U
NIVERSITÀ DI
P
ISA
S
CUOLA DI
I
NGEGNERIA
Master Thesis
Automotive Engineering
Vehicle Sideslip Angle estimation
using Kalman Filters: analysis,
modelling and simulation
Department of Civil
and Industrial Engineering
AUTHOR Cristiano Pieralice SUPERVISORS Francesco Bucchi Basilio Lenzo Marco Gabiccini Francesco Frendo ACADEMIC YEAR 2017/2018
I
Abstract
The present work is developed in the Material and Engineering Research Institute of the Sheffield Hallam University with the aim to develop a method to estimate the sideslip angle and the tire forces of a car using Kalman Filter. The estimation method based on these filters is useful because it allows to reach the goal using common and cheap sensors equipped on normal vehicles. After a first review of the methods listed in literature, the Kalman Filter is studied in both its formulations: the Kalman filter (KF) and the Extended Kalman Filter (EKF) for non-linear systems. The differences between these methods are shown analysing telemetry of a Range Rover Evoque, from the Project iCompose, equipped with a Datron sensor that provides reliable data about the Sideslip angle, useful for the validation of the model. At the end of the document is proposed a procedure for the vehicle model Pacejka Coefficient adaptation to those of the real car.
Keywords: Kalman Filter, Sideslip Angle, Estimation, Vehicle Dynamics.
Copyright information: All the non-referenced figures were created by the author.
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Index
1. Introduction ... 6
2. Vehicle Model Definition ... 8
3. What is a Kalman Filter? ... 12
3.1 Overview ... 12
3.2 Types of Kalman Filters ... 17
3.3 Extended Kalman Filter... 18
4. State of the Art: Sideslip estimation with KF ... 20
5. Validation Data: Equipment and Procedure ... 27
6. Estimation using a Kinematic Vehicle Model ... 31
6.1 Vehicle equations ... 31
6.2 Sideslip Estimation: Kinematic model with one Measurement ... 32
6.2.1 System Observability ... 35
6.2.2 Simulink Model ... 35
6.2.3 Sideslip Estimation: Kinematic model (Two Measurements) . 36 6.2.4 Sideslip Estimation: Kinematic model with a novel Measurement Vector ... 37
6.3 Results ... 37
7. Sideslip Estimation: Dynamic Model ... 39
7.1 Dynamic Two-Track model: Linear tire model ... 39
7.1.1 Observability of the system ... 44
7.1.2 Results... 45
7.2 Dynamic Two Track: Pacejka tire model ... 47
7.2.1 System Observability ... 49
7.2.2 Results... 49
8. Novel Tire Lateral Forces estimation method ... 58
8.1.1 First Fit: Classic Pacejka Model (complete) ... 61
8.1.2 Second fit: Complete Pacejka model formulation ... 62
8.1.3 Third Fit: Complex Pacejka Formulation ... 64
8.1.4 New Pacejka model: Implementation ... 65
9. Sideslip and Tire Forces Estimation: Pseudo Random Walk method ... 67
10. Conclusions ... 69 11. Appendix 1 ... 71 12. Appendix 2 ... 75 13. Appendix 3 ... 76 14. References ... 78 15. List of symbols ... 80 16. List of Figures ... 83 17. Acknowledgment ... 85
Section 1
1.
Introduction
Over the last few decades, with the rapid development of assisted and automated driving, industrial and academic research has dedicated great effort towards safer and better performing vehicles.
A lot of active safe systems are normally equipped on common cars as Anti-Brake Systems (ABS) or Electronic Stability Control (ESC) systems and all of them are based on Inertial and velocity sensors but none of them usually adopt the Sideslip angle as reference.
The Sideslip angle plays a crucial role in the study of the lateral dynamic of a car, especially in the dynamics control. Its estimation, by the use of control systems and starting from common installed car's sensors, may improve the efficiency and the performance of safety systems without an entailing of the production cost.
In addition, also during a testing process, the estimation system may lead to a rapid and precise analysis of the behaviour of the car and to the estimation of the forces acting on the vehicle independently of the use of expensive sensors. The method, developed at the Sheffield Hallam University Material and Engineering Research Institute and described in this document, is based on the use of Kalman Filters and in particular on its Extended formulation. The Extended Kalman Filter (EKF) is useful for the implementation into non-linear models as, for example, a dynamic model of a two-track car, where the non-linearity of the model is due to the aerodynamic force and to the tire characteristic, especially during a transient or a strict manoeuver.
Moreover, the uses of Kalman Filters merge the following objectives:
Estimate quantities that cannot be normally measured (if not with expensive sensors)
Fuse all the sensors information to reach a better result
Filter the input signal
7 o Kinematic (Based on kinematic equations without tire characteristics and
load transfers);
o Dynamic (Based on dynamic equations with tire characteristics and load transfers).
In both cases the models and the results are validated using real on-road data tests from the European project iCompose.
The document is organized as follows: In Section 2 are described the general model of a standard vehicle and the main quantities. In section 3 Kalman filters are introduced in both formulations. In section 4 a State of Art of the estimation methods from the literature is reported. In section 5 the equipment and data to validate the model are presented. In section 6 is addressed the estimation problem using a KF based on a Kinematic vehicle model. The same problem is addressed in Section 7 but with an EKF based on a Dynamic model. In section 8 a method for Pacejka’s coefficients estimation based on on-board data sensors is presented. In Section 9 a different approach for the lateral tire forces estimation is addressed.
8
Section 2
2.
Vehicle Model Definition
As described in [1] a vehicle model may be defined as an object with a clear heading direction and the capability to correct it thanks to the driver actions on the brake pedal, wheel steer and acceleration pedal.
It can be defined a basic reference system on the vehicle as shown in Fig.1:
Figure 1: Basic Vehicle Scheme and Reference System [1] Figure 1 defines some quantities:
The vehicle longitudinal axis x, and hence the vehicle direction i;
The height ℎ from the road plane of the centre of gravity G of the whole vehicle;
The longitudinal distances 𝑎1 and 𝑎2 of G from the front and rear axles, respectively;
The lateral position b of G from the axis;
9
The front and rear tracks t1 and t2;
The geometry of the linkages of the front and rear suspensions;
The position of the steering axis for each wheel.
For a deeper analysis, some kinematic quantities may be defined for the vehicle and for the tire as in [1] and [2]:
Figure 2: Kinematic Quantities Definition [1]
10 Where:
o 𝛽 is the sideslip angle of the vehicle defined as the angle between the speed vector 𝑽̅̅̅̅ and i. In particular: 𝒈
𝛽 = atan(𝑣/𝑢)
where u and v are respectively the longitudinal and lateral velocity of the vehicle. For small angles the formulation may be approximated with the arctangent argument but in this document this approximation is not used.
For the reference system adopted, the sideslip angle is defined positive if anticlockwise;
o 𝛼𝑖𝑗 is the tire slip angle defined as [1] the angle between the rolling
velocity Vr and the speed of travel of the tire Vc .
If 𝛾̇ ≃ 0 (where 𝛾̇ is the camber velocity variation), it may be approximated with:
tan 𝛼 = −𝑉𝑐𝑦
𝑉𝑐𝑥
o r is the yaw rate of the car;
o 𝛿11 and 𝛿12 are the steering angles of the front tires.
Each tire is denominated using the numerical identification ij where: i = axle index (1 for front axle, 2 for rear axle)
j = tire index (1 for left tire, 2 for right tire)
The sideslip angle can be defined for each point of the vehicle and, in particular, for each tire as shown below:
11 It is simple to demonstrate that the relationships between the sideslip angles and the tire slip angles are:
tan(𝛽11) = 𝑢 − 𝑟𝑡𝑣 + 𝑟𝑎1 1/2= tan(𝛿11− 𝛼11) tan(𝛽12) = 𝑢 + 𝑟𝑡𝑣 + 𝑟𝑎1 1/2= tan(𝛿12− 𝛼12) tan(𝛽21) = 𝑣 − 𝑟𝑎2 𝑢 − 𝑟𝑡2/2= tan(𝛿21− 𝛼21) tan(𝛽22) = 𝑢 + 𝑟𝑡𝑣 − 𝑟𝑎2 2/2= tan(𝛿22− 𝛼22)
This formulation will be useful further in the document, when the experimental part will be addressed.
