Vehicle virtual sensing : estimation of the longitudinal velocity and tyre force

Dissertation Degree in Industrial Technologies Engineering (GETI) Vehicle virtual sensing: estimation of the longitudinal velocity and tyre forces Report Autor: Maria Garabatos Saboya Director: Arnau Doria Cerezo Co-Supervisor: Stefano de Pinto Summons: June 2021 Escola Tècnica Superior d’Enginyeria Industrial de Barcelona Acknowledgements First of all I would like to thank Stefano De Pinto and McLaren Automotive LTD for their predisposition on taking part on a final degree project as this one. Appreciate Stefano for all the work he has done since November 2020 by proportioning me all sort of information and knowledge of a so important topic as virtual sensing. For introducing me to vehicle control world, for such dedication during all these months and for accepting to co-supervise my project. I am really pleased for all the support received and for teaching me extra knowledge on this topic, recommending me really interesting books and being on the lookout of me understanding everything. Furthermore, I would like to thank Arnau Doria for proposing me the incredible and unique opportunity of taking part in such a project together with McLaren Automotive LTD. For these eight months of complete dedication and support, giving me advice and suggestions based on his knowledge and experience. For all the meets we have had discussing about how everything should be scheduled and all the theoretical classes he has given to me to help me understand such a topic I was not familiar with before. Finally, I would like to thank my family and friends for all the support during the realization of the project and all the years of university studies. 1 Abstract In this project, a first contact with virtual sensing is explained. Denoting the importance of sensors information, of understanding vehicle dynamics and the influence the estimators have when modeling a vehicle. Two different vehicle models are presented, Bicycle and Four-wheel model, together with a linear and a nonlinear estimators, Kalman Filter and Extended Kalman Filter, respectively. At the end of the project, two approaches have been carried out in order to determine the correct functionality of those estimators. Finally, a more detailed approach has been done in which the Extended Kalman Filter is evaluated. In this case, the vehicle is not modelled, and those estimated IMU parameters have been replaced by real data provided by McLaren Automotive LTD. 2 Contents Nomenclature 6 Acronyms 8 1 Introduction 9 1.1 Motivation......................................... 9 1.2 Stateoftheart....................................... 10 1.3 Objectives ......................................... 11 1.3.1 Long-termobjectives............................... 11 1.4 Outlineoftheproject................................... 12 2 Kalman Filter 14 2.1 Kalmanfilter........................................ 14 2.2 ExtendedKalmanFilter ................................. 15 3 Vehicle modelling 17 3.1 Vehicledynamics ..................................... 17 3.2 Tyrebehaviour....................................... 18 3.3 MagicFormulatyremodel................................ 19 3.4 Four-wheelmodel..................................... 21 3.5 Bicyclemodel ....................................... 22 3.6 Summary.......................................... 23 4 Longitudinal speed estimation 25 4.1 Kinematicmodel ..................................... 25 4.2 Kinematic four-wheel model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 4.3 Summary.......................................... 33 5 Estimator verification 35 5.1 Steeringrepresentation.................................. 35 5.2 Longitudinal wheel velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 5.3 Summary.......................................... 41 6 Economical analysis 42 7 Conclusions 44 Bibliography 46 3 List of Figures 1.1 Schematization of the project. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.1 Observability scheme. [8] ................................. 15 3.1 Representation of vehicle coordinate frame. Courtesy from McLaren Automotive LTD. . . . 17 3.2 Tyre coordinate frame.