\left(s\right)$ of the test bench loading motor, so that the speed of the tested component in the bench always approaches the speed of the vehicle’s driving wheel.

The algorithm flow is shown in Figure 19.

The gantry load transfer function is

$$G_1\left(s\right)=\frac{\omega \left(s\right)}{T\left(s\right)}=\frac{G\left(s\right)\left[1-G_{em}\left(s\right)C\left(s\right)\right]}{1-G\left(s\right)C\left(s\right)}$$

(60)

4.2.2. Torque Feedforward and Compensation Module

The gantry load transfer function obtained by the forward speed tracking algorithm is $G_1\left(s\right)$, which needs to be improved for $G_1\left(s\right)\ne G_{em}\left(s\right)$ and the algorithm flow is shown in Figure 20.

The numbers 1 and 2 in Figure 20 represent the steps of improvement. As shown in the first step, the torque feedforward is added and then the gantry load transfer function becomes

$$\frac{\omega \left(s\right)}{T\left(s\right)}=G_{em}\left(s\right)\cdot \frac{G\left(s\right)C\left(s\right)}{1+G\left(s\right)C\left(s\right)}$$

(61)

The second step is adding a compensation module to the system.

$$\begin{array}{ll}{G}_2\left(s\right)& =\frac{\omega \left(s\right)}{T\left(s\right)}\\ & =C_{comp}\left(s\right)\cdot G_{em}\left(s\right)\cdot \frac{G\left(s\right)C\left(s\right)}{1+G\left(s\right)C\left(s\right)}\\ & =G_{em}\left(s\right)\end{array}$$

(62)

Then

$$C_{comp}\left(s\right)=1+\frac{1}{G\left(s\right)C\left(s\right)}$$

(63)

It can be known from the form of compensation module $C_{comp}\left(s\right)$ that its design is affected by the form of controller $C\left(s\right)$.

4.2.3. Forward Speed Tracking Algorithm with Compensator

Adding a compensator $G^{-1}\left(s\right)$ can make $G_1\left(s\right)=\frac{\omega \left(s\right)}{T\left(s\right)}=G_{em}\left(s\right)$. Since the rotational speed in $G^{-1}\left(s\right)$ has been calculated in the forward dynamics $G\left(s\right)$, there will be no disturbance to the system. This method is used to simulate the load in this paper, and the specific algorithm flow chart is as Figure 21.

Compared with adding a feedforward compensation module $C_{comp}\left(s\right)$, the form of the compensator $G^{-1}\left(s\right)$ is simpler and will not be limited by the form of the controller $C\left(s\right)$. When the controller adopts fuzzy control or a more intelligent algorithm, the compensation module is difficult to design and implement.

5. Results and Discussion

5.1. Motor Load Simulation

Figure 22a shows the force on the main shaft of the tracked vehicle brake test bench under ideal conditions in which the test inertia of the bench is equal to the equivalent inertia of the tracked vehicle at the steering wheel. So,

$$T_{bem}=J_0\frac{d{\omega }_{em}}{dt}$$

(64)

$J_0$ is the ideal state bench test inertia and the bench foundation inertia.

In the real situation, the basic inertia of the bench is much smaller than the equivalent inertia of the actual vehicle, and it is necessary to increase and adjust the inertia through a flywheel group or a motor. Adjusting inertia through the flywheel group has high real-time and accuracy, but it cannot be adjusted steplessly. The simulated inertia through the motor has a time delay, but the inertia can be adjusted step by step. During the test in this paper, the flywheel group and the motor are used to jointly simulate the load.

As shown in Figure 22b, $M_0$ is the resistance torque of the stand itself, $T_m$ is the output torque of the loading motor and $\omega$ is the actual spindle speed of the stand. The resistance torque $M_0$ of the gantry is opposite to the direction of the spindle speed $\omega$, and the direction of the output torque $T_m$ of the loading motor may be the same as or opposite to the direction of the spindle speed. Taking the direction of spindle speed as positive, then

$$T_{bem}+M_0+T_m=J\frac{d\omega }{dt}$$

(65)

where $J$ is the actual bench test inertia, which is composed of the base inertia $J_0$ of the bench and the inertia $J_{FW}$ of the flywheel group:

$$J=J_0+J_{FW}$$

(66)

In Figure 22a,b, when the braking torque and braking deceleration are respectively the same, there is

$$T_m=\frac{J_{FW}}{J_0}{T}_{bem}-M_0$$

(67)

It can be seen from the above formula that the output torque $T_m$ of the loading motor is positively correlated with the braking torque $T_{bem}$.

