Estimation of the Vehicle Sideslip Angle by Means of the State Dependent Riccati Equation Technique

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1 Proceedings o the World Congress on Engineering 7 Vol II WCE 7, Jul 5-7, 7, London, U.K. Estimation o the Vehicle Sideslip Angle b Means o the State Dependent Riccati Equation Technique S. Strano, M. Terzo Abstract The nowledge o the vehicle sideslip angle represents a undamental condition or all the actual vehicle dnamics control sstems. Since the measurement o the sideslip angle is expensive and unsuitable or common vehicles, its estimation is nowadas an important tas. To this aim, several techniques have been adopted and in man cases their limits are emerged due to the nonlinear nature o the vehicle sstem. In order to overcome these limits, this paper ocuses on an alternative nonlinear estimation method based on the State- Dependent-Riccati-Equation (SDRE). Simulations have been conducted and comparisons with the largel used Extended Kalman Filter (EKF) are illustrated. Index Terms Vehicle dnamics, Sideslip angle, Model based estimation, State dependent Kalman ilter, Extended Kalman ilter. I. INTRODUCTION The vehicle sideslip angle is an important phsical variable strongl lined to the directional behaviour and stabilit o the vehicle. As a consequence, the nowledge o the sideslip angle is requested rom the vehicle dnamics control sstems that establish their intervention on the basis o a dierence between a target and a current value []. The measurement o the vehicle sideslip angle can be obtained b means o devices that are ver expensive and not unctional or an eas installation on the car. The wellnown industrial solutions tpicall rel on observers, which are based on heavil simpliied dnamic vehicle models in combination with inematic models. Methodologies or vehicle sideslip angle estimation can be ound in literature. Man o them are generall based on the extended Kalman ilter (EKF) algorithm []. In an case, the nonlinear nature o the vehicle sstem strongl limits the perormance o linear and linearization based approaches [3] that inevitabl give estimation errors that aect the perormance o the vehicle dnamics controllers emploing the estimated variables as eedbac. In order to overcome this limit, this paper investigates on a nonlinear technique S. Strano is with the Department o Industrial Engineering, Universit o Naples Federico II, 85 ITALY (corresponding author, phone: ; salvatore.strano@unina.it). M. Terzo is with the Department o Industrial Engineering, Universit o Naples Federico II, 85 ITALY ( mario.terzo@unina.it). or the estimation o the vehicle sideslip angle. The approach is based on the State-Dependent-Riccati-Equation (SDRE) nonlinear iltering ormulation. The SDRE techniques are recentl emerging or optimal nonlinear control and iltering techniques. The SDRE ilter (SDREF) originates rom a suboptimal nonlinear regulator technique that uses parameterization to bring the nonlinear sstem into a linear-lie structure with state-dependent coeicients (SDC). II. VEHICLE MODEL The model adopted in a state observer has to be simple enough in order to limit the computational load but, at the same time, it has to tae into account the not negligible dnamics that aect the real sstem and that are crucial or a good estimation. A single-trac model has been assumed or the vehicle. It is characterized b two states reerred to the in plane vehicle bod motions (lateral and aw motions) with the steering angle and longitudinal velocit representing the sstem parameters. Fig. shows the vehicle model in the inertial reerence rame Ox and deines the bod ixed reerence rame Bx. With reerence to the same igure, v is the centre o mass absolute velocit reerred to the earth-ixed axis sstem, and u (longitudinal velocit) and v (lateral velocit) are its components in the vehicle axis sstem; r is the aw rate evaluated in the reerence rame Ox, F and F are the lateral interaction orces o the ront and rear axle, respectivel. Fig.. Reerence rames and vehicle model. The distances rom the ront and rear axle to the centre o gravit are represented b a and b, respectivel. The steering wheel angle o the ront tres is denoted b δ, while the rear tres are assumed to be no-steering. The in-plane motion equations are the ollowing: ISBN: ISSN: (Print); ISSN: (Online) WCE 7