12
Section 3
3.
What is a Kalman Filter?
3.1 Overview
The Discrete Kalman Filter is a probabilistic estimation technique. It may be typically described as a linear quadratic estimator (LQE) based on an algorithm that uses a series of measurements observed over time, containing statistical noise and other inaccuracies. It produces estimates of unknown variables that, in respect to the estimates based on a single measurement, tend to be more accurate by estimating a joint probability distribution over the variables for each timeframe.
Historically, the algorithm was developed by Rudolf E. Kalman and Richard S. Bucy (hence the name Kalman-Bucy) between 1958 and 1961 and applied for the first time on the Apollo computers by the MIT engineers to solve the nonlinear problem of trajectory estimation [3].
The Kalman filter is simultaneously an observer, a filter and an estimator. More in depth the Kalman filter is a particular Luenberger observer. In fact, consider to study the dynamics of the system:
𝑥̇𝑘 = 𝐴𝑥𝑘−1+ 𝐵𝑢
Where:
𝑥𝑘 is the system state; A is the dynamic matrix
B is the input matrix
If the goal is to modify the dynamics of the system, it is useful to add a term that is proportional to the error as:
𝑥̇𝑘 = 𝐴𝑥𝑘−1+ 𝐵𝑢 − 𝐾(𝑧𝑘−1− 𝐻𝑥𝑘−1) Where:
K is the gain matrix
H is the matrix that relates the measurements to the states
13 The error dynamics is influenced by the Gain matrix, as expressed by the following expression:
𝑒̇ = 𝐴𝑒 + 𝐾(𝑧 − 𝐶𝑥𝑘−1) = (𝐴 + 𝐾𝐶)𝑒
If the Gain Matrix K is time-variant and chosen to optimize the estimation, the observer is a Kalman Filter.
A general system governed by the linear stochastic difference equation can be considered: {𝑥̇𝑘= 𝐴𝑥𝑧𝑘−1+ 𝐵𝑢𝑘−1+ 𝑤𝑘−1 𝑘 = 𝐻𝑥𝑘+ 𝑣𝑘 And: 𝑥𝑘0= 𝑥0 Where:
𝑤𝑘−1 is the process noise; 𝑣𝑘−1 is the measurement noise;
𝑥𝑘0 = 𝑥0 is the initial state of the system.
The process noise represents the committed error modelling a system, specifically, the part that, for some reasons, cannot be modelled. In a generic system, for example, the process noise may represent the idea that the state of the system changes over time, but it is not known the exact details of when or how those changes occur, and thus it’s necessary to model them as a random variable.
On the other hand, the measurement noise, in an easier way, represents the error committed by the sensor during a measurement.
The measurement and process noises can be modelled as stochastic processes, or rather as casual variables that are characterize by some parameters, like mean and variance, which change in time. The system uncertainties, in Kalman, are modelled as Gaussian processes which in turn are based on random variables. These are defined as variables that assume casual values into a grouping of admissible numbers and for each of them it can be associated a probability to be assumed.
Two parameters may be defined:
14 𝐸[𝑋] = ∫ 𝑥𝑝(𝑥)𝑑𝑥
+∞
−∞
That is the generalisation of the of mean value in a random process. In the case of a continuous random variable 𝑝(𝑥) is the probability density.
Variance of a signal around its Expected Value:
𝜎2 = 𝐸[(𝑋 − 𝐸[𝑋])2] = ∫ (𝑋 − 𝐸[𝑋])2𝑝(𝑥)𝑑𝑥 +∞
−∞
That represent the random dispersion of a signal around its Expected value. A smaller value of 𝜎2 states a lower dispersion as shown in figure 5
where on the left is 𝜎2 = 1 and on the right is 𝜎2 = 0.1 (obviously
𝐸[𝑋] = 1 in both diagrams):
Figure 5: Random Variable Dispersion
A random continuous variable can be defined Gaussian if its diagram has the classical shape of a Gauss bell and if it has:
15
𝐸[𝑋] = µ (also called mean value)
𝐸[𝑋] = 𝐸[(𝑋 − µ)2]
Consequently, tighter the bell lower is the variance.
It can be stated that a Gaussian random variable is completely described by its Expected value and Variance.
This concept can be extended to the space where now the two parameters are defined as:
𝐸[𝑋] = µ
𝐸[𝑋] = 𝐸[(𝑋 − µ)(𝑋 − µ)𝑇] = ∑
In this case the new name of the variance is “Covariance” and in particular the matrix ∑ in discrete time is equal to 𝑃𝑘 that is the Riccati’s equation solution.
Let assume, in discrete time, that 𝑤 and 𝑣 are white Gaussian stochastic processes with zero mean and that between them there is no dependence. In addition, the initial state is a Gaussian random variable with known mean and covariance values.
As stated in [4], for a system governed by the equations:
{𝑥̇𝑘= 𝐴𝑥𝑧𝑘−1+ 𝐵𝑢𝑘−1+ 𝑤𝑘−1
𝑘 = 𝐻𝑥𝑘+ 𝑣𝑘
The structure of the discrete Kalman Filter is defined as:
Time update equations: 𝑥̂𝑘− = 𝐴𝑥̂ 𝑘−1− + 𝐵𝑢𝑘−1 𝑃𝑘− = 𝐴𝑃 𝑘−1𝐴𝑇+ 𝑄
Measurement update equations: 𝐾𝑘= 𝑃𝑘−𝐻𝑇(𝐻𝑃𝑘−𝐻𝑇+ 𝑅)−1 𝑥̂𝑘 = 𝑥̂𝑘−+ 𝐾 𝑘(𝑧𝑘− 𝐻𝑥̂𝑘−) 𝑃𝑘 = (𝐼 − 𝐾𝑘𝐻)𝑃𝑘− Where:
16 o 𝑥̂𝑘 is the a posteriori state estimate at the step k;
o 𝑃𝑘− is the a priori covariance matrix of the a priori estimate state;
o 𝑃𝑘 is the a posteriori covariance matrix of the a posteriori estimate state; o 𝑅is the covariance matrix of measurement error;
o 𝑄is the covariance matrix of the process error;
The method can be interpreted as follows: considering a real system with some sensors installed to measure a definite number of variables; moreover, a similar system modelled by equations has to be considered into the algorithm. The Kalman filter, starting from the modelled system and the measurements, tries to estimate the state vector comparing, with a particular algorithm, this two information. In order to do that, both information are associated to an analogue number of covariance matrices (again, the covariance matrix of the measurement error 𝑅 and the covariance matrix of the process error 𝑄).
The typical feature of the EKF, thanks to the presence of the ideal model inside the algorithm, is that the estimation can be done also on state whose the direct measurements are not available. At each time step the state estimation is associated to a covariance matrix 𝑃𝑘 that gives a measure of how much the estimation is accurate. This parameter affects the Kalman Gain 𝐾 that in turn affects the new a posteriori estimate state. The product between this gain and the residual (𝑧𝑘− 𝐻𝑥̂𝑘−) allows to the method to take into account the difference
between the estimate states and the measurements, computing also a filtering of the measurements signals. Thanks to this product, the algorithm changes its gain step by step and the estimation results, in the initial phase, results affected by adaptation noise that sometimes can reach high values but then converge. On the other hand, there is also the possibility that the adaptation fails, principally when wrong covariance matrices are settled.
The R and Q matrices may change with each time step but they have been assumed constant in this study. From the definition of process and measurement noise, the calculation of R is easier than Q, indeed the value of R can be calculated using the information about the sensor accuracy (i.e. the power spectral density of the signal). These information can be collected from sensor manuals (as shown after). On the other hand, the process noise covariance Q is difficult to calculate because there is no possibility to observe directly the process that has to be estimated.
Considering that these matrices are involved into the Kalman gain 𝐾𝑘, in order
17
If 𝑅tend to zero, that implies that the Gauss measurement error probability distribution becomes more tight, therefore the Kalman filter model relies more in measurements. In this case 𝑥̂𝑘tend to 𝑧𝑘;
If 𝑄 tend to zero, that implies that the Gauss process error probability distribution become more tight, therefore the Kalman filter model relies less in measurements and more in itself. In this case 𝑥̂𝑘 tend to 𝑥̂𝑘−.
