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공학박사 학위논문

스키드 조향 인휠 구동 차량의 험로 주행 제어 알고리즘

Terrain Driving Control Algorithm

for Skid-steered In-wheel Driving Vehicles

2014년 8월

서울대학교 대학원

융합과학기술대학원

나 재 원

스키드 조향 인휠 구동 차량의 험로 주행 제어 알고리즘

Terrain Driving Control Algorithm for Skid-steered In-wheel Driving Vehicles

지도 교수 이 경 수

이 논문을 공학박사 학위논문으로 제출함 2014 년 6 월

서울대학교 대학원 융합과학기술대학원 나 재 원

나재원의 박사 학위논문을 인준함 2014 년 6 월

위 원 장 기 창 돈 (인)

부위원장 이 경 수 (인)

위 원 박 재 흥 (인)

위 원 이 강 원 (인)

위 원 임 성 진 (인)

i

Abstract Terrain Driving Control Algorithm for Skid-steered In-wheel Driving Vehicles

Jaewon, Nah

Intelligent Convergence System

Graduate School of Convergence Science and Technology

Seoul National University

This thesis describes torque distribution control of six-wheeled skid-steered

in-wheel motor vehicles with consideration of friction circle of each wheel to

maximize terrain driving and maneuvering performance. To decide desired

yaw rate according to driver’s steering command, the maximum performance

of yaw rate in accordance with vehicle speed and lateral tire force disturbance

have been analyzed. In order to satisfy both desired net longitudinal force and

desired yaw moment, which are decided in accordance with driver’s intension,

the torque distribution algorithm determines torque command to each wheel,

in consideration of friction circles of all wheels, slip condition and motor

torque limitation, based on control allocation method. Vehicle speed

estimation algorithm for six-wheeled independent driving vehicles is designed

to estimate accurate speed using six wheel speed, acceleration and yaw rate

signals. The friction circle of each wheel is estimated using linear

parametrized tire model with two threshold values, based on recursive least

square method. The response of the six-wheeled and skid-steered vehicle with

the proposed torque distribution algorithm and friction circle estimation

ii

algorithm has been evaluated via computer simulations using TruckSim and

Matlab/Simulink co-simulation. The simulation studies show that the

proposed friction circle estimation algorithm is sufficiently accurate even

when a wheel is lifting under terrain-driving condition. Hill-climbing and

terrain driving performance with the proposed torque distribution and friction

circle estimation is enhanced in comparison with proportional torque

distribution. Maneuvering performance will be verified via comparison with

Ackerman steered vehicles in the near future.

Keywords: terrain driving, skid-steer, six-wheel, in-wheel motor, torque distribution, control allocation Student Number: 2010-30742

iii

Contents

Chapter 1 Introduction ............................................................ 1 1.1 Background and Motivations .................................................. 1

1.2 Previous Researches ................................................................ 7

1.3 Thesis Objectives and Contribution ...................................... 10

1.4 Thesis Outline ....................................................................... 12

Chapter 2 Six-wheeled Vehicle Dynamic Model .................. 142.1 Vehicle Dynamics ................................................................. 14

2.2 Driving Control System Architecture ................................... 19

2.3 Power Train and Actuators .................................................... 20

Chapter 3 State Estimation Algorithm .................................. 203.1 Vehicle Speed Estimation ..................................................... 22

3.2 Longitudinal Tire Force Estimation ...................................... 38

3.3 Friction Circle Estimation ..................................................... 39

Chapter 4 Torque Distribution Algorithm ............................. 514.1 Driver’s Command ................................................................ 53

4.2 Upper Level Controller ......................................................... 54

4.3 Lower Level Controller ......................................................... 63

Chapter 5 Simulation Results ................................................ 71

iv

5.1 Friction Circle Estimation ..................................................... 72

5.2 Slip Control ........................................................................... 76

5.3 Terrain Driving Performance Verification ............................ 80

5.4 Step-steering Response Verification ..................................... 84

5.5 U-turn Maneuver ................................................................... 91

Chapter 6 Conclusions .......................................................... 95

Bibliography .......................................................................... 97

Abstract ............................................................................... 104

v

List of Tables

Table 1 Parameters of the six-wheeled vehicle........................................ 17

Table 2 Outline of speed estimation simulations ..................................... 30

Table 3 Outline of smooth bump simulation for friction circle estimation72

Table 4 Outline of split-mu hill-climbing simulation .............................. 76

Table 5 Outline of terrain driving and hill-climbing simulation .............. 80

Table 6 Outline of step-steering simulation ............................................. 86

Table 7 95% settling time of yaw rate of step steering simulation .......... 88

Table 8 Outline of U-turn maneuvering simulation ................................. 92

vi

List of Figures

Figure 1 Six-wheeled systems and skid-steered systems for terrain driving2

Figure 2 Torque split of Porsche 959’s PSK system .................................. 4

Figure 3 Idea of friction circle of a six-wheeled vehicle ............................. 6

Figure 4 Configuration of vehicle dynamics ............................................ 15

Figure 5 Longitudinal and lateral tire force map for several fixed values of

vertical tire force .............................................................................. 16

Figure 6 Vehicle dynamic modeling of six-wheeled skid-steered vehicle

using TruckSim .............................................................................. 18

Figure 7 Driving control system architecture including control units and

actuators ........................................................................................... 19

Figure 8 Motor torque-speed characteristics .................................................. 20

Figure 9 Power flow of the hybrid power train system .................................. 21

Figure 10 Block diagram of state estimation algorithm .............................. 23

Figure 11 Block diagram of vehicle speed estimation algorithm ................ 25

Figure 12 KSdensity function and probability of wheel speed data for

deciding threshold ............................................................................ 29

Figure 13 Simulation results of vehicle speed estimation under off-road

condition .............................................................................. 32

Figure 14 Simulation results of vehicle speed estimation under fish-hook

test .............................................................................. 33

Figure 15 Trajectory of the test vehicle which performs drift maneuver .... 35

Figure 16 Wheel speed of the test vehicle which performs drift maneuver 36

Figure 17 Test results of slip ratio calculation and speed estimation of the

test vehicle ............................................................................... 37

Figure 18 Longitudinal tire force characteristic as a function of slip ratio,

depending on surface ....................................................................... 40

vii

Figure 19 Thresholds of slip ratio and estimated slip ratio and longitudinal

tire force ......................................................................... 44

Figure 20 Simulation results in the case when the road surface changes.... 47

Figure 21 An example of the results of polynomial estimation of

longitudinal tire force curve ............................................................. 50

Figure 22 Block diagram of torque distribution algorithm ......................... 52

Figure 23 Desired value in accordance with driver’s command ................. 53

Figure 24 The maximum yaw rate curve .................................................... 54

Figure 25 The maximum value of longitudinal tire force as a function of

vehicle speed ............................................................................ 55

Figure 26 Disturbance from lateral tire forces ............................................ 58

Figure 27 Proportional relationship between cornering stiffness and lateral

tire force ............................................................................... 60

Figure 28 Comparison between steady-state circular turning and the

maximum yawrate analysis .............................................................. 61

Figure 29 Wheel slip control strategy ......................................................... 66

Figure 30 Torque distribution using Control Allocation and Proportion

distribution ............................................................................... 70

Figure 31 Road profile of a bump simulation ............................................. 74

Figure 32 Simulation result of friction circle estimation for the rear wheel

with the proposed friction circle estimation algorithm and with the

polynomial estimation method......................................................... 75

Figure 33 Simulation environment of climbing a 30deg hill with split mu 78

Figure 34 Results of slip control simulation ............................................... 79

Figure 35 Simulation environment of climbing a 30deg hill with road

profile ............................................................................... 81

Figure 36 Results of terrain hill driving performance simulations ............. 83

Figure 37 Ackerman steered vehicle layout in comparison with skid-steered

vehicle .............................................................................. 85

viii

Figure 38 Comparison Results of step steering input simulation : yaw rate

at each speed ............................................................................ 88

Figure 39 b g- phase plane of step steering input simulation at 30kph .... 89

Figure 40 The maximum yaw rate curve for both the skid-steered vehicle

and the Ackerman steered vehicle.................................................... 90

Figure 41 Results of U-turn simulations ..................................................... 94

ix

Nomenclature

, ,x y za a a : Longitudinal, lateral, and vertical acceleration of the vehicle [m/s2]

g : Acceleration of gravity [m/s2]

, ,f m rl : Distance from the center of gravity to the front, middle and rear axle [m]

m : Mass of the vehicle [kg] r : Radius of tire [m]

wl : Track width of the vehicle [m]

, ,x y zV V V , : Longitudinal, lateral, and vertical speed of the vehicle [m/s]

desV : Desired longitudinal velocity of the vehicle [m/s]

x : Longitudinal global position of the vehicle [m] y : Lateral global position of the vehicle [m]

, ,f q y : Roll, pitch, and yaw angle of the vehicle [rad]

, ,f m rC : Cornering stiffness for front, middle and rear wheel [N/rad]

_x desF : Desired longitudinal net tire force of the vehicle [N]

