İTÜ-EEF Dept.

Of Control Engineering

Wheeled Mobile Robots

KON 318E : Introduction to Robotics 11

 İTÜ-EEF Dept. Of Control Engineering

General Interest
• Increasing interest in autonomous robots
operating outdoors on difficult terrains

KON 318E : Introduction to Robotics 22

 İTÜ-EEF Dept. Of Control Engineering

Terminology
• Locomotion — the process of causing an robot to
move.
– In order to produce motion, forces must be applied to the
robot
– Motor output, payload

• Kinematics – study of the mathematics of motion

without considering the forces that affect the motion.
– Deals with the geometric relationships that govern the
system
– Deals with the relationship between control parameters
and the behavior of a system.

• Dynamics – study of motion in which these forces are

modeled
– Deals with the relationship between force and motions.
KON 318E : Introduction to Robotics 33
 İTÜ-EEF Dept. Of Control Engineering

Notation :
Posture: position (x, y)
and orientation θ

RS (a )R T = S (Ra )
{Xm, Ym} : Moving Frame
{Xb, Yb} : Base Frame
 x
q =  y : Robot posture in base frame
 
θ 
 cos θ sin θ 0  : Rotaion matrix expressing the orientaion
R (θ )  − sin θ cos θ 0  of the base frame with respect to the
 0 0 1  moving frame
KON 318E : Introduction to Robotics 44
 İTÜ-EEF Dept. Of Control Engineering

Idealized Rolling Wheel

• Assumptions:
– No slip occurs in the orthogonal
direction of rolling (non-
slipping).
– No translation slip occurs
between the wheel and the floor
(pure rolling).
Non-slipping and pure rolling – At most one steering link per
wheel with the steering axis
perpendicular to the floor.
• Wheel parameters:
– r : wheel radius
– v : wheel linear velocity
– ω : wheel angular velocity
Lateral slip – t : steering velocity

KON 318E : Introduction to Robotics 55

 İTÜ-EEF Dept. Of Control Engineering

Wheel Types
Fixed wheel Centered orientable wheel

Off-centered orientable wheel

(Castor wheel)
Swedish wheel:omnidirectional
property

KON 318E : Introduction to Robotics 66

 İTÜ-EEF Dept. Of Control Engineering

Fixed Wheel
• Velocity of Point P
V = ( r × ω ) ax where, ax : A unit vector to X axis

• Restriction to the robot mobility

– Point P cannot move to the direction perpendicular to
plane of the wheel.

KON 318E : Introduction to Robotics 77

 İTÜ-EEF Dept. Of Control Engineering

Centered orientable wheels

• Velocity of Point P
V = ( r × ω ) ax where, ax : A unit vector to X axis

• Restriction to the robot mobility

– Point P cannot move to the direction perpendicular to
plane of the wheel.

KON 318E : Introduction to Robotics 88

 İTÜ-EEF Dept. Of Control Engineering

Off-Centered Orientable Wheels

• Velocity of Point P
V = ( r × ω ) ax + ( d × t ) a y
ax : A unit vector of x axis
ay : A unit vector of y axis

• Restriction to the robot mobility

KON 318E : Introduction to Robotics 99

 İTÜ-EEF Dept. Of Control Engineering

Swedish Wheel
• Velocity of Point P
V = ( r × ω ) ax + Uas
ax : A unit vector of x axis
as : A unit vector to the motion of roller

• Omnidirectional

KON 318E : Introduction to Robotics 10

10
İTÜ-EEF Dept. Of Control Engineering

Examples of WMR
Example
• Smooth motion
• Risk of slipping
• Some times use roller-ball to
Bi-wheel type robot make balance

• Exact straight motion

• Robust to slipping
• Inexact modeling of turning
Caterpillar type robot

• Free motion
• Complex structure
Omnidirectional robot
• Weakness of the frame

KON 318E : Introduction to Robotics 11

11
İTÜ-EEF Dept. Of Control Engineering

Mobile Robot Locomotion

• Instantaneous center of rotation (ICR) or

Instantaneous center of curvature (ICC)

