- Email: [email protected]
or brake forces distribution
Alexandria Engineering Journal (2015) xxx, xxx–xxx
H O S T E D BY
Alexandria University
Alexandria Engineering Journal www.elsevier.com/locate/aej www.sciencedirect.com
ORIGINAL ARTICLE
Stabilization of car-caravan combination using independent steer and drive/or brake forces distribution Ossama Mokhiamar
*,1
Mechanical Engineering Department, Faculty of Engineering, Alexandria University, El-Chatby, Alexandria 21544, Egypt Received 28 August 2014; revised 19 April 2015; accepted 11 May 2015
KEYWORDS Vehicle dynamics; Nonlinear tire characteristics; Optimum control; Sliding control
Abstract Once a combined vehicle becomes unstable, it is very difficult for a driver to stabilize it especially under severe driving conditions, such as turning with braking. This is mainly due to the effect of the towed vehicle on the towing vehicle through the hitch jackknifing. This effect makes the handling characteristics of a car-caravan combination different from those of a single vehicle. Therefore, this paper proposes a control design concept for an optimum distribution of longitudinal and lateral forces of the four tires of a towing vehicle. The mean objectives of the control system were to stabilize the motion of an articulated vehicle utilizing the tires entire ability in both longitudinal and lateral directions as well as to make the handling characteristics of an articulated vehicle similar to those of a single one. The sliding control law based on vehicle planar equations of motion is used to derive the control laws. The proposed control system is evaluated under severe driving conditions and compared with the results of integrated control systems. The robustness of the articulated vehicle motion with the proposed control against the coefficient of friction variation is discussed. ª 2015 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 three decades, various chassis control methods have been developed for improvement of the vehicle stability, * Now on leave to Beirut Arab University, Faculty of Engineering, Mechanical Engineering Department, P.O. Box 11-5020 Reyad El Solh, Beirut 1107-2809, Lebanon. E-mail addresses: [email protected], usamam@yahoo. com. 1 Tel.: +20 1220473850. Peer review under responsibility of Faculty of Engineering, Alexandria University.
handling characteristics and ride comfort. For example fourwheel steering system (4WS), direct yaw moment (DYC), anti-lock braking system (ABS), active front steering system (AFS), and active suspension have been developed and implemented on some passenger cars in the market. More recently, significant researches have been reported on integration of two or more controllers. Direct yaw moment and active steering control have been frequently employed in order to influence vehicle handling characteristics. Direct yaw moment control system relies on the remaining margin of tire longitudinal force. On the other hand, active steering system relies on remaining margin of tire lateral force. In active steering control technique an appropriate wheel sideslip angle is assigned to
http://dx.doi.org/10.1016/j.aej.2015.05.006 1110-0168 ª 2015 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/). Please cite this article in press as: O. Mokhiamar, Stabilization of car-caravan combination using independent steer and drive/or brake forces distribution, Alexandria Eng. J. (2015), http://dx.doi.org/10.1016/j.aej.2015.05.006
2
O. Mokhiamar
Nomenclature ax ax ay Ci Iz Ki Kf Kr l lf;r Lx Ly M Mz m n r s t
longitudinal acceleration traction/braking command from the driver lateral acceleration weighting coefficient yaw moment of inertia cornering stiffness of tire number i cornering stiffness of front tire cornering stiffness of rear tire wheel base distance between mass center and axle hitch point longitudinal force hitch point lateral force total yaw moment direct yaw moment control vehicle mass gear ratio yaw rate Laplace transform vehicle tread
wheel to develop the needed lateral force to follow the control law. Nowadays, x-by-wire technology in the automotive industry replaces the traditional mechanical and hydraulic control systems with electronic control systems using electromechanical actuators and human–machine interfaces such as pedal and steering feel emulators. Hence, the traditional components such as the steering column, intermediate shafts, pumps, hoses, fluids, belts, coolers and brake boosters and master cylinders are eliminated from the vehicle. Therefore, many active steering control and traction/braking control studies have emerged based on the new subject of x-by-wire system. The author [1] analyzed the effect of optimum distribution of tire lateral and longitudinal forces on small passenger car performance. The results were compared to those obtained in the case without control and the case with combined control type direct yaw moment and rear and front active steering. Followed to this theoretical study, the author [2] has conducted experimental validation of the effect of the proposed system on the performance of small passenger car using driving simulator. Both theoretical and experimental results show that both controllers can be used effectively in situations close to the limit. However, in more severe driving situations, the combined control type direct yaw moment and rear and front active steering failed to achieve the desired aim while the optimum tire force distribution control successively achieved the desired responses. At the mean time, the author [3], investigated the effectiveness of weighting coefficients adaptation in the proposed simultaneous optimum distribution of tire lateral and longitudinal forces. Three different cases of weighting coefficients adaptation were considered. The computational results of this study showed that weighting coefficients adaptation can be used as design parameters for the designer to compromise between vehicle stability and responsiveness. Most of these systems are available for single passenger cars, but not for heavier or complex vehicles, such as an
V X Xi Y Yf;r Yi Zf;r Zi b be d dw df;r di l
vehicle forward speed total longitudinal force required longitudinal force at tire number i total lateral force required lateral force at front/rear axle required lateral force at tire number i estimated vertical load estimated vertical load of tire number i vehicle slip angle estimated slip angle steering wheel angle wheel steering angle active wheel steering angle active steering angle of tire number i friction coefficient
Subscripts f front r rear
articulated heavy-duty vehicle, or lighter vehicle configuration (e.g. car-caravan combination). However, when driving a multi-unit vehicle, the driver does not have enough information on the behavior of the rear unit(s), thus his/her action (steering, braking, acceleration) mainly depends on the actual state of the towing vehicle. From the forgoing, the stability problems of light and heavy articulated vehicles are in the center of interest of the vehicle dynamicists. In [4,5] the system was designed to produce a stability torque acting on one unit of the vehicle configuration. In [6] a new concept called Active Unilateral Brake Control (AUBC) is applied for a car-caravan combination. The AUBC is applied for the towed vehicle part showing considerable improvement in the lateral stability. The effect of active rear wheel steering on the articulated vehicle stability is investigated in [5] while the effect of the all-wheel steering is investigated in [7]. It was concluded that these system can improve the vehicle stability. The author [8] examined the effect of direct yaw moment control system on the handling characteristics of a car-caravan combination. The yaw rate response of the two-degree of freedom vehicle motion (bicycle model) is chosen as a model response for the model following control. A computer simulation of a closed loop driver-vehicle system subjected to evasive lane change with braking was carried out in order to prove the effect of the control. It is proved that the effect of DYC is reasonable to prevent the combined vehicle from falling into unstable motion due to nonlinear tire characteristics as well as the combination of the car and the caravan. The work done in [8] has been extended to examine different models following types of yaw moment control strategy for improving handling safety of a car-caravan combination [9]. In these systems direct yaw moment control is generated by intentional distribution of tire longitudinal forces, where axle wheels bear the same amount of traction/braking force. The results show that the influence of the
Please cite this article in press as: O. Mokhiamar, Stabilization of car-caravan combination using independent steer and drive/or brake forces distribution, Alexandria Eng. J. (2015), http://dx.doi.org/10.1016/j.aej.2015.05.006
Stabilization of car-caravan combination
3
direct yaw moment control on car-caravan combination stability is significantly apparent using both model responses. However, the side-slip control type of DYC is more effective to compensate loss of stability due to non-linear tire characteristics as well as the combination of the car and the caravan. The author [10] proposed an integrated control type of direct yaw moment plus active rear wheel steering plus active front wheel steering (DYC + RWS + FWS) aiming at utilizing overall tire ability to maximize both stability limit and responsiveness of a car-caravan combination. The total lateral force and the total yaw moment were introduced using model following control. DYC + RWS + FWS control system is an intelligent and sophisticated control system. In this system, tire longitudinal force is calculated directly from direct yaw moment while tire lateral force is calculated based on simple force sharing tire load as well explained later in Section 3. However, it is not easy to determine how much longitudinal and lateral force is required for each tire in order to obtain the target lateral force and yaw moment. In another words, how to tune the entire set of longitudinal and lateral forces required to be generated at the four tires to achieve the ideal performance of a combined vehicle is still under investigation. So, the purpose of this paper was to present an optimization technique to find out how the tires should share longitudinal and lateral forces in order to achieve the optimum performance of a tire under the assumption that all the four wheels can be individually steered and driven/braked using a complete steer, brake and drive by wire system (i.e. to achieve vehicle optimal performance, it is necessary to control each tire according to its capacity). It is essential to know the forces acting at the hitch point to introduce the control laws. It is also essential to know the side slip angle for the control. Under the assumption that the force acting at the hitch point can be measured, the model observer is used for the side-slip angle estimation.
