Stability of Motion and Mobility Analysis of a 4x4 Hybrid-Electric Vehicle with Passive Drivelines.
A vehicle's wheel power distribution (determined by the driveline's method of dividing power among the axles and driving wheels) has a strong effect on the circumferential forces of the front wheels. In regards to vehicle mobility, the circumferential forces are tied to the slippage at each wheel which limits its forward motion capability. Lateral forces in turn are connected with the side slip angles which affect the stability in the lateral direction. Much research has gone into mathematical and experimental studies on vehicle longitudinal and lateral dynamics and their effect on vehicle motion qualities. A goal of many of these research studies is to better control the interaction of forces using active control systems. Keller, Gorelov, and Anchukov  modeled the motion of a truck with a lockable power dividing unit moving on a sloped road. An active differential control algorithm was developed which evaluates the difference in output shaft rotational velocities in order to lock the differential at high slippage when at a steep slope angle.  (Oraby et al.) simulated the lateral and longitudinal forces on a vehicle using an active steering system which controls the rear steer angle for its improvement in the handling performance. In , Karogal and Ayalew studied four types of torque distribution strategies for stability control using different patterns of corrective torques added to or subtracted from the left or right wheels. K. Isa  developed a simulation of the lateral and longitudinal dynamics of a vehicle using a vision-based autonomous control using an image-processing lane detection algorithm. J. Pytka in  presented experimental lateral dynamics results from a test vehicle instrumented with rotating wheel dynamometers and inertia GPS platform driven over deformable surfaces. In , Senatore and Sandu modeled the effect of torque distribution on the mobility, handling, and tractive efficiency of an off-road vehicle on moist loam and dry sand. It has been shown that the circumferential force distribution significantly influences the magnitude and direction of the front wheels' wheel lateral forces . Therefore, lateral dynamics properties (forces and side slip) affect the mobility estimate in the longitudinal direction while, at the same time, the circumferential forces contribute to changes in the lateral stability. This means that there is a coupling effect in the vehicle's lateral and longitudinal forces, the driveline power split, and the vehicle's mobility and stability characteristics. In this paper, a mathematical model is derived for a simulation study of the coupled effect of longitudinal and lateral forces on a 4x4 vehicle's mobility and stability of motion. The paper is organized as follows: first, a mathematical model of the 4x4 vehicle in planar motion is described, by which the determination of tire forces and slippage is made. The torque distribution from the transfer case to the tires is then modeled for the two types of passive drivelines. Next, the indices for estimating the vehicle's mobility and stability is provided. Lastly, computational results are provided for the vehicle model performed with two steering maneuvers and changing terrain conditions.
COUPLED DYNAMICS MODEL
The mathematical model of the 4x4 vehicle models forces at each of the vehicle's four driving wheels. Figure 1 demonstrates the vehicle's forces, steering angles, and geometry .
The motion of the vehicle can be described using Appel's equations :
[mathematical expression not reproducible] (1)
where S is the acceleration energy, [[pi].sub.i] is the quasi-acceleration, [Q.sub.i] is a generalized force corresponding to the ith quasi-coordinate, and k is the number of quasi-coordinates. In Fig. 1, the vehicle motion is described by three generalized coordinates of the center of mass: [X.sub.a], [Y.sub.a] and [[beta].sub.a]. Three quasi-velocities [[pi].sub.1], [[pi].sub.2], and [[pi].sub.3] related to [X.sub.a], [Y.sub.a] and [[beta].sub.a] are introduced using the expressions in Equations (2). (3). (4) .
[X.sub.a] = [[pi].sub.1] cos [[beta].sub.a]-[[pi].sub.2] sin [[beta].sub.a] (2)
[Y.sub.a] = [[pi].sub.1] sin [[beta].sub.a]+[[pi].sub.2] cos [[beta].sub.a] (3)
[[beta].sub.a] = [[pi].sub.3] (4)
Using Equations (2), (3), (4), the acceleration energy S is determined for the energy of acceleration that arises from translational motion of the center of mass and rotational motion about this center .
