# 3D Kinematics

Mechanics

Nov 14, 2013 (4 years and 5 months ago)

116 views

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

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

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

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
...
~