Alexandria Engineering Journal (2017) xxx, xxx-xxx

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ORIGINAL ARTICLE

Lateral motion control of skid steering vehicles using full drive-by-wire system

Ossama Mokhiamar a,b,% Semaan Amine a

a Beirut Arab University, Faculty of Engineering, Mechanical Engineering Department, P.O. Box 11-5020, Riad El Solh, Beirut 1107-2809, Lebanon

b Alexandria University, Faculty of Engineering, Mechanical Engineering Department, El-Chatby, Alexandria 21544, Egypt Received 22 December 2016; revised 16 February 2017; accepted 13 March 2017

KEYWORDS

Vehicle dynamics; Traction control; Nonlinear control; Skid steering vehicles

Abstract This paper introduces a control system to stabilize the motion of skid steering vehicles, extends their stability limit and makes their handling performance similar to that of the conventional two-wheel steering vehicle. For this purpose, the yaw rate response of the two degree of freedom (2DOF) linear model of the conventional two-wheel steering vehicle is chosen as a model response. The model following control theory is used to introduce the direct yaw moment needed to stabilize and steer the skid steering vehicle. The direct yaw moment has been split into the four tires based on two methods. The first method is based on a simple distribution technique (SD), whereas the second method is based on an independent distribution technique (ID) where the four wheels can be driven individually using a full drive-by-wire system. A comprehensive nonlinear dynamic model of the skid steering vehicle has been simulated using Matlab/Simulink in order to examine the effectiveness of the proposed control system. The results of both open and closed loop tests show that the proposed control system has a significant effect on stabilizing the lateral motion of skid steering vehicles as well as improving their handling characteristics.

© 2017 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

Over the past decade, a few researches have addressed the increase in the stability limit of conventional two-wheel steering vehicles based on an optimal distribution of both longitu-

* Corresponding author at: Beirut Arab University, Faculty of Engineering, Mechanical Engineering Department, P.O. Box 11-5020 Riad El Solh, Beirut 1107-2809, Lebanon.

E-mail addresses: ossama.mokhiamar@bau.edu.lb (O. Mokhiamar), s.amin@bau.edu.lb (S. Amine).

Peer review under responsibility of Faculty of Engineering, Alexandria University.

dinal and lateral tire forces. Suzuki et al. [1] studied feedforward types of tire force distribution controls for full drive-by-wire electric vehicles based on two strategies. The first strategy was to minimize the tire workload whereas the second strategy was to minimize the tire energy dissipation. The effect of the control system was investigated experimentally on a proving ground where it was concluded that the first strategy has a great influence on the vehicle stability. Goodarzi and Mohammadi [2] designed an integrated control of three layers for improving the stability and the fuel economy of a 4-wheel-drive hybrid vehicle. One of the three layers was to determine the optimum values of longitudinal and lateral forces of the four tires that allow achieving a desired vehicle response.

http://dx.doi.org/10.1016/j.aej.2017.03.024

1110-0168 © 2017 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Nomenclature

a distance from the mass center of the vehicle to front axle P pitch angular velocity of the vehicle about the y-axis

ax longitudinal acceleration p pitch angular acceleration of the vehicle about the

axdes traction/braking command from the driver y-axis

ay lateral acceleration q roll angular velocity of the vehicle about the x-axis

A stability factor q roll angular acceleration of the vehicle about the x-

b distance from the mass center of the vehicle to rear axis

axle r steering gear ratio

Bpitch suspension pitch damping coefficient R wheel radius

Broll suspension roll damping coefficient s Laplace transform

Cf cornering stiffness of front wheel S sliding surface

Cr cornering stiffness of rear wheel t vehicle tread

dr distribution ratio of the direct yaw moment Ti traction/braking torque at wheel i (i = 1-4)

Fxi friction force in the longitudinal direction of wheel V speed of the vehicle

i (i = 1-4) Vx speed of the vehicle in the longitudinal direction

F*. XI required longitudinal force at wheel i (i = 1-4) Vx acceleration of the vehicle in the longitudinal

Fyf linear lateral force produced by front wheels direction

Fyf estimated lateral force produced by front wheels Vy speed of the vehicle in the lateral direction

Fyi friction force in the lateral direction of wheel i Vy acceleration of the vehicle in the lateral direction

