Human Modeling: Kinematics - Human Motion Simulation: Predictive Dynamics - page 13

2.3 Concept of extended vectors and homogeneous
coordinates
The concept of an extended vector is introduced to facilitate vector and matrix
operations. The extended vector of

T

x 5 ½ xyz is the (4 1) vector given by

3

T where a is an arbitrary number. The concept of homoge-
neous coordinates is introduced indicating that the values become homogeneous
when adding a new coordinate to the vector. In this case, it is homogeneous
because the vector will have the same meaning even if multiplied by a constant.
Formulations involving homogeneous coordinates are often simpler and more sym-
metric than their Cartesian counterparts. Homogeneous coordinates have many
applications, including computer graphics and 3D computer vision, where affine
transformations are allowed and projective transformations are easily represented
by a matrix. We shall use the concept of a homogeneous transformation to repre-
sent the rotation and translation into one homogeneous matrix transformation.
In human modeling, the importance of using homogeneous coordinates and
the concept of an extended vector stem from the representation of Equation (2.2) ,
which will become fundamental to the formulation of a systematic method for
representing the motion of one segmental link with respect to another. It is possi-
ble to write Equation (2.2) in terms of a (4

x 5 ½ ax ay az a

4) matrix as

3

~ ~

⎡

A

x ⎤ ⎡

A

R

A

p ⎤ ⎡

B

x ⎤

≡

B

(2.5)

⎣ ⎦ ⎣ ⎦ ⎣ ⎦

1 0 1 1

x x

⎡

A

⎤ ⎡

B

⎤

where the extended vectors

A

x = and

B

x = can be used to rewrite

⎣ ⎦ ⎣ ⎦

1 1

Equation (2.3) as

⎡

A A

⎤

R p

A B B

x = x

(2.6)

⎣ ⎦

0 1

or as

A A

T B B x 5 x (2.7)

where

⎡

A A

⎤

R p

A B

T =

(2.8)

⎣ ⎦ B

0 1

This A

T B matrix is called the Homogenous Transformation matrix and is read
as the transformation from B to A . It can be seen as acting on a rigid body causing
a transformation, i.e., changing its configuration. On the other hand, and this is
the concept that will be used throughout this text, it is seen as an operator acting
on a vector B

(which is resolved in the B -coordinate system), and resolving the
resulting vector in the A -coordinate system. This is similar in action to the rota-
tion matrix but includes the translation as well.

x