Parameter Identiﬁcation of Magic Formula Tire Model Based on Fibonacci Tree Optimization Algorithm

doi:10.1007/s12204-021-2354-9

Abstract

Abstract: The magic formula (MF) tire model is a semi-empirical tire model that can precisely simulate tire behavior. The heuristic optimization algorithm is typically used for parameter identification of the MF tire model. To avoid the defect of the traditional heuristic optimization algorithm that can easily fall into the local optimum, a parameter identification method based on the Fibonacci tree optimization (FTO) algorithm is proposed, which is used to identify the parameters of the MF tire model. The proposed method establishes the basic structure of the Fibonacci tree alternately through global and local searches and completes optimization accordingly. The global search rule in the original FTO was modified to improve its efficiency. The results of independent repeated experiments on two typical multimodal function optimizations and the parameter identification results showed that FTO was not sensitive to the initial values. In addition, it had a better global optimization performance than genetic algorithm (GA) and particle swarm optimization (PSO). The root mean square error values optimized with FTO were 5.09%, 10.22%, and 3.98% less than the GA, and 6.04%, 4.47%, and 16.42% less than the PSO in pure lateral and longitudinal forces, and pure aligning torque parameter identi?cation. The parameter identification method based on FTO was found to be effective.

Key words: magic formula, tire model, parameter identi?cation, Fibonacci tree optimization (FTO) algorithm

CLC Number:

U 463.341

FENG Shilin (冯世林), ZHAO Youqun (赵又群), DENG Huifan (邓汇凡), WANG Qiuwei(王秋伟), CHEN Tingting (陈婷婷). Parameter Identiﬁcation of Magic Formula Tire Model Based on Fibonacci Tree Optimization Algorithm[J]. J Shanghai Jiaotong Univ Sci, 2021, 26(5): 647-657.

References 26

1]	PENG D, YIN C L, ZHANG J W. An investigation into regenerative braking control strategy for hybrid electric vehicle [J]. Journal of Shanghai Jiao Tong Uni-versity (Science), 2005, 10(4): 407-412. [2] JAZAR R N. Vehicle dynamics: Theory and applica-tion [M]. Boston: Springer, 2008.
[3]	BESSELINK I J M, SCHMEITZ A J C, PACEJKA H B. An improved Magic Formula/Swift tyre model that can handle in?ation pressure changes [J]. Vehicle System Dynamics, 2010, 48(sup1): 337-352.
[4]	ORTIZ A, CABRERA J A, GUERRA A J, et al. An easy procedure to determine Magic Formula parame-ters: A comparative study between the starting value optimization technique and the IMMa optimization al-gorithm [J]. Vehicle System Dynamics, 2006, 44(9):
68	9- 718.
[5]	VANOOSTENJJM,BAKKERE.Determination of magic tyre model parameters [J]. Vehicle System Dynamics, 1992, 21(sup001): 19-29.
[6]	KIM J. Identi?cation of lateral tyre force dynamics using an extended Kalman ?lter from experimental road test data [J]. Control Engineering Practice, 2009, 17(3): 357-367.
[7]	FARRONI F, LAMBERTI R, MANCINELLI N, et al. TRIP-ID: A tool for a smart and interactive identi?-cation of Magic Formula tyre model parameters from experimental data acquired on track or test rig [J]. Mechanical Systems and Signal Processing, 2018, 102:
1-	22.
[8]	CHENG Z, LU Z. Nonlinear research and e?cient pa-rameter identi?cation of magic formula tire model [J]. Mathematical Problems in Engineering, 2017, 2017: 6924506.
[9]	CABRERA J A, ORTIZ A, CARABIAS E, et al. An alternative method to determine the magic tyre model parameters using genetic algorithms [J]. Vehicle Sys-tem Dynamics, 2004, 41(2): 109-127.
[10]	ZHUO G, WANG J, ZHANG F. Parameter iden-ti?cationoftiremodel basedonimprovedparticle swarm optimization algorithm [J]. SAE Technical Pa-per, 2015: 2015-01-1586.
[11]	TALEBITOOTI R, TORABI M. Identi?cation of tire force characteristics usingaHybridmethod[J]. Ap-plied Soft Computing, 2016, 40: 70-85. [12] ALAGAPPAN A V, RAO K V N, KUMAR R K. A comparison of various algorithms to extract Magic For-mula tyre model coe?cients for vehicle dynamics sim-ulations [J]. Vehicle System Dynamics, 2015, 53(2):
15	4- 178.
[13]	GUO L, MENG Z, SUN Y, et al. Parameter identi?ca-tion and sensitivity analysis of solar cell models with cat swarm optimization algorithm [J]. Energy Conver-sion and Management, 2016, 108: 520-528.
[14]	SHIEH H L, KUO C C, CHIANG C M. Modi?ed particle swarm optimization algorithm with simulated annealing behavior and its numerical veri?cation [J]. Applied Mathematics and Computation, 2011, 218(8): 4365-4383.
[15]	ZHANG W F. Simpli?ed group search optimizer algo-rithm for large scale global optimization [J]. Journal of Shanghai Jiao Tong University (Science), 2015, 20(1):
38	- 43.
[16]	WANG X, WANG Y, WU H, et al. Fibonacci multi-modal optimization algorithm in noisy environment [J]. Applied Soft Computing, 2020, 88: 105874.
[17]	ETMINANIESFAHANI A, GHANBARZADEH A, MARASHI Z. Fibonacci indicator algorithm: A novel tool for complex optimization problems [J]. Engineer-ing Applications of Arti?cial Intelligence, 2018, 74:1-9.
[18]	SUBASI M, YILDIRIM N, YILDIZ B. An improve-ment on Fibonacci search method in optimization the-ory [J]. Applied Mathematics and Computation, 2004, 147(3): 893-901. [19] KIM H S, NEGGERS J. Fibonacci mean and golden section mean [J]. Computers & Mathematics with Ap-plications, 2008, 56(1): 228-232.
[20]	HORADAM A F. A generalized ?bonacci sequence [J]. The American Mathematical Monthly, 1961, 68(5):
45	5- 459.
[21]	CHIPPERFIELD A J, FLEMING P J. The MAT-LAB genetic algorithm toolbox [C]//IEE Colloquium on Applied Control Techniques Using MATLAB.Lon-don, UK: IEE, 1995: 1-4.
[22]	SHI Y, EBERHART R. A modi?ed particle swarm optimizer [C]//1998 IEEE International Conference on Evolutionary Computation Proceedings. Anchorage, AK, USA: IEEE, 1998: 69-73.
[23]	ADORIO E P, DILIMAN U P. MVF: Multi-variate test functions library in C for uncon-strained global optimization [EB/OL]. (2005-01-14). http://www.geocities.ws/eadorio/mvf.pdf.
[24]	DAMAVANDI N, SAFAVI-NAEINI S. A hybrid evo-lutionary programming method for circuit optimiza-tion [J]. IEEE Transactions on Circuits and Systems I : Regular Papers, 2005, 52(5): 902-910.
[25]	PACEJKA H B, BAKKER E. The magic formula tyre model [J]. Vehicle System Dynamics, 1992, 21(sup001): 1-18.
[26]	Formula SAE Tire Test Consortium. Calspan Tire Research Facility Project Number: 1654 [EB/OL].[2020-12-01]. https://github.com/shiliu520/Formula-SAE-Tire-Test-Data.