A cross point of all axes of the wheels

KON 318E : Introduction to Robotics 12

12
İTÜ-EEF Dept. Of Control Engineering

Degree of Mobility
• The degree of freedom of the robot motion

Cannot move Fixed arc motion

anywhere (No ICR) (Only one ICR)

• Degree of mobility : 0 • Degree of mobility : 1

Fully free motion

Variable arc motion
(line of ICRs) ( ICR can be located
at any position)
• Degree of mobility : 2 • Degree of mobility : 3

KON 318E : Introduction to Robotics 13

13
 İTÜ-EEF Dept. Of Control Engineering

Degree of Steerability
• The number of centered orientable wheels that can be steered
independently in order to steer the robot

No centered orientable wheels

• Degree of steerability : 0

One centered orientable

wheel

Two mutually Two mutually

dependent centered independent
orientable wheels centered orientable
wheels

• Degree of steerability : 1 • Degree of steerability : 2

KON 318E : Introduction to Robotics 14

14
 İTÜ-EEF Dept. Of Control Engineering

Degree of Maneuverability
• The overall degrees of freedom that a robot can manipulate:
δM = δm + δs
Degree of Mobility 3 2 2 1 1
Degree of Steerability 0 0 1 1 2

• Examples of robot types (degree of mobility, degree of steerability)

KON 318E : Introduction to Robotics 15

15
İTÜ-EEF Dept. Of Control Engineering

Degree of Maneuverability

δM = δm + δs

KON 318E : Introduction to Robotics 16

16
 İTÜ-EEF Dept. Of Control Engineering

Non-holonomic constraint
A non-holonomic constraint is a constraint on the
feasible velocities of a body
So what does that mean?
Your robot can move in some directions (forward
and backward), but not others (sideward).
The robot can instantly
move forward and backward,
but can not move sideward Parallel parking,
Series of maneuvers

KON 318E : Introduction to Robotics 17

17
 İTÜ-EEF Dept. Of Control Engineering

Mobile Robot Locomotion

• Differential Drive
– two driving wheels (plus roller-ball for balance)
– simplest drive mechanism
– sensitive to the relative velocity of the two wheels (small error result in
different trajectories, not just speed)
• Steered wheels (tricycle, bicycles, wagon)
– Steering wheel + rear wheels
– cannot turn ±90º
– limited radius of curvature
• Synchronous Drive
• Omni-directional
• Car Drive (Ackerman Steering)
KON 318E : Introduction to Robotics 18
18
İTÜ-EEF Dept. Of Control Engineering

Differential Drive
İTÜ Control Eng. Mobile Robot (Diff. Drive)

KON 318E : Introduction to Robotics 19

19
 İTÜ-EEF Dept. Of Control Engineering

Differential Drive

• Posture of the robot • Control input

(x,y) : Position of the robot v : Linear velocity of the robot

: Orientation of the robot w : Angular velocity of the robot
(notice: not for each wheel)

KON 318E : Introduction to Robotics 20

20
İTÜ-EEF Dept. Of Control Engineering

Differential Drive
VR (t ) – linear velocity of right wheel
VL (t ) – linear velocity of left wheel
r – nominal radius of each wheel
R – instantaneous curvature radius of the robot trajectory
(distance from ICC to the midpoint between the two wheels).

