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

Abstract

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.

Nonparametric identification of nonlinear dynamic systems using a synchronisation-based method

“…A stochastic approximation discussed in [6] belongs to observerbased methods and optimal estimation algorithms such as the Kalman filter. Nonlinear extensions of the Kalman filter have been successfully implemented for multiple DoF mechanical structures in [7,8] to estimate coefficients of strongly nonlinear characteristics, and in [9] to estimate inertia properties of ground vehicles. An interesting off-line time domain procedure applied in [10] combines nonlinear optimisations with optimal estimations to achieve an objective function surface with fewer local minima.…”

“…The solution to this problem is the so-called state augmentation, which introduces the unknown parameters in (1) and (2) as additional state variables to achieve a model that is assumed correctly and to let the Kalman filter estimate the parameters. With this approach, the augmented state vector will be of size n a , where the number of unknown parameters equals n a À n. This technique was successfully implemented for multiple DoF mechanical structures to observe unknown excitation forces in [21], to estimate coefficients of strongly nonlinear characteristics in [7,8] and to estimate inertia properties in [9]. For a 1DoF nonlinear oscillator derived from (4), the unknown restoring force is introduced as a state variable.…”

“…For the estimation of constant or slowly varying parameters, the Kalman filter is typically tuned in order for the synchronisation error to converge to the measurement error, which provides a smooth noise-free estimate of augmented state variables. This practice can be observed in [7][8][9], where constant parameters of mechanical systems are estimated using different types of nonlinear Kalman filters. In this study, a different tuning strategy is proposed to handle the high-frequency oscillations of the augmented states introduced in Section 4.1.…”

Methods in Vehicle Mass and Road Grade Estimation

“…In this paper a discrete-time filter is formulated using an Euler approximation for the following state vector at time k. (6) v x is longitudinal vehicle velocity while θ 1 and θ 2 are defined in equation (5). The system propagates in one time step as (7) with differentiate state transition model (8) and process noise (9) T s is the sampling period, and process noise w(k) is assumed to be zero-mean with diagonal covariance matrix Q. Velocity is the observable quantity, thus the observation model is (10) The observation matrix H is [1 0 0] and v(k + 1) represents observation noise with zero mean and covariance R. The model described in equations (7,8,9,10,11) may be implemented with the extended Kalman filter. A thorough explanation of the filter, including linearization and equations for the prediction and update step, is found in [13].…”

“…This avoids redundant computations and conflicting parameter estimates, which is crucial when a supervisory controller is coordinating the behavior of multiple control systems [6]. While there have been numerous studies for inertial load estimation which employ various prediction methods and modeling approaches [3,7,8,9,10], many previous efforts toward on-line mass and grade estimation using existing on-board sensor systems have been based on vehicle longitudinal dynamics models due to the many common driving scenarios for which such models apply. Estimation approaches include recursive least squares (RLS) with multiple forgetting factors [11,12,13], extended Kalman filtering (EKF) [5], a dynamic grade observer (DGO) [14] requiring only longitudinal acceleration and an estimate of powertrain torque, and grade estimation using kinematic information provided by a longitudinal accelerometer [15].…”