Development of a Vehicle Stability Detection Signal

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Development of a Vehicle Stability Detection Signal

Francesco Siciliani

Automotive Engineering, bachelor's level 2019

Luleå University of Technology

Department of Engineering Sciences and Mathematics

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Acknowledgment

I would like to express my sincere gratitude towards Mr. Fred Schmidt for providing me with the opportunity to write my thesis at ZF Active Safety GmbH (ZF Group). I would like to express my deep appreciation for all the support and valuable

knowledge provided by my colleagues within the Customer Applications and Systems Engineering (CA&SE) team, especially Frank Reindl, for supervising me and always being willing to help me to improve.

Many thanks to my thesis supervisor Kim Berglund for his assistance. Also, I would like to thank my professor Jan Van Deventer and Mr. Bodo Barth, for making my career at ZF possible.

Finally, I would like to thank my wife Elin, for her incredible patience and encouragement I got from her throughout my studies.

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Abstract

It is possible to obtain information about the stability conditions of a vehicle by observing and comparing existing signals involved in the rotational motion of the vehicle around the vertical axis.

Accurate information about the current state of a vehicle is critical for the development and function of new active safety features in a vehicle.

Therefore, the goal of this thesis is to create a new signal based on already existing signals from the vehicle electronic control unit for detecting understeering and oversteering of a vehicle.

The signal should consider all the factors that affect the evaluation of the vehicle´s stability conditions.

The results show that the developed signal can, in certain conditions, detect understeering and oversteering. Issues arise in situations such as banked curves or low-mu surfaces. In those cases, the signal is not fully able to describe the vehicle behavior.

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List of Symbols

Symbol Description (Unit) 𝒂 vehicle acceleration vector (m/s²)

𝑎_𝑐 lateral acceleration (m/s²)

𝑎𝑐−𝐸𝑠𝑡 estimated lateral acceleration (m/s²) 𝑎_𝑥 acceleration vector x-component (m/s²)

𝑎_𝑦 acceleration vector y-component (m/s²) 𝑎𝑧 acceleration vector z-component (m/s²) B point B

BbDcrsFact banked bend decreasing factor B𝑥𝑦𝑧 non-inertial frame of reference

𝒃_𝒙 unit vector (non-inertial frame of reference, x-coordinate) 𝒃_𝒚 unit vector (non-inertial frame of reference, y-coordinate) 𝒃_𝒛 unit vector (non-inertial frame of reference, z-coordinate) 𝐶_𝑓 front wheel cornering stiffness (N/rad)

𝐶_𝑟 rear wheel cornering stiffness (N/rad)

𝒆_X unit vector (fixed frame of reference, X-coordinate) 𝒆_Y unit vector (fixed frame of reference, Y-coordinate) 𝒆_Z unit vector (fixed frame of reference, Z-coordinate) 𝐹_𝑓 side force front wheel (N)

𝐹_𝑟 side force rear wheel (N) 𝑰 identity matrix

intEstBankAngle integral of banking angle 𝑖_𝑠 steering ratio

𝐽_𝑧 moment of inertia (kg · m²) 𝐾 DCDS proportional parameter 𝐾_𝑑 DCDS derivative parameter

𝑙 vehicle´s length from Front to rear wheel´s center (m) 𝑙_𝑓 vehicle length from front wheel to vehicle´s mass center (m) 𝑙_𝑟 vehicle length from rear wheel to vehicle´s mass center (m) O origo

O_XYZ fixed frame of reference P point P

𝑹 rotation matrix 𝑹̇ rate of change of rotation matrix

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7 𝑅 curve radius (m)

𝑅_𝐵 distance between point O and point B (m) 𝑅_𝑓 curve radius front wheel (m)

𝑅_𝑃 distance between point O and point P (m) 𝑅_𝑟 curve radius rear wheel (m)

𝑹^𝑻 transpose of rotation matrix

𝑹^𝑻̇ rate of change of rotation matrix transpose 𝑡 time (s)

𝑽 vehicle velocity vector (m/s) 𝑽̇ derivative of velocity vector (m/s²)

𝑉 vehicle velocity (m/s)

|𝑉| vehicle speed (m/s)

𝑣_𝑐ℎ characteristic speed (m/s) 𝑉_𝑓 front wheel velocity (m/s) 𝑉𝑟 rear wheel velocity (m/s)

𝑉_X velocity vector X-component (m/s)

𝑉_Ẋ rate of change of velocity vector X-component (m/s) 𝑉_𝑥 velocity vector x-component (m/s)

𝑉_𝑥̇ rate of change of velocity vector x-component (m/s) 𝑉_Y velocity vector Y-component (m/s)

𝑉_Ẏ rate of change of velocity vector Y-component (m/s) 𝑉_𝑦 velocity vector y-component (m/s)

𝑉_𝑦̇ rate of change of velocity vector y-component (m/s) 𝑉_Z velocity vector Z-component (m/s)

𝑉_Ż rate of change of velocity vector Z-component (m/s) 𝑉_𝑧 velocity vector z-component (m/s)

𝑉_𝑧̇ rate of change of velocity vector z-component (m/s) 𝑋_𝑛 banked-bend lookup table input

YscMetric Ysc-Metric (signal that is currently used by ZF) 𝑌_𝑛 banked-bend lookup table output

𝛼_𝑟 rear-wheel-slip angle (deg) 𝛼_𝑓 front-wheel-slip angle (deg) 𝛽 side-slip angle (deg)

𝛽̇ rate of change of side-slip angle (deg/s)

𝛽₀ side-slip angle (motion without wheel-slip angle) (deg)

∆𝑡 sampling time (ms)

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𝛿_𝐴 Ackermann angle (deg)

𝛿_𝑓 𝛿_𝑠

wheel angle (deg) steering angle (deg)

𝜓̇ yaw-rate (deg/s)

𝜓̈ rate of change of yaw-rate (deg/s²)

𝛹̇_𝑎𝑐 target yaw-rate estimated from the lateral acceleration (deg/s) 𝛹̇𝑒 yaw-rate error (deg/s)

𝛹̇_{𝑒(𝑎𝑐𝑡𝑢𝑎𝑙)} yaw-rate error from the actual iteration (deg/s)

𝛹̇_{𝑒(𝑝𝑎𝑠𝑡)} yaw-rate error from the previous iteration (deg/s) 𝛹̇_𝑒𝑜𝑠 oversteer error (deg/s)

𝛹̇_𝑒𝑢𝑠 understeer error (deg/s) 𝛹̇_𝑚 measured yaw-rate (deg/s) 𝛹̇_𝑟𝑒𝑓 yaw-rate reference (deg/s)

𝛹̇_𝑡 target yaw-rate estimated from the steering wheel angle (deg/s) 𝛚 angular velocity vector (deg/s)

𝐵𝑑𝑉

𝑑𝑡 rate of change of velocity vector in relation to fixed frame of reference (m/s²)

∑ 𝐹_𝑌 sum of forces acting along y-axis (N)

∑ 𝜏_𝑍 sum of torques acting around z-axis (N)

Abbreviations

ABS anti-lock braking system

DCDS driving condition detection signal

ECU electronic control unit

MC mass centrum TC traction control YSC yaw stability control

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1 Introduction

During vehicle testing, it is of major importance to be able to monitor the stability of the vehicle at any given time. Having accurate data to rely on is crucial when it comes to evaluating the quality and the functionality of a dynamic safety system.

