Background Information and Techniques - Computer Animation: Algorithms and Techniques - page 446

Link i 1

y i 1

z i 1

d i 1

i

Link i

x i 1

z i

y i

x i

i 1

a i

Joint i 1

Joint i

FIGURE B.28
DH parameters.

Table B.1 DH Joint Parameters for Joint i
Name Symbol Description

Link offset d i Distance from x i-1 to x i along z i

Joint angle y i Angle between x i-1 and x i about z i

Link length a i Distance from z i to z iþ1 along x i

Link twist a i Angle between z i and z iþ1 about x i

Stated another way, the parameters that describe the relationship of the i þ 1 frame to the i th frame

are a combination of i th joint parameters and i þ 1 joint parameters. The parameters can be paired off to

define two screw transformations , each of which consists of a translation and rotation relative to a sin-
gle axis. The offset ( d iþ 1 ) and angle ( y i þ 1 joint relative to

the i th joint with respect to the i th joint's z -axis. The length ( a i ) and twist ( a i ) are the translation and
rotation of the i þ

1) are the translation and rotation of the i þ

1 joint with respect to the i th joint's x -axis (see Table B.2 ). The transformation of the

i þ 1 joint's frame from the i th frame can be constructed from a series of transformations, each of which

corresponds to one of the DH parameters. As an example, consider a point, V iþ 1 , whose coordinates are
given in the coordinate system of joint i þ 1. To determine the point's coordinates in terms of the coor-

dinate system of joint i , the transformation shown in Equation B.63 is applied.
In Equation B.63 , T and R represent translation and rotation transformation matrices respectively;
the parameter specifies the amount of rotation or translation, and the subscript specifies the axis
involved. The matrix M maps a point defined in the i þ

1 frame into a point in the i th frame. By forming
the M matrix and its inverse associated with each pair of joints, one can convert points from one frame