Human Modeling: Kinematics - Human Motion Simulation: Predictive Dynamics - page 31

Careful attention must be given when the following cases occur:

1. When two consecutive axes are parallel, the common normal between them is
not uniquely defined, i.e., the direction of x i must be perpendicular to both

x i is arbitrary.
2. When two consecutive axes intersect, the direction of

axes; however, the position of

x i is arbitrary.

a i ; α i are typically entered into a
table known as the DH table. A 10-DOF model will have a table with 10 rows.
Each row is used to generate the homogeneous transformation matrix for a DOF.

The four values for the DH parameters θ i ; d i ;

2.10.1 Example: a 9-DOF model of an upper limb
A 9-DOF model of the upper extremity is shown in Figure 2.20 . Sketch the coor-
dinate systems for each joint according to the DH representation method (note
that the first two joints are prismatic (translational).
Each z-axis, starting with the first joint and denoted by

z 0 , is located along the
joint ( Figure 2.21 ). The x-axis is also located but the y-axis is not shown to avoid
cluttering the figure.
The position and location of each axis determines the parameters a i ; α i ,

and hence determine the resulting (4 3 4) homogeneous transformation matrix
from Equation (2.49) , and repeated here for continuity.

θ i ; d i ;

2
4

3
5

cos θ i cos α i sin θ i sin α i sin θ i a i cos θ i

2

sin θ i cos α i cos θ i sin α i cos θ i a i sin θ i

2

i 2 1

T i 5 (2.50)

0 sin α i cos α i d i

0 0 0 1

For any sequence of consecutive transformations or for any serial mechanism,
any two reference frames can be represented by the multiplication of the transfor-
mations between them. To relate the coordinates frames m

1 and m n , the fol-

2 1

lowing transformations are computed

m 2 1

T m m

m 1 n 2 1 m 2 1

T m 1 1 ...: T m 1 n 5 T m 1 n (2.51)

FIGURE 2.20
A 9-DOF model of the upper extremity.