[mathematical expression not reproducible] (5)
where [m.sub.a] and [p.sub.a] are the vehicle mass and radius of inertia. Generalized forces [Q.sub.i] which correspond to the quasi-coordinates are determined from the expression of virtual work :
[delta][A.sub.1] = [Q.sub.1] [delta][[pi].sub.i], i = 1,3 (6)
Using Equations (1), (5), and (6), the planar motion equations of the vehicle become
[m.sub.a] ([[pi].sub.1] - [[pi].sub.2][[pi].sub.3] = [F.sub.[SIGMA]x] (7)
[m.sub.a] ([[pi].sub.2] - [[pi].sub.1][[pi].sub.3] = [F.sub.[SIGMA]x] (8)
[I.sub.a][[pi].sub.3] = [M.sub.[SIGMA]](9)
The planar motion of the vehicle is described by these three equations (7), (8), (9): these equations model forces and acceleration along the x-axis (longitudinal), forces and acceleration along the y-axis (lateral), and moments and rotational acceleration (yaw).. [F.sub.[SIGMA]x] is the sum of all forces projected on the longitudinal axis and [F.sub.[SIGMA]y] is the sum of all forces projected on the lateral axis. [M.sub.[SIGMA]] is the sum of moments. [t.sub.bi] is the lateral distance between left and right tire centers. [l.sub.a] is the longitudinal distance from the front axle to the center of gravity. [l.sub.i] are the distance between axles ([l.sub.1] = 0 and [l.sub.2] is the wheelbase).
[[pi].sub.1], [[pi].sub.2], and [[pi].sub.3] are quasi-velocities of the vehicle's center of gravity: [[pi].sub.1] in the longitudinal direction, [[pi].sub.2] in the lateral direction, and yaw rate [[pi].sub.3]. [[pi].sub.i] (i = 1,3) are quasi-accelerations. [??] are the wheel steer angles. The front axle's steer angle, [[delta].sub.1], is set by the vehicle steering system. The individual steer angles of the front wheels to give each wheel the same turn center, [??], are derived from Fig. 2, showing the separate wheel steer angles and turn radii.
Three equations derive from to the geometry in Fig. 2:
[mathematical expression not reproducible] (10)
[mathematical expression not reproducible] (11)
[mathematical expression not reproducible] (12)
Equation (13) is the result of summing Equations (11) and (12):
[mathematical expression not reproducible] (13)
By solving Equation (10) for cot [[sigma]".sub.1] or cot [[sigma]'.sub.1], then substituting it in Equation (13), [[sigma]'.sub.1] and [[sigma]".sub.1] can then be solved for resulting in
[mathematical expression not reproducible] (14)
The rear wheels are not steered, making [[sigma]".sub.2]. Forces contributing to the vehicle motion are the rolling resistance forces [R'.sub.xi], lateral forces [F'(").sub.li], and circumferential forces [F'(").sub.xi]. Rolling resistance forces are proportional to the normal load on the tire by a rolling resistance coefficient, f. This coefficient is a property of the tire and terrain. The tire model is the same for all tires and does not change throughout the tests, however the terrain condition changes as the vehicle moves. To model the effects of changing ground conditions, the terrain properties are modeled as a continuous random process using a method described in . Normal reactions are modeled using a vibration multi-mass model which takes into account dynamic shifting in the vertical and lateral directions. The loads without vertical vibrations are first calculated as
[mathematical expression not reproducible] (15)
[R.sub.st_zi] are the static loads on each axle, calculated according to the vehicle's front/rear mass distribution. [h.sub.y] is the height of the wheel gravity center. Forces [F.sub.yi] act at the height of the gravity center for the front and rear wheel, loading the outer wheels and reducing the load of the inner wheels during a curvilinear maneuver. The forces are computed proportionally to the front and rear masses [m.sub.i] and the lateral acceleration with Equation (16).
[F.sub.yi] = [m.sub.i]/2[a.sub.yi] (16)
[a.sub.yi] is the lateral acceleration projected to each axle, calculated with
[a.sub.yi] = [[pi].sub.2] + ([l.sub.a]-[l.sub.i])[[pi].sub.3] (17)
The displacements of the sprung and unsprung masses are used to calculate new values of the normal loads with dynamic through Equation (18) .