3.2 Types of Kalman Filters
In literature there are a lot of typologies of Kalman Filters, each for a specific application. The major types are listed as follows:
Kalman Filter (KF): is the type of algorithm described above. Is suitable to address the general problem of trying to estimate the state 𝑥 ∈ 𝔎𝑛 of a
discrete-time controlled process governed by a linear stochastic difference equation;
Extended Kalman Filter (EKF): is the extension of the algorithm described above to address the estimation of systems dominated by non-linear processes. It will be described in depth further in the document;
Ensemble Kalman Filter (EnKF): this method is an approximation of the classical (linear) Kalman filter, introduced to reduce the computational effort; this characteristic gets the filter adapt to the estimation of complicated and big phenomena like weather forecast or image reconstruction. E.g.: if the classical KF is used to weather forecasts, taking a 3D grid covering Europe in horizontal resolution of 10km may have dimensions between 100 and 200 in each direction and e.g. 30-50 levels. With six state variables this causes, that length of the model state vector 𝑥̂𝑘is about 5 x 106 . In such case it is not possible to store covariance
matrices in any computer memory. The fundamental idea of the EnKF is to store in the memory at each step an ensemble of random samples of state vectors and after that propagates, using a specific criterion, only a small part of them allowing also a reduction of the covariance matrix. [5]
Unscented Kalman Filter (UKF): this method is addressed to solve the approximation errors of the EKF due to the introduction of a Gaussian random variable into the first-order linearization, that may conduct to have problems in the a posteriori estimation of the state and covariance matrix and, in some cases, also to divergence of the filter [6]. In order to do that the UKF uses a deterministic sampling approach, where the state
18 distribution is always approximated by a Gaussian random variable but represented by using a minimal set of chosen sample points. Therefore, since these sample points capture precisely the mean and covariance of the Gaussian random variable up to the 3rd order (instead of the 1st of the EKF) for the same computational complexity, this method is more precise of the EKF (i.e. it’s often used for important analysis as the re-entry problem of a vehicle from the space).
In next subsection, the EKF is explained in detail and will be the most used for the experimentation.
3.3 Extended Kalman Filter
The Extended Kalman Filter addresses the problem of the estimation of the state 𝑥 ∈ 𝔎𝑛 of a discrete-time controlled process that is governed by a non-linear stochastic difference equation. This Kalman filter non-linearizes the estimation around the current mean estimate through the partial derivatives of the process and measurement function, computing estimates even if non-linear equations are present.
The structure of the EKF presents some differences in respect to the KF, as shown below.
The non-linear stochastic equations that regulate the system are: 𝑥𝑘 = 𝑓(𝑥𝑘−1, 𝑢𝑘−1, 𝑤𝑘−1)
𝑧𝑘 = ℎ(𝑥𝑘, 𝑣𝑘)
Where: 𝑧 ∈ 𝔎𝑚
Since it is impossible to know the values of the random variables 𝑤𝑘 and 𝑣𝑘 (that are, again, the process and measurement noises respectively), it can be supposed to approximate the state and the measurement vector without them like:
𝑥̃𝑘 = 𝑓(𝑥̂𝑘−1, 𝑢𝑘−1, 0)
𝑧̃𝑘 = ℎ(𝑥̃𝑘, 0)
The set of the equations for EKF is [4]:
19 𝑥̂𝑘− = 𝑓(𝑥̂ 𝑘−1, 𝑢𝑘−1, 0) 𝑃𝑘− = 𝐴 𝑘𝑃𝑘−1𝐴𝑇𝑘+ 𝑊𝑘𝑄𝑘−1𝑊𝑘𝑇
Measurement update equations 𝐾𝑘 = 𝑃𝑘−𝐻𝑇(𝐻𝑘𝑃𝑘−𝐻𝑇𝑘+ 𝑉𝑘𝑅𝑉𝑘𝑇)−1 𝑥̂𝑘 = 𝑥̂𝑘−+ 𝐾 𝑘(𝑧𝑘− ℎ(𝑥̂𝑘−, 0)) 𝑃𝑘 = (𝐼 − 𝐾𝑘𝐻)𝑃𝑘− Where:
o A is the Jacobian matrix of partial derivatives of 𝑓with respect to 𝑥 like: 𝐴[𝑖,𝑗]= 𝜕𝑓[𝑖]
𝜕𝑥[𝑗](𝑥̂𝑘−1, 𝑢𝑘−1, 0)
o W is the Jacobian matrix of partial derivatives of 𝑓 with respect to 𝑤 like: 𝑊[𝑖,𝑗]= 𝜕𝑓[𝑖]
𝜕𝑤[𝑗](𝑥̂𝑘−1, 𝑢𝑘−1, 0)
o H is the Jacobian matrix of partial derivatives of ℎ with respect to 𝑥 like: 𝐻[𝑖,𝑗]= 𝜕𝑓[𝑖]
𝜕𝑥[𝑗](𝑥̃𝑘, 0)
o V is the Jacobian matrix of partial derivatives of ℎ with respect to 𝑣 like: 𝑉[𝑖,𝑗]=
𝜕𝑓[𝑖]
𝜕𝑣[𝑗](𝑥̃𝑘, 0)
The matrices are updating at each time step.
The EKF presents two features: firstly the distributions of the various random variables, after being computed by the algorithm are no longer normal. This is related to the passage from a nonlinear model to a linear one. The second is that 𝐻𝑘, in the formulation, magnify only the relevant component of the measurement (if the process is observable), filtering the non-useful part.
20
Section 4
4.
State of the Art: Sideslip estimation with KF
The state estimation is the base for the future development of the autonomous driving and vehicle control. This is why in literature there are a lot of researches about the states estimation for a vehicle and, in particular, one of the most addressed argument of these document is the sideslip angle estimation, that sometimes is abbreviated as VSA (vehicle sideslip angle). The reason of this interest about this angle is linked to the difficulty of its direct measurement, which can be done only using expensive sensors, unsuitable for commercial and mass use and frequently forbidden by the rules of motorsport.
The literature researches can be divided based on the type of model equations used that may be kinematic or dynamic. Usually kinematical models are implemented firstly because it permits to go around the problem of the estimation of the non-linarites of the systems; secondly this models don't need vehicle parameters (as stiffness of the axles, geometry dimensions, etc.). Some studies introduce, in order to get even more simple model, the small angle hypothesis for calculation of the VSP, that states:
For: 𝛽 ≪ 1 it is atan(𝛽) ≃ 𝛽
This hypothesis introduces another approximation error into the model. Moreover, many of these methods do not provide the best estimation of the sideslip angle.
On the other hand, the dynamic models are more complex but also more complete and precise. The level of the details that is representable in a dynamic model can be chosen based on the proposed goal of the specific study requested. The dynamic models are based on the 2nd law of motion for the translation of the
vehicle along the two axes x,y and for the rotation around z. It is possible to adopt different solution for the vehicle implementation, choosing, for example, between a single track model or a two track model. In a single-track model the system is simplified and can be analysed taking into account only the axles contribution but losing in precision because it is not possible to consider some important car's behaviour aspects like, for example, the weight transferts.
Some studies adopt algorithms to update the parameters of the vehicle as done in [7] where the tire parameter is updated through the use of fuzzy logics. Again, thanks to the dynamic models, it is possible to use different tire model
21 types like linear model, Pacejka model, Rational Tire model, Burckhardt model, Dugoff tire model, etc.
In [8] is addressed the problem of the estimation of the states 𝑢, 𝑣 using the accelerations 𝑎𝑥, 𝑎𝑦as inputs and measurements. The estimation is based on the use of a non-linear state observer with a kinematic model. Usually the kinematic methods are affected by the divergence of the solution during a straight-line travel and this is not an exception. Therefore, the author proposes a solution based on comparison between a mean value of the longitudinal velocity of the four wheels and two thresholds: Ax and -Ax.