(i = 1-4) W vehicle weight

F yr linear lateral force produced by rear wheels Ws vehicle sprung weight

F* yr Fzi estimated lateral force produced by rear wheels x vehicle longitudinal direction

estimated vertical load of wheel i (i = 1-4) y vehicle lateral direction

g gravity acceleration z vehicle vertical direction

Gr yaw rate gain a sideslip angle of the vehicle

hs sprung mass height aes estimated vehicle sideslip angle

Ix mass moment of inertia about the x-axis a time derivative of sideslip angle

Iy mass moment of inertia about the y-axis c yaw rate of the vehicle

Iz yaw moment of inertia Css steady state yaw rate response

Ji moment of inertia of wheel i (i = 1-4) yaw acceleration of the vehicle

k time constant f damping factor

kf lateral weight-shift distribution on the front wheel h pitch angle of the vehicle

kr lateral weight-shift distribution on the rear wheel m sum of squared normalized longitudinal forces

Kpitch suspension pitch stiffness produced at the four wheels

Kroll suspension pitch stiffness u roll angle of the vehicle

Mz direct yaw moment control w driver's steering wheel angle

Mzf direct yaw moment to be generated from front Xi angular velocity of wheel i

wheels Xn natural frequency

Mzr direct yaw moment to be generated from rear wheels

The simulation results showed noticeable improvements on the stability and fuel consumption of the vehicle. Following the driver command, Wang et al. [3] found the optimum tire forces to follow a given target path. Computer simulations were conducted to validate the effectiveness of the proposed control. In the simulation, a 7 degrees of freedom vehicle model was used while the magic formula model was used to detect the tire forces. Based on proportional, integral and derivative (PID) control theory, Kim et al. [4] determined the yaw moment required to stabilize the lateral vehicle motion of rear wheel drive electric vehicles. Then, they distributed the yaw moment to the wheels in order to calculate the needed longitudinal tire force by taking into account the desired longitudinal force according to the driver input through the accelerator pedal. CarSim software was used to verify the effectiveness of the proposed method. During cornering with high speed, the sim-

ulation results showed a saving in the electric energy in addition to an improvement of the vehicle stability. Naraghi et al. [5] presented an integration of the vehicle longitudinal and lateral motion control in which the tire forces were distributed based on an adaptive optimal approach. In the proposed system, each tire was assumed to be steered and driven and/or braked individually. Digital simulations were executed based on a comprehensive nonlinear vehicle model. The results proved the positive influence of the proposed control. Mokhiamar [6] and Mokhiamar and Abe [7] proposed an optimum distribution of lateral and longitudinal tire forces based on minimizing the tire workload. The 2DOF state variables, yaw rate and sideslip angle, were chosen as desired responses for the model following control. A comprehensive nonlinear vehicle model was used to predict the dynamics of the controlled vehicle via computer simulations. The results

of the simulations showed a great effect of the control system on the vehicle handling characteristics. Once again, the idea of minimizing the tire workload was presented by Ono et al. [8] based on nonlinear optimization technique to find out the steering angle and traction/braking torque at the four wheels.

Skid steering vehicles are mainly used in off-road vehicles in addition to some four and six wheeled vehicles. On the other hand, a few skid steering vehicles run at medium/high speed. Instead of turning the vehicle by steering the front wheels, as in the conventional two-wheel steering vehicle, the wheels on each side can be driven at various speeds to turn the skid steering vehicle. This can be achieved by intentionally applying traction/braking force on the wheels to produce the direct yaw moment needed to turn the vehicle. Accordingly, the mechanical steering system will be eliminated which results in more interior space. The caster wheel effect will be eliminated as well. As stated by Sasaki and Abe [9], it is not easy to drive skid steering vehicles especially for drivers who are used to drive conventional two-wheel steering vehicle. This is due to the over-steer characteristics of skid steering vehicles mainly when the vehicle speed increases.