[mathematical expression not reproducible] (18)
The next set of forces in the planar motion equations are the lateral forces. Lateral forces are proportional to the tire's side slip angles [[alpha].sub.i] and the tire's cornering stiffness, [??], which is dependent on the tire normal reaction, tire traction force, and surface of motion properties .
[mathematical expression not reproducible] (19)
The angles of the side slip can be calculated from Equation (20) by comparing the projection of the velocity of the wheels on the vehicle's longitudinal axis, [V'(").sub.xi], and the velocity projected on the lateral axis, [V'(").sub.xi]. Tire sideslip angles are computed with
[mathematical expression not reproducible] (20)
Where [V'(").sub.yi] and [V'(").sub.xi] are velocity projections on the vehicle's longitudinal and lateral axes:
[mathematical expression not reproducible] (21)
[mathematical expression not reproducible] (22)
The last force, [F'(").sub.xi] is the circumferential wheel force developed by the wheel torque. The tire radius which relates the torque and circumferential force is rolling radius in the driven mode [r.sup.0'(").sub.wi].
[mathematical expression not reproducible] (23)
[r.sup.0'(").sub.wi] is accepted as a reference radius for determining theoretical velocity at zero slip ratio under any type of road and terrain condition . This tire slip ratio [s'(").sub.[delta]i] is modeled using an exponential function :
[mathematical expression not reproducible] (24)
[[micro]'(").sub.pxi] is the peak coefficient of friction, which determines the maximum circumferential forces that can be developed under the current terrain condition. Equation (24) was derived mathematically based on experimental results showing that at a constant [F'(").sub.xi], the wheel has a higher slippage when it is running with a higher side slip angle [alpha]'(") . This may immobilize the vehicle if the slippage is very high when the side slip angle goes up. This illustrates coupled dynamics between the longitudinal and lateral dynamics and between vehicle mobility and stability of motion.
INTERAXLE OPEN SYMMETRICAL DIFFERENTIAL MODEL
The model of both drivelines dictates how the torque provided to its input shaft, [T.sub.0], is split to two axle torques [T.sub.a1] and [T.sub.a2]. For the driveline with an open differential in the transfer case, the torque split is a constant ratio to both axles. The torque split ratio can be expressed using its internal gear ratio [u.sub.d]. [u.sub.d] is defined as the ratio of the number of teeth of the output-side gears coupled to the output shafts with the housing (carrier) stationary . In this model, a ratio of 1 is used. When [u.sub.d]= 1, it is called a symmetrical differential and an equal split of torque is provided to both axles. The axle torque split is given by 
[mathematical expression not reproducible] (25)
[u.sub.1] and [u.sub.2] are the fixed gear ratios from the output shafts of the differential to the front and rear axles. Open symmetrical differentials are also used in each axle's interwheel torque split. This makes each wheel torque half of the axle torque:
[mathematical expression not reproducible] (26)
To obtain an expression for the total torque, [T.sub.0], the projection of all forces on the vehicle's x-axis is used.
[mathematical expression not reproducible] (27)
Equation (27) results when making [[pi].sub.1] equal to zero in Equation (1). The vehicle motion is modeled at a constant speed in the longitudinal direction and thus acceleration [[pi].sub.1] becomes zero. After using Equation (23) to substitute wheel torque for circumferential force and solving (25), (26), (27) for [T.sub.0], the total torque at the transfer case input is
[mathematical expression not reproducible] (28)
The above Equations (25), (26), (27), (28) characterize the torque distribution of the vehicle with an open symmetrical differential.
LOCKED TRANSFER CASE MODEL
A driveline with a locked up power dividing unit, also called a positively engaged power dividing unit, is the second type of driveline model that is analyzed. This type of PDU differs from the open differential in that it provides equal angular velocities (or a constant ratio of angular velocities) rather than an equal torque ratio. While the angular velocities are equal, the theoretical linear velocities of the wheels and axles will differ. This kinematic discrepancy arises from the different values of the rolling radius in the driven mode [r'(").sub.wi], different turn radii of each wheel when the vehicle is turning, and any difference in front/rear gear ratios [u.sub.i]. The kinematic discrepancy at each axle due to these factors can be calculated with 
[mathematical expression not reproducible] (29)
where [m.sub.Hi] is the kinematic discrepancy of axle i = 1,2. The kinematic discrepancy is the main factor in calculating the wheel power distribution for the locked drive. This is true because the kinematic discrepancy results in different slip ratios at the axles.