The main velocity is defined as:
𝑉𝑚𝑒𝑎𝑛 = 𝑊𝑚𝑒𝑎𝑛(𝑉11, 𝑉12, 𝑉21, 𝑉22) Where: 𝑉11 = 𝑉11𝑚𝑒𝑎𝑠𝑐𝑜𝑠𝛿11− 𝜔𝑧 𝑙 2 𝑉12 = 𝑉12𝑚𝑒𝑎𝑠𝑐𝑜𝑠𝛿12+ 𝜔𝑧 𝑙 2 𝑉21= 𝑉21𝑚𝑒𝑎𝑠− 𝜔𝑧 𝑙 2 𝑉22= 𝑉22𝑚𝑒𝑎𝑠+ 𝜔 𝑧 𝑙 2
and 𝑉𝑖𝑗𝑚𝑒𝑎𝑠are the measured equivalent linear velocities of the wheels.
If the mean value is greater than the Ax threshold, the algorithm considers the
minimum of the four velocities. On the other hand, if the mean value is lower than the –Ax threshold the algorithm considers the maximum value of the four
velocities. If the mean velocity is between the two values –Ax, Ax the algorithm
uses its value for the estimation.
In addition, the method takes into account the gravitational acceleration influence on the measurements due to the roll.
Also in [9] the estimation method is based on a Kalman Filter with a kinematic model of the vehicle. In this case, the available measurements are the tire forces provided by special bearing sensors from SKF. The author proposes also an observability test for the system for a straight-line manoeuvre and demonstrates that the system is unobservable when the yaw-rate assumes zero value. This peculiarity is common to all the kinematical systems. In [10] a non-linear observer is considered with, again, a kinematic vehicle model and further in the
22 document is implemented a linear-tire model for the estimation of the tire forces. In particular, the lateral force can be calculated as:
𝐹𝑦 = 𝐶𝛼(𝐹𝑧)𝛼
Where:
𝐶𝛼(𝐹𝑧) =
𝜕𝐹𝑦
𝜕𝛼 |𝛼=0
which is called cornering stiffness.
In addition, the article provides a method to take into account the problem of the passage between a cornering or a straight travel, using a new parameter into an inequality. Under precise conditions, the inequality decides between them and chooses the right model system to approximate the manoeuvre. At the end of the article, is described a method for the estimation of both inclination and bank angles. In [11] the sideslip estimation is based on the adoption of an Extended Kalman filter with a kinematic model. To take into account the sensor noises that normally contaminate the measurements, that, in this case, are 𝑟, 𝑎𝑥, 𝑎𝑦, the authors propose to model them as follows. Suppose that the measurement values can be divided into a mean and a fluctuant part (the noise) as:
𝑟𝑚 = 𝑟 + 𝑟𝑛, 𝑎𝑥,𝑚 = 𝑎𝑥+ 𝑎𝑥,𝑛, 𝑎𝑦,𝑚 = 𝑎𝑦+ 𝑎𝑦,𝑛
Where 𝑎𝑥,𝑎𝑦 are respectively the longitudinal and lateral accelerations and 𝑚, 𝑛
subscripts respectively denote the measurements and the noises. Therefore, the system can be rewritten as:
𝑥̇ = 𝐴𝑥 + 𝐵𝑢 + 𝐺𝑤
Where 𝐺 and 𝑤 are respectively the input matrix and the process noise vector. In the document, there is an explanation of the calculation method for the matrix covariance Q.
In [12] is addressed the problem to estimate the vehicle side slip angle through an Extended Kalman filter with a dynamdic single track vehicle model.
23 𝑥 = { 𝑟̇ 𝛽 𝐹(𝑦,1) 𝐹(𝑦,2)} Where:
- 𝑟̇ is the derivative of the yaw rate;
- 𝐹(𝑦,𝑗) is the total lateral force acting on each axle. The input and output vectors are:
𝑢 = [𝑉𝑥
𝛿] 𝑧 = [𝑟̇ 𝑎𝑦]
In the paper it is used a set of tuning parameters, inserted to make the estimation work properly, that are modified using a tuning procedure based on a four-parameter optimization in which one parameter is employed to adapt the Pacejka tire model to the actual tires, while the others to adapt the system noise covariance as follows: 𝑄 = 𝜓 [ 𝑞 0 0 0 0 𝑠 0 0 0 0 1 0 0 0 0 1 ]
The tuning procedure provides for a numerical optimization evaluating the best combination of the four parameters in order to minimise the two-norm:
‖[𝑎𝑟̇𝑦] − 𝐻𝑥̂‖
2
In order to obtain the best parameters set, the optimization is done before the real estimation starting from standard high-dynamic manoeuvers. Then the results are implemented into the estimator process.
In [13] is introduced an estimation method for the sideslip angle and for the tire parameters employing an Extended Kalman Filter with a single-track vehicle model as shown below:
24 Figure 7: Scheme of a Single-Track Vehicle Model
Where 𝑙𝑓 and 𝑙𝑟 are used instead of 𝑎1 and 𝑎2.
The model equations are represented by the equilibrium equations of the vehicle along the three axles 𝑥, 𝑦, 𝑧. The inputs considered for the system are the longitudinal acceleration 𝑎𝑥 and the steering angle 𝛿𝑑𝑟𝑖𝑣𝑒𝑟combined with a
kinematical map of the steering system in order to obtain the real left and right steering angles of the front wheels. The input angle considered for the model is the average angle of both 𝛿11, 𝛿12 like:
𝛿 =𝛿11+ 𝛿12
2
The authors effort the estimation using two approaches with two different types of tire model: the first one is the direct estimation of the tire forces while the second is use of the linear tire model to calculate them.
In the second part of the paper, a deep observability analysis and a Pacejka's parameters estimation method are proposed.
In [14] is proposed to estimate the lateral tire force and the VSA through a method derived from EKF and based on the dynamic response of a vehicle instrumented with innovative sensors. The vehicle is modelled as a two-track vehicle like:
25 Figure 8: Scheme of a Two-Track Vehicle Model
The authors, in order to simplify the complexity of the process, choose for the tires the Dugoff model that, assuming pure slip with neglegible longitudinal slip, is formulated, in a simple way, as:
𝐹𝑦𝑖𝑗 ̅̅̅̅̅ = −𝐶𝛼𝑖𝑗tan(𝛼) 𝑓(𝜆), Where: 𝑓(𝜆) = {(2 − 𝜆)𝜆 1 𝑖𝑓𝜆 < 1 𝑖𝑓𝜆 ≥ 1 And: 𝜆 = 𝜇𝑦𝐹𝑧𝑖𝑗 2𝐶𝛼𝑖𝑗|tan𝛼𝑖𝑗|
The present study applies two different estimation methods to reach the aim of the sideslip angle and lateral tire forces estimation.
In the first part of the document is presented a Kalman filter based on a kinematic vehicle model. The method will be developed following two different implementations changing the measurement input typologies. Thanks to the kinematic model, no tire models are required for the lateral force estimation.
26 In the second part an Extended Kalman filter is presented to estimate the sideslip angle and the lateral tire forces. A two-track vehicle is implemented in conjunction with the Pacejka tire model in a reduced form.
In addition, it is presented a new adaptive estimation approach of the Magic Formula's coefficients starting from the results of the EKF estimation and from on-road test data.
All the methods are validated with real vehicle experimental data. The reference vehicle is a Range Rover Evoque prototype, from the European Commission project iCompose, equipped, with inertial and gyroscopic sensors and an optical sensor able to give a direct measurement of the vehicle sideslip angle .
27
Section 6
5.
Validation Data: Equipment and Procedure
The present study has the purpose to develop a new method for the states estimation of a vehicle. The method has to be tested and validated on real vehicle data so, in order to do that, thanks to the collaboration with the European Commission project iCompose, it is possible to exploit the parameters of a special Range Rover Evoque.
In particular, as stated in [15], the abovementioned Range Rover is equipped with a four identical on-board drivetrains consisting of switched reluctance electric motors with double-stage single-speed transmission systems, constant velocity joints and a half-shaft for each. This drivetrain configuration allows to choose the type of drive mode between three typologies: Front wheel drive (FWD), Rear wheel drive (RWD), All wheel drive (AWD).
