2002 - © Pedro Lima, M. Isabel Ribeiro

Robótica Móvel

Kinematics Models

KINEMATICS MODELS OF

KINEMATICS MODELS OF

MOBILE ROBOTS

MOBILE ROBOTS

Maria Isabel Ribeiro

Pedro Lima

mir@isr.ist.utl.pt

pal@isr.ist.utl.pt

Instituto Superior Técnico (IST)

Instituto de Sistemas e Robótica (ISR)

Av.Rovisco Pais, 1

1049-001 Lisboa

PORTUGAL

April.2002

All the rights reserved

MOBILE ROBOTICS course

2002 - © Pedro Lima, M. Isabel Ribeiro

Robótica Móvel

Kinematics Models

References

• Gregory Dudek, Michael Jenkin, “Computational Principles of

Mobile Robotics”, Cambridge University Press, 2000 (Chapter 1).

• Carlos Canudas de Wit, Bruno Siciliano, Georges Bastin (eds),

“Theory of Robot Control”, Springer 1996.

2002 - © Pedro Lima, M. Isabel Ribeiro

Robótica Móvel

Kinematics Models

Kinematics for Mobile Robots

• What is a

kinematic

kinematic

model ?

• What is a

dynamic

dynamic

model ?

• Which is the difference between kinematics and

dynamics?

•

Locomotion

is the process of causing an autonomous

robot to move.

– In order to produce motion, forces must be applied to the

vehicle

•

Dynamics

– the study of motion in which these forces are

modeled

– Includes the energies and speeds associated with these

motions

•

Kinematics

– study of the mathematics of motion withouth

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 beahvior of a system in state space.

2002 - © Pedro Lima, M. Isabel Ribeiro

Robótica Móvel

Kinematics Models

Notation

• {X

m

,Y

m

} – moving frame

• {X

b

, Y

b

} – base frame

X

m

Y

m

P

X

b

Y

b

x

y

θ

θ

=

y

x

q

robot posture in base frame

θθ−

θθ

=θ

100

0cossin

0sincos

)(R

Rotation matrix expressing

the orientation of the base

frame with respect to the

moving frame

2002 - © Pedro Lima, M. Isabel Ribeiro

Robótica Móvel

Kinematics Models

Wheeled Mobile Robots

• Idealized rolling wheel

y axis

y axis

x axis

z motion

• If the wheel is free to rotate about its axis (x axis), the

robot exhibits preferencial rollong motion in one direction

(y axis) and a certain amount of lateral slip.

• For low velocities, rolling is a reasonable wheel model.

– This is the model that will be considered in the kinematics

models of WMR

Wheel parameters:

• r = wheel radius

• v = wheel linear velocity

• w = wheel angular velocity

2002 - © Pedro Lima, M. Isabel Ribeiro

Robótica Móvel

Kinematics Models

Differential Drive

• v

r

(t) – linear velocity of right wheel

• v

l

(t) – linear velocity of left wheel

• r – nominal radius of each wheel

• R – instantaneous curvature radius of the robot trajectory, relative

to the mid-point axis

L

R

ICC

x

y

θ

2

L

R

−

2

L

R

+

Curvature radius of trajectory

described by LEFT WHEEL

Curvature radius of trajectory

described by RIGHT WHEEL

2

L

R

)t(v

)t(w

r

+

=

2

L

R

)t(v

)t(w

l

−

=

L

)t(v)t(v

)t(w

lr

−

=

))t(v)t(v(

))t(v)t(v(

2

L

R

rl

rl

−

+

=

))t(v)t(v(

2

1

R)t(w)t(v

lr

+==

• 2 drive rolling wheels

)Rcos y,sinRx(ICC

θ+θ−=

control variables

2002 - © Pedro Lima, M. Isabel Ribeiro

Robótica Móvel

Kinematics Models

Differential Drive

•

Kinematic model in the robot frame

−

=

θ

)t(w

)t(w

LrLr

00

2r2r

)t(

)t(v

)t(v

r

l

y

x

• w

r

(t) – angular velocity of right wheel

• w

l

(t) – angular velocity of left wheel

Useful for velocity control

2002 - © Pedro Lima, M. Isabel Ribeiro

Robótica Móvel

Kinematics Models

Differential Drive

θ

θ

=

θ

)t(w

)t(v

10

0)t(sin

0)t(cos

)t(

)t(y

)t(x

)t()q(S)t(q

ξ

=

control

variables

))t(v)t(v(

2

1

R)t(w)t(v

lr

+==

( )

