Extended model predictive control scheme for smooth path following of autonomous vehicles

Abstract

This paper presents an extended model predictive control (MPC) scheme for implementing optimal path following of autonomous vehicles, which has multiple constraints and an integrated model of vehicle and road dynamics. Road curvature and inclination factors are used in the construction of the vehicle dynamic model to describe its lateral and roll dynamics accurately. Sideslip, rollover, and vehicle envelopes are used as multiple constraints in the MPC controller formulation. Then, an extended MPC method solved by differential evolution optimization algorithm is proposed to realize optimal smooth path following based on driving path features. Finally, simulation and real experiments are carried out to evaluate the feasibility and the effectiveness of the extended MPC scheme. Results indicate that the proposed method can obtain the smooth transition to follow the optimal drivable path and satisfy the lateral dynamic stability and environmental constraints, which can improve the path following quality for better ride comfort and road availability of autonomous vehicles.

Graphical abstract

Keywords

autonomous vehicles / vehicle dynamic modeling / model predictive control / path following / optimization algorithm

References

[1] LiK Q, GaoF, LiS E, ZhengY, GaoH. Robust cooperation of connected vehicle systems with eigenvalue-bounded interaction topologies in the presence of uncertain dynamics. Frontiers of Mechanical Engineering, 2018, 13( 3): 354– 367

[2] SpielbergN A, BrownM, KapaniaN R, KegelmanJ C, GerdesJ C. Neural network vehicle models for high-performance automated driving. Science Robotics, 2019, 4( 28): aaw1975–

[3] BadueC, GuidoliniR, CarneiroR V, AzevedoP, CardosoV B, ForechiA, JesusL, BerrielR, PaixãoT M, MutzF, dePaula Veronese L, Oliveira-SantosT, DeSouza A F. Self-driving cars: a survey. Expert Systems with Applications, 2021, 165 : 113816–

[4] GonzálezD, PérezJ, MilanésV, NashashibiF. A review of motion planning techniques for automated vehicles. IEEE Transactions on Intelligent Transportation Systems, 2016, 17( 4): 1135– 1145

[5] GuoJ H, LuoY G, LiK Q, DaiY. Coordinated path-following and direct yaw-moment control of autonomous electric vehicles with sideslip angle estimation. Mechanical Systems and Signal Processing, 2018, 105 : 183– 199

[6] ThrunS, MontemerloM, DahlkampH, StavensD, AronA, DiebelJ, FongP, GaleJ, HalpennyM, HoffmannG, LauK, OakleyC, PalatucciM, PrattV, StangP, StrohbandS, DupontC, JendrossekL E, KoelenC, MarkeyC, RummelC, van NiekerkJ, JensenE, AlessandriniP, BradskiG, DaviesB, EttingerS, KaehlerA, NefianA, MahoneyP. Stanley: the robot that won the DARPA Grand Challenge. Journal of Field Robotics, 2006, 23( 9): 661– 692

[7] SniderJ M. Automatic steering methods for autonomous automobile path tracking. Dissertation for the Doctoral Degree. Pittsburgh: Carnegie Mellon University Pittsburgh, 2009

[8] Morales J, Martínez J L, Martínez M A, Mandow A. Pure-pursuit reactive path tracking for nonholonomic mobile robots with a 2D laser scanner. EURASIP Journal on Advances in Signal Processing, 2009, (1): 935237

[9] ChenQ Y, SunZ P, LiuD X, LiX. Ribbon model based path tracking method for autonomous ground vehicles. Journal of Central South University, 2014, 21( 5): 1816– 1826

[10] ZhangC Z, HuJ F, QiuJ B, YangW, SunH, ChenQ. A novel fuzzy observer-based steering control approach for path tracking in autonomous vehicles. IEEE Transactions on Fuzzy Systems, 2019, 27( 2): 278– 290

[11] MarinoR, ScalziS, NettoM. Nested PID steering control for lane keeping in autonomous vehicles. Control Engineering Practice, 2011, 19( 12): 1459– 1467

[12] MohammadzadehA, TaghavifarH. A robust fuzzy control approach for path-following control of autonomous vehicles. Soft Computing, 2020, 24( 5): 3223– 3235

