# Research on Steering Stability of High-Speed Tracked Vehicles

Authors: Weijian Jia; Xixia Liu; Chuanqing Zhang; Mianhao Qiu; Yu Zhang; Qingda Quan; Hong Sun
Journal: Mathematical Problems in Engineering
 (2022)

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## 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 sprocketΔumax, the minimum steering radius Rmin, the maximum sideslip angle βmax, and the theoretical centroid speed vth 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.


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## Body

## 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 sprocketΔui, the minimum steering radius Rmin, the maximum sideslip angle βmax, and the theoretical centroid speed vth 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 stressτ at any point on the track is related to the shear displacement j at that point, according to the shear stress model. The expression is(1)τ=c+σtanϕ1−e−j/K,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.


### 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 Figure1 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 vx and vy 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=arctanvy/vx. 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. O1 and O2 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 offset of steering center Oc relative to x-axis, d=vy/θ˙.Figure 1
Kinematics of tracked vehicle steering.The relationship between the centroid velocityv of the tracked vehicle and the accelerations ax and ay along x-axis and y-axis, respectively, can be expressed as follows:(2)v=vx2+vy2,ax=v˙x+vyθ˙,ay=v˙y−vxθ˙.During the steering process of the tracked vehicle, the trajectory ofXOY can be expressed as follows:(3)X=−∫0tvcosφdt,Y=∫0tvsinφdt,φ=θ−β.The velocities of each point on the same track are the same in thex-direction. The velocity of a point xp,yp on the track in the y-direction is determined by the yaw rate θ˙ and the x-axis coordinate of the point xp. The shear velocities vsxi and vsyi of this point relative to the ground in x-direction and y-direction can be expressed as follows:(4)vsx1=vx−B2θ˙−u1,vsx2=vx+B2θ˙−u2,vsyi=vy−xpθ˙,i=1,2,where ui is the circumferential velocity of the i-side sprocket, ui=ωir, ωi is the angular velocity of the i-side sprocket, and r is the radius of the sprocket.The angleψi between the speed of a point on the i-side track and the x-axis can be expressed as follows:(5)ψi=arccosvsxivsxi2+vsyi2,i=1,2.The slip ratioδi of the i-side track on both sides can be expressed as follows:(6)δi=vsximaxui+vsxi,ui,i=1,2.The shear displacementj 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 ji of a point xp,yp on the i-side track can be obtained by integrating the shear velocity, and it can be expressed as follows:(7)jxi=∫0tvsxidt=vsxiωirL2−xp,i=1,2,jyi=∫0tvsyidt=1ωirvyL2−xp+12θ˙xp2−L24,i=1,2,ji=jxi2+jyi2,i=1,2.


