ECE 450 Introduction to Robotics

ECE 450 Introduction to Robotics Section: 50883 Instructor: Linda A. Gee 9/09/99 Lecture 04

Homogeneous Coordinate Representation • When an N-component position vector is represented by an (N+1)-component vector, the representation is known as the homogeneous coordinate representation. • The transformation of an N-dimensional vector is performed in (N+1)-space • The physical N-dimensional vector is obtained by dividing the homogeneous coordinates by the (N+1)th coordinate

Vector Representation • For a position vector in 3-D, P= (PX PY PZ)T column vector the homogeneous coordinate representation is given by (wPX wPY wPZ w)T

Physical Coordinates • The physical coordinates are related to the homogeneous coordinates by PX = wPX /w PY = wPY /w PZ = wPZ /w

Homogeneous Transformation Matrix • Definition 4x4 matrix that maps a position vector in homogeneous coordinates from one coordinate system to another.

Representation of Homogeneous Transformation Matrix T = position vector Rotation matrix 3x3 = R3x3 p3X1 f 1x3 1x1 perspective transformation scaling Use T to represent the geometric relationship between the body-attached frame OUVW and the reference coordinate system OXYZ

Scaling • Produced from principal diagonal elements of the homogeneous transformation matrix a 0 0 0 0 b 0 0 0 0 c 0 0 0 0 1 x y z 1 ax by cz 1 =

Global Scaling • 4th diagonal element in the homogeneous transformation matrix globally scales Physical cartesian coordinates of a vector: PX = x/s PY = y/s PZ = z /s w = s/s if s > 1, global reducing if s < 1, global enlarging

Basic Homogeneous Rotation Matrices = TX, 1 0 0 0 angle  rotated about x-axis 0 cos -sin 0 0 sin cos 0 0 0 0 1 cos 0 sin 0 = TY, angle  rotated about y-axis 0 1 0 0 -sin 0 cos 0 0 0 0 1

Basic Homogeneous Rotation Matrices cont’d cos -sin 0 0 angle  rotated about z-axis TZ, = sin cos 0 0 0 0 1 0 0 0 0 1 1 0 0 dx Ttran = 0 1 0 dy 0 0 1 dz 0 0 0 1

Inverse of the Homogeneous Transformation Matrix nx sx ax px T = ny sy ay py nz sz az pz 0 0 0 1 -nTp nx ny nz -nTp T-1 = RT3x3 -sTp sx sy sz -sTp = -aTp ax ay az -aTp 0 0 0 1 0 0 0 1

Object Description • Use homogeneous transformations to describe objects and locations with 4xN matrices where N represents the number of vertices • Each vertex is described T x y z 1

Example of a cuboid representation P7 P4 General description for an object: P6 P5 c . . . z x0 y0 z0 1 x1 y1 z1 1 xN-1 yN-1 zN-1 1 = object x . . . b P0 P3 . . . y . . . P1 P2 a

Cuboid Representation in Homogeneous Coordinates T = P0 0 0 0 1 T = P1 0 b 0 1 T = P2 a b 0 1 P0 P1 P2 P3 P4 P5 P6 P7 . . . 0 0 0 1 0 b 0 1 a b 0 1 a 0 0 1 0 0 c 1 0 b c 1 a b c 1 a 0 c 1 P7

Composite Homogeneous Transformation Matrix • Homogeneous rotation and translation matrices can be multiplied together to yield a composite homogeneous transformation matrix

Composite Homogeneous Transformation Matrix Rules • Rules: • If both systems are coincident, the homogeneous transformation matrix is given by I4 • If rotating the coordinate system OUVW about the principal axes OXYZ frame, premultiply the homogeneous transformation matrix with the basic homogeneous rotation matrix • If rotating the coordinate system OUVW about its own principal axes, postmultiply the homogeneous transformation matrix with the basic homogeneous rotation matrix

Examples • Pure translation • Rotation • Rotation and translation

ECE 450 Introduction to Robotics