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Wheeled Locomotion

Introduction toMobile Robotics

Wolfram Burgard, Cyrill Stachniss, Maren

Bennewitz, Diego Tipaldi, Luciano Spinello

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Locomotion of Wheeled Robots

Locomotion (Oxford Dict.):Power of motion from place to place

! Differential drive (AmigoBot, Pioneer 2-DX)! Car drive (Ackerman steering)! Synchronous drive (B21)! XR4000! Mecanum wheels

y

roll

z motion

x

y

we also allow wheels torotate around the z axis

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Instantaneous Center of Curvature

ICC

! For rolling motion to occur, each wheel has tomove along its y-axis

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Differential Drive

R

ICC !

(x,y)

y

l /2

" x

v l

v r

! ( R + l / 2) = v r

! ( R ! l / 2) = v l

R =l

2

(v l + v r )

(v r ! v l )

! =v r ! v l

l

v =v r + v l

2

]cos,sin[ICC ! ! R y R x +"=

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Differential Drive: Forward Kinematics

ICC

R

P(t)

P(t+ #t)

!!!

"

#

$$$

%

&+

!!!

"

#

$$$

%

&''

!!!

"

#

$$$

%

& '=

!!!

"

#

$$$

%

&

t

y x

t t t t

y x

y

x

y

x

() *

() () () ()

*

ICCICC

ICCICC

100

0)cos()sin(0)sin()cos(

'

''

')'()(

')]'(sin[)'()(

')]'(cos[)'()(

0

0

0

! !

!

=

=

=

t

t

t

dt t t

dt t t vt y

dt t t vt x

" #

#

#

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Differential Drive: Forward Kinematics

ICC

R

P(t)

P(t+ #t)

!!!

"

#

$$$

%

&+

!!!

"

#

$$$

%

&''

!!!

"

#

$$$

%

& '=

!!!

"

#

$$$

%

&

t

y x

t t t t

y x

y

x

y

x

() *

() () () ()

*

ICCICC

ICCICC

100

0)cos()sin(0)sin()cos(

'

''

')]'()'([1

)(

')]'(sin[)]'()'([21

)(

')]'(cos[)]'()'([21)(

0

0

0

!

!

!

"=

+=

+=

t

l r

t

l r

t l r

dt t vt vl

t

dt t t vt vt y

dt t t vt vt x

#

#

#

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!

" "

tan

]cos,sin[ICC

d R

R y R x

=

+#=Ackermann Drive

R

ICC

(x,y)

y

l /2

" x

v l

v r l vv

vv

vvl R

vl R

vl R

l r

l r

r l

l

r

!=

!+

=

=!

=+

"

"

"

)(

)(

2

)2/(

)2/(! $

d

$

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Synchronous Drive

"

y

x

v(t)

! ( ) t ')'()(

')]'(sin[)'()(

')]'(cos[)'()(

0

0

0

!

!

!

=

=

=

t

t

t

dt t t

dt t t vt y

dt t t vt x

" #

#

#

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XR4000 Drive

"

y

x

vi(t) ! i(t)

')'()(

')]'(sin[)'()(

')]'(cos[)'()(

0

0

0

!

!

!

=

=

=

t

t

t

dt t t

dt t t vt y

dt t t vt x

" #

#

#

ICC

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XR4000

[courtesy by Oliver Brock & Oussama Khatib]

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Mecanum Wheels

4

4

4

4

3210

3210

3210

3210

/ )vvvv( v

/ )vvvv( v / )vvvv( v

/ )vvvv( v

error

x

y

+!!=

!!+=

!+!=

+++=

"

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Example: Priamos (Karlsruhe)

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Example

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Example: KUKA youBot

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Example: Segway Omni

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Tracked Vehicles

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Other Robots: OmniTread

[courtesy by Johann Borenstein]

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Other Robots: Humanoids

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Non-Holonomic Constraints!

Non-holonomic constraints limit the possibleincremental movements within theconfiguration space of the robot.

! Robots with differential drive or synchro-drive move on a circular trajectory andcannot move sideways.

! XR-4000 or Mecanum-wheeled robots canmove sideways (they have no non-holonomic constraints).

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Drives with Non-HolonomicConstraints

! Synchro-drive! Differential drive! Ackermann drive

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Drives without Non-HolonomicConstraints

! XR4000 drive! Mecanum wheels

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Dead Reckoning and Odometry!

Estimating the motion based on the issuedcontrols/wheel encoder readings! Integrated over time

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Summary!

Introduced different types of drives forwheeled robots! Math to describe the motion of the basic

drives given the speed of the wheels! Non-holonomic Constraints! Odometry and dead reckoning