[13] HuangW, WongP K, WongK I, VongC M, ZhaoJ. Adaptive neural control of vehicle yaw stability with active front steering using an improved random projection neural network. Vehicle System Dynamics, 2021, 59( 3): 396– 414

[14] ZhaoP, ChenJ J, SongY, TaoX, XuT, MeiT. Design of a control system for an autonomous vehicle based on adaptive-PID. International Journal of Advanced Robotic Systems, 2012, 9( 2): 44–

[15] AbdelhakimG, AbdelouahabH. A new approach for controlling a trajectory tracking using intelligent methods. Journal of Electrical Engineering & Technology, 2019, 14( 3): 1347– 1356

[16] GuoJ H, HuP, LiL H, WangR. Design of automatic steering controller for trajectory tracking of unmanned vehicles using genetic algorithms. IEEE Transactions on Vehicular Technology, 2012, 61( 7): 2913– 2924

[17] KritayakiranaK, GerdesJ C. Using the centre of percussion to design a steering controller for an autonomous race car. Vehicle System Dynamics, 2012, 50(sup1 S1): 33– 51

[18] KapaniaN R, GerdesJ C. Design of a feedback-feedforward steering controller for accurate path tracking and stability at the limits of handling. Vehicle System Dynamics, 2015, 53( 12): 1687– 1704

[19] ChenW W, ZhaoL F, WangH R, HuangY. Parallel distributed compensation/H_∞ control of lane-keeping system based on the Takagi-Sugeno fuzzy model. Chinese Journal of Mechanical Engineering, 2020, 33( 1): 61–

[20] CalzolariD, SchurmannB, AlthoffM. Comparison of trajectory tracking controllers for autonomous vehicles. In: Proceedings of IEEE 20th International Conference on Intelligent Transportation Systems (ITSC). Yokohama: IEEE, 2017

[21] FalconeP, BorrelliF, AsgariJ, TsengH E, HrovatD. Predictive active steering control for autonomous vehicle systems. IEEE Transactions on Control Systems Technology, 2007, 15( 3): 566– 580

[22] FalconeP, BorrelliF, TsengH E, AsgariJ, HrovatD. Linear time-varying model predictive control and its application to active steering systems: stability analysis and experimental validation. International Journal of Robust and Nonlinear Control, 2008, 18( 8): 862– 875

[23] PengH N, WangW D, AnQ, XiangC, LiL. Path tracking and direct yaw moment coordinated control based on robust MPC with the finite time horizon for autonomous independent-drive vehicles. IEEE Transactions on Vehicular Technology, 2020, 69( 6): 6053– 6066

[24] SuhJ, ChaeH, YiK. Stochastic model-predictive control for lane change decision of automated driving vehicles. IEEE Transactions on Vehicular Technology, 2018, 67( 6): 4771– 4782

[25] LiZ J, DengJ, LuR Q, XuY, BaiJ, SuC Y. Trajectory-tracking control of mobile robot systems incorporating neural-dynamic optimized model predictive approach. IEEE Transactions on Systems, Man, and Cybernetics: Systems, 2016, 46( 6): 740– 749

[26] AmerN H, ZamzuriH, HudhaK, KadirZ A. Modelling and control strategies in path tracking control for autonomous ground vehicles: a review of state of the art and challenges. Journal of Intelligent & Robotic Systems, 2017, 86( 2): 225– 254

[27] SongY T, ShuH Y, ChenX B. Chassis integrated control for 4WIS distributed drive EVs with model predictive control based on the UKF observer. Science China. Technological Sciences, 2020, 63( 3): 397– 409

[28] TagneG, TaljR, ChararaA. Design and comparison of robust nonlinear controllers for the lateral dynamics of intelligent vehicles. IEEE Transactions on Intelligent Transportation Systems, 2016, 17( 3): 796– 809

[29] ChowdhriN, FerrantiL, IribarrenF S, ShyrokauB. Integrated nonlinear model predictive control for automated driving. Control Engineering Practice, 2021, 106 : 104654–

[30] GuoH Y, LiuJ, CaoD P, ChenH, YuR, LvC. Dual-envelop-oriented moving horizon path tracking control for fully automated vehicles. Mechatronics, 2018, 50 : 422– 433