The vehicle is fitted with different sensors:
Steering wheel sensor : it provides the measurement of the steering wheel input applied by the driver, 𝛿𝑑𝑟𝑖𝑣𝑒𝑟;
Inertial Measurement Unit (IMU): this module, from SBG systems, provides measurements about accelerations and rotational velocities of the vehicle in particular the two accelerations 𝑎𝑥,𝑎𝑦 of the CG and the vehicle yaw rate 𝑟. It is based on MEMS technology and it merges all the sensor information through an internal specially designed Extended Kalman Filter;
Wheel speed sensors: these encoders, from Continental Teves, provide the angular speed measurements of wheels 𝜔𝑖𝑗;
Corrsys Datron sensor: this optical sensor provides the vehicle sideslip angle and the vehicle speed. These quantities are respectively indicated as: 𝛽𝑑𝑎𝑡 and 𝑉𝑑𝑎𝑡 (that in Fig.8 is called 𝑉𝑠𝑝𝑒𝑒𝑑).
The prototype is shown in Fig. 8. where it is possible to individuate the Corrsys Datron sensor positioned in front of the car. It is clear that, because of the positioning, the output data for the sideslip angle has to be corrected and reported to the CG of the vehicle.
28 Figure 9: Range Rover Evoque Prototype
Figure 10: Range Rover Evoque Prototype Scheme
In particular, it is possible to individuate the position of the Datron Sensor, which is installed ahead the front axle of the quantity 𝑎𝑑𝑎𝑡.
29 ---
Symbol Name Unit Value
--- 𝑚 Mass [kg] 2290 𝑎1 Front semi-wheelbase [m] 1.365 𝑙 Wheelbase [m] 2.665 𝜏 Transmission ratio [-] 10.56 𝑅𝑤 Wheel Radius [m] 0.364 𝑤 Track width [m] 1.616 h CG height [m] 0.55
d no roll axis height [m] 0.15
𝐾𝜙 Roll stiffness total [Nm/rad] 190000
𝐾𝜙1 Roll stiffness front [Nm/rad] 102600
𝐾𝜙2 Roll stiffness front [Nm/rad] 87400
𝑎𝑑𝑎𝑡 Datron Distance [m] 0.95
𝜂𝑡𝑞 Motor efficiency [-] 0.95
𝐽𝑧 Moment of intertia [kg/m2] 2761
𝑆𝑥 Frontal Section Area [m2] 2.69
𝜌𝑎𝑖𝑟 Air Density [kg/m3] 1.225
---
Table 1: Prototype Parameters
The correlation between the sideslip angle measured at the sensor with the with the sideslip at the CG is:
tan(𝛽𝐶𝐺) = tan(𝛽𝑑𝑎𝑡) −
𝑟
𝑢(𝑎𝑑𝑎𝑡+ 𝑎1)
The steering angles of the front tires are not equal, but fortunately it is available the law for each tire under the form of non-linear map that uses as input the 𝛿𝑠𝑤of the driver. The diagrams are shown in Fig.10.
30 Figure 11: Steering Law Tire 11 (left) and Tire 12 (right)
The vehicle data are obtained from steering pad manoeuvres executed at different velocities, using all the possible layout combinations of the drivetrain. Summarizing, it is possible to obtain 9 combinations mixing the following cases:
- Velocities: 30km/h, 60km/h, 80km/h;
- Layouts: FWD, RWD, AWD (in all cases with equal distribution for left and right tires).
Each specific data set will be indicated with a specific code containing the velocity and the drivetrain layout configuration names. For example, the code v60rwd identifies the test at 60km/h with the Rear Wheel Drive layout configuration.
The target velocity of the vehicle is reached and maintained during the manoeuvres thanks to a Proportional Integral speed tracking feedback controller, that compares the target velocity with the actual one.
The procedure followed for each test is:
1. The vehicle is accelerated from zero to the target velocity on a straight line through the use of the PI;
2. Once the target velocity is reached, the driver executes a steering ramp applying a constant (almost) rate of 2 deg/s;
3. The test is ended when the vehicle yaw rate reaches its saturation.
The cornering is conducted on the left direction so the tires of the right side of the vehicle, which are the 12 and 22, are considered external.
The outputs of the tests are sampled at a sample time 𝑡𝑠 equal to:
𝑡𝑠 = 0.002𝑠
The comparison between the data tests and the estimation outputs are shown in the next document sections.
31
Section 6
6.
Estimation using a Kinematic Vehicle Model
The present experiment it’s introduced with the purpose to show the logic on which the Kalman is based on and its limits when used with a low number of measurements. In particular, three different systems are presented with different typologies of measurement inputs.
6.1 Vehicle equations
As stated in [1], the vehicle kinematic congruence equations can be deduced from the Fig. 2. Considering the vehicle reference system fixed to the body and centred on its CG and assuming a planar motion of the vehicle, the CG velocity 𝐕g has horizontal direction. For the same reason the rotational velocity 𝛀 has vertical direction.
In particular:
𝐕g = 𝑢𝐢 + 𝑣𝐣
and:
𝛀 = 𝑟𝐤
where u is the longitudinal velocity, v is the lateral velocity and r is the yaw rate of the vehicle.
The speed of every point of a rigid body can be described as follows:
𝐕p = 𝐕g + 𝛀xGP
The kinematics of the body is completely defined by three quantities: u(t), v(t) and r(t).
The CG acceleration can be obtained deriving the velocity 𝐕g as: 𝑎𝑔 =𝑑𝐕𝑔
𝑑𝑡 = 𝑢̇𝐢 + 𝑢𝑟𝐣 + 𝑣̇𝐣 − 𝑣𝑟𝐢 = (𝑢̇ − 𝑣𝑟)𝐢 + (𝑣̇ + 𝑢𝑟)𝐣 = 𝑎𝑥𝐢 +𝑎𝑦𝐣 Therefore, the longitudinal acceleration is defined by (24):
32
𝑎𝑥= 𝑢̇ − 𝑣𝑟
And similarly, the lateral acceleration is by (25):
𝑎𝑦 = 𝑣̇ + 𝑢𝑟
From these equations, the vehicle model for the Kalman Filter estimator is defined.
Thanks to their simplicity, the computational effort is low if compared with more complex formulations that can describe vehicle behaviour.
6.2 Sideslip Estimation: Kinematic model with one Measurement
In this experiment, the systems states to be considered are the longitudinal and lateral velocities 𝑢 and 𝑣.Considering to use the longitudinal velocity as measurement for the KF, the system can be rewritten in state form as in 26 and 27:
[𝑢̇𝑣̇] = [ 0−𝑟 0𝑟] [𝑢𝑣] + [1 00 1] [𝑎𝑎𝑥
𝑦]
𝑦 = [1 0] [𝑢𝑣]
Where:
- The yaw rate is considered as input of the system and it is placed into the dynamic matrix 𝐴 of the model;
- The two accelerations 𝑎𝑥 , 𝑎𝑦 , similarly to the yaw rate, are considered as input for the system. It is possible to recognize the input matrix 𝐵 that multiply the input vector.
The matrices are:
𝐴 = [ 0−𝑟 0𝑟] 𝐵 = [1 00 1] 𝐻 = [1 0]
With H that represent the output matrix.
The measurement of the longitudinal velocity u is derived from the speed sensors (encoders) of the wheels. In order to consider the value of 𝑢 that
33 describes in the best way the real velocity of the vehicle, different ways to consider the 𝑢 measurements have been developed in literature.
In [16] the authors consider the algebraic mean of the four wheels without taking care of the differences between them due to the different positions of each in the car. The solution here adopted is similar to the one applied in [8] and consist in a mean velocity 𝑢𝑚𝑒𝑎𝑛 that considers the contribution of the yaw rate on each tire. In addition, for the front wheels only the component along the x axle of the longitudinal velocity is considered, multiplied for the cosine of the steering angle. This angle is derived from an encoder placed on the steering column and has to be reported to the wheels through the steering ratio. Thus, the new mean longitudinal velocity is calculated like:
𝑢11= 𝑢11𝑚𝑒𝑎𝑠𝑐𝑜𝑠𝛿 − 𝜔 𝑧𝐿/2 𝑢12= 𝑢12𝑚𝑒𝑎𝑠𝑐𝑜𝑠𝛿 + 𝜔 𝑧𝐿/2 𝑢21 = 𝑢21𝑚𝑒𝑎𝑠− 𝜔 𝑧𝐿/2 𝑢22 = 𝑢22𝑚𝑒𝑎𝑠+ 𝜔𝑧𝐿/2 and: 𝑢𝑚𝑒𝑎𝑛 = 𝑢11+ 𝑢12+ 𝑢21+ 𝑢22 4
Where: 𝑢𝑖𝑗𝑚𝑒𝑎𝑠 are the longitudinal velocity measurements; 𝛿 is the arithmetic mean value of the steering angles.