### 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 pis 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 ptis′. Thus, the ground pressure can be expressed as(8)pis=p¯is+ptis′.Figure2 shows the change in the ground pressure of each road wheel under the influence 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. tis is the vertical component of the i-side track tension acting on the s-th road wheel. Tfi is the front track tension of the i-th track. Tri is the tension force of the rear track of the i-side track. γf is the approach angle of the tracked vehicle. γr is the departure angle of the tracked vehicle. l is the track plate pitch. M1−M2 is the ground auxiliary line, which is a distance L/n+1 from the front edge of the track grounding area. This is used to solve the track ground pressure. Tfi and Tri are equal to the force Fxi on the i-side track in the x-direction, which can be expressed as follows:(9)Tfi=Fxi,Tri=0,ifvsxi>0,i=1,2,Tfi=0,Tri=Fxi,ifvsxi<0,i=1,2.Figure 2
Normal load distribution of road wheel due to track tension.The component of track tension in the vertical direction directly acts on the first and last road wheels. The ground pressure variationtis of the s-th road wheel of the i-side track can be expressed as follows:(10)ti1=Tfisinγf,i=1,2,ti2=ti3=⋯tin−1=0,i=1,2,tin=Trisinγr.i=1,2.Based on previous work [15], it was assumed that the pressure variation αis 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. αis can be expressed as follows:(11)αis=ksLn−1lb+Δi,i=1,2,where Δi is the value of αis at M1−M2.From Equations (9)–(11), the variation of the ground pressure pits′ of the s-th road wheel caused by the track tension on the i-th side can be expressed as follows:(12)ptis′=αis,i=1,2,s≠1,s≠n,pti1′=αi1−ti1/bl,i=1,2,ptin′=αin−tin/bl,i=1,2.The torque balance equations were established for the ground pressure of each road wheel about the auxiliary lineM1–M2. These can be expressed as follows:(13)∑i=12∑s=1nαis+pis⋅sLlbn−1−∑i=12ti1⋅Ln−1−∑i=12tin⋅nLn−1=mg⋅L2+Ln−1,∑s=1nα1s+p1s⋅lb−t11−t1n=∑s=1nα2s+p2s⋅lb−t21−t2n=mg2,∑s=1nαis=ti1+tin,i=1,2.The ground pressure variationptis′ of the s-th road wheel under track tension can be obtained by solving the equilibrium equation, as follows:(14)pt1s′=5t11−t1n−3t21−3t2n2nlb+3t1n+t2n−t11−t21snn+1bl−t1sbl,pt2s′=3t11−3t1n−5t21−t2n2nlb+3t1n+t2n−t11−t21snn+1bl−t2sbl.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. Figure3 shows the distribution of the track ground pressure under the influence of the inertial force.Figure 3
Normal load distribution of track on both sides under influence of inertial force. (a) Normal force distribution of track grounding. (b) Ground pressure distribution of road wheels on the unilateral track.
(a)(b)The component of the inertial force in they-direction may increases the normal force N2 of the high-speed track in contact with the ground and decreases the normal force N1 of the low-speed track. The normal force Ni of the i-side track grounding can be expressed as follows:(15)Ni=mg2±mayhB,i=1,2,where h is the height of the track center of gravity.The inertial force component in thex-direction max redistributes the pressure of each road wheel on the unilateral track. The ground pressure p¯is per unit area of the s-th road wheel of the i-side track can be expressed as(16)p¯is=Ni−32s−n−1maxhnn+1blL,i=1,2.Figure4 shows the track force when the tracked vehicle turns. Fxi and Fyi are the components of the shear force between the i-side track and the ground in the x-direction and y-direction, respectively. MRi is the steering resistance torque of the i-side track. FRi is the rolling resistance of the i-side track. Based on (1) and (14)–(16), the shear forces Fxis and Fyis of the s-th road wheel of the i-side track were obtained. Fxi and Fyi were obtained by summing Fxis and Fyis, respectively, which can be expressed as follows:(17)Fxis=∫3−sL4−l23−sL4+l2b⋅c+pistanϕ1−e−jxi/Kcosψi+πdx,i=1,2,Fxi=∑s=1nFxis,i=1,2,Fyis=∫3−sL4−l23−sL4+l2b⋅c+pistanϕ1−e−jyi/Ksinψi+πdx,i=1,2,Fyi=∑s=1nFyis,i=1,2.Figure 4
Steering force of tracked vehicle.The rolling resistanceFRi of the i-side track can be expressed as follows:(18)FRi=∑s=1npisfbl,where f is the coefficient of rolling resistance.The steering resistance torqueMRi of the i-side track was also obtained by summing the steering resistance torque MRis of each road wheel, which can be expressed as follows:(19)MRis=∫3−sL4−l23−sL4+l2b⋅c+pistanϕ1−e−jyi/Ksinψi+πxdx,i=1,2,MRi=∑s=1nMRis,i=1,2.The dynamic steering balance equation of tracked vehicles can be expressed as follows:(20)max=Fx1+Fx2−FR1−FR2,may=Fy1+Fy2,Izθ¨=B2Fx2−Fx1+B2FR2−FR1−MR1−MR2,where Iz is the yaw inertia of the tracked vehicle around the z-axis.The accelerations of the vehicle in thex- and y-directions and the yaw angular acceleration by (2) and (20) can be expressed as follows:(21)v˙x=Fx1+Fx2−FR1−FR2m−vyθ˙,v˙y=Fy1+Fy2m+vxθ˙,θ¨=B2Fx2−Fx1+B2FR2−FR1−MR1−MR2/Iz.