[mathematical expression not reproducible] (30)
[s.sub.[delta]ai] are the generalized slippages of axle, i = 1,2; [s.sub.[delta]a] is the generalized slippage of the vehicle. The generalized slippage of an axle is
[mathematical expression not reproducible] (31)
where [R.sub.t] is the vehicle's theoretical turn radii (without a lateral slip angle) of the wheels or axles. Expressions for each can be determined from vehicle geometry in Fig. 2:
[mathematical expression not reproducible] (32)
[mathematical expression not reproducible] (33)
[mathematical expression not reproducible] (34)
[mathematical expression not reproducible] (35)
Fully locked interwheel drives are not used in ordinary vehicles because each wheel needs to travel a different path when taking a turn, resulting in slipping, skidding, and loss of mobility when turning. Therefore, the locked drive is used as the interaxle differential while the interwheel differentials are open symmetrical. Different interwheel power splits are outside the scope of this study. This results in the left/right wheel power split being
[mathematical expression not reproducible] (36)
The above Equations (29), (30), (31), (32), (33), (34), (35), (36) provide sufficient information to model the power split of the driveline with positive engagement using a set of five equations with five unknown variables: four individual wheel slips [s'(").sub.[delta]i] and the generalized slippage of the vehicle [s.sub.[delta]a]. Equation (36) provides two equations for the left/right power split; the circumferential forces [F'(").sub.xi] are replaced with (24). (31) provides another equation for each axle with axle slips [s.sub.[delta]ai] replaced with Equation (30). The fifth equation is the force projection on the x-axis (Equation (27)) with circumferential forces [F'(").sub.xi] again replaced with (24).
MOBILITY AND STABILITY MODELING
The mathematical model demonstrates that the tire side forces impact the relation between the circumferential forces and tire slippages. The driveline power distribution is the primary factor influencing the distribution of circumferential forces. These forces, by referring to Equation (24). produces different tire slippage values. By this equation, the tire slippage is also impacted by the lateral tire slip angle and normal loading. The normal load is in turn affected by the lateral forces. This is an important concern for vehicle mobility, which decreases when a tire is slipping. A mobility estimation based on individual wheel power distribution can be expressed using 
[mathematical expression not reproducible] (37)
The wheel mobility index (WMI) decreases with increasing circumferential force, reaching zero when the circumferential force reaches its maximum value which would cause the wheel to be fully spinning with no traction. This maximum value is equal to [R'(").sub.zi] [[micro]'(").sub.pxi], meaning that available mobility capability is a function of changing terrain conditions and the normal load distribution. Each driving wheel's individual mobility contributes to the overall vehicle mobility index (VMI) .
[mathematical expression not reproducible] (38)
VMI evaluates the overall mobility of the vehicle by counting the mobility of each wheel relative to the number of driving wheels. At zero VMI, the vehicle is fully immobilized an unable to move forward.
In the lateral direction, high circumferential and lateral forces can lead to skidding of the tires. The tires have a limited potential to grip the road, which can be estimated by a gripping force utilization factor, [K.sub.[micro]] 
[mathematical expression not reproducible] (39)
At lower values, the wheels have a higher gripping potential and are therefore more able to follow the path set by the vehicle's steering system. When [K.sub.[micro]] reaches high values (above 0.5), the vehicle loses lateral stability and skidding can occur. The reason for this is that the lateral reaction force is limited by the deformation properties of the tire and the soil. If the wheel is subjected to both longitudinal and lateral forces, then the longitudinal reaction (traction force) [F.sub.w] and lateral reaction [F.sub.l] are inter-releated by 
[mathematical expression not reproducible] (40)
When tangential reaction [F.sub.w] increases with increasing numerical value of wheel torque [T.sub.w], then the lateral reaction should decrease. This makes a decrease in the available cornering force that can be absorbed by the wheel. If [F.sub.w] = [[micro].sub.p][R.sub.z], then [F.sub.l],= 0 and the wheel is unable to take up any lateral forces. At high traction forces, a side force such as a side wind or rolling onto a surface irregularity can cause the wheel to slip in the lateral direction, especially if the vehicle is in a situation where it is accelerating on an icy road .
