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Application of stochastic model predictive control to modeling driver steering skills.

ABSTRACT

With the development of the advanced driver assistance system and autonomous vehicle techniques, a precise description of the driver's steering behavior with mathematical models has attracted a great attention. However, the driver's steering maneuver demonstrates the stochastic characteristic due to a series of complex and uncertain factors, such as the weather, road, and driver's physiological and psychological limits, generating negative effects on the performance of the vehicle or the driver assistance system. Hence, this paper explores the stochastic characteristic of driver's steering behavior and a novel steering controller considering this stochastic characteristic is proposed based on stochastic model predictive control (SMPC). Firstly, a search algorithm is derived to describe the driver's road preview behavior. Then, the internal vehicle model including driver's knowledge of the vehicle lateral dynamics is derived by a nonlinear 2-DOF model, and a sideslip angle perception model is proposed in the state feedback process to describe the stochastic characteristic of driver's steering behavior, which is characterized by a gauss stochastic variable, whose mean represents the real sideslip angle and variance is associated with driver's steering skills and vehicle lateral dynamics. Moreover, the steering control rule is formulated to generate the trajectory track steering commands according to the trajectory error and steering input. Finally, the proposed SMPC driver model is validated. The results demonstrate the SMPC driver model has an excellent trajectory track capability and describes the stochastic characteristic of driver's steering behavior precisely.

CITATION: Wang, C., Zhang, X., Guo, K., Ma, F. et al., "Application of Stochastic Model Predictive Control to Modeling Driver Steering Skills," SAE Int. J. Passeng. Cars - Mech. Syst. 9(1):2016.

INTRODUCTION

It is widely recognized that driver's vehicle manipulation behavior demonstrates a critical influence on the driving safety, and the interaction among the driver, vehicle and environment has been widely explored in recent years [1]. Especially, understanding and modeling the steering behavior of a human driver is becoming a popular topic to enable the development of the advanced driver assistance system and autonomous vehicle techniques [2].

Over the past decades, various driver models are developed including the compensation tracking models [3, 4, 5], preview tracking models [6, 7, 8, 9, 10] and adaptive intelligent models [11, 12, 13]. The previous driver modeling mainly focuses on trajectory track capability in the driver-vehicle-environment closed-loop system, but lacks a thorough understanding for the human physiological characteristic. Therefore, many researchers have studied driver modeling based on the neuromuscular dynamics, cognitive processes and the learning behavior [14]. Pick and Cole [15] proposed a method for identifying the arms' dynamic properties of the human driver and developed a model including the neuromuscular dynamics to illustrate the driver's steering behavior [16]. Will and David [17] derived a neuromuscular driver model with the feed-forward and feedback controls simultaneously. Salvucci [18] proposed a driver model based on the cognitive architecture and developed a rapid prototype to evaluate the in-vehicle interfaces [19]. Kim and Cole [20] improved the steering manipulation of a human driver under constant or varying road frictions through the steering torque feedback control. In addition, tremendous research efforts on driver skills modeling have contributed to a full development of the autonomous driving techniques [21]. Dolgov et al. [22] derived a path-planning algorithm for autonomous vehicles driving in an unknown environment. Liu et al. [23] proposed a real-time trajectory planning framework and developed an integrated local trajectory planning and tracking control approach regarding autonomous vehicles [24].

In recent years, human driver's steering manipulation behavior is widely explored via model predictive control (MPC) [25, 26]. MacAdam [27, 28] derived an optimal preview steering controller based on the MPC theory. Cole et al. [29] developed and compared two optimal linear preview steering controllers with the linear quadratic regulator (LQR) and MPC approaches. Additionally, it is widely accepted that different drivers generally demonstrate different steering manipulation behaviors [14], whereas a driver cannot repeat his steering maneuver on a same road and a set of complex and uncertain factors that include the external disturbance, internal physiological and psychological limits enable the stochastic characteristic of driver' steering behavior, limiting the performance of the vehicle or the driver assistance system. Qu et al. [30] studied the driver's cognition behavior of the uncertain road friction and roughness, modeling the stochastic characteristic of driver's steering behavior via SMPC. Ploechl and Edelmann [31] gave an excellent review of driver model and pointed out that the human factors including the stochastic characteristic in the vehicle-driver coupled system should be invested more deeply.

