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Yaw Stability Enhancement of Articulated Commercial Vehicles via Gain-Scheduling Optimal Control Approach.

INTRODUCTION

Commercial vehicles continue to be the dominant mode of freight transportation via roads (about 70%) in North America [1]. Owing to their large sizes and inertia, and high mass center, heavy commercial vehicles exhibit relatively lower roll and yaw stability limits compared with other road vehicles, and thereby pose greater risks of property damages and fatalities due to their disproportionally higher inertia. What's more, the freight carried by commercial vehicle is forecast to increase by around 40% from 2012 to 2040 according to U.S Department of Transportation [2]. Therefore, there is an increasing need for further improving the stability and safety of commercial vehicles and reducing the traffic crashes caused by loss of control.

A typical articulated commercial vehicle consists of two units: tractor and semitrailer. The tractor unit is controlled directly by the driver while the semitrailer, connected to tractor unit through the fifth wheel, is used to transport goods and materials. Handling dynamics and directional control properties of articulated commercial vehicles differ significantly from those of other road vehicles such as passenger cars and light trucks. These vehicles may experience two types of instability behaviors [3]: rollover and loss of control (LOC). Rollover is known to be the most dangerous type of instability for heavy vehicles, which is not the focus of this paper. Loss of control, also referred to the loss of directional/ yaw stability, jackknifing and trailer swing, is another significant safety concern. Jackknifing accident is defined as a loss of stability in the yaw motion of the articulated commercial vehicle, which is one of the main causes of accidents resulting from loss of control.

Recently, some active control strategies have been studied to improve the lateral/yaw stability of articulated commercial vehicles, such as PID (Proportional integration derivative) control [4, 5], LQR (Linear quadratic regulator) [6, 7, 8, 9], SMC (Slide mode control) [10], LMIs (Linear matrix inequalities) [11] or MPC (Model predictive control) [12]. For example, Palkovics and EI-Gindy [6] developed a LQR controller based on differential braking to improve the directional behavior of articulated commercial vehicle. Zong et al. [7] proposed an optimal yaw controller based on a single-track linear vehicle model to reduce the risks of rollover and jackknifing. Yang [10] designed a two-level control structure consisting of sliding-mode yaw moment controller and optimal braking force distributor to stabilize the yaw dynamics of the tractor-semitrailer vehicle. Of all the proposed control algorithms in literature, LQR method based on the linear vehicle model is the widely-implemented control algorithm with fixed control gains.

However, vehicle lateral/yaw dynamics varies with respect to vehicle/tire parameters and road conditions, such as vehicle mass, center of mass, cornering stiffness, and so on. On the other hand, vehicle velocity may vary within a wide range and thereby lateral/yaw dynamics is changed accordingly. So, it is not realistic to employ a fixed-gain linear controller in real time because the characteristics of vehicle dynamics are nonlinear in nature. Gain scheduling method is a common used controller design approach for nonlinear systems. Inspired by the work [11, 13] which proposed gain-scheduling control methods for roll control of articulated vehicles, a gain-scheduling optimal nonlinear controller is proposed to enhance the yaw stability of articulated vehicles in this paper. Gain scheduling is realized using a linear optimal approach for the simplified linear vehicle model with respect to vehicle velocity. The effectiveness of the proposed controller is verified through software-in-the-loop co-simulations in Matlab/Simulink and TruckSim under the lane change maneuver.

The rest of the paper is organized as follows. Vehicle models for controller design and its verification are described in Section 2. The gain-scheduling optimal controller design is presented in Section 3. Section 4 present the simulation results. Finally, the concluding marks are drawn in Section 5.

VEHICLE MODELS

In this section, two vehicle models are presented including a single-track linear vehicle model for controller design and a nonlinear vehicle model used for validating the proposed control system.

Single-Track Vehicle Model

For designing a gain-scheduling optimal yaw moment controller, a single-track linear vehicle model shown in Figure 1 is derived by linearizing the nonlinear model for lateral dynamics with constant forward velocity [5, 9]. This model is particularly well-suited to generate desired vehicle states for the control system design in a highly efficient manner. To derive this model, some assumptions must be made. The articulation angle between tractor and semitrailer units and the tire slip angles are assumed to be small. In addition, the lateral tire forces are modeled as a linear function of tire slip angles. Furthermore, the roll dynamics, load transfer and the effects of lateral winds and road camber are not considered.

