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Robust Vehicle Sideslip Estimation Based on Kinematic Considerations
Proceedings of the 20th World Congress Proceedings of 20th The International Federation of Congress Automatic Control Proceedings of the the 20th World World Congress Proceedings of the 20th9-14, World The Federation of Automatic Control Toulouse, France, July 2017 The International International Federation of Congress Automatic Control Available online at www.sciencedirect.com The International of Automatic Control Toulouse, France, July Toulouse, France,Federation July 9-14, 9-14, 2017 2017 Toulouse, France, July 9-14, 2017
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IFAC PapersOnLine 50-1 (2017) 14855–14860
Robust Vehicle Sideslip Estimation Robust Vehicle Sideslip Estimation Robust Vehicle Sideslip Estimation RobustonVehicle Sideslip Estimation Kinematic Considerations on Kinematic Considerations on Kinematic Considerations on Kinematic Considerations ∗ ∗∗ ∗∗
Based Based Based Based
D. Selmanaj ∗ M. Corno ∗∗ G. Panzani ∗∗ S.M. Savaresi ∗∗ ∗∗ D. G. Panzani ∗∗ S.M. Savaresi ∗∗ D. Selmanaj Selmanaj ∗∗ M. M. Corno Corno ∗∗ ∗∗ G. Panzani ∗∗ S.M. Savaresi ∗∗ D. Selmanaj M. Corno G. Panzani S.M. Savaresi ∗ ∗ Institute for Dynamic Systems and Control, ETH Zurich, Zurich ∗ Institute for Dynamic Systems and Control, ETH Zurich, Zurich for Dynamic and Control, ETH Zurich, Zurich 8092, Switzerland andSystems Department of Automation, Polytechnic ∗ Institute Institute for Dynamic Systems and Control, ETH Zurich, Zurich 8092, and Department of Polytechnic 8092, Switzerland Switzerland and Department of Automation, Automation, Polytechnic University of Tirana, Tirana 1001, Albania 8092, Switzerland and Department of Automation, Polytechnic University of Tirana, Tirana 1001, Albania ∗∗ University of Tirana, Tirana 1001, Albania di Elettronica, Informazione e Bioingegneria, ∗∗ Dipartimento University of Tirana, Tirana 1001, Albania ∗∗ Dipartimento di Informazione ee32, Bioingegneria, di Elettronica, Elettronica, Informazione Bioingegneria, Politecnico di Milano. Piazza Leonardo da Vinci 20132 Milano, ∗∗ Dipartimento Dipartimento di Elettronica, Informazione e32, Bioingegneria, Politecnico di Milano. Piazza Leonardo da Vinci Politecnico di Milano. Piazza Leonardo da Vinci 32, 20132 20132 Milano, Milano, Italy Politecnico di Milano. Piazza Leonardo da Vinci 32, 20132 Milano, Italy Italy Italy Abstract – This paper proposes an inertial measurement based sideslip angle estimation Abstract – paper an based sideslip angle Abstract – This Thisapplications. paper proposes proposes an inertial inertial measurement based sideslip angle estimation for automotive Measuring sideslipmeasurement angle is costly; in most cases, in estimation most case Abstract – Thisapplications. paper proposes an inertial measurement based sideslip angle estimation for automotive Measuring sideslip angle is costly; in most cases, in case for automotive applications. Measuring sideslip angle is costly; in mostoncases, in most most case estimation is a preferred choice. Most estimation solutions are based dynamical models for automotive applications. Measuring sideslip angle is costly; in mostoncases, in most case estimation is a preferred choice. Most estimation solutions are based dynamical models estimation isToa properly preferredwork, choice. Most estimation solutions are basedthe on road dynamical models of the car. these solutions need to also estimate friction. This estimation isToa properly preferredwork, choice. Most estimation solutions are basedthe on road dynamical models of the car. these solutions need to also estimate friction. This of the car. To complexity properly work, these solutions need toThe alsopresent estimate the proposes road friction. This increases their and affects their robustness. study an inertial of the car. To properly work, these solutions need toThe alsopresent estimate the road friction. This increases their and affects robustness. study an increases their complexity complexity and require affects their their robustness.nor The present studyofproposes proposes an inertial inertial based approach that does not the knowledge the estimation road friction. This increases their complexity and require affects their robustness.nor The present studyofproposes an inertial based approach that does not the knowledge the estimation road friction. This based approach that does not require the and knowledge nor of thethe estimation ofFocus road friction. This considerably simplifies the design, tuning validation approach. is devoted to based approach that does not require the and knowledge nor of the estimation of road friction. This considerably simplifies the design, approach. is to considerably simplifies the design, tuning tuningerrors. and validation validation of the the approach. Focus Focus is devoted devoted to the study of the effect of measurement An extensive experimental validation confirms considerably simplifies the design, tuningerrors. and validation of the approach. Focus is devoted to the of of An extensive experimental validation the study of the the effect effect of measurement measurement An range extensive experimental validation confirms confirms thatstudy the estimate is accurate and robust errors. to a wide of driving scenarios. the study of the effect of measurement errors. An range extensive experimental validation confirms that the estimate is accurate and robust to a wide of driving scenarios. that the estimate is accurate and robust to a wide range of driving scenarios. that theIFAC estimate is accurate and robust to a wide rangeHosting of driving scenarios. © 2017, (International Federation of Automatic Control) by Elsevier Ltd. All rights reserved. Keywords: Side Slip Estimation, Vehicle Dynamics Control, Automotive Keywords: Side Slip Estimation, Vehicle Dynamics Control, Automotive Keywords: Side Slip Estimation, Vehicle Dynamics Control, Automotive Keywords: Side Slip Estimation, Vehicle Dynamics Control, Automotive 1. INTRODUCTION and lateral velocities with longitudinal and lateral acceler1. and lateral velocities with acceler1. INTRODUCTION INTRODUCTION and lateral velocities with longitudinal longitudinal and lateral accelerations and yaw rate. Consequently, theyand do lateral not depend on 1. INTRODUCTION and lateral velocities with longitudinal and lateral accelerations and yaw rate. Consequently, they do not depend on ations and yaw rate. Consequently, they do not depend on vehicle or tyre models. One of the first studies on kinematic Sideslip angle, i.e. the angle between the longitudinal vehicle ations and yaw rate. Consequently, they do not depend on tyre One the first on vehicle or tyre models. models. One of ofand theWellstead first studies studies on kinematic kinematic Sideslip angle, i.e. the angle the longitudinal sideslipor observers is Farrelly (1996) in which Sideslip angle, i.e. the and anglethebetween between the longitudinal direction of the vehicle velocity vector, heavily vehicle or tyre models. One of the first studies on kinematic sideslip observers is Farrelly and Wellstead (1996) in which Sideslip angle, i.e. the and anglethebetween the longitudinal sideslip observers is Farrelly and Wellstead (1996) in which direction of the vehicle velocity vector, heavily a kinematic observer is proven to be asymptotically stable direction of the vehicleofand the velocity vector, heavily influences of thethe dynamics wheeled vehicles.vector, Large side slip sideslip observers is Farrelly andtoWellstead (1996) in stable which a kinematic observer is proven be asymptotically direction vehicle and the velocity heavily a kinematic observer is proven to be asymptotically stable influences the dynamics of wheeled vehicles. Large side slip for all cornering conditions (non-zero yaw rate). From influences the dynamics of wheeled vehicles. Large side slip angle maythe lead to instability. Thevehicles. knowledge of side sideslip kinematic observer is proven(non-zero to be asymptotically stable all cornering conditions yaw rate). influences dynamics of wheeled Large slip afor for all cornering conditions (non-zero yawdeveloped rate). From From angle may lead to instability. The knowledge of sideslip that original paper, the approach has been and angle is may lead to instability. The knowledge of sideslip therefore extremely important in vehicle dynamall cornering conditions (non-zero yawdeveloped rate). From that original has been and angle is may lead toextremely instability. The knowledge of dynamsideslip for that original paper, theetapproach approach hasthe been developed and important in enriched. In paper, Panzanithe al. (2009), authors propose angle is therefore therefore