It is noteworthy that a very few studies were conducted to find out the proper distribution of the longitudinal forces that should be produced at the four wheels in order to stabilize the motion of skid steering vehicles. Sasaki and Abe [10] proposed an approximate first order lag of the yaw rate response of the 2DOF linear vehicle model as a target response for the model following control. The direct yaw moment control, needed to be generated from the four wheels in order to stabilize the skid steering vehicle motion and follow the target yaw rate response, was distributed as traction/braking torque at each tire. The distribution technique was based on a very simple strategy without considering the tire capacity during motion. The effectiveness of the proposed control system was evaluated theoretically using computer simulations and experimentally using a driving simulator. The strategy used by Juyong et al. [11] aimed at minimizing the sum of longitudinal tire forces to find the proper longitudinal tire force distribution. The idea was applied on a skid steering for a robot vehicle with articulated suspension and a maximum speed of 15 km/h.

Based on the foregoing, it is clear that: (i) the most of the work done focused on optimizing the tire usage in longitudinal and lateral direction for the conventional two-wheel steering and (ii) there is an essential need for more researches on the optimal distribution of longitudinal tire forces for skid steering vehicles. Therefore, the objective of this research was to propose a control system to:

• stabilize the motion of skid steering vehicles,

• make the performance of the controlled skid steering vehicle similar to that of conventional one,

• extend the stability limit of skid steering vehicles and allow them to be driven at a relatively high speed.

In order to achieve the aforementioned goals, the exact second order yaw rate response of the conventional two-wheel steering is chosen as a model response for the model following control. The nonlinear sliding control theory is used to calculate the required direct yaw moment by forcing the error between the desired and the actual yaw rate response of the skid steering vehicle to be zero. This yaw moment is required

not only to stabilize the skid steering vehicle motion but also to make its handling characteristics close to those of the conventional one. The direct yaw moment is distributed to the four wheels as traction/braking torque using two different strategies. The first one is based on a simple sharing of longitudinal forces among the four wheels whereas the second one is based on the individual ability of each tire to produce longitudinal force using full drive-by-wire system. The effects of the suggested two strategies are investigated and compared using computer simulations program. In addition, both open loop and closed loop evaluations are executed.

2. Nonlinear dynamic model of the skid steering vehicle

This section presents the nonlinear dynamic model of the skid steering vehicle used in this research. The model is described by 9 degrees of freedom. Five degrees are considered for the vehicle body: longitudinal motion, lateral motion, rotation about the z-axis (yaw motion), rotation about the x-axis (roll motion), and rotation about the y-axis (pitch motion), since the main focus of this research was to study the effect of the proposed control system on the skid steering vehicle handling characteristics and not to study the ride comfort. Moreover, the vertical load at each wheel is calculated taking into account the weight shift due to roll and pitch motions as it will be seen later in Section 5.2. Based on this justification, the vertical motion in z direction can be ignored without affecting the simulation results predicted from the proposed dynamic model and related to the handling characteristics. Referring to Fig. 1, the five equations for the vehicle body motion are as follows:

— {Vx - yVy)+— hsp — Fx1 + Fx2 + Fx3 + Fx4 g g

W. • , Ws

— {Vy + yVx )--hsq — Fy1 + Fy2 + Fy3 + Fy4

IzC + {ly - Ix)qp — a[Fyi + Fy2) — b{Fy3 + Fy4) t

+ 2 {Fx2 + Fx4 — Fx3 — Fx1 )

Ixq + {Iz — Iy)yp — —hs(Vy + yVx ) g

= — hsgu — Km„y — BroUq g

IyP + {Ix — Iz)yq + —hs{Vx — yVy ) g

— —hsgh — KpitchO — BpitchP

The remaining four degrees correspond to the dynamics of the four wheels. From Fig. 2, the wheel rotational equation of motion can be written as follows:

Jr^- = T, - FXiR dt

The longitudinal and lateral friction forces generated at the tires were estimated using the nonlinear tire model described by Abe [12].

Figure 1 Schematic for the mathematical skid steering vehicle model.

3. Linear dynamics of the skid steering vehicle

Referring to Fig. 1 and following Abe [12], the equations of motion of the 2DOF vehicle planar model with constant speed can be written as follows:

V(a + y)=Fyf + Fyr

(9) (10)

Substituting Eqs. (9) and (10) into Eqs. (7) and (8) yields

I2y — aFyf - bFyr + Mz

Fyf — -2Cf (a + y c)

Fyr — -2CA a - — c

— Va + 2(Cf + Cr)a + g

^V + -V (aCf - bCr; gV

2(aCf - bCr)a + Izc + 2-V(a2Cf + b2Cr) — Mz

(11) (12)

Figure 2 Wheel rotation and torques.

