ASL Autonomous Systems Lab

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Kinematics | Basics of Rigid Body Kinematics

Autonomous Mobile Robots

Marco Hutter

Margarita Chli, Paul Furgale, Martin Rufli, Davide Scaramuzza, Roland Siegwart

ASL Autonomous Systems Lab

Definition of kinematics

“Description of the motion of points, bodies, or systems of bodies”

… without consideration of the causes of motion (=> dynamics)

Required for kinematic simulation and control

Types of motion of single bodies

Translation

Rotation

Combined motion

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Introduction

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Reference frames

Coordinate system I (inertial, not moving)

Coordinate system B (body-fixed, moving)

Translation

Vector from O to P, expressed in I:

Rotation

Matrix from frame B to frame I:

Homogeneous transformation

Combination of translation and rotation

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Vectors and coordinate transformation

I OP I OB I BP r r r1 0 1 1

I OP IB I OB B BP

r R r rI OB IB B BP r R r

3 1

I OP

r

3 3

IB

R

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Rotation from frame B to I such that

Inversion

Composition of rotations

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Rotation

, C CI I CI CB BI r R r R R R

, , T

I IB B B BI I BI IB r R r r R r R ROrientation in 3D

Euler angles (z-x-z)

Tait-Bryan/Cardan angles (z-y-x)

cos sin 0

sin cos 0

0 0 1

IB

R

I IB Br R r

ASL Autonomous Systems Lab

Reference frames

Coordinate system I (inertial, not moving)

Coordinate system B (body-fixed, moving)

Translation

vector from O to P, expressed in I:

Rotation

matrix from frame B to frame I:

Homogeneous transformation

Combination of translation and rotation

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Vectors and coordinate transformation

3 1

I P I OP

v r

3 1

I IB

ω

I P I OP I OB I IB I BP v r r ω r

3 1

I OP

r

3 3

IB

R

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Angular velocity of frame B with respect to I expressed in I

Relation to rotation matrix

Transformation

Consecutive rotations

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Angular velocity

I IC I IB I BC ω ω ω

0

0I IB

ω

I IB IB B IBω R ω

0

ˆ ˆ, 0 ,

0

z y x

IB IB IB

T z x y

I IB IB IB IB IB IB IB IB

y x z

IB IB IB

ω R R ω ω

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Position and velocity in different frames

Inertial frame

Moving frame

Example: single pendulum,

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Vector differentiation

II I

d

dt

rr r

BB B IB BB

d

dt

rr r ω r

sin

cos

0

I OP

l

l

r

0 0 0

0 0 0

0 0 0

B OP

B P B IB B OP

ld

ldt

rv ω r

?B P v

cos

sin

0

I OP

I P

ld

ldt

rv

cos sin 0 cos

sin cos 0 sin 0

0 0 1 0 0

B P BI I P

l l

l

v R v

0

0

B OP l

r

ASL Autonomous Systems Lab

We learned the basics for describing the motion

Translations

Rotations

Homogeneous transformation

Anglar velocities

Differentiation of (position) vectors

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Recapitulation

1 1I OP I OB I BP r r r

1 1B OP BI I BPr R r

1 1

0 11 1

I OP B BPIB I OB

r rR r

BB B IB BB

d

dt

rr r ω r

I IC I IB I BC ω ω ω