extremely important in vehicle vehicle dynamics control Selmanaj et al. (2013). Sideslip angle can be that original paper, the approach has been developed and enriched. Panzani et (2009), angle is therefore extremely important in vehicle dynamenriched. In Panzani et al. al.Kim (2009), the authors propose ics control Selmanaj al. Sideslip angle an online In bias estimation. and the Ryuauthors (2011) propose designs ics control Selmanaj etoptical al. (2013). (2013). Sideslip angle can can be be enriched. directly measured viaet sensors and dual-antenna In Panzani et al.Kim (2009), the authors propose an online bias estimation. and Ryu (2011) designs ics control Selmanaj et al. (2013). Sideslip angle can be an online bias estimation. Kim and Ryu (2011) designs directly measured via optical sensors and dual-antenna and validates an Extended Kalman Filter (EKF) based directly measured viasolutions optical sensors and dual-antenna GPS systems; these are notand amenable to in- an online bias an estimation. Kim and Filter Ryu (2011) designs and validates Extended Kalman (EKF) based directly measured viasolutions optical sensors dual-antenna and validates an Extended Kalman Filter (EKF) based GPS systems; these are not amenable to inon the kinematic model; the validation is restricted to GPS systems; these solutions are not amenable to industrialization because of cost and dependability reasons. validates an Extended Kalman Filteris (EKF) based on the kinematic model; the validation restricted to GPS systems; because these solutions are dependability not amenablereasons. to in- and on the kinematic model; the validation is restricted to dustrialization of cost and short dynamic maneuvers (around 10s). Another group dustrialization because of cost and dependability reasons. The availabilitybecause of a good cost-effective and robust side on the kinematic model; the validation is restricted to short maneuvers (around group dustrialization of cost and dependability reasons. short dynamic maneuvers (around 10s). Another group The availability of cost-effective and side adoptsdynamic a sliding mode observer in Wei10s). et al. Another (2012). Results The availability of aa good good cost-effective and robust robust side slip angle estimation can make the difference between a short dynamic maneuvers (around 10s). Another group aa sliding mode in et The availability of a good cost-effective and robust side adopts sliding mode observer observer in Wei Weimethods et al. al. (2012). (2012). Results slip angle estimation can make difference aa adopts show that kinematic model-based are Results reliable slip angle active estimation candynamics make the the difference between successful vehicle control and abetween mediocre adopts a sliding mode observer in Weimethods et al. (2012). Results that kinematic model-based are reliable slip angle active estimation candynamics make the difference between a show show that kinematic model-based methods are reliable successful vehicle control and a mediocre for transient maneuvers, but they suffer from estimation successful active vehicle dynamics control and a mediocre one. that kinematic model-based methods are reliable for transient maneuvers, but they estimation successful active vehicle dynamics control and a mediocre show for transient maneuvers, but conditions they suffer sufferasfrom from estimation one. errors on nearly steady-state the integration one. for transient maneuvers, but conditions they sufferas from estimation errors on nearly steady-state the integration one. on nearly steady-state conditions as the integration Sideslip estimation has a long history in automotive publi- errors of measurement errors causesconditions the estimate to diverge. on nearly steady-state as the integration Sideslip estimation has aa long in publiof Sideslip estimation hasapproaches long history history in automotive automotive publi- errors of measurement measurement errors errors causes causes the the estimate estimate to to diverge. diverge. cations. The available are classified in dynamic Sideslip estimation hasapproaches a long history in automotive publi- Researchers measurement errors the causes the estimate to diverge. apcations. The are in dynamic merged dynamic and kinematic cations. The available available approaches are classified classified in Dynamic dynamic of model based, kinematic model based or hybrid. merged the dynamic and kinematic apcations. The available approaches are classified in Dynamic dynamic Researchers Researchers merged the dynamic and kinematic apmodel based, kinematic model based or hybrid. proaches yielding hybrid solutions, where depending on the model based, kinematicBaffet modeletbased or hybrid. Dynamic based methods al. (2009); Coyte et al. Researchers merged the dynamic anddepending kinematic approaches yielding hybrid solutions, where on the model based, kinematic model based or hybrid. Dynamic proaches yielding hybrid solutions, where depending on the based methods Baffet et al. (2009); Coyte et al. conditions the results of the kinematic and dynamics modmodel based methods Baffet et al. (2009); Coyte et al. (2014); based Dakhlallah et al. (2008); Ray (1995); Stephant yielding hybrid solutions, where depending on the conditions the of kinematic and model methods Baffet et al.Ray (2009); Coyte et al. proaches conditions the results results of the the kinematic and dynamics dynamics modmod(2014); Dakhlallah et (1995); Stephant els are combined Oh and Choi (2012); Piyabongkarn et al. (2014); Dakhlallah etetal. al.al.(2008); (2008); Ray (1995); Stephant et al. (2007); Wenzel (2006) are based on dynamic conditions the results of the kinematic and dynamics models are combined Oh and Choi (2012); Piyabongkarn et al. (2014); Dakhlallah et al.al.(2008); Ray (1995); Stephant els are combined Oh and Choi (2012); Piyabongkarn etand al. et al. Wenzel (2006) are based (2009). In particular, Grip et (2012); al. (2009) accuratelyet et al. (2007); (2007); Wenzel et et al.forces (2006) areact based on dynamic models that describe theal. that on on thedynamic vehicle. els are combined Oh and Choi Piyabongkarn al. (2009). In particular, Grip et al. (2009) accurately and et al. (2007); Wenzel et (2006) are based on dynamic (2009). In particular, Grip et al. (2009) accurately and models that describe the forces that act on the vehicle. jointly estimates the sideslip angle, friction coefficient models that them describe the forces that act onbut the vehicle. This makes potentially verythat accurate, are (2009). In particular, Grip et al. (2009) accurately and jointly estimates the friction models that them describe the forces act onbut the they vehicle. jointly estimates the sideslip sideslip angle, frictiontocoefficient coefficient and This makes very are road banking. These methodsangle, are shown be extremely This makes theminpotentially potentially very accurate, accurate, but they they are jointly complex systems which the interplay of different compoestimates the sideslip angle, friction coefficient and road banking. These methods are shown to be extremely This makes theminpotentially very accurate, but they are road banking. These methods are shown to be extremely complex systems which the interplay of different compoaccurate, but they are still subject to the considerable cost complex systems in which theFor interplay of different components is difficulty validated. example, in order compoto fully road banking. These methods are shown to be extremely accurate, but are subject complex systems in which theFor interplay of different accurate, but they they are still still model. subject to to the the considerable considerable cost cost nents is difficulty validated. in to fully of developing an accurate nents istheir difficulty validated. For example, example, in order orderthe to road fully accurate, exploit potential, they need to also estimate but they are still subject to the considerable cost developing an accurate model. nents istheir difficulty validated. For example, in orderthe to road fully of of developing an accurate model. exploit potential, they need to also estimate exploit their potential, they need toconditions also estimate the road The conditions online. Estimating roadto is inthe general developing an accurate present work proposesmodel. a reliable and industrially viexploit their potential, they need also estimate road of conditions online. Estimating road conditions is general present work proposes aa reliable and viconditions online. Estimating road conditions is in independs general The difficult and the quality of the estimate heavily The present work proposes reliable and industrially industrially viable solution to the sideslip angle estimation problem, that