Eqs. (11) and (12) can be written in matrix form as follows:

'gVs + 2(Cf + Cr ) WV + V (aCf - bCr)' ' a.(s) ' " 0 "

2(aCf - bCr) Izs + V (a2Cf + bLCr) .C(s). .Mz.

From the above matrix form, the steady state yaw rate response of the skid steering vehicle can be expressed as follows:

(Cf + Cr)

g 2(a+b)2 CfC,

(a + b) 2CfCr (a + b)

Abe [12] presented the steady state yaw rate response of the conventional two-wheel steering vehicle as follows:

1 _ W V2 (fCr) (a ^ b) g 2(a+b)2 CfCr /

Comparing Eqs. (13) and (14), an expression of the direct yaw moment needed to steer the uncontrolled skid steering vehicle can be obtained as follows:

2CfCr(a + b) W

Mz —

Cf + Cr r

4. Control method

4.1. Model response

The handling characteristics of a skid steering vehicle are different from those of a conventional two-wheel steering vehicle due to the over steer characteristics of the first. The motion of a skid steering vehicle becomes unstable when it reaches a specific speed. Indeed, driving a conventional two-wheel steering vehicle is convenient to the most of the drivers. Therefore, the yaw rate response of the 2DOF linear plane model of the conventional two-wheel steering vehicle is considered as model response for the model following control theory. It was adopted in order to stabilize the motion of skid steering vehicle and make its handling characteristics close to those of conventional one. The model response of yaw rate was written by Abe [12] as follows:

!L — r 1 + Trs

w — R 1 + C + ¿s2

1 + AV2\ a + b

2{a + b)Cr

WW {a2Cf + b2 Cr ) + Iz{ Cf + C ; 2{a + b)yjWWIzCfCr{1 + AV2)

- /V Y WI,

bCr — aC,

2{a + b)2 V CfC,

4.2. Control law of the direct yaw moment

The nonlinear sliding control theory presented by JeanJacques and Weiping [13] was used to calculate the required direct yaw moment needed to steer the skid steering vehicle and stabilize its motion. Accordingly, the sliding surface S can be defined as follows:

S — y — y, — 0

5. Determination of traction/braking torque

The direct yaw moment given by Eq. (21) is distributed to the four wheels as wheel longitudinal forces. Then, the traction/ braking torque that should be applied at each wheel in order to follow the desired yaw rate response and also stabilize the skid steering vehicle motion is calculated. In this paper, an independent distribution of wheel longitudinal forces is proposed and compared to the simple distribution method.

5.1. Simple distribution method

In the simple distribution method, the direct yaw moment is distributed to front and rear axles in such a way that the distribution ratio is constant as follows:

MZf — drMz Mzr — {1 — dr)Mz

{22) {23)

Meanwhile, the right and left wheels on front/rear axles bear the same amount of traction/braking force. As a result, the traction/braking torque on each wheel can be calculated as follows:

T1 — T2 — MzfR

T3 ~ T^ — MzrR

5.2. Independent distribution method

Eq. (17) states that the difference between the yaw rate response of the skid steering vehicle and that of the conventional two-wheel steering is converged to zero. In other words, the yaw rate response of the skid steering vehicle is forced to follow that of the conventional vehicle. The sliding condition is as follows:

S + kS — 0

Substituting Eq. (16) into Eq. (17), and then substituting the resultant equation into Eq. (18) the following equation can be obtained:

y + ky — Gr-4+T^ — 0

■1 + f s + -1 s2

xn a>n

And, from Fig. 1, the equation of yaw motion is

Izj = 2(aFf - bFw) + Mz (20)

From Eqs. (19) and (20), the control law of the direct yaw moment can be formulated as follows:

Mz = 2(bF, - aF*yf) - kc I, + gri, ft^)

^ xn ~ afn \ J

The lateral forces (Ff and FJr) generated by front and rear wheels were estimated using the simple tire model presented by Abe et al. [14].