Property: At each time instant, the

left and right wheels must follow a
trajectory that moves around the
ICC at the same angular rate ω, i.e.,
L L
ω ( R + ) = VR ω ( R − ) = VL
2 2

KON 318E : Introduction to Robotics 21

21
İTÜ-EEF Dept. Of Control Engineering

Differential Drive
Posture Kinematics Model: Kinematics model in world frame
• Relation between the control input and speed of wheels

• Kinematic equation

θ
90
− θ

• Nonholonomic Constraint
 xɺ 
[sin θ − cos θ ]   = xɺ sin θ − yɺ cos θ = 0
 yɺ 
Physical Meaning?
KON 318E : Introduction to Robotics 22
22
İTÜ-EEF Dept. Of Control Engineering

Differential Drive

Kinematics model in robot frame

⇒ configuration kinematics model

KON 318E : Introduction to Robotics 23

23
İTÜ-EEF Dept. Of Control Engineering

Basic Motion Control

• Instantaneous center of rotation

R : Radius of
rotation

• Straight motion
R = Infinity VR = VL

• Rotational motion
R= 0 VR = -VL

KON 318E : Introduction to Robotics 24

24
İTÜ-EEF Dept. Of Control Engineering

Basic Motion Control

• Velocity Profile

3 0 2 1

3 0 2 1

: Radius of rotation
: Length of path
: Angle of rotation

KON 318E : Introduction to Robotics 25

25
 İTÜ-EEF Dept. Of Control Engineering

Tricycle
• Three wheels and odometers on the two rear wheels
• Steering and power are provided through the front wheel
• control variables:
– steering direction α(t)
– angular velocity of steering wheel ws(t)

The ICC must lie on

the line that passes
through, and is
perpendicular to, the
fixed rear wheels

KON 318E : Introduction to Robotics 26

26
 İTÜ-EEF Dept. Of Control Engineering

Tricycle

• If the steering wheel is

set to an angle α(t) from
the straight-line
direction, the tricycle
will rotate with angular
velocity ω(t) about ICC
lying a distance R along
the line perpendicular to
and passing through the
rear wheels.

KON 318E : Introduction to Robotics 27

27
İTÜ-EEF Dept. Of Control Engineering

Tricycle

d: distance from the front wheel to the rear axle

KON 318E : Introduction to Robotics 28

28
 İTÜ-EEF Dept. Of Control Engineering

Tricycle

Kinematics model in the robot frame

⇒ configuration kinematics model

KON 318E : Introduction to Robotics 29

29
 İTÜ-EEF Dept. Of Control Engineering

Tricycle
Kinematics model in the World frame
⇒ Posture kinematics model

KON 318E : Introduction to Robotics 30

30
 İTÜ-EEF Dept. Of Control Engineering

Synchronous Drive
• In a synchronous drive robot (synchronous
drive) each wheel is capable of being driven
and steered.
• Typical configurations
– Three steered wheels arranged as vertices of an
equilateral
– triangle often surmounted by a cylindrical
platform
– All the wheels turn and drive in unison
• This leads to a holonomic behavior
KON 318E : Introduction to Robotics 31
31
İTÜ-EEF Dept. Of Control Engineering

Synchronous Drive

KON 318E : Introduction to Robotics 32

32
 İTÜ-EEF Dept. Of Control Engineering

Synchronous Drive
• All the wheels turn in unison
• All of the three wheels point in the same direction and
turn at the same rate
– This is typically achieved through the use of a complex collection of
belts that physically link the wheels together
– Two independent motors, one rolls all wheels forward, one rotate
them for turning
• The vehicle controls the direction in which the wheels
point and the rate at which they roll
• Because all the wheels remain parallel the synchro drive
always rotate about the center of the robot
• The synchro drive robot has the ability to control the
orientation θ of their pose directly.
KON 318E : Introduction to Robotics 33
33
 İTÜ-EEF Dept. Of Control Engineering

Synchronous Drive
• Control variables (independent)
– v(t), ω(t)

KON 318E : Introduction to Robotics 34

34
 İTÜ-EEF Dept. Of Control Engineering

Synchronous Drive
• Particular cases:
– v(t)=0, w(t)=w during a
time interval ∆t, The robot
rotates in place by an
amount w ∆t .
– v(t)=v, w(t)=0 during a time
interval ∆t , the robot
moves in the direction its
pointing a distance v ∆t.