conditions online.quality Estimatingthe road conditions is independs general The present work proposes a reliable and industrially vidifficult and estimate able solution to the sideslip angle estimation problem, that difficult and the the qualityofof ofthe the maneuvers. estimate heavily heavily depends on the characteristics Moreover the able solution to the sideslip angle estimation problem, that does not employ any dynamic model. Its tuning is thus difficult and the qualityofofthe the estimate heavily depends solution to the sideslip anglemodel. estimation problem, that on the characteristics the does not dynamic Its is on thealso characteristics of the maneuvers. maneuvers. Moreover the able model requires the knowledge of vehicle Moreover mass and yaw does not employ employ any any dynamic model. Its tuning tuning is thus thus very cost-effective. The algorithm is based on kinematic on thealso characteristics of the maneuvers. Moreover the does not employ any dynamic model. Its tuning is thus model requires the knowledge of vehicle mass and yaw very cost-effective. The algorithm is based on kinematic model also requires the knowledge of vehicle mass and yaw very inertia,also parameters thatknowledge can experience largemass variations. cost-effective. The algorithm is basedThe on heuristics, kinematic considerations augmented with heuristics. model requires the of vehicle and yaw very cost-effective. The algorithm is basedThe on heuristics, kinematic inertia, considerations augmented with heuristics. inertia, parameters parameters that that can can experience experience large large variations. variations. considerations augmented with heuristics. The heuristics, by adding an artificial damping term, prevents the diinertia, parameters that approaches can experience large variations. Kinematic model based are simpler. They are considerations augmented with heuristics. The heuristics, by adding an artificial damping term, prevents the diby adding an artificial damping term, prevents the diKinematic model based approaches are simpler. They are vergence of the estimation for low yaw rate driving. The Kinematic model based approaches arevehicle simpler. They are vergence based on the correlation between the longitudinal addingof an artificial damping term, prevents theThe dithe estimation for low yaw rate driving. Kinematic model based approaches arevehicle simpler. They are by vergence of the estimation for low yaw rate driving. The based on the correlation between the longitudinal based on the correlation between the vehicle longitudinal based on the correlation between the vehicle longitudinal vergence of the estimation for low yaw rate driving. The
Copyright 15420Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © © 2017 2017, IFAC IFAC (International Federation of Automatic Control) Copyright © 2017 15420 Copyright © under 2017 IFAC IFAC 15420Control. Peer review responsibility of International Federation of Automatic Copyright © 2017 IFAC 15420 10.1016/j.ifacol.2017.08.2513
Proceedings of the 20th IFAC World Congress 14856 D. Selmanaj et al. / IFAC PapersOnLine 50-1 (2017) 14855–14860 Toulouse, France, July 9-14, 2017
algorithm is validated in realistic driving tests representing hours worth of driving. These tests include both highly dynamic and nearly steady state maneuvers. Results on different road conditions demonstrate the robustness of the method to varying road conditions. Although dynamic model based approaches are more accurate in ideal conditions, we believe that the more predictable behavior of the proposed approach when facing unknown road conditions, represents a considerable advance that is well appreciated by the industry. The paper is organized as follows. In Section 2, the kinematic observer and the longitudinal vehicle speed estimation are described. Section 3 presents the detailed experimental validation. The paper ends with some concluding remarks. Part of the present work is protected by the patent Selmanaj et al. (2014). 2. KINEMATIC MODEL-BASED OBSERVER Equations (1) summarize the kinematic model according to the reference frame shown in Figure 1. Ax and Ay are the longitudinal and lateral body accelerations, V˙ x and V˙ y the time derivatives of the vehicle velocity and ωz is the yaw rate angle. ωz CG
Vx β
δ
Vy
Figure 1. Single track model. Ax (t) = V˙ x (t) − ωz (t) Vy (t) Ay (t) = V˙ y (t) + ωz (t) Vx (t)
(1)
Rewriting the model in the state space form, one obtains 0 ωz (t) Vx (t) 1 0 Ax (t) V˙ x (t) = + −ωz (t) 0 Vy (t) 0 1 Ay (t) V˙ y (t) [A] [B] (2) Vx (t) y(t) = [1 0] . Vy (t) [C]
In the above model, the yaw rate is modeled as a time varying parameter and the vehicle accelerations as inputs. Given the above symbols and definitions, the sideslip angle is computed as: Vy (t) . (3) β (t) = arctan Vx (t) Based on the model Farrelly and Wellstead (1996) proposed the following nonlinear state observer ˙ Vˆx (t) Vˆx (t) Ax (t) = (A − KC) ˆ +B + KVx , (4) ˙ Ay (t) Vy (t) Vˆy (t) where K, the observer gain T matrix, is defined as K = α2 − 1 ωz (t) . The original formulation 2α|ωz (t)| does not address two issues: (1) Based on K dependency on ωz (t), the longitudinal velocity estimation is updated only in corners. (2) During straight driving, the model looses its full observability and as a consequence, the observer simply
integrates the accelerations. In these conditions, small measurement errors will cause the estimate to diverge. The latter problem is intrinsic and structural with the choice of sensors. It does not depend on the algorithm. A possible solution is to incorporate additional information on the dynamics in the observer. Usually, this is done using a dynamic model, with the difficulties described in the Introduction. Here, we prefer to use a rougher but simpler description of the dynamics. The kinematics described by model (2) describe a free body on a plane; cars are not completely free. Some movements, if not impossible, are very unlikely. For example, it is extremely unlikely for a car to laterally drift without exhibiting any yaw rate; especially if one considers that sideslip angle estimators are often part of an electronic stability system designed to prevent the vehicle from reaching those conditions. This intuitive understanding (or heuristic) can be included in the model using an artificial, non-physical stabilizing term (−F (t)Vy (t)) in the second equation of (2). If F (t) is designed to be positive and large when the vehicle is driving straight, the lateral velocity estimate is prevented to drift due to integration errors. The first issue listed above is improved by modifying the gain term K. The gain matrix term affecting the longitudinal velocity estimate is now composed of yaw rate dependent and constant terms. α1 |ωz | update the observer state on cornering, while α0 guarantees the Vˆx update on straight maneuvers. Vx is the vehicle longitudinal velocity and is obtained from the equivalent linear wheel velocity. The complete observer expression is given in (5): ˙ Vˆx (t) −α0 − α1 |ωz (t)| ωz (t) Vˆx (t) = ˙ −(α2 + 1)ωz (t) −F (t) Vˆy (t) Vˆy (t) [A−K(ωz )C]
1 0 Ax (t) α + α1 |ωz (t)| + + 0 Vx . 0 1 Ay (t) α2 ωz (t) [B]
(5)
[K(ωz )]
The observer implements the heuristic according to which, if the yaw rate is small and constant, the sideslip angle will likely be small. In other words: if F is small (i.e. close to 0) the vehicle state is estimated from the kinematic model and the measurements; if F is large the estimated lateral velocity is driven to a small value, while the longitudinal velocity is updated by Vx . The key of this approach is the scheduling of F (t), described next. In principle, the same results can be achieved by turning off the observer when the vehicle is moving straight and turning it on when a curve is approached. . The solution proposed here guarantees a smoother transition than a sudden switching. 2.1 Heuristic scheduling F determines the strength of the artificial stabilizing term. Figure 2 represents the scheduling rationale. In the figure, δ is the steering wheel angle and β˙ is the sideslip angle derivative, directly estimated from measurements: Ay (t) − ωz (t)Vx (t) ˙ β(t) = . (6) Vx (t) F is scheduled based on indexes that describe whether the vehicle is cornering: sideslip rate, steering angle and yaw
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ωz
d dt
Fδ δ
β˙
d dt
d dt
meas meas meas and VRR are the measured where VFmeas L ,VF R ,VRL equivalent linear velocities of the wheels.
Fωz
ω˙z
×
δ˙
F
Fβ
β¨
Figure 2. Heuristic scheduling. rate and their derivatives. Fωz , Fδ and Fβ are based on bivariate gaussian functions that smoothly vary from zero to a maximum value; and each function outputs 0 when either input is nonzero.
