Theoretical Model of High-Speed Tracked Vehicle General Motion

This paper presents a systematically derived theoretical model for the motion of tracked vehicles, encompassing various scenarios such as soft terrain and slopes. The model considers all the resistances encountered by tracked vehicles, including track-terrain interaction, grade and air resistance, centrifugal force, inertial forces, and turning resistance. The heading angle determines the geometrical orientation of the vehicle on the slope. A MATLAB model of tracked vehicle general motion was established based on theoretical discussions. The paper discusses the steering performance of tracked vehicles on slopes of different inclinations. A simulation of steering with different turning radiuses was also presented. It is shown that the proposed model can be used for predicting and evaluating the steering performance of tracked vehicles with adequate accuracy. Furthermore, due to its high computational efficiency, the proposed model can be utilized for power demand modeling in research on the hybridization or electrification of tracked vehicles.


INTRODUCTION
Tracked vehicles have a wide application ranging from construction and agriculture to military and mining. Steering of a tracked vehicle can be done using various methods, such as curved track steering and articulated steering, but the most widespread and notable is skidsteering. In skid-steering, the turning motion of the tracked vehicle is performed by achieving a difference in the speed of the inner and outer track. In this way, steering affects traction performance, i.e., forces on tracks, resultant resisting force, and the moment of the turning resistance.
Many studies on the mobility of tracked vehicles have been performed so far. The initial research on steering characteristics was presented by Steeds [1], who clarified the skid-steering mechanism, derived equations of motion, and set a foundation for subsequent studies on this topic, but made an assumption that the forces generated between the tracks and terrain obey the Coulomb law of friction which turned out incorrect. Bekker [2] established a connection between pressure and sinkage using the bevameter technique. Later on, relaying on Bekker's work, an empirical relation establishing the shear stress -shear displacement relationship was derived [3]. Using this relationship, a general theory for steady-state skid-steering on firm ground was established [4]. Using the general theory, the lateral coefficient of friction can be quantitatively determined as a function of the turning radius without the need for the use of empirical relations. Transient skid-steering models were developed relying on general theory [5,6]. In these papers, equations for track slip were derived for discretized tracks and numerically integrated to obtain track forces for vehicles on firm ground.
However, to the best of author's knowledge, only research that has studied lateral and longitudinal dynamics of tracked vehicles on slopes [7] deals with steering of articulated tracked vehicle and, as such, is very complex. Also, most other flat terrain skid-steering models are very complex, some of which deal with track tension and wheel-track interaction besides track-terrain interaction [8]. Numerical calculation of such models is computationally very burdensome. Different simplifications are presented in the literature, e.g., introducing imaginary wheels and eliminating track chain [9], considering steady-state turns only [10], or approximating the multibody model with single body model [11], but these skid-steering models remained computationally demanding.
While hybrid propulsion systems have been very common in wheeled vehicles for many years [12], only recently has the hybridization of tracked vehicles gained interest [13][14][15]. With this increased interest in the hybridization and electrification of tracked vehicles and emerging autonomous tracked mobile robots, the need for computationally inexpensive yet high-fidelity skidsteering models is increasing. Many researchers create their own models with the goal of analyzing the impact of hybridization on fuel consumption. Since tracked vehicles operate over a variety of terrains, for such models, it is crucial to include all aspects of tracked vehicle movement, including traveling over a slope and soft terrain, while being computationally efficient for long and repetitive simulations. The motivation for this work stems from the mentioned fact and the general lack of research on all-round models encompassing all terrain conditions while being computationally inexpensive.
This paper aims to derive general motion equations that would be efficient, simple, and applicable to all driving conditions, including skid-steering on a slope, which hasn't been well presented in the literature.

