#### Abstract

The high-speed steering control of tracked vehicles has been studied thoroughly, but the high-speed steering stability boundary of tracked vehicles has rarely been investigated. The establishment of a dynamics model with a high computational accuracy is the premise of studying the high-speed steering stability boundary of tracked vehicles. In this work, the track forces of tracked vehicles during steering were determined based on a shear stress model, and the steering dynamics model of tracked vehicles was established. A zero-differential-steering tracked vehicle was used as the object of study. The calculation results of the steering kinematics model and dynamics model of tracked vehicles under different steering conditions were studied in detail on sandy road surface environment, as well as the change laws of the steering trajectory, steering radius, vehicle centroid velocity, and steering slip angle. The steering trajectories of tracked vehicles on five kinds of roads under the same steering conditions were determined by simulations. By setting the critical condition for unstable steering, the corresponding relationship between the maximum circumferential velocity difference of the sprocket , the minimum steering radius , the maximum sideslip angle , and the theoretical centroid speed was obtained when the vehicle turned on five kinds of roads, and it was used to determine the high-speed steering stability boundary of tracked vehicles. The simulation results can provide a reference for the design of tracked vehicle steering mechanisms and high-speed steering control.

#### 1. Introduction

The design and control of high-speed tracked vehicles require not only the study of the stable steering performance of the vehicle but also the understanding of the variations of the motion parameters during the unstable steering process. Unstable steering conditions of a tracked vehicle are conditions in which the tracked vehicle cannot maintain the desired stable steering radius due to factors such as the relative ground sliding of the track and centrifugal forces during steering. The motion parameters, such as the steering radius, yaw rate, and sideslip angle, change rapidly, and the vehicle will “spin out” at large lateral speeds [1]. Under unstable steering conditions, the steering controllability of tracked vehicles is reduced, and it is difficult to ensure driving safety [2]. Determining the high-speed steering stability boundary of tracked vehicles can provide theoretical support for the design and control of high-speed tracked vehicles and for improving driving safety [3]. For unmanned high-speed tracked vehicles, it is also necessary to consider the matching of the road parameters, planning speed, and planned trajectory curvature during trajectory planning to ensure that the vehicle can track the trajectory safely and accurately [4]. The kinematics model is widely used in the analysis and control of the low-speed steering conditions of tracked vehicles. However, the model calculation error is large when ignoring the influence of factors such as the relative ground slip of the track, vehicle lateral slip, and centrifugal force, and thus, it is difficult to meet the model calculation accuracy requirements for unstable steering conditions [5]. An effective means to study the high-speed steering stability of tracked vehicles is to establish a steering dynamics model with high accuracy, analyze the variations of the motion parameters in the process of high-speed steering, and determine the high-speed steering stability boundary of the tracked vehicles.

Researchers have conducted in-depth research on how to improve the calculation accuracy of tracked vehicle steering dynamics models. Steeds [6] assumed that the slip relationship between the track and the ground conforms to Coulomb’s law, and the track ground pressure is uniform. A steering model of tracked vehicles considering the slip of the track relative to the ground was established. Purdy [7] further extended the work of Steeds to establish a four-degree-of-freedom (4-DOF) tracking vehicle steering dynamics model including roll and applied the model to the steering yaw stability control of tracked vehicles. However, under the assumption that the slip relationship between the track and the ground conforms to Coulomb’s law, the track traction force, braking force, and steering resistance torque do not change with the change of the steering radius when the tracked vehicle turns at high speeds. This does not correspond to the track forces during the actual steering of the tracked vehicle [8]. Zhang et al. [9] regarded the lateral and longitudinal sliding friction coefficients of tracked vehicles as functions of the steering radius and established a steering dynamics model to analyze the slope steering of the tracked vehicle. Janarthanan et al. [10] assumed that there was a difference between the lateral and longitudinal friction coefficients in the steering process of tracked vehicles and developed a 6-DOF high-speed unstable steering dynamics model considering the anisotropy of the sliding friction coefficient. However, the above two models were based on empirical formulas, and the correlation between the soil properties and traction force of the track was not considered. Wong and Chiang [11] considered the relationship between the shear stress and shear displacement and deduced a steady-state steering dynamics model of tracked vehicles on solid ground. Based on Wong’s work, Said et al. [12] developed a steady-state steering dynamics model of tracked vehicles on soft ground and applied it to the study of steering trafficability and maneuverability of tracked vehicles. Rui et al. [13] systematically designed the experimental scheme of tracked vehicle steering and verified the accuracy of using the shear stress model to calculate the track force through experiments.