Accounting for the effect of longitudinal slip is more complex. The proposed solution carries out a kinematicbased wheel velocities sensor fusion. The rationale is to identify the lowest slip wheel by exploiting the acceleration of the vehicle. If the vehicle is accelerating the wheels tend to have higher velocity than the vehicle, thus the lowest wheel velocity is the closest to the vehicle velocity. Vice versa, if the vehicle is decelerating, the wheels tend to have lower velocity and the highest wheel velocity is the closest to the vehicle velocity. Acceleration and deceleration maneuvers are identified through thresholds on the longitudinal acceleration measurement (Figure 4). Ax Vx = min (VF L, VF R, VRL, VRR) Athreshold x
2.2 Longitudinal velocity estimation
Vx = Wmean (VF L, VF R, VRL, VRR)
The estimator uses the vehicle longitudinal velocity measurement (Vx ) as the feedback term that, using Kalman filtering nomenclature, drives the aposteriori correction of the estimate. The wheels rotational rates are the main source of information on the vehicle longitudinal velocity. The equivalent linear wheel velocities (VF L ,VF R ,VRL ,VRR ) are obtained multiplying the wheel angular rates, measured using encoders, by a constant rolling radius. However, wheels tend to slip longitudinally during acceleration and braking; moreover when the front wheels are steered their longitudinal direction is not aligned with the vehicle and the yaw rate adds an additional term. Here, we introduce a kinematic based correction mechanism to improve the estimator accuracy with respect to these errors. Figure 3 shows the main kinematic effects influencing the wheel velocities. The front wheel velocities measurements are δ
δ ωz L2
ωz L2
X
Y
CG
ωz
−Athreshold x Vx = max (VF L, VF R, VRL, VRR)
Figure 4. Vx estimation algorithm. For nearly zero longitudinal acceleration the vehicle velocity is computed as a weighted mean of the wheel velocities according to the following equation: 1 Wmean = Wi Vi i = F L, F R, RL, RR. (8) i Wi i The weight (Wi ) indicates the reliability of each wheel velocity measurement and its influence on the estimated velocity. If the derivative of a wheel velocity differs considerably from the vehicle acceleration Ax or the wheel velocity itself is different from the previously available estimated velocity, it is likely that the wheel is subject to slip and thus the weight is driven to zero not to be considered in the average. The weights are analytically computed as: dVi 2 2 ˆ t−1 −V ) Ax− V ( i dt x + −1 2
ωz L2 L
Figure 3. Steer and yaw rate effects on the wheel velocity measurements. affected by the front wheel steer angle and the vehicle yaw rate affects all the wheels as they are distant from the vehicle center of mass. These phenomena are easily compensated, using readily available measurements, as follows: L L VF L = VFmeas VF R = VFmeas L cos (δ) − ωz R cos (δ) + ωz 2 2 L L meas meas VRR = VRR + ωz , VRL = VRL − ωz 2 2 (7)
σ2 w1
σ2 w2
i = F L, F R, RL, RR (9) and illustrated in Figure 5. Vˆxt−1 is the estimated vehicle velocity at the previous time step. Since the bandwidth of Vˆx is much lower than the bandwidth of wheel velocities (Vi ), considering the vehicle velocity estimated at the previous time instant introduces a negligible effect. Although the method is not limited to a particular driving wheel configuration, it should be noted that the performance are expected to decrease on all-wheel-driving vehicles as all wheels may experience slip. Wi = e
ωz L2
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This section presented the estimation approach and showed that the method depends on a number of tuning parameters. These parameters are tuned by minimizing the Root Mean Squared (RMS) estimation error over the experimental tests described in the following section. All the available data is used in the tuning and a single set of parameters produced.
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second test is performed in more dynamical conditions and reaches larger sideslip angles. The first test is characterized by two long straight paths, a smooth and long curve and several sharp corners. Both experiments are run on real roads that exhibit banking and slope. Figure 7 shows the longitudinal velocity estimation validation, comparing the results of the algorithm of Section 2.2 against the real velocity and an estimate obtained by simply averaging the four wheel velocities. From the figure, it is clear that
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Figure 5. Wheel velocity weight function. 3. EXPERIMENTAL VALIDATION
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This section presents an extensive validation of the estimation algorithm on experimental data. For all the trials, a rear wheel driven vehicle is equipped with a 6DOF inertial measurement unit (IMU), wheels speed sensors, steer angle measurement and an optical sideslip angle sensor that provides the reference value for validation. The algorithm has been implemented on a series electronic control unit in its discrete form and runs at 100Hz. The pre-processing of the data, not reported in this paper, includes the online estimation of the sensor offsets, the vehicle roll angle, and the compensation of the gravity effect on the measurements. The experiments are run both on high and low adherence surfaces. 3.1 Step steer maneuver The step steer maneuver is considered an objective and repeatable test. The maneuver consists of a steer step while the speed is kept constant. Here, the front wheel steer angle reaches 7◦ (Figure 6). The longitudinal acceleration is nearly zero while the lateral acceleration and the yaw rate reach 9m/s2 and 25◦ /s respectively. The estimation results are shown in Figure 6; in these conditions, the algorithm performs well with an RMS of the estimate error of 0.48◦ and a maximum estimation error of 1◦ .
50 0
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Figure 7. High adherence low sideslip test: longitudinal speed estimate. RMS(estimate)=1.41km/h; RMS(average)=2.09km/h the average-based estimation is subject to errors during acceleration and braking. See for example t≈120s, when the vehicle brakes right before entering a corner. In this condition, the proposed longitudinal velocity estimation method yields a better accuracy than the average-based, both in terms of RMS and maximum error. 20 10 0 -10 200
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Figure 6. Step steer RMS(εβ )=0.48◦
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Figure 8. High adherence low sideslip test: sideslip estimate. RMS(εβ )=0.85◦
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estimate.
3.2 High µ test Two different high adherence tests (µ ≈ 1) are considered: the first test is representative of normal driving; the
Figure 8 plots the lateral dynamics estimation variables. Lateral acceleration and yaw rate are small during straight and nearly straight driving (t<20s, 90s160s). During curves Ay and ωz reach large values, respectively 10m/s2 and 50◦ /s. In all conditions, the observer performs well and the accuracy is satisfactory. Overall this test in real conditions shows a
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lower accuracy than the step steer test with an RMS error of 0.85◦ . The loss of performance with respect to the objective test is due to: (1) effect of road banking and slope; (2) severe acceleration that cause large tyre longitudinal slip and deteriorates the longitudinal speed estimation; (3) vehicle pitch caused by strong acceleration that generates additional terms on Ax that are not accounted for in the model.
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The results show also the main advantage of the heuristic; when driving straight the estimated sideslip angle is stable and accurate. The stabilizing term prevents the drift due to the unobservability. To better appreciate the role of the heuristic, Figure 9 plots the estimation results on the same data using different tunings of the F factor. Without
Rear-right Rear-left
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Figure 10. High adherence high sideslip test: longitudinal speed estimate. RMS(estimate)=4.01 km/h; RMS(average)=8.36 km/h. 20 0 -20 100 0 20 0 -20 5 0 -5 0
Figure 9. Details on the effect of the heuristic correction. the heuristic the performance of the observer considerably deteriorates to the point of rendering it useless. Note that the observer without the heuristic correction is capable of estimating the sideslip angle during dynamic maneuvers (98s140s) the estimation tends to diverge as the kinematic model loses observability for ωz ≈ 0. The second test is used to validate the method driving at high speed on a track mostly composed of narrow corners. The vehicle speed estimation and the wheel speeds are shown in Figure 10. Note how, during accelerations, such as 55s
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Figure 11. High adherence high sideslip test: sideslip estimate. RMS(εβ )=1◦ . consequence of the low friction, the vehicle drives with large sideslip angles for most of the test, despite reaching lower lateral and longitudinal acceleration compared to the previous tests. The low friction also influences the wheel velocities behavior (Figure 12), with lower grip the velocities are noisier than in previous cases (see for example 30s
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Figure 12. Low adherence test: longitudinal speed estimate. RMS(estimate)=2.45km/h; RMS(average)=4.24km/h 20 0 -20 100 50 0 30 20 10 0 -10 5 0 -5 0
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Figure 13. Low adherence test: side slip estimate. RMS(εβ )=1.10◦ overcomes the issue with the non-observability of the kinematic method excluding vehicle behaviors that are non likely. (2) A better longitudinal speed estimation algorithm that corrects the wheel measurements accounting for the main kinematic effects and the longitudinal wheel slip. The method is unaffected by vehicle parameters and tireroad friction. In fact, all methods relying on a dynamic model require an online estimate of the grip in order to work properly. This is possible only if excited driving maneuvers are executed; The method is tested on a large number of scenarios: step steer maneuver, high grip track and low β conditions, high grip track and high β conditions and low grip track. The estimation results are satisfactory (maxRMS =1.1◦ ). REFERENCES Baffet, G., Charara, A., and Lechner, D. (2009). Estimation of vehicle sideslip, tire force and wheel cornering stiffness. Control Engineering Practice, 17(11), 1255– 1264. Coyte, J.L., Li, B., Du, H., Li, W., Stirling, D., and Ros, M. (2014). Decision tree assisted ekf for vehicle slip angle estimation using inertial motion sensors. In
Neural Networks (IJCNN), 2014 International Joint Conference on, 940–946. IEEE. Dakhlallah, J., Glaser, S., Mammar, S., and Sebsadji, Y. (2008). Tire-road forces estimation using extended kalman filter and sideslip angle evaluation. In American Control Conference, 2008, 4597–4602. IEEE. Farrelly, J. and Wellstead, P. (1996). Estimation of vehicle lateral velocity. In Control Applications, 1996., Proceedings of the 1996 IEEE International Conference on, 552–557. IEEE. Grip, H.F., Imsland, L., Johansen, T.A., Kalkkuhl, J.C., and Suissa, A. (2009). Vehicle sideslip estimation. IIEEE Control Systems Magazine, 29(5), 36–52. Kim, H. and Ryu, J. (2011). Sideslip angle estimation considering short-duration longitudinal velocity variation. International Journal of Automotive Technology, 12(4), 545–553. Oh, J.J. and Choi, S.B. (2012). Vehicle velocity observer design using 6-d imu and multiple-observer approach. Intelligent Transportation Systems, IEEE Transactions on, 13(4), 1865–1879. Panzani, G., Corno, M., Tanelli, M., Savaresi, S.M., Fortina, A., and Campo, S. (2009). Control-oriented vehicle attitude estimation with online sensors bias compensation. In ASME 2009 Dynamic Systems and Control Conference, 819–826. American Society of Mechanical Engineers. Piyabongkarn, D., Rajamani, R., Grogg, J.A., and Lew, J.Y. (2009). Development and experimental evaluation of a slip angle estimator for vehicle stability control. Control Systems Technology, IEEE Transactions on, 17(1), 78–88. Ray, L.R. (1995). Nonlinear state and tire force estimation for advanced vehicle control. IEEE Transactions on Control Systems Technology, 3(1), 117–124. doi: 10.1109/87.370717. Selmanaj, D., Corno, M., Savaresi, S.M., Panzani, G., Bussalai, G., and Girardin, C. (2014). Method for estimating a vehicle side slip angle, computer program implementing said method, control unit having said computer program loaded, and vehicle comprising said control unit. Selmanaj, D., Corno, M., Sename, O., and Savaresi, S. (2013). Advantages of rear steer in lti and lpv vehicle stability control. In Decision and Control (CDC), 2013 IEEE 52nd Annual Conference on, 3523–3528. IEEE. Stephant, J., Charara, A., and Meizel, D. (2007). Evaluation of a sliding mode observer for vehicle sideslip angle. Control Engineering Practice, 15(7), 803–812. Wei, L., Wenying, L., Haitao, D., and Konghui, G. (2012). Side-slip angle estimation for vehicle electronic stability control based on sliding mode observer. In Measurement, Information and Control (MIC), 2012 International Conference on, volume 2, 992–995. IEEE. Wenzel, T.A., Burnham, K., Blundell, M., and Williams, R. (2006). Dual extended kalman filter for vehicle state and parameter estimation. Vehicle System Dynamics, 44(2), 153–171.
