Nonlinear optimal control for the five-axle and three-steering coupled-vehicle system

Abstract

Transportation of heavy loads is often performed by multi-axle multi-steered heavy duty vehicles In this article a novel nonlinear optimal control method is applied to the kinematic model of the five-axle and three-steering coupled vehicle system. First, it is proven that the dynamic model of this articulated multi-vehicle system is differentially flat. Next. the state-space model of the five-axle and three-steering vehicle system undergoes approximate linearization around a temporary operating point that is recomputed at each time-step of the control method. The linearization is based on Taylor series expansion and on the associated Jacobian matrices. For the linearized state-space model of the five-axle and three-steering vehicle system a stabilizing optimal (H-infinity) feedback controller is designed. This controller stands for the solution of the nonlinear optimal control problem under model uncertainty and external perturbations. To compute the controller’s feedback gains an algebraic Riccati equation is repetitively solved at each iteration of the control algorithm. The stability properties of the control method are proven through Lyapunov analysis. The proposed nonlinear optimal control approach achieves fast and accurate tracking of setpoints under moderate variations of the control inputs and minimal dispersion of energy by the propulsion and steering system of the five-axle and three-steering vehicle system.

Keywords

Five axle and three-steering vehicle / Heavy-duty vehicles / Cooperative load transportation / Differential flatness properties / Nonlinear H-infinity control / Taylor series expansion / Riccati equation / Global stability

References

[1.] Y. Tan, X. Su, O. Shen, J. Huang, Motion tracking control of six-wheel skid-steering electric vehicles via two-level sequence-based integrated algorithm. IEEE Trans. Ind. Electron., 1–9 (2024)

[2.] Shi X., Wang H., Chen L., Sun X., Yang C., Cai X.. Robust path-tracking control of distributed driving six-wheel steering commercial vehicle based on active disturbance rejection. IEEE Trans. Veh. Technol., 2023, 72(11):13940-13952.

[3.] Z. Li, J. Li, Y. Li, An integrated path-planning framework for multi-obstacle avoidance of the multi-axle autonomous vehicle with enhanced safety and stability. IEEE Trans. Veh. Technol., 1–15 (2024)

[4.] Z. Li, J. Li, Q. Min, B. Lin, A fast-computing path-tracking control strategy for autonomous multi-axle electric vehicle considering safety and stability. IEEE Trans. Transp. Electr., 1–14 (2024)

[5.] Zhang Y., Huang Y., Wang H., Khajepour A.. A comparative study of equivalent modelling for multi-axle vehicle. Veh. Syst. Dyn., 2018, 56(3):443-460

[6.] Li Z., Li J., Wang W.. Path planning and obstacles’ avoidance control for continuous multi-axis distributed vehicles based on dynamic constraints. IEEE Trans. Veh. Technol., 2023, 72(4):4342-4356

[7.] Bu Z., Fengod G., Tian L.. Normalized coupling method for speed synchronization of multi-axis driving vehicles. Int. J. Adv. Robot. Syst., 2012, 9(50):1-10.

[8.] Kim J.S., Lea D.K., Ahn C.K.. Receding horizon directional unscented filter for heavy-duty vehicles incorporating sensor modelling constraints. Measurement, 2021, 183: 109874-109881

[9.] Jiang Y., Meng H., Chen C., Yang C., Xu X., Zhang L., Xu H.. Differential steering-based path tracking control and energy-saving torque distribution strategy of 6WID unmanned ground vehicle. Energy, 2022, 254: 124209-124223

[10.] Beyerdorfer S., Wagner S.. Novel model-based path planning for multi-axle steered heavy load vehicles. IEEE ITSC 2013, IEEE 46th Annual Conference on Intelligent Transportation Systems, Hague, The Netherlands, 2013

[11.] Ahmed M., El-Girdy M., Liang H.. Path-following enhancement of an $8{\times}8$ combat vehicle using active real axles steering strategies. Proc. Inst. Mech. Eng. Part K, J. Multi-Body Dyn., 2021, 235(4):539-552.