2 Proceedings o the World Congress on Engineering 7 Vol II WCE 7, Jul 5-7, 7, London, U.K. m( v ur) F cos F J zr Fa cos Fb where m is the vehicle total mass, J z is its moment o inertia respect to z axis. Taing into account that in the hpothesis o constant u v u tan v u tan the sstem () becomes: r F cos F um r Fa cos Fb J z tan The estimation model o the vehicle dnamics includes the equations o motion (); the tre orces are considered as variables to be estimated and the sstem o equations () is augmented to include dierential equations or each orce. So, a random wal model [4] is considered to model each orce to be determined: w () () (3) III. EXTENDED KALMAN FILTER The linearization procedure is at the basis o the EKF approach, which is briel recalled in the ollowing, since it has been adopted or a comparative analsis. The sstem and the measurement equations can be genericall represented b: x ( t) ( x( ψ (5) x( t g z( t) h ) (6) being x the state vector, and h non-linear unctions, u the input vector, ψ the process noise with covariance Q, z the measurement vector, and g the Gaussian white measurement noise with covariance R. The estimator can be implemented in a discrete time orm, integrating the sstem equation rom time t to time t. The EKF methodolog is conceptuall based on two undamental steps, namel estimates and updates steps. Denoting the estimates as ˆ, the ollowing initializing conditions are applied to the state estimates (7) and to the error covariance (8): where represents the orce to be estimated, its irst time derivative, and w is the random white noise. E( x) (7) The model adopted in the estimators is: r F cos F um r Fa cos Fb J z F F F F F F tan (4) T x x E x (8) P ˆ being E the expected value. The state estimates and the estimation o the error covariance are given b (9) and () respectivel: x ˆ (ˆ x, u ) (9) T T A P A L Q L P () where F and F are the irst time derivative o the lateral interaction orces o the ront and rear axle, respectivel. The state and the input vectors o equations (4) are deined respectivel b T and u, T u. x, r, F, F, F, F The nonlinear equations o the sstem (4) have been adopted in order to derive the EKF and the SDREF starting rom the measurements o signals lie the lateral acceleration a and the aw rate r (common on current commercial vehicles) with the objective o estimating the sideslip angle o the vehicle. ISBN: ISSN: (Print); ISSN: (Online) with A () L () ψ With the computation o the ilter gain (3) and evaluating the measurement residual, the updates o the state estimates (4) and o the estimation o the error covariance (5) can be determined: WCE 7

3 Proceedings o the World Congress on Engineering 7 Vol II WCE 7, Jul 5-7, 7, London, U.K. T T T K P H ( H P H M R M ) (3) K z h (ˆ x, u ) (4) where K T x PxH x,u ˆ R ˆ () P ( I K H ) P (5) where: H M h h g IV. STATE DEPENDENT RICCATI EQUATION FILTER (6) (7) The SDRE techniques are used as control and iltering design methods and are based on state dependent coeicient (SDC) actorization [5]. Ininite-horizon nonlinear regulator problem is a generalization o time invariant ininite horizon linear quadratic regulator problem where all sstem coeicient matrices are state-dependent. When the coeicient matrices are constant, the SDRE control method changes into the stead-state linear regulator. Filtering counterpart o the SDRE control algorithm is obtained b taing the dual sstem o the stead-state linear regulator and then allowing coeicient matrices o the dual sstem to be state-dependent. Starting rom the nonlinear sstem (4), there are ininite solutions to transorm this nonlinear sstem into an SDC orm as: x ( t) F( x( x ψ (8) ( x g z( t ) H x t) (9) where ( x( F( x( x and x( ), t) Hx( t x h t ) () ψ is the process noise with covariance Q, and g is the Gaussian white measurement noise with covariance R. and P is the positive deinite solution o the algebraic Riccati equation (3). F T x,u ˆ P P F x,u ˆ T P H x,u ˆ R Hx,u ˆ P Q (3) The nonlinear equations () have been parameterized in SDC orm with the ollowing choice [6, 7]: F(x,u) tan cos tan um tan um acos J b z J z (4) Moreover, with reerence to the measurement equation, taing into account that a F cos F v ur u tan ur (5) m it results: cos H(x, u) H u m m (6) V. SIMULATION TEST Simulation tests have