5.2. Simulation Analysis of Motor Load

Taking a certain type of 30 t tracked vehicle as the research object, the forward speed tracking method in the form of adding a compensator is used for simulation calculation. The simulation parameters of the motor load simulation are shown in

Table 1. Main parameters of the motor load simulation.

Parameter	Notation	Unit	Value
Vehicle mass	m	t	30
Drive wheel radius	r	m	0.31
Frontal area of the vehicle	A	m2	4.6
Air density	ρ	Ns2/m4	1
Ratio of the side reducer	ir	/	3.8

.

Different signals are used as input to verify the algorithm. Figure 23a,b shows the results of torque ramp signal input and the torque square wave signal input, respectively. The gantry speed curve is close to the vehicle speed curve and has a high degree of consistency, indicating that the load simulation algorithm can effectively track the speed.

5.3. Bench Test of Braking Performance on Mu-Split Road

Set the initial braking speed to 1500 rpm and the final braking speed to 0. Then brake under the braking pressures of 1 Mpa, 2 MPa, and 3 MPa, respectively; collect the braking torque, the rotational speed of the left and right sides, and the torque and rotational speed of the shaft in the test chamber, and analyze the collected data. For convenience of observation and clear presentation, the following analysis is based on data collected under the braking pressure of 1 MPa. The laws under 2 MPa and 3 MPa pressure are similar.

Figure 24 shows the simulation results of bench tests and the actual vehicle movement. The two graphs above show the variation of the braking torque on the left and right sides of the bench tests and the actual vehicle, respectively. The bottom left picture shows the torque change between the two brakes and the right one shows the change in the speed of the driving wheel. Figure 24 also shows the error between the bench tests and the actual vehicle.

The load simulation algorithm was applied to the bench test, and good results were also obtained which is shown in Figure 24. The bench tests reproduced the actual vehicle operating conditions very well. The torque and the speed of the driving wheel can be well matched and the error is small, within the allowable range. In Figure 24, we can see some abrupt values which are caused by acquisition disturbances during bench testing and friction from the bench spindle. The results of the torque between the driving wheels on both sides have a large deviation, because the coupling mechanism is used between the two wheels on the real vehicle, and the bench test adopts a hard connection.

The main parameters of different types of mu-split road surfaces are shown in

Table 2. Main parameters of mu-split road surfaces.

Parameter	Notation	Unit	Value
Left tcrack road adhesion coefficient	${\phi }_1$	/	0.45
Right tcrack road adhesion coefficient	${\phi }_2$	/	0.8
Slope angle of horizontal road (longitudinal, lateral)	$\left(\alpha ,\beta \right)$	°	(0, 0)
Slope angle of longitudinal road (longitudinal, lateral)	$\left(\alpha ,\beta \right)$	°	(10, 0)
Slope angle of lateral road (longitudinal, lateral)	$\left(\alpha ,\beta \right)$	°	(0, 10)

.

The error $E$ is defined as follows

$$E=\frac{x_{bench\_test}-x_{vehicle}}{x_{vehicle}}\times 100\%$$

(68)

where, $x$ represents for torque and velocity data.

When the maximum error $E_{\max }$ is less than 5%, the bench tests are considered to be able to reproduce the actual vehicle conditions well. On the basis of removing singular values,

Table 3. The error between bench tests and actual vehicle on different types of the road.