[12.] Zhang Y., Khajepour A., Huang Y.. Multi-axle/articulated bus dynamics modelling: a reconfigurable approach. Veh. Syst. Dyn., 2018, 56(9):1315-1343

[13.] Jiang Y., Xu X., Zhang L.. Heading tracking of 6WID/4WIS unmanned ground vehicles with variable wheelbase based on model-free adaptive control. Mech. Syst. Signal Process., 2021, 159: 107715-107732

[14.] Wen Deng Z., Zha Q.X., Zhao Y.Q., Hun Wang B., Gao W., Kong X.X.. Active LQR multi-axle-steering method for improving maneuverability and stability of multi-trailer articulated heavy vehicles. Int. J. Automot. Technol., 2022, 23(4):939-955

[15.] Kim S.H., Kim D.H., Kim C.J., Kim Y.R., Kim Y.J., Han C.S.. A study on motion control of 6WD/6WS vehicle using optimum tire force distribution method. Intl. Conf. On Control, Automation and Systems, 2010. Korea: Gyeonygi-do

[16.] Shaohua L., Shaopu Y., Ligan C.. Investigation on cornering brake stability of a heavy-duty vehicle based on a nonlinear three-dimensional coupled model. Appl. Math. Model., 2016, 40: 6310-6323

[17.] Liu M., Huang J., Cao M.. Handling stability improvement for a four-axle hybrid-electric ground vehicle driven by in-wheel motors. IEEE Access, 2018, 6: 2668-2684

[18.] Juan J., Sun F., Huang Y.. Trajectory generation and tracking control for double-steering tractor-trailer mobile robot with an axle-hitching. IEEE Trans. Ind. Electron., 2015, 62(12):7665-7677

[19.] Hu J., Li J., Hu Z., Zhang B., Xin L., Ouyang M.. Energy-efficient torque allocation strategy for a $6{\times}6$ vehicle using electric wheels. eTransportation, 2021, 10: 100136-100150

[20.] Zhang Y., Khajepour A., Ataei M.. A universal and reconfigurable stability control methodology for articulated vehicles with any configurations. IEEE Trans. Veh. Technol., 2020, 69(4):3748-3759

[21.] Li Y., He L., Yang L.. Path following control for multi-axle car-like wheeled mobile robot with nonholonomic constraints, 2013. Australia: Wollowong

[22.] Kim C., Ashfaq A.M., Kim S., Back S., Kim Y., Hwang S., Jung J., Han C.. Motion control of a 6WD/6WS wheeled platform with in-wheel motors to improve its maneuverability. Int. J. Control Autom. Syst., 2015, 13(2):434-442

[23.] Yamaguchi H., Nishijima A., Kawakami A.. Control of two manipulation points of a cooperative transportation system with two car-like vehicles following parametric curve paths. Robotic and Autonomous Systems, 2015. Amsterdam: Elsevier 165-178 63

[24.] Yamaguchi H., Kameyama R., Kawakami A.. A path-following feedback control law for a five-axle three-steering coupled vehicle system. J. Robot. Mechatron., 2014, 26(5):551-565

[25.] Yamaguchi H., Mori M., Kawakami A.. Control of a five-axle three-steering coupled vehicle system and its experimental verification. IFAC WC 2011, Proc. Of the 18th World Congress of the International Federation of Automatic Control, Milan, Italy, 2011

[26.] Yamaguchi H., Sakagami Y., Yanezawa N.. Control of a five-axle three-steering articulated vehicle system - deformation of paths adapting to geometrical constraints of external and internal states, 2023. Tsu City: Society of Instrumentation and Control Engineers

[27.] Wang B., Sun Y., Zhang J., Wang W.. A motion decoupling control based on differential geometry for distributed drive articulated heavy vehicle. Int. J. Automot. Technol., 2024, 25: 381-398

[28.] Kim W., Yi K., Lee J.. Drive control algorithm for an independent 8 in-wheel motor drive vehicle. J. Mech. Sci. Technol., 2011, 25(6):1573-1581

