OutlineCoordinate Transformations

EE 451 - Coordinate Transformations

H.I. Bozma

Electric Electronic EngineeringBogazici University

September 21, 2016

H.I. Bozma EE 451 - Coordinate Transformations

OutlineCoordinate Transformations

Coordinate TransformationsCoordinate SystemsRotationsSO(n) RotationsComposition of RotationsParametrization of RotationsRigid Motions

H.I. Bozma EE 451 - Coordinate Transformations

OutlineCoordinate Transformations

Coordinate SystemsRotationsSO(n) RotationsComposition of RotationsParametrization of RotationsRigid Motions

Coordinate System

◮ Origin (A single point in space)

◮ 2 (2D) or 3 (3D) orthogonal coordinate axes

H.I. Bozma EE 451 - Coordinate Transformations

OutlineCoordinate Transformations

Coordinate SystemsRotationsSO(n) RotationsComposition of RotationsParametrization of RotationsRigid Motions

Coordinate Transformations

Kinematics → Coordinate frames to representpositions/orientations of robot joints and transformations amongcoordinate frames.

H.I. Bozma EE 451 - Coordinate Transformations

OutlineCoordinate Transformations

Coordinate SystemsRotationsSO(n) RotationsComposition of RotationsParametrization of RotationsRigid Motions

Euclidean Space

H.I. Bozma EE 451 - Coordinate Transformations

OutlineCoordinate Transformations

Coordinate SystemsRotationsSO(n) RotationsComposition of RotationsParametrization of RotationsRigid Motions

Points vs Vectors

◮ Point: A specific location (Independent of coordinate system)

◮ Coordinate vector in a coordinate system that is assigned torepresent the point

◮ Vector: A direction and magnitude◮ Free vector – Not constrained to be located at any position◮ v1 = v2 iff ‖v1| = ‖v3| and direction(v1)=direction(v2)

H.I. Bozma EE 451 - Coordinate Transformations

OutlineCoordinate Transformations

Coordinate SystemsRotationsSO(n) RotationsComposition of RotationsParametrization of RotationsRigid Motions

Rotational Transformations

◮ n × n orthonormal matrices R◮ Column vectors - of unit length◮ Column vectors - mutually orthogonal◮ Projection technique

◮ SO(n) - Special Orthogonal group of order n

H.I. Bozma EE 451 - Coordinate Transformations

OutlineCoordinate Transformations

Coordinate SystemsRotationsSO(n) RotationsComposition of RotationsParametrization of RotationsRigid Motions

Additional Properties of SO(n)

For any R ∈ SO(n), the following hold true:

◮ det R = 1

◮ R−1 = RT

H.I. Bozma EE 451 - Coordinate Transformations

OutlineCoordinate Transformations

Coordinate SystemsRotationsSO(n) RotationsComposition of RotationsParametrization of RotationsRigid Motions

Rotational Transformations

R ∈ SO(3) or R ∈ SO(2)

◮ Case 1: A coordinate transformation relating the coordinatesof a point p in two different coordinate systems

◮ Case 2: Orientation of a transformed coordinate wrt a fixedcoordinate system

◮ Case 3: An operator taking a vector and rotating it to a newvector in the same coordinate system

H.I. Bozma EE 451 - Coordinate Transformations

OutlineCoordinate Transformations

Coordinate SystemsRotationsSO(n) RotationsComposition of RotationsParametrization of RotationsRigid Motions

SO(2) Rotations

H.I. Bozma EE 451 - Coordinate Transformations

OutlineCoordinate Transformations

Coordinate SystemsRotationsSO(n) RotationsComposition of RotationsParametrization of RotationsRigid Motions

SO(3) Examples - Rotation around z-axis

H.I. Bozma EE 451 - Coordinate Transformations

OutlineCoordinate Transformations

Coordinate SystemsRotationsSO(n) RotationsComposition of RotationsParametrization of RotationsRigid Motions

Relative Orientations

H.I. Bozma EE 451 - Coordinate Transformations

OutlineCoordinate Transformations

Coordinate SystemsRotationsSO(n) RotationsComposition of RotationsParametrization of RotationsRigid Motions

Rotational Transformations

H.I. Bozma EE 451 - Coordinate Transformations

OutlineCoordinate Transformations

Coordinate SystemsRotationsSO(n) RotationsComposition of RotationsParametrization of RotationsRigid Motions

Representation of Purely Rotational Rigid Motion

H.I. Bozma EE 451 - Coordinate Transformations

OutlineCoordinate Transformations

Coordinate SystemsRotationsSO(n) RotationsComposition of RotationsParametrization of RotationsRigid Motions

Rotating a vector – Case 3

H.I. Bozma EE 451 - Coordinate Transformations

OutlineCoordinate Transformations

Coordinate SystemsRotationsSO(n) RotationsComposition of RotationsParametrization of RotationsRigid Motions

Composition of rotations

◮ Current axis

◮ Fixed axis

H.I. Bozma EE 451 - Coordinate Transformations

OutlineCoordinate Transformations

Coordinate SystemsRotationsSO(n) RotationsComposition of RotationsParametrization of RotationsRigid Motions