BASICS OF TERRAMECHANICS
For the purpose of determining the shear stresses in the zone of track-terrain interaction, the shear stress-shear displacement relationship is used. Regardless of the size of the displacement, the size of the shear stress is a finite value and has its maximum, which is determined by the Mohr-Coulomb law, more often called just Coulomb's law, which states: where τ max is the maximum shear stress, c is the terrain cohesion, σ is the normal stress, and is the angle of internal shearing resistance. Coulomb law implies that the maximum shear stress will be reached instantaneously as soon as the smallest relative movement happens. However, experimental data have shown that this is not the case. In reality, the maximum shear stress will be reached when the displacement reaches a certain value, as shown in Fig. 1. Therefore, the empirical relation widely used today was defined as: where j is the shear displacement, and K is the shear deformation modulus. The shear stress-shear displacement relationship also has its vertical counterpart called the pressure-sinkage relationship. The most commonly used equation has the following form [2]: where z 0 is the sinkage, p is the pressure, b is the cone radius, and n, k , and k c are pressure-sinkage parameters.

Resistances caused by track-terrain interaction
Resistance to motion due to track-terrain interaction includes three types of resistances: rolling resistance, compaction resistance and bulldozing resistance. Rolling resistance can be expressed: where Q is the ground normal reaction, and f r is the rolling resistance coefficient defined as [16]: where f 0 and f s are empirical coefficients that depend on the design of tracks, and V is the vehicle speed. The other two components are based on the established pressure-sinkage relationship. If (3) is written as: and assuming that track links are analogous to penetrating bevameter and that pressure is uniformly distributed, this equation can be written down as: where b and l are the width and length of the track contact surface. By integrating the pressure over depth, the work done to compact the soil is obtained: This equation combined with (6) gives: If the work from (9) is represented as the product of the traction force and the length of the track's contact surface, the traction force will equal the resistance force due to the terrain compaction due to the sinking. Therefore, the resistance due to compaction can be expressed as: Bulldozing resistance occurs on deformable soil with significant sinkage. An accumulation of soil is formed in front of the tracks, which acts as resistance to motion. This force is expressed as [17]: where c is the terrain cohesion, γ s is the specific weight of terrain, and K pc and K pγ are terrain capacity factors. Finally, total resistance due to the track-terrain interaction, i.e., the total road resistance, is obtained as:

LONGITUDINAL MOTION
Air resistance is often neglected in tracked vehicles due to low speeds. However, for high-speed vehicles with maximum movement speeds of over 60 km/h, air resistance significantly impacts total resistance. Air resistance is defined as: where C D is the aerodynamic drag coefficient, ρ is the air density, and A is the vehicle frontal area.
In addition to air resistance, grade resistance can have a very large influence due to the large mass of tracked vehicles. Grade resistance is defined as: where α is the road slope angle, when considering a turn on a slope, the grade resistance will be multiplied by a factor that takes into account the heading of the vehicle (heading angle), which defines the geometric orientation of the vehicle on the slope. The vehicle's inertia and all rotating elements cause the appearance of inertial forces that oppose propulsive forces per the Lagrange-D'Alembert principle. The inertial force arising from the vehicle's mass is inherently included in the equation of motion. On the other hand, the inertial forces of the rotating elements in the kinematic connection with the sprocket wheels are not included. They can be modeled as special submodels of transmission. Still, in vehicle dynamics theory studies, they are generally introduced together with translation inertial forces and, most often, through the mass coefficient [18].
where δ is the mass coefficient. An empirical formula related to tracked vehicles that define mass coefficient is proposed [19]: where i tr is the total transmission ratio. Note that the previous calculation of inertial forces is simplified because it was assumed that there are no transmission losses and that there is only one power flow. For energy efficiency analysis, other models could be created.

LATERAL MOTION
The turning behavior of a tracked vehicle depends on the size of the propulsive forces of the outer F 0 and inner F i track, the resultant resistance force R tot , and the moment of turning resistance M r created as a result of the soil's resistance to deformation. Along the entire contact surface of the tracks in the lateral direction, there are turning resistance forces that are proportional to the weight of the vehicle (Figure 2): where μ is the coefficient of lateral resistance which depends on the type of soil, design of the track chain, and turning radius.