, ,xi yi ziF F F : Longitudinal, lateral, and vertical tire force of i-th wheel [N]

samt : Sampling time [sec]

yawratet : Time delaying constant of yaw rate [sec]

zI : Moment of inertia of the vehicle [kgm2]

Jw : Wheel moment of inertia of i-th wheel [kgm2]

, ,P I DK K K : Proportional, integral and derivative control gain for PID control

kg : Sliding control gain

_z desM : Desired net yaw moment [Nm]

iT : Input wheel torque command of i-th wheel [Nm]

,RLS iJ : Index for recursive least square estimation i-th wheel [N]

ia : Tire slip angle of i-th wheel [rad]

b : Side slip angle of the vehicle [rad]

x

g : Yaw rate of the vehicle [ rad/s]

desg : Desired yaw rate [ rad/s]

driverd : Manual steering wheel angle (by the driver) [rad]

,i ia b : Parameters for estimation of longitudinal tire force

,a bh h : Forgetting factor of parameter a and b

il : Slip ratio of i-th wheel

thl : Threshold of slip ratio

, ,,th low th highl l : Threshold of linear slip region and nonlinear slip region

ziFm : Friction circle of i-th wheel [N]

_ maxxiF : The maximum value of longitudinal tire force of i-th wheel [N]

iw : Wheel speed (angular velocity) of i-th wheel [rad/s]

_des iw : Desired wheel speed for wheel speed control of i-th wheel [rad/s]

_ ,z stiatic iF : Vertical static force of i-th wheel [N]

1

Chapter 1. Introduction

1.1 Background and Motivations

Six-wheeled terrain driving vehicles with independent driving motors are being

developed for military purpose, surface exploration and leisure facilities, as

shown in Fig.1. Six-wheeled vehicles with independent driving system is capable

of generating variation in traction forces, compared to that of conventional ones

with trans-axles and differential gears. Thus, driving performance on off-road

surfaces can be enhanced using independent driving torque control, with

consideration for longitudinal tire force (traction) usage.

In this research, a driving control architecture for six-wheeled skid-steered

independent driving vehicles is treated. The target vehicle which weighs

6000kg and is equipped with six in-wheel driving motors has been designed to

drive on terrain. Hence, the driving control architecture has to decide and

distribute torque command appropriate to skid-steered vehicles driving on

terrain.

2

Figure 1: Six-wheeled systems and skid-steered systems for terrain driving : Mars pathfinder (six-wheeled)/ Loader (skid-steered)/ ATV(six-wheeled and

skid-steered)/ Crusher robot vehicle (six-wheeled and skid-steered)

1.1.1 Skid-steering System

Skid-steered vehicle system is adopted to off-road terrain driving vehicles

such as military, surface exploration and industrial vehicles, such as loaders

shown in Fig.1. Skid-steered vehicle system is not equipped with steering

linkages, unlike conventional Ackerman-steered vehicles.[RTO(2004)] This

system has advantages of maneuverability on off-road surfaces and small

3

volume in the front hull. Instead, it needs differential traction forces to be

steered, coping with disturbance from lateral tire forces. Also, skid steering

reduces considerable life cycle of pneumatics particularly on road and it

shows quite poor drivability at high speed. For this reason, skid-steer driving

control system must consider characteristic of tire forces and limitation of

turning, to maximize its drivability.

1.1.2 Torque Vectoring System

To enhance terrain driving performance of skid-steered independent driving

vehicles, driving control architecture should be designed to maximize traction

force of each wheel. For example, four-wheel-drive sport car, Porsche 959's

PSK (Porsche-Steuer Kupplung) system was designed for best use of traction,

using multi-plate cluch [Autozine]. In most of the time, torque split between

front and rear was 40:60, that is the same as the car's weight distribution,

while 20:80 in hard acceleration, because hard acceleration leads to rearward

weight transfer, as shown in Fig.2. This made the best use of traction. For the

latest AWD vehicles, this can be realized using torque vectoring control

system [Wheals(2004)]. Torque vectoring can be achieved using redesigned

differential gears that can distribute power to each wheel. Therefore, the target

4

of torque vectoring system is to optimally utilize the different road-tire

adhesion at each wheel and thus making the cornering more stable and

increasing agility of the vehicle [Croft-Whitea(2006)]. From this idea, desired

longitudinal net force and yaw moment to follow driver’s command or

autonomous driving control are generated using distribution of independent

driving torque of each wheel [Kang (2009)]. This idea can maximize driving

performance of skid-steered independent driving vehicles on terrain.

60%40%

(a) usual driving

20% 80%

(b) hard accelerating

Figure 2: Torque split of Porsche 959’s PSK system

5

1.1.3 Tire Force Usage on Terrain

Tire force of each wheel generated by torque vectoring is limited by the

product of friction coefficient and vertical load. This idea, i.e., a friction circle

can be used in the computation of usable tire forces and also the contact

condition of each wheel, as follows :

( )22 2xi yi ziF F Fm+ = (1.1)

where, xiF and yiF are tire forces of x- (longitudinal) and y- (lateral)

axis of i-th wheel.

Similar to the “g-g diagram [Rice(1970)]”, the maximum tire forces are

essentially limited to a circle in Fx-Fy plane, as shown in Fig.3. The friction

circle represents the force-producing limit of the tire for a given set of

operating conditions (load, surface, temperature, etc.). [Milliken(1995)]. Also,

the size of friction circle depends on weight transfer, because vertical load of

each wheel is changed. For this reason, the limitation of tire force usage of

each wheel is changed in real time during driving on terrain.

6

ZMxF

1zFm1xF

1yF

Figure 3: Idea of friction circle of a six-wheeled vehicle

7

1.2 Previous Researches

Driving controller for skid-steered vehicles should be designed to

optimize maneuver performance using torque distribution of each wheel.

Several skid-steering control methods have been studied and actively

developed to improve maneuverability of the skid-steered vehicle. Dawson,

et al. were investigated nonlinear control of wheeled mobile robots.

[Dawson(2001)]. But in this research, only pivot turning case has been

considered in skid-steer control strategy. Economou and Colyer proposed

fuzzy logic control of wheeled skid-steer electric vehicles [Colyer(2000)].

Their fuzzy logic controller prioritizes in favor of the yaw demand, by

limiting the speed demand. However, vehicle can be unstable in case of

severe turning because the only feedback for fuzzy control is wheel speed

sensor signal. Also, their simulation result improves performance of steady

state yaw-rate only. S.Golconda presented the steering controller of a six-

wheeled vehicle based on skid steering [Golconda (2005)]. The steering

controller consists of a PID controller with two filters, a prediction filter and

a safety filter. However, their skid-steering control input is on/off signal of

left and right brake. Those three researches did not cover how to distribute

torque command to each wheel and control wheel slip.

8

Recently, a driving control algorithm based on skid steering for a Robotic

Vehicle with Articulated Suspension (RVAS) has been designed [Kang

(2009)]. The driving controller is designed to optimize longitudinal tire

forces and to keep a slip ratio below a limit value as well as to track the

desired longitudinal tire force. However, their optimal tire force distribution

strategy considered magnitude of vertical tire force and wheel slip control

only and those two factors could not be treated conjunctly. They did not

consider characteristic of lateral dynamics of skid-steered vehicle and

dynamic model parameters such as cornering stiffness were regarded as

specific values.

To maximize terrain driving performance of the six-wheeled independent

driving vehicles, information of the friction circle of each wheel has to be known.

Several studies on the estimation of the friction circles or vertical loads have been

performed. Hoseinnezhad treats friction circle estimation method using the

relationship between longitudinal tire force and wheel slip-ratio

[Hoseinnezhad(2011)]. Ono presented estimation of friction force between tires

and the road using the relationship between self-aligning torque and

lateral/longitudinal tire forces [Ono(2005)]. However, those researches

[Hoseinnezhad(2011)] [Ono(2005)] did not present any practical solution of the

estimation because they used tire stiffness value and did not cover nonlinear slip

9

ratio or slip angle region. Estimation methods using stiffness value for linear

condition can be guaranteed only under regulated slip condition. Research

[Dakhlallah(2008)] proposed a method to estimate the tire/road forces in order to

evaluate sideslip angle and the mobilized friction coefficient that are among the

most important parameters that influence run-off-road risk and vehicle stability.

But to use this method, Dugoff tire model must be guaranteed.

In recent researches, the idea of friction circle estimation using relationship

between longitudinal tire force and slip ratio is adopted to distribute torque

command. Research [Brad(2006)] presents braking force distribution strategy

using control allocation method for rollover prevention, based on the maximum

tire force approximations. Driving control algorithms for optimized

maneuverability and stability based on vertical tire force estimation and friction

circle estimation are given in research [Kang(2009)] and [Kim(2011)],

respectively. However, their idea of friction circle estimation is not practical for

various driving condition and they did not cover driving control problem for off-

road driving condition. Under severe driving or terrain driving condition, friction

circle estimation using existing methods, hence maneuver and terrain driving

performance cannot be maximized. To overcome these problems, practical

friction circle estimation method under dynamic slip condition and for various

surfaces should be designed.