Composition of rotations around current axes

H.I. Bozma EE 451 - Coordinate Transformations

OutlineCoordinate Transformations

Coordinate SystemsRotationsSO(n) RotationsComposition of RotationsParametrization of RotationsRigid Motions

Composition of rotations around current axes

◮ Frame 0 → Frame 1 → Frame 2

◮ Postmultiply R02 = R0

1R12

H.I. Bozma EE 451 - Coordinate Transformations

OutlineCoordinate Transformations

Coordinate SystemsRotationsSO(n) RotationsComposition of RotationsParametrization of RotationsRigid Motions

Composition of rotations around fixed axes

H.I. Bozma EE 451 - Coordinate Transformations

OutlineCoordinate Transformations

Coordinate SystemsRotationsSO(n) RotationsComposition of RotationsParametrization of RotationsRigid Motions

Representation of Arbitrary Rotations

◮ Euler Angles – 3 Successive rotations around current axes

◮ Roll-pitch-yaw angles – 3 successive rotations around fixedaxes

◮ Axis/Angle representation – Arbitrary axis

H.I. Bozma EE 451 - Coordinate Transformations

OutlineCoordinate Transformations

Coordinate SystemsRotationsSO(n) RotationsComposition of RotationsParametrization of RotationsRigid Motions

Euler Angles

H.I. Bozma EE 451 - Coordinate Transformations

OutlineCoordinate Transformations

Coordinate SystemsRotationsSO(n) RotationsComposition of RotationsParametrization of RotationsRigid Motions

Euler Angle Transformation

RZYZ =

cφ −sφ 0sφ cφ 00 0 1

cθ 0 sθ

0 1 0−sθ 0 cθ

cψ −sψ 0sψ cψ 00 0 1

=

cφcθcψ − sφsψ −cφcθsψ − sφcψ cφsθ

sφcθcψ + cφsψ −sφcθsψ + cφcψ sφsθ

−sθcψ sθsψ cθ

H.I. Bozma EE 451 - Coordinate Transformations

OutlineCoordinate Transformations

Coordinate SystemsRotationsSO(n) RotationsComposition of RotationsParametrization of RotationsRigid Motions

Roll (Z) ,Pitch (Y) & Yaw (X) Angles

H.I. Bozma EE 451 - Coordinate Transformations

OutlineCoordinate Transformations

Coordinate SystemsRotationsSO(n) RotationsComposition of RotationsParametrization of RotationsRigid Motions

Rotation around arbitrary axis kT = [kxkykz ]

H.I. Bozma EE 451 - Coordinate Transformations

OutlineCoordinate Transformations

Coordinate SystemsRotationsSO(n) RotationsComposition of RotationsParametrization of RotationsRigid Motions

Rotation θ around arbitrary axis kT = [kxkykz ]

◮ Two rotations to align the k direction with z-axis with -(RZ ,αRY ,β)

◮ Rk,θ = RZ ,αRY ,βRZ ,θ (RZ ,αRY ,β)−1

H.I. Bozma EE 451 - Coordinate Transformations

OutlineCoordinate Transformations

Coordinate SystemsRotationsSO(n) RotationsComposition of RotationsParametrization of RotationsRigid Motions

SO(3) - Equivalence to rotation by θ around k axis

Given R ∈ SO(3) → There exists (k , θ)

◮ k - Axis of rotation

◮ θ - Angle of rotation around k axis

θ = cos−1(r11 + r22 + r33 − 1

2) k =

1

2sinθ

r32 − r23r13 − r31r21 − r12

H.I. Bozma EE 451 - Coordinate Transformations

OutlineCoordinate Transformations

Coordinate SystemsRotationsSO(n) RotationsComposition of RotationsParametrization of RotationsRigid Motions

SE (3) - The Set of Rigid Motions

◮ Translation d ∈ R3

◮ Rotation R ∈ SO(3)

◮ Rigid motion: (d ,R)

◮ The set of all rigid motionsSE (3) =

{

(d ,R) | d ∈ R3,R ∈ SO(3)}

H.I. Bozma EE 451 - Coordinate Transformations

OutlineCoordinate Transformations

Coordinate SystemsRotationsSO(n) RotationsComposition of RotationsParametrization of RotationsRigid Motions

Homogeneous Coordinate

◮ p0 = R10p

1 + d01 - Not linear !

◮ Homogeneous transformation H =

[

R d

0 1

]

where

0 =[

0 0 0]

◮ H−1 =

[

RT −RTd

0 1

]

H.I. Bozma EE 451 - Coordinate Transformations

OutlineCoordinate Transformations

Coordinate SystemsRotationsSO(n) RotationsComposition of RotationsParametrization of RotationsRigid Motions

Homogeneous Coordinate System

◮ P =

[

p

1

]

◮ P0 = H01P

1

◮ Composition rules – Similar to 3× 3 rotations!

H.I. Bozma EE 451 - Coordinate Transformations