SA-1SA-1

Probabilistic Robotics

Wheeled Locomotion

2

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)• Mecanum wheels, XR4000

y

roll

z motion

x

y

3

Instantaneous Center of Curvature

ICC

• For rolling motion to occur, each wheel has to move along its y-axis

4

Differential Drive

R

ICC

(x,y)

y

l/2

x

vl

vr

l

vv

vv

vvlR

vlR

vlR

lr

lr

rl

l

r

)(

)(

2

)2/(

)2/(

]cos,sin[ICC RyRx

5

Differential Drive: Forward Kinematics

ICC

R

P(t)

P(t+t)

t

y

x

tt

tt

y

x

y

x

y

x

ICC

ICC

ICC

ICC

100

0)cos()sin(

0)sin()cos(

'

'

'

')'()(

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

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

0

0

0

t

t

t

dttt

dtttvty

dtttvtx

6

Differential Drive: Forward Kinematics

ICC

R

P(t)

P(t+t)

t

y

x

tt

tt

y

x

y

x

y

x

ICC

ICC

ICC

ICC

100

0)cos()sin(

0)sin()cos(

'

'

'

')]'()'([1

)(

')]'(sin[)]'()'([2

1)(

')]'(cos[)]'()'([2

1)(

0

0

0

t

lr

t

lr

t

lr

dttvtvl

t

dtttvtvty

dtttvtvtx

7

tan

]cos,sin[ICC

dR

RyRx

Ackermann Drive

R

ICC

(x,y)

y

l/2

x

vl

vr

l

vv

vv

vvlR

vlR

vlR

lr

lr

rl

l

r

)(

)(

2

)2/(

)2/(

d

8

Synchonous Drive

y

x

v(t)

t')'()(

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

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

0

0

0

t

t

t

dttt

dtttvty

dtttvtx

9

Synchro-Drive Robot

10

XR4000 Drive

y

x

vi(t)

i(t)

')'()(

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

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

0

0

0

t

t

t

dttt

dtttvty

dtttvtx

ICC

11

XR4000

[courtesy by Oliver Brock & Oussama Khatib]

12

Mecanum Wheels

4

4

4

4

3210

3210

3210

3210

/)vvvv(v

/)vvvv(v

/)vvvv(v

/)vvvv(v

error

x

y

13

Example: Priamos (Karlsruhe)

14

Example

15

Odometry

16

Non-Holonomic Constraints

•Non-holonomic constraints limit the possible incremental movements within the configuration space of the robot.

•Robots with differential drive or synchro-drive move on a circular trajectory and cannot move sideways.

•XR-4000 or Mecanum-wheeled robots can move sideways.

17

Holonomic vs. Non-Holonomic

•Non-holonomic constraints reduce the control space with respect to the current configuration (e.g., moving sideways is impossible).

•Holonomic constraints reduce the configuration space.