10

1.3 Thesis Objectives and Contribution

In this thesis, torque distribution strategy based on friction circle

estimation for six-wheeled skid-steered vehicles equipped with independent

driving motors is discussed. The goal of this research is to maximize

maneuverability and terrain driving performance of six-wheeled skid-steered

vehicles. To accomplish this goal, first, speed estimation and friction circle

estimation algorithm is proposed. Because there is no driving shaft or

transmission, vehicle speed cannot be measured and is estimated using wheel

speed sensors, acceleration sensor and yaw-rate sensor. Friction circles are

estimated using linear parameterized longitudinal tire force characteristic

with two thresholds of slip ratio, based on recursive least square method.

Second, torque distribution algorithm to satisfy both desired net longitudinal

force and desired yaw moment with consideration of friction circles and

wheel slip condition is designed. The target six-wheeled skid-steered vehicle

in this research is driven by both a human driver and remote control, hence

driving control architecture has to deal with driver’s steering, accelerating

and braking commands. Based on analysis of yaw-rate performance and

sliding mode control theory, desired yaw moment control is generated. Using

control allocation method, torque distribution to each wheel is decided with

11

consideration of friction circles, torque limitation of a motor and slip control.

Finally, maneuverability and terrain driving performance of six-wheeled

skid-steered vehicles is investigated via computer simulation using

TruckSim and Matlab Simulink.

The principal contribution of this thesis is the development of practical

method of vehicle speed estimation and friction circle estimation for

independent driving vehicle on various surfaces. These estimation

algorithms have been designed for adaptation to rough road profile and

verified via various simulations and vehicle test. On the basis of these

estimation methods, torque distribution algorithm has been designed with

consideration for the limitation of turning performance of the skid-steered

vehicle system and slip control strategy.

12

1.4 Thesis Outline

This thesis can be organized in the following manner. Description of the

six-wheeled independent driving vehicle model is presented in Section 2.

Vehicle dynamics of the six-wheeled skid-steered vehicle and motor

characteristic are modeled to design the proposed torque distribution

algorithm and investigate via computer simulations. The proposed driving

control architecture to enhance maneuverability and terrain driving

performance of the six-wheeled skid-steered vehicle consists of state

estimation algorithm and torque distribution algorithm. In Section 3, vehicle

speed estimation and friction circle estimation algorithms are described. A

practical vehicle speed and friction circle estimation algorithms are designed

to give accurate information of vehicle state to torque distribution algorithm.

In section 4, torque distribution strategy including dealing with driver’s

command, generation of desired longitudinal force and yaw moment, torque

distribution based on control allocation method and slip control is designed.

Results of the computer simulations using TruckSIM for evaluating the

proposed torque distribution and friction circle estimation are presented in

Section 5. To investigate maneuverability and terrain driving performance of

the six-wheeled skid-steered vehicle with the proposed torque distribution

13

algorithm, a bump, hill climbing and severe turning simulations have been

conducted. Finally, the conclusion of this thesis including summary of the

proposed algorithms and future works to be done is discussed in Section 6.

14

Chapter 2. Six-wheeled Vehicle Dynamic Model

The six-wheeled skid-steered vehicle is able to be steered by differential

torque distribution to in-wheel motor of each wheel. In this chapter, modeling

of vehicle dynamics of the six-wheeled skid-steered vehicle, driving control

system architecture and actuator characteristics are proposed to design torque

distribution algorithm and investigate via computer simulations.

2.1 Vehicle Dynamics

The subject vehicle has been designed to drive on a rough terrain, climb

hills and cross obstacles for military or exploratory purpose. The vehicle is

equipped with six independent driving motors, six independent brakes and six

independent suspensions. Fig. 4 shows configuration of vehicle dynamics.

Translational and rotational body dynamics of the vehicle can be expressed

using Newton and Euler equations, respectively, as follows :

( )

( )

( )

x x z y

y y x z

z z y x

F m a V V

F m a V V

F m a V V

q y

y f

f q

S = + -

S = + -

S = + -

& &&&

& & (2.1)

15

( )

( )

( )

x x x z y

y y y x y

z z z y x

M I a I I

M I a I I

M I a I I

qy

yf

fq

S = + -

S = + -

S = + -

& &&&& &

(2.2)

yz

x

flwl

ml rl

sh

1zF 3zF 5zF

y2xF

1xF

4xF 6xF

z

3xF 5xF

f q

xy

Figure 4 : Configuration of vehicle dynamics

The six-wheeled skid-steered vehicle model is equipped with 325/65R20

XLT tires. Tire forces can be calculated using Magic Formula tire model

which provides a method to calculate longitudinal and lateral tire force for a

wide range of operating conditions, including combined longitudinal and

lateral tire force characteristic [Pacejka(2002)]. Fig. 5 shows longitudinal and

lateral tire force map as functions of slip ratio and slip angle respectively, and

16

also vertical load.

Figure 5 : Longitudinal and lateral tire force map for several fixed values of vertical tire force

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

0.5

1

1.5

2

2.5

3

3.5

4x 10

4

Slip ratio [-]

Long

itudi

nal t

ire fo

rce

[N]

Fz=8583NFz=17167NFz=34335NFz=51502N

0 5 10 15 20 25 30 35 40 450

0.5

1

1.5

2

2.5

3

3.5

4x 10

4

Slip angle [deg]

Late

ral t

ire fo

rce

[N]

Fz=8583NFz=17167NFz=34335NFz=51502N

17

Table 1 shows specifications of the six-wheeled vehicle such as a sprung

mass, moment of inertia, tread and distance from c.g. to each axle.

Table1: Parameters of the six-wheeled vehicle

Description Symbol Value Unit

Distance from C.G. to the front axle lf 1.75 [m]

Distance from C.G. to the middle axle lm 0.25 [m]

Distance from C.G. to the rear axle lr -1.25 [m]

Wheel base lf +lr 3.00 [m]

Tread lw 2.50 [m]

Total vehicle mass m 6000 [kg]

Roll moment of inertia Ix 3300 [kgm2]

Pitch moment of inertia Iy, 45000 [kgm2]

Yaw moment of inertia Iz 42600 [kgm2]

Radius of tire (325/65R20 XLT tire) rf 0.465 [m]

18

In this research, the vehicle dynamic model of the six-wheeled skid-steered

vehicle is developed using “TruckSim” in order to analyze dynamic behavior

of the six-wheeled vehicle and to conduct a numerical simulation studies, as

shown in Fig.6.

Figure 6 : Vehicle dynamic modeling of six-wheeled skid-steered vehicle using TruckSim

19

2.2 Driving Control System Architecture

The subject vehicle of the proposed torque distribution algorithm is a six-

wheeled skid-steered in-wheel driving vehicle, which is driven by both a

human driver and remote control system. Driving control architecture gives

command to each wheel through a hydraulic brake control unit and a motor

control unit, in accordance with steering, throttle and braking commands from

a human driver and remote control system, as shown in Fig.7.

Figure 7 : Driving control system architecture including control units and

actuators

20

2.3 Power Train and Actuators

The skid-steered independent driving vehicle system is equipped with six

driving motors and six mechanical brakes as actuators for driving. Motor

torque-speed characteristics and delaying time of mechanical brake systems

are modeled using MATLAB/Simulink. Fig.8 shows the motor torque-speed

characteristics of a 40kW in-wheel motor for in-wheel driving system. Each

in-wheel motor is directly connected to the wheel with 4:1 reduction gears.

Brake actuator of each wheel has been simply modeled using a first-order

transfer function with a time constant, 0.2 sec.

Figure 8: Motor torque-speed characteristics

0 500 1000 1500 2000 2500 3000 3500 4000 4500 500050

100

150

200

250

300

350

400

450

500

Motor velocity [rpm]

Torq

ue [N

m]

21

The six-wheeled skid-steered vehicle is equipped with a series hybrid

power system, including an engine, a generator, a battery and an ultra

capacitor. Hybrid power system including an engine, an inverter and a battery

should be modeled to give allowed driving and regenerative power

information to the driving controller [Wang(2008)]. Because driving

performance of the vehicle is not related to hybrid power system on the

assumption of plenty of battery capacity, the hybrid power system is not

included in the vehicle modeling. Fig.9 shows power flow of the hybrid power

train system.

Figure 9: Power flow of the hybrid power train system

22

Chapter 3. State Estimation Algorithm

To give accurate information of vehicle state to torque distribution

algorithm, a practical vehicle speed and friction circle estimation algorithms

must be designed. Vehicle speed of independent driving vehicles cannot be

measured because there is no driving axle. Friction circles also cannot be

measured directly, because tire-road contact condition and friction change

continually. In this chapter, vehicle speed and friction circle estimation

algorithms using wheel torque, wheel speed, acceleration and yaw-rate sensor

signals are designed.

The friction circle represents the force-producing limit of the tire for a

given set of operating conditions. However, to measure the value of the

friction circle directly is impossible. Estimating the friction circle is more

convenient than estimating the vertical tire force and the friction coefficient

on dynamic driving conditions. Required information is vehicle speed

estimation, wheel speed sensor signals, wheel angular acceleration estimation

and wheel torque. Vehicle speed of a six-independent driving vehicle is

estimated based on wheel speed, yaw rate and acceleration sensor data, with

selection and filtering of wheel speed data to cope with even off-road

maneuver. Fig.10 shows a block diagram of state estimation algorithm.

