Rigid Body Transformations
Two distinct positions and orientations of the same rigid body
1
Reference Frames
We associate with any position and orientation a reference frame
In reference frame {A}, we can find three linearly
independent vectors a1, a2, and a3 that are basis vectors.
We can write any vector as a linear
combination of the basis vectors in either
frame.
v = v1 a1 + v2 a2 + v3a3
a3
a2
b3
a1
A
b2
b1
B
2
Notation
Vectors
● x,
y, a, …
Reference Frames
● Ax
● u,
Potential for Confusion!
v, p, q, …
● A,
B, C, …
● a, b, c, …
Matrices
Transformations
● A,
● AAB ARB AξB
B, C, …
● Aab
Rab
● gab, hab, …
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Rigid Body Displacement
Object
O ⊂ R3
R3
t
Rigid Body Displacement
€
Map
g : O → R3
O
Rigid Body Motion
Continuous family of maps
g (t ) : O → R
3
4
Example
5
Rigid Body Displacement
A displacement is a transformations of points
● Transformation
(g) of points induces an action
(g*) on vectors
g
p
v
q
g (p)
g* (v)
g (q)
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What makes g a rigid body
displacement?
g
v
p
q
g (p)
g* (v)
g (q)
kg(p)
g(q)k = kp
qk
1. Lengths are preserved
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What makes g a rigid body
displacement?
g
p
v
w
r
g (p)
q
g* (w)
g* (v)
g (q)
g? (v) ⇥ g? (w) = g? (v ⇥ w)
2. Cross products are preserved
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g is a rigid body displacement
a3
a2
b3
b1
a1
b2
mutually orthogonal unit
vectors get mapped to
mutually orthogonal unit
vectors
You should be able to prove
- orthogonal vectors are mapped to orthogonal vectors
- g* preserves inner products
g? (v) · g? (w) = g? (v · w)
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Summary
Rigid body displacements are
transformations (maps) that satisfy two
important properties
1. The map preserves lengths
2. Cross products are preserved by
the induced map
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Note
Rigid body displacements and rigid body
transformations are used interchangeably
1. Transformations generally used to describe relationship
between reference frames attached to different rigid bodies.
2. Displacements describe relationships between two positions
and orientation of a frame attached to a displaced rigid body
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g is a rigid body displacement
g
a3
mutually
orthogonal,
unit vectors
a2
a1
{A}
b1 = R11 a1 + R12 a2 + R13 a3
b2 = R21 a1 + R22 a2 + R23 a3
b3 = R31 a1 + R32 a2 + R33 a3
b3
b1
{B}
mutually
orthogonal,
unit vectors
b2
2
R11
R = 4R21
R31
R12
R22
R32
3
R13
R23 5
R33
rotation matrix
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Properties of a Rotation Matrix
● Orthogonal
▼ Matrix times its transpose equals the identity
● Special orthogonal
▼ Determinant is +1
● Closed under multiplication
▼ The product of any two rotation matrices is another rotation matrix
● The
inverse of a rotation matrix is also a rotation
matrix
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g is a rigid body displacement
g
a3
Q
b3
{A}
P is a reference point
fixed to the object, Q is
a generic point attached
to the object.
P
a2
b1
Q
P
a1
!
P Q = q1 a1 + q2 a2 + q3 a3
{B}
b2
!
P Q = q1 b1 + q2 b2 + q3 b3
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g is a rigid body displacement
a3
Q’
{A}
P
a1
2 03 2
?
q1
4q20 5 = 4?
?
q30
Q
?
?
?
a2
!
P Q = q1 a1 + q2 a2 + q3 a3
!0
P Q = q10 a1 + q20 a2 + q30 a3
32 3 2 3 2
q1
?
?
q1
?5 4q2 5 or 4q2 5 = 4?
q3
?
?
q3
?
?
?
32 03
?
q1
?5 4q20 5
?
q30
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g is a rigid body displacement
a3
{A}
Q
!
P Q = q1 a1 + q2 a2 + q3 a3
!0
P Q = q10 a1 + q20 a2 + q30 a3
Q’
P
a2
a1
Verify
2 3 2
q1
R11
4q2 5 = 4R21
q3
R31
R12
R22
R32
32 03
R13
q1
R23 5 4q20 5
R33
q30
rotation matrix
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Example: Rotation
● Rotation
about the x-axis through θ
a3
" 1
0
$
Rot ( x, θ ) = $ 0 cosθ
$ 0 sin θ
#
z
z
a2
0
−sin θ
cosθ
%
'
'
'
&
θ
y
y
x
a1
x
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Example: Rotation
Rotation about the y-axis through θ
" cosθ
$
Rot ( y, θ ) = $ 0
$ −sin θ
#
x'
0 sin θ
1
0
0 cosθ
" cosθ
$
Rot ( z, θ ) = $ sin θ
$ 0
#
x
0
0
1
−sin θ
cosθ
0
%
'
'
'
&
x
z'
θ
y
%
'
'
'
&
Rotation about the z-axis through θ
x'
θ
z
z
y
y'
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