The structure of the Kalman Filter, recalling the equations (7) and (8), is: Time update equations:
𝑥̂𝑘− = 𝐴𝑥̂
𝑘−1− + 𝐵𝑢𝑘−1
𝑃𝑘− = 𝐴𝑃
𝑘−1− 𝐴𝑇+ 𝑄
Measurement update equations: 𝐾𝑘= 𝑃𝑘−𝐻𝑇(𝐻𝑃
𝑘−𝐻𝑇+ 𝑅)−1
𝑥̂𝑘 = 𝑥̂𝑘−+ 𝐾𝑘(𝑧𝑘− 𝐻𝑥̂𝑘−)
34 The elements of the measurement covariance matrix R can be calculated
providing information from sensor manuals. The typology of the information can be different.
For both acceleration and gyroscopic sensors, the information provided is the Bandwidth of the signal. The covariance value can be calculated as [17]:
𝑉𝑟𝑚𝑠= 𝑉𝑁𝑆𝐷√𝐵𝑎𝑛𝑑𝑤𝑖𝑑𝑡ℎ
And:
𝑆𝑢 = 𝜎2 = (𝑉 𝑟𝑚𝑠)2
Where:
- 𝑉𝑟𝑚𝑠 is the Root Mean Square value of the signal; - 𝑉𝑁𝑆𝐷 is the Noise Spectral Density of the signal;
- 𝐵𝑎𝑛𝑑𝑤𝑖𝑑𝑡ℎ is the Spectral of Amplitude of the signal; - 𝑆𝑢 is the variance of a continuous signal.
For the velocity u, the sensor (encoder) manual does not provides information about the error that affects the signal. A practical way is to apply the variance definition:
𝜎2 = 𝐸[(𝑋 − 𝐸[𝑋])2] = ∫(𝑋 − 𝐸[𝑋])2𝑝(𝑥)𝑑𝑥 𝑏
𝑎
A measurement of a constant value can be used for the calculation. The initial part of one of the tests is perfect for this typology of approach because the vehicle velocity is constant and the steering angle is zero.
The results are:
𝜎𝑢2 = 0.0170[(𝑚/𝑠)2]
𝜎𝑟2 = 𝜎
𝑎𝑐𝑐2 = 2.99𝑒 − 4[(𝑚/𝑠)2]
The elements of the process covariance matrix Q can be calculated as suggested in [11]:
𝑄 = 𝐸[𝑤𝑘𝑤𝑘𝑡]
Where 𝑤𝑘 is the noise vector. The element values are (for each):
𝜎𝑄2 = 2.9929𝑒 − 04
35 𝑃𝑘0 = 𝑄;
These values are in line with other studies (i.e. [8]) but have to be adapted to each specific model every time, in order to obtain the best estimations.
6.2.1 System Observability
Similarly to the method proposed in [19], the system observability is guaranteed if the matrix 𝑂 : 𝑂 = ( 𝐶 𝐶𝐴 ⋮ 𝐶𝐴𝑛−1 )
has full Rank.
For the present document the matrix C coincides with H. The calculation can be done as follows:
𝐴 = [ 0 𝑟 −𝑟 0] 𝐻𝐴 = [1 0] [ 0−𝑟 0𝑟] = [−𝑟 𝑟] [ 𝐶𝐶𝐴] = [ 1 0 −𝑟 𝑟] so: ker [ 1−𝑟 𝑟0] = 0𝑖𝑓𝑟 ≠ 0
Therefore, during a straight line the system becomes unobservable. Fortunately, for the experimental tests under examination, this case will never occur.
If a straight line manoeuvre is being carried out, it would be necessary to project an appropriate correction system to take into account the system unobservability issue.
6.2.2 Simulink Model
The model described above is built into the tool Simulink® of Matlab.
It is possible to individuate four main blocks:
36 2. The Kalman Filter: where the logic of the estimation method is developed. It is used for all the vehicle models here shown, with just minor adaptations.;
3. The Measurement input block;
4. The scopes block: inside are designed, specifically for each model, the proper diagrams of the result outputs.
Figure 11 shows the scheme for the Kalman Filter algorithm.
The block receives in input the value of the matrices: A, H, R, Q, the difference between the measurement and the product between matrix H and state vector values 𝑥̂𝑘−. This difference is called “Residual” in control theory.
The principal output of the block is the result of the product between the Gain matrix and the residual.
Inside the system it is possible to set up the initial value of the Covariance matrix 𝑃𝑘0 (the value at the first step). To do that, an initial condition block it’s
used.
Figure 12: Kaman Filter Simulink Model
The scheme of the entire system model for this subsection is shown in Appendix 1.
6.2.3 Sideslip Estimation: Kinematic model (Two Measurements)
The second system built is based on the model of the previous subsection but it is fed with a different measurement vector that presents two elements. The first
37 element is always the mean velocity 𝑢𝑚𝑒𝑎𝑛 as described above. The second is the output value of the direct integration of the lateral acceleration 𝑎𝑦.
So it is:
𝑧2 = [
𝑢𝑚𝑒𝑎𝑛 𝑣𝑖𝑛𝑡 ]
6.2.4 Sideslip Estimation: Kinematic model with a novel Measurement Vector
The third system is based on the Kinematic formulation but, again, is fed with a different measurement vector composed by two elements. The first is the 𝑢𝑚𝑒𝑎𝑛.
For the second element the solution adopted is the use of the Kinematic Beta described in [1]. So this represents the measurement input for the lateral velocity into the KF model. The Kinematic Beta can be seen as the sideslip angle of the vehicle under the hypotheses of steady-state condition and no-sliding tires. Obviously, both hypotheses are not real for the test under examination so the expected results are not good but they can provide some useful observations. Therefore, the new measurement vector is composed as follows:
𝑧3 = [𝑢𝑣𝑤ℎ𝑒𝑒𝑙𝑠
𝛽𝑘𝑖𝑛 ]
The new measurement can be obtained as follows: 𝛽 =𝑣 𝑢 = − 𝑎2 𝑅 𝜌 = 𝑟 𝑢= 1 𝑅 And finally: 𝑣𝛽𝑘𝑖𝑛 = atan(−𝑎2/(𝑢/𝑟))
6.3 Results
The results of the experiments are shown in Fig.13. As expected the results for this model are not good but, beyond the numerical values, the diagrams show that in absence of sufficient measurement information, the KF provides an estimation that might be affected by big errors. For example in the model 1, without any indication about the vehicle lateral behaviour, the EKF integrate
38 the 𝑎𝑦 to obtain the 𝑣. This is demonstrated by the model 2 where the EKF
provides the same solution for 𝑣 following the measurement 𝑣𝑖𝑛𝑡𝑒𝑔𝑟𝑎𝑡𝑒𝑑. On the
other hand, if the KF has a good number of measurements it provides better estimations and their accuracy depends on the typology of the state to be estimated (that is linked to the typology of the system) and of the available measurements. This is demonstrated by the model three where the EKF provides a solution closer to the real one, even if wrong, following the measurement 𝑣𝛽𝑘𝑖𝑛.
39
Section 7
7.
Sideslip Estimation: Dynamic Model
With the aim of improving the sidelsip estimation method, a more detailed model is developed considering a two-track description of the vehicle. The equilibrium equations describe the entire vehicle and provide the option to choise between the formulation for the single or two track vehicle. In both cases it is possible to implement the caractheristic of the tire, obtained previously from other studies. This caractheristic can be modelled in depending on the objectives of the project, choosing between different typologies of tire model: Linear, Pacejka, Rational, Burckhardt, Dugoff, etc. In this document, the attention has been pointed on the two-track vehicle with, respectively, a linear tire model and a Pacejka tire model.
7.1 Dynamic Two-Track model: Linear tire model
The two-track model dynamic model is shown in Fig. 7 and here reported again:
Figure 14: Two-Track model
This model allows to take into account the tire characteristics and the load transfers of the vehicle. Both these features cannot be modelled using a single-track model, which considers the tires of each axle collapsed in the middle of the structure. In addition, the two-track vehicle model allows to consider the
left-40 right tire steering angles different. The two angles are linked together by a function that includes the kinematic characteristics of the steering system. This function is provided with the diagrams of Fig.10.
