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Archive of Mechanical Engineering

The Journal of Committee on Machine Building of Polish Academy of Sciences

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Real-Time Parameter Estimation Study for Inertia Properties of Ground Vehicles

Jeremy Kolansky
  • Mechanical Engineering Department AVDL, Virginia Tech 9L Randolph Hall, Blacksburg, VA 24061
  • Email:
/ Corina Sandu
  • Mechanical Engineering Department AVDL, Virginia Tech 104 Randolph Hall, Blacksburg, VA 24061
  • Email:
Published Online: 2013-03-27 | DOI: https://doi.org/10.2478/meceng-2013-0001

Vehicle parameters have a significant impact on handling, stability, and rollover propensity. This study demonstrates two methods that estimate the inertia values of a ground vehicle in real-time.

Through the use of the Generalized Polynomial Chaos (gPC) technique for propagating the uncertainties, the uncertain vehicle model outputs a probability density function for each of the variables. These probability density functions (PDFs) can be used to estimate the values of the parameters through several statistical methods. The method used here is the Maximum A-Posteriori (MAP) estimate. The MAP estimate maximizes the distribution of P(β | z) where β is the vector of the PDFs of the parameters and z is the measurable sensor comparison.

An alternative method is the application of an adaptive filtering method. The Kalman Filter is an example of an adaptive filter. This method, when blended with the gPC theory is capable at each time step of updating the PDFs of the parameter distributions. These PDF’s have their median values shifted by the filter to approximate the actual values.


Parametry pojazdu maja znaczny wpływ na jego własciwosci, takie jak sterowalnosc, stabilnosc i odpornosc na wywrócenie. W pracy przedstawiono dwie metody estymacji parametrów inercyjnych pojazdu terenowego w czasie rzeczywistym.

W modelu pojazdu z niepewnosciami wyznacza sie funkcje gestosci prawdopodobienstwa (PDF) dla kazdej wielkosci opisujac propagacje niepewnosci przez zastosowanie techniki uogólnionego chaosu wielomianowego (gPC). Funkcje te moga byc uzyte do estymacji wartosci parametrów przy wykorzystaniu róznych metod statystycznych. W pracy zastosowano metode maksymalnego estymatora a posteriori (MAP). Estymator MAP maksymalizuje funkcje rozkładu P(β | z), gdzie β jest wektorem funkcji gestosci prawdopodobienstwa parametrów, a z jest wielkoscia mierzalna, otrzymana z porównania wyjsc czujników.

Metoda alternatywna jest zastosowanie filtru adaptacyjnego, którego przykładem jest filtr Kalmana. Metoda ta, w połaczeniu z technika uogólnionego chaosu wielomianowego (gPC), umozliwia, w kazdym kroku adaptacji, uaktualnianie funkcji gestosci prawdopodobienstwa (PDF) parametrów systemu. Działanie filtru powoduje, ze mediany tych funkcji zmieniaja sie dazac do rzeczywistych wartosci poszukiwanych parametrów.

Keywords : Parameter Estimation; EKF; Polynomial Chaos; Bayesian Statistics

  • [1] Pence B.L., e.a.: Recursive Bayesian Parameter Estimation Using Polynomial Chaos Theory, 2010.

  • [2] Blanchard E.D., Sandu A., Sandu C.: Polynomial Chaos-Based Parameter Estimation Methods Applied to Vehicle System. IMechE, Vol. 223, 2009.

  • [3] Blanchard E.D., Sandu A., Sandu C.: PSM: A Polynomial Chaos-Based Kalman Filter Approach for Parameter Estimation of Mechanical Systems. Journal of Dynamic Systems, Measurement, and Control, Vol. 132, 2010. [Web of Science]

  • [4] Cheng H., Sandu A.: Efficient uncertainty quantification with the polynomial chaos method for stiff systems. Mathematics and Computers in Simulation, Vol. 79, pp. 3278-3295, 2009. [Web of Science]

  • [5] Cheng H., Sandu A.: Uncertainty Quantification in 3D Air Quality Models using Polynomial Chaoses. Environmental Modeling and Software, 2009.