## 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 Figure1 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 vx and vy 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=arctanvy/vx. 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. O1 and O2 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 offset of steering center Oc relative to x-axis, d=vy/θ˙.Figure 1
Kinematics of tracked vehicle steering.The relationship between the centroid velocityv of the tracked vehicle and the accelerations ax and ay along x-axis and y-axis, respectively, can be expressed as follows:(2)v=vx2+vy2,ax=v˙x+vyθ˙,ay=v˙y−vxθ˙.During the steering process of the tracked vehicle, the trajectory ofXOY can be expressed as follows:(3)X=−∫0tvcosφdt,Y=∫0tvsinφdt,φ=θ−β.The velocities of each point on the same track are the same in thex-direction. The velocity of a point xp,yp on the track in the y-direction is determined by the yaw rate θ˙ and the x-axis coordinate of the point xp. The shear velocities vsxi and vsyi of this point relative to the ground in x-direction and y-direction can be expressed as follows:(4)vsx1=vx−B2θ˙−u1,vsx2=vx+B2θ˙−u2,vsyi=vy−xpθ˙,i=1,2,where ui is the circumferential velocity of the i-side sprocket, ui=ωir, ωi is the angular velocity of the i-side sprocket, and r is the radius of the sprocket.The angleψi between the speed of a point on the i-side track and the x-axis can be expressed as follows:(5)ψi=arccosvsxivsxi2+vsyi2,i=1,2.The slip ratioδi of the i-side track on both sides can be expressed as follows:(6)δi=vsximaxui+vsxi,ui,i=1,2.The shear displacementj 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 ji of a point xp,yp on the i-side track can be obtained by integrating the shear velocity, and it can be expressed as follows:(7)jxi=∫0tvsxidt=vsxiωirL2−xp,i=1,2,jyi=∫0tvsyidt=1ωirvyL2−xp+12θ˙xp2−L24,i=1,2,ji=jxi2+jyi2,i=1,2.