Analysis of the lateral acceleration and yaw rate also are used in estimation of lateral stability. Lateral acceleration is [[phi].sub.2] from Equation (8). The yaw rate, [[phi].sub.3], is calculated by integration of [[phi].sub.3] from Equation (9).
ANALYSIS OF RESULTS
Computational results of the two vehicle driveline models are provided for two simulated maneuvers: a maneuver with increasing steering input and a fishhook maneuver. For the increasing steer maneuver, the steer angle of the front axle, [[delta].sub.1] (and two front wheels, [[delta].sub.1.sup.'] and [[delta].sub.1.sup."]) follows a driver's input in the form of a linear function of time. A rate of 4 deg/sec was used; the maximum steer angle is 45 degrees. This rate results in slow changes in the steer angle but a large magnitude by the end of the test. Angles [[delta].sub.1.sup.'] and [[delta].sub.1.sup."] are computed using a value of [[delta].sub.1] from Equation (14). The fishhook turn incorporates a quick left turn at the start of the maneuver, followed by a pause, then a turn to the right. The steer to the right is held for 3 sec, then returned to the zero degree position. Figure 3 shows the steer angles for both maneuvers. The two maneuvers allow study of vehicle behavior under low and high dynamic loads.
Stochastic terrain conditions influence the normal dynamics of the vehicle and its coupled planar dynamics. Terrain stochastic conditions are introduced by the terrain profile heights, peak friction coefficient and rolling resistance coefficient. They are modeled as a continuous random variable whose value changes between a maximum and minimum value defined by the terrain type. Therefore, uneven terrain is simulated as changes in terrain values with respect to distance along the road. This results in variation over time as the vehicle position on road profile changes. As an example, Fig. 4 illustrates stochastic changes of the rolling resistance coefficient and peak friction coefficient, which mirror each other; when peak friction is high, rolling resistance is low. When the vehicle is turning, the front and rear tires do not travel in the same track. Therefore, each tire has separately modeled terrain profile changes.
Figure 5 is the distribution of circumferential forces at each wheel during an increasing steer test on two types of terrain. With the open differential, the forces are relatively even; small differences in each are due to differences in [r'(").sub.wi] at each tire. Going from asphalt to off-road soil conditions, the required forces are higher to move with the same velocity. For the positive engagement driveline, the forces are close at the start of the maneuver, but diverge as the steer angle increases. The effect of the force distribution on vehicle mobility is seen in Fig. 6. The open differential provides high mobility on asphalt and overall lower mobility on soil. Positive engagement experiences a higher decease in mobility as a result of the steering maneuver. Furthermore, on soil the fluctuations in mobility are greater due to the higher variations in the quality of the ground conditions.
On snow (Fig. 7), the variations and drops in mobility are more pronounced. Near the end of the maneuver, the vehicle is nearly becoming immobilized. This occurs despite the forces not being higher than on soil. However, the lower values of [[micro]'(").sub.pxi] on snow allow for lower maximum values of the circumferential force before the vehicle loses mobility.
Changing the steering maneuver also gives a different mobility profile. Figure 8 shows circumferential forces and mobility in a fishhook turn maneuver on asphalt. The swift changes in the steering angle during the first and last two seconds of the maneuver produce cause corresponding swings in the force distribution; this leads to fast time changes in the vehicle's mobility.
The lateral forces are shown in Fig. 9 for the increasing steer test. High lateral forces develop at the front tires as the steer angle increases. At the rear (unsteered) wheels, the lateral forces at the left and right tires remain close to each other.
The gripping utilization factors for the increasing steer test are shown in Fig. 10. Comparing Figs. 9 and 10, the tire under the highest lateral forces increases its gripping factor faster than the others. In the positive engagement driveline, high gripping factors which reach sliding conditions (> 0.5) can occur at the rear wheels as well.