23

1~6w

gxa

1~6T

ˆxV

îl

ˆziFm

ˆxiF

,driver desVd

1~6T

ˆxV îl

Figure 10: Block diagram of state estimation algorithm

24

3.1 Speed Estimation

In previous researches, vehicle speed can be estimated using acceleration

sensor and wheel speed sensor based on Fuzzy logic or Kalman filter

[Kobayashi(1995)][Gao(2012)]. Otherwise, roadside traffic management

cameras or optical sensors can be used [Schoepflin(2003)]

[Litzenberger(2006)]. In this research, vehicle speed is estimated based on

wheel speed, yaw rate and acceleration sensor data, with selection and

filtering of wheel speed data to cope with even off-road maneuvering. From

calculation of average wheel speed and acceleration of each wheel, the

wheel speed in severe slip circumstance is filtered and vehicle acceleration

information is used to compensation. Following steps represent the proposed

vehicle speed estimation strategy. Fig.11 shows following four steps; Wheel

angular acceleration estimation, Vehicle speed estimation in terms of

longitudinal speed of each wheel, Selection and filtering, and Vehicle speed

estimation.

25

Vehicle Speed Estimation

Wheel accel. estimation

Vehicle speed estimation in

terms of longi. speed of each

wheel

Count the number of the wheels over the

thresholds

Vehicle speed estimation

1~6ŵ&

1~6txV

xa t×D

Step 1.

Step 2.

Step 3.

Step 4.

tD

1~6w

g

xa

_ˆx totalV

n

Figure 11 : Block diagram of vehicle speed estimation algorithm

Step 1. Wheel angular acceleration estimation

The state equation for the estimation of the angular acceleration of the wheel

is obtained from the Taylor formula for the angular velocity of the wheel as

follows [Zhang(2004)] :

2

1

2

3

( ) ( ) ( ) ( ) ( )2!

( ) ( ) ( ) ( )

( ) ( ) ( )

tt t t t t t w t

t t t t t w t

t t t w t

w w w w

w w w

w w

D+ D = + D × + × +

+ D = + D × +

+ D = +

& &&

& & &&

&& &&

(3.1)

where tD is the sampling time and ( )w t denotes disturbance. Discretized

state equation of Eqn.(3.1) can be written as follows :

( 1) ( ) ( )( ) ( ) ( )

X k AX k GW kY k HX k e k

+ = += +

(3.2)

26

[ ]2( ) 1 / 2

( ) ( ) , 0 1 , 1 0 0( ) 0 0 1

sam sam

sam

kwhere X k k A H

k

w t tw tw

é ùé ùê úê ú= = =ê úê úê úê úë û ë û

&&&

Covariance matrices of noise term ( )W k and ( )e k , with assuming zero-

mean white noise can be written as follows :

( ) ( )1

2

3

0 0cov ( ) 0 0 , cov ( )

0 0

qW k Q q e k R r

q

é ùê ú= = = =ê úê úë û

(3.3)

where 1 2,q q and 3q are covariance of each state, respectively. Using

Kalman filter method, the estimation of wheel angular acceleration can be

obtained as follows :

ˆ ˆ ˆ( ) ( 1 1) ( ) ( 1 1)X k k AX k k L Y k HAX k ké ù= - - + - - -ë û (3.4)

Step 2. Vehicle speed estimation in terms of longitudinal speed of each

wheel

Length of speed vector on each wheel can be shown as follows :

( ) ( )

( ) ( )

2 21,3,5

2 22,4,6

0.5 0.5

0.5 0.5

y x w x wtx fmr

y x w x wtx fmr

V V l V l V l

V V l V l V l

g g g

g g g

= + + - × - ×

= + + + × + ×

;

; (3.5)

27

In (16), influence of lateral speed ( )is ignored. From (3.5), vehicle

speed estimation in terms of each wheel speed can be calculated as follows :

( ) 1,3,52,4,6

0.5 0.5 ( 1,3,5)ˆ0.5 0.5 ( 2,4,6)

ii

i

w wtxx

w wtx

V l r l iV

V l r l ig w g

wg w g

ìïíïî

+ × = + × ==

- × = - × = (3.6)

Step 3. Selection and filtering

Average value of the six estimations obtained in ‘Step 2’ has to be calculated. If

the estimation of the i-th wheel exceeds the threshold th1 or the wheel angular

acceleration of the i-th wheel exceeds the threshold th2, n(i) is set to 0, otherwise

n(i) is set to 1. The estimation value of the wheel that n(i) is 0 is regarded as

under excessive-slip condition. The threshold th1 is decided based on several

severe turning and full braking simulation data. We simulate severe accelerating

and braking simulation and calculate Kernel Smoothing Density [Wand(2004)]

[Chen(2000)] of error between each wheel speed and average wheel speed

( avgi ie r rw w= × - × ), especially about 40kph to 0kph, 60kph to 0kph, and

80kph to 0kph braking, as shown in Fig.12 (a). After integrating Kernel

Smoothing Density of error between each wheel speed and average wheel speed,

probability can be obtained as shown in Fig.12 (b). Marking dots where

probability is 5% and 95% and finally the threshold th1 can be decided as

y fV l g+

28

follows :

1_

1_

5.25 / sec ( ,3.5 / sec (

3.5 / sec (5.25 / sec (

low

high

m deceleratingthm accelerating

m deceleratingthm accelerating

ìïíïîìïíïî

-=

-

=

)

)

)

) (3.7)

(a) KSdensity function

-10 -5 0 5 100

0.5

1

1.5

2

2.5

Velocity Error [m/sec]

40 to 060 to 080 to 0

29

(b) Probability

Figure 12: KSdensity function and probability of wheel speed data for deciding threshold

The threshold th2 has to be decided in consideration of physical limitation of

motor-torque and wheel speed.

Step 4. Vehicle speed estimation

Using n(i) for filtering speed estimation in terms of each wheel speed and

weighting that in terms of acceleration, final estimation can be obtained as

follows :

-10 -5 0 5 100

0.2

0.4

0.6

0.8

1

X: -5.25Y: 0.05068

Velocity Error [m/sec]

Pro

babi

lity

X: 3.5Y: 0.9506

30

( ) ( ) 6_ _1

ˆ ˆ ˆ( ) ( ) ( ) ( ) ( )xx total x totaln

V t V i n i V t t a t n iw=

= × + -D + ×D ×å (3.8)

To investigate performance of speed estimation under an off-road driving

condition, speed estimation simulation studies have been conducted. Vehicle

dynamics and environment have been modeled using TruckSim, while the

speed estimation algorithm has been implemented via MATLAB Simulink.

Table 2 shows outline of speed estimation simulations.

Table 2. Outline of speed estimation simulations

Acceleration on Off-road Fish-hook

Friction Mu=0.85 constant Mu = 0.5 constant

Profile X-Z axis Random road profile RMS = 0.056m

Flat

Profile X-Y axis Straight Straight

Scenario Accelerating from 60kph to 90kph

Braking and turning at 5sec while driving at 60kph

Comparison Target

Average of six wheel speed signals

Average of six wheel speed signals

31

The scenario of the first speed estimation simulation is acceleration on

off-road condition as shown in Fig. 13. To materialize a bad condition for

speed estimation, root mean square value of the road profile is set to 56mm

and desired vehicle speed is increased from 60kph to 90kph. Fig.13 (a)

shows slip ratio of each wheel. Because the maximum slip ratio is larger

than 0.3 which stands for very high slip condition, there are wide variations

in wheel speed of each wheel, as shown in Fig.13 (b). Hence, wheel speed

average is much higher than the actual value given by TruckSim, while the

proposed vehicle speed estimation is closed to the actual value, as shown in

Fig.13 (c). Average of speed estimation error with the proposed estimation

algorithm is 1.52kph, while that with wheel speed average is 4.90kph.

(a) Slip ratio [ ]

0 1 2 3 4 5 6 7 8 9 10-0.2

0

0.2

0.4

0.6

Time [sec]

Slip

ratio

[ ]

FLFRMLMR

Wheel speed

32

(b) Wheel speed [rad/s]

(c) Vehicle speed [kph] Figure 13 : Simulation results of vehicle speed estimation under off-road

condition

Fig. 14 shows results of speed estimation under fish-hook turning scenario,

involving wheel-locking circumstance. In Fig.14 (a), slip ratio of wheel 3

and 5 are fall to -1, which stands for wheel-locking. As shown in Fig.14 (b),

wheel No.3 and 5 are locked and wheel speed average is reduced, while

estimated speed is closed to the actual value, as shown in Fig.14 (c). As a

results, average of speed estimation error in this simulation with the

proposed estimation algorithm is 0.69kph, while that with wheel speed

0 1 2 3 4 5 6 7 8 9 1020

30

40

50

60

Time [sec]

Whe

el s

peed

[rad

/s]

FLFRMLMR

Wheel speed

0 1 2 3 4 5 6 7 8 9 1060

70

80

90

100Wheel speed

Time [sec]

Veh

icle

spe

ed [k

ph]

RealWheelspeed avgEstimation

33

average is 2.99kph.

