Research on Steering Stability of High-Speed Tracked Vehicles

,


Introduction
Te 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. Te 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 difcult 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]. Te 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 infuence of factors such as the relative ground slip of the track, vehicle lateral slip, and centrifugal force, and thus, it is difcult to meet the model calculation accuracy requirements for unstable steering conditions [5]. An efective 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 fourdegree-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. Tis 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 coefcients 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 diference between the lateral and longitudinal friction coefcients 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 coefcient. 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 trafcability and maneuverability of tracked vehicles. Rui et al. [13] systematically designed the experimental scheme of tracked vehicle steering and verifed 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-diferential 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. Ten, the infuence of road parameters on the steering performance of tracked vehicles was analyzed. Te critical conditions of the unstable steering of tracked vehicles when steering on diferent roads were determined. Te corresponding relationship between the maximum circumferential velocity diference of the sprocket Δu i , the minimum steering radius R min , the maximum sideslip angle β max , and the theoretical centroid speed v th was obtained and used to determine the high-speed steering stability boundary of tracked vehicles.

Steering Dynamics Model of Tracked Vehicle Based on Shear Stress Model
Te establishment of the tracked vehicle dynamics model requires the following assumptions: (1) Te centroid of the tracked vehicle coincides with the geometric center (2) Track subsidence and side pushing efects of the track slab are neglected (3) Te track is not stretchable, and the infuence of the track width on the ground pressure can be neglected (4) During the steering process of the tracked vehicle, the shear stress τ at any point on the track is related to the shear displacement j at that point, according to the shear stress model. Te expression is where c is the soil cohesion, σ is the normal stress, ϕ is the angle of internal shearing resistance of the terrain, and K is the shear deformation modulus.

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. XOY is the geodetic coordinate system. xoy is the vehicle body coordinate system, the origin of coordinate o coincides with the geometric center of the vehicle, and the x-direction is the vehicle's forward direction. v is the vehicle centroid speed, and v x and v y are the components of v in the x-direction and y-direction, respectively. θ is the yaw angle of the tracked vehicle. φ is the angle between vehicle speed direction and the X-axis. β is the sideslip angle, β � arctan(v y /v x ). L is the length of the track contact with the ground. B is the distance of the track center line on both sides. b is the track widths. ψ is the angle between the direction of the velocity at a point on the track and the x-axis. O 1 and O 2 are the instantaneous steering centers of low-speed side track and high-speed side track, respectively. In this paper, the subscript i � 1 represents the low-speed side track, and i � 2 represents the high-speed side track. d is the ofset of steering center O c relative to x -axis, d � v y / _ θ. Te relationship between the centroid velocity v of the tracked vehicle and the accelerations a x and a y along x-axis and y-axis, respectively, can be expressed as follows: During the steering process of the tracked vehicle, the trajectory of XOY can be expressed as follows: 2 Mathematical Problems in Engineering Te velocities of each point on the same track are the same in the x-direction. Te velocity of a point (x p , y p ) on the track in the y-direction is determined by the yaw rate _ θ and the x-axis coordinate of the point x p . Te shear velocities v sxi and v syi of this point relative to the ground in x -direction and y-direction can be expressed as follows: where u i is the circumferential velocity of the i-side sprocket, u i � ω i r, ω i is the angular velocity of the i-side sprocket, and r is the radius of the sprocket. Te angle ψ i between the speed of a point on the i-side track and the x-axis can be expressed as follows: Te slip ratio δ i of the i-side track on both sides can be expressed as follows: Te shear displacement j 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. Te shear displacement j i of a point (x p , y p ) on the i-side track can be obtained by integrating the shear velocity, and it can be expressed as follows:

Steering Force Analysis of Tracked Vehicle.
During the steering process of the tracked vehicle, the distribution of the track ground pressure directly afects 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. Te ground pressure on the unit grounding area p is of the s-th road wheel on the i-side track consists of two parts: one part is the ground pressure on the unit grounding area p is 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 s-th road wheel on the i-side track caused by the tension of the track, which is denoted as p tis ′ . Tus, the ground pressure can be expressed as Figure 2 shows the change in the ground pressure of each road wheel under the infuence of track tension. m is the vehicle weight. g is the acceleration of gravity. n is the number of road wheels of the one-sided track. t is is the vertical component of the i-side track tension acting on the s-th road wheel. T fi is the front track tension of the i-th track. T ri is the tension force of the rear track of the i-side track. c f is the approach angle of the tracked vehicle. c r is the departure angle of the tracked vehicle. l is the track plate pitch. M 1 − M 2 is the ground auxiliary line, which is a distance L/(n + 1) from the front edge of the track grounding area. Tis is used to solve the track ground pressure. T fi and T ri are equal to the force F xi on the i-side track in the x-direction, which can be expressed as follows: Te component of track tension in the vertical direction directly acts on the frst and last road wheels. Te ground pressure variation t is of the s-th road wheel of the i-side track can be expressed as follows: Based on previous work [15], it was assumed that the pressure variation α is of the unit grounding area of the s-th road wheel caused by the track tension on the i-side track changed linearly, and the rate of change was k. α is can be expressed as follows: From Equations (9)-(11), the variation of the ground pressure p its ′ of the s-th road wheel caused by the track tension on the i-th side can be expressed as follows: Te torque balance equations were established for the ground pressure of each road wheel about the auxiliary line M 1 -M 2 . Tese can be expressed as follows: Te ground pressure variation p tis ′ of the s-th road wheel under track tension can be obtained by solving the equilibrium equation, as follows: Te 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 signifcant in high-speed steering. Figure 3 shows the distribution of the track ground pressure under the infuence of the inertial force.
Te component of the inertial force in the y-direction ma y increases the normal force N 2 of the high-speed track in contact with the ground and decreases the normal force N 1 of the low-speed track. Te normal force N i of the i-side track grounding can be expressed as follows: where h is the height of the track center of gravity. Te inertial force component in the x-direction ma x redistributes the pressure of each road wheel on the unilateral track. Te ground pressure p is per unit area of the s-th road wheel of the i-side track can be expressed as (16) Figure 4 shows the track force when the tracked vehicle turns. F xi and F yi are the components of the shear force between the i-side track and the ground in the x-direction and y-direction, respectively. M Ri is the steering resistance torque of the i-side track. F Ri is the rolling resistance of the i -side track. Based on (1) and (14)-(16), the shear forces F xis and F yis of the s-th road wheel of the i-side track were obtained. F xi and F yi were obtained by summing F xis and F yis , respectively, which can be expressed as follows: . Figure 2: Normal load distribution of road wheel due to track tension.

Mathematical Problems in Engineering
Te rolling resistance F Ri of the i-side track can be expressed as follows: where f is the coefcient of rolling resistance. Te steering resistance torque M Ri of the i-side track was also obtained by summing the steering resistance torque M Ris of each road wheel, which can be expressed as follows:   Mathematical Problems in Engineering Te dynamic steering balance equation of tracked vehicles can be expressed as follows: where I z is the yaw inertia of the tracked vehicle around the z-axis.
Te accelerations of the vehicle in the x-and y-directions and the yaw angular acceleration by (2) and (20) can be expressed as follows:

Simulation Analysis of High-Speed Steering Process of Tracked Vehicles
Te 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. Te simulation step size was Δt � 0.1 s. Te simulation inputs at time t were the circumferential velocities of the sprockets u 1 t and u 2 t on both sides of the tracked vehicle. Te solution process of the steering dynamics and kinematics simulation model is shown in Figure 5. Te simulation object was a zero-diferential-steering tracked vehicle. For a zero-diferential-steering tracked vehicle, the theoretical centroid speed v th was determined by the circumferential velocity of the sprocket u i on both sides, v th � (u 1 + u 2 )/2. Te sand road was selected for the simulation experiment. Te tracked vehicles and road parameters are shown in Table 1.
During the simulation, the initial centroid velocity v 0 � 6 m/s, the initial yaw angle θ � π ra d, the initial centroid sideslip angle β 0 � 0 ra d, and the initial centroid acceleration a x0 � a y0 � 0 m/s 2 were set. Tree sets of steering conditions were set by changing u i . At the beginning of the simulation, the velocities u 1 � u 2 � 6 m/s were set. u 2 was increased to 7, 8, and 9 m/s in 0-3 s. At the same time, u 1 was reduced to 5, 4, and 3 m/s. Te relationship between u 1 and u 2 is shown in Figure 6. Te simulation experiment time was 14 s, and the vehicle position was marked once every 2 s. 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 signifcantly 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 v th � 6 m/s and could be considered to be out of control. When the tracked vehicle turned under steering condition 3, the spinning out phenomenon was more signifcant with the increase in the circumferential velocity diference of the sprocket Δu. However, with the increase in Δu, 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 R � 2.64 m was calculated by the kinematics model. Te simulation results of the kinematics model showed that the tracked vehicle could achieve stable steering with a radius R � 2.64 m under condition 3, which was inconsistent with the actual situation. 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 Δu in 0-3 s. Δu did not change in 3-14s, and the turning radius of the vehicle remained unchanged. Te steering dynamics model calculated a steering radius of R � 9.62 m. It can be seen that the tracked vehicle could achieve stable steering at R � 9.62 m 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 R � 1.22 m at t � 5.23 s. Ten, the steering radius began to increase, and it reached the maximum steering radius of R � 4.81 m at t � 7.69 s. During the whole steering process, the turning radius of the tracked vehicle fuctuated up and down, and the fuctuation range gradually decreased with the steering. Figure 9 shows the change of the track centroid velocity v under three working conditions calculated by the dynamics model. Te centroid velocity v of the zero-diferentialsteering tracked vehicle was less than the theoretical centroid velocity v th when steering at a small radius at high speeds, and it decreased with the increase in Δu. Tis phenomenon was caused by an increase in v sxi due to the increase in Δu.  Step 1:Kinematics Analysis Step 2:State Update , , , ,