( )

σσ=θ

σσθσ=

σσθσ=

∫

∫

∫

d)(w)t(

d)(sin)(v)t(y

d)(cos)(v)t(x

t

0

t

0

t

0

L

)t(v)t(v

)t(w

lr

−

=

)t(w)t(

)t(sin)t(v)t(y

)t(cos)t(v)t(x

=θ

θ=

θ=

Kinematic model in the world frame

2002 - © Pedro Lima, M. Isabel Ribeiro

Robótica Móvel

Kinematics Models

Differential Drive

• Particular cases:

– v

l

(t)=v

r

(t)

•

Straight line trajectory

– v

l

(t)=-v

r

(t)

•

Circular path with ICC (instantaneous center of curvature) on

the mid-point between drive wheels

.cte)t( 0)t( 0)t(w

)t(v)t(v)t(v

lr

=θ⇒=θ⇒=

==

)t(v

L

2

)t(w

0)t(v

R

=

=

2002 - © Pedro Lima, M. Isabel Ribeiro

Robótica Móvel

Kinematics Models

Synchronous drive

• In a synchronous drive robot (synchro 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

•

Steered wheel

– The orientation of the rotation axis can be controlled

y axis

2002 - © Pedro Lima, M. Isabel Ribeiro

Robótica Móvel

Kinematics Models

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

• 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 diretly.

• Control variables (independent)

– v(t), w(t)

• The ICC is always at infinity

• Changing the orientation of the wheels

manipulates the direction of ICC

( )

( )

σσ=θ

σσθσ=

σσθσ=

∫

∫

∫

d)(w)t(

d)(sin)(v)t(y

d)(cos)(v)t(x

t

0

t

0

t

0

2002 - © Pedro Lima, M. Isabel Ribeiro

Robótica Móvel

Kinematics Models

Synchronous Drive

• Particular cases:

– v(t)=0, w(t)=w=cte. during a time interval

•

The robot rotates in place by an amount

– v(t)=v, w(t)=0 during a time interval

•

The robot moves in the direction its pointing a distance

t

∆

t w

∆

t

∆

t v

∆

2002 - © Pedro Lima, M. Isabel Ribeiro

Robótica Móvel

Kinematics Models

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 w

s

(t)

ICC

ICC

The ICC must lie on

the line that passes

through, and is

perpendicular to, the

fixed rear wheels

R

2002 - © Pedro Lima, M. Isabel Ribeiro

Robótica Móvel

Kinematics Models

Tricycle

If the steering wheel is set to

an angle

α

(t) from the

straight-line direction, the

tricycle will rotate with

angular velocity w(t) about a

point lying a distance R along

the line perpendicular to and

passing through the rear

wheels.

d

x

y

θ

α

R

(

)

)t(

2

tg d)t(R

α−

π

=

22

s

)t(Rd

r )t(w

)t(w

+

=

X

b

Y

b

r = steering wheel radius

angular velocity of the moving frame

relative to the base frame

r )t(w)t(v

ss

=

linear velocity of steering wheel

)t(sin

d

)t(v

)t(w

s

α=

2002 - © Pedro Lima, M. Isabel Ribeiro

Robótica Móvel

Kinematics Models

Tricycle

Kinematic model in the robot frame

Kinematic model in the world frame

)t(sin

d

)t(v

)t(

)t(sin)t(cos)t(v)t(y

)t(cos)t(cos)t(v)t(x

s

s

s

α=θ

θα=

θα=

)t(sin

d

)t(v

)t(

0)t(v

)t(cos )t(v)t(v

s

y

sx

α=θ

=

α=

with no splippage

)t(sin

d

)t(v

)t(w

)t(cos)t(v)t(v

s

s

α=

α=

θ

θ

=

θ

)t(w

)t(v

10

0)t(sin

0)t(cos

)t(

)t(y

)t(x

2002 - © Pedro Lima, M. Isabel Ribeiro

Robótica Móvel

Kinematics Models

Omnidireccional

X

m

Y

f

θ

30º

1

2

3

L

Y

m

X

f

−

−

=

θ

3

2

1

y

x

w

w

w

L3

r

L3

r

L3

r

r

3

1

r

3

1

r

3

2

r

3

1

r

3

1

0

V

V

Kinematic model in the robot frame

w

1

, w

2

, w

3

–

angular

velocities of the three

swedish wheels

Swedish wheel

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