(a) Slip ratio [ ]

(b) Wheel speed [rad/s]

(c) Vehicle speed [kph]

Figure 14 : Simulation results of vehicle speed estimation under fish-hook test with wheel-locking

0 1 2 3 4 5 6 7 8 9 10-1

-0.5

0

Time [sec]

Slip

ratio

[-]

wheel1wheel2wheel3wheel4wheel5wheel6

0 1 2 3 4 5 6 7 8 9 10-100

0

100

200

300

Time [sec]

Vehi

cle

spee

d [ra

d/s]

wheel1wheel2wheel3wheel4wheel5wheel6

0 1 2 3 4 5 6 7 8 9 100

20

40

60

Time [sec]

Vehi

cle

spee

d [k

ph]

RealWheelspeed avgEstimation

34

To verify the proposed speed estimation algorithm, conventional vehicle

test has been performed. The test vehicle is a rear-wheel-drive sport car

Genesis Coupe with 300 horsepower, of which sensor signals can be

acquired, equipped with GPS/IMU to acquire actual speed value, driven by a

highly skilled driver to perform drift maneuver. Fig.15 shows the trajectory

of the test vehicle, which performs drift with large side slip angle. Fig.16

shows wheel speed and slip ratio of each wheel, respectively. Fig. 17 shows

results of speed estimation of the test vehicle which performs drift maneuver.

Because the test vehicle is a rear-wheel-drive car and the parking brake

operates on the rear wheels only, slip ratio of rear wheels decreases to -1 and

increases to +1 rapidly, as shown in Fig.17 (a) and (b). For this reason, speed

sensor signal of the test vehicle is not precise when the rear wheels are

slippery, while estimation with the proposed speed estimation algorithm is

closed to actual value given by GPS/IMU as shown in Fig.17 (c).

35

Figure 15 : Trajectory of the test vehicle which performs drift maneuver

(a) Wheel speed of the front wheels

-220 -200 -180 -160 -140 -120 -100 -80

-20

0

20

40

60

80

X : Longitudinal Position [m]

Y :

Late

ral P

ositi

on [m

]

0 2 4 6 8 10 12 14 16 180

50

100

Time [sec]

Whe

el S

peed

[kph

]

Front LeftFront Right

36

(b) Wheel speed of the rear wheels

Figure 16 : Wheel speed of the test vehicle which performs drift maneuver

(a) Slip ratio of the front wheels

(b) Slip ratio of the rear wheels

0 2 4 6 8 10 12 14 16 180

50

100

Time [sec]

Whe

el S

peed

[kph

]

Rear LeftRear Right

0 2 4 6 8 10 12 14 16 18-1

-0.5

0

0.5

1

Time [sec]

Slip

ratio

[ ]

Front LeftFront Right

0 2 4 6 8 10 12 14 16 18-1

-0.5

0

0.5

1

Time [sec]

Slip

ratio

[ ]

Rear LeftRear Right

37

(c) Vehicle speed estimation, actual speed and speed sensor signal

Figure 17 : Test results of slip ratio calculation and speed estimation of the test vehicle which performs drift maneuver, compared to speed sensor signal and the actual value given by GPS/IMU

0 2 4 6 8 10 12 14 16 180

20

40

60

80

100

Time [sec]

Long

itudi

nal s

peed

[kph

]

EstimationActualSpeed Sensor

38

3.2 Longitudinal Tire Force Estimation

Because friction circles are estimated using the maximum value of

longitudinal tire force and slip ratio of each wheel, longitudinal tire force

must be estimated. Slip ratio is estimated using the longitudinal vehicle

velocity and wheel angular velocity. The slip ratio is defined as follows:

( )

( )

ˆ0

ˆˆ

0ˆ

i xxdes

ii

i xxdes

x

r VF

r

r VF

V

ww

lw

ì -³ï

ï= í- +ï

39

3.3 Friction Circle Estimation

Friction circles can be estimated using the relationship between

longitudinal tire force, slip ratio and the friction circle, as follows :

( ) ( ) _ nominalnominal: :zi zi xi xiestF F F Fm m = (3.10)

In this relationship, the nominal value of longitudinal tire force over slip

ratio is assumed that in case of general on-road condition. Some studies show

that nonlinear tire force and road friction can be identified with assuming that

vehicles are on asphalt surfaces only [Yi(1999)]. But in a specific condition,

the nominal value can be different, depending on the characteristic of the

surface. In this research, friction circle is estimated using linear parameterized

longitudinal tire force model. The maximum value of longitudinal tire force is

close to friction circle when lateral tire force is small, as follows :

( )( )

2 2 2

2_ max 0

zi xi yi

xi yi

F F F

F F

m = +

; ; (3.11)

Fig.18 shows the longitudinal tire force characteristic as a function of slip

ratio, depending on the surface. In this figure, we define that ˆia is

longitudinal tire force coefficient, îb is saturation value of longitudinal tire

40

force, and t̂hl is threshold of slip ratio. As shown in this figure, shape of

longitudinal tire force characteristic curve varies with type of the surface. For

this reason, coefficient ˆia , îb of parameterized longitudinal tire force and

slip ratio threshold t̂hl at k-th step must be estimated for approximating

longitudinal tire force curve and its maximum value.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

1000

2000

3000

4000

5000

6000

7000

8000

9000

Slipratio [ ]

Tire

forc

e [N

]

StandardGravel/SandSnow

îa

îb

thl

Figure 18. Longitudinal tire force characteristic as a function of slip ratio, depending on the surface

Because longitudinal tire force characteristic as a function of slip ratio is

nonlinear, relationship written in (3.10) cannot be used to estimate friction

41

circle. For this reason, longitudinal tire force curve is approximated to a

simplified function using parameters, as follows :

( )( ) ( )

( )ˆ ˆ ˆˆ ( ) ( ) ( )

ˆˆ ˆ ˆ( ) ( ) ( )

i i i th

i

i i th

a k k k kf k

b k k k

l l l

l l

ì × £ï= í>ïî

(3.12)

In least square estimation method, unknown parameters of a linear model

are chosen in such a way that the sum of the squares of difference between the

actually observed and the computed values is the minimum [Vahidi(2005)].

To find the unknown parameters ˆia and îb , the index function which

should be minimized is defined as follows :

( ) ( ){ }2, ˆ ˆ( )k

RLS i i xik N

J k f k F k-

= -å,

( ), ( ) 0ˆ

RLS i

i

J kkl

¶=

¶ (3.13)

Most of tire force curves in accordance with terrain surface characteristics

have linear region below 0.05 and constant region above 0.18 of slip ratio.

[Stephant(2002)] Here, threshold of linear region ,th lowl and of nonlinear

region ,th highl are set to 0.05 and 0.18, respectively. When ,ˆ ( )i th lowkl l£ ,

42

the unknown paramter ˆia that minimizes the index function can be achieved

using the following formulation.

( ) ( ){ }( ) ( ){ }( ){ }

12

ˆˆˆ ˆ ˆ( ) ( 1) ( ) ( 1)

ˆ ˆ, ( ) ( 1) ( 1) ,

1ˆ( ) 1 ( ) ( 1)

i i xi i i

a a i a a i

a a i aa

a k a k L k F k a k k

where L k P k k P k k

P k L k k P k

l

l h l

lh

-

= - + - - ×

= - + -

= - - ×

(3.14)

Where h is a forgetting factor reflecting the rate of change of ˆ ( )ia k . ( )aL k

and ( )aP k are update gain and covariance, respectively. When

,ˆ ( )i th highkl l> , the unknown paramter îb that minimizes the index

function also can be achieved using the following formulation.

( ){ }{ }

{ }

1

ˆ ˆ ˆˆ( ) ( 1) ( ) ( 1)

, ( ) ( 1) ( 1) ,1( ) 1 ( ) ( 1)

i i b xi i

b b b b

b b bb

b k b k L k F k b k

where L k P k P k

P k L k P k

h

h

-

= - + - -

= - + -

= - - ×

(3.15)

Fig.19 shows thresholds of slip ratio and simulation results of estimated

slip ratio and longitudinal tire force. As shown in this figure, linear region of

longitudinal tire force is shown within 0.05± of slip ratio. However,

43

saturation region appears over ,th highl and transient region is shown between

,th lowl and ,th highl . Hence, the maximum longitudinal tire force when slip

ratio is below ,th lowl can be approximated using ˆ ( )ia k , using ˆ ( )ib k when

the slip ratio is over ,th highl , and using the value of pre-step when the slip

ratio is between ,th lowl and ,th highl . If the threshold of slip ratio t̂hl can

found, the maximum of longitudinal tire force can be approximated as follows:

( ) ( )( )

( )

,

_ max ,

_ max , ,

ˆ ˆˆ ( ) ( )

ˆ ˆˆ ( ) ( ) ( )

ˆˆ ( 1) ( )

i th i th low

xi i i th high

xi th low i th high

a k k k

F k b k k

F k k

l l l

l l

l l l

ì × £ïï

= >íïï - < £î

(3.16)

