Human Modeling: Kinematics - Human Motion Simulation: Predictive Dynamics - page 17

2.5 Composite transformations
In this section, we show that two consecutive transformation matrices can be
characterized by the multiplication of two independent transformation matrices
that yield the same result. In fact, we will show that consecutive transformations
can be represented by the multiplication of their respective transformation matri-
ces. Indeed, consider the transformation from B to A given by the matrix A

T B .
Consider also another transformation from B to C represented by the transforma-
tion matrix B T C . The resulting transformation can be obtained by multiplying the

two transformation matrices as

⎡

A A

⎤ ⎡

B B

⎤

R p R p

A B B AB C BC

T T =

(2.21)

⎣ ⎦ ⎣ ⎦

B C

000 1 000 1

A B A B A

⎡ R R + 0 R p + p ⎤

A B B C B B C AB

T T =

(2.22)

⎣ ⎦

B C

000 1

A

R B B

The multiplication of two consecutive rotation matrices R C yields the

rotation matrix A R C . In addition, the position vector to the origin of the B coordi-

nate system is seen by the A coordinate system as

A

R B B

A

p BC 5 p BC (2.23)

and the summation of the two position vectors yields

A A A

p BC 1 p AB 5 p AC (2.24)

Substituting Equations (2.23 and 2.24) into Equation (2.22) yields

⎡

A A

⎤

R p

A

T

B

T =

C AC

=

A

T

(2.25)

⎣ ⎦

B C C

000 1

which indicates that the chain rule applied to rotation matrices is also applica-
ble to transformation matrices. In analogy with rotation matrices, the concept
of extending the chain rule to a sequence of transformations can be applied
such that the resulting transformation matrix characterizes the combined motion
from A to C . This is a very important result, which will be used extensively
throughout this text.

2.5.1 Example: composite transformations
Consider the arm shown in Figure 2.8 with three coordinate systems. For this
model, there exists one coordinate system at the shoulder called A , one at the
elbow called B , and one at the wrist called C. Note that all coordinate systems are
restricted and can only rotate about their own z-axis.