Four motion states are considered in this model : tractor lateral velocity, tractor yaw rate, articulation angle between the tractor and semitrailer and articulation rate. The equations of motion for the linear model of the tractor-semitrailer combination are expressed as:

[m.sub.1](v.sub.y1] + [v.sub.x][[gamma].sub.1]) = [F.sub.yf] + [F.sub.yr] - [F.sub.yh] (1)

[I.sub.z1][Y.sub.1] = a[F.sub.yf] - b[F.sub.yr] + [F.sub.yh]e + [DELTA][M.sub.1] (2)

[m.sub.2][a.sub.y2] = [F.sub.yh] + [F.sub.yt] (3)

[I.sub.z2][[gamma].sub.z] = [F.sub.yh]c - [F.sub.yt]d + [DELTA][M.sub.2] (4)

Considering the following kinematic relationship,

[[gamma].sub.2] = [[gamma].sub.1] + q (5)

[a.sub.y2] = [v.sub.y1] + [v.sub.x][[gamma].sub.1] - [e[gamma].sub.1] - c([[gamma].sub.1] + q) (6)

The lateral forces developed at the tractor front and rear axle, and semitrailer axle wheels are assumed to be proportional to the tire slip angle, such that:

[mathematical expression not reproducible] (7)

[mathematical expression not reproducible] (8)

[mathematical expression not reproducible] (9)

Where [F.sub.yf], [F.sub.yr] and [F.sub.yt] are the lateral forces developed at tractor front and rear, and trailer axle tires, respectively, which are evaluated assuming constant cornering stiffness ([C.sub.af], [C.sub.ar], [C.sub.at]) and [a.sub.f], [a.sub.r] and [a.sub.t], are the respective side-slip angles.

The description for the main parameters of tractor semitrailer system is given in Table 1 below.

Rewrite the motion equations in matrix form as below:

Mx = Kx + [B.sub.1][delta] + [B.sub.2]u (10)

where M is the mass matrix; K is the stiffness matrix; [B.sub.1] and [B.sub.2] are the input matrices. [delta] is the front steering angle. The control input u, and the system state vector x are defined by

x = [[v.sub.y] [gamma] q [theta].sup.T], u = [[[DELTA][M.sub.1] [DELTA][M.sub.2]].sup.T] (11)

To observe the dynamics of the model, the equation above can be converted into state space format, shown as

x = Ax + Bu + E[delta] (12)

with

A = [M.sup.-1] K

B = [M.sup.1] [B.sub.2]

E = [M.sup.-1] [B.sub.1]

The matrices M, K and [B.sub.1] are given in the Appendix. It should be noticed that matrix K is a function of vehicle velocity [v.sub.x]. And the validity of single-track linear vehicle models can be found in Reference [5].

Nonlinear Vehicle Model

TruckSim is a multi-body dynamics software used widely for simulating and analyzing the dynamic characteristics of medium to heavy trucks [14]. It can model multiple bodies such as a tractor, trailer and dolly. The tractor-semitrailer nonlinear vehicle model is formulated in TruckSim platform to verify the effectiveness of the proposed controller under specified maneuvers considering the model uncertainties, as shown in Figure 2.

CONTROL SYSTEM DESIGN

The block diagram of the proposed control system is shown in Figure 3. This control system structure includes reference model, gain-scheduling optimal yaw moment controller and braking torque distribution. [x.sub.d] and x are the desired vehicle state and actual vehicle states, respectively. 8 is the front steering wheel angle. e is the tracking error between the actual states and the desired states. [DELTA][M.sub.1] and [DELTA][M.sub.2] are the corrective yaw moments for tractor and semitrailer, respectively. [T.sub.bLi] and [T.sub.bRi] are the braking torques for left and right wheels. The main goal of the proposed control system is to make the actual tractor yaw rate and articulation angle to follow the desired values obtained from reference model. The reference model is developed to generate the desire vehicle states that keep the vehicle within the linear region. The gain-scheduling optimal yaw moment controller responds to the inputs: vehicle state errors for optimal controller design and vehicle velocity as a scheduling variable, and the outputs are the corrective yaw moments for tractor and semitrailer. Vehicle states such as vehicle velocity, tractor yaw rate, articulation angle, articulation rate and lateral velocity are assumed to be available in this study. In the following sections, each block is presented in details.