44

-0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-4000

-2000

0

2000

4000

6000

8000

10000

Slip ratio [ ]

Tire

forc

e [N

]

( )îa k

,th highl

ˆ ( )ib k

,th lowl ( )ˆth kl

Figure 19 : Thresholds of slip ratio and estimated slip ratio and longitudinal tire force

To avoid divergences of the friction circle estimation near non-slip

condition, friction circle calculation update is restricted when the slip ratio is

smaller than 0.01, as follows :

( )( ) ( )( ) ( )

_ maxnominal_ max_ nominal

1 ˆˆ ( ) ( ) 0.01( )

ˆ( 1) ( ) ( ) 0.01

z xi ixi

z est

z iest

F F k kFF k

F k no update k

m lm

m l

ì × × ³ïï= íï -

45

Because the shape of slip ratio - longitudinal tire force curve is various

according to road surface condition, the threshold of slip ratio should be also

updated. If difference between longitudinal tire force coefficient

( ˆia )multiplied by threshold of slip ratio and simplified longitudinal tire force

in nonlinear region ( îb ) increases over a small value e , threshold of slip

ratio t̂hl is updated to the value satisfies simplified longitudinal tire force

function, as follows :

( )

2 1ˆ ˆ( ) ( )ˆ ( )

ˆ ˆˆˆ ( )( ) ˆ ( ) ( 1) ( )ˆ ( 1) ( )

i th i thi

ith i th i

th

k or kb ka kk and a k k b k

k no update else

l l l l

l l e

l

> £

= - - >

-

ì æ öç ÷ïï ç ÷í è øï

ïî (3.18)

In Eqn (3.18), e should be determined considering nonlinearity of

longitudinal tire force curve and difference between terrain surface

characteristics. Fig.20 shows Simulation results in the case when the road

surface changes from standard terrain to gravel terrain at 20 sec. To generate

variation of slip ratio, drastic changes in vehicle speed and its estimation as

shown in Fig.20 (a). Fig.20 (b) shows estimation of slip ratio tracks actual slip

ratio though the road surface changes. Fig.20 (c) shows update of slip ratio

46

threshold t̂hl . Due to high slip ratio from 20 to 23 sec, slip ratio threshold

update error exists. After slip ratio is reduced, updated slip ratio threshold is

quite close to the actual value. Fig.20 (d) shows estimated and actual

coefficient ˆia of parameterized longitudinal tire force. After slip ratio rises

at 5 sec, the proposed estimation algorithm starts to update estimation of

coefficient ˆia with error due to recursive least square. The estimation error

is reduced in 2 sec. At 20 sec, estimation error is occurred due to high slip

ratio and slip ratio update error. The estimation error is reduced

simultaneously with slip ratio threshold correction.

(a) Vehicle speed

0 5 10 15 20 25 30 3525

30

35

40

45

50

Time [sec]

Spe

ed [k

ph]

EstimatedActual

47

(b) Slip ratio

(c) Slip ratio threshold

(d) Longitudinal tire force coefficient

Figure 20 : Simulation results in the case when the road surface changes from standard terrain to gravel terrain

0 5 10 15 20 25 30 35-0.2

-0.1

0

0.1

0.2

Time [sec]

Slip

ratio

[ ]

EstimatedActual

0 5 10 15 20 25 30 350

0.05

0.1

0.15

0.2

Time [sec]

Thre

shol

d [ ]

EstimatedActual

4

0 5 10 15 20 25 30 350

1

2

3

4x 10

4

Time [sec]

Long

i. C

oeff.

[N]

EstimatedActual

48

Polynomial Estimation Method : for comparison

Comparison target of the proposed friction circle estimation algorithm is a

polynomial estimation of tire force curve, to find a parametric polynomial

function of slip ratio using recursive least square method. The polynomials

describing each segment are given by,

5 4 3 2y ax bx cx dx ex= + + + +

( )

5

4

3

2

, , , ,x

xab x

where y F x c xd xe x

l l q f

é ùé ù ê úê ú ê úê ú ê úê ú= = = = ê úê ú ê úê ú ê úê úë û ê úë û

(3.19)

To find the parameter a, b, c, d and e, the index function which should be

minimized is defined as follows :

( ){ }2, ( ) ( ) ( )k

TRLS i

k NJ k y k k kf q

-

= -å

(3.20)

The unknown parameters a, b, c, d and e that minimize the index function

can be achieved using the following formulation.

49

( ) ( ){ }( ) ( ) ( ){ }( ){ }

1

ˆ ˆ ˆ( ) ( 1) ( ) ( 1)

, ( ) ( 1) ( 1) ,

1( ) 1 ( ) ( 1)

Ti i i i

Ti i i

Ti

k k L k y k k k

where L k P k k k P k k

P k L k k P k

q q q f

f h f f

fh

-

= - + - - ×

= - + -

= - - ×

(3.21)

Fig. 21 shows an example of the results of polynomial estimation of

longitudinal tire force curve. As shown in the figure, the maximum value of

longitudinal tire force can be found at the point where the first of the relative

maximum points. For this reason, the point i.e, threshold of slip ratio can be

found using the first order derivative of the polynomials, as follows :

4 3 20 5 4 3 2

min , , 0 0.2th

dy ax bx cx dx edx

x where xl

= = + + + +

= < <

(3.22)

Finally, friction circle estimation using polynomial estimation method can

be achieved as follows:

( ) ( )

5 4 3 2( ) ( ) ( ) ( ) ( )( )

( 1) ( ( ) 0.01)th th th th th

z estz thest

a k b k c k d k e kF k

F k kl l l l l

mm l

ì + + + +ï= í -

50

Figure 21 : An example of the results of polynomial estimation of

longitudinal tire force curve

-0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3-4000

-2000

0

2000

4000

6000

8000

10000

Slip ratio [ ]

Long

itudi

nal t

ire fo

rce

[N]

DataPolynomial estimation

51

Chapter 4. Torque Distribution Control

Algorithm

A six-wheeled vehicle equipped with 6 in-wheel-motors is able to drive

over terrain using independent wheel torque control. The proposed driving

control algorithm is designed to maximize terrain driving and hill-climbing

performance of the independent driving vehicle under its physical limitation.

The driving control algorithm consists of upper level control layer and lower

level control layer, as shown in Fig. 22. The upper level control layer is

designed to determine desired net longitudinal force and desired yaw

moment. The desired net longitudinal force and desired yaw moment are

calculated to track the desired speed and reference yaw rate respectively, on

the basis of sliding mode control theory. The lower level control layer

determines driving and braking torques of each wheel, satisfying both

desired net longitudinal force and desired yaw moment, with consideration

of friction circles and wheel slip condition.

52

a

_z desM

xdesF

ˆˆ ˆ, ,x ziV Fl m

, ig w

x̂V

g

driverd

Brake

xdesFdesV

1~6T

Figure 22: Block diagram of torque distribution algorithm

53

4.1 Driver’s Command

A conventional vehicle with a combustion engine reaches a certain speed in

accordance with its throttle angle. For realization of this characteristic,

desired speed ( desV ) is regarded as steady state vehicle speed ( ssV ), as a

function of throttle pedal, as shown in Fig.23 (a). When the human driver

intends deceleration, desired longitudinal force as a function of brake pedal

is decided similar to characteristic of a conventional vehicle, as shown in

Fig.23 (b). Separation of weak and strong region of the Fig.23 (b) is for

reducing sensitivity of the brake pedal.

ssV

throttlea pedalBrake

weak strong

_ [ ]x desF N

Figure 23: Desired value in accordance with driver’s command : (a) Steady state vehicle speed as a function of throttle pedal (b) desired longitudinal

force as a function of brake pedal

54

4.2 Upper Level Controller

The upper level control layer takes the driver’s steering, accelerating and

braking inputs to decide desired longitudinal force and desired yaw moment.

Desired longitudinal force control is designed to satisfy a desired velocity

controlled by the driver’s acceleration pedal, using PID control as follows:

( )

( ) ( )_

_

ˆ ˆ( ) ( )0ˆ( )

0

p x xIdes des

pedalxdes

x des d

x des pedal pedal

K V V K V V dtm Braked V VF K dtF Brake Brake

ì ì üï ï ïï ïï í ýï ï ïí

ï ïï î þïïî

- + -× =-= +

>

ò

(4.1)

Desired yaw rate which depends on driver’s steering angle and vehicle

speed is expressed as a first order transfer function.

ˆ( )1

steer x steerdes

yawrate f r

k Vs l l

dgt

= ×+ +

(4.2)

where yawratet is time-delay constant of yaw rate. For skid-steered vehicles,

stroke range of driver’s steering wheel is restricted, despite the maximum

yaw rate is according to vehicle speed. For this reason, desired yaw rate is

decided with consideration of vehicle speed. The steering gain steerk is a

function of estimated vehicle speed x̂V , as shown in Fig. 24. For example,

55

the maximum value of desired yaw rate is 1.2 rad/sec when the vehicle speed

is under 10kph, while 0.4 rad/sec when the vehicle speed is 50kph. This idea

is realization of AFS(Active Front Steering) [Klier(2004)], which considers

the mechanical limitation of the skid-steered in-wheel driving system.