TB ¼
IV 1 m 2llr Kr 1 lf V2 2l lr Kr
Tr ¼
mlf V 2lKr
GR ¼
1 V 1 þ AV2 l
GB ¼
A¼
lf lr Kr
V 2 lr 1 þ AV2 l
1m 2l
m lr K r lf K f Kf Kr 2l2
For the model following control, a sliding control theory [13] is used to derive the control laws of total lateral force (YÞ and total yaw moment (MÞ required for the controlled vehicle to follow the model responses. Following [14], the total lateral force and the total yaw moment control laws can be obtained as follows: c2 sd þ c3 d c4 be ð3Þ Y ¼ mV þ c1 d þ r s2 þ c5 s þ c6 M ¼ Iz
c8 sd þ c9 d c10 r þ c d 7 s2 þ c11 s þ c12
ð4Þ
From the above equations, it is clear that estimation of the vehicle side slip angle is essential to use in the control. Abe et al. [15] have proposed the estimation of vehicle side-slip angle by a model observer not only to be used in the control but also used to determine the control variable, active wheel steering angle. 3. DYC + RWS + FWS combined control
2. Control law of total lateral force and total yaw moment In order to compensate for loss of stability due to nonlinear tire characteristics and to keep the articulated vehicle handling characteristics similar to those of the single vehicle, a model following control to follow the response of a linear two degree of freedom vehicle plane model with constant speed to front wheel steer, well known as bicycle model, is proposed. These responses represent the ideal responses of vehicle state variables, side-slip angle and yaw rate, where linear tire model involving cornering stiffness and tire slip angle only is considered. Then the model responses of side-slip angle and yaw rate described as follows [11,12]. b 1 þ TB s ðsÞ ¼ G B d 1 þ QP s þ P1 s2
ð1Þ
r 1 þ Tr s ðsÞ ¼ G R d 1 þ QP s þ P1 s2
ð2Þ
where
Referring to Fig. 1, the required yaw moment control to be generated by transversally distributed tire longitudinal forces in order to follow the yaw rate response (DYC) is calculated using Eq. (4) as follows: Mz ¼ M Yf lf þ Yr lr þ Ly d
ð5Þ
where Yf and Yr are the estimated lateral forces produced by front and rear axles using the simple tire model [15]. In the simple tire model, tire lateral force is described by a second-order polynomial of tire slip angle with initial slope equal to tire cornering stiffness and peak value equal to lZ while considering the effect of its longitudinal force by the concept of friction-circle, see Appendix A. The total lateral force required from rear and front tires to follow the side-slip angle model response, Y, has been split between the front and rear tires, and the split ratio is directly proportional to the estimated vertical load Yf ¼ Y
Zf Zf þ Zr
ð6Þ
Zr Zf þ Zr
ð7Þ
P¼
4l2 Kf Kr 1 þ AV2 mIz V2
Yr ¼ Y
Q¼
i 2 h 2 m lf Kf þ l2r Kr þ Iz Kf þ Kr mVIz
Once the lateral force required from the tire Yf;r is known, the tire side-slip angle can be calculated by inverse use of the
Please cite this article in press as: O. Mokhiamar, Stabilization of car-caravan combination using independent steer and drive/or brake forces distribution, Alexandria Eng. J. (2015), http://dx.doi.org/10.1016/j.aej.2015.05.006
4
O. Mokhiamar y Y1 Y3 X1
X3 δ3
δ1
M
3 Lx
V r
t
1
β
Y4
x Y2
X4
Ly
X2
δ4
δ2
4
2 lf
lr d
Figure 1
Tractor part of the articulated vehicle free body diagram in the x–y plane.
simple tire model. Consequently, the front and/or rear active steering angle, df;r , can be determined, see Appendix A. The main idea of the above control method is shown in Fig. 2(a). 4. The proposed optimum tire force distribution method
The tires’ forces (X14 and Y14 Þ have to satisfy the following constraints under the assumption that all the four wheels can be individually steered and driven or braked. The sum of the generated lateral forces on all four wheels should be equal to the required total lateral force to follow the side-slip angle model response. Y1 þ Y2 þ Y3 þ Y4 þ Ly ¼ Y
In the above integrated control method the required total lateral force has been split between front and rear axles based upon simple force sharing tire load. So far, direct yaw moment control (DYC) is generated by intentional distribution of tire longitudinal forces, where axle wheels bear the same amount of traction/braking force. Therefore, the objective of the proposed optimum tire force distribution method was to determine how much force should be generated at each tire to obtain the target total lateral force and yaw moment required to follow the model responses as well as to meet with the driver’s traction/braking command. (i.e. to find X14 and Y14 as shown in Fig. 1). The inputs to the optimization process are the driver’s commands, while the outputs are longitudinal and lateral forces on all four wheels, X14 and Y14 . 4.1. Cost function There are many ways to choose the cost function. However, how to optimize tire usage and find out the optimum tire force distribution is not obvious. From the concept of friction circle, the resultant lateral force, or traction/braking force, or a combination of the two, acting on a tire will be determined by the coefficient of friction l times its vertical load. From the foregoing concept and to simplify the optimization problem mathematically as well as to obtain a linear equation system, a weighted sum of the absolute normalized forces produced at the tires is chosen as the cost function J¼
4 4 X X X2 þ Y2 Ci l2i ¼ Ci i 2 i Zi i¼1 i¼1
ð8Þ
ð9Þ
The sum of the generated longitudinal forces on all four wheels should be equal to the required total longitudinal force to meet driver’s traction and/or braking command. X1 þ X2 þ X3 þ X4 þ Lx ¼ X
ð10Þ
where X ¼ ax m ax is due to foot-brake pressure and/or accelerator pedal pressure command from the driver and assumed to be measured. Finally, the sum of the generated yaw moment by tire longitudinal and lateral forces should be equal to the required total yaw moment to follow the yaw rate model response. t ðX2 X1 þ X4 X3 Þ þ lf ðY1 þ Y2 Þ lr ðY3 þ Y4 Þ dLy ¼ M 2
ð11Þ
This optimization problem has eight variables and three equality constraints. Hence the equality constraints can be used to eliminate any three of the eight variables as follows: Y4 ¼ Y Y1 Y2 Y3 Ly X3 ¼
X4 ¼
1 2lf 2lr Y Y1 Y2 L y 2X1 þ ðY1 þ Y2 Þ 2 t t 2 2d þ X M Lx Ly t t 1 2lf 2lr Y Y1 Y2 L y 2X2 ðY1 þ Y2 Þþ t t 2 2 2d þ X þ M Lx þ Ly t t
ð12Þ
ð13Þ
ð14Þ
Please cite this article in press as: O. Mokhiamar, Stabilization of car-caravan combination using independent steer and drive/or brake forces distribution, Alexandria Eng. J. (2015), http://dx.doi.org/10.1016/j.aej.2015.05.006
Stabilization of car-caravan combination
5
δ w /n V
ax
V
Ly
V
δf Front Tire Model
Y
Yf
V
ax
1 mV
+
+
+
V
+ -
1 s
βe
Calculation of Y and M
M
Calculation of Control Variables
Real Vehicle
MZ
β r
δr
Yr
Rear Tire Model
(a) δw
1/n
ax
estimation Z i of tires vertical loads
δ
V
V
front tire model Y +Y f1 f2
V sliding control to calculate estimation of side-slip angle
V
optimum distribution technique M
rear tire model
Xi
Y
Y and M
βe
Yr3+Yr4 ay
β
V
δ1 Yi
inverse use of tire model
actual vehicle
δ2 δ3 δ4
r
Ly
(b) Figure 2 Block diagram of the control systems: (a) DYC + RWS + FWS combined control and (b) independent steer and drive/brake tire forces distribution.