[31] BealC E, GerdesJ C. Model predictive control for vehicle stabilization at the limits of handling. IEEE Transactions on Control Systems Technology, 2013, 21( 4): 1258– 1269

[32] JiJ, KhajepourA, MelekW W, HuangY. Path planning and tracking for vehicle collision avoidance based on model predictive control with multiconstraints. IEEE Transactions on Vehicular Technology, 2017, 66( 2): 952– 964

[33] LiX H, SunZ P, CaoD P, LiuD, HeH. Development of a new integrated local trajectory planning and tracking control framework for autonomous ground vehicles. Mechanical Systems and Signal Processing, 2017, 87 : 118– 137

[34] MerabtiH, BelarbiK, BouchemalB. Nonlinear predictive control of a mobile robot: a solution using metaheuristcs. Journal of the Chinese Institute of Engineers, 2016, 39( 3): 282– 290

[35] BrownM, FunkeJ, ErlienS, GerdesJ C. Safe driving envelopes for path tracking in autonomous vehicles. Control Engineering Practice, 2017, 61 : 307– 316

[36] GuoH Y, LiuF, XuF, ChenH, CaoD, JiY. Nonlinear model predictive lateral stability control of active chassis for intelligent vehicles and its FPGA implementation. IEEE Transactions on Systems, Man, and Cybernetics: Systems, 2019, 49( 1): 2– 13

[37] DuX X, HtetK K K, TanK K. Development of a genetic-algorithm-based nonlinear model predictive control scheme on velocity and steering of autonomous vehicles. IEEE Transactions on Industrial Electronics, 2016, 63( 11): 6970– 6977

[38] GuoH Y, ShenC, ZhangH, ChenH, JiaR. Simultaneous trajectory planning and tracking using an MPC method for cyber-physical systems: a case study of obstacle avoidance for an intelligent vehicle. IEEE Transactions on Industrial Informatics, 2018, 14( 9): 4273– 4283

[39] CaoH T, SongX L, ZhaoS, BaoS, HuangZ. An optimal model-based trajectory following architecture synthesising the lateral adaptive preview strategy and longitudinal velocity planning for highly automated vehicle. Vehicle System Dynamics, 2017, 55( 8): 1143– 1188

[40] Di CairanoS, TsengH E, BernardiniD, BemporadA. Vehicle yaw stability control by coordinated active front steering and differential braking in the tire sideslip angles domain. IEEE Transactions on Control Systems Technology, 2013, 21( 4): 1236– 1248

[41] XiaoL J, WangM, ZhangB J, ZhongZ. Vehicle roll stability control with active roll-resistant electro-hydraulic suspension. Frontiers of Mechanical Engineering, 2020, 15( 1): 43– 54

[42] LiuQ J, ChenW, HuH S, ZhuQ, XieZ. An optimal NARX neural network identification model for a magnetorheological damper with force-distortion behavior. Frontiers in Materials, 2020, 7 : 10–

Nomenclature

Cdj, Cj, C0j	Cornering stiffness coefficient
Cφ	Combined roll damping coefficient
CR	Crossover probability within [0, 1]
ds	Desired minimum safe distance
D	Objective control input size
ey	Lateral error
eyl	Left road boundary
eyr	Right road boundary
eψ	Heading error
Envmax	Maximum feasible road region
Envmin	Minimum feasible road region
f	Fitness function
Fyf	Lateral force of front tire
Fyr	Lateral force of rear tire
g	Gravitational acceleration
h	Distance from the spring mass to the roll center
Ix	Roll moment of inertia
Iz	Yaw moment of inertia
k	Discrete time
Kφ	Combined roll stiffness coefficient
lf	Distance from center to front axle
lr	Distance from center to rear axle
LTR	Lateral load transfer rate
LTRd	Equivalent LTR
LTRmax	Maximum value of LTRd
m	Total vehicle mass
ms	Sprung vehicle mass
Nc	Control horizon
Np	Prediction horizon
P	Population size
r	Vehicle yaw rate
rand	Random number with a uniform probability distribution in [0, 1]
rd	Randomly generated integral number in [1, 2, ..., D]
S	Driving path length
Tr	Vehicle track
Ts	Sample time
uij	New experimental input individual
vij	Mutation individual of Δδfij
vx	Longitudinal velocity
vy	Lateral velocity
w	Vehicle width
x0	Longitudinal position of vehicle origin
∆x	Distance between two adjacent points in x direction
y0	Lateral position of vehicle origin
yF	Lateral position of front axle