[1] ................................. 18 3.3 Pure longitudinal slip characteristics according to longitudinal load. . . . . . . . 20 3.4 Pure lateral slip characteristics according to lateral load. . . . . . . . . . . . . . . . 20 3.5 The four wheel model. Courtesy from McLaren Automotive LTD. ................ 21 3.6 Thebicyclemodel..................................... 23 4.1 Longitudinal speed estimation. Comparison with KF. . . . . . . . . . . . . . . . . 26 4.2 Lateral speed estimation. Comparison with KF. . . . . . . . . . . . . . . . . . . . . 27 4.3 Longitudinal velocity of the four wheels. . . . . . . . . . . . . . . . . . . . . . . . 29 4.4 Longitudinal velocity of the vehicle. Comparison with the EKF. . . . . . . . . . . 30 4.5 Longitudinal velocity of front left wheel. Comparison with EFK. . . . . . . . . . . 31 4.6 Longitudinal velocity of front right wheel. Comparison with EFK. . . . . . . . . . 32 4.7 Longitudinal velocity of rear left wheel. Comparison with EFK. . . . . . . . . . . 32 4.8 Longitudinal velocity of rear right wheel. Comparison with EFK. . . . . . . . . . 33 5.1 Steering representation of both front wheels. . . . . . . . . . . . . . . . . . . . . . 36 5.2 Wheels longitudinal velocity estimation of a high gamma vehicle of McLaren AutomotiveLTD........................................ 37 5.3 Wheels longitudinal velocity of a high gamma vehicle of McLaren Automotive LTD.............................................. 37 5.4 Front left wheel velocity. Comparison with EKF. . . . . . . . . . . . . . . . . . . . 38 5.5 Front right wheel velocity. Comparison with EKF. . . . . . . . . . . . . . . . . . . 39 5.6 Rear left wheel velocity. Comparison with EKF. . . . . . . . . . . . . . . . . . . . . 40 5.7 Rear right wheel velocity. Comparison with EKF. . . . . . . . . . . . . . . . . . . . 40 4 5 page 6 Report Nomenclature Symbols Units Meaning A - System matrix B - Input matrix C - Output matrix D - Feedthrought matrix xInput vector yMeasurement vector KkKalman gain PkError covariance matrix Q - Process noise covariance matrix R - Measurement noise covariance matrix fdState update function hMeasurement function ˙ ψrad/s Yaw rate ¨ ψrad/s2Yaw acceleration FxN Longitudinal force FyN Lateral force FzN Vertical force MxNm Overturning moment MyNm Rolling resistance moment MzNm Aligning moment mkg Vehicle mass vxm/s Longitudinal velocity vym/s Lateral velocity γrad Camber angle αij rad Tyre sideslip angle (front or rear, right or left) βrad Vehicle sideslip angle Vehicle virtual sensing: estimation of the longitudinal velocity and tyre forces page 7 Symbols Units Meaning CxN/rad Longitudinal cornering stiffness CyN/rad Lateral cornering stiffness CiN/rad Cornering stiffness of each wheel (front or rear) λLongitudinal slip Lim Length of the vehicle (front or rear) tim Width of the vehicle (front or rear) δrad Steering angle vx,wh m/s Longitudinal velocity of the wheels vy,wh m/s Lateral velocity of the wheels axm/s2Longitudinal acceleration ˙vxm/s2Longitudinal acceleration aym/s2Lateral acceleration ˙vym/s2Lateral acceleration Chapter 2 Kalman Filter 2.1 Kalman filter The Kalman Filter consists of the estimation of a model value, the state vector, of the previous instant which is obtained by the measured value in the actual instant. This value can be estimated according to the system dynamics and the noise measurements of the model [7]. This filter is linear, meaning that the uncertainty accounts on a Gaussian noise and the covariance errors. In this linear configuration, this filter is the optimal to use, but when nonlinearity appears, other Kalman filters should be studied, such as the Unscented and the Extended Kalman filters. This report will focus on the linear Kalman filter and the Extended Kalman filter for nonlinear systems [1]. Starting with the simplest, it is a linear and autonomous system described by the equations (2.2). ˙x=Ax +Bu (2.1) y=Cx +Du (2.2) Being a linear system means that matrices A, B and C above are constant, and D is non-existent. The filter is divided in two parts, the prediction represents the estimation of the state and the error covariance according to the posteriori state and the system’s