KON 318E : Introduction to Robotics 35

35
İTÜ-EEF Dept. Of Control Engineering

Omidirectional

KON 318E : Introduction to Robotics 36

36
İTÜ-EEF Dept. Of Control Engineering

Omidirectional

KON 318E : Introduction to Robotics 37

37
İTÜ-EEF Dept. Of Control Engineering

KON 318E : Introduction to Robotics 38

38
İTÜ-EEF Dept. Of Control Engineering

Car Drive (Ackerman Steering)

• Used in motor vehicles, the
inside front wheel is rotated
slightly sharper than the
outside wheel (reduces tire
slippage).
R • Ackerman steering provides
a fairly accurate dead-
reckoning solution while
supporting traction and
ground clearance.
• Generally the method of
choice for outdoor
autonomous vehicles.
KON 318E : Introduction to Robotics 39
39
İTÜ-EEF Dept. Of Control Engineering

Car Drive (Ackerman Steering)

where
d = lateral wheel separation
l = longitudinal wheel separation
θi = relative steering angle of inside wheel
θo = relative steering angle of outside wheel
R=distance between ICC to centerline of the vehicle
KON 318E : Introduction to Robotics 40
40
 İTÜ-EEF Dept. Of Control Engineering

Car Drive (Ackerman Steering)

• The Ackerman Steering equation:
d cos θ
– :
cot θ i − cot θ o = cot θ =
sin θ
l

cot θ i − cot θ o
R−d /2 R+d /2
= −
l l
d
=−
l
R

KON 318E : Introduction to Robotics 41

41
İTÜ-EEF Dept. Of Control Engineering

Car Drive (Ackerman Steering)

Equivalent:

KON 318E : Introduction to Robotics 42

42
İTÜ-EEF Dept. Of Control Engineering

Kinematic model for car-like robot

• Control Input
• Driving type: Forward wheel drive

Y x, y
ϕ { x, y , θ , ϕ }
u1 : forward vel
θ {u1 , u2 } u : steering vel
2
{τ 1 , τ 2 }

X
KON 318E : Introduction to Robotics 43
43
 İTÜ-EEF Dept. Of Control Engineering

Kinematic model for car-like robot

xɺ = u1 cosθ
x, y
yɺ = u1 sin θ Y
ϕ
u1
θ = tan ϕ
ɺ
l θ
ϕɺ = u2
non-holonomic constraint:
X
xɺ sin θ − yɺ cosθ = 0 u1
u2
: forward velocity
: steering velocity

KON 318E : Introduction to Robotics 44

44
 İTÜ-EEF Dept. Of Control Engineering

Mobile Manipulators
Mobile Manipulator = Mobile Robot + Robot Manipulator(s)

KON 318E : Introduction to Robotics 45

45
İTÜ-EEF Dept. Of Control Engineering

Mobile Manipulators
İTÜ Control Eng. Mobile Manipulator

KON 318E : Introduction to Robotics 46

46
 İTÜ-EEF Dept. Of Control Engineering

Dynamic model for car-like robot

• Dynamic model Y x, y
ϕ

X
m 0 0   ɺxɺ   sin θ   cos θ 0
       f1 
0 m 0   ɺyɺ  =  cos θ  λ +  sin θ 0   
f
0
 0 I   θɺɺ   0 

0
 1   2 

KON 318E : Introduction to Robotics 47

47
İTÜ-EEF Dept. Of Control Engineering

Summary
• Mobot: Mobile Robot
• Classification of wheels
– Fixed wheel
– Centered orientable wheel
– Off-centered orientable wheel (Caster Wheel)
– Swedish wheel
• Mobile Robot Locomotion
– Degrees of mobility
– 5 types of driving (steering) methods
• Kinematics of WMR
• Basic Control
• Mobile Manipulators

KON 318E : Introduction to Robotics 48

48