3D Kinematics

taupeselectionMechanics

Nov 14, 2013 (3 years and 9 months ago)

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3D Kinematics

Eric Whitman

1/24/2010

Rigid Body State: 2D

p


x
ˆ
y
ˆ
Rigid Body State: 3D

p

x
ˆ
y
ˆ
z
ˆ
Add a Reference Frame

p

'
ˆ
x
'
ˆ
y
'
ˆ
z
x
ˆ
y
ˆ
z
ˆ
Rotation Matrix


Linear Algebra definition


Orthogonal matrix: R
-
1

= R
T


square


d
et
(R) = 1


2D: 4 numbers


3D: 9 numbers

Unit Vectors

p

z
z
y
z
x
z
z
z
y
y
y
x
y
y
z
x
y
x
x
x
x
z
y
x
z
y
x
z
y
x
ˆ
'
ˆ
'
ˆ
'
'
ˆ
ˆ
'
ˆ
'
ˆ
'
'
ˆ
ˆ
'
ˆ
'
ˆ
'
'
ˆ




















z
z
z
y
y
y
x
x
x
z
y
x
z
y
x
z
y
x
R
'
'
'
'
'
'
'
'
'
'
ˆ
x
'
ˆ
y
'
ˆ
z
x
ˆ
y
ˆ
z
ˆ
Using the Rotation Matrix

p

'
ˆ
x
'
ˆ
y
'
ˆ
z
x
ˆ
y
ˆ
z
ˆ
A

a

a
R
p
A





Pros

and
Cons


Rotates Vectors Directly


Easy composition


9 numbers


Difficult to enforce
constraints

Simple Rotation Matrices









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





R
2D

3
D





































)
cos(
)
sin(
0
)
sin(
)
cos(
0
0
0
1
)
(
)
cos(
0
)
sin(
0
1
0
)
sin(
0
)
cos(
)
(
1
0
0
0
)
cos(
)
sin(
0
)
sin(
)
cos(
)
(















x
y
z
R
R
R
Degrees of Freedom


2D


2x2 matrix has 4
numbers


Only one
DoF


3D


3x3 matrix has 9
numbers


6 constraints


3
DoF

Euler Angle Combinations


Can use body or world coordinates


2 consecutive angles must be different


Can alternate (3
-
1
-
3) or be all different (3
-
1
-
2)


24 possibilities (12 pairs of equivalent)


For aircraft, 3
-
2
-
1 body is common


Yaw, pitch, roll


For spacecraft, 3
-
1
-
3 body is common


Construct a Rotation Matrix



















































c
c
s
s
s
s
c
c
c
c
s
s
s
c
c
c
s
s
s
c
c
s
s
c
s
c
s
c
c
R
)
,
,
(
3
-
1
-
3 Body Convention


Common for spacecraft

Recover Euler Angles



















































c
c
s
s
s
s
c
c
c
c
s
s
s
c
c
c
s
s
s
c
c
s
s
c
s
c
s
c
c
R
)
,
,
(
)
arctan(
)
arctan(
)
arccos(
23
,
13
32
,
31
33
R
R
R
R
R







Gimbal

Lock


Physically: two
gimbal

axes line up, making
movement in one direction impossible


Mathematically describes a singularity in Euler
angle systems


For the 3
-
1
-
3 body convention, this occurs
when angle 2 equals 0 or pi


For the 3
-
1
-
2 body convention, this occurs
when angle 2 +/
-

pi/2


Switching helps


Pros

and
Cons


Minimal Representation


Human readable


Gimbal

Lock


Must convert to RM to
rotate a vector


No easy composition

Axis Angle (4 numbers)


A special case of Euler’s Rotation Theorem:


any combination of rotations can be
represented as a single rotation


3 numbers to represent the axis of rotation


1 number to represent the angle of rotation


Has singularity for small rotations




,
a

Rotation Vector (3 numbers)


The axis can be a unit vector (only 2
DoF
)


Multiply axis by angle of rotation


Can easily extract axis angle


Axis = rotation vector


Normalize if desired


Angle = ||rotation vector||


Same singularity


small rotations



a
ˆ

Pros

and
Cons


Minimal Representation


Human readable (sort
of)


Singularity for small
rotations


Must convert to RM to
rotate a vector


No easy composition

(Unit)
Quaternions


All schemes with 3 numbers will have a
singularity


So says math (topology)

)
2
/
cos(
ˆ
)
2
/
sin(
4
3
2
1
4
4
3
2
1



































q
a
q
q
q
q
q
q
q
q
q
q
q
Constraint


Easy to enforce

1
2
4
2
3
2
2
2
1





q
q
q
q
q
q
T
Conversion with RM





























2
4
2
3
2
2
2
1
1
4
2
3
2
4
1
3
1
4
3
2
2
4
2
3
2
2
2
1
3
4
1
2
2
4
3
1
3
4
2
1
2
4
2
3
2
2
2
1
)
(
2
)
(
2
)
(
2
)
(
2
)
(
2
)
(
2
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
R
)
(
4
1
)
(
4
1
)
(
4
1
1
2
1
21
12
4
3
13
31
4
2
32
23
4
1
33
22
11
4
R
R
q
q
R
R
q
q
R
R
q
q
R
R
R
q










Composition













q
q
q
q
q
q
'
'
'
'
'
'
4
4
4
4
q
q
q
q
q
q
Pros

and
Cons


No Singularity


Almost minimal
representation


Easy to enforce
constraint


Easy composition


Interpolation possible


Not quite minimal


Somewhat confusing

Summary of Rotation Representations


Need rotation matrix to rotate vectors


Often more convenient to use something else
and convert to rotation matrix


Euler angles good for small angular deviations


Quaternions

good for free rotation

Homogeneous Transformations

1
0
1
0
1
0
p
R
o
p


Define
:














1
~
p
p













1
0
0
1
0
1
0
1
T
o
R
A
Composition

n
n
n
p
A
A
A
p
1
1
2
0
1
0
...
~