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IFAC PapersOnLine 50-1 (2017) 14855–14860
Robust Vehicle Sideslip Estimation Robust Vehicle Sideslip Estimation Robust Vehicle Sideslip Estimation RobustonVehicle Sideslip Estimation Kinematic Considerations on Kinematic Considerations on Kinematic Considerations on Kinematic Considerations ∗ ∗∗ ∗∗
Based Based Based Based
D. Selmanaj ∗ M. Corno ∗∗ G. Panzani ∗∗ S.M. Savaresi ∗∗ ∗∗ D. G. Panzani ∗∗ S.M. Savaresi ∗∗ D. Selmanaj Selmanaj ∗∗ M. M. Corno Corno ∗∗ ∗∗ G. Panzani ∗∗ S.M. Savaresi ∗∗ D. Selmanaj M. Corno G. Panzani S.M. Savaresi ∗ ∗ Institute for Dynamic Systems and Control, ETH Zurich, Zurich ∗ Institute for Dynamic Systems and Control, ETH Zurich, Zurich for Dynamic and Control, ETH Zurich, Zurich 8092, Switzerland andSystems Department of Automation, Polytechnic ∗ Institute Institute for Dynamic Systems and Control, ETH Zurich, Zurich 8092, and Department of Polytechnic 8092, Switzerland Switzerland and Department of Automation, Automation, Polytechnic University of Tirana, Tirana 1001, Albania 8092, Switzerland and Department of Automation, Polytechnic University of Tirana, Tirana 1001, Albania ∗∗ University of Tirana, Tirana 1001, Albania di Elettronica, Informazione e Bioingegneria, ∗∗ Dipartimento University of Tirana, Tirana 1001, Albania ∗∗ Dipartimento di Informazione ee32, Bioingegneria, di Elettronica, Elettronica, Informazione Bioingegneria, Politecnico di Milano. Piazza Leonardo da Vinci 20132 Milano, ∗∗ Dipartimento Dipartimento di Elettronica, Informazione e32, Bioingegneria, Politecnico di Milano. Piazza Leonardo da Vinci Politecnico di Milano. Piazza Leonardo da Vinci 32, 20132 20132 Milano, Milano, Italy Politecnico di Milano. Piazza Leonardo da Vinci 32, 20132 Milano, Italy Italy Italy Abstract – This paper proposes an inertial measurement based sideslip angle estimation Abstract – paper an based sideslip angle Abstract – This Thisapplications. paper proposes proposes an inertial inertial measurement based sideslip angle estimation for automotive Measuring sideslipmeasurement angle is costly; in most cases, in estimation most case Abstract – Thisapplications. paper proposes an inertial measurement based sideslip angle estimation for automotive Measuring sideslip angle is costly; in most cases, in case for automotive applications. Measuring sideslip angle is costly; in mostoncases, in most most case estimation is a preferred choice. Most estimation solutions are based dynamical models for automotive applications. Measuring sideslip angle is costly; in mostoncases, in most case estimation is a preferred choice. Most estimation solutions are based dynamical models estimation isToa properly preferredwork, choice. Most estimation solutions are basedthe on road dynamical models of the car. these solutions need to also estimate friction. This estimation isToa properly preferredwork, choice. Most estimation solutions are basedthe on road dynamical models of the car. these solutions need to also estimate friction. This of the car. To complexity properly work, these solutions need toThe alsopresent estimate the proposes road friction. This increases their and affects their robustness. study an inertial of the car. To properly work, these solutions need toThe alsopresent estimate the road friction. This increases their and affects robustness. study an increases their complexity complexity and require affects their their robustness.nor The present studyofproposes proposes an inertial inertial based approach that does not the knowledge the estimation road friction. This increases their complexity and require affects their robustness.nor The present studyofproposes an inertial based approach that does not the knowledge the estimation road friction. This based approach that does not require the and knowledge nor of thethe estimation ofFocus road friction. This considerably simplifies the design, tuning validation approach. is devoted to based approach that does not require the and knowledge nor of the estimation of road friction. This considerably simplifies the design, approach. is to considerably simplifies the design, tuning tuningerrors. and validation validation of the the approach. Focus Focus is devoted devoted to the study of the effect of measurement An extensive experimental validation confirms considerably simplifies the design, tuningerrors. and validation of the approach. Focus is devoted to the of of An extensive experimental validation the study of the the effect effect of measurement measurement An range extensive experimental validation confirms confirms thatstudy the estimate is accurate and robust errors. to a wide of driving scenarios. the study of the effect of measurement errors. An range extensive experimental validation confirms that the estimate is accurate and robust to a wide of driving scenarios. that the estimate is accurate and robust to a wide range of driving scenarios. that theIFAC estimate is accurate and robust to a wide rangeHosting of driving scenarios. © 2017, (International Federation of Automatic Control) by Elsevier Ltd. All rights reserved. Keywords: Side Slip Estimation, Vehicle Dynamics Control, Automotive Keywords: Side Slip Estimation, Vehicle Dynamics Control, Automotive Keywords: Side Slip Estimation, Vehicle Dynamics Control, Automotive Keywords: Side Slip Estimation, Vehicle Dynamics Control, Automotive 1. INTRODUCTION and lateral velocities with longitudinal and lateral acceler1. and lateral velocities with acceler1. INTRODUCTION INTRODUCTION and lateral velocities with longitudinal longitudinal and lateral accelerations and yaw rate. Consequently, theyand do lateral not depend on 1. INTRODUCTION and lateral velocities with longitudinal and lateral accelerations and yaw rate. Consequently, they do not depend on ations and yaw rate. Consequently, they do not depend on vehicle or tyre models. One of the first studies on kinematic Sideslip angle, i.e. the angle between the longitudinal vehicle ations and yaw rate. Consequently, they do not depend on tyre One the first on vehicle or tyre models. models. One of ofand theWellstead first studies studies on kinematic kinematic Sideslip angle, i.e. the angle the longitudinal sideslipor observers is Farrelly (1996) in which Sideslip angle, i.e. the and anglethebetween between the longitudinal direction of the vehicle velocity vector, heavily vehicle or tyre models. One of the first studies on kinematic sideslip observers is Farrelly and Wellstead (1996) in which Sideslip angle, i.e. the and anglethebetween the longitudinal sideslip observers is Farrelly and Wellstead (1996) in which direction of the vehicle velocity vector, heavily a kinematic observer is proven to be asymptotically stable direction of the vehicleofand the velocity vector, heavily influences of thethe dynamics wheeled vehicles.vector, Large side slip sideslip observers is Farrelly andtoWellstead (1996) in stable which a kinematic observer is proven be asymptotically direction vehicle and the velocity heavily a kinematic observer is proven to be asymptotically stable influences the dynamics of wheeled vehicles. Large side slip for all cornering conditions (non-zero yaw rate). From influences the dynamics of wheeled vehicles. Large side slip angle maythe lead to instability. Thevehicles. knowledge of side sideslip kinematic observer is proven(non-zero to be asymptotically stable all cornering conditions yaw rate). influences dynamics of wheeled Large slip afor for all cornering conditions (non-zero yawdeveloped rate). From From angle may lead to instability. The knowledge of sideslip that original paper, the approach has been and angle is may lead to instability. The knowledge of sideslip therefore extremely important in vehicle dynamall cornering conditions (non-zero yawdeveloped rate). From that original has been and angle is may lead toextremely instability. The knowledge of dynamsideslip for that original paper, theetapproach approach hasthe been developed and important in enriched. In paper, Panzanithe al. (2009), authors propose angle is therefore therefore extremely important in vehicle vehicle dynamics control Selmanaj et al. (2013). Sideslip angle can be that original paper, the approach has been developed and enriched. Panzani et (2009), angle is therefore extremely important in vehicle dynamenriched. In Panzani et al. al.Kim (2009), the authors propose ics control Selmanaj al. Sideslip angle an online In bias estimation. and the Ryuauthors (2011) propose designs ics control Selmanaj etoptical al. (2013). (2013). Sideslip angle can can be be enriched. directly measured viaet sensors and dual-antenna In Panzani et al.Kim (2009), the authors propose an online bias estimation. and Ryu (2011) designs ics control Selmanaj et al. (2013). Sideslip angle can be an online bias estimation. Kim and Ryu (2011) designs directly measured via optical sensors and dual-antenna and validates an Extended Kalman Filter (EKF) based directly measured viasolutions optical sensors and dual-antenna GPS systems; these are notand amenable to in- an online bias an estimation. Kim and Filter Ryu (2011) designs and validates Extended Kalman (EKF) based directly measured viasolutions optical sensors dual-antenna and validates an Extended Kalman Filter (EKF) based GPS systems; these are not amenable to inon the kinematic model; the validation is restricted to GPS systems; these solutions are not amenable to industrialization because of cost and dependability reasons. validates an Extended Kalman Filteris (EKF) based on the kinematic model; the validation restricted to GPS systems; because these solutions are dependability not amenablereasons. to in- and on the kinematic model; the validation is restricted to dustrialization of cost and short dynamic maneuvers (around 10s). Another group dustrialization because of cost and dependability reasons. The availabilitybecause of a good cost-effective and robust side on the kinematic model; the validation is restricted to short maneuvers (around group dustrialization of cost and dependability reasons. short dynamic maneuvers (around 10s). Another group The availability of cost-effective and side adoptsdynamic a sliding mode observer in Wei10s). et al. Another (2012). Results The availability of aa good good cost-effective and robust robust side slip angle estimation can make the difference between a short dynamic maneuvers (around 10s). Another group aa sliding mode in et The availability of a good cost-effective and robust side adopts sliding mode observer observer in Wei Weimethods et al. al. (2012). (2012). Results slip angle estimation can make difference aa adopts show that kinematic model-based are Results reliable slip angle active estimation candynamics make the the difference between successful vehicle control and abetween mediocre adopts a sliding mode observer in Weimethods et al. (2012). Results that kinematic model-based are reliable slip angle active estimation candynamics make the difference between a show show that kinematic model-based methods are reliable successful vehicle control and a mediocre for transient maneuvers, but they suffer from estimation successful active vehicle dynamics control and a mediocre one. that kinematic model-based methods are reliable for transient maneuvers, but they estimation successful active vehicle dynamics control and a mediocre show for transient maneuvers, but conditions they suffer sufferasfrom from estimation one. errors on nearly steady-state the integration one. for transient maneuvers, but conditions they sufferas from estimation errors on nearly steady-state the integration one. on nearly steady-state conditions as the integration Sideslip estimation has a long history in automotive publi- errors of measurement errors causesconditions the estimate to diverge. on nearly steady-state as the integration Sideslip estimation has aa long in publiof Sideslip estimation hasapproaches long history history in automotive automotive publi- errors of measurement measurement errors errors causes causes the the estimate estimate to to diverge. diverge. cations. The available are classified in dynamic Sideslip estimation hasapproaches a long history in automotive publi- Researchers measurement errors the causes the estimate to diverge. apcations. The are in dynamic merged dynamic and kinematic cations. The available available approaches are classified classified in Dynamic dynamic of model based, kinematic model based or hybrid. merged the dynamic and kinematic apcations. The available approaches are classified in Dynamic dynamic Researchers Researchers merged the dynamic and kinematic apmodel based, kinematic model based or hybrid. proaches yielding hybrid solutions, where depending on the model based, kinematicBaffet modeletbased or hybrid. Dynamic based methods al. (2009); Coyte et al. Researchers merged the dynamic anddepending kinematic approaches yielding hybrid solutions, where on the model based, kinematic model based or hybrid. Dynamic proaches yielding hybrid solutions, where depending on the based methods Baffet et al. (2009); Coyte et al. conditions the results of the kinematic and dynamics modmodel based methods Baffet et al. (2009); Coyte et al. (2014); based Dakhlallah et al. (2008); Ray (1995); Stephant yielding hybrid solutions, where depending on the conditions the of kinematic and model methods Baffet et al.Ray (2009); Coyte et al. proaches conditions the results results of the the kinematic and dynamics dynamics modmod(2014); Dakhlallah et (1995); Stephant els are combined Oh and Choi (2012); Piyabongkarn et al. (2014); Dakhlallah etetal. al.al.(2008); (2008); Ray (1995); Stephant et al. (2007); Wenzel (2006) are based on dynamic conditions the results of the kinematic and dynamics models are combined Oh and Choi (2012); Piyabongkarn et al. (2014); Dakhlallah et al.al.(2008); Ray (1995); Stephant els are combined Oh and Choi (2012); Piyabongkarn etand al. et al. Wenzel (2006) are based (2009). In particular, Grip et (2012); al. (2009) accuratelyet et al. (2007); (2007); Wenzel et et al.forces (2006) areact based on dynamic models that describe theal. that on on thedynamic vehicle. els are combined Oh and Choi Piyabongkarn al. (2009). In particular, Grip et al. (2009) accurately and et al. (2007); Wenzel et (2006) are based on dynamic (2009). In particular, Grip et al. (2009) accurately and models that describe the forces that act on the vehicle. jointly estimates the sideslip angle, friction coefficient models that them describe the forces that act onbut the vehicle. This makes potentially verythat accurate, are (2009). In particular, Grip et al. (2009) accurately and jointly estimates the friction models that them describe the forces act onbut the they vehicle. jointly estimates the sideslip sideslip angle, frictiontocoefficient coefficient and This makes very are road banking. These methodsangle, are shown be extremely This makes theminpotentially potentially very accurate, accurate, but they they are jointly complex systems which the interplay of different compoestimates the sideslip angle, friction coefficient and road banking. These methods are shown to be extremely This makes theminpotentially very accurate, but they are road banking. These methods are shown to be extremely complex systems which the interplay of different compoaccurate, but they are still subject to the considerable cost complex systems in which theFor interplay of different components is difficulty validated. example, in order compoto fully road banking. These methods are shown to be extremely accurate, but are subject complex systems in which theFor interplay of different accurate, but they they are still still model. subject to to the the considerable considerable cost cost nents is difficulty validated. in to fully of developing an accurate nents istheir difficulty validated. For example, example, in order orderthe to road fully accurate, exploit potential, they need to also estimate but they are still subject to the considerable cost developing an accurate model. nents istheir difficulty validated. For example, in orderthe to road fully of of developing an accurate model. exploit potential, they need to also estimate exploit their potential, they need toconditions also estimate the road The conditions online. Estimating roadto is inthe general developing an accurate present work proposesmodel. a reliable and industrially viexploit their potential, they need also estimate road of conditions online. Estimating road conditions is general present work proposes aa reliable and viconditions online. Estimating road conditions is in independs general The difficult and the quality of the estimate heavily The present work proposes reliable and industrially industrially viable solution to the sideslip angle estimation problem, that conditions online.quality Estimatingthe road conditions is independs general The present work proposes a reliable and industrially vidifficult and