In this study, a steering dynamics model of tracked vehicles was established based on a shear stress model, and the model was applied to the study of the high-speed steering stability of tracked vehicles. A zero-differential steering tracked vehicle was selected as the object of study, and a simulation model was established using MATLAB/Simulink to simulate and analyze the changes of the steering trajectory, steering radius, vehicle centroid velocity, and sideslip angle in the process of high-speed steering. Then, the influence of road parameters on the steering performance of tracked vehicles was analyzed. The critical conditions of the unstable steering of tracked vehicles when steering on different roads were determined. The corresponding relationship between the maximum circumferential velocity difference of the sprocket , the minimum steering radius , the maximum sideslip angle , and the theoretical centroid speed was obtained and used to determine the high-speed steering stability boundary of tracked vehicles.

#### 2. Steering Dynamics Model of Tracked Vehicle Based on Shear Stress Model

The establishment of the tracked vehicle dynamics model requires the following assumptions:(1)The centroid of the tracked vehicle coincides with the geometric center(2)Track subsidence and side pushing effects of the track slab are neglected(3)The track is not stretchable, and the influence of the track width on the ground pressure can be neglected(4)During the steering process of the tracked vehicle, the shear stress at any point on the track is related to the shear displacement at that point, according to the shear stress model. The expression iswhere is the soil cohesion, is the normal stress, is the angle of internal shearing resistance of the terrain, and is the shear deformation modulus.

##### 2.1. Kinematics Analysis of Tracked Vehicle Steering

To facilitate the analysis of the horizontal ground steering performance of tracked vehicles, the coordinate system shown in Figure 1 was established. is the geodetic coordinate system. is the vehicle body coordinate system, the origin of coordinate coincides with the geometric center of the vehicle, and the *x*-direction is the vehicle’s forward direction. is the vehicle centroid speed, and and are the components of in the -direction and -direction, respectively. is the yaw angle of the tracked vehicle. is the angle between vehicle speed direction and the -axis. is the sideslip angle, . is the length of the track contact with the ground. is the distance of the track center line on both sides. is the track widths. is the angle between the direction of the velocity at a point on the track and the -axis. and are the instantaneous steering centers of low-speed side track and high-speed side track, respectively. In this paper, the subscript represents the low-speed side track, and represents the high-speed side track. is the offset of steering center relative to -axis, .

The relationship between the centroid velocity of the tracked vehicle and the accelerations and along -axis and -axis, respectively, can be expressed as follows:

During the steering process of the tracked vehicle, the trajectory of can be expressed as follows:

The velocities of each point on the same track are the same in the -direction. The velocity of a point on the track in the -direction is determined by the yaw rate and the -axis coordinate of the point . The shear velocities and of this point relative to the ground in -direction and -direction can be expressed as follows:where is the circumferential velocity of the -side sprocket, , is the angular velocity of the -side sprocket, and is the radius of the sprocket.

