3d kinematics eric whitman 1/24/2010. rigid body state: 2d p
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Rotation Matrix
• Linear Algebra definition– Orthogonal matrix: R-1 = RT
• square
– det(R) = 1• 2D: 4 numbers• 3D: 9 numbers
Unit Vectors
p
zzyzxzz
zyyyxyy
zxyxxxx
zyx
zyx
zyx
ˆ'ˆ'ˆ''ˆ
ˆ'ˆ'ˆ''ˆ
ˆ'ˆ'ˆ''ˆ
zzz
yyy
xxx
zyx
zyx
zyx
R
'''
'''
'''
'x
'y
'z
x
y
z
Pros and Cons
• Rotates Vectors Directly• Easy composition
• 9 numbers• Difficult to enforce
constraints
Simple Rotation Matrices
)cos()sin(
)sin()cos()(
R
2D 3D
)cos()sin(0
)sin()cos(0
001
)(
)cos(0)sin(
010
)sin(0)cos(
)(
100
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
ccsss
sccccssscccs
ssccsscscscc
R ),,(
3-1-3 Body Convention – Common for spacecraft
Recover Euler Angles
ccsss
sccccssscccs
ssccsscscscc
R ),,(
)arctan(
)arctan(
)arccos(
23,13
32,31
33
RR
RR
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
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
Conversion with RM
24
23
22
2114232413
143224
23
22
213412
2431342124
23
22
21
)(2)(2
)(2)(2
)(2)(2
qqqqqqqqqqqq
qqqqqqqqqqqq
qqqqqqqqqqqq
R
)(4
1
)(4
1
)(4
1
12
1
21124
3
13314
2
32234
1
3322114
RRq
q
RRq
q
RRq
q
RRRq
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
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