In the simple distribution method presented above, the direct yaw moment was distributed to the four wheels without taking into account the ability of each wheel to produce the required longitudinal force. Therefore, the main objective of the independent distribution method proposed in this paper was to figure out how much traction/braking torque should be applied at each wheel to: (a) steer the skid steering vehicle, (b) stabilize the skid steering vehicle lateral motion, and finally (c) force the skid steering vehicle yaw rate to follow the desired yaw rate response. From the concept of friction circle, the traction/ braking force acting on a wheel is proportional to its vertical load. Accordingly, the sum of squared normalized longitudinal forces produced at the four wheels is adopted as the objective function for the proposed independent distribution technique.

F*2 Fx3

F*2 rx4

_ X1 X2 X3 X4

1 — F2 + F2 + F2 + F2 Fz1 F z2 Fz3 Fz'

The independent distribution of F£i-4 is obtained by minimizing the objective function m taking into account the following constraints: (a) the direct yaw moment must be equal to the yaw moment produced by the longitudinal forces of wheels, Eq. (27) and (b) the total longitudinal force following the driver's command must be equal to the sum of longitudinal forces produced by the four wheels, Eq. (28).

Mz — 2 (Fa-

-FX, + FX4 — F.3

g axdes — + + FFX3 + Fx4 g

The equality constraints presented in Eqs. (27) and (28) can be used to eliminate two of the four variables in the objective function given by Eq. (26). From Eqs. (27) and (28), the following relations can be obtained:

Fx4 ~ ~~axdes ~ P*xl ~ Fx2 ~ Fx3 S

F3 — 2g°xdes Fx1 t

Substituting the above expression into Eq. (26), the new objective function of two variables Fx1 and Fx2 is obtained

F2 F2 1 (w

1 4 + Fx2 + FlX2gaxdeS Fx1 T

(W _ F , Mi

+ F\\2gaxdes Fx2 + t

The expressions of Fxl and Fx2 can be determined using the following minimizing conditions:

^ 2 F2a Fl\2gaxde Fx1 t) 0

W axdes 2Mz

g F2 tF2

F* _ g z3 z3

Fx1 — 2,2

F21+ F3

z4 \2gC

2 F2 p2 ^ ^ ~axdes

-K2 + MM =o

F2 ^ p2

Once the wheel longitudinal force is obtained, the traction/ braking torque that should be applied at each wheel in the full drive-by-wire system is given by the following:

T, — RFxl{i— 1 - 4) (34)

It is noticed that the independent distribution technique depends on the wheel vertical load, Fzi. Therefore, it is essential to estimate the vertical load at each wheel by taking into account the effect of load shift due to roll and pitch motions. The load shift can be estimated by measuring both the longitudinal and lateral acceleration of the vehicle. The estimated vertical loads at the four wheels are obtained as follows:

2(a + b) g 2(a + b) J g t

Wb Ws axhs

+ k/-^

2(a + b) g 2(a + b) f g t

Wa Ws axhs Ws a„hs

---k,---—

2(a + b) g 2(a + b) r g t

2(a + b) g 2(a + b) r g t

The block diagram presented in Fig. 3 summarizes the proposed control system.

6. Simulation results and discussion

The dynamic model of the skid steering vehicle (SSV) described in Section 2 was simulated using Matlab/Simulink software. The influence of the proposed independent distribution method (ID) was investigated through both open loop and closed loop tests. Small passenger car with data as shown in Table 1, Appendix A, was used in the computer simulations. In the closed loop test, the simple first order preview model described by Mokhiamar and Abe [7] modeled the driver. In this model, the driver steered the steering wheel in such way to eliminate the difference between the target path and the vehicle position look-ahead distance. The geometry of the target path is shown in Fig. 4. On the other hand, in case of open loop test, the driver steered the steering wheel with a specific and prescribed steering input that did not change during the simulation regardless of the vehicle response. Moreover, the handling performance of the skid steering vehicle of the proposed control system (SSV/ID) is compared with (a) the performance of skid steering vehicle without control (SSV/NC); (b) the performance of controlled skid steering vehicle based on the simple distribution method (SSV/SD); and (c) the handling characteristics of the conventional two-wheel steering vehicle (CFWSV).

The SSV/SD response had been evaluated for various values of dr and the most acceptable response was found when dr equals to 0.5. As a first phase of this research, the vehicle speed was maintained constant during the tests. In other words, the foot-brake pressure provided by the driver was equal to zero, whereas the acceleration pedal pressure is constant. The vehicle running speed was equal to 30 m/s and the tests were executed on dry road condition.