In this paper, the driver's perception to vehicle states is investigated and a novel steering controller considering the stochastic characteristic is proposed with structure in Figure 1 based on SMPC. Especially, a sideslip angle perception model described by a gauss stochastic variable is proposed in the state feedback process to characterize the stochastic characteristic of driver's steering behavior, whose mean represents the real sideslip angle and variance characterizes driver's steering skills and vehicle lateral dynamics. The structure of this paper is organized as follow. Section 2 describes the driver's road preview behavior. Section 3 formulates the internal vehicle dynamics model. The steering control rule is derived in Section 4. The proposed SMPC driver model is validated in Section 5. Eventually, the conclusions are drawn in Section 6.

ROAD PREVIEW

The road preview indicates the driver's ability to observe the road conditions ahead of the vehicle [7], and the controller previews the incremental lateral displacement of the desired trajectory with respect to the vehicle frame of reference to generate a steering command. In this paper, the desired trajectory is described with a series of discrete coordinate points as shown in Table 1. In addition, the multiple-point preview assumption is formulated and a search algorithm [32] is adopted here to characterize the driver's preview behavior as shown in Figure 2, whose detailed process is illustrated as follows.

Firstly, the vehicle states including the position ([X.sub.0](k), [Y.sub.0](k)) and the heading angle [psi](k) at the present sample time k are obtained via state feedback, and the transform matrix between the Cartesian coordinate system X-Y and the Vehicle coordinate system [x.sub.v] -[y.sub.v] is derived as Equation (1).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

Then, the closest point s behind the present vehicle position is searched in the Table 1 according to the constraint condition in Equation (2), which is used for updating the next point.

[x.sub.v] (s) x [x.sub.v] (s +1) [less than or equal to] 0 (2)

Finally, the closest two points m, m+1 nearby the preview one are obtained based on the constraint condition formulated in Equation (3), and the incremental lateral displacement r (k) of the preview point relative to the present vehicle position is obtained as Equation (4).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

INTERNAL VEHICLE DYNAMICS MODEL

The internal vehicle dynamics model reflects the driver's cognition characteristic to the vehicle lateral dynamics, and generates the predicted trajectory under the present vehicle states. In addition, the driver generally shows more concerns with the vehicle's lateral and yaw motion during the steering process [29], and a 2-DOF vehicle model is adopted as the internal model.

Tire Model

In this paper, it is assumed that the nonlinear characteristic of the 2-DOF vehicle model is mainly from the nonlinear tire performance, which is described by the UniTire model [33] as the Appendix.

Sideslip Angle Perception Model

In this paper, the stochastic characteristic of driver's steering behavior is explored based on the human driver's cognition behavior to the vehicle lateral dynamics, which is regarded as a stochastic process and associated with the driver's steering skills and vehicle lateral dynamics. Generally, a skilled driver shows a more precise perception of the vehicle states, contributing to a better driving performance; a novice driver usually shows a poor perception to the vehicle lateral dynamics, leading to an inferior driving performance. Furthermore, the driver's cognition behavior normally presents a decreasing tendency when the vehicle lateral dynamics transforms from the linear to the nonlinear region.

Based on the discussions above, a detailed cognition process of the nonlinear tire lateral force is studied in this paper and a sideslip angle perception model described by a gauss stochastic variable Z is proposed as Equation (5), whose mean [[alpha].sub.real] represents the real sideslip angle and the variance [[sigma].sup.2] characterizes the human driver's steering skills and the vehicle lateral dynamics.

Z~N([[alpha].sub.real],[[sigma].sup.2]) [sigma] = [p.sub.1][|[[alpha].sub.real]|.sup.2] + [p.sub.2]|[[alpha].sub.real]| (5)

where, [p.sub.1] and [p.sub.2] are the identified parameters associated with steering skills.

For simplification, the common drivers are categorized into the novice one with less driving experience and the skilled one with abundant driving experience, and their sideslip angle perception characteristic with various real sideslip angles derived as the Equation (5) are presented in Figure 3 respectively.

Regarding the skilled driver, the sideslip angle perception characteristic with various real sideslip angles is derived by a "steep mountain". Considering a small real sideslip angle [alpha]', the perceived sideslip angle demonstrates a Gauss possibility distribution with a small variance nearby its real value, leading to an excellent steering manipulation performance via the accurate perception to the real sideslip angle. Conversely, the sideslip angle perception characteristic of the novice driver is described by a relatively "fat mountain". With the same real sideslip angle [alpha]' as the skilled one, the Gauss distribution demonstrates a larger variance, and the perceived sideslip angles are distributed more discretely relative to its real value, which characterizes a poor perception to tire sideslip angle and results in an inferior steering manipulation performance.