The angle between the speed of a point on the -side track and the -axis can be expressed as follows:

The slip ratio of the -side track on both sides can be expressed as follows:

The shear displacement of a point on the track begins to accumulate from the front edge of the plane where the track touches the ground, and it reaches the maximum at the rear edge of the plane. The shear displacement of a point on the -side track can be obtained by integrating the shear velocity, and it can be expressed as follows:

##### 2.2. Steering Force Analysis of Tracked Vehicle

During the steering process of the tracked vehicle, the distribution of the track ground pressure directly affects the track force. Wang et al. [14] found that the ground pressure of the track is mainly concentrated under the road wheel. To simplify the calculation, it is assumed that the ground pressure of the road wheel is rectangularly distributed directly below the road wheel. The ground pressure on the unit grounding area of the -th road wheel on the -side track consists of two parts: one part is the ground pressure on the unit grounding area under the action of the inertial force and gravity, and the other is the change in the ground pressure on the unit grounding area of the -th road wheel on the -side track caused by the tension of the track, which is denoted as . Thus, the ground pressure can be expressed as

Figure 2 shows the change in the ground pressure of each road wheel under the influence of track tension. is the vehicle weight. is the acceleration of gravity. is the number of road wheels of the one-sided track. is the vertical component of the -side track tension acting on the -th road wheel. is the front track tension of the -th track. is the tension force of the rear track of the -side track. is the approach angle of the tracked vehicle. is the departure angle of the tracked vehicle. is the track plate pitch. is the ground auxiliary line, which is a distance from the front edge of the track grounding area. This is used to solve the track ground pressure. and are equal to the force on the -side track in the -direction, which can be expressed as follows:

The component of track tension in the vertical direction directly acts on the first and last road wheels. The ground pressure variation of the -th road wheel of the -side track can be expressed as follows:

Based on previous work [15], it was assumed that the pressure variation of the unit grounding area of the -th road wheel caused by the track tension on the -side track changed linearly, and the rate of change was . can be expressed as follows:where is the value of at .

From Equations (9)–(11), the variation of the ground pressure of the -th road wheel caused by the track tension on the -th side can be expressed as follows:

The torque balance equations were established for the ground pressure of each road wheel about the auxiliary line –. These can be expressed as follows:

The ground pressure variation of the -th road wheel under track tension can be obtained by solving the equilibrium equation, as follows:

The inertial force generated by the steering of tracked vehicles will change the distribution of the ground pressure on both sides of the tracked vehicles, which is more significant in high-speed steering. Figure 3 shows the distribution of the track ground pressure under the influence of the inertial force.

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**(b)**

The component of the inertial force in the -direction increases the normal force of the high-speed track in contact with the ground and decreases the normal force of the low-speed track. The normal force of the -side track grounding can be expressed as follows:where is the height of the track center of gravity.

The inertial force component in the -direction redistributes the pressure of each road wheel on the unilateral track. The ground pressure per unit area of the -th road wheel of the -side track can be expressed as

Figure 4 shows the track force when the tracked vehicle turns. and are the components of the shear force between the -side track and the ground in the -direction and -direction, respectively. is the steering resistance torque of the -side track. is the rolling resistance of the -side track. Based on (1) and (14)–(16), the shear forces and of the -th road wheel of the -side track were obtained. and were obtained by summing and , respectively, which can be expressed as follows:

The rolling resistance of the -side track can be expressed as follows:where is the coefficient of rolling resistance.

The steering resistance torque of the -side track was also obtained by summing the steering resistance torque of each road wheel, which can be expressed as follows:

The dynamic steering balance equation of tracked vehicles can be expressed as follows:where is the yaw inertia of the tracked vehicle around the -axis.

The accelerations of the vehicle in the *x*- and *y*-directions and the yaw angular acceleration by (2) and (20) can be expressed as follows:

#### 3. Simulation Analysis of High-Speed Steering Process of Tracked Vehicles

The steering dynamics simulation model of tracked vehicles was established based on MATLAB/Simulink, and the steering process was simulated and analyzed. At the same time, the steering kinematics model was established as the simulation experiment control group. The simulation step size was . The simulation inputs at time were the circumferential velocities of the sprockets and on both sides of the tracked vehicle. The solution process of the steering dynamics and kinematics simulation model is shown in Figure 5. The simulation object was a zero-differential-steering tracked vehicle. For a zero-differential-steering tracked vehicle, the theoretical centroid speed was determined by the circumferential velocity of the sprocket on both sides, . The sand road was selected for the simulation experiment. The tracked vehicles and road parameters are shown in Table 1.