Figure 2. Forces acting on tracked vehicles during turning
The total moment of turning resistance is obtained as follows: This equation is valid for uniform pressure distribution and flat terrain. Other cases will be derived in later text.

Coefficient of lateral resistance
Based on (17), it can be concluded that the coefficient of lateral resistance μ represents the proportionality coefficient between M r and Gl/4, i.e., it is equivalent to the lateral coefficient of adhesion (friction). In the past, a constant value was often adopted for a certain type of base; however, constant values of the coefficient of resistance to turning did not give realistic solutions [20]. In this paper, an empirical formula will be used [21]: where μ max is the maximum μ at R = B/2, R is the theoretical turning radius, and a is the empirical coefficient (0,85 or 0,925 depending on the steering mechanism). Using (19), the coefficient of lateral resistance can be calculated for different values of the turning radius. Using the general theory and experimental data obtained from the literature [1,4], Figures 3  and 4 were created, which show a comparison of the proposed coefficient of lateral resistance to data obtained from the literature. The proposed empirical formula achieves a high agreement with the experimental data. The general theory gives a slightly higher degree of matching for greater speeds (Figure 4), but the mathematical model is significantly more complicated and computationally burdensome.

Turning resistances
In the case of high-speed tracked vehicles, such as military-tracked vehicles, turning can be performed at very high speeds, and the centrifugal force cannot be ignored. Accordingly, the derivation of the mathematical model will be performed with the following assumptions: -The ground is firm -consequently, compaction and bulldozing resistances in the lateral direction are negligible. -Pressure distribution is uniform -assuming another pressure distribution (trapezoid, dualtrapezoid, etc.) is possible, but this results in an unnecessary complication of the mathematical model. -Centrifugal force is not negligible and is taken into account -this assumption implies taking into account the centrifugal force, which can have a significant effect on the moment of turning resistance, especially at high speeds or very small turning radiuses. Suppose that the tracked vehicle travels on the slope that forms angle α with the horizontal plane. When executing a turn, the vehicle changes the heading (direction) defined by the angle ψ = [0°, 360°]. The zero-heading angle is the initial movement heading along the slope, as shown in Figure 5. In addition to the already defined resistance forces, the vehicle is also affected by the centrifugal and weight forces. These components depend on the angle of slope and the angle of course. In accordance with the introduced assumptions, it is considered that the pressure distribution is always uniform.
By changing the heading angle ψ, the longitudinal and lateral components of the weight force appear, which are calculated as: In the case of a turn, as illustrated in Figure 5, i.e., to the left (negative side) of the heading angle, the weight force components have directions as in Figure 6.
Since the turning radius R is usually significantly larger than contact length l, β can be approximated as cosβ = 1. After calculation and rearranging: 2 0 sin sin 2 cos 2 cos

Figure 6. The general case of motion -back and top view
As a consequence of turn center offset, the corresponding moment of turning resistance has three components -the moment created by forces of turning resis-tance, the moment created by centrifugal force, and the moment created by the lateral component of the weight force: Combining this equation with (22) gives: When considering the transient motion, the moment of turning resistance needs to be reduced for rotational inertia factor -I z · ε where I z is the yaw moment of inertia, and ε is the angular acceleration.

GENERAL MOTION EQUATIONS
Generalizing the equations, i.e., assuming that the motion is carried out on soft terrain and at a variable speed, it is possible to derive general motion equations. The force on the outer and inner tracks are given as (25)   In equations (25) and (26), the first three terms on the right side represent the resistances arising from track-terrain interaction. Note that the normal reaction is modified due to the load transfer. The second and third terms do not depend on the slope of the terrain and the heading angle. The fourth and fifth terms are the longitudinal forces resulting from the effect of centrifugal force and the slope of the terrain, and the sixth and seventh terms are the acceleration and air resistance. The eighth term is the moment of turning resistance.
Derived equations enable describing the motion of a tracked vehicle in all possible conditions, including transient motion, deformable ground, and motion on a slope which, coupled with computational inexpensiveness, makes these equations unique in the literature.

