Introduction to Mobile Robotics Wheeled Locomotion Wolfram - - PowerPoint PPT Presentation

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Wolfram Burgard, Cyrill Stachniss, Maren Bennewitz, Kai Arras

Wheeled Locomotion Introduction to Mobile Robotics

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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 to rotate around the z axis

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Instantaneous Center of Curvature

ICC § For rolling motion to occur, each wheel has to move along its y-axis

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Differential Drive

R

ICC ω

(x,y)

y l/2

θ

x v

l

v

r

l v v v v v v l R v l R v l R

l r l r r l l r

− = − + = = − = + ω ω ω ) ( ) ( 2 ) 2 / ( ) 2 / (

] cos , sin [ ICC θ θ R y R x + − =

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Differential Drive: Forward Kinematics

ICC R P(t) P(t+δt)

⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ + ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ − − ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ − = ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ t y x t t t t y x

y x y x

ωδ θ ωδ ωδ ωδ ωδ θ ICC ICC ICC ICC 1 ) cos( ) sin( ) sin( ) cos( ' ' '

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

∫ ∫ ∫

= = =

t t t

dt t t dt t t v t y dt t t v t x ω θ θ θ

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Differential Drive: Forward Kinematics

ICC R P(t) P(t+δt)

⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ + ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ − − ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ − = ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ t y x t t t t y x

y x y x

ωδ θ ωδ ωδ ωδ ωδ θ ICC ICC ICC ICC 1 ) cos( ) sin( ) sin( ) cos( ' ' ' ' )] ' ( ) ' ( [ 1 ) ( ' )] ' ( sin[ )] ' ( ) ' ( [ 2 1 ) ( ' )] ' ( cos[ )] ' ( ) ' ( [ 2 1 ) (

∫ ∫ ∫

− = + = + =

t l r t l r t l r

dt t v t v l t dt t t v t v t y dt t t v t v t x θ θ θ

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ϕ θ θ tan ] cos , sin [ ICC d R R y R x = + − =

Ackermann Drive

R

ICC

(x,y)

y l/2

θ

x v

l

v

r

l v v v v v v l R v l R v l R

l r l r r l l r

− = − + = = − = + ω ω ω ) ( ) ( 2 ) 2 / ( ) 2 / (

ω

ϕ

d

ϕ

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Synchonous Drive

θ

y x

v(t) ω( ) t

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

∫ ∫ ∫

= = =

t t t

dt t t dt t t v t y dt t t v t x ω θ θ θ

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XR4000 Drive

θ

y x

vi(t) ωi(t)

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

∫ ∫ ∫

= = =

t t t

dt t t dt t t v t y dt t t v t x ω θ θ θ

ICC

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Mecanum Wheels

4 4 4 4

3 2 1 3 2 1 3 2 1 3 2 1

/ ) v v v v ( v / ) v v v v ( v / ) v v v v ( v / ) v v v v ( v

error x y

+ − − = − − + = − + − = + + + =

θ

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Example: Priamos (Karlsruhe)

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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 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).

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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 car and a trailer (not all angles between car and trailer are possible)

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Drives with Non-Holonomic Constraints

§ Synchro-drive § Differential drive § Ackerman drive

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Drives without Non-Holonomic Constraints

§ XR4000 drive § Mechanum wheels

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Dead Reckoning and Odometry

§ Estimating the motion based on the issues

controls/wheel encoder readings

§ Integrated over time

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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