The signal that is currently used by ZF, denoted Ysc-Metric, is based on existing raw-signals from the vehicle´s own electronic control unit (ECU), and its task is to accurately detect oversteering and understeering. In case of oversteering or

understeering, the value provided by the signal will take on a positive, respectively a negative value.

The problem is that these values are often misleading: when a departure occurs, the signal would remain in the stable range or vice versa. This could be truly problematic if an accurate analysis is to be made. Moreover, during dynamic maneuvers at high speeds, the signal can´t handle rapid changes from oversteering to understeering.

Additionally, issues arise when driving in banked curves.

A reliable stability estimation tool is fundamental for the development and

troubleshooting of many safety systems such as antilock braking system (ABS) and traction control system (TC).

The goal with this project is to build a more reliable measurement signal that can detect understeering and oversteering.

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2 Background: The Physics Behind It

The logic employed in the calculations is based in part upon the data collected from the vehicle´s measurement equipment, and in part on estimations from the same data.

It is beneficial to estimate the stability of the vehicle, using existing systems, as much as possible. It would have been much easier to have a sensor for each signal, but the costs and the complexity of the systems would increase very rapidly.

The new signal will make large use of the variables involved in the vehicle´s rotational motion, (i.e. the vehicle´s angular velocity around the vertical axis).

Figure 2.1. Vehicle interaction with surroundings and the driver.

The variables that are to be used are the target yaw-rate estimated from the steering wheel angle 𝛹̇_𝑡, the target yaw-rate estimated from the lateral acceleration 𝛹̇_𝑎𝑐 and the measured yaw-rate 𝛹̇_𝑚. These three signals give us information about the actual state of the vehicle (i.e. how the vehicle is actually rotating), the driver´s intentions and information from the surface, (i.e. the friction forces that the surface can sustain).

Before delving into the solution´s algorithms, it would be beneficial to have a look at some theoretical aspects behind the vehicle´s behavior in a curve. The following section covers the basics of the physics behind the yaw-rate estimations adopted throughout the project. Also, we will briefly describe how Ysc-Metric works and how it is defined. For more information on how to derive the yaw-rate estimations, see [1].

Measured Yaw-Rate Vehicle

Target Yaw-Rate Estimated from Steering Wheel

Driver

Target Yaw-Rate Estimated from the Lateral Acceleration

Environment

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2.1 Target Yaw-Rate Estimated from the Steering-Wheel Angle

Let´s look more closely at how the target yaw-rate estimated from the steering-wheel angle is defined. When driving in a curve, the angle between the wheel´s direction of travel and the direction toward the point onto which the wheel points is defined as the wheel-slip angle. In particular the front-wheel-slip angle, denoted 𝛼_𝑓, and the rear- wheel-slip angle, denoted 𝛼_𝑟. (When we write “wheel-slip angle” we refer to both front and rear wheel-slip angles).

Hereafter we are going to provide a geometrical illustration for two cases, that is motion without the presence of wheel-slip angle and motion with the presence of wheel-slip angle: in the former case scenario 𝛼_𝑓 and 𝛼_𝑟 are equal to zero, whereas in the latter 𝛼_𝑓 and 𝛼_𝑟 are greater than zero.

The model used in the illustrations is the bicycle model.

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12 2.1.1 Vehicle Motion without Wheel-Slip Angle

Figure 2.2. Vehicle plane motion without wheel-slip angle

The vehicle can be viewed as a rigid body, given that it rolls without wheel-slip angle.

𝑡𝑎𝑛𝛿_𝑓 = 𝑙

𝑅_𝑟 = 𝑙

√𝑅²− 𝑙_𝑟²

(2.1)

With the assumption that 𝛿_𝑓 is very small and R>>l, the wheel angle becomes 𝛿_𝑓= 𝑙

𝑅 = 𝛿_𝐴 (2.2)

where 𝛿_𝐴 is defined as the Ackermann-angle. The side-slip angle (motion without wheel slip angle) can be obtained in the same fashion:

𝑡𝑎𝑛𝛽₀ = 𝑙_𝑟

𝑅_𝑟 𝛽₀ = 𝑙_𝑟

𝑅_𝑟 (2.3)

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13 2.1.2 Vehicle Motion with Wheel-Slip Angle

Figure 2.3. Vehicle plane motion without wheel-slip angle

Using the law of sine, 𝑅_𝑓

𝑠𝑖𝑛(90° − 𝛼_𝑟)= 𝑙

sin (𝛿_𝑓− (𝛼_𝑓− 𝛼_𝑟)) (2.4)

The Ackermann-angle and the side-slip angle become

𝛿_𝑓= 𝑙

𝑅+ (𝛼_𝑓− 𝛼_𝑟) ; 𝛽 = 𝑙_𝑟

𝑅 − 𝛼_𝑟 (2.5)

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For both cases, motion with and without wheel-slip angle, we can express the

relationship between the vehicle velocity and the angular velocity around the vertical axis through

𝑉 = 𝑅𝜓̇ 𝑉_𝑓 = 𝑅_𝑓𝜓̇ (2.6) 𝑉_𝑟 = 𝑅_𝑟𝜓̇

where 𝜓̇ is defined as the yaw-rate, the angular velocity at which the vehicle rotates around the vertical axes. Also, from equation 2.6 it is possible to obtain the vehicle acceleration vector. Since the vehicle is represented as a body moving in a non- inertial system of motion, a non-inertial frame of reference must be adopted.

The vehicle acceleration vector is defined as

𝒂 = 𝑽̇ + 𝝎 × 𝑽 (2.7) Where

𝝎 = ( 0 0

𝜓̇

) (2.8)

The aim with the following section is to derive eq. 2.7.

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15 2.1.3 Vehicle Relative Motion

Figure 2.4. Vehicle relative motion.

A vehicle can be viewed as a body undergoing rotational and translation motion. As depicted in figure 2.4 the vehicle is represented as the point B moving toward point P.

The fixed frame of reference O_XYZ, with coordinates X, Y and Z, is at rest. 𝒆_X,𝒆_Yand 𝒆_Z are the unit vectors with length 1.