When the above expressions are substituted into the original objective function, Eq. (8), this gives a new objective function involving only five independent variables (X1 ; X2 ; Y1 ; Y2 and Y3 Þ. The new objective function is not
subjected to any constraints, and hence its optimum can be found using the unconstrained optimization technique, multivariable optimization with no constraints, [16]. The necessary conditions for the minimum of J give
Please cite this article in press as: O. Mokhiamar, Stabilization of car-caravan combination using independent steer and drive/or brake forces distribution, Alexandria Eng. J. (2015), http://dx.doi.org/10.1016/j.aej.2015.05.006
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O. Mokhiamar
@J C1 C3 C 3 lf þ lr C 3 lf þ lr ¼ þ X Y Y2 1 1 @X1 t t Z21 Z23 Z23 Z23 C3 Xt 2M Ylr lr d Lx Ly ¼0 2 þ 2t t t 2 Z3 @J ¼ @X2
C2 C4 C 4 lf þ lr C 4 lf þ lr X Y Y2 þ þ þ 2 1 t t Z22 Z24 Z24 Z24 C4 Xt þ 2M Ylr d lr Lx Ly ¼0 2 þ þ t t 2 2t Z4
ð15Þ
5. Computer simulation results
ð16Þ
@J C 3 lf þ lr C4 lf þ lr X1 þ 2 2 X2 ¼ 2 2 t t @Y1 Z Z4 " 3 2 2 # C1 C 3 lf þ lr C4 C4 lf þ lr Y1 þ 2 2 þ2 2 þ2 2 þ2 2 t t Z1 Z3 Z4 Z4 " # 2 2 C 3 lf þ lr C 4 lf þ lr C4 þ 2 2 þ2 2 þ 2 2 Y2 t t Z3 Z4 Z4 C4 C 3 lf þ lr Ylr Xt 2M d lr Ly þ Lx þ 2 2 Y3 2 2 þ t t t 2t Z4 Z 3 C4 Ylr lf þ lr lf þ lr 2M 2 2 þ Xþ t t t t Z4 lf lr d lr lf þ lr þ Ly þ 2 Y Ly Lx ¼ 0 ð17Þ t t t @J C 3 lf þ lr C4 lf þ lr X1 þ 2 2 X2 ¼ 2 2 t t @Y2 Z Z4 " 3 2 2 # C 3 lf þ lr C4 C4 lf þ lr Y1 þ 2 2 þ2 2 þ2 2 t t Z3 Z4 Z4 " # 2 2 C 3 lf þ lr C 4 lf þ lr C2 C4 þ 2 2 þ2 2 þ 2 2 þ 2 2 Y2 t t Z3 Z4 Z2 Z4 C4 C 3 lf þ lr Ylr Xt 2M d lr Ly þ Lx þ 2 2 Y3 2 2 þ 2t t t t Z4 Z 3 C4 Ylr lf þ lr lf þ lr 2M 2 2 þ Xþ t t t t Z4 lf lr d lr lf þ lr Ly þ 2 Y Ly Lx ¼ 0 ð18Þ þ t t t @J C4 C4 C3 C4 C4 Y3 2 Y Ly ¼ 0 ¼ 2 Y1 þ 2 Y2 þ þ @Y3 Z4 Z4 Z23 Z24 Z4
at each wheel to follow the model responses and to meet the driver’s traction and/or braking command. Fig. 2(b) is the block diagram, which summarizes the main idea of the forgoing method.
ð19Þ
The above equations are linear and can be more conveniently expressed using matrix form, which is suitable for linear system analysis 2 32 3 2 3 a1 0 a2 a3 0 X1 b1 6 76 7 6 7 0 a a a 0 X 6 76 2 7 6 b2 7 4 5 6 6 76 7 6 7 6 a7 a8 a9 a10 a11 76 Y1 7 ¼ 6 b3 7 6 76 7 6 7 6 76 7 6 7 4 a12 a13 a14 a15 a16 54 Y2 5 4 b4 5 0 0 a17 a18 a19 Y3 b5 where ai and bj (i = 1–19 and j = 1–5) are expressed by C14 ; Z14 ; t; lf ; lr ; d; X; Y; Lx ; Ly , and M. This linear system is solved at each time step to find the optimum tire force distribution (X14 and Y14 Þ. Once the lateral and longitudinal tire force are obtained the active steering angle can be determined (using inverse use of simple tire model) as well as the traction and/or braking torque required
In order to analyze the effect of the independent steer and drive/or brake forces distribution method on the car-caravan handling performance, computer simulations of the articulated vehicle handling response subjected to evasive lane change with braking were carried out. These responses are compared to those obtained in the case without control and the case with combined control type DYC + RWS + FWS. The simulation model is created using Matlab/Simulink and composed of a 15-degree-of-freedom non-linear model, nine degrees for the vehicle units while the remaining six degrees for the wheels. The towing vehicle motion has six independent degrees of freedom; vertical motion, lateral motion, longitudinal motion, rolling motion, pitching motion and yawing motion while the towed vehicle has three independent degrees of freedom; rolling motion, pitching motion and yawing motion. The simulation concerns small passenger car (1350 kg) pulling trailer (630 kg) at different running conditions. The tire forces were obtained by integrating the distributed tire deformations in the contact-patch in lateral and longitudinal directions respectively using a brush type tire model. The tire forces were obtained by integrating the distributed tire deformations in the contact-patch in lateral and longitudinal directions, respectively, using a brush-type tire model. In the tire model, the coefficient of friction was treated as a function of the vertical load as well as the slip velocity. All detailed information about this tire model can be found in [11]. In the simulated cases, the vehicle runs at constant speed for 0.5 s and then braking is conducted. The deceleration is increased linearly from zero to its maximum value over a period of 0.5 s (driver brake pedal operation, ax ). Near the limit region, the articulated vehicle shows oscillatory handling response in the case without control as shown in Fig. 3, (a) for the tractor part while (b) for the trailer part. On the other hand, the controllers presented here were both found to produce desirable effects on the handling dynamics of the car-caravan combination, Figs. 4 and 5. In the figures, the dashed lines represent the target response of yaw rate and side-slip angle. 5.1. Robustness against road friction variation In order to investigate the effects of the proposed independent steer and drive/or brake force distribution method on articulated vehicle handling response on a low friction road, computer simulations of vehicle responses on a road of which the nominal value of the friction coefficient is equal to 0.3 are presented in Figs. 6–8. However, the friction coefficient used in the simple tire model used in the control is fixed as 1.0 (dry road). The computational results indicate that it is essential to know precisely the friction coefficient between the tire and road surface for use in the simple tire model in the case of the DYC + RWS + FWS combined control. On the contrary, in the case of optimum tire forces distribution control, the friction coefficient does not need to be known
Please cite this article in press as: O. Mokhiamar, Stabilization of car-caravan combination using independent steer and drive/or brake forces distribution, Alexandria Eng. J. (2015), http://dx.doi.org/10.1016/j.aej.2015.05.006
2
3
-5 5
4
-15
-0.8
-2.5
βt
-30 0
1
2
Time (sec)
-0.8
r
0
-30 0
2.5 0
-15
-2.5
β
1
2
3
-5 5
4
0.4 0 -0.4
30
rt
15 ψ
0 -15
-0.8
0 -2.5
βt
-30 0
-0.8
15
(a) for the tractor part V = 100 km/h ax = -0.2 g
r
0
0
β
-15 -30 0
2.5
-2.5
1
2
3
-5 5
4
0.8
1
2
0.4 0 -0.4 -0.8
-3 -6
(b) for the trailer part V = 100 km/h ax = -0.2 g
rt
15 ψ
0
-15
0 -2.5
ayt
βt
-30 0
2.5
1
2
3
-5 5
4
6 3 0 -3 -6
ay
0
0
-15 -30 0
2.5
(a) for the tractor part V = 70 km/h ax = -0.1 g
β
1
2
3
Time (sec)
Figure 6
4
-2.5 -5 5
0.8 0.4 0 -0.4 -0.8
30 Yaw rate (deg/sec)
15
Side-slip angle (deg.)
5 r
target response. This is simply due to the saturation property of the tire which limits the lateral and longitudinal forces generated from the tire especially at such severe driving condition
Lateral acceleration (g)
-0.8
30
Yaw rate (deg/sec)
Lateral acceleration (g)
-0.4
0
Car-caravan handling response with independent steer and drive/brake tire forces distribution, on dry surface.
0.8
0
-5 5
3
Time (sec)
precisely. This proves the robustness of optimum distribution technique against road friction variations even though the controlled responses do not seem to follow or come close to the
0.4
4
5
Time (sec)
Figure 5
3
30
Yaw rate (deg/sec)
-0.4
5
ay
Lateral acceleration (g)
0
2.5
Time (sec)
Side-slip angle (deg.)
0.4
(b) for the trailer part V = 100 km/h ax = -0.2 g
6
Car-caravan handling response with DYC + RWS + FWS combined control, on dry surface.
30
Yaw rate (deg/sec)
Lateral acceleration (g)
0.8
-6
5
ayt
Time (sec)
Figure 4
-5 5
4
5 ψ
(b) for the trailer part V = 70 km/h ax = -0.1 g
ayt
15
2.5
rt
0
0
βt
-15 -30 0
-2.5
1
2
3
4
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6 3 0 -3 -6
Articulation angle (deg.)