Figure 24: The maximum yaw rate curve

The maximum yaw rate curve has been drawn using results of steady-state

circular turning simulations at several constant speeds, where the value is the

limitation that the vehicle turns stably under 5 deg of side slip angle

( )atan( / )y xV Vb = . Sharp drop of the curve is due to the motor torque

characteristic. The maximum value of yaw moment generated by

longitudinal tire force distribution ( 1~6( )z xM F ) is limited by motor torque

characteristic. Fig. 25 shows the maximum value of longitudinal tire force as

56

a function of vehicle speed, according to motor torque characteristic. The

gap between longitudinal tire forces of right wheels and left wheels can

make the maximum value of yaw moment, with assuming that longitudinal

disturbance is neglectable. The maximum longitudinal tire force during

steady state turning and the maximum yaw moment can be written as

follows :

( )( )

( ) ( )

1,3,5 1,3,5

1,3,5

1,3,5

11

max1 1

1 1

z motor zth x

x

motor motor zth x th x

F P FV

F

P P FV V

m ml

ml l

-

- -

ì æ öï - × ³ç ÷ç ÷+ïï è ø= í

æ öï- × ×

57

Figure 25: The maximum value of longitudinal tire force as a function of

vehicle speed

In skid-steered vehicle system, lateral tire force of each wheel is

disturbance to cornering, as shown in Fig.26. Equation (4.6) shows a

bicycle model of the skid-steered vehicle.

0 10 20 30 40 50 60 70 80 90 100-1.5

-1

-0.5

0

0.5

1

1.5x 10

4

Speed [kph]

Fx [N

]

Right wheelLeft wheel

58

Figure 26: Disturbance from lateral tire forces

( ) ( )

( ) ( )2

1~62 2 2

2 21

0 1 ( )122

f m r f f m m r r

x xz x

zf f m m r rf f m m r r

z z x

C C C l C l C l C

mV mVd M Fdt Il C l C l Cl C l C l C

I I V

b bg g

é ù- + + - + -ê ú-ê úé ù é ù é ù

= + ×ê úê ú ê ú ê ú- + +ë û ë û ë û- + -ê ú

ê úë û

(4.6)

where, ,f mC C and rC are cornering stiffness of front, mid and rear

wheels. Cornering stiffness of each wheel changes according to friction

circle and longitudinal tire force. For example, high slip ratio can generate

59

large longitudinal tire force and reduce lateral tire force [Nah(2012)]. When

a wheel generates the maximum value of longitudinal tire force, maximum

lateral tire force of the wheel can be defined as follows :

( ) ( ) ( )2 2max maxyi zi xiF F Fm= - (4.7)

Cornering stiffness can be estimated using proportional relationship

between nominal value and actual value, as follows:

( ) ( )_ , , _ , ,max : max :yiyi nominal f m r nominal f m rF F C C= (4.8)

Fig.27 shows proportional relationship between cornering stiffness and

lateral tire force as a function of slip angle. When state of tire changes from

nominal case to case 2, the maximum value of lateral tire force can be

estimated and cornering stiffness can be calculated using Eqn. (4.7) and (4.8),

respectively.

60

, , _f m r nominalC

, ,f m rC

_max yi nominalFæ öç ÷è ø

max yiFæ öç ÷è ø

Figure 27: Proportional relationship between cornering stiffness and lateral

tire force

From Eqn (4.6), with assuming that side slip angle is limited to 5deg, the

maximum value of steady state yaw rate can be written as below :

( ) ( )

( )

( )2 2 2

2( 5deg)

max2 1 max ( )

f f m m r r

z x z

f f m m r rz x

z

l C l C l CI V I

l C l C l CM F

I

bg

ì ü- + -ï ï× £ï ï= ×í ý

+ + ï ï+ï ïî þ (4.9)

Fig.28 shows comparison between results of steady-state circular turning

simulations at several constant speeds and of calculation using the Eqn (4.9).

61

Figure 28: Comparison between simulation results of steady-state circular turning simulations at several constant speeds and the maximum yaw rate

analysis

To track proposed desired yaw rate, a yaw moment generation is designed

based on the bicycle model. From Eqn.(4.6) and (4.9), yaw moment to be

generated by longitudinal tire force differential is calculated as follows :

( ) ( )2 2 2

_

22 m m r rf fz m m r rz des f f

x

l C l C l CM I l C l C l C

Vg b g

+ += + + - × + ×&

(4.10)

The desired yaw moment is calculated to satisfy the desired net yaw rate

by yaw rate feedback control method based on sliding mode control theory

0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

1.2

1.4

Speed [m/sec]

Yaw

rate

[rad

/sec

]

AnalysisSimulation

62

[Mahdi(2003)] [Van Zanten(1999)]. The sliding surface is defined by lateral

acceleration error as follows :

yawrate desS g g= -

2yawrate yawrate

d S Sdt

h£ - (4.11)

Eqn.(4.11) can be differentiate as follows :

( )sgndes desg g h g g- £ - × -& & (4.12)

By substituting (4.12) for g& in Eqn (4.10), desired yaw moment is

decided as follows :

( ){ }

( ) ( )_

2 2 2

sgn

22

des slidingzz des des

m m r rf fm m r rf f

x

kM I

l C l C l Cl C l C l C

V

g g g

b g

-= × -

+ ++ + - × + ×

&

(4.13)

where, slidingk is sliding control gain. From Eqn.(4.6), side slip angle ( b )

in Eqn.(4.13) can be calculated using g as follows :

( ) ( ) 22

2 2m r m m r r xf f fx x

C C C l C l C l C mVmV mV

b b g+ + + - +

= - × - ×& (4.14)

63

4.3 Lower Level Controller

In lower level control layer, torque command to each wheel is decided for

the purpose of generating the desired net longitudinal force and desired yaw

moment, based on the fixed-point control allocation method. Torque

command to each wheel is bounded under the motor torque-speed curve, as

shown in Fig.7. When slip ratio of a wheel becomes larger than threshold, slip

control is activated to keep the slip ratio of the wheel below the threshold.

Torque Distribution Algorithm

Torque distribution algorithm is designed to distribute wheel torque inputs

to generate desired net longitudinal force and desired yaw moment, using

control allocation method. The fixed-point control allocation (CA) method

originally proposed by Burcken [Burken(2001)], and then Wang

[Wang(2006)] applied this method to optimal distribution for ground

vehicles. Control inputs are driving torque (Ti, i=1,…,6) of in-wheel motors

and can generate the desired net longitudinal force and yaw moment which

is determined by the upper level control layer. The desired dynamics and

control inputs are related as follows :

64

( )( ) ( )_

xdes

z des

F kB u k

M ké ù

= ×ê úë û

where, [ ]1 2 3 4 5 6( ) ( ) ( ) ( ) ( ) ( ) ( )Tu k T k T k T k T k T k T k=

1 1 1 1 1 11

2 2 2 2 2 2w w w w w w

B l l l l l lr

=- - -

é ùê úê úë û

(4.13)

The control input u of the control allocation is determined to minimize the

performance index as follows: [20]

[ ] [ ]1 1min ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )2 2

T Td v d uJ k Bu k v k W Bu k v k u k W k u ke= - - +

subject to min max( ) ( ) ( )u k u k u k£ £ (4.14)

where, ( )( )

max, max

min, max

( ) ( )

( ) ( )i i

i i

u k T k

u k T k

w

w

=

= - , _( ) ( ) ( )

Td xdes z desv k F k M k= é ùë û

minu and maxu denote the lower and upper bounds of control input limits,

respectively. These limits depend on motor torque limitation in general,

wheel slip condition and failure information of in-wheel motors in addition.