Analysis of High-Speed Steering of Tracked Vehicle on Five Kinds of Roads
Te steering performances of the tracked vehicles on different roads were signifcantly diferent. We selected fve typical roads to analyze the steering performances of tracked vehicles: gault, snow, grit, marsh, and sand. Te road parameters are shown in Table 2.
Te steering processes of the tracked vehicle on the fve terrain types under steering condition 2 were simulated by the simulation model. Figure 11 shows the steering trajectories of the tracked vehicle on fve kinds of roads. Te 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 signifcant on the snow road. Te reason may have been that the steering resistance coefcient and road adhesion coefcient of the snow road were low, and the road surface could provide sufcient adhesion for the tracked vehicle to turn.
Te 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 v th is constant when a zero-diferential steering tracked vehicle is turning, the unstable steering critical conditions can be determined by v th and Δu. Te steering safety of tracked vehicles was improved by limiting the maximum circumferential velocity diference of sprocket Δu max when the tracked vehicle turned at v th . Only Δu max ≤ 2v th was considered in this study. In other words, u i ≥ 0 during the steering of a tracked vehicle. By taking Δu max /2v th as the evaluation standard, the steering ability of the tracked vehicle was evaluated. When Δu max /2v th � 1, it meant that the tracked vehicle was allowed to perform pivot steering at v th , and the circumferential velocity of the low-speed side track u 1 � 0 m/s. A small Δu max /2v th indicated that when a tracked vehicle was turning at v th , the adjustment interval of Δu was small, the steering selection was lower, and the steering ability was poor. Figure 12 shows the corresponding relationship between v th and Δu max /2v th when the tracked vehicle turned on diferent roads. Figures 13 and 14 show the variations of the minimum steering radius R min and the maximum sideslip angle β max with v th , respectively. When v th ≤ 3 m/s, the tracked vehicle could spot turn with u 1 � 0 m/s on the fve kinds of roads. When v th > 3 m/s, Δu max /2v th decreased as v th increased. On the snow road, when v th � 10 m/s,     Figure 14 that when v th ≥ 3 m/s, the β max of the tracked vehicle under each critical condition did not change signifcantly with the change in v th and the ground parameters, and β max was mostly in the range of [0.45, 0.55]. Te variation range of β max might be afected by the vehicle structural parameters. Whether a certain β max 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.

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) Te steering dynamics model takes into account the infuence 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 refect the actual steering conditions. (2) By simulating the steering process of tracked vehicles on diferent roads, the relationship between the maximum circumferential velocity diference of the sprocket Δu max , the minimum steering radius R min , the maximum sideslip angle β max , and the theoretical centroid speed v th 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 ω, v, _ φ, and β can be measured by sensors. It is feasible to use this method to predict the high-speed steering trajectories of tracked vehicles. Tis also creates conditions for the practical application of steering dynamics models in high-speed tracked vehicle steering control.

Data Availability
Te numerical data used to support the fndings of this study are available from the corresponding author upon request.

Conflicts of Interest
Te authors declare that there are no conficts of interest regarding the publication of this article.