estimate able solution to the sideslip angle estimation problem, that difficult and the the qualityofof ofthe the maneuvers. estimate heavily heavily depends on the characteristics Moreover the able solution to the sideslip angle estimation problem, that does not employ any dynamic model. Its tuning is thus difficult and the qualityofofthe the estimate heavily depends solution to the sideslip anglemodel. estimation problem, that on the characteristics the does not dynamic Its is on thealso characteristics of the maneuvers. maneuvers. Moreover the able model requires the knowledge of vehicle Moreover mass and yaw does not employ employ any any dynamic model. Its tuning tuning is thus thus very cost-effective. The algorithm is based on kinematic on thealso characteristics of the maneuvers. Moreover the does not employ any dynamic model. Its tuning is thus model requires the knowledge of vehicle mass and yaw very cost-effective. The algorithm is based on kinematic model also requires the knowledge of vehicle mass and yaw very inertia,also parameters thatknowledge can experience largemass variations. cost-effective. The algorithm is basedThe on heuristics, kinematic considerations augmented with heuristics. model requires the of vehicle and yaw very cost-effective. The algorithm is basedThe on heuristics, kinematic inertia, considerations augmented with heuristics. inertia, parameters parameters that that can can experience experience large large variations. variations. considerations augmented with heuristics. The heuristics, by adding an artificial damping term, prevents the diinertia, parameters that approaches can experience large variations. Kinematic model based are simpler. They are considerations augmented with heuristics. The heuristics, by adding an artificial damping term, prevents the diby adding an artificial damping term, prevents the diKinematic model based approaches are simpler. They are vergence of the estimation for low yaw rate driving. The Kinematic model based approaches arevehicle simpler. They are vergence based on the correlation between the longitudinal addingof an artificial damping term, prevents theThe dithe estimation for low yaw rate driving. Kinematic model based approaches arevehicle simpler. They are by vergence of the estimation for low yaw rate driving. The based on the correlation between the longitudinal based on the correlation between the vehicle longitudinal based on the correlation between the vehicle longitudinal vergence of the estimation for low yaw rate driving. The
Copyright 15420Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © © 2017 2017, IFAC IFAC (International Federation of Automatic Control) Copyright © 2017 15420 Copyright © under 2017 IFAC IFAC 15420Control. Peer review responsibility of International Federation of Automatic Copyright © 2017 IFAC 15420 10.1016/j.ifacol.2017.08.2513
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algorithm is validated in realistic driving tests representing hours worth of driving. These tests include both highly dynamic and nearly steady state maneuvers. Results on different road conditions demonstrate the robustness of the method to varying road conditions. Although dynamic model based approaches are more accurate in ideal conditions, we believe that the more predictable behavior of the proposed approach when facing unknown road conditions, represents a considerable advance that is well appreciated by the industry. The paper is organized as follows. In Section 2, the kinematic observer and the longitudinal vehicle speed estimation are described. Section 3 presents the detailed experimental validation. The paper ends with some concluding remarks. Part of the present work is protected by the patent Selmanaj et al. (2014). 2. KINEMATIC MODEL-BASED OBSERVER Equations (1) summarize the kinematic model according to the reference frame shown in Figure 1. Ax and Ay are the longitudinal and lateral body accelerations, V˙ x and V˙ y the time derivatives of the vehicle velocity and ωz is the yaw rate angle. ωz CG
Vx β
δ
Vy
Figure 1. Single track model. Ax (t) = V˙ x (t) − ωz (t) Vy (t) Ay (t) = V˙ y (t) + ωz (t) Vx (t)
(1)
Rewriting the model in the state space form, one obtains 0 ωz (t) Vx (t) 1 0 Ax (t) V˙ x (t) = + −ωz (t) 0 Vy (t) 0 1 Ay (t) V˙ y (t) [A] [B] (2) Vx (t) y(t) = [1 0] . Vy (t) [C]
In the above model, the yaw rate is modeled as a time varying parameter and the vehicle accelerations as inputs. Given the above symbols and definitions, the sideslip angle is computed as: Vy (t) . (3) β (t) = arctan Vx (t) Based on the model Farrelly and Wellstead (1996) proposed the following nonlinear state observer ˙ Vˆx (t) Vˆx (t) Ax (t) = (A − KC) ˆ +B + KVx , (4) ˙ Ay (t) Vy (t) Vˆy (t) where K, the observer gain T matrix, is defined as K = α2 − 1 ωz (t) . The original formulation 2α|ωz (t)| does not address two issues: (1) Based on K dependency on ωz (t), the longitudinal velocity estimation is updated only in corners. (2) During straight driving, the model looses its full observability and as a consequence, the observer simply
integrates the accelerations. In these conditions, small measurement errors will cause the estimate to diverge. The latter problem is intrinsic and structural with the choice of sensors. It does not depend on the algorithm. A possible solution is to incorporate additional information on the dynamics in the observer. Usually, this is done using a dynamic model, with the difficulties described in the Introduction. Here, we prefer to use a rougher but simpler description of the dynamics. The kinematics described by model (2) describe a free body on a plane; cars are not completely free. Some movements, if not impossible, are very unlikely. For example, it is extremely unlikely for a car to laterally drift without exhibiting any yaw rate; especially if one considers that sideslip angle estimators are often part of an electronic stability system designed to prevent the vehicle from reaching those conditions. This intuitive understanding (or heuristic) can be included in the model using an artificial, non-physical stabilizing term (−F (t)Vy (t)) in the second equation of (2). If F (t) is designed to be positive and large when the vehicle is driving straight, the lateral velocity estimate is prevented to drift due to integration errors. The first issue listed above is improved by modifying the gain term K. The gain matrix term affecting the longitudinal velocity estimate is now composed of yaw rate dependent and constant terms. α1 |ωz | update the observer state on cornering, while α0 guarantees the Vˆx update on straight maneuvers. Vx is the vehicle longitudinal velocity and is obtained from the equivalent linear wheel velocity. The complete observer expression is given in (5): ˙ Vˆx (t) −α0 − α1 |ωz (t)| ωz (t) Vˆx (t) = ˙ −(α2 + 1)ωz (t) −F (t) Vˆy (t) Vˆy (t) [A−K(ωz )C]
1 0 Ax (t) α + α1 |ωz (t)| + + 0 Vx . 0 1 Ay (t) α2 ωz (t) [B]
(5)
[K(ωz )]
The observer implements the heuristic according to which, if the yaw rate is small and constant, the sideslip angle will likely be small. In other words: if F is small (i.e. close to 0) the vehicle state is estimated from the kinematic model and the measurements; if F is large the estimated lateral velocity is driven to a small value, while the longitudinal velocity is updated by Vx . The key of this approach is the scheduling of F (t), described next. In principle, the same results can be achieved by turning off the observer when the vehicle is moving straight and turning it on when a curve is approached. . The solution proposed here guarantees a smoother transition than a sudden switching. 2.1 Heuristic scheduling F determines the strength of the artificial stabilizing term. Figure 2 represents the scheduling rationale. In the figure, δ is the steering wheel angle and β˙ is the sideslip angle derivative, directly estimated from measurements: Ay (t) − ωz (t)Vx (t) ˙ β(t) = . (6) Vx (t) F is scheduled based on indexes that describe whether the vehicle is cornering: sideslip rate, steering angle and yaw
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ωz
d dt
Fδ δ
β˙
d dt
d dt
meas meas meas and VRR are the measured where VFmeas L ,VF R ,VRL equivalent linear velocities of the wheels.
Fωz
ω˙z
×
δ˙
F
Fβ
β¨
Figure 2. Heuristic scheduling. rate and their derivatives. Fωz , Fδ and Fβ are based on bivariate gaussian functions that smoothly vary from zero to a maximum value; and each function outputs 0 when either input is nonzero.
