Vehicle sideslip estimation via kernel-based LPV identification: Theory and experiments

Abstract

Many vehicle control systems depend on the body sideslip angle, but robust and cost-effective direct measurement of this angle is yet to be achieved for production vehicles. Estimation from indirect measurements is thus the only viable option. In the paper, a sideslip estimator is obtained through the identification of a linear parameter varying (LPV) model. Although inspired by physical insights into the vehicle lateral dynamics, the structure of the LPV estimator is not parametrized beforehand. Instead, the estimator is learned by means of a state-of-the-art non-parametric method for linear parameter varying identification, namely least-squares support vector machines (LS-SVM). Its performance is assessed over an extensive and heterogeneous set of experimental data, showing the effectiveness of the proposed estimator.

Introduction

Active vehicle dynamics control is playing a pivotal role in the reduction of road accidents (Ferguson, 2007), as proven by the fact that Electronic Stability Controls (ESCs), which stabilize the vehicle by acting on the hydraulic brakes, are standard on all commercial cars. One of its primary objectives is to avoid an excessive body sideslip angle (the planar angle between the vehicle’s longitudinal axis and its velocity vector) (Liebemann, Meder, Schuh, & Nenninger, 2004). Unfortunately, this angle is not easily measurable (no low-cost direct instrumentation exists so far) and it needs to be estimated on-line through commonly available sensors. Given the role played by this angle, both industry and academic researchers have dedicated a great effort to its estimation.

Sideslip estimators can be classified into two families: physics-based and black-box approaches. The first class of estimators exploits the available knowledge on the vehicle dynamics by relying on existing physical models, whose parameters can be retrieved from experimental data (Fergani et al., 2017, Gaspar et al., 2007). On the contrary, black-box approaches rely on (usually nonlinear) models of the vehicle dynamics that are structured and optimized only to match experimental data. Physics-based estimators can be divided in two sub-categories, namely kinematic-based and dynamic-based estimators (Singh et al., 2019, Ungoren et al., 2004). The first use simple vehicle models that correlate kinematic quantities (Panzani et al., 2009, Selmanaj et al., 2017). As a result, these methods do not depend on the specific vehicle or tire friction parameters, but are prone to drift during straight-driving, when the kinematic model is non observable (Selmanaj et al., 2017). Conversely, dynamic-based approaches assume some prior knowledge on the parameters characterizing the model, which are generally vehicle and road dependent. So, although they can be very accurate, these estimators usually require a parallel road friction estimation (Chen & Hsieh, 2008) or adaptive tire models (van Aalst, Naets, Boulkroune, De Nijs, & Desmet, 2018). Some of these limitations are overcome by hybrid approaches, where both kinematic and dynamic information are employed (Galluppi et al., 2018, Imsland et al., 2006, Liao and Borrelli, 2019, Piyabongkarn et al., 2009). Indeed, these methods mix a general kinematic approach, which does not include any vehicle-dependent parameters, with a very simple lateral dynamic model of the vehicle. However, some assumptions are still required that might not be verified at different operating conditions. In turn, these might result in an inaccurate reconstruction of the sideslip angle.

If experimental data are sufficiently informative, these limitations can be overcome by black-box approaches, which rely almost completely on data to design the sideslip angle estimator. However, many black-box models, e.g., neural networks (Bonfitto et al., 2019, Melzi and Sabbioni, 2011), lack of an intuitive physical interpretation, which might be essential for practitioners.

Inspired by Cerone, Piga, and Regruto (2011), in this paper we introduce a data-driven estimator for the sideslip angle $\beta$, designed via the identification of an input–output (IO) linear parameter varying (LPV) model for its dynamics. Real-time estimates of the sideslip angle can then be retrieved by simulating the obtained LPV model. Due to the chosen model class, the estimator is characterized by a linear input–output relationship, but its coefficients are functions of one or more time-varying measurable signals, the so-called scheduling signals, so that the resulting model is non-stationary (Shamma, 2012). Since the dependence of the model coefficients on the scheduling signal is unknown, one of the main tuning knobs in constructing the estimator lies in the selection of their structure, which can heavily influence the resulting accuracy. In the foundational paper (Cerone et al., 2011), a set-membership approach is exploited to retrieve the estimator from data, with the problem of structure selection dealt with by defining the model coefficients as combinations of prefixed basis functions. However, it is well known that the use of a large set of basis functions might lead to an over-parametrized model. At the same time, the choice of the wrong basis functions leads to a potential structural bias (Vapnik, 1998). To overcome these problems, we learn the sideslip estimator via the support vector machine (SVM) based approach presented in Tóth, Laurain, Zheng, and Poolla (2012). The estimator is tested against a large dataset acquired by performing a considerable variety of maneuvers, resulting in an exhaustive overview of its performance.

Although the structure of the proposed estimator is inspired to the one presented in (Cerone et al., 2011), since we still rely on a simple mathematical model for the lateral dynamics to choose its architecture, the scheduling signals are selected to account for additional indicators on possible nonlinear operating regimes of the tires, error propagations and numerical issues. Moreover, by using a non-parametric approach, we do not select any basis function beforehand. Instead, the coefficients of the LPV estimator are directly retrieved from data, thus reducing the required tuning effort, while improving the robustness of the estimator to improper structural choices. The tuning effort is further reduced by replacing traditional grid optimization with Bayesian Optimization (BO). Because of its nature, similarly to Cerone et al. (2011) and differently from Fergani et al. (2017) and Gaspar et al. (2007), the estimator shares the appealing features of black-box ones, while maintaining a fairly straightforward connection to physics.

The paper is organized as follows. The structure of the estimator is discussed in Section 2. The nonparametric method used to design the LPV sideslip angle estimator is outlined in Section 3, along with the approach used for the automatic selection of the tunable parameters. Extensive experimental results are discussed in Section 4. Concluding remarks and directions for future research are finally reported in Section 5.