-0.4
15
0.8
Yaw rate (deg/sec)
0
5 (a) for the tractor part V = 100 km/h ax = -0.2 g
Lateral acceleration (g)
0.4
-3
Car-caravan handling response without control, on dry surface.
30 ay
3
0
Time (sec)
Side-slip angle (deg.)
0.8
Yaw rate (deg/sec)
Lateral acceleration (g)
Figure 3
0
Articulation angle (deg.)
1
-0.4
0
3
Articulation angle (deg.)
-30 0
-2.5
β
0
(b) for the trailer part V = 100 km/h ax = -0.2 g
angle (deg.)
-15
0
2.5 rt
Side-slip
0
ψ
15
6
Articulation angle (deg.)
(a) for the tractor part V = 100 km/h ax = -0.2 g
0.4
5
ayt
Side-slip angle (deg.)
2.5
r
30
Side-slip angle (deg.)
-0.8
15
0.8
Side-slip angle (deg.)
-0.4
5 ay
Yaw rate (deg/sec)
0
30
Lateral acceleration (g)
0.4
7
Side-slip angle (deg.)
0.8 Yaw rate (deg/sec)
Lateral acceleration (g)
Stabilization of car-caravan combination
Time (sec)
Car-caravan handling response without control, on slippery surface.
Please cite this article in press as: O. Mokhiamar, Stabilization of car-caravan combination using independent steer and drive/or brake forces distribution, Alexandria Eng. J. (2015), http://dx.doi.org/10.1016/j.aej.2015.05.006
β
0
-15
-2.5
-30 0
1
2
3
-5 5
4
0 -0.4 -0.8
15
ayt
rt
0
βt
-15
-2.5
-30 0
1
2
ay
2.5
r
0
0
β
-15
-2.5
-30 0
1
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3
-5 5
4
0.8 0.4 0 -0.4
30
Yaw rate (deg/sec)
15
Lateral acceleration (g)
-0.8
5 (a) for the tractor part V = 70 km/h ax = -0.1 g
Side-slip angle (deg.)
-0.4
4
-3 -6
-0.8
15
5
βt
rt
ψ
2.5
ayt
0 -15 -30 0
0 -2.5
(b) for the trailer part V = 70 km/h ax = -0.1 g
1
2
3
4
-5 5
6 3 0 -3 -6
Time (sec)
Time (sec)
Car-caravan handling response with independent steer and drive/brake tire forces distribution, on slippery surface.
Figure 8
due to large tire slip in both lateral and longitudinal directions, Fig. 5. On the other hand, the difference between the controlled responses and the target ones became more obvious
in case of slippery surface, Fig. 8, because the same previous reason as well as because the simulation was conducted while the combined vehicle was running on slippery surface and the
850
850 V=70 km/h, ax=-0.1g
V=70 km/h, ax=-0.1g
tire 1
tire 1
tire 2
425
tire 2
tire 3
Tire longitudinal force (N)
Tire longitudinal force (N)
-5 5
0
Time (sec)
30
Yaw rate (deg/sec)
Lateral acceleration (g)
0
3
3
Car-caravan handling response with DYC + RWS + FWS combined control, on slippery surface.
Figure 7
0.4
0
ψ
Time (sec)
0.8
2.5
6
Articulation angle (deg.)
ay
0
0.4
Side-slip angle (deg.)
2.5
5 (b) for the trailer part V = 70 km/h ax = -0.1 g
Articulation angle (deg.)
-0.8
r
30
Side-slip angle (deg.)
-0.4
15
0.8
Yaw rate (deg/sec)
0
5 (a) for the tractor part V = 70 km/h ax = -0.1 g
Side-slip angle (deg.)
0.4
30
Yaw rate (deg/sec)
0.8
Lateral acceleration (g)
O. Mokhiamar Lateral acceleration (g)
8
tire 4
2
0 3 4 -425
425
tire 3 tire 4
0 3 1 -425 4
1
2
-850
-850
0
1
2
3
4
5
0
1
2
3
Time (sec)
Time (sec)
(a)
(b)
4
5
Figure 9 Tires longitudinal forces time history, on slippery surface: (a) DYC + RWS + FWS combined control and (b) independent steer and drive/brake tire forces.
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Stabilization of car-caravan combination
9
1200
1200 1
V=70 km/h, ax=-0.1g
2
4
tire 1
V=70 km/h, ax=-0.1g
tire 1
3
tire 2
4
tire 2
600
tire 3
3
Tire lateral force (N)
Tire lateral force (N)
600
tire 4
2 1
0
-600
tire 3 tire 4
0
-600
-1200
-1200 0
1
2
3
4
5
0
1
2
3
Time (sec)
Time (sec)
(a)
(b)
4
5
Figure 10 Tires lateral forces time history, on slippery surface: (a) DYC + RWS + FWS combined control and (b) independent steer and drive/brake tire forces.
the longitudinal and lateral dynamics of the combined vehicle as shown before in Figs. 7 and 11. On the other hand, the proposed optimum control succeed to tune the entire set of longitudinal and lateral forces required to be generated at the four tires to achieve smooth and reasonable performance of the combined vehicle, Figs. 9(b) and 10 (b). As a result, this will affect longitudinal and lateral dynamics of the combined vehicle as shown in Figs. 8 and 11, where combined vehicle with the proposed control shows smooth handling dynamics.
0.5 V=70 km/h, ax=-0.1g
Longitudinal acceleration (m/s2)
driver operation DYC+RWS+FWS
0
optimum distribution
-0.5
6. Summary and conclusion -1
-1.5
0
1
2
3
4
5
Time (sec)
Figure 11 Longitudinal acceleration time history of the carcaravan combination, on slippery surface.
coefficient of friction between tire and ground in the control algorithm was fixed as on dry surface. Going deep in the dynamics of the car-caravan combination with the combined control type DYC + RWS + FWS and the proposed optimum control, Figs. 9–11 are presented. Fig. 9 represents wheel longitudinal force; Fig. 9(a) for DYC + RWS + FWS combined control while Fig. 9(b) is for the proposed control. From the figure, in case of DYC + RWS + FWS, it is noted that large traction force is generated at tire number 1 while large braking force is generated at tire number 2 in order to generate large direct yaw moment control, Mz , needed to stabilize the combined vehicle motion. Again due to saturation property of the tire, small lateral force generated at tires 1 and 2 as shown in Fig. 10(b). Consequently this will affect
The influence of independent steer and drive/or brake force distribution on the articulated vehicle handling characteristics is examined. Near the limit region, the two controllers stabilize the combined vehicle motion. However, the effect of the proposed optimum control is more obvious especially on the response of the trailer part. In more severe situation, articulated vehicle running on low friction coefficient road, the combined control-type DYC + RWS + FWS failed to achieve a desirable response. On the other hand, the proposed optimum control successively achieves smooth and reasonable responses.
Appendix A. Simple tire model sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ffi K2i 2 Xi 1 Yi ¼ Ki bi signðbi Þ b 4lZi i lZi i when: j bi j< 2lZ . Ki
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O. Mokhiamar
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ffi Xi Yi ¼ signðbi ÞlZi 1 lZi
[3]
2lZi . Ki
when: j bi jP Inverse use of the simple tire model i (a) when: j bi j< 2lZ . Ki In this case the tire side-slip angle can be calculated from the following equation: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b b2 4a c bi ¼ signðYi Þ 2a
where
[4]
[5]
[6]
[7]
K2 a¼ i 4lZi
[8]
b ¼ Ki jYi j c ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1
Xi lZi
2
i (b) when: j bi jP 2lZ .The following condition is used to Ki account for the saturation property of the tire, if the tire lateral force is greater than the saturated value in the tire model.