MODEL PERFORMANCE AND ANALYSIS
A general motion model was established in MATLAB based on the derived equations. A case of turning on the slope was considered in accordance with Figure 5. The tracked vehicle starts to turn from the reference position, i.e., from the straight uphill direction. Figure 7 and Figure 8 show forces on tracks during a 360-degree turn. The ground is assumed to be firm, the speed is constant, and the turning radius is constant and smaller than the free turning radius R 0 ( Figure 10). As shown in Figure 7 and Figure 8, when turning on a slope, the forces on the tracks change significantly. At the maximum slope angle of α = 30°, both tracks exhibit braking forces in the range ψ = [100° to 250°]. Force on the inner track is negative for smaller α for the entire range of ψ which is in agreement with the vehicle dynamics theory, which states that at small R the force on the inner track is negative. For α > 20° force on inner track is positive for ψ = [-70°, 70°] due to grade resistance force, which increases in intensity with α.
The decrease in force intensity when turning down a slope is due to the lateral force that creates a moment opposite to the moment of turning resistance, while the opposite case is present when turning upwards. A steering mechanism constructed with friction elements would not be able to achieve turns similar to the presented one due to the variability of the forces on the tracks. The sharpest rate of change of inner and outer track forces is exhibited in the range of ψ = [100° to 250°] due to the vehicle's weight, namely its lateral component, which increases sharply in that range. The closer the heading angle is to ψ = 180°, the lower the force on the outer track, even becoming negative for very large slope angles. This happens due to the grade resistance force, which, with such orientation of the vehicle, acts as a propulsive force. However, Figure 9 shows that at an angle of α = 20°, slippage of the tracks is possible. With an increase in the slope angle, turning becomes impossible, which coincides with empirical findings from the literature [21]. When simulating a turn with different turning radiuses, forces on tracks change accordingly, as shown in Figure 10. Force on the outer track increases in intensity, while the force on the inner track decreases, and at radiuses smaller than free turning radius (R < R 0 ) becomes negative (braking force). This is due to the moment of turning resistance, or the coefficient of turning resistance, to be more specific. As the turning radius gets smaller, the moment of turning resistance increases in intensity, which causes the mentioned phenomenon.

CONCLUSION
This paper presents a systematically derived theoretical model for describing the motion of a tracked vehicle applicable to all possible conditions, including transient motion, soft terrain, and slopes. Special emphasis was given to modeling steering on slopes due to the existing research gap.
The theoretical model encompasses resistances arising from track-terrain interaction, grade and air resistance, resistance arising from centrifugal force, inertial forces, and moment of turning resistance. For the coefficient of lateral resistance, an empirical formula was used, the validity of which has been confirmed by comparison with experimental data from the literature and widely recognized Wong's general theory. The heading angle determines the geometrical orientation of the vehicle on the slope. A general motion model was established in MATLAB based on the derived equations. The steering performance of a tracked vehicle on the slope of varying inclination is discussed using the established model. It is shown that the proposed model can be used for predicting and evaluating the steering performance of tracked vehicles with adequate accuracy. Moreover, due to being computationally very efficient, the proposed model can be used for power demand modeling in research on the hybridization or electrification of tracked vehicles. By utilizing information about the soil's basic characteristics and the intended route of the vehicle, the required power can easily be determined. Such fast simulation will significantly reduce the time for drivetrain design iterations or the ability to implement online algorithms for power demand prediction.
Further work will focus on the unified modeling of hybrid and electric tracked vehicles and the creation of complete, scalable, and efficient models for analyzing and designing such vehicles.