The non-inertial frame of reference B_𝑥𝑦𝑧 with coordinates 𝑥, 𝑦 and 𝑧 on the other hand, is rotating. It origins at point B and vectors 𝒃_𝒙, 𝒃_𝒚 and 𝒃_𝒛 are the unit vectors. The point B is fixed on the vehicle. The system B_𝑥𝑦𝑧 changes its direction. To describe the relation between the two systems it is appropriate to use a rotation matrix. So, the relation can be described as

𝒃_𝒙(𝑡) = 𝑹 (𝑡)𝒆_X , 𝒃_𝒚(𝑡) = 𝑹 (𝑡)𝒆_Y , 𝒃_𝒛(𝑡) = 𝑹 (𝑡)𝒆_Z (2.9)

Since the rotational matrices are square matrix they can be represented as orthogonal matrices

𝑹^𝑻 (𝑡)𝑹 (𝑡) = 𝑹(𝑡)𝑹^𝑻 (𝑡) = 𝑰 (2.10)

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where I is the identity matrix. Equation 2.10 leads to

𝒆_X = 𝑹^𝑻(𝑡 )𝒃_𝒙(𝑡) , 𝒆_Y = 𝑹^𝑻(𝑡)𝒃_𝒚(𝑡) , 𝒆_Z = 𝑹^𝑻(𝑡)𝒃_𝒛(𝑡) (2.11)

The velocity vector can be defined as

𝑽(𝑡) = 𝑉_X(𝑡)𝒃_𝒙(𝑡) + 𝑉_Y(𝑡)𝒃_𝒚(𝑡) + 𝑉_Z(𝑡)𝒃_𝒛(𝑡) (2.12)

Its derivative is defined as

𝑽̇(𝑡) = 𝑉_Ẋ (𝑡)𝒃_𝒙(𝑡) + 𝑉_Ẏ (𝑡)𝒃_𝒚(𝑡) + 𝑉_Ż (𝑡)𝒃_𝒛(𝑡) +

𝑉_X(𝑡)𝒃_𝒙̇ (𝑡) + 𝑉_Y(𝑡)𝒃_𝒚̇ (𝑡) + 𝑉_Z(𝑡)𝒃_𝒛̇ (𝑡) (2.13)

For an observer in the moving system the derivative of the velocity vector would be

𝐵𝑑𝑉

𝑑𝑡 = 𝑉_Ẋ (𝑡)𝒃_𝒙(𝑡) + 𝑉_Ẏ (𝑡)𝒃_𝒚(𝑡) + 𝑉_Ż (𝑡)𝒃_𝒛(𝑡) (2.14)

which it is defined as the rate of change of the velocity vector in relation to B_𝑥𝑦𝑧.

The remaining part of equation 2.13 is

𝒃_𝒙̇ (𝑡) = 𝑹̇(𝑡)𝒆_X = 𝑹̇(𝑡)𝑹^𝑻(𝑡 )𝒃_𝒙(𝑡) 𝒃_𝒚̇ (𝑡) = 𝑹̇(𝑡)𝒆_Y = 𝑹̇(𝑡)𝑹^𝑻(𝑡 )𝒃_𝒚(𝑡) (2.15)

𝒃_𝒛̇ (𝑡) = 𝑹̇(𝑡)𝒆_Z= 𝑹̇(𝑡)𝑹^𝑻(𝑡 )𝒃_𝒛(𝑡)

From eq. 2.15 we get

𝑽̇(𝑡) = ^𝐵𝑑𝑉

𝑑𝑡 + 𝑹̇𝑹^𝑻𝑽(𝑡) (2.16)

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17 From equation

𝑹^𝑻(𝑡)𝑹 (𝑡) = 𝑹(𝑡)𝑹^𝑻 (𝑡) = 𝑰

𝑑

𝑑𝑡(𝑹𝑹^𝑻) = 𝑹̇𝑹^𝑻+ 𝑹𝑹^𝑻̇ = 𝟎 → 𝑹̇𝑹^𝑻 = −𝑹𝑹^𝑻̇ (2.17)

Equation 2.17 shows that 𝑹̇𝑹^𝑻 is antisymmetric. Hence every vector 𝑽 can be defined as the cross product

𝑹̇𝑹^𝑻𝑽 = 𝝎 × 𝑽 (2.18)

where 𝝎 b is the angular velocity at which the car rotates. So, the acceleration vector can be expressed as

𝒂 = ^𝐵𝑑𝑉

𝑑𝑡 + 𝝎 × 𝑽

or

𝒂 = 𝑽̇ + 𝝎 × 𝑽 (2.19)

For more information about vehicle relative motion, see [2].

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2.1.4 Definition of the Target Yaw-Rate Estimated from the Steering Wheel Angle

The following figure depicts the effects of the side forces acting on the wheels. The motion around the y and x-axis, respectively pitch and roll are ignored. The vehicle is treated as a rigid body and 𝑉 , 𝛽, 𝜓̇ are assumed to be constant.

Figure 2.5. The effects of side forces impacting a vehicle during circular motion.

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The acceleration-vector components from equation 2.19 are

𝑎_𝑥= 𝑉_𝑥̇ − 𝑉_𝑦𝜓̇ 𝑎_𝑦 = 𝑉_𝑦̇ − 𝑉_𝑥𝜓̇ (2.20) 𝑎_𝑧= 0

given that that no motion along the z-axes occurs. The velocity vector components can be rewritten as

𝑉_𝑥 = 𝑉𝑐𝑜𝑠𝛽 𝑉_𝑦 = 𝑉𝑠𝑖𝑛𝛽 (2.21) 𝑉_𝑦̇ = 𝑉̇ ∙ 𝑠𝑖𝑛𝛽 + 𝑉 ∙ 𝛽̇𝑐𝑜𝑠𝛽

Which can be inserted into equation 2.20. That provides

𝑎_𝑦 = 𝑉̇ ∙ 𝑠𝑖𝑛𝛽 + ∙ 𝛽̇𝑐𝑜𝑠𝛽 +𝜓̇ 𝑐𝑜𝑠𝛽 (2.22)

Under the assumption that the vehicle´s velocity along the y-axis is constant, and the side slip angle is very small equation 2.22 becomes

𝑎_𝑦 = 0 + 𝑉 ∙ 𝛽̇(1) +𝜓̇ (1) (2.23)

Given that

• The wheel-slip angle is very small

• The wheel load remains constant throughout the curve

• No slip occurs

The side force on front and rear wheel can be defined as

𝐹_𝑓= 𝐶_𝑓∙ 𝛼_𝑓 (2.24) 𝐹_𝑟 = 𝐶_𝑟∙ 𝛼_𝑟 (2.25)

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20 Using Newtons II Law´s equations

