Corina Sandu

A Polynomial Chaos-Based Kalman Filter Approach for Parameter Estimation of Mechanical Systems

Background. Mechanical systems operate under parametric and external excitation uncertainties. The polynomial chaos approach has been shown to be more efficient than Monte Carlo for quantifying the effects of such uncertainties on the system response. Many uncertain parameters cannot be measured accurately, especially in real time applications. Information about them is obtained via parameter estimation techniques. Parameter estimation for large systems is a difficult problem, and the solution approaches are computationally expensive. Method of Approach. This paper proposes a new computational approach for parameter estimation based on the Extended Kalman Filter (EKF) and the polynomial chaos theory for parameter estimation. The error covariances needed by EKF are computed from polynomial chaos expansions, and the EKF is used to update the polynomial chaos representation of the uncertain states and the uncertain parameters. The proposed method is applied to a nonlinear four degree of freedom roll plane model of a vehicle, in which an uncertain mass with an uncertain position is added on the roll bar. Results. The main advantages of this method are an accurate representation of uncertainties via polynomial chaoses, a computationally efficient update formula based on EKF, and the ability to provide aposteriori probability densities of the estimated parameters. The method is able to deal with non-Gaussian parametric uncertainties. The paper identifies and theoretically explains a possible weakness of the EKF with approximate covariances: numerical errors due to the truncation in the polynomial chaos expansions can accumulate quickly when measurements are taken at a fast sampling rate. To prevent filter divergence we propose to lower the sampling rate, and to take a smoother approach where a set of time-distributed observations are all processed at once. Conclusions. We propose a parameter estimation approach that uses polynomial chaoses to propagate uncertainties and estimate error covariances in the EKF framework. Parameter estimates are obtained in the form of a polynomial chaos expansion which carries information about the aposteriori probability density function. The method is illustrated on a roll plane vehicle model.

Parameter estimation for mechanical systems via an explicit representation of uncertainty

Purpose – The purpose of this paper is to propose a new computational approach for parameter estimation in the Bayesian framework. A posteriori probability density functions are obtained using the polynomial chaos theory for propagating uncertainties through system dynamics. The new method has the advantage of being able to deal with large parametric uncertainties, non‐Gaussian probability densities and nonlinear dynamics.Design/methodology/approach – The maximum likelihood estimates are obtained by minimizing a cost function derived from the Bayesian theorem. Direct stochastic collocation is used as a less computationally expensive alternative to the traditional Galerkin approach to propagate the uncertainties through the system in the polynomial chaos framework.Findings – The new approach is explained and is applied to very simple mechanical systems in order to illustrate how the Bayesian cost function can be affected by the noise level in the measurements, by undersampling, non‐identifiablily of the sy...

A POLYNOMIAL­CHAOS­BASED BAYESIAN APPROACH FOR ESTIMATING UNCERTAIN PARAMETERS OF MECHANICAL SYSTEMS

This is the second part of a two-part article. In the first part, a new computational approach for parameter estimation was proposed based on the application of the polynomial chaos theory. The maximum likelihood estimates are obtained by minimizing a cost function derived from the Bayesian theorem. In this part, the new parameter estimation method is illustrated on a nonlinear four-degree-of-freedom roll plane model of a vehicle in which an uncertain mass with an uncertain position is added on the roll bar. The value of the mass and its position are estimated from periodic observations of the displacements and velocities across the suspensions. Appropriate excitations are needed in order to obtain accurate results. For some excitations, different combinations of uncertain parameters lead to essentially the same time responses, and no estimation method can work without additional information. Regularization techniques can still yield most likely values among the possible combinations of uncertain parameters resulting in the same time responses than the ones observed. When using appropriate excitations, the results obtained with this approach are close to the actual values of the parameters. The accuracy of the estimations has been shown to be sensitive to the number of terms used in the polynomial expressions and to the number of collocation points, and thus it may become computationally expensive when a very high accuracy of the results is desired. However, the noise level in the measurements affects the accuracy of the estimations as well. Therefore, it is usually not necessary to use a large number of terms in the polynomial expressions and a very large number of collocation points since the addition of extra precision eventually affects the results less than the effect of the measurement noise. Possible applications of this theory to the field of vehicle dynamics simulations include the estimation of mass, inertia properties, as well as other parameters of interest.

Multibody dynamics modelling and system identification of a quarter-car test rig with McPherson strut suspension

In this paper, we develop a multibody dynamics model of a quarter-car test-rig equipped with a McPherson strut suspension and we apply a system identification technique on it. Constrained equations of motion in the Lagrange multiplier form are derived and employed to characterise the dynamic behaviour of the test rig modelled once as a linear system and once as a non-linear system. The system of differential algebraic equations is integrated using a Hilber–Hughes–Taylor integrator. The responses of both models (linear and non-linear) to a given displacement input are obtained and compared with the experimental response recorded using the physical quarter-car test rig equipped with a McPherson strut suspension. The system identification is performed for control purposes. The results, as well as the performance and area of applicability of the test rig models derived, are discussed.