knowledge and it is represented by the equations (2.3)-(2.5): ˆxk=Adˆxk−1+Bduk−1(2.3) ˆyk=Cˆxk+Duk(2.4) ˆ Pk=Adˆ Pk−1AT d+Q(2.5) While the correction part uses the output of the system to estimate the posteriory state, and it is presented by the equations (2.6)-(2.8): Kk=PkCT(CP − kCT+R)−1(2.6) ˆxk=ˆx− k+Kk(yk−ˆyk)(2.7) Pk=(I−KkC)Pk(2.8) 14 Vehicle virtual sensing: estimation of the longitudinal velocity and tyre forces page 15 It is important noticing that Q and R are matrices which depend on the dimensions of the state and the outputs of the system, and are the covariance matrices of the model and measurement noise, respectively. For this filter to start operating, an initial value for the state and the error covariance must be defined. Only then, the Kalman Filter will be initialized and will provide all the necessary measurements to obtain the minimum error of the estimated states. The Kalman Filter can be seen as an optimal observer. The latest occurs when the current state of the process is estimated, x, and this happens thanks to the measurement and the input vectors, yand u, respectively. Figure 2.1: Observability scheme. [8] Furthermore, observability can be defined as a property of the system for both model and measurement equations, which is independent of the type of estimator used [1]. Therefore, a system is observable when the initial state can be determined. As a consequence, the whole system can be determined. For a Kalman Filter to be correctly applied, the system has to be observable. 2.2 Extended Kalman Filter The Extended Kalman Filter is a nonlinear system which depends on the state, and its linearization is accomplished by means of a successive approximation of each point from the change of matrices A, B and C values. The most general form of the discretized state space representation is: ˆxk=fd(ˆxk−1, uk−1)(2.9) ˆyk=h(ˆxk, uk)(2.10) As the initial filter is described by a linear system, first of all a linearization of the system must occur [1]. This will happen by the transformation of the matrices A and C towards F and H. This transformation is already considered in the following equations (2.11)-(2.12) of the Extended Kalman Filter. Fk−1=∂fd(x, u) ∂x |ˆxk−1,uk−1(2.11) Hk=∂h(x, u) ∂x |ˆxk,uk(2.12) page 16 Report This filter is also divided in two parts. The first one, which corresponds to the prediction part of the linear system and is based on the replica of the nonlinear system. The discretization of this filter is denoted by (2.13)-(2.15). ˆxk= ˆxk−1+4tf(ˆxk−1, uk−1)(2.13) ˆyk=h(ˆxk−1, uk−1)(2.14) Pk=Fk−1Pk−1+Q(2.15) The equations corresponding to the correction part are described in (2.16)-(2.18): Kk=P− kHT k(HkP−1 kHT k+R)−1(2.16) ˆx+ k=ˆx− k+Kk(yk−ˆy− k)(2.17) P+ k=(I−KkHk)P− k(2.18) The old A matrix now corresponds to Fk−1while the old C matrix to Hk. Subsequently, those matrices value will be obtained according to the Kalman filter required in each case. Chapter 3 Vehicle modelling 3.1 Vehicle dynamics Before starting with the accurate definition of each tyre and its effect towards the vehicle behaviour, it is necessary to define the more important dynamic characteristics in order to model correctly the vehicle. This is why the next coordinate frames must be introduced. Those coordinates make reference on three rotational-degrees-of-freedom, but also three different velocities according to the system, as seen in Figure 3.1. To see more information about the translational and rotational equations of motion of this frame see [1]. Figure 3.1: Representation of vehicle coordinate frame. Courtesy from McLaren Automotive LTD. 17 page 18 Report In this report, as only the longitudinal motion of the vehicle will be studied, the only moment that will be relevant will be the yaw, as will be seen in the following chapters. Also longitudinal and lateral velocities must be considered. 