Robotics: Science and Systems

Zhibin (Alex) LiSchool of Informatics

University of Edinburgh

Coordinate Transformation

Slide Credit: Prof. Sethu Vijayakumar

Recap: What Tools Do We Need?

2

Coordinate Transformation(Where am I in relation to the

world?)

World Frame

Local Frame

Coordinate definitionThe common definition of the x-y-z coordinate used in robotics: a Cartesian coordinate system that defines our 3-dimensional space (right-hand rule).A position in space is therefore defined by a vector:

The distance between points and is: x y

z

Representation of positionA position is space is therefore defined by a vector:

Hence, two points and can be represented by vectors:

Quiz: what is a mathematically neatway to represent the spatial relation between these two points?

x y

z

r1

r2

Coordinate transformationLet two vectors and be the origins of two coordinates , what the coordinate transformation would be with respect to the global coordinate ? For example, how to represent a vector in coordinate in the global coordinate ?

x y

z

r1

r2

O2 O1

O

Coordinate transformationExample: represent a vector in local coordinate in the global coordinate . Position of a point in local coordinate Position of local coordinate in global coordinate

So position of in

O2

O

v

x y

z

r2

Representation of spatial relation

The coordinates in previous examples have all axes aligned. So what is still missing in a more general case?

Rotational representation

Rigid Body Position & Pose

Pose = Position + OrientationPosition

Prof. Sethu Vijayakumar // R:SS 2017 8

Rotation Matrices• Properties

orthonormal matrix(orthogonal vectors stay orthogonal, normal vectors stay normal)

columns and rows are orthogonal unit vectors

Let the new basis vectors be (e.g.)Then, the coordinate transformation from frame is:

Prof. Sethu Vijayakumar // R:SS 2017

9

Coordinate Transform

Prof. Sethu Vijayakumar // R:SS 2017 10

Simple Rotation Matrices• 2D • 3D

Prof. Sethu Vijayakumar // R:SS 2017

11

Example: a 2D Rotation MatrixHow the rotational matrix is formulated?

How this matrix is used?

or Note: is in a global coordinate, is in a local coordinate.Quiz: a whiteboard exercise to derive these equations

x

y

x’

y’

Rotation Matrix: Good & Bad

• Pros– Rotates vectors

directly– Easy

composition

• Cons– 9 numbers– Difficult to

enforce constraints

Degrees of Freedom (DOF) of a Rotation Matrix• R3x3 has 9 numbers• 6 constraints ( 3 orthogonal, 3 normal)• only 3 degrees of freedom (DOF)Can we represent with minimal (=3) independent parameters?

Prof. Sethu Vijayakumar // R:SS 2017

13

Rotation: Euler Angles• Describe rotations by consecutive rotations about

different axes:

• Z-Y-Z (3-1-3) representation• yaw-pitch-roll or Z-Y-X (3-2-1) ….used in flight!

Prof. Sethu Vijayakumar // R:SS 2017 14

Euler Angles and Gimbal Lock

• Euler angles have a severe problem:– If two axes align: blocks 1 DOF – ‘singularity’ of Euler angles

• Pros– minimal

representation– human readable

• Cons– Gimbal lock– must convert to

matrix to rotate vector

– no easy compositionProf. Sethu Vijayakumar // R:SS 2017 15

Coordinate definitionA GUI that helps to understand the properties, click here

Rotation: Rotation Vector• Using 3 numbers…

• Pros– minimal

representation– human readable

• Cons– singularity for small

rotations– must convert to matrix to

rotate vector– no easy composition

Prof. Sethu Vijayakumar // R:SS 2017 17

Quaternions: extension of complex numbers from 2 dimensions, into 4 dimensions.Complex numbers: Quaternions:

-Recall representation by the rotational Vector: 1) 3d rotation axis is define by a vector (position in [m]); 2) rotational angle is defined by radians. → discrepancy and ambiguity.Quaternions, however, represent all 4dimensions coherently.

Quarternion

Rotation: Quarternion• Math tells: all schemes with 3 numbers will have a

singularity

Prof. Sethu Vijayakumar // R:SS 2017 19

Quarternion: Composition• Conversion to/from matrix

• CompositionProf. Sethu Vijayakumar // R:SS 2017 20

Quaternions: Pros and Cons• Pros

– no singularity– almost minimal

representation– easy to enforce constraints– easy composition– easy interpolation

• Cons– somewhat

confusing– not quite minimal– must convert to

matrix to rotate vector

• Summary of Rotation representations– need rotation matrix to rotate vectors– Quarternions good for free rotations– Euler angles OK for small angular deviations

– but beware singularities!Prof. Sethu Vijayakumar // R:SS 2017 21

Homogeneous Transformations

Definition: position vector in a global coordinate; position vector in a local coordinate;

: rotational matrix

Hence, the transformation is

: offset vector between origins : vector is local frametransformed by

x’

y’

x

y p

r’

Homogeneous Transformations

A compact way of representing coordinate transformations between two frames

in homogeneous transformations, we append 1 to all coordinate vectorsProf. Sethu Vijayakumar // R:SS 2017 23

Composition of transforms

Prof. Sethu Vijayakumar // R:SS 2017 24