∑ 𝐹_𝑌 = 𝑚 ∙ 𝑎_𝑦 = 𝐹_𝑓∙ 𝑐𝑜𝑠𝛿_𝑓+ 𝐹_𝑟

(2.26)

∑ 𝜏_𝑍 = 𝐽_𝑧∙𝜓̈_𝑧= 𝐹_𝑓∙ 𝑐𝑜𝑠𝛿_𝑓∙ 𝑙_𝑓− 𝐹_𝑟∙ 𝑙_𝑟

Equations 2.24 and 2.25 lead to

0 = 𝐶_𝑓∙ 𝛼_𝑓∙ 𝑐𝑜𝑠𝛿_𝑓+ 𝐶_𝑟∙ 𝛼_𝑟− 𝑚 ∙ 𝑎_𝑦

0 = 𝐶_𝑓∙ 𝛼_𝑓∙ 𝑐𝑜𝑠𝛿_𝑓∙ 𝑙_𝑓− 𝐶_𝑟∙ 𝛼_𝑟∙ 𝑙_𝑟− 𝐽_𝑧∙ 𝜓̈

Figure 2.6. Velocity vector addition.

It is possible to cancel out the front and rear slip angles by taking advantage of the vector addition on the velocity vectors.

The above leads to

𝛼_𝑟 = − 𝛽 +𝑙_𝑟∙𝜓̇ 𝑉

(2.27) 𝛼_𝑓 = − 𝛽 +𝑙_𝑓∙𝜓̇

𝑉 + 𝛿_𝑓

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From the wheel angle we obtain the steering-wheel angle

𝛿_𝑠 =𝛿_𝑓

𝑖_𝑠 (2.28)

which included in equation 2.26 provides the equation of motion:

[𝛽̇

𝜓̈] = [

−𝐶_𝑓+ 𝐶_𝑟 𝑚𝑉

𝐶_𝑟𝑙_𝑟− 𝐶_𝑓𝑙_𝑓 𝑚𝑉² 𝐶_𝑟𝑙_𝑟− 𝐶_𝑓𝑙_𝑓

𝐽_𝑧

𝐶_𝑟𝑙_𝑟²− 𝐶_𝑓𝑙_𝑓² 𝐽_𝑧𝑉 ]

∙ [𝛽

𝜓̇] [

𝐶_𝑟 𝑚𝑉𝑖_𝑠

𝐶_𝑓𝑙_𝑓 𝐽_𝑧𝑖_𝑠 ]

∙ 𝛿_𝑠 (2.29)

Assumed that the vehicle is in stationary motion 𝛽̇ and 𝜓̈ would be equal to zero, and the equation of motion becomes

[0 0] =

[

−𝐶_𝑓+ 𝐶_𝑟 𝑚𝑉

𝐶_𝑟𝑙_𝑟− 𝐶_𝑓𝑙_𝑓 𝑚𝑉² 𝐶_𝑟𝑙_𝑟− 𝐶_𝑓𝑙_𝑓

𝐽_𝑧

𝐶_𝑟𝑙_𝑟²− 𝐶_𝑓𝑙_𝑓² 𝐽_𝑧𝑉 ]

∙ [𝛽

𝜓̇] [

𝐶_𝑟 𝑚𝑉𝑖_𝑠

𝐶_𝑓𝑙_𝑓 𝐽_𝑧𝑖_𝑠 ]

∙ 𝛿_𝑠 (2.30)

Some algebraic manipulations together with the assumption that 𝐶_𝑟𝑙_𝑟 ≥ 𝐶_𝑓𝑙_𝑓 provide the definition of the target yaw-rate estimated from the steering-wheel angle,

𝛹̇_𝑡 = ^𝑉∙𝛿^𝑠

𝑖_𝑠∙𝑙∙(1+(^𝑉 𝑣𝑐ℎ)

2 )

(2.31)

Where the characteristic velocity is defined as

𝑣_𝑐ℎ = √ 𝑙²𝐶_𝑓𝐶_𝑟

𝑚 ∙ (𝐶_𝑟𝑙_𝑟− 𝐶_𝑓𝑙_𝑓) (2.32)

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Figure 2.7. Characteristic speed.

Figure 2.7 illustrates what the characteristic speed of a vehicle is ( 𝑣_𝑐ℎ ). It is defined as the speed at which the ratio 𝛹̇_𝑡/𝛿_𝑠 is greatest. More precisely the characteristic speed is defined as the speed for an understeer vehicle where the steering angle needed to negotiate the curve is twice the Ackermann angle.

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23

2.2 Target Yaw-Rate Estimated from the Lateral Acceleration

The target yaw-rate estimated from the lateral acceleration provides information about the vehicle´s capability to negotiate the curve. It is used to establish what the surface limitations are.

It is possible to make an estimation of the yaw-rate estimated from the later acceleration.

We know that

𝑎_𝑐 = 𝑉²

𝑅 (2.33)

which is the definition of the lateral acceleration. Moreover

𝛹̇_𝑎𝑐 = 𝑉

𝑅 (2.34)

Combining the two above equations we obtain the target yaw-rate estimated from the lateral acceleration,

𝛹̇_𝑎𝑐 =𝑎_𝑐

𝑉 (2.35)

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24 2.3 Yaw-Rate Reference 𝛹̇_𝑟𝑒𝑓

The yaw-rate reference 𝛹̇𝑟𝑒𝑓is built upon 𝛹̇𝑡 and 𝛹̇𝑎𝑐. Depending on the situation, this signal is weighted between these two. In the situations in which the driver steers fast, 𝛹̇_𝑟𝑒𝑓 will be weighted toward the 𝛹̇_𝑡.

By doing so it is possible to avoid false interventions. In the situations in which the driver steers excessively (e.g. driving on low friction surfaces) 𝛹̇_𝑟𝑒𝑓will be weighted toward 𝛹̇_𝑎𝑐. In this fashion we can take into consideration the surface limitations. The measurement trace below has been taken at the test facility of Arvidsjaur (Sweden).

The driver is performing a right turn on an icy surface, causing understeering. See figure 2.8.

Figure 2.8. Measurement trace of an understeering vehicle on ice.

We can observe how 𝛹̇_𝑡-signal behaves while cornering: the driver attempts to steer, as shown by 𝛹̇_𝑡, but the vehicle does not fully respond to the command. At first 𝛹̇_𝑟𝑒𝑓, detecting the steering input, tracks 𝛹̇𝑡 accordingly. Although, the steering input does not have the desired effect, therefore 𝛹̇_𝑟𝑒𝑓 changes direction and starts taking after 𝛹̇_𝑎𝑐. That occurs at the point where the green cursor is located.