The results obtained using the simulations program are illustrated in Figs. 5-13. In all these figures, there are two sub-figures: sub-figure (a) corresponds to open loop test, whereas sub-figure (b) corresponds to closed loop test. Fig. 5 shows the steering wheel angle provided by the driver. Of course, in case of open loop test, there is only one plot for all the cases treated in this paper. On the other hand, in case of closed loop test, there are four different plots one for each case. Clearly, skid steering vehicles without control, for which Mz is calculated using Eq. (15), have unstable motion. However, each of the proposed systems allows achieving stable motion. In addition, there is a good matching between the handling performance of the controlled skid steering vehicle and the performance of the conventional two-wheel steering vehicle. The yaw rate, the vehicle sideslip angle and the lateral acceleration responses are illustrated in Figs. 6-8. Once again, both the simple and the independent distribution methods resulted in a stable motion. Moreover, the proposed control method produces desirable vehicle responses when compared with the target yaw rate response and the response of the conventional vehicle. It is noteworthy that at such a high speed, the uncontrolled SSV shows unstable motion. This is mainly because the stabilizing yaw moment generated by the wheel lateral forces could not balance the yaw moment, Mz, generated by the wheel longitudinal forces and eventually the vehicle falls into an over steering state. Therefore, there is an essential need to control the value of the input direct yaw moment in order to stabilize the SSV motion.

Wb Ws axhs

Figure 3 Block diagram of the proposed control systems.

• • •

Ï • • •

T In 3.5 m ^----------'

40 m -

Figure 4 Geometry of the single lane change task.

80 60 40 20 0 -20 -40 -60 -80

Time (sec)

Time (sec) (b)

Figure 5 Driver steering wheel input time history (dry road, V = 30m/s): (a) open loop evaluation, and (b) closed loop evaluation.

Time (sec)

Time (sec) (b)

Figure 6 Comparison of vehicle yaw rate response (dry road, V =30 m/s): (a) open loop evaluation, and (b) closed loop evaluation.

Time (sec)

Time (sec) (b)

Figure 7 Comparison of vehicle sideslip angle response (dry road, V =30 m/s): (a) open loop evaluation, and (b) closed loop evaluation.

Time (sec)

Figure 8 Comparison of vehicle lateral acceleration response (dry road, V = 30 m/s): (a) open loop evaluation, and (b) closed loop evaluation.

-0.5 -

Regarding the difference between the controlled skid steering vehicle based on simple distribution and independent distribution methods, it can be said that both methods control the motion of SSV. However, the SSV with independent distri-

bution shows a better response compared to simple distribution method. The evidence of good matching between controlled SSV and CFWSV performances is also clearly spotted in Figs. 6-8. In addition, the matching in case of SSV/ID is

Longitudinal position of the vehicle (m)

« ►J

50 100 150

Longitudinal position of the vehicle (m) (b)

Figure 9 Vehicle trajectory (dry road, V = 30m/s): (a) open loop evaluation, and (b) closed loop evaluation.

•3 3

-2 -2000

• S H

SSV/SD

tire 1 tire 2 tire 3 tire 4

Time (sec)

•3 3

SSV/ID

-•- tire 1

_ _ _ r\ _ -•- tire 2 _

-♦- tire 3

tire 4

1 1 -f H 1 1 1

Time (sec)

• S H

-4000 (b)

Time (sec)

SSV/ID

•- tire 1

•- tire 2

+- tire 3

tire 4

Time (sec)

Figure 10 Time history of the tires' longitudinal forces (dry road, V =30 m/s): (a) open loop evaluation, and (b) closed loop evaluation.

better than in case of SSV/SD. The trajectories of controlled SSV and CFSWV are compared in Fig. 9. In the closed loop test, Fig. 9(b), the initial condition of the longitudinal position of the vehicle is set to zero in order to force the driver to apply a sudden steering in order to simulate the case of severe evasive lane change. This explains the relatively large difference between the actual vehicle position and the target path especially at the beginning of the maneuvering. The presented results show a significant effect of the proposed control system on the trajectory of SSV.