## 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 pis 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 ptis′. Thus, the ground pressure can be expressed as(8)pis=p¯is+ptis′.Figure2 shows the change in the ground pressure of each road wheel under the influence 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. tis is the vertical component of the i-side track tension acting on the s-th road wheel. Tfi is the front track tension of the i-th track. Tri is the tension force of the rear track of the i-side track. γf is the approach angle of the tracked vehicle. γr is the departure angle of the tracked vehicle. l is the track plate pitch. M1−M2 is the ground auxiliary line, which is a distance L/n+1 from the front edge of the track grounding area. This is used to solve the track ground pressure. Tfi and Tri are equal to the force Fxi on the i-side track in the x-direction, which can be expressed as follows:(9)Tfi=Fxi,Tri=0,ifvsxi>0,i=1,2,Tfi=0,Tri=Fxi,ifvsxi<0,i=1,2.Figure 2
Normal load distribution of road wheel due to track tension.The component of track tension in the vertical direction directly acts on the first and last road wheels. The ground pressure variationtis of the s-th road wheel of the i-side track can be expressed as follows:(10)ti1=Tfisinγf,i=1,2,ti2=ti3=⋯tin−1=0,i=1,2,tin=Trisinγr.i=1,2.Based on previous work [15], it was assumed that the pressure variation αis 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. αis can be expressed as follows:(11)αis=ksLn−1lb+Δi,i=1,2,where Δi is the value of αis at M1−M2.From Equations (9)–(11), the variation of the ground pressure pits′ of the s-th road wheel caused by the track tension on the i-th side can be expressed as follows:(12)ptis′=αis,i=1,2,s≠1,s≠n,pti1′=αi1−ti1/bl,i=1,2,ptin′=αin−tin/bl,i=1,2.The torque balance equations were established for the ground pressure of each road wheel about the auxiliary lineM1–M2. These can be expressed as follows:(13)∑i=12∑s=1nαis+pis⋅sLlbn−1−∑i=12ti1⋅Ln−1−∑i=12tin⋅nLn−1=mg⋅L2+Ln−1,∑s=1nα1s+p1s⋅lb−t11−t1n=∑s=1nα2s+p2s⋅lb−t21−t2n=mg2,∑s=1nαis=ti1+tin,i=1,2.The ground pressure variationptis′ of the s-th road wheel under track tension can be obtained by solving the equilibrium equation, as follows:(14)pt1s′=5t11−t1n−3t21−3t2n2nlb+3t1n+t2n−t11−t21snn+1bl−t1sbl,pt2s′=3t11−3t1n−5t21−t2n2nlb+3t1n+t2n−t11−t21snn+1bl−t2sbl.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. Figure3 shows the distribution of the track ground pressure under the influence of the inertial force.Figure 3
Normal load distribution of track on both sides under influence of inertial force. (a) Normal force distribution of track grounding. (b) Ground pressure distribution of road wheels on the unilateral track.
(a)(b)The component of the inertial force in they-direction may increases the normal force N2 of the high-speed track in contact with the ground and decreases the normal force N1 of the low-speed track. The normal force Ni of the i-side track grounding can be expressed as follows:(15)Ni=mg2±mayhB,i=1,2,where h is the height of the track center of gravity.The inertial force component in thex-direction max redistributes the pressure of each road wheel on the unilateral track. The ground pressure p¯is per unit area of the s-th road wheel of the i-side track can be expressed as(16)p¯is=Ni−32s−n−1maxhnn+1blL,i=1,2.Figure4 shows the track force when the tracked vehicle turns. Fxi and Fyi are the components of the shear force between the i-side track and the ground in the x-direction and y-direction, respectively. MRi is the steering resistance torque of the i-side track. FRi is the rolling resistance of the i-side track. Based on (1) and (14)–(16), the shear forces Fxis and Fyis of the s-th road wheel of the i-side track were obtained. Fxi and Fyi were obtained by summing Fxis and Fyis, respectively, which can be expressed as follows:(17)Fxis=∫3−sL4−l23−sL4+l2b⋅c+pistanϕ1−e−jxi/Kcosψi+πdx,i=1,2,Fxi=∑s=1nFxis,i=1,2,Fyis=∫3−sL4−l23−sL4+l2b⋅c+pistanϕ1−e−jyi/Ksinψi+πdx,i=1,2,Fyi=∑s=1nFyis,i=1,2.Figure 4
Steering force of tracked vehicle.The rolling resistanceFRi of the i-side track can be expressed as follows:(18)FRi=∑s=1npisfbl,where f is the coefficient of rolling resistance.The steering resistance torqueMRi of the i-side track was also obtained by summing the steering resistance torque MRis of each road wheel, which can be expressed as follows:(19)MRis=∫3−sL4−l23−sL4+l2b⋅c+pistanϕ1−e−jyi/Ksinψi+πxdx,i=1,2,MRi=∑s=1nMRis,i=1,2.The dynamic steering balance equation of tracked vehicles can be expressed as follows:(20)max=Fx1+Fx2−FR1−FR2,may=Fy1+Fy2,Izθ¨=B2Fx2−Fx1+B2FR2−FR1−MR1−MR2,where Iz is the yaw inertia of the tracked vehicle around the z-axis.The accelerations of the vehicle in thex- and y-directions and the yaw angular acceleration by (2) and (20) can be expressed as follows:(21)v˙x=Fx1+Fx2−FR1−FR2m−vyθ˙,v˙y=Fy1+Fy2m+vxθ˙,θ¨=B2Fx2−Fx1+B2FR2−FR1−MR1−MR2/Iz.


## 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 wasΔt=0.1s. The simulation inputs at time t were the circumferential velocities of the sprockets u1t and u2t 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 vth was determined by the circumferential velocity of the sprocket ui on both sides, vth=u1+u2/2. The sand road was selected for the simulation experiment. The tracked vehicles and road parameters are shown in Table 1.Figure 5
Solution process of steering dynamics and kinematics simulation model. (a) Solution process of steering dynamics simulation model. (b) Solution process of steering kinematics simulation model.
(a)(b)Table 1
Vehicle and road parameters.
c (Pa)1040Terrainϕ (°)28K (cm)2.5f0.065Vehiclem (kg)36500Izkg⋅m213000L (m)3.78B (m)2.64b (m)0.58l (m)0.18r (m)0.283h (m)1.159n5During the simulation, the initial centroid velocityv0=6m/s, the initial yaw angle θ=πrad, the initial centroid sideslip angle β0=0rad, and the initial centroid acceleration ax0=ay0=0m/s2 were set. Three sets of steering conditions were set by changing ui. At the beginning of the simulation, the velocities u1=u2=6m/s were set. u2 was increased to 7, 8, and 9 m/s in 0–3 s. At the same time, u1 was reduced to 5, 4, and 3 m/s. The relationship between u1 and u2 is shown in Figure 6.Figure 6
Variations ofui on both sides.The simulation experiment time was 14s, 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 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 vth=6m/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 significant with the increase in the circumferential velocity difference 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=2.64m 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 R=2.64m under condition 3, which was inconsistent with the actual situation.Figure 7
Steering trajectory of the tracked vehicle under three steering conditions. (a) Steering condition 1. (b) Steering condition 2. (c) Steering condition 3.
(a)(b)(c)Figure8 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. The steering dynamics model calculated a steering radius of R=9.62m. It can be seen that the tracked vehicle could achieve stable steering at R=9.62m 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=1.22m at t=5.23s. Then, the steering radius began to increase, and it reached the maximum steering radius of R=4.81m at t=7.69s. 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 8
Change ofR under three steering conditions calculated by dynamics model.Figure9 shows the change of the track centroid velocity v under three working conditions calculated by the dynamics model. The centroid velocity v of the zero-differential-steering tracked vehicle was less than the theoretical centroid velocity vth when steering at a small radius at high speeds, and it decreased with the increase in Δu. This phenomenon was caused by an increase in vsxi due to the increase in Δu. Figure 10 shows the change of the sideslip angle β under the three working conditions. β increased with the increase in Δu. Under condition 3, the centroid sideslip angle fluctuated up and down during the whole steering process. The maximum sideslip angle βmax=1.43rad. At this time, the vehicle centroid speed v was almost along the y−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.Figure 9
Change ofv under three steering conditions calculated by dynamics model.Figure 10
Change ofβ under three steering conditions calculated by dynamics model.


