Human Modeling: Kinematics - Human Motion Simulation: Predictive Dynamics - page 15

with
no displacement yields (i.e., the first three elements of the last column are all
zeros)

Inserting this basic rotation matrix into the transformation matrix T

2
4

3
5

cos θ 2 sin θ 00

sin θ cos θ 00

T z;θ 5 (2.13)

0 0 1 0

0 0 0 1

which is a basic homogeneous transformation matrix for rotation about the

z-axis.
Similarly, two other basic homogeneous transformation matrices for rotation
about the x - and y -axes, respectively, are given by

2 3

10 00
0

4 5

cos α 2 sin α 0

T x;α 5 (2.14)

0 0

00 01

sin α cos α

and

2
4

3
5

cos 0

0100

ϕ 0 sin ϕ

T y;ϕ 5 (2.15)

sin 0

0001

ϕ 0 cos ϕ

2

A pure translation matrix, called a basic transformation matrix for translation,
can be written not by specifying rotations (i.e., an identity matrix for the rotation
matrix), but by specifying the coordinates of the three elements in the last col-
umn. A transformation matrix for translation along the x-axis by a -units can be
written as

2
4

3
5

100 a
0100
0010
0001

T x;a 5 (2.16)

Similarly, a basic transformation matrix translating along x -, y -, and z -axes is
written as

2
4

3
5

100 a
010 b
001 c
0001

T translation 5 (2.17)

which can be thought of as translating the x 2 y 2 z 2 a distance a along the x-axis, a
distance b along the y-axis, and a distance c along the z -axis as shown in
Figure 2.6(C) .