**(a)**

**(b)**

During the simulation, the initial centroid velocity , the initial yaw angle , the initial centroid sideslip angle , and the initial centroid acceleration were set. Three sets of steering conditions were set by changing . At the beginning of the simulation, the velocities were set. was increased to 7, 8, and 9 in 0–3 . At the same time, was reduced to 5, 4, and 3 . The relationship between and is shown in Figure 6.

The simulation experiment time was 14 , and the vehicle position was marked once every 2 . Figure 7 shows the steering trajectory of the tracked vehicle. When the tracked vehicle turned under steering condition 1, the steering trajectories calculated by the dynamics model and the kinematics model were regular circles. Compared with the calculation results of the kinematics model, the radius of the steering trajectory calculated by the steering dynamics model was significantly increased. When the tracked vehicle turned under steering conditions 2 and 3, the steering trajectory calculated by the dynamics model was no longer a regular circle. Instead, the vehicle spun out at and could be considered to be out of control. When the tracked vehicle turned under steering condition 3, the spinning out phenomenon was more significant with the increase in the circumferential velocity difference of the sprocket . However, with the increase in , the turning trajectory calculated by the kinematics model was still a regular circle, and the turning radius was reduced. Under steering condition 3, the vehicle steering radius was calculated by the kinematics model. The simulation results of the kinematics model showed that the tracked vehicle could achieve stable steering with a radius under condition 3, which was inconsistent with the actual situation.

**(a)**

**(b)**

**(c)**

Figure 8 shows the change in the steering radius of the tracked vehicle under three steering conditions calculated by the dynamics model. Under steering condition 1, the turning radius decreased with the increase in in 0–3 . did not change in 3–14, and the turning radius of the vehicle remained unchanged. The steering dynamics model calculated a steering radius of . It can be seen that the tracked vehicle could achieve stable steering at under this steering condition. Under conditions 2 and 3, it can be seen from the results of the dynamics model that the steering process of the tracked vehicle with a small turning radius at high speeds was unstable. Under steering condition 3, the vehicle reached the minimum steering radius of at . Then, the steering radius began to increase, and it reached the maximum steering radius of at . During the whole steering process, the turning radius of the tracked vehicle fluctuated up and down, and the fluctuation range gradually decreased with the steering.

Figure 9 shows the change of the track centroid velocity under three working conditions calculated by the dynamics model. The centroid velocity of the zero-differential-steering tracked vehicle was less than the theoretical centroid velocity when steering at a small radius at high speeds, and it decreased with the increase in . This phenomenon was caused by an increase in due to the increase in . Figure 10 shows the change of the sideslip angle under the three working conditions. increased with the increase in . Under condition 3, the centroid sideslip angle fluctuated up and down during the whole steering process. The maximum sideslip angle . At this time, the vehicle centroid speed was almost along the direction, and the tracked vehicle underwent a significant lateral slip. Under unstable steering conditions, it would be very dangerous for the driver to perform accurate trajectory control of the vehicle through normal handling. In normal vehicle driving, in order to ensure safety, the vehicle should try to avoid entering unstable steering conditions.

#### 4. Analysis of High-Speed Steering of Tracked Vehicle on Five Kinds of Roads

The steering performances of the tracked vehicles on different roads were significantly different. We selected five typical roads to analyze the steering performances of tracked vehicles: gault, snow, grit, marsh, and sand. The road parameters are shown in Table 2.