Reference Model

Vehicle active control systems are synthesized to track the desired responses generated by a reference model. In this section, the reference vehicle model is presented in details to produce four desired vehicle states: tractor yaw rate, tractor lateral velocity, articulation angle and articulation rate.

The desired tractor yaw rate is the steady-state yaw rate response of a linear single-track vehicle model to a steering command at steady state cornering, which is given by

[mathematical expression not reproducible] (13)

[k.sub.t] is understeer coefficient for tractor and defined as blow

[mathematical expression not reproducible] (14)

where [l.sub.1] = a + b, [l.sub.2] = c + d.

The desired yaw rate mentioned above, however, is not necessarily achievable under all possible driving conditions. Hence, it should be bounded by reasonable values. For instance, tire saturation on a low friction road may not support a high yaw rate demand due to limited tire forces. The lateral acceleration is restricted by the tire-road friction limit and the lateral acceleration threshold for rollover [a.sub.yroll] which is calculated based on the static rollover threshold.

[mathematical expression not reproducible] (15)

The steady state value of tractor lateral velocity can be derived from the linear single-track vehicle model. In many cases, for simplicity, the desired lateral velocity is chosen to be zero, i.e.,

[v.sub.1yd] = 0 (16)

The steady state response of articulation angle can be also derived from linear vehicle model as below

[mathematical expression not reproducible] (17)

where

[mathematical expression not reproducible]

Then, the desire articulation rate can be obtained below by differentiating the equation (17),

[mathematical expression not reproducible] (18)

Where, [a.sub.x] is the deceleration, [k.sub.s] is the understeer coefficient for the semitrailer expressed below.

[k.sub.s] = [m.sup.1][al.sub.2] + [m.sub.2]d(a + e)/[l.sub.2][l.sub.2][C2ar] - [m.sub.2]c/[l.sub.2][C.sub.[alpha]s

Gain-Scheduling Optimal Controller

From the equation (12). it can be observed that the system matrix A is a function of vehicle velocity [v.sub.x], i.e,A = A([v.sub.x]). Therefore, the dynamic characteristic of tractor semitrailer vehicle is strongly dependent on the vehicle velocity. Since vehicle velocity can be measured directly, it is possible to apply vehicle velocity based gain-scheduling approach to design yaw moment controller, which is helpful to improve the performance of yaw stability control at a wide vehicle speed range.

LQR controller is usually designed at an operation point and provide a good performance around the operation point. But when the operating point is changed, the control gain obtained from LQR optimal controller cannot be changed accordingly. To reduce the dependency, LQR optimal controller is scheduled with vehicle velocity in this study.

The gain-scheduling optimal controller design is divided into three steps as below.

1. Determine six specified operation points, i.e., six different vehicle velocity points: [40, 60, 80, 100, 120, 140]km/h.

2. Design the local LQR optimal controller at each specified vehicle velocity.

3. Gain-scheduling optimal controller synthesis by interpolating the local LQR optimal controllers.

In the following section, local LQR optimal controller design and gain- scheduling optimal controller synthesis are described in details.

Define the tracking error between the actual states and the desired states as follows:

e = x - [x.sub.d] = [[[v.sub.y1 - [v.sub.y1d] [[gamma].sub.1] - [[gamma].sub.1d] q - [q.sub.d] [theta] - [[theta].sub.d]].sup.T] (19)

where [x.sub.d] is the desired vehicle states in 'Reference Model' section.

The tracking error dynamics can be derived below by differentiating the equation (19):

e = x - [x.sub.d] = Ae + Bu + [Ax.sub.d] + E[delta] (20)

where the third and fourth terms on the right side of equation (20) are considered as the disturbances which depends on the front steering angle. So, the tracking error dynamics equation can be reduced to the equation below:

e = x - [x.sub.d] = Ae + Bu + w (21)

Neglecting the disturbance term, LQR optimal control method can be employed to develop the feedback control system. The corrective yaw moments are calculated as follows:

[mathematical expression not reproducible] (22)

where [k.sub.ij] (i,j = 1, 2,3,4) denotes the feedback gain of the tracking error defined in equation (19). These gains can be determined by minimizing the following cost function:

[mathematical expression not reproducible] (23)

with

[mathematical expression not reproducible]

where the matrices Q and R are the weighting matrices for tracking error e and control inputs u, respectively. The optimal feedback gain matrix K can be calculated by solving Riccati equation with the predefined weighting matrices Q and R.