The lateral acceleration and yaw rates are shown in Figs. 11 and 12. When the lateral forces and gripping factors are high, the vehicle's lateral stability reduces. Oscillations occur in the lateral acceleration in the later part of the maneuver, starting between 6-10 seconds. The increase in the yaw rate also becomes uneven.
The lateral dynamics properties for the fishhook test are plotted in Figs. 13 and 14. Here, large lateral forces are produced at the front wheels during the initial sharp turning motion. This results in a large spike in the gripping factors, which drop down when the steer angle is held constant. Lateral acceleration reaches its highest peak during the first 2 sec of motion; it becomes nearly zero during the constant phase of the test.
The results of the vehicle maneuvers reveal several challenges to vehicle mobility and stability. Drops in mobility occur during high-angle steering motion, variation in terrain quality, and is limited by changes from one type of terrain to another. Loss of motion stability can occur during transient maneuvers and at high steering angles, but can also be temporarily decreased due to terrain changes. Additionally, different patterns of the mobility and stability indices emerge for the two driveline power split types even on identical maneuvers and terrains. The driveline's impact on the vehicle forces is coupled to its mobility and stability of motion due to the interrelation of forces [F'(").sub.xi], [R'(").sub.[delta]i], [R'(").sub.zi] and [F'(").sub.ii] in the vehicle dynamics. This demonstrates the advantage of an active driveline capable of dynamically changing the circumferential force distribution of [F'(").sub.xi] through redistributing the torque to each axle. Decoupling the power distribution (making it under active control) would allow for
1. Dynamically adjusting power distribution to counter fluctuation in tire slippage which results in drops in mobility.
2. Changing the power distribution in steering maneuvers, influencing the front tires' lateral forces in order to maintain lateral stability.
This analysis was performed on a vehicle model incorporating a hybrid-electric powertrain. While the results of this study are generally applicable to 4x4 offroad vehicles, a hybrid powertrain raises the potential to incorporate electric power dividing units into the driveline for active control of the power split. The paper presented a part of a comprehensive analytical analysis to propose a component of a hybrid powertrain to improve energy efficiency, mobility and stability of HE vehicles. The next step includes design and testing of this system for active control of the front/rear power split.
In this paper, a mathematical model is developed of the lateral dynamics of a 4x4 vehicle with two passive driveline systems: an interaxle open symmetrical differential and locked transfer case. The model demonstrates the coupling effect of the lateral dynamics with the circumferential tire forces. These circumferential forces are interdependent with the driveline system characteristics, the tire side forces, and the ground surface under the tires. In a simulated increasing steer maneuver, gripping factors became over 0.5 after 6 sec for the open differential on asphalt at one tire. For positive engagement, values were greater than 0.5 for all tires by 8 sec. Deformable soil terrain produced oscillations in the mobility and gripping factors. A fishhook test resulted in peaks in the gripping factors greater than 0.6 at one front tire between 1 and 2 sec. Simulation results show the coupled dynamics effect on the vehicle's mobility, which decreases due to tire slippage and affected by circumferential force, and stability, which decreases at high values of the side forces. This demonstrates the advantage of decoupling the driveline forces from the vehicle dynamics, which would allow the mobility and stability to be improved by actively controlling the tire force distributions.
[1.] Keller A.V., Gorelov V.A., Anchukov V.V., Modeling Truck Driveline Dynamic Loads at Differential Locking Unit Engagement, Procedia Engineering, Volume 129, 2015, Pages 280-287, ISSN 1877-7058, http://dx.doi.org/10.1016/i.proeng.2015.12.063.
[2.] Oraby, W., Aly, M., El-demerdash, S., and El-Nashar, M., "On The Integration of Actively Controlled Longitudinal/Lateral Dynamics Chassis Systems," SAE Technical Paper 2014-01-0864, 2014, doi :10.4271/2014-01-0864.
[3.] Karogal, I. and Ayalew, B., "Independent Torque Distribution Strategies for Vehicle Stability Control," SAE Technical Paper 2009-01-0456, 2009, doi: 10.4271/2009-01-0456.