yR	Lateral position of rear axle
∆y	Distance between two adjacent points in y direction
αhj	Saturation slip angle
αj	Tire slip angle
αf	Front tire slip angle
αr	Rear tire slip angle
αt	Limit tire slip angle
β	Vehicle sideslip angle
δf	Front steer angle
δfmax	Maximum value of δf
δfmin	Minimum value of δf
ε	Slack variable
φ	Roll angle of vehicle
φr	Road bank angle
ηr	Robust mutation factor
\mathit{\text{κ}}	Reference road curvature
λ	Slack weight
σ	Penalty factor
ρ	Driving path curvature
ψ	Yaw angle
ψref	Reference heading angle
∆δfmax	Maximum of steer angle increment
∆δfmin	Minimum of steer angle increment
Δδfij	Initial individual
{\text{Г}}_{\mathit{\text{κ}}}	Weight factor of ρ
Гs	Weight factor of S
Гu	Weight factor of ∆u
Гψ	Weight factor of eψ
A	Augmented state matrix
Ac	State matrix
Ad	Discrete state matrix
Am	System state matrix
\tilde{\mathit{A}}	Predicted state matrix for output system
{\tilde{\mathit{A}}}_{\text{h}}, {\tilde{\mathit{B}}}_{\text{uh}}, {\tilde{\mathit{B}}}_{\text{vh}}	Coefficient matrices for hard constraints
{\tilde{\mathit{A}}}_{\text{s}}, {\tilde{\mathit{B}}}_{\text{us}}, {\tilde{\mathit{B}}}_{\text{vs}}	Coefficient matrices for soft constraints
{\tilde{\mathit{A}}}_x	Predicted state matrix
Bu	Augmented input matrix
Buc	Input matrix
Bud	Discrete input matrix
Bum	System input matrix
Bv	Augmented disturbance matrix
Bvc	Disturbance matrix
Bvd	Discrete disturbance matrix
Bvm	System disturbance matrix
{\tilde{\mathit{B}}}_{\text{u}}	Predicted input matrix for output system
{\tilde{\mathit{B}}}_{\text{u}x}	Predicted input matrix
{\tilde{\mathit{B}}}_{\text{v}}	Predicted disturbance matrix for output system
{\tilde{\mathit{B}}}_{\text{v}x}	Predicted disturbance matrix

C	Augmented output matrix
Cc	Output matrix
Cd	Discrete output matrix
E	Output deviation
E1, E2, Ed2	Constraint state matrix
F1, F2	Constraint input matrix
Hk, Vk, Gk, Pk	Quadratic transformation matrices
Jc	Conventional cost function
Je	Optimized cost function
J(Env)	Penalty function of feasible road region
M	System matrix
M1, M2	Constraint output matrix
Q	Output weight matrix
R	Input weight matrix
u	Control input
umax	Maximum value of u
umin	Minimum value of u
Ui	Experimental control input sequence
∆u	Input increment
∆umax	Maximum value of ∆u
∆umin	Minimum value of ∆u
∆Uc	Input variation
Vi	Mutation control input
Wi	Initial population
x	State vector
Xp	State prediction
y	Output vector
yhmax	Hard constraint of upper output bound
yhmin	Hard constraint of lower output bound
ymax	Maximum value of y
ymin	Minimum value of y
yref	Desired reference output
ysmax	Soft constraint of upper output bound
ysmin	Soft constraint of lower output bound
Yp	Output prediction
Yref	Reference output matrix
γ	Disturbance input
ξ	Augmented state vector

Acknowledgements

This work was funded by the National Natural Science Foundation of China (Grant No. 52075461), the Key Project in Science and Technology Plan of Xiamen, China (Grant No. 3502Z20201015), and the Science and Technology Plan of Fujian Province of China (Grant No. 2021H6019).

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