[29.] Zhang Y., Khajepour A., Hashemi E., Qin Y., Huang Y.. Reconfigurable model predictive control for articulated vehicle stability with experimental validation. IEEE Trans. Transp. Electr., 2020, 6(1):308-317

[30.] Xu T., Liu X., Li Z., Feng B., Ji X., Wu F.. A sliding-mode control scheme for steering flexibility and stability in all-wheel-steering multi-axle vehicles. Int. J. Control Autom. Syst., 2023, 21: 1926-1938

[31.] Ljungqvist O., Bergman K., Axehill D.. Optimization-based motion planning for multi-steered articulated vehicles, 2020

[32.] Rigatos G.. Nonlinear Control and Filtering Using Differential Flatness Theory Approaches: Applications to Electromechanical Systems, 2015. Berlin: Springer

[33.] Rigatos G., Abbaszadeh N., Hamida M., Siano P.. Intelligent Control for Electric Power Systems and Electric Vehicles, 2024 CRC Press, Boca Raton

[34.] G. Rini, V. Mazzilli, F. Bottiglione, P. Gruber, M. Dhaens, A. Sorniotti, On nonlinear model predictive vehicle dynamics control for multi-actuated car-semitrailer configurations. IEEE Access, 1–19 (2024)

[35.] Rigo D., Sansonetto N., Muradore R.. Geometric optimal filtering for an articulated n-trailer vehicle with unknown parameters. Int. J. Robust Nonlinear Control, 2024, 34: 11868-11886

[36.] Rigatos G., Busawon K.. Robotic Manipulators and Vehicles: Control, Estimation and Filtering, 2018. Berlin: Springer

[37.] Rigatos G., Karapanou E.. Advances in Applied Nonlinear Optimal Control, 2020 Cambridge Scholars Publishing, Berlin

[38.] Rigatos G., Busawon K., Abbaszadeh M.. A nonlinear optimal control approach for the truck and N-trailer robotic system. IFAC J. Syst. Control, 2022, 20: 100191.

[39.] Rigatos G., Siano P., Wira P., Busawon K., Binns R.. A nonlinear H-infinity control approach for autonomous truck and trailer systems. Unmanned Syst., 2020, 8(1):49-69

[40.] Rigatos G., Abbaszadeh M., Siano P.. Control and Estimation of Dynamical Nonlinear and Partial Differential Equation Systems: Theory and Applications, 2022 IET Publications

[41.] Rigatos G.G., Tzafestas S.G.. Extended Kalman Filtering for Fuzzy modelling and multi-sensor fusion. Mathematical and Computer Modelling of Dynamical Systems, 2007. London: Taylor & Francis 251-266 13

[42.] Basseville M., Nikiforov I.. Detection of Abrupt Changes: Theory and Applications, 1993. New York: Prentice Hall

[43.] Rigatos G., Zhang Q.. Fuzzy model validation using the local statistical approach. Fuzzy Sets Syst., 2009, 60(7):882-904

[44.] Toussaint G.J., Basar T., Bullo F.. $H_{\infty}$ optimal tracking control techniques for nonlinear underactuated systems. Proc. IEEE CDC 2000, 39th IEEE Conference on Decision and Control, Sydney Australia, 2000

[45.] Grimble M.J., Majeski P.. Nonlinear Industrial Control Systems: Optimal Polynomial Systems and State-Space Approach, 2020. Berlin: Springer

[46.] Corriou J.P.. Numerical Methods and Optimization: Theory and Practice for Engineers, 2021. Berlin: Springer

[47.] Tao K.L., Li B., Rehbock Y.. Applied and Computational Optimal Control: A Control Parametrization Approach, 2021. Berlin: Springer

[48.] Kim W.G., Kang J.Y., Yi K.. Drive control system design for stability and maneuverability of a 6WD/6WS vehicle. Int. J. Automot. Technol., 2011, 12(1):67-74

Funding

Unit of Industrial Automation / Industrial Systems Institute(Ref 301022 - Nonlinear optimal and flatness-based control methods and applications); Deanship of Scientific Research, King Saud University http://dx.doi.org/10.13039/501100011665(Researchers Supporting Project Number (RSP2024R150))