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25 2.4 Banked Bend Detection

The banking angle of a curve impacts directly the vehicle´s lateral acceleration sensor, which is fixed on the chassis. Indeed, is the lateral acceleration reduced during

cornering in a banked curve, which subsequently, affects the behavior of the 𝛹̇_𝑟𝑒𝑓. See figure 2.9.

Figure 2.9. Measurement trace of a vehicle travelling, in stable conditions, in a banked bend.

We can notice how 𝛹̇_𝑎𝑐behaves as the driver approaches the banked curve. See the area delimited by the green and black cursors.

As we can see 𝛹̇_𝑟𝑒𝑓diverges from 𝛹̇_𝑚, and starts tracking 𝛹̇_𝑎𝑐. Soon we will see how that can result in a false oversteering detection. We will also see, in the next section, that the difference between the 𝛹̇_𝑟𝑒𝑓and 𝛹̇_𝑚 is defined as the oversteering error.

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26 2.5 Error definition

The yaw-rate error 𝛹̇_𝑒is here next defined as

𝛹̇_𝑒= { (𝑂𝑣𝑒𝑟𝑠𝑡𝑒𝑒𝑟𝑖𝑛𝑔) → 𝛹̇_𝑒𝑜𝑠 = 𝛹̇_𝑚− 𝛹̇_𝑟𝑒𝑓

(𝑈𝑛𝑑𝑒𝑟𝑠𝑡𝑒𝑒𝑟𝑖𝑛𝑔) → 𝛹̇𝑒𝑢𝑠 = 𝛹̇𝑚− 𝛹̇𝑡 } (2.36)

During oversteering, the vehicle´s actual rotation is greater than the driver´s input rotation.

The difference between 𝛹̇_𝑟𝑒𝑓and 𝛹̇_𝑚 gives our definition of the oversteering error.

During understeering 𝛹̇𝑡is greater than 𝛹̇𝑚.

For the sake of simplicity, we have made the calculation direction independent, i.e.

the yaw-signals switch signs as the vehicle changes direction. The above calculation refers to left curves. For right curves the logic reverses the position of the terms in both oversteering and understeering expressions.

As a direction indicator we utilize 𝛹̇_𝑚.

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27 2.6 Ysc-Metric

The logic employed by ZF is based on the yaw-rate error 𝛹̇_𝑒 defined in equation 2.36.

The Ysc-Metric is not parametrizable and it is heavily weighted toward oversteering, which means the signal is more sensible to oversteering errors than it is to

understeering errors. We will soon see how that can become an issue.

The Ysc-Metric is defined as

𝑌𝑠𝑐𝑀𝑒𝑡𝑟𝑖𝑐 = 𝛹̇_𝑒𝑢𝑠 + 𝛹̇_𝑒𝑜𝑠· ( 1 + ^𝛹̇^𝑒𝑢𝑠

100 ) (2.37)

where ( 1 + ^𝛹̇^𝑒𝑜𝑠

100 )is the oversteering weighting factor.

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28

3 Methodology

The new signal is denoted driving condition detection signal (DCDS). The requirements are

• It must be trimmable

• It must provide information about how far from the stability threshold the vehicle is

• It must be robust

As for YscMetric, DCDS is also based on the yaw-rate error 𝛹̇_𝑒 defined in equation 2.36.

The idea is to develop a tool that calculates the yaw-rate error and its derivative. The structure of the tool reminds of a PD-controller, but it is not. The only task that is to be taken on by the tool is, to evaluate (not to control) the error. Most importantly, the components of the tool are parameterizable: That makes it possible to alter the

behavior of the signal, to adapt it to different situations (I.e. different vehicles, driving modes, environments etc.).

For this project we are going to be using Vector Canape©, a measurement tool that is used for calibration and analysis of automotive systems. The vehicles used are mostly BMW models. All of the vehicles are equipped with a measurement system. A deeper explanation of how the measurement system works is beyond the scope of this

project.

The tests are performed in part in the ZF test facility of Arvidsjaur (Sweden) and in part in the ZF Test Facility of Wüscheim. Also, some tests take place at the BMW Test Facility of Miramas (France). The DCDS-logic is written in C.

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29

OS

US

Figure 3.1. DCDS´s flowchart.

As depicted in the flowchart in figure 3.1 this tool is made of two main parts. One for the oversteering estimation, and one for the understeering estimation. Depending on the values that the yaw-rate error 𝛹̇_𝑒takes on, the logic will arbitrate which block is to be employed, and the calculation will be executed accordingly. We do not need to iterate in the logic as an iteration process is performed automatically by the equipment.

𝐾 ∙ 𝛹̇_𝑒𝑜𝑠

DCDS

𝛹̇

_𝑒

𝐾_𝑑∙ 𝛹̈_𝑒𝑢𝑠 𝐾 ∙ 𝛹̇_𝑒𝑢𝑠

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30

For the proportional component we simply multiply 𝛹̇_𝑒by a parameter 𝐾.

For the derivative component of the tool we use

𝐾_𝑑∙ 𝛹̈_𝑒= 𝐾_𝑑 𝛹̇_{𝑒(𝑎𝑐𝑡𝑢𝑎𝑙)}− 𝛹̇_{𝑒(𝑝𝑎𝑠𝑡)}

∆𝑡 (3.1)

where, 𝛹̇_{𝑒(𝑎𝑐𝑡𝑢𝑎𝑙)} is the yaw-rate error from the actual iteration and 𝛹̇_{𝑒(𝑝𝑎𝑠𝑡)} is the yaw-rate error from the previous iteration. Subsequently, the obtained value is multiplied by a parameter denoted 𝐾_𝑑, see figure 3.1.

∆𝑡 is the sampling time, which, in our system, it is set up to be 0,2 milliseconds.

At the end of every iteration, the tool calculates the sum of the two entries (the error and its derivative) and before starting a new calculation, we store the value from the last loop into 𝛹̇_{𝑒(𝑝𝑎𝑠𝑡)}.

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31

3.1 Oversteer-Error Desensitization for Banked Bends

As stated in section 2, detecting banked bends was a major problem. For this project we are going to implement a routine for detecting banked bends and consequently desensitize the oversteer error.

In this section we will briefly describe how the banked bend detection tool works. The logic is complex and a detailed analysis of how it works is beyond the scope of this project. Only the part of the logic that has been utilized during the project will be treated.

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32 3.1.1 Banked-Bend Detection Tool

If we go back to fig. 2.9 we can notice how, in a banked curve, 𝛹̇_𝑎𝑐 , 𝛹̇_𝑚 and 𝛹̇_𝑡 diverge from each other; that is due to the effect that the banking angle has on the lateral acceleration´s vector, as stated in section 2.4.