bi ¼ signðYi Þ
[9]
2lZi : Ki
[10]
[11] [12] [13] [14]
References [1] O. Mokhiamar, M. Abe, Simultaneous optimal distribution of lateral and longitudinal tire forces for the model following control, J. Dyn. Syst., Meas. Control, Trans. ASME 126 (4) (2004) 753–763. [2] O. Mokhiamar, M. Abe, An experimental validation of the effect of combined lateral force and yaw moment control to maximize vehicle stability limit as well as responsiveness for active vehicle handling control, in: Proc. of the 9th Mini
[15]
[16]
Conference on Vehicle Systems Dynamics, Identification and Anomalies, Budapest, Hungary, 2004, pp. 435–443. O. Mokhiamar, M. Abe, How the four wheels should share forces in an optimum cooperative chassis control, Control Eng. Pract. 14 (March) (2006) 295–304 (n. 2 SPEC. ISS). S. Kimbrough, W. VanMoorhem, A control strategy for stabilizing trailers via selective actuation of brakes, ASME Adv. Automot. Technol., DSC 44 (1992) 413–428. L. Palkovics, M. El-Gindy, L. Ilosvai, Examination of Different Control Strategies of Heavy-vehicle Performance, ASME Advanced Automotive Technologies, Lousiana, New Orleans, 1993, pp. 349–362. L. Palkovics, J. Bokor, Stabilization of a car-caravan combination using active unilateral brake control, in: Proc. of AVEC’94, Tsukuba, Japan, 1994, pp. 141–146. S. Chikamori, S. Kawasawa, Stability analysis of articulated vehicles with all-wheel steering, in: Proc. of AVEC’96, Aachen, Germany, 1996, pp. 395–407. O. Mokhiamar, M. Abe, Improvement of handling safety of carcaravan combination by direct yaw moment control, in: Proc. of AVEC’02, Hiroshima, Japan, 2002, pp. 57–62. O. Mokhiamar, M. Abe, Examination of different models following types of yaw moment control strategy for improving handling safety of a car-caravan combination, Proc. Inst. Mech. Eng., Part D: J. Automob. Eng. 217 (7) (2003) 561–571. O. Mokhiamar, Integrated control of lateral force and direct yaw moment to improve handling safety of car-caravan combination, in: Proc. of AVEC’08, Kobe, Japan, 2008, pp. 385–390. M. Abe, Vehicle Dynamics: Theory and Application, first ed., Elsevier Ltd, 2009. J.R. Ellis, Vehicles Dynamics, London Business Book LTD, London, 1969. E.J.-J. Slotine, W. Li, Applied Nonlinear Control, Prentice-Hall, Englewood Cliffs, New Jersey, 1991. O. Mokhiamar, M. Abe, Active wheel steering and direct yaw moment control combination to maximize stability as well as vehicle responsiveness during quick lane change for active vehicle handling safety, Proc. Inst. Mech. Eng., Part D: J. Automob. Eng. 216 (part D) (2002) 115–123. M. Abe, A. Kato, K. Suzuki, Y. Kano, Estimation of vehicle side-slip angle for DYC by using on-board-tire-model, in: Proc. of AVEC’98, Nagoya, Japan, 1998, pp. 437–442. S.R. Singiresu, Engineering Optimization, John Wily &Sons, Inc., New York, 1996.
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H O S T E D BY
Alexandria University
Alexandria Engineering Journal www.elsevier.com/locate/aej www.sciencedirect.com
ORIGINAL ARTICLE
Stabilization of car-caravan combination using independent steer and drive/or brake forces distribution Ossama Mokhiamar
*,1
Mechanical Engineering Department, Faculty of Engineering, Alexandria University, El-Chatby, Alexandria 21544, Egypt Received 28 August 2014; revised 19 April 2015; accepted 11 May 2015
KEYWORDS Vehicle dynamics; Nonlinear tire characteristics; Optimum control; Sliding control
Abstract Once a combined vehicle becomes unstable, it is very difficult for a driver to stabilize it especially under severe driving conditions, such as turning with braking. This is mainly due to the effect of the towed vehicle on the towing vehicle through the hitch jackknifing. This effect makes the handling characteristics of a car-caravan combination different from those of a single vehicle. Therefore, this paper proposes a control design concept for an optimum distribution of longitudinal and lateral forces of the four tires of a towing vehicle. The mean objectives of the control system were to stabilize the motion of an articulated vehicle utilizing the tires entire ability in both longitudinal and lateral directions as well as to make the handling characteristics of an articulated vehicle similar to those of a single one. The sliding control law based on vehicle planar equations of motion is used to derive the control laws. The proposed control system is evaluated under severe driving conditions and compared with the results of integrated control systems. The robustness of the articulated vehicle motion with the proposed control against the coefficient of friction variation is discussed. ª 2015 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 three decades, various chassis control methods have been developed for improvement of the vehicle stability, * Now on leave to Beirut Arab University, Faculty of Engineering, Mechanical Engineering Department, P.O. Box 11-5020 Reyad El Solh, Beirut 1107-2809, Lebanon. E-mail addresses: [email protected], usamam@yahoo. com. 1 Tel.: +20 1220473850. Peer review under responsibility of Faculty of Engineering, Alexandria University.
handling characteristics and ride comfort. For example fourwheel steering system (4WS), direct yaw moment (DYC), anti-lock braking system (ABS), active front steering system (AFS), and active suspension have been developed and implemented on some passenger cars in the market. More recently, significant researches have been reported on integration of two or more controllers. Direct yaw moment and active steering control have been frequently employed in order to influence vehicle handling characteristics. Direct yaw moment control system relies on the remaining margin of tire longitudinal force. On the other hand, active steering system relies on remaining margin of tire lateral force. In active steering control technique an appropriate wheel sideslip angle is assigned to
http://dx.doi.org/10.1016/j.aej.2015.05.006 1110-0168 ª 2015 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/). Please cite this article in press as: O. Mokhiamar, Stabilization of car-caravan combination using independent steer and drive/or brake forces distribution, Alexandria Eng. J. (2015), http://dx.doi.org/10.1016/j.aej.2015.05.006
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O. Mokhiamar
Nomenclature ax ax ay Ci Iz Ki Kf Kr l lf;r Lx Ly M Mz m n r s t
longitudinal acceleration traction/braking command from the driver lateral acceleration weighting coefficient yaw moment of inertia cornering stiffness of tire number i cornering stiffness of front tire cornering stiffness of rear tire wheel base distance between mass center and axle hitch point longitudinal force hitch point lateral force total yaw moment direct yaw moment control vehicle mass gear ratio yaw rate Laplace transform vehicle tread
wheel to develop the needed lateral force to follow the control law. Nowadays, x-by-wire technology in the automotive industry replaces the traditional mechanical and hydraulic control systems with electronic control systems using electromechanical actuators and human–machine interfaces such as pedal and steering feel emulators. Hence, the traditional components such as the steering column, intermediate shafts, pumps, hoses, fluids, belts, coolers and brake boosters and master cylinders are eliminated from the vehicle. Therefore, many active steering control and traction/braking control studies have emerged based on the new subject of x-by-wire system. The author [1] analyzed the effect of optimum distribution of tire lateral and longitudinal forces on small passenger car performance. The results were compared to those obtained in the case without control and the case with combined control type direct yaw moment and rear and front active steering. Followed to this theoretical study, the author [2] has conducted experimental validation of the effect of the proposed system on the performance of small passenger car using driving simulator. Both theoretical and experimental results show that both controllers can be used effectively in situations close to the limit. However, in more severe driving situations, the combined control type direct yaw moment and rear and front active steering failed to achieve the desired aim while the optimum tire force distribution control successively achieved the desired responses. At the mean time, the author [3], investigated the effectiveness of weighting coefficients adaptation in the proposed simultaneous optimum distribution of tire lateral and longitudinal forces. Three different cases of weighting coefficients adaptation were considered. The computational results of this study showed that weighting coefficients adaptation can be used as design parameters for the designer to compromise between vehicle stability and responsiveness. Most of these systems are available for single passenger cars, but not for heavier or complex vehicles, such as an
V X Xi Y Yf;r Yi Zf;r Zi b be d dw df;r di l
vehicle forward speed total longitudinal force required longitudinal force at tire number i total lateral force required lateral force at front/rear axle required lateral force at tire number i estimated vertical load estimated vertical load of tire number i vehicle slip angle estimated slip angle steering wheel angle wheel steering angle active wheel steering angle active steering angle of tire number i friction coefficient
Subscripts f front r rear
articulated heavy-duty vehicle, or lighter vehicle configuration (e.g. car-caravan combination). However, when driving a multi-unit vehicle, the driver does not have enough information on the behavior of the rear unit(s), thus his/her action (steering, braking, acceleration) mainly depends on the actual state of the towing vehicle. From the forgoing, the stability problems of light and heavy articulated vehicles are in the center of interest of the vehicle dynamicists. In [4,5] the system was designed to produce a stability torque acting on one unit of the vehicle configuration. In [6] a new concept called Active Unilateral Brake Control (AUBC) is applied for a car-caravan combination. The AUBC is applied for the towed vehicle part showing considerable improvement in the lateral stability. The effect of active rear wheel steering on the articulated vehicle stability is investigated in [5] while the effect of the all-wheel steering is investigated in [7]. It was concluded that these system can improve the vehicle stability. The author [8] examined the effect of direct yaw moment control system on the handling characteristics of a car-caravan combination. The yaw rate response of the two-degree of freedom vehicle motion (bicycle model) is chosen as a model response for the model following control. A computer simulation of a closed loop driver-vehicle system subjected to evasive lane change with braking was carried out in order to prove the effect of the control. It is proved that the effect of DYC is reasonable to prevent the combined vehicle from falling into unstable motion due to nonlinear tire characteristics as well as the combination of the car and the caravan. The work done in [8] has been extended to examine different models following types of yaw moment control strategy for improving handling safety of a car-caravan combination [9]. In these systems direct yaw moment control is generated by intentional distribution of tire longitudinal forces, where axle wheels bear the same amount of traction/braking force. The results show that the influence of the
Please cite this article in press as: O. Mokhiamar, Stabilization of car-caravan combination using independent steer and drive/or brake forces distribution, Alexandria Eng. J. (2015), http://dx.doi.org/10.1016/j.aej.2015.05.006
Stabilization of car-caravan combination
3
direct yaw moment control on car-caravan combination stability is significantly apparent using both model responses. However, the side-slip control type of DYC is more effective to compensate loss of stability due to non-linear tire characteristics as well as the combination of the car and the caravan. The author [10] proposed an integrated control type of direct yaw moment plus active rear wheel steering plus active front wheel steering (DYC + RWS + FWS) aiming at utilizing overall tire ability to maximize both stability limit and responsiveness of a car-caravan combination. The total lateral force and the total yaw moment were introduced using model following control. DYC + RWS + FWS control system is an intelligent and sophisticated control system. In this system, tire longitudinal force is calculated directly from direct yaw moment while tire lateral force is calculated based on simple force sharing tire load as well explained later in Section 3. However, it is not easy to determine how much longitudinal and lateral force is required for each tire in order to obtain the target lateral force and yaw moment. In another words, how to tune the entire set of longitudinal and lateral forces required to be generated at the four tires to achieve the ideal performance of a combined vehicle is still under investigation. So, the purpose of this paper was to present an optimization technique to find out how the tires should share longitudinal and lateral forces in order to achieve the optimum performance of a tire under the assumption that all the four wheels can be individually steered and driven/braked using a complete steer, brake and drive by wire system (i.e. to achieve vehicle optimal performance, it is necessary to control each tire according to its capacity). It is essential to know the forces acting at the hitch point to introduce the control laws. It is also essential to know the side slip angle for the control. Under the assumption that the force acting at the hitch point can be measured, the model observer is used for the side-slip angle estimation.