3.2 Tyre behaviour This section is based on studying the different properties and characteristics of a vehicle and its pneumatic tyres. Providing the most common tyre models, and making emphasis on the simple ones. Additionally, this section goes into detail about the distinctive vehicle models. Tyre forces and moments are responsible of the vehicle motion, and are extremely important when it comes to represent the dynamic behaviour of the vehicle. For the correct modeling of the vehicle it is important to understand the vehicle dynamics of it, but also the effect of the forces to the vehicle’s behaviour. Describing the tyre road interaction begins with describing a coordinate frame attached to the centre of the contact patch as seen in Figure 3.2. Figure 3.2: Tyre coordinate frame.[1] According to the representation of Figure 3.2, the x-axis is the intersection between the tyre plane line and the ground; the z-axis is perpendicular to the ground plane and will always point upwards; finally, the y-axis is on the ground plane and points according to the righthanded agreement. The tyre orientation is represented by the camber angle γand the sideslip angle α. The first one defines the inclination of the tyre plane according to the x-axis, while the second one, the z-axis rotation of the velocity vector and the x-axis. It can be measured according to: tan(α) = vy vx (3.1) being vx, vythe x and the y coordinates of the velocity vector v. Vehicle virtual sensing: estimation of the longitudinal velocity and tyre forces page 19 Straightaway, forces and moments represented in Figure 3.2 will be briefly explained, just making emphasis in those parameters that will be relevant for the models that will be studied further on. Those forces are represented as: •Fx−Longitudinal force :is applied along the x-axis. When the car is accelerating, a positive torque is applied, this force is defined as positive. And when the car is braking, vice versa. •Fy−Lateral force :is applied along the y-axis. When left-cornering it is defined positive, the wheel rotates in anti-clockwise, and vice versa when right-cornering. Those moments are represented as: •Mz−Aligning moment :about the z-axis. On account of the lateral tyre force displaced backwards from the centre of the contact patch. As a planar motion is considered, the vertical forces and the overturning and rolling resistance moment will not be considered. To the contrary, longitudinal and lateral forces are fundamental when it comes to descibring vehicle dynamics because are the ones responsible of the appearance of a shear mechanism, generating a tyre-road friction coefficient that will prevent slip from happening. 3.3 Magic Formula tyre model The Magic Formula tyre model was first created by Hans Baastian Pacejka [6], a Dutch professor expert in vehicle dynamics, and consists of the modelling of the tyre model. It is based in an empirical model that let us study the tyre forces behaviour according to the slip. The main objective is to represent the tyre and vehicle behaviour for small slip values according to a linear approximation of the slip. Y(x) = Dsin(Carctan(Bx −E(Bx −arctan(Bx)))) + SV(3.2) being, B (Stiffness factor), C (Shape factor), D(Peak value), E (Curvature factor) and SV(Vertical shift). The linear approximation mentioned above represents the linear part of both figures 3.3 and 3.4, respectively, and it is described by the following equations: Fx=Cxλ(3.3) Fy=−Cyα(3.4) To make Figure 3.3 easier to understand, ABS is introduced. ABS is the Braking Control System, and its main objective is to prevent the wheel from blocking when braking. The wheel blocks when the slip ratio is not operating in the optimal values of the figure below, in the pic of it. But when this happens, vehicle stability and steerability are retained so the slip starts operating in the optimal values and the wheel will not block. In conclusion, operating in the optimal values allows having a better performance of the wheel. page 20 Report -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 -2000 -1500 -1000 -500 0 500 1000 1500 2000 Figure 3.3: Pure longitudinal slip characteristics according to longitudinal load. -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 -2000 -1500 -1000 -500 0 500 1000 1500 2000 Figure 3.4: Pure lateral slip characteristics according to lateral load. When it comes to study the behaviour according to the lateral force, the objective of the lateral slip is the same as the slip ratio, to operate near the optimal values, but