In addition, the distribution variance presents an increasing tendency with the real sideslip angle's increase, in other words, the perception accuracy is decreasing when the vehicle lateral dynamics transforms from the linear to the nonlinear region, causing a decreasing steering manipulation performance.

Vehicle Model

In general, the dynamics equation of a typical 2-DOF bicycle model as shown in Figure 4 can be derived as Equation (6) with a small steering angle assumption.

m[??] = -m[v.sub.x][??] + 2[F.sub.yf] + 2[F.sub.yr] I[??] = 2a[F.sub.yf]-2b[F.sub.yr] (6)

In addition, the front and rear tire sideslip angles [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] can be derived approximately as Equation (7).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

Substituting the Equations (5), (7) and the perceived angle [[alpha].sub.perceived] =z into the Equation (6), the Equation (6) can be rewritten as Equation (8).

m[??] = f([??], [??] ,[[delta].sub.f]) m[??] = g([??], [??], [[delta].sub.f]) (8)

Generally, the cognition behavior of human driver for the nonlinear vehicle lateral dynamics is limited by the driving conditions and the driver's physiological delay [15], and the linearization and discretization for the derived nonlinear 2-DOF vehicle model is performed. To define the state variable [??]= [[[??], [??]].sup.T], the input [??]=[lambda] x [[delta].sub.f] and the output [??] = [[y, [psi]].sup.T], the state space expression of the Equation (8) is obtained as Equation (9).

[??](k+l) = G[??](k) + H[??](k) [??](k) = C[??](k) (9)

where, matrix G=[e.sup.AT], matrix [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; matrixes A, B and C are derived as Equation (10)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

where, the perceived cornering stiffness of the front and rear wheels [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are derived as Equation (11) based on the UniTire model [33].

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

STEERING CONTROLLER

Based on the predictive model of the vehicle lateral dynamics, namely the internal 2-DOF vehicle model derived in Section 3, the steering controller generates the steering angle change commands at each SMPC control cycle via minimizing cost function, to track the desired trajectory obtained by the road preview presented in Section 2.

Response Matrix

First of all, the SMPC steering controller calculates the system states within time [k, k+[N.sub.p]], where k is the sample time and [N.sub.p] is the prediction horizon. In addition, it is assumed that the control horizon [N.sub.u] [less than or equal to] [N.sub.p], and the control input keeps constant from time [N.sub.u], i.e. [??](k+[N.sub.u]-1)=[??](k+[N.sub.u])= ... [??](k+[N.sub.p]-1), hence, the control matrix [U.sub.Nu] |k at the sample time k is defined as Equation (12).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

Meanwhile, the response matrix [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is defined as Equation (13).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

Substituting the Equations (9) and (12) into the Equation (13), the Equation (13) is rewritten as Equation (14)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

where,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

Cost Function

It is commonly expected that the driving task can be well-performed with a less physical exhaustion, and the cost function J considering the trajectory error [J.sub.1] and the steering input [J.sub.2] is adopted as Equation (16) [30].

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)

where, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the desired trajectory matrix at the sample time k derived as Equation (17); [DELTA] [U.sub.Nu]|k is the incremental control matrix derived as Equation (18).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)

Eventually, the optimal sequence of the steering angle change commands [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] can be obtained via minimizing the cost function J, and the first element of the sequence is transmitted to the controlled vehicle via a physiological and mechanical delay element [32], which is derived as Equation (19).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)

VALIDATION AND DISCUSSION

First of all, the sideslip angle perception model parameters are identified shown in Table 2 via the real human driver test data from the published results [14, 30], and all the other model parameters used in this paper are shown in Table 3.

In addition, a simulation is studied via MATLAB /Simulink-CarSim to validate the proposed SMPC driver model, and three driving conditions, including [v.sub.x]=65 km/h, [mu]=0.8 (linear region), [v.sub.x]=80 km/h, [mu]=0.75 (weak nonlinear region) and [v.sub.x]=95 km/h, [mu]=0.7 (strong nonlinear region), are used to explore the control performance of the proposed SMPC driver model with various vehicle lateral dynamics. Moreover, a vehicle trajectory database, built via simulation based on the SMPC driver model, is compared with another real human driver test database from the published results [20, 32, 34], shown in Figures 5, 6, 7.

The envelop regions in the Figures 5, 6 and 7 are distributed nearby the desired trajectory. With the same DLC cases, the skilled driver shows a smaller envelop region than the novice one, describing a better steering maneuver. In addition, the envelop regions for both the skilled and novice drivers are increasingly enlarged with the increasing [v.sub.x] and decreasing [mu], describing a decreasing trajectory track performance of the driver when the vehicle lateral dynamics transforms from the linear region to the nonlinear region. Moreover, the detailed vehicle trajectory error [T.sub.error] and correlative mathematical statistics are calculated in Table 4.