The banked-bend detection tool estimates the lateral acceleration based on the value of 𝛹̇𝑚through

𝛹̇_𝑚· |𝑉| = 𝑎_{𝑐−𝐸𝑠𝑡} (3.2)

where |𝑉| isthe vehicle speed, and𝑎_{𝑐−𝐸𝑠𝑡} is the estimated lateral acceleration.In a banked curve, the estimated lateral acceleration can be viewed as the value that the lateral acceleration would have taken on, in absence of a banking angle.

The banked-bend detection tool uses the difference between 𝑎_{𝑐−𝐸𝑠𝑡} and 𝑎_𝑐 to provide an estimation of the banking angle.

The estimation of the banking angle is performed if the following conditions are met.

• The difference between 𝑎_{𝑐−𝐸𝑠𝑡}and 𝑎_𝑐 is greater than a predefined threshold.

(This threshold is calculated in the banked-bend detection tool.)

• 𝛹̇_𝑚 and 𝛹̇_𝑡 excessively diverge from each other beyond predefined boundaries.

(The boundaries are defined in the banked-bend detection tool.)

• The vehicle is in stable conditions

As soon as these conditions are met, the banked-band detection logic starts a timer.

As soon as the timer becomes active, the banked-bend detection tool starts estimating the banking angle and integrates its value.

The value of this integral (denoted intEstBankAngle) will increase as long as the timer is active. If one or more of the above-mentioned conditions do not longer hold true, the timer will reverse the counting, and intEstBankAngle will decrease.

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33

Let´s look at the measurement from section 2.5. The goal is to illustrate how the banked-bend tool works.

Figure 3.2. Vehicle travelling in a banked bend

The trace depicted in figure 3.2 represents a vehicle travelling in a banked bend. In this case, the vehicle is in stable conditions. Although, as we can see in the

measurement, 𝛹̇_𝑚 deviates from 𝛹̇_𝑡 (See area between the green and black cursors).

Moreover, the difference between 𝑎_{𝑐−𝐸𝑠𝑡}and 𝑎_𝑐 is greater than its threshold (Unfortunately it was not possible to plot this difference and cannot be viewed in figure 3.2).

In this case all the conditions for the banked bend detection mentioned in the previous section (see 3.1.1) are met, and the timer signal starts counting accordingly.

Consequently, intEstBankAngle increases.

We will take advantage of this routine to desensitize the value of 𝛹̇_𝑒𝑜𝑠, so to avoid misleading data from DCDS.

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34

3.1.2 Oversteering-Error Desensitization Tool for Banked Bends

The way we are going to advantage of the bank bend detection logic is, we are going to be using a 2-dimensional look-up table (refer to table 3.1). Let´s have a closer look.

𝑋_𝑛 𝑌_𝑛

5 0.9

10 0.75

40 0.70

80 0.65

100 0.55

Table 3.1. The oversteering desensitizing factor´s look-up table.

The X-column represents the input, which in our logic is a scaled version of intEstBankAngle. The reason we scaled it is to reduce its size (so to prevent the system from crashing etc.).

The Y-column is the output, which is our 𝛹̇_𝑒𝑜𝑠-desensitization factor which we denote BbDcrsFact (Banked-Bend Decreasing Factor). The routine´s task is to search the table and then to perform interpolation to extract the appropriate factor. Every time

intEstBankAngle reaches one of the 𝑋_𝑛-value defined in the table the routine will assign BbDcrsFact to its respective 𝑌_𝑛-value.

The greater the value of intEstBankAngle, the smaller the value of BbDcrsFactwill get.

BbDcrsFact will in turn be multiplied by the oversteering-error value 𝛹̇_𝑒𝑜𝑠 to erase the effects that banked curves have on our evaluation of the oversteering-error.

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35

Figure 3.3. DCDS flowchart. It represents the DCDS-logic including the banked bend detection routine.

Establish Direction

Banked Bend?

OS Desens.

US Error Os Error

or Us Error

OS Error Reduced OS

DCDS

𝑂𝑆: [𝛹̇_𝑚 ; 𝛹̇_𝑟𝑒𝑓 ] 𝑈𝑆: [𝛹̇_𝑚 ; 𝛹̇_𝑡]

FALSE

TRUE

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36 3.2 Parameters

The tool offers the possibility to adapt the signal behavior based on the situations in which the evaluation of the yaw-rate error takes place. That is, the tool is

parameterizable. In this fashion we can “shape” the behavior of the signal by choosing the parameters K, and 𝐾_𝑑 that best suits the situation.

The weighting factor in YscMetric only allows to affect the scaling of the oversteering component of the tool.

DCDS on the other hand, offers the opportunity to parametrize its components one by one affecting different aspects of the signal behavior: For example, if we wanted to change the signal´s rapidity we would tune 𝐾_𝑑. If we simply needed to scale it, we would tune K.

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37

4 Results and Discussions

In this section we are going to analyze the results from our tests. The tests were performed both in cold-weather conditions, at the ZF test facility of Arvidsjaur, and warm-weather conditions, at the ZF test facilities of Wücheim and the BMW test facility of Miramas. In some of the measurements we made a comparison between the YscMetric and DCDS, to see which one provides the best analysis. During some of the tests we experienced software-related issues. Consequently, we could not always test the signal with different parameters or compare with YscMetric.

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38 4.1 Understeering

The following test is performed at the BMW´s test facility in Miramas in France. The maneuver is performed on dry asphalt. In this test we are going to analyze the

behavior of DCDS in an understeering situation.

The vehicle travels in a left curve. The steering wheel angle is kept relatively constant throughout the maneuver. At the same time, the driver gradually accelerates until the vehicle understeers. We chose a driving scenario in which YscMetric normally fails to provide reliable data. The biggest issues in this case is the fact that YscMetric was heavily weighted toward oversteering. That makes the signal too sensible to the vehicle´s oversteering reactions.

A schematic of the curve where the tests were performed can be seen in Fig. 4.1.

Figure 4.1 The figure depict a curve (marked in red) in the handling track located in the BMW Test Facility of Miramas (France).

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39

Figure 4.2. Measurement of an understeering vehicle on dry asphalt. DCDS-Parameters K=1.5, 𝐾𝑑= 1.

The measurement in figure 4.2 represents the vehicle´s understeering behavior in the curve represented in figure 4.1. We can see in the measurement that the magnitude of 𝛹̇_𝑡 becomes greater than 𝛹̇_𝑚, which indicates that the vehicle is understeering (See section 2.5).

At the point where the vertical cursor is positioned, YscMetric is in the oversteering range, although the vehicle is understeering. DCDS on the other hand, remains in the understeering range.

If we have a closer look, we can see that DCDS, even if remaining in the

understeering range, is able to detect a tendency to oversteering. That is indicated by the peaks that DCDS shows. A good example is the peak that appears where the cursor is positioned. The way we can interpret that peak in this case is, the vehicle tends to switch behavior, from understeering to oversteering. Although, the understeer-error is too high to allow the oversteering error to take over.