in this case, its behaviour is described in the TCS system. Vehicle virtual sensing: estimation of the longitudinal velocity and tyre forces page 21 Its functionality can be appreciated in Figure 3.4. The similarity to Figure 3.3 can be easily appreciated, but the main difference is, apart from the fact that lateral tyre characteristics are taken intro consideration, the lateral force of the tyres is always denoted as a negative force because it is opposed to the movement of the tyre. 3.4 Four-wheel model Now that the tyre model has already been introduced, this project will present the different models a vehicle can be modelled by. The first one is the four-wheel model, this is a complete representation of the vehicle considering the effect of the four wheels, their steering angle, the longitudinal and lateral velocities of all of them and the effect of their forces. Even-though it does not consider the suspensions of any wheel. Figure 3.5: The four wheel model. Courtesy from McLaren Automotive LTD. For these model, as all the wheels of the vehicle are considered, there is no need the slip angles of all four wheel are identical. Will be seen later on but, when these happens there is already and existent model which will model it. The main difference between them is that in this case, all slip angles can have and will have a different value. This is on account of the four wheel model is a more realistic one adjusted in reality, and the other one is just and approximation of it. The equations of motion of this model are: m(˙vx−vy˙ ψ) =Fx,fl cos(δfl) + Fx,fr cos(δfr)−Fy,fl sin(δfl) −Fy,fr sin(δfr) + Fx,rl +Fx,rr (3.5) m(˙vy+vx˙ ψ) =Fy,fl cos(δfl) + Fy,fr cos(δfr) + Fx,fl sin(δfl) +Fx,fr sin(δfr) + Fy,rl +Fy,rr (3.6) 2Izz ¨ ψ= 2lf(Fx, fl sin(δfl) + Fy,fl cos(δfl) + Fx,fr sin(δfr) + Fy,fr cos(δfr)) +tf(Fy,fl sin(δfl)−Fx,fl cos(δfl)−Fy,fr sin(δfr) + Fx,fr cos(δfr)) −2lf(Fy,rl +Fy,rr) + tr(Fx,rr −Fx,rl)(3.7) page 22 Report Noticing that the only part of the vehicle which is in contact with the road are the tyres, it is easy to find out that a correct behaviour of that one comes from a correct study and representation of the tyres and their parameters. This is why when modelling a car is as important modeling correctly the equations of motion, as it is to model correctly the tyres. For this model, they are represented with the equation (3.8) below: "vx,wh,ij vy,wh,ij #=   cos δij sin δij 0 −sin δij cos δij 0 0 0 1        vx vy 0  +   0 0 ˙ ψ  ×   ±li ±tj 2 0      (3.8) The wheel’s velocity is necessary when it comes to finding the value of the lateral forces represented in the equations of motion. This lateral force is obtained by a linear approximation on the cornering stiffness. Fy,ij =Ciαij (3.9) Notice that idenotes front and rear wheels and jdenotes right and left wheels. The necessary sideslip angle comes from finding the longitudinal and lateral velocities of each wheel according to the equation (3.10). αij = arctan vy,wh,ij vx,wh,ij !(3.10) For more information about these model and its repercussion to the vertical forces see section 2.1.2 of [1]. 3.5 Bicycle model After introducing the complete model of the vehicle, four-wheel model, a more simple one will be presented. This one is the bicycle model, which is a simplification of the previous, by assuming identical slip angles on both left and right wheels and also ignoring the lateral load transfer. It is based in the study of the longitudinal and lateral vehicle dynamics. Although it is not a suitable model for simulation, it is useful for estimation and control purposes. As this report focuses on the study of the longitudinal vehicle dynamics, this model will be modelled in order to study the vehicle behaviour towards these characteristics and can reflect on whether the simulation obtained is specific and exact or can be improved. Figure 3.6 is a representation of this, from where can be extracted the longitudinal, lateral and yaw equations of motion. Vehicle virtual sensing: estimation of the longitudinal velocity and tyre forces page 23 lrlf Vy Fx,r Fy,r v v v β αr αf δ Ψ Vx Fy,f Fx,f Figure 3.6: The bicycle model Those equations of motions are: ˙vx=1 m(Fx,f cos(δf)−Fy,r sin(δf) + Fx,r) + vy˙ ψ(3.11) ˙vy=1 m(Fy,f cos(δf) + Fx,f sin(δf) + Fy,r)−vx˙ ψ(3.12) ¨ ψ=1 Izz (lfFx,f sin(δf) + lfFy,f cos(δf)−lrFy,r)(3.13) Considering only front traction and a cornering stiffness linear approximation, equation (3.14) is described. Fy,i =Ciαi(3.14) being i, front or rear, respectively. Notice that, to find the sideslip angle, the wheel velocities in the x-axis and y-axis are necessary. Those equations are find below: αi= arctan vy,wh,i vx,wh,i !