Types of the Road		$E_{\max }$
Horizontal mu-split road	Torque in left side	0.017
	Torque in right side	0.016
	Rotating speed of driving wheel	0.009
Longitudinal mu-split road	Torque in left side	0.023
	Torque in right side	0.032
	Rotating speed of driving wheel	0.029
Lateral mu-split road	Torque in left side	0.035
	Torque in right side	0.027
	Rotating speed of driving wheel	0.035

shows the error between bench tests and actual vehicle on different types of the road.

In the bench tests, we obtained the rotation speed of the drive wheels and the torque variation on both sides of the tracked vehicle, which is shown in Figure 25.

We add the left and right end torque and then compare it to the torque in the test chamber.

As we see in Figure 26, the sum of the left and right end torques has a double relationship with the torque in the test chamber.

We use the left end to subtract the right end torque, and then we put the difference between left and right torque, retarder torque, and capstan speed into the same graph.

As we can see in Figure 27, the speed is divided into three stages which are marked with the numbers 1, 2, and 3 in the figure during the braking phase.

In the first stage, the initial speed of braking is 1500 rpm. When the speed is above 1000 rpm, the eddy current retarder brakes alone. In the second stage, when the speed drops to 1000 rpm, the mechanical brake participates in the braking and cooperates with the retarder for joint braking. In the third stage, when the speed drops to 500 rpm, the mechanical brake brakes alone. At the same time, we can also observe that, before braking, when the speed of the driving wheel changes, the torque will change accordingly, which is mainly generated by the friction of the main shaft of the test bench.

In addition, the instantaneous radius of curvature of the tracked vehicle and the change in the transfer torque over the entire time course are shown in Figure 28.

Combining Figure 27 and Figure 28, it can be found that the change trend of the transfer torque is similar to the difference between the left and right end torques, but the magnitude and direction of the torque are different.

There are many studies on the mu-split roads, but they are all based on wheeled vehicles. Ahn et al. [22] proposed a design of a cooperative control between a steering system and a brake system on a split-mu surface and designed the steering angle controller. The controller effectively improved the braking stability of the vehicle on the mu-split road. Ping et al. [21] constructed a strong tracking unscented Kalman filter and achieved good results on both the joint road and the bisectional road.

However, there is almost no literature that studies the operation of tracked vehicles on split roads. Adhesion degradation of tracked vehicles is generally not considered since tracked vehicles by definition have high levels of adhesion. But the working conditions of tracked vehicles are very complex and harsh, especially when moving on the mu-split roads, the possibility of rollover is high. In this paper, we established the dynamic model of the tracked vehicle on three different types of mu-split roads. We performed load simulations on the permanent magnet synchronous motors and carried out bench tests with good results. Our research fills a gap in the field of tracked vehicles driving on the mu-split roads.

6. Conclusions and Future Developments

In this paper, we established the longitudinal dynamics model of the electric-driven tracked vehicle and divided the split road into three types: horizontal, longitudinal, and transverse ramp, depending on the driving state of the vehicle. The track–ground mechanical properties were studied based on these three bisected roads; the steering stability of the tracked vehicle on different bisected roads was analyzed. Due to the different road adhesion coefficients on both sides of the mu-split road, the braking torque on both sides of the tracked vehicle is different. Based on this, we propose the concept of transmission torque.

Additionally, this paper builds a load simulation system and adopts a dynamic load simulation algorithm based on forward speed tracking, which is verified that the algorithm has the characteristics of stability and fast dynamic response through the motor load simulation analysis. The motor load simulation also provides a basis for bench testing.

Bench tests were carried out to reproduce the actual driving conditions of the electric-driven tracked vehicle on a mu-split road surface, and the change diagram is drawn over time. During the whole test process, the actual speed can follow the reference speed well, and the change process of the transfer torque can be clearly observed.

There are still some areas for improvement in this paper and the developments for the future are as follows:

(1)

The load simulation method based on the forward speed tracking algorithm has a delay, and it still has room for optimization.

(2)

This paper only considered the case where the adhesion coefficient is large on one side and small on the other side. Subsequent studies can consider the case where the adhesion coefficients of the pavement on both sides are very small or large.

(3)

The load simulation of the mu-split road surfaces can be further extended to the research related to the road load spectrum, which makes the coverage of the load simulation more extensive.