  • [6] Cui Y., Kurfess T.R.: Influence of Parameter Variation for System Identification of Pitch-heave Car Model, 2010.

  • [7] Fathy H.K., Kang D., Stein J.L.: Online Vehicle Mass Estimation Using Recursive Least Squares and Supervisory Data Extraction. American Control Conference, 2008.

  • [8] Hays J.: Parametric Optimal Design Of Uncertain Dynamical Systems, 2011.

  • [9] III, S.K.S.: Vehicle Sprung Mass Parameter Estimation Using an Adaptive Polynomial-Chaos Method.

  • [10] Julier S.J., Uhlmann J.K.: A New Extension of the Kalman Filter to Nonlinear Systems.

  • [11] Lankarany M., Rezazade A.: Parameter estimation optimization based on genetic algorithm applied to DC motor. 2007 International Conference On Electrical and Electronics, 2007.

  • [12] Marzouk Y.M., Najm H.N., Rahn L.A.: Stochastic spectral methods for efficient Bayesian solution of inverse problems. J. of Computational Physics, Vol. 224, pp. 560-586, 2007. [Web of Science]

  • [13] McIntyre M.L., Ghotikar T.J., Vahidi A., Song X., Darren M. Dawson: A Two-Stage Lyapunov-Based Estimator for Estimation of Vehicle Mass and Road Grade. IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, Vol. 58, 2009. [Web of Science]

  • [14] Merwe R.v.d., Wan E.A., Julier S.I.: Sigma-Point Kalman Filters for Nonlinear Estimation and Sensor-Fusion - Applications to Integrated Navigation.

  • [15] Pence B., Hays J., Fathy H., Sandu C., Stein J.: Vehicle Sprung Mass Estimation for Rough Terrain, 2011.

  • [16] Pence B.L., Fathy H.K., Stein J.L.: A Base -Excitation Approach to Polynomial Chaos-Based Estimation of Sprung Mass for Off-Road Vehicles.

  • [17] Rozyn M., Zhang N.: A method for estimation of vehicle inertial parameters. Vehicle System Dynamics, Vol. 48, pp. 547-565, 2010. [Web of Science]

  • [18] Sandu A., Sandu C., Ahmadian M.: Modeling Multibody Systems With Uncertainties. Part I: Theoretical And Computational Aspects. Multibody Syst Dyn, 2006.

  • [19] Sandu C., Sandu A., Ahmadian M.: Modeling multibody systems with uncertainties. Part II: Numerical applications. Multibody Syst Dyn, Vol. 15, 2006.

  • [20] Southward S.C.: REAL-TIME PARAMETER ID USING POLYNOMIAL CHAOS EXPANSIONS. ASME International Mechanical Engineering Congress and Exposition, 2007.

  • [21] Vahidi A., Stefanopoulou A., Peng H.: Recursive Least Squares with Forgetting for Online Estimation of Vehicle Mass and Road Grade: Theory and Experiments.

  • [22] Welch G., Bishop G.: An Introduction to the kalman Filter. 2006.

  • [23] Xiu D.: Efficient collocational approach for parametric uncertainty analysis. Communications in Computational Physics, Vol. 2, pp. 293-309, 2007.

  • [24] Xiu D.: Fast numerical methods for stochastic computations: a review. Communications in Computational Physics, Vol. 5, pp. 242-272, 2009.

  • [25] Xiu D., Hesthaven J.S.: High-Order Collocation Methods for Differential Equations with Random Inputs. SIAM Journal of Scientific Computing, Vol. 27, pp. 118-1139, 2005.

  • [26] Zarchan P., Musoff H.: Fundamentals of Kalman Filtering: A practical Approach vol. 232: American Institute of Aeronautics and Astronautics, Inc., 2009.

About the article

Published Online: 2013-03-27

Published in Print: 2013-03-01

Citation Information: Archive of Mechanical Engineering, ISSN (Print) 0004-0738, DOI: https://doi.org/10.2478/meceng-2013-0001. Export Citation

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