All in all, we would rather rely on the information provided by DCDS for this particular driving situation.

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40

The test depicted in figure 4.3 is performed at the test facility of Arvidjaur (cold- weather conditions). The vehicle travels at a constant speed of approximately 60 km/h. The driver increases the steering wheel angle until the vehicle understeers. The goal is to investigate how DCDS reacts on an understeering error on low-friction.

Figure 4.3. Measurement of an understeering vehicle on snow. DCDS-Parameters K=1.5, 𝐾𝑑= 1.

We can notice the vehicle´s slower reaction to the driver´s input by analyzing 𝛹̇_𝑚, 𝛹̇_𝑡 and 𝛹̇_𝑎𝑐. The driver attempts to steer (magnitude of 𝛹̇_𝑡 increases), but the vehicle delays in following the driver´s command, due to the low friction (the magnitude of 𝛹̇_𝑚 and 𝛹̇_𝑎𝑐 remain relatively constant).

All in all, DCDS does a decent job detecting the understeer-error in this case. In this test DCDS and YscMetric behave similarly. YscMetric was not plotted.

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41 4.1 Oversteering

The test depicted in figure 4.4 is executed at the test facility of Arvidjaur, in cold- weather conditions. The chosen scenario is a roundabout on a frozen lake.

The driver approaches the curve increasing the vehicle speed. The steering angle is kept constant throughout the maneuver. Subsequently the driver releases the throttle.

As a result, a weight transfer occurs, and the car oversteers. The driver applies countersteering toward the end of the maneuver to prevent the car from spinning.

Let´s examine figure 4.4.

Figure 4.4. Measurement of an oversteering vehicle on dry asphalt. DCDS-Parameters K=1.5, 𝐾𝑑= 1.

The oversteering departure starts where the vertical cursor is positioned. At that point the 𝛹̇_𝑚 and 𝛹̇_𝑟𝑒𝑓 start to diverge significantly. The magnitude of 𝛹̇_𝑚 grows rapidly while the magnitude of 𝛹̇_𝑟𝑒𝑓 remains relatively constant at first. DCDS takes on a positive value, according to its definition, which was expected. (See section 2.5).

YscMetric is much quicker at detecting the error. This is because it is weighted toward oversteering.

Also, notice when the driver applies countersteering to prevent the car from spinning (t = 11s). When this happens 𝛹̇_𝑚 and 𝛹̇_𝑡 have opposite signs. (I.e. the driver steers toward the opposite direction compared to the vehicle). For this reason, DCDS might show misleading values. The fact that during countersteering 𝛹̇_𝑚 and 𝛹̇_𝑡 have

opposite signs, causes DCDS to take on extremely high values, which in most cases, do not correspond to the error that is actually occurring. On the other hand,

countersteering does not affect YscMetric in the same way, since YscMetric has a max-value, that never gets surpassed. For this test YscMetric is more reliable.

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42 4.2 DCDS-Parameters’ Effect on DCDS

As stated before, the idea is to build a measurement tool that can be parametrized. In this way we can find a substitute to the weighting method used in YscMetric and finally obtain a signal that works in the vast majority of driving scenarios.

In this section we are going to see how varying the DCDS-parameters (𝐾 and 𝐾_𝑑) is going to affect DCDS. Also, we are going to compare DCDS with YscMetric.

The driver performs an understeering and an oversteering maneuver at the test facility of Arvidsjaur. Both maneuvers are repeated several times using different DCDS- Parameters. We changed one parameter at a time to see how each parameter affects DCDS.

We have attempted to make the various factors involved in the maneuvers (e.g.

steering angle and vehicle velocity) as congruent with each other as possible.

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43 4.2.1 First Scenario: Oversteering Maneuver

This oversteering maneuver takes place on ice. The steering angle is kept constant.

The driver, once having reached a speed of 60 km/h, releases the throttle causing a weight transfer. The driver let the car spin until coming to a complete stop. Figures 4.5 and 4.6 depict the same maneuver, performed two times using different DCDS- parameters. We have included 𝛹̇_𝑚 and 𝛹̇_𝑟𝑒𝑓 in the trace, since they are related to the definition of 𝛹̇_𝑒𝑜𝑠 (See equation 2.36).

Figure 4.5. DCDS-Parameters K=15, 𝐾𝑑= 1

For the test depicted in figure 4.5 we attempted to increase 𝐾. Based on how much the yaw-rates differ from each other, we can see that DCDS shows a relatively high value compared to YscMetric. By increasing 𝐾 we drastically affected the scaling of DCDS.

The result was expected. All in all, DCDS and YscMetric deliver the same result.

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44

Figure 4.6. DCDS-Parameters K=1.5, 𝐾_𝑑=25

In the measurement depicted in figure 4.6, we attempted to drastically increase the 𝐾_𝑑-term to 25. As a result, DCDS shows a slightly noisier form. Also, this parameter affects how quick the DCDS reacts on the errors. As we can see DCDS detects the oversteering error quicker than YscMetric. (It can be seen a t = 1m34s).

This was what we hoped. We want to be able to control how quick the signal reacts on the yaw-rate error.

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45 4.2.2 Second Scenario: Understeering Maneuver

For the next maneuver we are going to analyze how changing the DCDS-parameters affects the behavior of the signal in understeering situations.

Figures 4.7 and 4.8 depict the same maneuver, performed two times using different DCDS-parameters. Firstly, the driver reaches a speed of approximately 65 km/h, subsequently increasing the steering wheel angle to 180 degrees. The vehicle, travelling on snow, cannot deal with the curve and understeers. We repeated this maneuver two times. We have included 𝛹̇_𝑚 and 𝛹̇_𝑡 in the trace, since they are related to the definition of 𝛹̇_𝑒𝑢𝑠 (See equation 2.36).

Let us see what happens when we increase the 𝐾-parameter. (See figure 4.7)

Figure 4.7 DCDS-Parameters: K=7.5, 𝐾𝑑=1

As in section 4.1.1, changing K affected the scaling of DCDS. The result was expected. We can notice that YscMetric shows the same issue mentioned in section 4.1, falsely detecting an oversteering error, even though the understeering error is predominant. That is visible at t = 57.5s and t = 59s.

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46

Let´s examine the trace in figure 4.8 at the point where the cursor is positioned.

Figure 4.8 DCDS-Parameters: K=1.5, 𝐾𝑑=6

A higher value of 𝐾_𝑑 makes the signal react promptly on an understeering error. 𝛹̇_𝑚 and 𝛹̇_𝑡 start diverging and the magnitudes of DCDS and YscMetric start to increase.