(3.15) "vx,wh,i vy,wh,i#=   cos δisin δi0 −sin δicos δi0 0 0 1        vx vy 0  +   0 0 ˙ ψ  ×   ±li 0 0      (3.16) 3.6 Summary In this chapter a brief introduction to vehicle dynamics is presented, together with the importance of tyre behaviour and modelling. For this, all the necessary parameters that will be used to model the tyre and vehicle behaviour are described. After that, the Magic Formula tyre model is defined. Straightaway, the tyre’s behaviour according to the Pacejka formula is represented in figures 3.3 and 3.4, and its relation with the security systems explained in Chapter 1 is defined. Furthermore, not only tyres are important to be modelled, but so is the vehicle. This is why the next sections of this chapter represent two different vehicle models, the first one, considers page 30 Report As can be seen in Figure 4.4, this problem is solved. This has occurred because the vehicle model considered for this estimation is more complex and, consequently, provides more and more accurate information. In reality, this information will also be obtained by the IMU sensor, the same as with the other model, but in the case of simulations, this information will be more similar to reality. The comparison of the lateral velocity obtained in Figure 4.2 with the one obtained from an Extended Kalman Filter is not object of this simulation, but can be found in [5]. Even though, it is important knowing that the convergence problem obtained in the previous section will be solved. 0 5 10 15 20 25 30 35 40 45 50 0 10 20 30 40 50 60 70 80 90 estimated velocity four-wheel model Figure 4.4: Longitudinal velocity of the vehicle. Comparison with the EKF. Establishing a color reference according to Figure 4.3, a front left, front right, rear left, and rear right real velocity is compared to their respective estimations after applying and Extended Kalman Filter. This can be found in figures 4.5, 4.6, 4.7 and 4.8. The discussion of the velocity estimation done for Figure 4.5 will apply to the other wheels represented in the following figures. They will not be discussed as the explanation is the same. Vehicle virtual sensing: estimation of the longitudinal velocity and tyre forces page 31 0 5 10 15 20 25 30 35 40 45 50 66 68 70 72 74 76 78 80 82 84 86 estimated real Figure 4.5: Longitudinal velocity of front left wheel. Comparison with EFK. Notice that this estimation is quite accurate and is not much influenced by noise, except when lowering the velocity. Despite that, those simulations can be considered correct with an error calculated of 0,8715%. Following, Table 4.2 will present the error percentage of the estimated velocities obtained from the root square mean (RSME). For that, the transient state has not been considered. Table 4.2: RSME calculation between the real and the estimated velocity. Longitudinal velocity Error (%) Vehicle velocity 0,8501 Front left wheel 0,8715 Front right wheel 0,8715 Rear left wheel 0,8533 Rear right wheel 0,8534 page 32 Report 0 5 10 15 20 25 30 35 40 45 50 66 68 70 72 74 76 78 80 82 84 86 estimated real Figure 4.6: Longitudinal velocity of front right wheel. Comparison with EFK. 0 5 10 15 20 25 30 35 40 45 50 70 72 74 76 78 80 82 84 86 estimated real Figure 4.7: Longitudinal velocity of rear left wheel. Comparison with EFK. Vehicle virtual sensing: estimation of the longitudinal velocity and tyre forces page 33 0 5 10 15 20 25 30 35 40 45 50 70 72 74 76 78 80 82 84 86 estimated real Figure 4.8: Longitudinal velocity of rear right wheel. Comparison with EFK. For all of these simulations, the slip ratio has not been considered when describing the Extended Kalman Filter. As explains [1] in Chapter 4, some approaches have been made which consider the slip