Rigid body transformations
Intelligent Robotics
2014/15
Chris Burbridge
Euclidean Space
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N-dimensional Euclidean space with n-dimensional vectors or points.
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Vector.
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–
Expresses a relative displacement, change or direction.
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The sum and the difference of two vectors is a vector.
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Denoted by line segment connecting two points.
Point.
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Expresses an absolute location in some reference frame.
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The sum is undefined, the difference is a vector.
Rigid Bodies
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Point p can move on a trajectory represented by a
parametrised curve p(t) = (x(t), y(t)).
A rigid body is a collection of points (particles) which
mutual distances are fixed regardless of trajectories they
can move on.
For any pair of points p and q belonging to a rigid body:
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ǁp(0) – q(0)ǁ = ǁp(t) – q(t)ǁ, for any t.
Rotation Matrix
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Y
Rotate point p = (x, y) by angle θ.
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p’=(x’, y’)
From trigonometry:
p=(x, y)
θ
–
Equivalently in a matrix form:
X
Transforming a point
Y
–
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Rotation is anticlockwise by θ
Y'
X'
Rotate a coordinate frame by angle θ:
θ
y'
x
p=(x, y)
y p’=(x’, y’)
x'
X
Transforming coordinate frames
Rotation Matrix Properties
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Rotation matrices form a group with multiplication operation ◦ :
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Closure: If R1, R2 are rotation matrices R1◦R2 is also a rotation matrix.
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Identity: For each rotation matrix R, there exists an identity element I, such that
R◦I = I◦R.
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Inverse: For each rotation matrix R, there exists a (unique) inverse, R-1 such that
R◦R-1 = R-1◦R = I. Inverse is just the transpose!
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Associativity: For any rotation matrices R1, R2, R3 : (R1◦R2)◦R3 = R1◦(R2◦R3).
Other properties:
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Columns of R are mutually orthonormal.
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Not commutative R1◦R2 ǂ R2◦R1
Rotation Matrix Example
Rigid Body Transformations
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A rigid body transformation g is a mapping from R2 to R2 (or in
3D R3) which represents a rigid motion and preserves:
Distance between any points p and q:
∥g ( p) – g (q)∥=∥ p – q∥
Cross product of any vectors v and w:
g (v×w)= g (v )× g (w )
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It is a series of rotations and translations.
Homogeneous Representation
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Rotating, translating, rotating, translating....
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It is easier to transform, transform, transform..
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Can represent a two-step rotation & translation
as a single-step matrix multiplication using
homogeneous co-ordinates
Homogeneous Representation
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Any point p from R2 has a corresponding R3
vector:
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For any point p, the transformation
p’= g(p) = Rp + v can be rewritten as a 3x3
matrix M:
Moving to 3D
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Translation in R3 is no more complicated
Rotation in R3 is the same, but now we have a
rotation about an x, y, z axis
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Euler angles; Tait-bryan angles
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Which order, which axis?
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Often (and in ROS), Quaternion's
are used instead of rotation
matrices.
Quaternions for 3D Rotation
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A unit quaternion Q=[q x , q y , q z , q w ] represents a
rotation in 3D
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Can be converted to a rotation matrix
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Simplest to think of in terms of angle axis:
Q=[ x .sin (α/2) , y .sin (α/2) , z .sin (α/2) , cos(a/ 2)]
Is a quaternion representing a rotation α about the axis [x, y, z]
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For example, a 90 degree rotation about the z
axis:
Q=[0 .sin (90/ 2) , 0 . sin (90/ 2) ,1 .sin (90/2) , cos(90/2)]
=[0, 0, 0.707,0.707]
Quaternion Product
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Combining two rotations Q1 & Q2 to find an
overall rotation quaternion Q3:
[
w
x
x
w
y
z
z
y
Q 1 . Q 2 + Q 1 .Q 2 + Q 1 . Q 2 − Q 1 .Q 2
w
y
x
z
y
w
z
x
Q
.Q
−
Q
.Q
+
Q
.Q
−
Q
.Q
1
2
1
2
1
2
Q 3= 1w 2z
x
y
y
x
z
w
Q 1 . Q 2 − Q 1 .Q 2 + Q 1 .Q 2 + Q 1 .Q 2
Q 1w . Q w2 + Q 1x .Q 2x − Q 1y .Q 2y + Q 1z . Q 2z
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]
Quaternion products are not commutative
Why Quaternions?
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Smooth interpolation
Computationally efficient combination of
rotations
Less floating point rounding problems
Summary
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Rigid body transformations are important in
many areas of robotics
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Quaternions are nifty
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Reading:
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Mathematics for Computer Graphics, J. A. Vince,
ISBN 978-1-84996-022-9
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www.euclideanspace.com
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ROS TF Library: http://www.ros.org/wiki/tf/
including ROS tf.transformations python code.