Homogeneous Transformation MatricesExample: Puma 560

©2017 Max Donath

Position and Orientation of End Effector

Homogeneous Transformation

4 x 4 Matrix

Accounts for body Rotation

Translation

Columns Specify the directions of the body’s coordinate axes

Translation Vector

Calculation of Position and Orientation

in World Coordinates from the Joint Angles:

For a manipulator:Base A hand = Base T Hand Origin x Hand OriginAHand

For a six-jointed manipulator:Base T Hand Origin = BaseA1 x 1A2 x 2A3 x 3A4 x 4A5 x 5AHand origin

Where:N-1A n= Homogeneous transformation matrix which relates the coordinate frame of link n to the coordinate frame of link n-1

X2 behind Y2Z2 planeX3 behind Y3Z3 planeY4 behind X4Z4 plane

Homogeneous Transformation-combines rotation and translation

Definition: ref H loc = homogeneous transformation matrix which defines a location (position and orientation) with respect to a reference frame

Sequential TransformationsTranslate by x, y, zYaw: Rotate about Z, by (270˚ + q)Pitch: Rotate about Y’ by (a + 90˚)Roll: Rotate about Z” by t

,y

Step A: Translation by x, y, z

World Coordinate System

Step B: Rotation about Z (vertical axis) by 270˚

(shift coordinate frames to align X’Y’ for q = 0º orientation angle or yaw)

Step C: Rotation about Z (vertical axis) by q˚

Step D: Rotation about Y’ (orientation vector)by 90˚ (shift coordinate frames to align Z”X” for α = 0º approach angle or pitch)

Step E: Rotation about Y’ (Orientation Vector)by a˚ from horizontal plane to Z” for τ = 0° tool angle or roll

Step F: Rotation about Z” (approach vector)by t˚

Calculation of homogeneous transformation matrix from position and orientation in world coordinates

Homogeneous Transformation Matrices:Significance of columns and rows

Linear transformation which provides the geometric relationship between two coordinate systems:

Nx Ox Ax PxNy Oy Ay PyNz Oz Az Pz0 0 0 1

Nx - Cosine of the angle between N and X

Ny - Cosine of the angle between N and Y

Nz - Cosine of the angle between N and Z

NxNy = Unit vector describing direction of NNz

Similar definitions for and

P = = Position vector relating origins of coordinate systems

T =Initial coordinate system X Y Z

Final coordinate system N O A

Ox

Oy

Oz

Ax

Ay

AzPx

Py

Pz