The states and the measurements for this model are listed below:
𝑋 = { 𝑢 𝑣 𝑟} , 𝑋̇ = { 𝑢̇ 𝑣̇ 𝑟̇ }, 𝑧 = {𝑢𝑟𝑚𝑒𝑎𝑛 𝑚𝑒𝑎𝑠}
Through the use of the dynamic model, more aspects of the vehicle can be modelled as, for example, the tire forces exerted between the road and the tires and the load transfers due to the roll and the pitch. Also the aerodynamic drag force (longitudinal component) can be modelled following the expression (35):
𝐹𝐷 =
1
2𝜌𝑎𝐶𝑥𝑆𝑥𝑢2
Where:
- 𝜌𝑎 is the air density; - 𝐶𝑥 is the shape coefficient; - 𝑆𝑥 is the frontal section area;
- 𝑢 is the longitudinal velocity component of the car.
The aerodynamic force components along y and z are considered negligible. For the specific model described in this subsection, the tire model used is linear. This is based on the equation:
𝐹𝑦 = 𝐶𝛼𝛼
where the value of cornering stiffness 𝐶𝛼 is taken from manuals.
Considering the diagram of a generic tire characteristic depending on the vertical force and the tire slip angle, like in [20] and shown in Fig. 15:
41 Figure 15: Tire Characteristic vs. Fz and Alpha
From the diagram observation is possible to recognise the features that characterize the linear model. In particular it this tire model is ideal for first approximation test because of the following reasons:
It represents well the characteristic of the tire in the linear zone (near the ordinate axle);
Unlike other models, as the most famous Pacejka's one, this formulation does not depend on the vertical load on the tire;
Thanks to its simple formulation, it is easy to implement and computationally light.
The cornering stiffness is chosen in such a way to either give a good representation of prototype tire data here analysed. The value is:
𝐶𝛼= 70000𝑁/𝑟𝑎𝑑
The two-track vehicle model is based on the dynamic equilibrium equations expressed in (37) and (38) as:
𝑚𝑎𝑥= 𝑋11cos 𝛿11+ 𝑋12cos 𝛿12+ 𝑋21+ 𝑋22− 𝑌11𝑠𝑖𝑛 𝛿11− 𝑌12𝑠𝑖𝑛𝛿12− 𝐹𝐷
𝑚𝑎𝑦 = 𝑌11cos 𝛿11+ 𝑌12cos 𝛿12+ 𝑌21+ 𝑌22+ 𝑋11sin 𝛿11+ 𝑋12sin 𝛿12
𝐽𝑧𝑟̇ = (𝑌11cos 𝛿11+ 𝑌12cos 𝛿12+ 𝑋11sin 𝛿11+ 𝑋12sin 𝛿12)𝑎1− (𝑌21+ 𝑌22)𝑎2+
+(𝑋12cos 𝛿12− 𝑋11cos 𝛿11) 𝑡1 2 + (𝑋22− 𝑋21) 𝑡2 2 + (𝑌11sin 𝛿11− 𝑌12sin 𝛿12) 𝑡1 2
42 and, from (24) and (25), it is:
𝑢̇ = 1
𝑚(𝑋11cos 𝛿11+ 𝑋12cos 𝛿12+ 𝑋21+ 𝑋22− 𝑌11𝑠𝑖𝑛 𝛿11− 𝑌12𝑠𝑖𝑛𝛿12− 𝐹𝐷) + 𝑣𝑟 𝑣̇ = 1
𝑚(𝑌11cos 𝛿11+ 𝑌12cos 𝛿12+ 𝑌21+ 𝑌22+ 𝑋11sin 𝛿11+ 𝑋12sin 𝛿12) − 𝑢𝑟 𝑟̇ = 1
𝐽𝑧((𝑌11cos 𝛿11+ 𝑌12cos 𝛿12+ 𝑋11sin 𝛿11+ 𝑋12sin 𝛿12)𝑎1− (𝑌21+ 𝑌22)𝑎2+ +(𝑋12cos 𝛿12− 𝑋11cos 𝛿11) 𝑡1 2 + (𝑋22− 𝑋21) 𝑡2 2 + (𝑌11sin 𝛿11− 𝑌12sin 𝛿12) 𝑡1 2) Where:
- 𝑋𝑖𝑗 are the longitudinal forces exchanged between the road surface and the tires;
- 𝑌𝑖𝑗 are the lateral forces exchanged between the road surface and the tires;
- 𝛿1𝑗 are the steering angles for each tire;
- 𝐹𝐷 is the aerodynamic force due to the drag;
- 𝑎1,𝑎2 are respectively the front and rear semi-wheelbases;
- 𝑡1,𝑡2 are respectively the front and rear tracks; - 𝑚 is the mass of the vehicle
- 𝐽𝑧 is the inertial moment of the vehicle at the CG point.
The lateral forces depend on tire slip angles as stated in the above formulation, so at each step they have to be updated with the new estimated values of 𝛼𝑖𝑗(𝑘).
43 𝛼11 = 𝛿11− 𝑎𝑡𝑎𝑛( 𝑣 + 𝑟𝑎1 𝑢 −𝑟𝑡1 2 ) 𝛼12 = 𝛿12− 𝑎𝑡𝑎𝑛 (𝑣 + 𝑟𝑎1 𝑢 +𝑟𝑡1 2 ) 𝛼21= 𝛿21− 𝑎𝑡𝑎𝑛 ( 𝑣 − 𝑟𝑎2 𝑢 −𝑟𝑡2 2 ) 𝛼22= 𝛿22− 𝑎𝑡𝑎𝑛 (𝑣 − 𝑟𝑎2 𝑢 +𝑟𝑡2 2 )
It is clear that each 𝛼𝑖𝑗(𝑘) depending on the estimation a posteriori of the longitudinal and lateral velocities at the previous step k-1.
The Kalman Filter estimator, due to the presence of the non-linearity of the aerodynamic force, it is not able to estimate the system states, so it is necessary to introduce the EKF. Recalling the equations (11) and (12), it is:
Time update equations 𝑥̂𝑘− = 𝑓(𝑥̂
𝑘−1, 𝑢𝑘−1, 0)
𝑃𝑘− = 𝐴
𝑘𝑃𝑘−1𝐴𝑇𝑘+ 𝑊𝑘𝑄𝑘−1𝑊𝑘𝑇
Measurement update equations 𝐾𝑘 = 𝑃𝑘−𝐻𝑇(𝐻
𝑘𝑃𝑘−𝐻𝑇𝑘+ 𝑉𝑘𝑅𝑉𝑘𝑇)−1
𝑥̂𝑘 = 𝑥̂𝑘−+ 𝐾
𝑘(𝑧𝑘− ℎ(𝑥̂𝑘−, 0))
𝑃𝑘 = (𝐼 − 𝐾𝑘𝐻)𝑃𝑘−
Following the definition stated in [4], because the computer works only in Discrete-Time, it is necessary to discretize and, after that, linearize the vehicle model equations. This linearization is useful to compute definition of the 𝑃𝑘 where, at each step, it is required the product between the a posteriori estimate state covariance matrix of the previous step 𝑃𝑘−1 and the dynamic matrix 𝐴𝑘 (updated every time with the new input for the step k).
44 1) Discretization
2) Linearization
The discretization method used for the dynamic system, unlike the one carried out for the Kinematic model, is necessary explicit, because of the structure of the equations that are too complicated for an implicit (is impossible to explicit the states in an easy way).
The procedure adopted for the discretization is the forward differences method, which states:
𝑓̇(𝑘) ≈𝑓(𝑘 + 1) − 𝑓(𝑘)
𝑇
so:
𝑓(𝑘 + 1) ≈ 𝑓(𝑘) + 𝑓̇(𝑘)𝑇
Where 𝑇 is equal to the sample time 𝑇𝑠 of the data logger. Therefore, rewriting the system equations, it is:
𝑢(𝑘 + 1) = 𝑢(𝑘) + 𝑢̇(𝑘)𝑇
𝑣(𝑘 + 1) = 𝑣(𝑘) + 𝑣̇(𝑘)𝑇
𝑟(𝑘 + 1) = 𝑟(𝑘) + 𝑟̇(𝑘)𝑇
The Jacobian Matrices of the model can be calculated as described for the parameters 𝐴[𝑖,𝑗] in (12).