Stochastic Modeling of Terrain Profiles and Soil Parameters

One fundamental difficulty in understanding the physics of the off-road traction and in predicting vehicle performance is the variability of the terrain profile and soil parameters. These operating conditions are uniquely defined at a given spatial location and a given time. It is not practically feasible, however, to measure them at a sufficiently large number of points to be able to accurately represent the terrain in models. This renders traditional analysis tools insufficient when dealing with rough deformable terrain. We employ stochastic analysis to capture the uncertain nature of this running support and the corresponding vehicle response. From a finite number of observations the terrain profile and soil properties can be modeled as random processes, with the actual operating conditions viewed as a particular realization of these processes. Soil parameters vary substantially from one type of soil to another. Moreover, for the same soil type, the parameters change with environmental conditions difficult to predict, for example the moisture content. Such uncertain soil parameters are modeled in this study as uniform or normal distributed random variables.

Modeling and Simulation of a Full Vehicle With Parametric and External Uncertainties

This paper presents the mathematical development of and simulation results for a full vehicle model with parametric uncertainties operating over unprepared terrain. The vehicle is modeled as a rigid multi-body dynamic system, consisting of chassis and four suspension and tire subsystems. The vehicle parameters considered uncertain are the suspension damping and the tire stiffness. The terrain profile is also modeled as a stochastic function. The uncertainties are explicitly represented using polynomial chaos decompositions. The computational technique presented in this study is more efficient than the traditional Monte Carlo approach, in modeling nonlinear multi-body dynamic systems with uncertainties. The numerical results presented here are very promising. The general computational tools discussed in this paper can be applied directly to any area that involves multi-body dynamic models, e.g., robotics, autonomous mechanical systems, actuator dynamics, and automatic control of systems with uncertainties.Copyright

Polynomial chaos-based parameter estimation methods applied to a vehicle system

Abstract Parameter estimation for large systems is a difficult problem, and the solution approaches are computationally expensive. The polynomial chaos approach has been shown to be more efficient than Monte Carlo for quantifying the effects of uncertainties on the system response. This article compares two new computational approaches for parameter estimation based on the polynomial chaos theory for parameter estimation: a Bayesian approach, and an approach using an extended Kalman filter (EKF) to obtain the polynomial chaos representation of the uncertain states and the uncertain parameters. The two methods are applied to a non-linear four-degree-of-freedom roll plane model of a vehicle, in which an uncertain mass with an uncertain position is added on the roll bar. When using appropriate excitations, the results obtained with both approaches are close to the actual values of the parameters, and both approaches can work with noisy measurements. The EKF approach has an advantage over the Bayesian approach: the estimation comes in the form of a posteriori probability densities of the estimated parameters. However, it can yield poor estimations when dealing with non-identifiable systems, and it is recommended to repeat the estimation with different sampling rates in order to verify the coherence of the results with the EKF approach. The Bayesian approach is more robust, can recognize non-identifiability, and use regularization techniques if necessary.

Treating Uncertainties in Multibody Dynamic Systems Using a Polynomial Chaos Spectral Decomposition

This study addresses the critical need for computational tools to model complex nonlinear multibody dynamic systems in the presence of parametric and external uncertainty. Polynomial chaos has been used extensively to model uncertainties in structural mechanics and in fluids, but to our knowledge they have yet to be applied to multibody dynamic simulations. We show that the method can be applied to quantify uncertainties in time domain and frequency domain.© 2004 ASME

Vehicle sprung mass estimation for rough terrain

This paper provides methods and experimental results for recursively estimating the sprung mass of a vehicle driving on rough terrain. A base-excitation model of vertical ride dynamics treats the unsprung vertical accelerations, instead of the terrain profile, as the input to ride dynamics. Recently developed methods based on polynomial chaos and maximum likelihood theory estimate the most likely value of the vehicle sprung mass. The polynomial chaos estimator is compared to least squares and Kalman filtering approaches. An experimental study suggests that the proposed approach provides accurate outputs and is less sensitive to tuning parameters than the benchmark algorithms.

Comparison of Linear, Nonlinear, Hysteretic, and Probabilistic Models for Magnetorheological Fluid Dampers

Magnetorheological (MR) fluid dampers have a semicontrollable damping force output that is dependent on the current input to the damper, as well as the relative velocity. The mechanical construction, fluid properties, and embedded electromagnet result in a dynamic damper response. This study evaluates four modeling approaches with respect to predicting the multi-input single-output behavior of an experimental MR damper when the inputs are band-limited random signals typically encountered in primary suspension applications. The first two models in this study are static in the sense that there is a unique output for any given set of inputs and no dynamics is present in either model. The third model incorporates a dynamic filter with the nonlinear model to exhibit hysteretic effects, which are known to exist in actual MR dampers. The fourth model is probabilistic and illustrates the dynamic nature of an actual MR damper. The results of this study clearly show the importance of nonlinear and dynamic effects in magnetorheological damper response. This study also highlights the importance of characterizing magnetorheological dampers using excitation signals that are representative of an actual implementation.