The steering processes of the tracked vehicle on the five terrain types under steering condition 2 were simulated by the simulation model. Figure 11 shows the steering trajectories of the tracked vehicle on five kinds of roads. The tracked vehicle turned smoothly on the gault road under steering condition 2, and the steering trajectory was a regular circular. When the tracked vehicle turned on the grit, marsh, and snow roads, the tracked vehicle spun out, which was particularly significant on the snow road. The reason may have been that the steering resistance coefficient and road adhesion coefficient of the snow road were low, and the road surface could provide sufficient adhesion for the tracked vehicle to turn.

The steering condition in which the tracked vehicle was about to spin out was considered to be the unstable steering critical condition. Since the theoretical centroid speed is constant when a zero-differential steering tracked vehicle is turning, the unstable steering critical conditions can be determined by and . The steering safety of tracked vehicles was improved by limiting the maximum circumferential velocity difference of sprocket when the tracked vehicle turned at . Only was considered in this study. In other words, during the steering of a tracked vehicle. By taking as the evaluation standard, the steering ability of the tracked vehicle was evaluated. When , it meant that the tracked vehicle was allowed to perform pivot steering at , and the circumferential velocity of the low-speed side track . A small indicated that when a tracked vehicle was turning at , the adjustment interval of was small, the steering selection was lower, and the steering ability was poor.

Figure 12 shows the corresponding relationship between and when the tracked vehicle turned on different roads. Figures 13 and 14 show the variations of the minimum steering radius and the maximum sideslip angle with , respectively. When , the tracked vehicle could spot turn with on the five kinds of roads. When , decreased as increased. On the snow road, when , , and . In this case, the tracked vehicle almost lost its steering ability. To avoid danger, the speed of tracked vehicles on snow roads cannot exceed . Since the steering resistance coefficient and the ground adhesion coefficient of the gault were high, when , . At this time, tracked vehicles on the gault still had some ability to turn. The and values of the tracked vehicle obtained from the simulation solution with steering on each road surface could be used as the high-speed steering stability boundary of the tracked vehicle, which could provide a reference for the matching of tracked vehicle steering parameters and autonomous driving trajectory planning and effectively reduce the real vehicle test work. It can be seen in Figure 14 that when , the of the tracked vehicle under each critical condition did not change significantly with the change in and the ground parameters, and was mostly in the range of [0.45, 0.55]. The variation range of might be affected by the vehicle structural parameters. Whether a certain could be determined as a characterization parameter of the unstable steering critical condition of tracked vehicles and applied to the steering stability control of vehicles requires further study.

#### 5. Conclusions

By establishing the steering dynamics model of a tracked vehicle, the following conclusions could be obtained based on the parameter changes in the high-speed steering process of the tracked vehicle:(1)The steering dynamics model takes into account the influence of track slip relative to the ground, vehicle lateral slip, and the centrifugal force. Compared with the steering kinematics model, the calculation results of the steering dynamics for unstable steering conditions of tracked vehicles could more accurately reflect the actual steering conditions.(2)By simulating the steering process of tracked vehicles on different roads, the relationship between the maximum circumferential velocity difference of the sprocket , the minimum steering radius , the maximum sideslip angle , and the theoretical centroid speed was obtained. Taking the simulation results as the high-speed steering stability boundary of tracked vehicles can provide a reference for the structure design, autonomous driving trajectory planning, and steering control of high-speed tracked vehicles, reduce the test work, and shorten the development cycle.(3)In the steering process of tracked vehicles, parameters such as , , , and can be measured by sensors. It is feasible to use this method to predict the high-speed steering trajectories of tracked vehicles. This also creates conditions for the practical application of steering dynamics models in high-speed tracked vehicle steering control.

#### Data Availability

The numerical data used to support the findings of this study are available from the corresponding author upon request.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this article.

#### Acknowledgments

The authors acknowledge the support of the National Key R&D Program of China (Grant no. 2018YFC0810500).