To develop the gain-scheduling optimal controller, a family of local LQR optimal controller are designed for each specified vehicle velocity. The chosen weighting matrices Q and R also vary accordingly. For each specified vehicle velocity, the feedback gains are calculated:

u = -K([v.sub.x])e (24)

Then the gains K([v.sub.x]), which depend on the vehicle velocity, are interpolated between all the specified vehicle velocity points. For example, when the vehicle runs between 80km/h and 100km/h, the feedback gains K is given by

K([v.sub.x]) = 1/20((100 - [v.sub.x])[K.sub.80] + {[v.sub.x] - 80)[K.sub.100]), 80 [less than or equal to] [v.sub.x] < 100 (25)

where [K.sub.80] and [K.sub.100] are the feedback gains at [v.sub.x] = 80km/h and [v.sub.x] = 100km/h, respectively.

Brake Torque Distribution

The corrective yaw moments produced from gain-scheduling optimal controller can be generated by applying proper braking torque on the wheels of tractor and semitrailer. Based on the relationship between yaw moment and braking torque, let us convert corrective yaw moments into braking torque first and then select the proper wheels to be braked. For simplicity, the quasi-static wheel dynamics is assumed, which yields:

[T.sub.bi]= [F.sub.xi][R.sub.w] (26)

where [T.sub.bi] is the braking torque, [F.sub.xi] is the braking force and [R.sub.w] is the effective tire radius.

The corrective yaw moment [DELTA]M is subsequently generated by active braking at each wheel, such that:

[DELTA]M = [t.sub.w]/2 [F.sub.xi] (27)

where [t.sub.w] is track width of the axle of which a wheel is braked; The track width is different for the tractor and semitrailer axles.

Following Equations (26) and (27). the braking torques applied to the wheel(s) of the tractor front, tractor rear tandem and semitrailer tandem axles assumes the following form:

[T.sub.bi]=2/[t.sub.w] [DELTA]M[R.sub.W] (28)

Considering the road friction and tire vertical load, the braking torque cannot be too large to avoid wheel lock-up and subsequent potential for skidding.

The brake torque distribution scheme is formulated to select the proper wheel(s) to be braked based on two different situations corresponding to a potential yaw instability, denoted as understeer ([e.sub.[gamma]] < 0) and over-steer ([e.sub.[gamma]] > 0) where [e.sub.[gamma] is the yaw rate error for tractor and semitrailer. Under understeer condition, the tractor or semitrailer's yaw rate is less than the desired value ([e.sub.[gamma] < 0). Therefore, the distribution strategy is designed to reduce the error magnitude by applying brake torques to wheels on the inner track. Under oversteer condition, i.e., the yaw rate is greater than the desired one, a negative yaw moment thus needs to be applied, which could be realized through braking of outer-track wheels. The details about the braking torque distribution strategy can be found in [5].

SIMULATION RESULTS AND DISCUSSIONS

The effectiveness of the proposed gain-scheduling optimal control system employing active braking control strategy is evaluated through co-simulations in the TruckSim and Matlab/Simulink. The vehicle is subject to a lane change maneuver idealized by a 90-degree sinusoidal steer input at a frequency of 0.2Hz, after 1s, as shown in Figure 4. The steer gear ratio is taken as 25. In this paper, the vehicle is tested taking three different vehicle velocities: 90, 100 and 110km/h as example. The road friction coefficient [mu] is 0.4. The feedback control gains at 90km/h and 110km/h are determined in local LQR controller by selecting the proper weighting matrices Q and R while the feedback gains at 100km/h is obtained by through gain-scheduling method. In the following sections, the simulation results at 100km/h are given to show the validity of proposed gain-scheduling control method. The simulation results are also obtained for the vehicle model without the control to be compared with that of the proposed gain scheduled optimal controller.