ratio so small that can be neglected, as can be appreciated in the following equations: vij =(λij + 1)vw,ij (4.13) vij =vw,ij (4.14) For this reason, each wheel velocity is approximated to the longitudinal velocity of each wheel in the contact patch. This contact patch is described in Figure 3.2. It is the contact point between the wheel and the ground plane, which in normal conditions, its velocity is described according to the following equations: vw,ij =   cos δij sin δij 0 −sin δij cos δij 0 0 0 1        vx vy 0  +   0 0 ˙ ψ  ×   ±li ±tj 2 0      (4.15) 4.3 Summary After having described the different Kalman filters and the different ways to model a vehicle, this chapter is introduced to determine the functionality of this estimators according to each vehicle model. page 34 Report The first section estimates, according to a linear system, the velocities of the center of gravity of a vehicle, just considering one front and one rear wheel (Bicycle Model in Section 3.5). Figures 4.1 and 4.2, represent the longitudinal and lateral velocity of a vehicle compared to the estimated velocities according to a linear Kalman Filter, respectively. It can be seen how the lateral velocity is not estimated correctly. In this section the motive from this is explained and a solution is proposed. The solution is to implement a nonlinear estimator which will consider the four wheels of the vehicle (Four-Wheel Model in Section 3.4). These estimations starts with the linearization of the system and then its discretization. As can be seen in figures from 4.5 to 4.8, and with the calculated error of Table 4.2, the four wheels are estimated correctly. So this filter is the suitable to be implemented. Chapter 5 Estimator verification As has been explained during the first chapter of this report, one of the main objectives was to design an estimator able to compare the real velocity of a McLaren Automotive LTD high gamma vehicle with the estimated velocity towards an Extended Kalman Filter. In this chapter, the simulations obtained will be presented and explained in detail. Before starting explaining the results obtained, it is important noticing a big difference in comparison with the simulations explained in Chapter 4. Previously, the vehicle was modelled according two different models, Bicycle or Four-Wheel model, depending on the information wanted. Remember the latest allows to obtain more information than the former because of having considered the four wheels of the vehicle and both steering angles of the front wheels, there is nothing simplified in there. On the contrary, to start this simulation, the model of the vehicle was not considered. This is because the information extracted from the model to enter as an input in the filter of Kalman implemented (in this case, Extended Kalman Filter) is directly provided by McLaren Automotive LTD as input signals. Having now clarified why the previously explained models will not be considered here, the results obtained in the simulations will be discussed in the following sections. 5.1 Steering representation Figure 5.1 represents the steering angle of both front wheels. This figure helps understanding the upcoming figures of the longitudinal velocities of the wheels. The information of the steering angle described when the vehicle is turning right or left, following the signal criteria denoted by Figure 3.5. In the estimator designed, it enters as an input which helps estimate the different wheels velocities and the longitudinal velocity of the vehicle. It can be seen how this trajectory is really complex, but, what is more important, is that both front wheels will be submitted to the same steering. This may not happen, and lets to a more complex study which is not object of this project, but more information about this different configurations can be found in [9]. 35 page 36 Report 0 0.5 1 1.5 2 2.5 3 3.5 4 104 -6 -4 -2 0 2 4 6 Figure 5.1: Steering representation of both front wheels. 