been carried out in CarSim, that constitutes a reerence in vehicle simulation environment. The irst simulated manoeuvre consists o an open loop steering pad, unctional to evaluate the stead state circular driving behaviour. It is based on a constant longitudinal speed o m/h, a steering angle smoothl and linearl increasing with the time and high adhesion conditions. The characteristics o the simulated vehicle are listed in Table. The perormance o the SDREF has been evaluated b means o comparison with the EKF estimator and the simulated state given b CarSim. The observers have been designed taing into account as input the longitudinal speed (u), the steering angle ( ), the measurements given b the lateral acceleration a and the aw rate r. Moreover, the two estimators have been identicall parameterized and a sampling time o ms has been selected or both. Starting rom the SDC orm, the derivative o the state estimate is given b: z x Hx,u ˆ x F(ˆ, x u)ˆ x K ˆ () ISBN: ISSN: (Print); ISSN: (Online) WCE 7

4 Proceedings o the World Congress on Engineering 7 Vol II WCE 7, Jul 5-7, 7, London, U.K. Table. Vehicle parameters or simulation test. Distance o the centre o gravit rom the ront axle (m) Distance o the centre o gravit rom the rear axle (m).5.5 Height o the centre o gravit (m).35 Front and rear trac width (m).6 Vehicle mass (g) Yaw moment o inertia (g m ) 6 8 Wheel radius (m).3 Wheel moment o inertia (g m ) Numerical simulations have perormed in Matlab/Simulin environment adopting an integration algorithm with a ixed step size. The irst important result is showed in Fig., where the capabilit o the SDREF is easil visible: indeed, the comparison with the EKF technique highlights substantial dierences due to the linearization process. The nonlinearities o the sstem are ull considered in the SDREF and, consequentl, the estimation gives a value that is superimposed to the simulated one. This result inds an important application in all the vehicle dnamics control sstems based on the sideslip angle adopted as eedbac. It has to be highlighted the better perormance o the SDREF respect to the EKF, as it can be observed in Fig. 3 that shows an estimation error or the EKF greater o one order o magnitude. B comparing the estimation error with the simulated value o (Figs., 3) it results that the EKF estimation causes an error corresponding to about % o the simulated value while the SDREF gives an error lower than %. This important result demonstrates the eectiveness o the proposed technique applied to the estimation o the vehicle sideslip angle. Fig. 3. Estimation error or sideslip angle steering pad manoeuvre. With the particular reerence to the vehicle dnamics, the nonlinearities involve the sideslip angle equation o (4). As a consequence, the dierences between the EKF and the SDREF are evident with the particular reerence to the estimate o. In an case, with the aim o completeness o the discussion, the comparisons with the estimated measurements are given because the are undamental to evaluate the coherence o the estimator with the sstem involved in the observation procedure. The results show the better perormance o the SDREF in terms o estimation o the lateral acceleration measurement (Figs. 4 and 5) and substantiall comparable results in terms o estimation o the aw rate measurement (Figs 6, 7). This is due to the presence o nonlinearities in the irst measurement equation (5). Fig. 4. Lateral acceleration steering pad manoeuvre. Fig.. Sideslip angle steering pad manoeuvre. Fig. 5. Estimation error or lateral acceleration steering pad manoeuvre. ISBN: ISSN: (Print); ISSN: (Online) WCE 7

5 Proceedings o the World Congress on Engineering 7 Vol II WCE 7, Jul 5-7, 7, London, U.K. Fig. 6. Yaw rate steering pad manoeuvre. Fig. 9. Estimation error or sideslip angle double lane change manoeuvre. Fig. represents the measurement a with its estimates, while Fig. shows the estimation error. Fig. 7. Estimation error or aw rate steering pad manoeuvre. A second simulated result concerns a transient manoeuvre given b a double lane change at a constant longitudinal speed o m/h, steering amplitude o 5 in high adhesion conditions. As in the previous test, the comparisons concerning the estimates o the vehicle sideslip angle are plotted. Also in this case, the goodness o the SDREF can be easil