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