5.2 Longitudinal wheel velocity Figure 5.2 represents the estimated velocity of the four wheels. To obtain them, was crucial understanding the measures provided by McLaren Automotive LTD, and the units with which they where measured. After that, the four wheels information enter the estimation as an input without considering white Gaussian noise. This is because as this data is real data obtained from the sensors of the vehicle when going testing, the noise is already included. In relation with Chapter 2, this data is the measured data ykof the discretized state space representation of the Extended Kalman Filter (equations (2.9) - (2.10)). Meanwhile, Figure 5.3 represents the exact same graphic as before but without having estimated the velocity, meaning, this is done directly from the data provided. At first sight, an important similitude between both of then can be appreciated. This is why, without further delay, those velocities will be compared in detail. Vehicle virtual sensing: estimation of the longitudinal velocity and tyre forces page 37 0 0.5 1 1.5 2 2.5 3 3.5 4 104 -20 0 20 40 60 80 100 120 140 160 180 front left front right rear left rear right Figure 5.2: Wheels longitudinal velocity estimation of a high gamma vehicle of McLaren Automotive LTD. 0 0.5 1 1.5 2 2.5 3 3.5 4 104 0 20 40 60 80 100 120 140 160 180 front left front right rear left rear right Figure 5.3: Wheels longitudinal velocity of a high gamma vehicle of McLaren Automotive LTD. page 38 Report To keep in mind the velocities defined following, remember Figure 3.5. Figures 5.4 and 5.5 compares front velocities from both, left and right, wheels, while Figures 5.6 and 5.7 compares the rear velocities from both wheels. As was previously appreciated, the estimation of each velocity is done correctly, meaning, the Extended Kalman Filter design is a good filter that complies with his functionality independently of the type of vehicle studied until now. The mathematical extraction of these velocities is the same as explained in Chapter 4, and are denoted by the equation (4.15). In this case, the slip is not still being considered, and because of the duration of the project, this will be one of the further objectives to achieve, as been explained in Section 1.3.1. 0 0.5 1 1.5 2 2.5 3 3.5 4 104 -20 0 20 40 60 80 100 120 140 160 180 estimated real Figure 5.4: Front left wheel velocity. Comparison with EKF. One of the main objectives of testing McLaren Automotive LTD data with the estimator designed in Chapter 4 was to determine whether that estimator was good calibrated or any parameter change should be done. Vehicle virtual sensing: estimation of the longitudinal velocity and tyre forces page 39 The conclusion to that objective is that there is a little parameter modification which will be object of the future objectives explained before (Section 1.3.1) in order to reduce the error percentage of the rear left wheel, which increases a lot in comparison with the other wheels. Those error percentages are shown in Table 5.1. As has been explained in the previous chapter, this error has been calculated according to the root mean square procedure (RSME). But, as real data is used in this simulations, function nanmeadian of Matlab has been necessary to delete the NaN values of the look-up table. Table 5.1: RMSE calculation of the longitudinal velocity of the wheels. Longitudinal velocity Error (%) Front left wheel 1,8833 Front right wheel 1,9007 Rear left wheel 6,1109 Rear right wheel 1,8767 0 0.5 1 1.5 2 2.5 3 3.5 4 104 -20 0 20 40 60 80 100 120 140 160 180 estimated real Figure 5.5: Front right wheel velocity. Comparison with EKF. Bibliography [1] Sebastiaan Van Aalst. “Virtual Sensing for Vehicle Dynamics.A model-based approach for indirect meadurement of the vehicle motion states and tyre forces.” PhD thesis. [2] Wenchang Li et al. “Adaptive Car-Following Control of Intelligent Electric Vehicles”. In: 2018 IEEE 4th International Conference on Control Science and Systems Engineering (ICCSSE). 2018, pp. 86–89. [3] A. Alloum, A. Charara, and M. Rombaut. “Vehicle dynamic safety system by nonlinear control”. In: Proceedings of Tenth International Symposium on Intelligent Control. 1995, pp. 525– 530. [4] Manuel Acosta, Stratis Kanarachos, and Michael E. Fitzpatrick. “Accurate virtual sensing of vertical tire forces for enhanced handling dynamics”. 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