[4.] Isa K., "Experimental Studies on Dynamics Performance of Lateral and Longitudinal Control for Autonomous Vehicle Using Image Processing," Computer and Information Technology Workshops, 2008. CIT Workshops 2008. IEEE 8th International Conference on, Sydney, QLD. 2008, pp. 411-416. doi: 10.1109/CIT2008.Workshops.89
[5.] Pytka, J., "Experimental Research on Stability of an Off-Road Vehicle on Deformable Surfaces," SAE Technical Paper 2010-01-1898, 2010, doi :10.4271/2010-01-1898.
[6.] Senatore C., Sandu C., Torque distribution influence on tractive efficiency and mobility of off-road wheeled vehicles, Journal of Terramechanics, Volume 48, Issue 5, October 2011, Pages 372-383. ISSN 0022-4898, http://dx.doi.org/10.1016/i.iterra.2011.06.008.
[7.] Vantsevich V. V., "AWD Vehicle Dynamics and Energy Efficiency Improvement by Means of Interaxle Driveline and Steering Active Fusion", ASME 15th International Conference on Advanced Vehicle Technologies, Portland, OR, August 4-7, 2013.
[8.] Andreev, A.F., Kabanau, VI., and Vantsevich, V. V., Driveline Systems of Ground Vehicles: Theory and Design. Vantsevich V.V., Scientific and Engineering Editor, Taylor and Francis Group/CRC Press, ISBN 978-1-4398-1727-8, 2010.
[9.] Gray, J.P, Vantsevich, V.V., Opeiko, A.F., and Hudas, G.R., 2013, "A Method for Unmanned Ground Wheeled Vehicle Mobility Estimation in Stochastic Terrain Conditions", Proc. of the 7th Americas Regional Conference of the ISTVS, Tampa, Florida, USA.
' - Left wheel (superscript)
" - Right wheel (superscript)
[C.sub.t] - Tire damping
[C.sub.a] - Tire cornering stiffness
f- Rolling resistance coefficient
[F.sub.l] - Wheel lateral force
[F.sub.x] - Wheel circumferential force
[F.sub.w] - Wheel traction force
i - Axle number, front to rear (subscript)
k - Factor from tire slippage exponential characteristic
[K.sub.a] - Tire longitudinal stiffness coefficient for axle
[K.sub.t] - Tire normal stiffness
[K.sub.[micro]] - Tire gripping factor
[l.sub.2] - Distance between axle 1 and axle 2
[l.sub.a] - Distance between axle 1 and center of gravity
[m.sub.a] - Vehicle mass
[m.sub.Hi] - Kinematic discrepancy
[r.sub.a.sup.0] - Generalized rolling radius of axle in driven mode
[r.sub.w] - Effective rolling radius of a wheel in driving mode
[r.sub.w.sup.0] - Effective rolling radius of a wheel in driven mode
[R.sub.t] - Theoretical turn radius
[R.sub.x] - Tire rolling resistance
[R.sub.z] - Ground normal reaction on wheel
[s.sub.[delta]] - Tire slip ratio (slippage)
[T.sub.a] - Axle torque
[t.sub.b] - Distance between left and right wheel centers
[T.sub.0] - Transfer case input torque
[T.sub.w] - Applied wheel torque
VMI - Vehicle mobility index
WMI - Wheel mobility index
[z.sub.r] - Vertical displacement of road surface
[z.sub.u] - Vertical displacement of unsprung masses
[u.sub.1] - Gear ratio from transfer case to front axle
[u.sub.2] - Gear ratio from transfer case to rear axle
[u.sub.d] - Open differential internal gear ratio
[alpha] - Side slip angle
[delta] - Steering angle
[[micro].sub.px] - Peak longitudinal tire friction coefficient
[phi] - Quasi-velocity of the gravity center
Jesse Paldan and Vladimir V. Vantsevich
University of Alabama at Birmingham
Vladimir V. Vantsevich
Mechanical Engineering Department
University of Alabama at Birmingham
Birmingham, Alabama, USA
Jesse R. Paldan
Mechanical Engineering Department
University of Alabama at Birmingham
Birmingham, Alabama, USA
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|Author:||Paldan, Jesse; Vantsevich, Vladimir V.|
|Publication:||SAE International Journal of Passenger Cars - Mechanical Systems|
|Date:||Apr 1, 2017|
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