We can notice that DCDS quickly detects the understeering error, has lower

oscillation in comparison to the YscMetric, and follows 𝛹̇_𝑒𝑢𝑠 more accurately. At the same time YscMetric falsely shows a tendency to oversteering. This result was expected.

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47

In both scenarios we could notice substantial differences by altering the 𝐾_𝑑 and 𝐾- parameters. The former affects primarily the rapidity at which DCDS detects an error.

The latter mostly impacts the magnitude of DCDS.

It is particularly interesting how the 𝐾_𝑑-parameter affects the behavior of the signal making it more reactive. The effects of altering the parameters had the effect we expected, i.e. it was possible to tune the behavior of the signal.

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48 4.3 Banked Bend Decreasing Factor

The test depicted in figure 4.9 takes place at the ZF test facility of Arvidjsaur. The curve is 20% banked and the maneuver is performed on snow. The driver is travelling at a constant speed and with constant steering wheel angle. We did not plot

YscMetric.

Figure 4.9. Vehicle travelling in a banked bend, on snow. DCDS-Parameters K=1.5, 𝐾_𝑑= 1.

Even though the vehicle is fully stable DCDS veers toward the oversteering range, especially in the very first section of the curve where the variation in banking angle occurs. (See section 2.4).

BbDcrsFact decreases with respect to the banking angle of the road, which in turn, is multiplied by 𝛹̇_𝑒𝑜𝑠. From this operation we obtain a reduced value of DCDS.

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49

The banked-bend-decreasing factor BbDcrsFact works as expected, and it can desensitize 𝛹̇_𝑒𝑜𝑠.

Unfortunately, the banked-bend detection tool employed in the logic cannot detect a banked bend quickly enough. In particular, it cannot detect the very first section of the banked curve. That is exactly where the false oversteering detection occurs, due to the change in banking angle. Moreover, as the banking angle goes back to zero, we face the reverse problem. The change in banking angle in this case is negative, which causes the oversteering error to take on a negative value. As a result, DCDS falsely detects understeering.

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50 4.4 Low-Mu Issue

A major issue arises when an oversteering error is to be detected on a low-mu surface (E.g. ice).

The test depicted in figure 4.10 is performed at the test facility of Wücheim, on a low- mu surface (the friction coefficient is roughly 0.2). The driver attempts to cause oversteering. The response at the steering wheel is reduced due to the low friction.

When the driver attempts to steer the vehicle goes into understeering, right before entering oversteering. Let´s analyze the trace in figure 4.10

Figure 4.10. Vehicle travelling on a very low-mu surface. DCDS-Parameters K=1.5, 𝐾𝑑= 1.

The area within the vertical cursors depicts the vehicle´s oversteering reaction right after the understeering error has occurred. Even though the car is oversteering DCDS shows understeering. This is because the understeering error have already built up and by the time the vehicle enters oversteering, the understeering error is too high in comparison with the oversteering error. This makes DCDS delay in detecting oversteering.

This issue impacted in YscMetric as well and was addressed by weighting the signal toward oversteering. In this case YscMetric provides a more reliable measurement.

For our signal, this issue can be addressed by increasing 𝐾_𝑑 , making the signal more reactive to oversteering.

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51 4.5 Lane-Change

The maneuver depicted in figure 4.11 is performed on snow, at the test facility of Arvidsjaur. The driver performs a lane-change. The scope of this maneuver is to simulate an emergency situation in which the driver needs to avoid an obstacle. On a two-way traffic street that would mean to temporarily invade the opposite lane. The test is also known as the Moose Test. The driver approaches the obstacle at 60 km/h and steers to the left. As the vehicle starts veering to the left, the car begins to understeer. As soon as the driver passes the obstacle he immediately attempts to return to the original lane. In doing so the driver triggers an oversteering reaction.

Issues arises when trying to attenuate the oscillations caused by the weight transfer while repositioning the vehicle in the right lane. Let´s have a closer look at the trace in figure 4.11.

Figure 4.11. Lane-change on low-mu. DCDS-Parameters K=1.5, 𝐾𝑑= 1.

In the area delimited by the black cursors we can notice that DCDS shows how the vehicle firstly understeers, subsequently ending in oversteering. It would have been better if the transition from understeering to oversteering was slightly quicker. This issue can be addressed by tuning the parameters. YscMetric is quicker at detecting the oversteering error because of its weighting factor (See also section 4.4).

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5 Conclusion

The fact that YscMetric is weighted toward oversteering provides a positive outcome in situations in which oversteering quickly needs to be detected. For example, when driving on low-friction surfaces. Although, that same weighting factor causes the YscMetric to be very sensitive to oversteering, causing false detections.

DCDS seems to work better in understeering situations because of the very fact that the signal is not weighted. When it comes to oversteering situations, the signal can be improved. By tuning the K-parameters of DCDS it is possible to alter the signal´s behavior. In particular, it is possible to increase the rapidity at which DCDS detects an error.

A very hard task is to appropriately detect banked curves. Our main issue is to detect the change in the banking angle that occurs right at the start of the curve and at the end of the curve.

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6 Appendix

6.1 Oversteering Evaluation Function

/*This function calculates the oversteering error*/

S32 yawOSEval (void) {

S32 PdOS;// Oversteer-error variable

/* Establish the direction of travel */

if(Yaw_Ref.Turn_Direction_Sign<0) //If steering to the left..

{

Ysc_Vehicle_State.yawOS = Yaw_Ref.R_Ref - Ysc_Veh_Input.Yaw_Rate;

}

else //If steering to the right {

Ysc_Vehicle_State.yawOS = Ysc_Veh_Input.Yaw_Rate - Yaw_Ref.R_Ref;

}

/* Calculate the error and its derivative */

//Make the Os-error value global and multiply it by the parameter K Ysc_Vehicle_State.yawOsP = (Ysc_Vehicle_State.yawOS * DCDS_OS_KP / PD_GAIN_ADJ_SF);

//Calculate the derivative of the error and multiply by its KD parameter Ysc_Vehicle_State.yawOsD = ((Ysc_Vehicle_State.yawOS -

Ysc_Vehicle_State.yawLastLoopOS) * DCDS_OS_KD / PD_GAIN_ADJ_SF);

PdOS = Ysc_Vehicle_State.yawOsP + Ysc_Vehicle_State.yawOsD;

//Calculate the oversteer error. In case of a banked curve this value will be reduced.

PdOS = PdOS * Ysc_Vehicle_State.Bb_PdOS_Red_Factor;

PdOS = PdOS / PD_GAIN_ADJ_SF;

Ysc_Vehicle_State.PdOS = PdOS;

//Store the value from the last loop. This is used in the derivative part.

Ysc_Vehicle_State.yawLastLoopOS = Ysc_Vehicle_State.yawOS;

//Return the oversteer-error return PdOS;

}

Development of a Vehicle Stability Detection Signal