1

Wheeled Locomotion

Introduction to Mobile Robotics

Wolfram Burgard, Michael Ruhnke, Bastian

Steder

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) XR4000 Mecanum wheels

y

ro ll

z m o tio n

x

y

we also allow wheels to rotate around the z axis

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 w

(x,y)

y

l/2

q x

v l

v r

w(R+ l / 2) = vr

w(R- l / 2) = vl

R =l

2

(vl + vr)

(vr - vl)

w =vr - vl

l

v =vr + vl

2

]cos,sin[ICC qq RyRx

5

Differential Drive: Forward Kinematics

ICC

R

P(t)

P(t+dt)

t

y

x

tt

tt

y

x

y

x

y

x

wdq

wdwd

wdwd

q

ICC

ICC

ICC

ICC

100

0)cos()sin(

0)sin()cos(

'

'

'

')'()(

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

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

0

0

0

t

t

t

dttt

dtttvty

dtttvtx

wq

q

q

6

Differential Drive: Forward Kinematics

ICC

R

P(t)

P(t+dt)

t

y

x

tt

tt

y

x

y

x

y

x

wdq

wdwd

wdwd

q

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

q

q

q

7

qq

tan

]cos,sin[ICC

dR

RyRx

Ackermann Drive

R

ICC

(x,y)

y

l/2

q x

v l

v r l

vv

vv

vvlR

vlR

vlR

lr

lr

rl

l

r

w

w

w

)(

)(

2

)2/(

)2/(w

d

8

Synchronous Drive

q

y

x

v(t)

w( ) t

')'()(

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

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

0

0

0

t

t

t

dttt

dtttvty

dtttvtx

wq

q

q

9

XR4000 Drive

q

y

x

vi(t)

wi(t)

')'()(

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

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

0

0

0

t

t

t

dttt

dtttvty

dtttvtx

wq

q

q

ICC

11

Mecanum Wheels

4

4

4

4

3210

3210

3210

3210

/)vvvv(v

/)vvvv(v

/)vvvv(v

/)vvvv(v

error

x

y

q

The Kuka OmniRob Platform

15

Example: KUKA youBot

17

Tracked Vehicles

18

Other Robots: OmniTread

[courtesy by Johann Borenstein]

20

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 (they have no non-holonomic constraints).

21

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.

E.g., a train on tracks (not all positions and orientations are possible)

22

Drives with Non-Holonomic Constraints

Synchro-drive

Differential drive

Ackermann drive

23

Drives without Non-Holonomic Constraints

XR4000 drive

Mecanum wheels

24

Dead Reckoning and Odometry

Estimating the motion based on the issued controls/wheel encoder readings

Integrated over time

25

Summary

Introduced different types of drives for wheeled robots

Math to describe the motion of the basic drives given the speed of the wheels

Non-holonomic constraints

Odometry and dead reckoning