TB ¼
IV 1 m 2llr Kr 1 lf V2 2l lr Kr
Tr ¼
mlf V 2lKr
GR ¼
1 V 1 þ AV2 l
GB ¼
A¼
lf lr Kr
V 2 lr 1 þ AV2 l
1m 2l
m lr K r lf K f Kf Kr 2l2
For the model following control, a sliding control theory [13] is used to derive the control laws of total lateral force (YÞ and total yaw moment (MÞ required for the controlled vehicle to follow the model responses. Following [14], the total lateral force and the total yaw moment control laws can be obtained as follows: c2 sd þ c3 d c4 be ð3Þ Y ¼ mV þ c1 d þ r s2 þ c5 s þ c6 M ¼ Iz
c8 sd þ c9 d c10 r þ c d 7 s2 þ c11 s þ c12
ð4Þ
From the above equations, it is clear that estimation of the vehicle side slip angle is essential to use in the control. Abe et al. [15] have proposed the estimation of vehicle side-slip angle by a model observer not only to be used in the control but also used to determine the control variable, active wheel steering angle. 3. DYC + RWS + FWS combined control
2. Control law of total lateral force and total yaw moment In order to compensate for loss of stability due to nonlinear tire characteristics and to keep the articulated vehicle handling characteristics similar to those of the single vehicle, a model following control to follow the response of a linear two degree of freedom vehicle plane model with constant speed to front wheel steer, well known as bicycle model, is proposed. These responses represent the ideal responses of vehicle state variables, side-slip angle and yaw rate, where linear tire model involving cornering stiffness and tire slip angle only is considered. Then the model responses of side-slip angle and yaw rate described as follows [11,12]. b 1 þ TB s ðsÞ ¼ G B d 1 þ QP s þ P1 s2
ð1Þ
r 1 þ Tr s ðsÞ ¼ G R d 1 þ QP s þ P1 s2
ð2Þ
where
Referring to Fig. 1, the required yaw moment control to be generated by transversally distributed tire longitudinal forces in order to follow the yaw rate response (DYC) is calculated using Eq. (4) as follows: Mz ¼ M Yf lf þ Yr lr þ Ly d
ð5Þ
where Yf and Yr are the estimated lateral forces produced by front and rear axles using the simple tire model [15]. In the simple tire model, tire lateral force is described by a second-order polynomial of tire slip angle with initial slope equal to tire cornering stiffness and peak value equal to lZ while considering the effect of its longitudinal force by the concept of friction-circle, see Appendix A. The total lateral force required from rear and front tires to follow the side-slip angle model response, Y, has been split between the front and rear tires, and the split ratio is directly proportional to the estimated vertical load Yf ¼ Y
Zf Zf þ Zr
ð6Þ
Zr Zf þ Zr
ð7Þ
P¼
4l2 Kf Kr 1 þ AV2 mIz V2
Yr ¼ Y
Q¼
i 2 h 2 m lf Kf þ l2r Kr þ Iz Kf þ Kr mVIz
Once the lateral force required from the tire Yf;r is known, the tire side-slip angle can be calculated by inverse use of the
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4
O. Mokhiamar y Y1 Y3 X1
X3 δ3
δ1
M
3 Lx
V r
t
1
β
Y4
x Y2
X4
Ly
X2
δ4
δ2
4
2 lf
lr d
Figure 1
Tractor part of the articulated vehicle free body diagram in the x–y plane.
simple tire model. Consequently, the front and/or rear active steering angle, df;r , can be determined, see Appendix A. The main idea of the above control method is shown in Fig. 2(a). 4. The proposed optimum tire force distribution method
The tires’ forces (X14 and Y14 Þ have to satisfy the following constraints under the assumption that all the four wheels can be individually steered and driven or braked. The sum of the generated lateral forces on all four wheels should be equal to the required total lateral force to follow the side-slip angle model response. Y1 þ Y2 þ Y3 þ Y4 þ Ly ¼ Y
In the above integrated control method the required total lateral force has been split between front and rear axles based upon simple force sharing tire load. So far, direct yaw moment control (DYC) is generated by intentional distribution of tire longitudinal forces, where axle wheels bear the same amount of traction/braking force. Therefore, the objective of the proposed optimum tire force distribution method was to determine how much force should be generated at each tire to obtain the target total lateral force and yaw moment required to follow the model responses as well as to meet with the driver’s traction/braking command. (i.e. to find X14 and Y14 as shown in Fig. 1). The inputs to the optimization process are the driver’s commands, while the outputs are longitudinal and lateral forces on all four wheels, X14 and Y14 . 4.1. Cost function There are many ways to choose the cost function. However, how to optimize tire usage and find out the optimum tire force distribution is not obvious. From the concept of friction circle, the resultant lateral force, or traction/braking force, or a combination of the two, acting on a tire will be determined by the coefficient of friction l times its vertical load. From the foregoing concept and to simplify the optimization problem mathematically as well as to obtain a linear equation system, a weighted sum of the absolute normalized forces produced at the tires is chosen as the cost function J¼
4 4 X X X2 þ Y2 Ci l2i ¼ Ci i 2 i Zi i¼1 i¼1
ð8Þ
ð9Þ
The sum of the generated longitudinal forces on all four wheels should be equal to the required total longitudinal force to meet driver’s traction and/or braking command. X1 þ X2 þ X3 þ X4 þ Lx ¼ X
ð10Þ
where X ¼ ax m ax is due to foot-brake pressure and/or accelerator pedal pressure command from the driver and assumed to be measured. Finally, the sum of the generated yaw moment by tire longitudinal and lateral forces should be equal to the required total yaw moment to follow the yaw rate model response. t ðX2 X1 þ X4 X3 Þ þ lf ðY1 þ Y2 Þ lr ðY3 þ Y4 Þ dLy ¼ M 2
ð11Þ
This optimization problem has eight variables and three equality constraints. Hence the equality constraints can be used to eliminate any three of the eight variables as follows: Y4 ¼ Y Y1 Y2 Y3 Ly X3 ¼
X4 ¼
1 2lf 2lr Y Y1 Y2 L y 2X1 þ ðY1 þ Y2 Þ 2 t t 2 2d þ X M Lx Ly t t 1 2lf 2lr Y Y1 Y2 L y 2X2 ðY1 þ Y2 Þþ t t 2 2 2d þ X þ M Lx þ Ly t t
ð12Þ
ð13Þ
ð14Þ
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Stabilization of car-caravan combination
5
δ w /n V
ax
V
Ly
V
δf Front Tire Model
Y
Yf
V
ax
1 mV
+
+
+
V
+ -
1 s
βe
Calculation of Y and M
M
Calculation of Control Variables
Real Vehicle
MZ
β r
δr
Yr
Rear Tire Model
(a) δw
1/n
ax
estimation Z i of tires vertical loads
δ
V
V
front tire model Y +Y f1 f2
V sliding control to calculate estimation of side-slip angle
V
optimum distribution technique M
rear tire model
Xi
Y
Y and M
βe
Yr3+Yr4 ay
β
V
δ1 Yi
inverse use of tire model
actual vehicle
δ2 δ3 δ4
r
Ly
(b) Figure 2 Block diagram of the control systems: (a) DYC + RWS + FWS combined control and (b) independent steer and drive/brake tire forces distribution.