The performance of the tractor and semitrailer units with the proposed gain-scheduling optimal controller are compared with those of the model without control, as shown in Figures 5 to 8. Figures 5 and 6 illustrates yaw rate responses of the tractor and semitrailer units, respectively. The results show that the proposed control ensures effective tracking of desired yaw rates of both the units. In Figure 5, the tractor yaw rate response increases rapidly to a large value after 6s in the absence of the control, suggesting possible loss of vehicle stability. Similarly, the yaw rate response for semitrailer also cannot track the desired value after around 5s and the yaw rate is over 0.2 rad/s. This suggests the yaw instability without control. The divergent articulation angle response of tractor semitrailer without control in Figure 7 also confirms the loss of vehicle stability, that's, jackknife accident occurs. From Figure 7, with the proposed gain-scheduling optimal controller, the actual articulation angle can follow the desired value with small tracking error. In Figure 8, the time history of sideslip angle response for tractor and semitrailer is given. The sideslip angles are divergent without control while the sideslip angles with the proposed control returns to zero after 7s. All the results above demonstrate that the proposed optimal controller provides a satisfactory tracking performance of the vehicle states and thus improves the yaw stability of tractor semitrailer. The corrective yaw moments obtained from the proposed optimal controller are presented in Figure 9, which can be achieved by applying braking to proper wheels of tractor and semitrailer.

It should be noted that the performance of optimal controller depends greatly on the selection of weighting factors which are a trade-off between different control goals and control inputs. So, the further tuning of the controller gains should be done for a better vehicle performance.

CONCLUSIONS

In this paper, a gain scheduled optimal yaw moment controller is proposed and analyzed to improve the yaw stability of heavy commercial vehicles. In the controller synthesis, a set of local optimal controllers at specific vehicle velocities are developed first, and then a gain scheduled optimal controller is constructed by interpolation. The corrective yaw moments obtained from gain-scheduling optimal controller are generated by braking torque distribution among the tractor and semitrailer wheels. Once the feedback control gains at specified vehicle velocities are determined, the control gains at any vehicle velocity can be calculated. Simulation results confirmed that the proposed gain scheduled optimal controller can force the vehicle states to track the desired values and thus improve the yaw stability of heavy commercial vehicles. In the future, the robustness to parameter uncertainties for the proposed controller should be analyzed.

REFERENCES

[1.] Gerdes J. C. "Safety performance and robustness of heavy vehicle AVCS," Stanford University: Dept. of Mechanical Engineering-Design Division, 2002.

[2.] Freight Facts and Figures 2013. U.S. Department of Transportation, Washington, D.C., 2014.

[3.] Kang, X. and Deng, W., "Vehicle-Trailer Handling Dynamics and Stability Control--an Engineering Review," SAE Technical Paper 2007-01-0822, 2007, doi:10.4271/2007-01-0822.

[4.] Chen C.-K., Dao T.-K. and Lin H.-P. "A compensated yaw moment-based vehicle stability controller," International lournal of Vehicle Design, 53:3, pp. 220-38, 2010.

[5.] Li, B. and Rakheja, S., "Jackknifing Prevention of Tractor-Semitrailer Combination Using Active Braking Control," SAE Technical Paper 2015-01-2746, 2015, doi:10.4271/2015-01-2746.

[6.] Palkovics L. and El-Gindy M. "Design of an Active Unilateral Brake Control System for Five-Axle Tractor-Semitrailer Based on Sensitivity Analysis," Vehicle System Dynamics, 24:10, pp. 725-758, 1995.

[7.] Zong C, Zhu T, Wang C and Liu H. "Multi-objective stability control algorithm of heavy tractor semi-trailer based on differential braking," Chinese lournal of Mechanical Engineering, 25, pp.88-97, 2012.

[8.] Tabatabaei Oreh, S., Kazemi, R., and Azadi, S., "Directional Control of Articulated Heavy Vehicles," SAE Int. J. Commer. Veh. 6(1): 143-149. 2013, doi:10.4271/2013-01-0711.

[9.] Hac, A., Fulk, D., and Chen, H, "Stability and Control Considerations of Vehicle-Trailer Combination," SAE Int. J. Passeng. Cars -Mech. Syst. 1(1):925-937, 2009, doi:10.4271/2008-01-1228.

[10.] Yang X. "Optimal reconfiguration control of the yaw stability of the tractor-semitrailer vehicle," Mathmatical Problem Engineering, Article ID 602502: 1-23, 2012.