The Table 4 indicates that the mean [[??].sub.error] and the mean square error [[sigma].sub.T] of [T.sub.error] for the skilled driver are much less than the novice one with the same DLC cases. The variance of the Gauss sideslip angle perception model for the skilled driver is smaller than the novice one with the same desired trajectory according to the Figure 3, contributing to the more precise and consistent trajectory track performance of the skilled driver. Besides, [[??].sub.error] and [[sigma].sub.T] demonstrate an increasing tendency regarding both the skilled and novice drivers with the increasing nonlinearity of vehicle. The real sideslip angle increases with the increasing v and the decreasing [muj], increasing the variance of the Gauss sideslip angle perception model and leading to the decreasing trajectory track performance eventually.

In short, the envelop regions, the calculated mean [[??].sub.error] and mean square error [[sigma].sub.T] from the SMPC driver model match the human driver very well with various steering skills and vehicle lateral dynamics characterized via different DLC cases, and the proposed SMPC driver model describes the stochastic characteristic of driver's steering skills precisely.

CONCLUSIONS

A study on modeling the driver steering skills considering the stochastic characteristic is performed and a steering controller based on SMPC is proposed. In addition, the vehicle trajectory error with various driver's steering skills and vehicle lateral dynamics is calculated and discussed via mathematical statistics including its mean and mean square error to explore their influence on the stochastic characteristic of driver's steering behavior. The results indicate the skilled driver performs the trajectory track task more precisely and consistently compared with the novice one, and both the skilled and novice drivers present a decreasing trajectory track performance with the increasing nonlinearity of vehicle due to the decreasing perception accuracy of vehicle states.

The proposed SMPC driver model is applicable for simple road conditions without any other traffic participants or obstacles, and its robustness will be explored in future study.

REFERENCES

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[26.] Maciejowski, J. M., "Predictive Control: With Constraints," (Harlow, 1st Pearson Education Limited, 2002), ISBN 978-0-20139-823-6.

[27.] MacAdam, C. C., "An optimal preview control for linear systems," Journal of Dynamic Systems, ASME: Vol. 102, No. 3, pp. 188-190, 1980.

[28.] MacAdam, C. C., "Application of an optimal preview control for simulation of closed-loop automobile driving," IEEE Transactions on Systems, Man and Cybernetics: Vol. 11, No. 6, pp. 393-399, 1981.

[29.] Cole, D. J., Pick, A. J. and Odhams, A. M. C., "Predictive and linear quadratic methods for modeling driver steering control," Vehicle System Dynamics: Vol. 44, No. 3, pp. 259-284, 2006.

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[32.] Guo, K. H., "Vehicle handling dynamics," (Nanjing, Jiang Su Phoenix Science Press, 2011), ISBN 978-7-53456-994-4.

[33.] Guo, K. H., Lu, D., Chen, S. K., et al., "The UniTire model: a nonlinear and non-steady-state tyre model for vehicle dynamics simulation," Vehicle System Dynamics: Vol. 43, No. Suppl1, pp. 341-358, 2005.

[34.] Bai, Y, Guan, H. and Chen, Y. C, "Vehicle ease of control evaluation via stochastic driver model", Proceedings of the FISITA 2012 World Automotive Congress, Beijing, China, November 27-30, 2012.

Chunlei Wang, Xinjie Zhang, Konghui Guo, Fangwu Ma, and Dong Chen

ASCL, Jilin University

CONTACT INFORMATION

Xinjie Zhang

State Key Laboratory of Automotive Simulation and Control

Jilin University, China

5988 Renmin Street, Changchun, Jilin Prov. China, 130022

Tel: +86 (431)85095090exit6108

xjzhang5885@gmail.com

ACKNOWLEDGMENTS

Special thanks are due to the National Natural Science Foundation of China (51205155), the Natural Science Foundation of Jilin Province (20140520136JH), the Natural Science Foundation of Jilin University for Distinguished Young Scholar (450060521093) and the China Automobile Industry Innovation and Development Joint Fund (U1564213) for supporting authors' research on "Application of Stochastic Model Predictive Control to Modeling Driver Steering Skills".