highlighted (Fig. 8) since the estimate is practicall superimposed to the simulated value. Fig.. Lateral acceleration double lane change manoeuvre. Fig.. Estimation error or lateral acceleration double lane change manoeuvre. The SDREF shows a better perormance, while comparable results can be seen or the aw rate measurement, highlighting a unctional parameterisation o the two ilters (Figs., 3). Fig. 8. Sideslip angle double lane change manoeuvre. The SDREF gives an estimation error (Fig. 9) reduced o an order o magnitude i compared to the EKF. This result strongl validates the nonlinear estimation technique also or this transient manoeuvre. Fig.. Yaw rate double lane change manoeuvre. ISBN: ISSN: (Print); ISSN: (Online) WCE 7

6 Proceedings o the World Congress on Engineering 7 Vol II WCE 7, Jul 5-7, 7, London, U.K. Fig. 3. Estimation error or aw rate double lane change manoeuvre. The illustrated results allow to appreciate the goodness o the SDREF or the estimation o the vehicle sideslip angle. VI. CONCLUSIONS A state-dependent-riccati-equation based Kalman ilter has been proposed or the estimate o the vehicle sideslip angle. The estimator has been designed taing into account a single trac vehicle model together with a random wal model or the lateral interaction orces. Consequentl, the proposed approach is not based on speciic interaction models that require the nowledge o parameters. The ilter can be easil adapted to dierent vehicles, tres and boundar conditions without a detailed nowledge o their characteristics and, dierentl rom common approaches, no tuning procedure is required. The results show the advantages o the SDREF, able to ull capture all nonlinearities. Nomenclature Vehicle model u Vehicle longitudinal velocit v Vehicle lateral velocit r Vehicle aw rate β Vehicle sideslip angle a Vehicle lateral acceleration F Lateral interaction orce o the ront axle F Lateral interaction orce o the rear axle F First time derivative o the lateral interaction orce o the ront axle F First time derivative o the lateral interaction a b δ m J z orce o the rear axle Distance rom the ront axle to the centre o gravit Distance rom the rear axle to the centre o gravit Steering wheel angle o the ront tres Vehicle total mass Vehicle moment o inertia respect to z axis Force to be estimated First time derivative o the orce to be estimated w Random white noise Q z g Covariance o the process noise or the EKF Measurement vector Gaussian white measurement noise or the EKF R Covariance o the measurement noise or the EKF K Filter gain o the EKF t Time ˆ Estimate E Expected value x Initial condition on the state vector P P Initial condition on the error covariance Error covariance or the EKF A -, L -, H, M Partial derivative matrices or the EKF F ( x(, H x( t) Input and state dependent matrices o the SDC orm Q Covariance o the process noise or the SDREF g Gaussian white measurement noise or the SDREF R Covariance o the measurement noise or the SDREF K Filter gain o the SDREF P Solution o the algebraic Riccati equation ψ Process noise or the EKF ψ Process noise or the SDREF Superscripts - a priori estimate + a posteriori estimate Subscripts - related to time t - related to time t REFERENCES [] Rajamani, R. (5). Vehicle Dnamics and Control. Springer- Verlag, New Yor. [] Chen, B.C. and Hsieh, F.C. (). Sideslip angle estimation using extended Kalman ilter. Vehicle Sstem Dnamics, 46 (Suppl.), [3] Strano, S., Terzo, M. (6). Actuator dnamics compensation or real-time hbrid simulation: an adaptive approach b means o a nonlinear estimator. Nonlinear Dnamics, 85 (4), [4] Russo, R., Strano, S., Terzo, M. (6). Enhancement o vehicle dnamics via an innovative magnetorheological luid limited slip dierential. Mechanical Sstems and Signal Processing, 7 7, [5] Mrace, C.P., Cloutier, J.R., D Souza, C.N. (996). A new technique or nonlinear estimation. Proc. IEEE Con. Control Appl. Dearborn, MI, [6] Çimen, T. (). Surve o State-Dependent Riccati Equation in Nonlinear Optimal Feedbac Control Snthesis. AIAA Journal o Guidance Control and Dnamics, 35 (4), [7] Strano, S. and Terzo, M. (6). Accurate state estimation or a hdraulic actuator via a SDRE nonlinear ilter. Mechanical Sstems and Signal Processing, 75, EKF and SDREF u Input vector, h Nonlinear unctions ISBN: ISSN: (Print); ISSN: (Online) WCE 7