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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