## 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 Table2.Table 2
Parameters of five typical roads.
ckPaφ°KcmGault68.92340.6Snow620.73.5Grit0332.5Marsh2.839.43.1Sand1.331.11.2The steering processes of the tracked vehicle on the five terrain types under steering condition 2 were simulated by the simulation model. Figure11 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.Figure 11
Steering trajectories of tracked vehicles on five roads.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 speedvth is constant when a zero-differential steering tracked vehicle is turning, the unstable steering critical conditions can be determined by vth and Δu. The steering safety of tracked vehicles was improved by limiting the maximum circumferential velocity difference of sprocket Δumax when the tracked vehicle turned at vth. Only Δumax≤2vth was considered in this study. In other words, ui≥0 during the steering of a tracked vehicle. By taking Δumax/2vth as the evaluation standard, the steering ability of the tracked vehicle was evaluated. When Δumax/2vth=1, it meant that the tracked vehicle was allowed to perform pivot steering at vth, and the circumferential velocity of the low-speed side track u1=0m/s. A small Δumax/2vth indicated that when a tracked vehicle was turning at vth, the adjustment interval of Δu was small, the steering selection was lower, and the steering ability was poor.Figure12 shows the corresponding relationship between vth and Δumax/2vth when the tracked vehicle turned on different roads. Figures 13 and 14 show the variations of the minimum steering radius Rmin and the maximum sideslip angle βmax with vth, respectively. When vth≤3m/s, the tracked vehicle could spot turn with u1=0m/s on the five kinds of roads. When vth>3m/s, Δumax/2vth decreased as vth increased. On the snow road, when vth=10m/s, Δumax/2vth=0.07, and Rmin=27.78m. In this case, the tracked vehicle almost lost its steering ability. To avoid danger, the speed of tracked vehicles on snow roads cannot exceed 10m/s. Since the steering resistance coefficient and the ground adhesion coefficient of the gault were high, when vth=10m/s, Rmin=12.5m. At this time, tracked vehicles on the gault still had some ability to turn. The Δumax/2vth and Rmin values of the tracked vehicle obtained from the simulation solution with vth 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 vth≥3m/s, the βmax of the tracked vehicle under each critical condition did not change significantly with the change in vth and the ground parameters, and βmax was mostly in the range of [0.45, 0.55]. The variation range of βmax might be affected by the vehicle structural parameters. Whether a certain βmax 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.Figure 12
Variation ofΔumax/2vth under the unstable steering critical condition.Figure 13
Variation ofRmin under the unstable steering critical condition.Figure 14
Variation ofβmax under the unstable steering critical condition.


## 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 sprocketΔumax, the minimum steering radius Rmin, the maximum sideslip angle βmax, and the theoretical centroid speed vth 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 asω, 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. This also creates conditions for the practical application of steering dynamics models in high-speed tracked vehicle steering control.



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*Source: 4850104-2022-11-08.xml*