Human Modeling: Kinematics - Human Motion Simulation: Predictive Dynamics - page 14

The transformation matrix is indeed a partitioned matrix where the upper left
corner sub-matrix is the rotation matrix and the upper right vector is the position
vector

rotation position

⎡ ⎤

⎢ ⎥

A

T B =

(2.9)

ma trix ve c tor

⎢ ⎥

0 1

⎣ ⎦

2.4 Basic transformations
Figure 2.6 shows two coordinate systems x 1 y 1 z 1 and x 2 y 2 z 2 that are coincident.
Consider the matrix generated by the rotation of the coordinates system x 2 y 2 z 2
about the z 1 axis.
By definition, this rotation can be written as the dot product of unit vectors as

2
4

3
5

x 1 U x 2 x 1 U y 2 x 1 U z 2

R z;θ 5 y 1 Ux 2 y 1 Uy 2 y 1 Uz 2 (2.10)

z 1 Ux 2 z 1 Uy 2 z 1 Uz 2

where the subscripts z and θ denote a rotation about z with an angle θ . From

Figure 2.6(B) , carrying out the dot product yields

2
4

3
5

cos θ cos ð 90

1 θÞ

0

R z;θ 5 cos ð 90 2 θÞ cos θ 0 (2.11)

0 0 1

Further simplification using the identity cos(90

1 θ

) sin θ and cos(90

2 θ

)

52 5

sin θ yields the basic rotation matrix

2
4

3
5

cos θ 2 sin θ 0

R z;θ 5 sin θ cos θ 0 (2.12)

0 0 1

z 1

z 1 z 1

z 2

y 2

z 2

z 2

y 1

y 1

y 1 y 2

y 2 c

x 2

a

x 2

b

θ

x 2

x 1

x 1

x 1

(A) (B) (C)

FIGURE 2.6