Background Information and Techniques - Computer Animation: Algorithms and Techniques - page 445

p

m 0 ; 0 þ 1 2 s

2

x ¼

2

p

m 1 ; 1 þ

2 s

2

1

y ¼

(B.62a)

2

p

m 2 ; 2 þ

2 s

2

1

z ¼

2

B.4 Denevit and Hartenberg representation for linked appendages
B.4.1 Denavit-Hartenberg notation
The Denavit-Hartenberg (DH) notation is a particular way of describing the relationship of a parent
coordinate frame to a child coordinate frame. This convention is commonly used in robotics and often
adopted for use in computer animation. Each frame is described relative to an adjacent frame by four
parameters that describe the position and orientation of a child frame in relation to its parent's frame.
For revolute joints, the z -axis of the joint's frame corresponds to the axis of rotation (prismatic joints are
discussed in the following paragraph). The link associated with the joint extends down the x -axis of the
frame. First, consider a simple configuration in which the joints and the axes of rotation are coplanar.
The distance down the x -axis fromone joint to the next is the link length , a i .The joint angle , y i þ 1 ,isspec-
ified by the rotation of the i þ 1 joint's x -axis, x i þ 1 , about its z -axis relative to the i th frame's x -axis direc-

tion, x i (see Figure B.27 ) .
Nonplanar configurations can be represented by including the two other DH parameters. For this gen-
eral case, the x -axis of the i th joint is defined as the line segment perpendicular to the z -axes of the i th and i
þ

1frame's z -axis about this per-
pendicular relative to the z -axis of the i th frame. The link offset parameter, d i þ 1 , specifies the distance
along the z -axis (rotated by a i )ofthe i þ

1frames.The link twist parameter, a i , describes the rotation of the i þ

1 x -axis (see Figure B.28 ).
Notice that the parameters associated with the i th joint do not all relate the i th frame to the i þ

1 frame from the i th x -axis to the i þ

1

frame. The link length and link twist relate the i th and i þ 1 frames; the link offset and joint rotation

relate the i 1 and i th frames (see Table B.1 ).

x i 1

z i z i 1 Link i 1

y i

y i 1

Link i 1

i 1

x i

Link i

a i

Joint i Joint i 1

FIGURE B.27