When the above expressions are substituted into the original objective function, Eq. (8), this gives a new objective function involving only five independent variables (X1 ; X2 ; Y1 ; Y2 and Y3 Þ. The new objective function is not
subjected to any constraints, and hence its optimum can be found using the unconstrained optimization technique, multivariable optimization with no constraints, [16]. The necessary conditions for the minimum of J give
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@J C1 C3 C 3 lf þ lr C 3 lf þ lr ¼ þ X Y Y2 1 1 @X1 t t Z21 Z23 Z23 Z23 C3 Xt 2M Ylr lr d Lx Ly ¼0 2 þ 2t t t 2 Z3 @J ¼ @X2
C2 C4 C 4 lf þ lr C 4 lf þ lr X Y Y2 þ þ þ 2 1 t t Z22 Z24 Z24 Z24 C4 Xt þ 2M Ylr d lr Lx Ly ¼0 2 þ þ t t 2 2t Z4
ð15Þ
5. Computer simulation results
ð16Þ
@J C 3 lf þ lr C4 lf þ lr X1 þ 2 2 X2 ¼ 2 2 t t @Y1 Z Z4 " 3 2 2 # C1 C 3 lf þ lr C4 C4 lf þ lr Y1 þ 2 2 þ2 2 þ2 2 þ2 2 t t Z1 Z3 Z4 Z4 " # 2 2 C 3 lf þ lr C 4 lf þ lr C4 þ 2 2 þ2 2 þ 2 2 Y2 t t Z3 Z4 Z4 C4 C 3 lf þ lr Ylr Xt 2M d lr Ly þ Lx þ 2 2 Y3 2 2 þ t t t 2t Z4 Z 3 C4 Ylr lf þ lr lf þ lr 2M 2 2 þ Xþ t t t t Z4 lf lr d lr lf þ lr þ Ly þ 2 Y Ly Lx ¼ 0 ð17Þ t t t @J C 3 lf þ lr C4 lf þ lr X1 þ 2 2 X2 ¼ 2 2 t t @Y2 Z Z4 " 3 2 2 # C 3 lf þ lr C4 C4 lf þ lr Y1 þ 2 2 þ2 2 þ2 2 t t Z3 Z4 Z4 " # 2 2 C 3 lf þ lr C 4 lf þ lr C2 C4 þ 2 2 þ2 2 þ 2 2 þ 2 2 Y2 t t Z3 Z4 Z2 Z4 C4 C 3 lf þ lr Ylr Xt 2M d lr Ly þ Lx þ 2 2 Y3 2 2 þ 2t t t t Z4 Z 3 C4 Ylr lf þ lr lf þ lr 2M 2 2 þ Xþ t t t t Z4 lf lr d lr lf þ lr Ly þ 2 Y Ly Lx ¼ 0 ð18Þ þ t t t @J C4 C4 C3 C4 C4 Y3 2 Y Ly ¼ 0 ¼ 2 Y1 þ 2 Y2 þ þ @Y3 Z4 Z4 Z23 Z24 Z4
at each wheel to follow the model responses and to meet the driver’s traction and/or braking command. Fig. 2(b) is the block diagram, which summarizes the main idea of the forgoing method.
ð19Þ
The above equations are linear and can be more conveniently expressed using matrix form, which is suitable for linear system analysis 2 32 3 2 3 a1 0 a2 a3 0 X1 b1 6 76 7 6 7 0 a a a 0 X 6 76 2 7 6 b2 7 4 5 6 6 76 7 6 7 6 a7 a8 a9 a10 a11 76 Y1 7 ¼ 6 b3 7 6 76 7 6 7 6 76 7 6 7 4 a12 a13 a14 a15 a16 54 Y2 5 4 b4 5 0 0 a17 a18 a19 Y3 b5 where ai and bj (i = 1–19 and j = 1–5) are expressed by C14 ; Z14 ; t; lf ; lr ; d; X; Y; Lx ; Ly , and M. This linear system is solved at each time step to find the optimum tire force distribution (X14 and Y14 Þ. Once the lateral and longitudinal tire force are obtained the active steering angle can be determined (using inverse use of simple tire model) as well as the traction and/or braking torque required
In order to analyze the effect of the independent steer and drive/or brake forces distribution method on the car-caravan handling performance, computer simulations of the articulated vehicle handling response subjected to evasive lane change with braking were carried out. These responses are compared to those obtained in the case without control and the case with combined control type DYC + RWS + FWS. The simulation model is created using Matlab/Simulink and composed of a 15-degree-of-freedom non-linear model, nine degrees for the vehicle units while the remaining six degrees for the wheels. The towing vehicle motion has six independent degrees of freedom; vertical motion, lateral motion, longitudinal motion, rolling motion, pitching motion and yawing motion while the towed vehicle has three independent degrees of freedom; rolling motion, pitching motion and yawing motion. The simulation concerns small passenger car (1350 kg) pulling trailer (630 kg) at different running conditions. The tire forces were obtained by integrating the distributed tire deformations in the contact-patch in lateral and longitudinal directions respectively using a brush type tire model. The tire forces were obtained by integrating the distributed tire deformations in the contact-patch in lateral and longitudinal directions, respectively, using a brush-type tire model. In the tire model, the coefficient of friction was treated as a function of the vertical load as well as the slip velocity. All detailed information about this tire model can be found in [11]. In the simulated cases, the vehicle runs at constant speed for 0.5 s and then braking is conducted. The deceleration is increased linearly from zero to its maximum value over a period of 0.5 s (driver brake pedal operation, ax ). Near the limit region, the articulated vehicle shows oscillatory handling response in the case without control as shown in Fig. 3, (a) for the tractor part while (b) for the trailer part. On the other hand, the controllers presented here were both found to produce desirable effects on the handling dynamics of the car-caravan combination, Figs. 4 and 5. In the figures, the dashed lines represent the target response of yaw rate and side-slip angle. 5.1. Robustness against road friction variation In order to investigate the effects of the proposed independent steer and drive/or brake force distribution method on articulated vehicle handling response on a low friction road, computer simulations of vehicle responses on a road of which the nominal value of the friction coefficient is equal to 0.3 are presented in Figs. 6–8. However, the friction coefficient used in the simple tire model used in the control is fixed as 1.0 (dry road). The computational results indicate that it is essential to know precisely the friction coefficient between the tire and road surface for use in the simple tire model in the case of the DYC + RWS + FWS combined control. On the contrary, in the case of optimum tire forces distribution control, the friction coefficient does not need to be known
Please cite this article in press as: O. Mokhiamar, Stabilization of car-caravan combination using independent steer and drive/or brake forces distribution, Alexandria Eng. J. (2015), http://dx.doi.org/10.1016/j.aej.2015.05.006
2
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(a) for the tractor part V = 100 km/h ax = -0.2 g
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ayt
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Figure 6
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0.8 0.4 0 -0.4 -0.8
30 Yaw rate (deg/sec)
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Side-slip angle (deg.)
5 r
target response. This is simply due to the saturation property of the tire which limits the lateral and longitudinal forces generated from the tire especially at such severe driving condition
Lateral acceleration (g)
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Yaw rate (deg/sec)
Lateral acceleration (g)
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Car-caravan handling response with independent steer and drive/brake tire forces distribution, on dry surface.
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Time (sec)
precisely. This proves the robustness of optimum distribution technique against road friction variations even though the controlled responses do not seem to follow or come close to the
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Figure 5
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(b) for the trailer part V = 100 km/h ax = -0.2 g
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Car-caravan handling response with DYC + RWS + FWS combined control, on dry surface.