[11.] Zhu T. and Zong C. "Research on Heavy Truck Rollover Prevention Based on LMI Robust Controller," IITA International Conference on Control, Automation and System Engineering, Zhangjiajie, China, 2009.

[12.] Chen L. K. and Shieh Y. A. "Jackknife prevention for articulated vehicles using model reference adaptive control," Pro IMechE Part D: J Automobile Engineering, 225:1, pp. 28-42, 2011.

[13.] Miege A.J.P and Cebon D. "Optimal roll control of an articulated vehicle: theory and model validation," Vehicle System Dynamics: International Journal of Vehicle Mechanics and Mobility, 43:12, pp.867-893.

[14.] TruckSim User Manual Version 8. Mechanical Simulation Corporation (MSC), Ann Arbor, MI, 2009.

Bin Li and Subhash Rakheja

Concordia University Montreal

APPENDIX

Matrices M, K and [B.sub.1] are given below.

[mathematical expression not reproducible]

where

[a.sub.11]=-([C.sub.af]+ [C.sub.ar] + [C.sub.at])

[a.sub.12] = -a[C.sub.af] + b[C.sub.ar] + ^[l.sub.2])[C.sub.at] - (m.sub.1] + [m.sub.2]) [V.sub.x.sup.2]

[a.sub.13] = e + [l.sub.2][f.sub.at]

[a.sub.14] = [v.sub.x] [C.sub.at]

[a.sub.14] = [v.sub.x][C.sub.at]

[a.sub.21] = -a[C.sub.af] + b[C.sub.ar] + ([l.sub.2] + e) [C.sub.at]

[a.sub.22] = -[a.sub.2][C.sub.af]- [b.sup.2][C.sub.ar] - [(e + [l.sub.2]).sup.2](C.sub.at] + (e + c) [v.sub.x.sup.2][m.sub.2]

[a.sub.23] = -(e + [l.sub.2])[l.sub.2][C.sub.at]

[a.sub.24] = -(e + [l.sub.2])[v.sub.f][C.sub.at]

[a.sub.31] = [l.sub.2][C.sub.at]

[a.sub.32] = -(e + [l.sub.2])[l.sub.2][C.sub.at] + [cv.sub.x.sup.2][m.sub.2]

[a.sub.33] =-[l.sub.2.sup.2][C.sub.at]

[a.sub.33] = [l.sub.2.sup.2] [C.sub.at]

[a.sub.34] - [l.sub.2][v.sub.x] [C.sub.at]

[a.sub.34] -[l.sub.2][v.sub.x][C.sub.at]

[l.sub.2] = c + d

doi:10.4271/2017-01-0437
Table 1. Primary parameters of tractor semitrailer

Symbol             Parameter

[m.sub.1]          Mass of tractor, kg
[m.sub.2]          Mass of semitrailer (empty), kg
[I.sub.z2]         Moment of inertia of tractor, kg-[m.sup.2]
[I.sub.z2]         Moment of inertia of semitrailer, kg-[m.sup.2]
a                  Distance from tractor CG to front axle, m
b                  Distance from tractor CG to rear axle, m
c                  Distance from trailer CG to fifth wheel, m
d                  Distance from trailer CG to trailer axle, m
e                  Distance from tractor CG to fifth wheel, m
[[gamma].sub.1],
[[gamma].sub.2],   Yaw rate for tractor and semitrailer, rad/s
[v.sub.x]          Vehicle velocity, m/s
[V.sub.y1]         Tractor's lateral velocity
[a.sub.y2]         Semitrailer's lateral velocity, [m.sup.2] /s
[theta]            Articulation angle, rad
q                  Articulation rate, rad/s
[F.sub.yf],
[F.sub.yr],
[F.sub.yt]         Lateral forces at tractor front, rear and trailer
                   axles, N
[F.sub.yh]         Lateral force action on the fifth wheel, N
[C.sub.[alpha]f],
[C.sub.[alpha]r],
[C.sub.[alpha]t]   Cornering stiffness at tractor front, rear and
                   trailer axles
[delta]            Front steering angle, rad
[DELTA][M.sub.1],
[[DELTA]M.sub.2]   Corrective yaw moments acting on tractor and
                   semitrailer
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Author:Li, Bin; Rakheja, Subhash
Publication:SAE International Journal of Commercial Vehicles
Article Type:Technical report
Date:May 1, 2017
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