Detail contents of wheel conditions contain angular velocity, tire normal

force and friction coefficient between tire and road. dv denotes the desired

dynamic matrix and B matrix represents relationship between the desired

65

dynamics and control inputs. Weighting factor matrix is a function of

friction circles as follows :

( ) ( ) ( )1 2 6

1

2

1 1 1( ) ,

( ) ( ) ( )

00

uz z zest est est

v

W k diagF k F k F k

W

m m m

r

r

=

=

é ùê úë û

é ùê úë û

L

(4.15)

where 1r and 2r are weighting factors for desired net longitudinal force

and desired yaw moment, respectively. The fixed-point control allocation

algorithm iterates according to the equation as follows :

( ) ( ) ( ) ( ) ( )1 1 ( ) 1 ( ) ( ) ( )Tc v d cu k sat k B k W v k k k I u ke h h+ = - - - T -é ùë û (4.16)

where, ( ) ( ) ( )( ) 1 ( ) 1 1 ( )Tc v uk k B k W B k W ke h eT = - - - + ,

1/ 22

1 1

( ) ( ) , ( ) 1/ ( )p p

ij cF Fi j

k k k kt h= =

T = = Tæ öç ÷è øåå

66

Slip Control Strategy

Wheel slip control strategy is to keep the slip ratio of each wheel below the

slip ratio threshold, in order to maintain stable region of both longitudinal tire

force and lateral tire force. Conventional slip control strategy is to make

torque command zero when slip ratio is over the threshold. In contrast,

calculate torque command to make slip ratio within two slip ratio threshold, to

make full use of tire force. Fig.29 shows wheel slip control strategy.

thl,th lowl

Figure 29 : Wheel slip control strategy

67

The slip control flag z is defined by :

( )( )( )

,

,

ˆ0 ( ) ( )

ˆ1 ( ) ( )

i th i

i

i th i

k kk

k k

l lz

l l

£=

>

ìïíïî (4.17)

To avoid chattering of the slip control, slip ratio threshold thl is decreased

to ,th lowl during the slip control flag is on, as follows :

( )( )( )( ), ,

1 0( )

1 1th i

th i

th low i

kk

k

l zl

l z

- ==

- =

ìïíïî (4.18)

Slip control gain slipK is designed considering that desired derivative of slip

ratio satisfies wheel dynamics ( w xJ T r Fw = - ×& ). The slip ratio shown in

Eqn.(1) can be differentiated, with assumption of constant speed as follows :

i

ix

drd dtdt V

wl

- ×=

(4.19)

Substituting this relationship for w& , the wheel dynamics can be re-written as

follows :

xw i i xiV d

J T r Fr dt

l× × = - × (4.20)

68

Desired derivative of slip ratio can be written as follows :

{ }, ,1des i th low iddt tl l l= -D (4.21)

Thus, torque command for reducing slip ratio can be decided as follows :

( ),1xi w th low i xiVT J r Fr t l l= × × - + ×D (4.22)

When the slip control flag is on, the input constraint of the control

allocation method is controlled to reduce the input torque of the wheel, as

follows :

( ) ( )( )( ) ( )( )

maxmax,

,

min, max,

( ) 0( )

ˆ ˆ( ) ( ) ( ) 1

( ) ( )

i ii

slip th low i xi i

i i

T k ku k

K k k r F k k

u k u k

w z

l l z

ì =ï= í- + × =ïî

= -

(4.23)

( ) 1, ( ) xslip wsam

V kwhere K k Jr t

= × ×

Proportional Torque Distribution: for comparison

Comparison target of the proposed driving control algorithm is a proportion

torque distribution logic, which decides wheel torque command evenly

distributed to the wheels of the same side, to satisfy desired yaw moment and

69

desired longitudinal force, as follows :

( )

( ) ( )

51 3 2 4 6_

51 3 2 4 6_

1

2 2w w

x des

z des

F T T T T T Tr

l lM T T T T T Tr r

=

=

+ + + + +

- + + + + + (4.21)

Considering static weight distribution to each wheel, torque distribution

proportional to static weight is decided as belows :

_ _

_ _

_ ,

_ 5_ 1 _ 3

_ ,

_ 2 _ 4 _ 6

2

2

( 1,3,5)

( 2,4,6)

x des z desw

x des z desw

z static i

z staticz static z statici

z static i

z static z static z static

r rF M

l

r rF M

l

Fi

F F FT

Fi

F F F

× +

× -

ì æ öï ç ÷ï è øïíï æ öï ç ÷ï è øî

=+ +

=

=+ +

(4.22)

Fig.30 shows figuration of torque distribution using (a) control allocation

and (b) proportional torque distribution. When the vehicle climbs a hill,

torque commands to rear wheels are larger than those to front wheel using

control allocation, while proportional to static weight using proportional

distribution. For this reason, terrain driving and hill-climbing performance of

the vehicle using control allocation torque distribution can be enhanced, in

comparison with that using proportional distribution or even-torque

distribution.

70

51 3 zz zF F Fm m m

_ 5_ 1 _ 3 z staticz static z staticF F F

Figure 30 : Torque distribution using (a) Control Allocation (b) Proportion

distribution

71

Chapter 5. Simulation and Test Results

To investigate performance of the six-wheeled and skid-steered vehicle

with the torque distribution algorithm and the friction circle estimation

algorithm, computer simulations have been performed. The proposed torque

distribution controller and the friction circle estimator were implemented

using MATLAB Simulink, while the vehicle model and road conditions

provided by Trucksim were used. In this chapter, five simulations have been

conducted : Friction circle estimation, Slip control, Terrain driving

performance verification, Step-steering response verification and U-turn

maneuver. For verification of the proposed friction circle estimation algorithm,

a driving simulation on a smooth bump has been performed. The proposed

slip control strategy is verified via split-mu hill-climbing simulation. After

verification of friction circle estimation and slip control strategy, terrain

driving and hill-climbing simulation has been conducted to investigate driving

performance with the proposed torque distribution based on friction circle

estimation. Step-steering and U-turn maneuver simulations have been

conducted to compare maneuver performance of skid-steered vehicle with the

proposed algorithms and Ackerman steered vehicles.

72

5.1 Friction circle estimation

To enhance terrain driving performance, accuracy of friction circle

estimation must be acceptable even when the vehicle faces a bump and a

wheel is lifting. To verify performance of the proposed friction circle

estimation algorithm and torque distribution according to the friction circle

estimation, a driving simulation at 20kph on a 90cm smooth bump has been

performed, as explained in Table 3.

Table 3. Outline of smooth bump simulation for friction circle estimation

Smooth Bump Estimation

Friction Mu=0.85 constant

Profile X-Z axis 90cm Smooth bump

Profile X-Y axis Straight

Scenario Driving at 20kph constant speed

Comparison Target

Actual friction circle value (given by TruckSim)

73

The length of the bump is 8m, height of the bump is 90cm, and road

friction is 0.85, as shown in Fig.31. Fig.32.(a) and (b) show that results of

friction circle estimation with the proposed estimation algorithm and with

the polynomial method, respectively, compared to the actual value given by

TruckSim. As shown in these figures, the proposed estimation algorithm can

estimate the effects of weight transfer and wheel-lifting circumstances e.g.

zero friction circle during driving on a bump. From Fig.32 (a), it is shown

that response of friction circle estimation of a wheel is accurate even when

the tire meets the bump and lands after lifting, compared to the result with

the polynomial method as shown in Fig.32 (b). However, at the moment that

the tire meets the bump, the actual value is grown up rapidly, while the

estimation cannot track the actual value, as shown in Fig. 32 (a), near 3.6 sec,

5.6 sec and 7,4 sec. For this reason, response of the proposed friction circle

estimation algorithm is acceptable for terrain driving condition, except for

disturbance with high frequency.

74

Figure 31: Road profile of a bump simulation

(a)

Time [sec]

2 3 4 5 6 7 80

1

2

3

4

5x 10

4

Time [sec]

muF

z [N

]

EstimatedActual

75

(b)

Figure 33: Simulation result of friction circle estimation for the rear wheel (a)

with the proposed friction circle estimation algorithm and (b) with the

polynomial estimation method

Using the proposed friction circle estimation method, accuracy of

estimation is 86.7%, while that using the polynomial estimation method is

76.2%.

Time [sec]

2 3 4 5 6 7 80

1

2

3

4

5x 10

4

Time [sec]

muF

z [N

]

EstimatedActual

76

5.2 Slip control

Before terrain driving simulation is conducted, the proposed slip control

strategy is verified via split-mu hill-climbing simulation. This simulation

shows that the proposed slip control strategy can help maintain stable region

of longitudinal tire force and slip ratio, compared to conventional slip control

which turns off torque command to the slippery wheel. Table 4 shows outline

of split-mu hill-climbing simulation.

Table 4. Outline of split-mu hill-climbing simulation

Split-mu hill-climbing simulation

Friction Mu=0.80 constant (left hand side)

Switching from 0.80 to 0.20 (right hand side)

Profile X-Z axis 30deg 12m hill

Profile X-Y axis Straight

Scenario Driving at 11kph constant speed

Comparison Target

Conventional slip control (On/Off)

77

Fig.33 shows simulation environment of climbing a 30deg hill with split

mu. The friction coefficient of left hand side changes from 0.8 to 0.2 and

target speed of the vehicle is 11kph. Fig.34 shows the results of slip control

simulation. When slip ratio rises over the threshold, slip flag becomes on and

wheel torque is controlled to make slip ratio within stable region, while

chattering with conventional on/off control. As a result, longitudinal tire force

can maintain stable region and also vehicle speed can keep the target value.

(a) Friction coefficient (Split mu)

(b) 30deg hill profile [m]

0 10 20 30 40 50 60 70 80 90 1000.2

0.4

0.6

0.8

1

X-axis[m]

Fric

tion

coef

f.[ ]

LeftRight

0 10 20 30 40 50 60 70 80 90 1000

5

10

15

X-axis[m]

Z-ax

is[m

]

78

Figure 33: Simulation environment of climbing a 30deg hill with split mu

(a) Vehicle speed [kph]

(b) Wheel speed of wheel 1 [rad/sec]

(c) Wheel torque of wheel 1 [Nm]

11 11.2 11.4 11.6 11.8 12 12.2 12.4 12.6 12.8 1310

10.5

11

11.5

Time [sec]

Spe

ed [k

ph]

On/off slip conAdv. slip con

11 11.2 11.4 11.6 11.8