Going deep inside the SSV dynamics, Figs. 10-13 are provided in order to understand and analyze the difference between the proposed independent longitudinal force distribution method and the simple one. Fig. 10 shows the longitudinal force at each wheel due to the traction/braking torque calculated from Eqs. (24) and (25)/or (34). Fig. 10(a), corresponding to open loop test, has two sub-figures: one for SSV/SD and another for SSV/ID. It is obvious that the SSV/ID utilizes the ability of each wheel properly in coordination with its capacity since the ability of the wheel to produce force depends

Time (sec) Time (sec)

Figure 11 Time history of the tires' longitudinal slip ratio (dry road, V = 30 m/s): (a) open loop evaluation, and (b) closed loop evaluation.

Î 3 ^ 2.5

1 0.5 -a

■I 0

<3 -0.5

Time (sec)

is 2.5

.¡a 0.5

° -0.5

SSV/ID SSV/SD CFWSV

1......1.....1.

Time (sec) (b)

Figure 12 Comparison of the longitudinal acceleration of the vehicle (dry road, V =30 m/s): (a) open loop evaluation, and (b) closed loop evaluation.

on the vertical load applied on it. On the other hand, an illogical time history is presented in the case of SSV/SD. Definitely, this will affect the longitudinal slip ratio of the four wheels as shown in Fig. 11. It should be noted from this figure that in case of SSV/SD, the longitudinal slip of wheels 3 and 4 exceeds —0.3. The negative sign appears since the wheels are subjected to a traction torque produced by a positive longitudinal force as already shown in Fig. 10. This means that the ability of these wheels to produce longitudinal force decreases. More-

over, these wheels may lock if an extra traction torque is required. On the other hand, the maximum value of the wheel longitudinal slip among the four wheels in case of SSV/ID is less than ±0.05. This will necessarily have an impact on the longitudinal dynamics of the SSV as shown in Fig. 12. On the other hand, the SSV/ID has a very reasonable longitudinal dynamics as well as a performance very close to that of the conventional two-wheel steering vehicle. On the contrary, SSV/SD shows unacceptable longitudinal dynamics. Finally,

Time (sec)

Time (sec) (b)

Figure 13 Comparison of direct yaw moment control (dry road, V =30 m/s): (a) open loop evaluation, and (b) closed loop evaluation.

Fig. 13 depicts the time history of the direct yaw moment for uncontrolled SSV and controlled SSV with the proposed two distribution techniques. As illustrated in this figure, unstable motion is predicted for the uncontrolled SSV, while smooth and acceptable direct yaw moment (without large fluctuation) is predicted in case of controlled SSV with independent distribution technique.

7. Summary and conclusions

In this paper, an independent distribution technique of wheel longitudinal forces taking into account the ability of the wheels to produce individual friction force based on full drive-by-wire system was proposed. A distribution method based on a simple strategy was presented for the sake of comparison. The control methods aimed at stabilizing the motion of skid steering vehicles at high speed and forcing the yaw rate response of the skid sheering vehicle to follow the yaw response of the conventional two-wheel steering vehicle. Simulations program of the comprehensive nonlinear dynamic model of the skid steering vehicle were used to investigate the effectiveness of the suggested controllers. Vehicle responses to both open and closed loop tests were studied. As a first phase of this research work, the simulations were executed assuming that the skid steering vehicle runs at constant speed.

The simulation results showed that both proposed distribution methods increased the stability limit of skid steering vehicles at relatively high speed. In addition, the methods allowed letting the yaw rate response of such vehicles follow the desired one. However, skid steering vehicles with independent distribution technique showed better and more acceptable longitudinal dynamics. Finally, it can be said that the proposed independent distribution technique, if implemented in off-road vehicles that install wheels with high performance, opens new insights to run skid steering vehicles at relatively high speeds. In addition, it allows generating stable motion and reasonable handling characteristics for common drivers.

Appendix A

See Table 1.

Table 1 Parameters of skid steering vehicle.

Parameter Value Unit

a 1.0 m

b 1.5 m

Cf, Cr 62,760 N/rad

Bpitch 12,950 N ms/rad

Broil 4402 N ms/rad

hs 0.52 m

Ix 500 kg m2

Iy 2300 kg m2

Iz 2200 kg m2

Ji 1.2 kg m2

Kpitch 140,000 N m/rad

Kroll 88,327 N m/rad

R 0.3 m

t 1.46 m

W 13,500 N

Ws 12,300 N

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