Accounting for the effect of longitudinal slip is more complex. The proposed solution carries out a kinematicbased wheel velocities sensor fusion. The rationale is to identify the lowest slip wheel by exploiting the acceleration of the vehicle. If the vehicle is accelerating the wheels tend to have higher velocity than the vehicle, thus the lowest wheel velocity is the closest to the vehicle velocity. Vice versa, if the vehicle is decelerating, the wheels tend to have lower velocity and the highest wheel velocity is the closest to the vehicle velocity. Acceleration and deceleration maneuvers are identified through thresholds on the longitudinal acceleration measurement (Figure 4). Ax Vx = min (VF L, VF R, VRL, VRR) Athreshold x
2.2 Longitudinal velocity estimation
Vx = Wmean (VF L, VF R, VRL, VRR)
The estimator uses the vehicle longitudinal velocity measurement (Vx ) as the feedback term that, using Kalman filtering nomenclature, drives the aposteriori correction of the estimate. The wheels rotational rates are the main source of information on the vehicle longitudinal velocity. The equivalent linear wheel velocities (VF L ,VF R ,VRL ,VRR ) are obtained multiplying the wheel angular rates, measured using encoders, by a constant rolling radius. However, wheels tend to slip longitudinally during acceleration and braking; moreover when the front wheels are steered their longitudinal direction is not aligned with the vehicle and the yaw rate adds an additional term. Here, we introduce a kinematic based correction mechanism to improve the estimator accuracy with respect to these errors. Figure 3 shows the main kinematic effects influencing the wheel velocities. The front wheel velocities measurements are δ
δ ωz L2
ωz L2
X
Y
CG
ωz
−Athreshold x Vx = max (VF L, VF R, VRL, VRR)
Figure 4. Vx estimation algorithm. For nearly zero longitudinal acceleration the vehicle velocity is computed as a weighted mean of the wheel velocities according to the following equation: 1 Wmean = Wi Vi i = F L, F R, RL, RR. (8) i Wi i The weight (Wi ) indicates the reliability of each wheel velocity measurement and its influence on the estimated velocity. If the derivative of a wheel velocity differs considerably from the vehicle acceleration Ax or the wheel velocity itself is different from the previously available estimated velocity, it is likely that the wheel is subject to slip and thus the weight is driven to zero not to be considered in the average. The weights are analytically computed as: dVi 2 2 ˆ t−1 −V ) Ax− V ( i dt x + −1 2
ωz L2 L
Figure 3. Steer and yaw rate effects on the wheel velocity measurements. affected by the front wheel steer angle and the vehicle yaw rate affects all the wheels as they are distant from the vehicle center of mass. These phenomena are easily compensated, using readily available measurements, as follows: L L VF L = VFmeas VF R = VFmeas L cos (δ) − ωz R cos (δ) + ωz 2 2 L L meas meas VRR = VRR + ωz , VRL = VRL − ωz 2 2 (7)
σ2 w1
σ2 w2
i = F L, F R, RL, RR (9) and illustrated in Figure 5. Vˆxt−1 is the estimated vehicle velocity at the previous time step. Since the bandwidth of Vˆx is much lower than the bandwidth of wheel velocities (Vi ), considering the vehicle velocity estimated at the previous time instant introduces a negligible effect. Although the method is not limited to a particular driving wheel configuration, it should be noted that the performance are expected to decrease on all-wheel-driving vehicles as all wheels may experience slip. Wi = e
ωz L2
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This section presented the estimation approach and showed that the method depends on a number of tuning parameters. These parameters are tuned by minimizing the Root Mean Squared (RMS) estimation error over the experimental tests described in the following section. All the available data is used in the tuning and a single set of parameters produced.
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second test is performed in more dynamical conditions and reaches larger sideslip angles. The first test is characterized by two long straight paths, a smooth and long curve and several sharp corners. Both experiments are run on real roads that exhibit banking and slope. Figure 7 shows the longitudinal velocity estimation validation, comparing the results of the algorithm of Section 2.2 against the real velocity and an estimate obtained by simply averaging the four wheel velocities. From the figure, it is clear that
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Figure 5. Wheel velocity weight function. 3. EXPERIMENTAL VALIDATION
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This section presents an extensive validation of the estimation algorithm on experimental data. For all the trials, a rear wheel driven vehicle is equipped with a 6DOF inertial measurement unit (IMU), wheels speed sensors, steer angle measurement and an optical sideslip angle sensor that provides the reference value for validation. The algorithm has been implemented on a series electronic control unit in its discrete form and runs at 100Hz. The pre-processing of the data, not reported in this paper, includes the online estimation of the sensor offsets, the vehicle roll angle, and the compensation of the gravity effect on the measurements. The experiments are run both on high and low adherence surfaces. 3.1 Step steer maneuver The step steer maneuver is considered an objective and repeatable test. The maneuver consists of a steer step while the speed is kept constant. Here, the front wheel steer angle reaches 7◦ (Figure 6). The longitudinal acceleration is nearly zero while the lateral acceleration and the yaw rate reach 9m/s2 and 25◦ /s respectively. The estimation results are shown in Figure 6; in these conditions, the algorithm performs well with an RMS of the estimate error of 0.48◦ and a maximum estimation error of 1◦ .
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Figure 7. High adherence low sideslip test: longitudinal speed estimate. RMS(estimate)=1.41km/h; RMS(average)=2.09km/h the average-based estimation is subject to errors during acceleration and braking. See for example t≈120s, when the vehicle brakes right before entering a corner. In this condition, the proposed longitudinal velocity estimation method yields a better accuracy than the average-based, both in terms of RMS and maximum error. 20 10 0 -10 200
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Figure 8. High adherence low sideslip test: sideslip estimate. RMS(εβ )=0.85◦
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Figure 8 plots the lateral dynamics estimation variables. Lateral acceleration and yaw rate are small during straight and nearly straight driving (t<20s, 90s
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lower accuracy than the step steer test with an RMS error of 0.85◦ . The loss of performance with respect to the objective test is due to: (1) effect of road banking and slope; (2) severe acceleration that cause large tyre longitudinal slip and deteriorates the longitudinal speed estimation; (3) vehicle pitch caused by strong acceleration that generates additional terms on Ax that are not accounted for in the model.
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The results show also the main advantage of the heuristic; when driving straight the estimated sideslip angle is stable and accurate. The stabilizing term prevents the drift due to the unobservability. To better appreciate the role of the heuristic, Figure 9 plots the estimation results on the same data using different tunings of the F factor. Without
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Figure 10. High adherence high sideslip test: longitudinal speed estimate. RMS(estimate)=4.01 km/h; RMS(average)=8.36 km/h. 20 0 -20 100 0 20 0 -20 5 0 -5 0
Figure 9. Details on the effect of the heuristic correction. the heuristic the performance of the observer considerably deteriorates to the point of rendering it useless. Note that the observer without the heuristic correction is capable of estimating the sideslip angle during dynamic maneuvers (98s
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Figure 11. High adherence high sideslip test: sideslip estimate. RMS(εβ )=1◦ . consequence of the low friction, the vehicle drives with large sideslip angles for most of the test, despite reaching lower lateral and longitudinal acceleration compared to the previous tests. The low friction also influences the wheel velocities behavior (Figure 12), with lower grip the velocities are noisier than in previous cases (see for example 30s
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Figure 12. Low adherence test: longitudinal speed estimate. RMS(estimate)=2.45km/h; RMS(average)=4.24km/h 20 0 -20 100 50 0 30 20 10 0 -10 5 0 -5 0
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Figure 13. Low adherence test: side slip estimate. RMS(εβ )=1.10◦ overcomes the issue with the non-observability of the kinematic method excluding vehicle behaviors that are non likely. (2) A better longitudinal speed estimation algorithm that corrects the wheel measurements accounting for the main kinematic effects and the longitudinal wheel slip. The method is unaffected by vehicle parameters and tireroad friction. In fact, all methods relying on a dynamic model require an online estimate of the grip in order to work properly. This is possible only if excited driving maneuvers are executed; The method is tested on a large number of scenarios: step steer maneuver, high grip track and low β conditions, high grip track and high β conditions and low grip track. The estimation results are satisfactory (maxRMS =1.1◦ ). REFERENCES Baffet, G., Charara, A., and Lechner, D. (2009). Estimation of vehicle sideslip, tire force and wheel cornering stiffness. Control Engineering Practice, 17(11), 1255– 1264. Coyte, J.L., Li, B., Du, H., Li, W., Stirling, D., and Ros, M. (2014). Decision tree assisted ekf for vehicle slip angle estimation using inertial motion sensors. In
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