In the usual case of revolute joint,
i
is called the joint variable, and the other three quantities would be fixed link parameters. For prismatic joints,
i
d is the joint variable and the other three quantities are fixed link parameters. The definition of
mechanism by means of these quantities is a convention usually called Denavit- Hartenberg DH notation [3].
2.1.1 Denavit Hartenberg Representation
Figure 2.1: Link Parameter.
A commonly used convention for selecting frames of reference in robotic applications is the Denavit-Hartenberg, or DH convention. In this convention, each
homogeneous transformation
i
A is represented as a product of four basic
transformations. The procedure based on the DH convention in the following algorithm for deriving the forward kinematics for any manipulator are summarize as below [4]:
Step l : Locate and label the joint axes
1
,....,
n
z z
. Step 2 : Establish the base frame. Set the origin anywhere on the
z -axis. The x and
y axes are chosen conveniently to form a right-hand frame. For
1,...., 1
i n
, perform Steps 3 to 5.
Step 3 : Locate the origin
i
O where the common normal to
i
z and
1 i
z
intersects
i
z . If
i
z intersects
1 i
z
locate
i
O at this intersection. If
i
z and
1 i
z
are parallel, locate
i
O in any convenient position along
i
z . Step 4: Establish xi along the common normal between
1 i
z
and
i
z through
i
O i, or in the direction normal to the
1 i
i
z z
plane if
1 i
z
and
i
z intersect. Step 5 : Establish
i
y to complete a right-hand frame. Step 6 : Establish the end-effector frame
n n
n n
o x y z . Assuming the n-th joint is revolute set
n
z a
along the direction
1 n
z
. Establish the origin
n
o conveniently along
n
z , preferably at the center of the gripper or at the tip of any tool that the
manipulator may be carrying. Set
n
y s
in the direction of the gripper closure and set
n
x n
as
s a . If the tool is not a simple gripper set
n
x and
n
y
conveniently to form a right-hand frame. Step 7 : Create a table of link parameters
i
a ,
i
d ,
i
,
i
.
i
a Distance along xi from
i
o to the intersection of the xi and
1 i
z
axes.
i
d Distance along
1 i
z
from
1 i
O
to the intersection of the
i
x and
1 i
z
axes.
i
d is variable if joint
i
is prismatic.
i
The angle between
1 i
z
and
i
z measured about
i
x see Figure 2.1.
i
The angle between
1 i
x
and
i
x measured about
1 i
z
see Figure 2.1.
i
is variable if joint
i
is revolute. Step 8 : Form the homogeneous transformation matrices
i
A by substituting the above parameters into equation 2.1.
Step 9 : Form
1
....
n n
T A
A
. This then gives the position and orientation of the tool frame expressed in base coordinates.
The overall transformation is obtained by post multiplication of individual transformations:
1
, 0, 0,
0, 0, ,
i i
i i
i i
T Rot z
Trans d Trans
a Rot x
2.1
1
1 1
1 1
0 1 1
1 1
0 1 1
1 1
i i
i i
i i
i i
i i
i i
c s
a s
c c
s T
d s
c
2.2
1
cos sin
cos sin
sin cos
sin cos
cos cos
sin sin
sin cos
1
i i
i i
i i
i i
i i
i i
i i
i i
i i
i a
a T
d
2.3
2.2 Inverse Kinematics