The Simulink system model is shown in Appendix 2.
7.1.1 Observability of the system
For nonlinear systems, a local criterion concerns is the rank condition of the observability matrix defined as the Jacobian of the observability function computed from the Lie derivative [21]. A necessary condition for the observability is the existence of a casual chain between the [inputs, outputs] and the state variables to be observed. This type of chain is shown in [22] and is here reported:
45 This condition is valid also for the next dynamic models, because for all of them the observability concerns only the three states 𝑢,𝑣,𝑟.
7.1.2 Results
For the following analysis, the v60rwd test is taken as reference; the covariance matrices 𝑅,𝑄,𝑃𝑘0,𝑋0 for this particular simulation are:
R = [10*0.0172 0; 0 0.0172];
Q = [2.99e-2 0 0;0 2.99e-2 0 ;0 0 3.0462e-1]; Pk0 =[2.99e-4 0 0;0 2.99e-2 0 ;0 0 3.0462e-1];
x_0 = [16.667;0;0]
The result diagrams presented below for the longitudinal velocity, lateral velocity, sideslip angle and covariance values are:
46 Figure 16: Linear Tire Model - Results: u (top left), v (top right), sideslip angle (centre) The values labelled with "Kalman" are referred to the estimation; values with "Datron" to the sensor measurements; value with "mean" is referred to the mean value of the longitudinal velocity as presented in (29) and (30). The final flat part of diagrams is related to an overdue time simulation necessary to confirm the right behaviour of the algorithm.
If the diagram for 𝑢 is considered, the graph of the Kalman estimation is not visible. This is caused by the overlapping of this curve with 𝑢𝑚𝑒𝑎𝑛(that is the measurement) as visible in the zoom of Fig. 17:
Figure 17: Velocity u - Zoom
Looking at the estimation results for 𝑣, it is possible to recognize the initial adaptation phase of the estimation process that tries to find the best solution. In order to do that, it associates (they results from the algorithm calculations) big values of the a posteriori covariance elements until it find the right values. This phase converge in few seconds.
The estimation achieves good results up to the time of almost 45 s, after that, also if there are associated good covariance values (the algorithm doesn't know
47 which is the right value but it tries to do its better with the available information), the estimation moves away from the Datron measurements. The reason of this difference is that the linear tire model does not provide the tire saturation of the saturation of the tire forces, so they have the capability to rise with the rising of the alpha angle with a linear relationship.
This relationship is plotted using the outputs of the EKF as showed in Fig. 18:
Figure 18: Tire Characteristic - Results
The solution to this problem is presented in the next subsection where a different tire model is implemented.
7.2 Dynamic Two Track: Pacejka tire model
In order to solve the problems of the precedent system, the new vehicle model is represented by a set of equations that includes a Pacejka tire model for a correct representation of the lateral tire forces.
The formulation for this particular equation can be found direclty in its author's book [23]:
𝑦(𝑥) = 𝐷𝑠𝑖𝑛{𝐶𝑎𝑡𝑎𝑛[𝐵𝑥 − 𝐸(𝐵𝑥 − atan(𝐵𝑥))]}
Where:
- 𝑦(𝑥) represents the lateral force 𝑌𝑖𝑗 [N]
48 - 𝐵 is the stiffness factor
- 𝐶 is the shape factor
- 𝐷 = (𝑎1𝑀𝑎𝑔𝑖𝑐𝐹𝑧+ 𝑎2𝑀𝑎𝑔𝑖𝑐)𝐹𝑧 is the peak value
- 𝐸 is the curvature factor
The coefficients B,C,D,E are provided from a precedent study conducted on the same vehicle. The values of the coefficients are:
- B = 9.81376; - C = 1.57817;
- 𝑎1𝑀𝑎𝑔𝑖𝑐 = -0.0000197392;
- 𝑎2𝑀𝑎𝑔𝑖𝑐 = 1.09568; - 𝐸 = 0
Due to the new formulation, the alpha angles have to be calculated at each step. This is possible using the formulation (43) where the values of 𝑢 and 𝑣 are achievable through the estimate states from the output of the EKF.
The vertical loads for each tire are necessary and in order to calculate them, the following equations are described.
The static longitudinal load transferts are: 𝑍10 = 𝑚𝑔𝑎2
𝑙 𝑍2
0 = 𝑚𝑔𝑎1
𝑙
The load transfert due to the longitudinal acceleration can be calculated as:
𝛥𝑧 = −𝑚𝑎𝑥ℎ 𝑙
where the contribution due to the inertial term is considered neglectable. Moreover the lateral load transfert can be calculated as follows:
𝛥𝐹𝑧1 = 𝑚 1 𝑡1( 𝑎2 𝑙 𝑑1+ 𝐾𝜙1(ℎ − 𝑑) 𝐾𝜙 ) 𝑎𝑦 𝛥𝐹𝑧2 = 𝑚 1 𝑡2( 𝑎1 𝑙 𝑑2+ 𝐾𝜙2(ℎ − 𝑑) 𝐾𝜙 ) 𝑎𝑦
Finally,considering that 𝑑1 = 𝑑2 = 𝑑, the vertical loads for each tire are:
𝐹𝑧11= 0.5(𝑍10+ 𝛥𝑧) − 𝛥𝐹𝑧1
𝐹𝑧12= 0.5(𝑍10+ 𝛥𝑧) + 𝛥𝐹 𝑧1
49 𝐹𝑧21= 0.5(𝑍20− 𝛥𝑧) − 𝛥𝐹
𝑧2
𝐹𝑧22= 0.5(𝑍20− 𝛥𝑧) + 𝛥𝐹 𝑧2
From the formulation shown above, it is possible to state that the vertical loads are not depending on the estimate states because they contain only the measurement values.
The two track model provides the possibility to take into account the toe angles of the tires. For the prototype examinated, the values are:
Toe 11 = 0.07 rad Toe 12 = 0.14 rad Toe 21 = 0.09 rad Toe 22 = 0.04 rad
Unfortunately, not all the preototype parameters are known like, i.e., the static camber angle, are not known.
At this point, the equations for the model can be implemented into Simulink blocks. The block equations are reported into the Appendix 2 in their entire formulation for representative purpose. For clarity, the Jacobian matrix is not reported (too big formulation) but it is obtained from the partial derivative of the equations listed.
Again, for this model the states and the measurements are listed below:
𝑋 = { 𝑢 𝑣 𝑟} , 𝑋̇ = { 𝑢̇ 𝑣̇ 𝑟̇ }, 𝑧 = {𝑢𝑟𝑚𝑒𝑎𝑛 𝑚𝑒𝑎𝑠}
Again, with representative purpose, the scheme for this new model is shown in Appendix 1.
7.2.1 System Observability
The system observability is already discussed in paragraph 7.1.1.
7.2.2 Results
For the following analysis, the v60rwd test is taken as reference; the covariance matrices 𝑅,𝑄,𝑃𝑘0,𝑋0 of the simulations are:
50 R = [0.0172 0; 0 0.0172];
Q = [2.99e-3 0 0;0 2.99e-3 0 ;0 0 3.0462e-1];
P_k_0 = [2.99e-5 0 0;0 2.99e-5 0 ;0 0 0.985*3.0462]; x_0 = [16.667;0;0]
It is possible to note that, in respect to the model of the subsection 7.1, the covariance values are changed, in order to be adapted to the new model with the goal to achieve a better estimation. This suggests a future development of the method, which may be improved with covariance matrices that adapt their values at each step, in order to increase the estimation precision.
The results are plotted in Fig. 19. In particular the diagrams report the estimations for 𝑢,𝑣,𝛽:
Figure 19: Pacejka tire model - Results: u (top left), v (top right), sideslip angle (centre) The graph of the estimated velocity 𝑢 is not visible because overlapped with the input graph 𝑢𝑚𝑒𝑎𝑛.
Talking about the results for 𝑣, it is possible to recognize the adaptation phase of the process that in the first part it is possible by the spikes. The results for this