NOMENCLATURE

a - Distance from vehicle mass center to front axle

b - Distance from vehicle mass center to rear axle

[F.sub.y] - Tire lateral force

[F.sub.yf] - Front wheel lateral force

[F.sub.yr] - Rear wheel lateral force

i - Point number of the desired trajectory

I - Vehicle rotational inertia

m - Vehicle mass

T - Sample time

[T.sub.d] - Delay time

[T.sub.p] - Preview time

y - Lateral displacement of vehicle mass center

[alpha] - Tire sideslip angle

[[GAMMA].sub.[DELTA]U] - Steering burden weighting factor

[[GAMMA].sub.Y] - Trajectory error weighting factor

[[delta].sub.f] - Front wheel angle

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] - Theoretical steering wheel angle

[[delta].sub.sw] - Practical steering wheel angle

[psi] - Vehicle heading angle

[lambda] - Steering angle ratio

[mu] - Road friction coefficient

APPENDIX

UNITIRE TIRE MODEL

A simple UniTire steady state tire model [33] under the pure cornering conditions is presented as follows.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In addition, all the parameter values of the UniTire model above are shown as follows.

[F.sub.z0]      5000 N

[PK.sub.y1]      0.1456
[PK.sub.y2]      0.0913
[PK.sub.y3]      0.0063
[PE.sub.y1]     74.0294
[PE.sub.y2]     27.7642
[P.sub.[mu]y1]   1.4274
[P.sub.[mu]y2]  -0.0671
[P.sub.[mu]y3]  -0.0925
[P.sub.[mu]y4]   0.8225
[P.sub.[mu]y5]   0.0605
[P.sub.[mu]y6]  -0.0276
[P.sub.[mu]y7]  -1.1629
[P.sub.[mu]y8]  -1.8269

Table 1. Desired trajectory information

Point number  X-coordinate  7-coordinate

1             [X.sub.1]     [Y.sub.1]
2             [X.sub.2]     [Y.sub.2]
n             [X.sub.n]     [Y.sub.n]

Table 2. Identification results (DLC, [v.sub.x] =80 km/h, [mu] =0.75)

           Skilled driver  Novice driver

[P.sub.1]   0.0144          0.0392
[P.sub.2]  -0.00981        -0.0346

Table 3. SMPC driver model parameters

Symbol                  Value (unit)

m                       1650 kg
I                       2450 kg*[m.sup.2]
a                          0.922 m
b                          1.723 m
[lambda]                  23.2
[T.sub.P]                  0.8 s
[T.sub.d]                  0.1 s
T                          0.01 s
[[GAMMA].sub.[DELTA]U]     0.4
[[GAMMA].sub.Y]            0.6
[N.sub.p]                 20
[N.sub.u]                 19

Table 4. Comparison on DLC vehicle trajectory error

Conditions  [v.sub.x]=65 km/h, [mu] =0.8

                     Skilled   Skilled SMPC  Novice    Novice SMPC
                     driver    driver model  driver    driver model

Min [T.sub.error]     0.015 m   0.011 m       0.023 m   0.020 m
Max [T.sub.error]     0.023 m   0.025 m       0.038 m   0.041 m
[[??].sub.error]      0.016 m   0.017m        0.033 m   0.033 m
[[sigma].sub.[tau]]  12.78%    11.53%        16.11%    17.35%

Conditions  [v.sub.x] = 80 km/h,,[mu] =0.75

Controller           Skilled   Skilled SMPC  Novice    Novice SMPC
                     driver    driver model  driver    driver model

Mm [T.sub.error]      0.010 m   0.012 m       0.040 m   0.032 m
Max [T.sub.error]     0.035 m   0.035 m       0.055 m   0.059 m
[[??].sub.error]      0.021 m   0.022 m       0.048 m   0.046 m
[[sigma].sub.[tau]]  16.58%    17.47%        23.55%    25.53%

conditions  [v.sub.x] = 95 km/h, [mu] =0.7

Controller           Skilled   Skilled SMPC  Novice    Novice SMPC
                     driver    driver model  driver    driver model

Mill [T.sub.error]    0.028     0.025 m       0.052 m   0.053 m
Max [T.sub.error]     0.059     0.060 m       0.160 m   0.199 m
[[??].sub.error]      0.045 m   0.040 m       0.094 m   0.129 m
[[sigma].sub.[tau]]  25.44%    26.33%        37.14%    42.19%

Note: [T.sub.error] = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN
ASCII], where x=155-225 m and sample number n=10.
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Author:Wang, Chunlei; Zhang, Xinjie; Guo, Konghui; Ma, Fangwu; Chen, Dong
Publication:SAE International Journal of Passenger Cars - Mechanical Systems
Article Type:Report
Date:Apr 1, 2016
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