30
Yaw rate (deg/sec)
Lateral acceleration (g)
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ayt
Time (sec)
Figure 4
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(b) for the trailer part V = 70 km/h ax = -0.1 g
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Car-caravan handling response without control, on dry surface.
30 ay
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Figure 3
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Side-slip angle (deg.)
0.8 Yaw rate (deg/sec)
Lateral acceleration (g)
Stabilization of car-caravan combination
Time (sec)
Car-caravan handling response without control, on slippery surface.
Please cite this article in press as: O. Mokhiamar, Stabilization of car-caravan combination using independent steer and drive/or brake forces distribution, Alexandria Eng. J. (2015), http://dx.doi.org/10.1016/j.aej.2015.05.006
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Time (sec)
Time (sec)
Car-caravan handling response with independent steer and drive/brake tire forces distribution, on slippery surface.
Figure 8
due to large tire slip in both lateral and longitudinal directions, Fig. 5. On the other hand, the difference between the controlled responses and the target ones became more obvious
in case of slippery surface, Fig. 8, because the same previous reason as well as because the simulation was conducted while the combined vehicle was running on slippery surface and the
850
850 V=70 km/h, ax=-0.1g
V=70 km/h, ax=-0.1g
tire 1
tire 1
tire 2
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tire 2
tire 3
Tire longitudinal force (N)
Tire longitudinal force (N)
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Car-caravan handling response with DYC + RWS + FWS combined control, on slippery surface.
Figure 7
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O. Mokhiamar Lateral acceleration (g)
8
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(a)
(b)
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Figure 9 Tires longitudinal forces time history, on slippery surface: (a) DYC + RWS + FWS combined control and (b) independent steer and drive/brake tire forces.
Please cite this article in press as: O. Mokhiamar, Stabilization of car-caravan combination using independent steer and drive/or brake forces distribution, Alexandria Eng. J. (2015), http://dx.doi.org/10.1016/j.aej.2015.05.006
Stabilization of car-caravan combination
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Figure 10 Tires lateral forces time history, on slippery surface: (a) DYC + RWS + FWS combined control and (b) independent steer and drive/brake tire forces.
the longitudinal and lateral dynamics of the combined vehicle as shown before in Figs. 7 and 11. On the other hand, the proposed optimum control succeed to tune the entire set of longitudinal and lateral forces required to be generated at the four tires to achieve smooth and reasonable performance of the combined vehicle, Figs. 9(b) and 10 (b). As a result, this will affect longitudinal and lateral dynamics of the combined vehicle as shown in Figs. 8 and 11, where combined vehicle with the proposed control shows smooth handling dynamics.
0.5 V=70 km/h, ax=-0.1g
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driver operation DYC+RWS+FWS
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optimum distribution
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6. Summary and conclusion -1
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Figure 11 Longitudinal acceleration time history of the carcaravan combination, on slippery surface.
coefficient of friction between tire and ground in the control algorithm was fixed as on dry surface. Going deep in the dynamics of the car-caravan combination with the combined control type DYC + RWS + FWS and the proposed optimum control, Figs. 9–11 are presented. Fig. 9 represents wheel longitudinal force; Fig. 9(a) for DYC + RWS + FWS combined control while Fig. 9(b) is for the proposed control. From the figure, in case of DYC + RWS + FWS, it is noted that large traction force is generated at tire number 1 while large braking force is generated at tire number 2 in order to generate large direct yaw moment control, Mz , needed to stabilize the combined vehicle motion. Again due to saturation property of the tire, small lateral force generated at tires 1 and 2 as shown in Fig. 10(b). Consequently this will affect
The influence of independent steer and drive/or brake force distribution on the articulated vehicle handling characteristics is examined. Near the limit region, the two controllers stabilize the combined vehicle motion. However, the effect of the proposed optimum control is more obvious especially on the response of the trailer part. In more severe situation, articulated vehicle running on low friction coefficient road, the combined control-type DYC + RWS + FWS failed to achieve a desirable response. On the other hand, the proposed optimum control successively achieves smooth and reasonable responses.
Appendix A. Simple tire model sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ffi K2i 2 Xi 1 Yi ¼ Ki bi signðbi Þ b 4lZi i lZi i when: j bi j< 2lZ . Ki
Please cite this article in press as: O. Mokhiamar, Stabilization of car-caravan combination using independent steer and drive/or brake forces distribution, Alexandria Eng. J. (2015), http://dx.doi.org/10.1016/j.aej.2015.05.006
10
O. Mokhiamar
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ffi Xi Yi ¼ signðbi ÞlZi 1 lZi
[3]
2lZi . Ki
when: j bi jP Inverse use of the simple tire model i (a) when: j bi j< 2lZ . Ki In this case the tire side-slip angle can be calculated from the following equation: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b b2 4a c bi ¼ signðYi Þ 2a
where
[4]
[5]
[6]
[7]
K2 a¼ i 4lZi
[8]
b ¼ Ki jYi j c ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1
Xi lZi
2
i (b) when: j bi jP 2lZ .The following condition is used to Ki account for the saturation property of the tire, if the tire lateral force is greater than the saturated value in the tire model.
bi ¼ signðYi Þ
[9]
2lZi : Ki
[10]
[11] [12] [13] [14]
References [1] O. Mokhiamar, M. Abe, Simultaneous optimal distribution of lateral and longitudinal tire forces for the model following control, J. Dyn. Syst., Meas. Control, Trans. ASME 126 (4) (2004) 753–763. [2] O. Mokhiamar, M. Abe, An experimental validation of the effect of combined lateral force and yaw moment control to maximize vehicle stability limit as well as responsiveness for active vehicle handling control, in: Proc. of the 9th Mini
[15]
[16]
Conference on Vehicle Systems Dynamics, Identification and Anomalies, Budapest, Hungary, 2004, pp. 435–443. O. Mokhiamar, M. Abe, How the four wheels should share forces in an optimum cooperative chassis control, Control Eng. Pract. 14 (March) (2006) 295–304 (n. 2 SPEC. ISS). S. Kimbrough, W. VanMoorhem, A control strategy for stabilizing trailers via selective actuation of brakes, ASME Adv. Automot. Technol., DSC 44 (1992) 413–428. L. Palkovics, M. El-Gindy, L. Ilosvai, Examination of Different Control Strategies of Heavy-vehicle Performance, ASME Advanced Automotive Technologies, Lousiana, New Orleans, 1993, pp. 349–362. L. Palkovics, J. Bokor, Stabilization of a car-caravan combination using active unilateral brake control, in: Proc. of AVEC’94, Tsukuba, Japan, 1994, pp. 141–146. S. Chikamori, S. Kawasawa, Stability analysis of articulated vehicles with all-wheel steering, in: Proc. of AVEC’96, Aachen, Germany, 1996, pp. 395–407. O. Mokhiamar, M. Abe, Improvement of handling safety of carcaravan combination by direct yaw moment control, in: Proc. of AVEC’02, Hiroshima, Japan, 2002, pp. 57–62. O. Mokhiamar, M. Abe, Examination of different models following types of yaw moment control strategy for improving handling safety of a car-caravan combination, Proc. Inst. Mech. Eng., Part D: J. Automob. Eng. 217 (7) (2003) 561–571. O. Mokhiamar, Integrated control of lateral force and direct yaw moment to improve handling safety of car-caravan combination, in: Proc. of AVEC’08, Kobe, Japan, 2008, pp. 385–390. M. Abe, Vehicle Dynamics: Theory and Application, first ed., Elsevier Ltd, 2009. J.R. Ellis, Vehicles Dynamics, London Business Book LTD, London, 1969. E.J.-J. Slotine, W. Li, Applied Nonlinear Control, Prentice-Hall, Englewood Cliffs, New Jersey, 1991. O. Mokhiamar, M. Abe, Active wheel steering and direct yaw moment control combination to maximize stability as well as vehicle responsiveness during quick lane change for active vehicle handling safety, Proc. Inst. Mech. Eng., Part D: J. Automob. Eng. 216 (part D) (2002) 115–123. M. Abe, A. Kato, K. Suzuki, Y. Kano, Estimation of vehicle side-slip angle for DYC by using on-board-tire-model, in: Proc. of AVEC’98, Nagoya, Japan, 1998, pp. 437–442. S.R. Singiresu, Engineering Optimization, John Wily &Sons, Inc., New York, 1996.
Please cite this article in press as: O. Mokhiamar, Stabilization of car-caravan combination using independent steer and drive/or brake forces distribution, Alexandria Eng. J. (2015), http://dx.doi.org/10.1016/j.aej.2015.05.006