Page 180 - Mechatronics with Experiments

166 MECHATRONICS
The Denavit-Hartenberg method [7] defines a standard way of attaching coordinate
frames to a robotic manipulator such that only four numbers (one variable, three con-
stant parameters) are needed per one degree of freedom joint to represent the kinematic
relationships.
Given the 4x4 transformation matrix description of a coordinate frame attached to the
tool, T , with respect to a reference coordinate frame 22, we can determine the position
0T
coordinates of the tool-coordinate frame (x , y , z ) as well as orientation angles (i.e., roll,
T
T
T
yaw, pitch) of the tool-coordinate frame with respect to the reference coordinate frame and
form the corresponding x vector.
In generic terms, the relationship between the coordinates of the tool and the joint
displacement variables can be expressed as,
x = f( ) (3.211)

The vector variable x represents the Cartesian coordinates of the tool (i.e., position coor-
dinates x , y , z in a given coordinate frame and oriention angles where three angles can
P
P
P
be used to describe the orientation). The description of the position coordinates of a point
with respect to a given reference frame is unique. However, the orientation of an object
with respect to a reference coordinate frame can be described by many different possible
combinations of angles. Hence, the orientation description is not unique. The vector vari-
able represents the joint variables of the robotic manipulator, that is for a six joint robot
T
= [ , , , , , ] .
6
5
4
1
2
3
The f( ) is called the forward kinematics of the mechanism, which is a vector
nonlinear function of joint variables,
f ( ) = [f ( ), f ( ), f ( ), f ( ), f ( ), f ( )] T (3.212)
3
2
1
6
4
5
The inverse relationship, that is the geometric function which defines the axis posi-
tions as functions of tool position and orientation, is called the inverse kinematics of the
mechanism,
= f −1 (x) (3.213)
The inverse kinematics function may not be possible to express in one analytical closed
form for every mechanism. It must be determined for each special mechanism on a case by
case basis. For a six revolution joint manipulator, a sufficient condition for the existence of
the inverse kinematic solution in analytical form is that three consecutive joint axes must
intersect at a point. Forward and inverse kinematic functions of a mechanism relate the
joint positions to the tool positions.
If a robotic manipulator is always taught (i.e., by using a “teach mode”) the points
the tool tip must go through in three-dimensional space, the controller can record the
corresponding joint angles at each point. Then, some form of interpolation can be used
to define intermediate points to generate the desired motion path. In this type of robotic
manipulator, where the desired motion of the manipulator is defined by a teach mode, inverse
kinematics are never needed. However, in applications where not all of the target points
are taught, inverse kinematics are needed. For instance, target tool tip position may need to
be determined in real-time based on a vision sensor measurement of the target workpiece
position and orientation. Then, the corresponding joint angles must be calculated using
inverse kinematic relations.
The differential relationships between joint axis variables and tool variables are
obtained by taking the differential of the forward kinematic function. The resultant matrix
that relates the differential values of joint and tool position variables (in other words, it
relates the velocities of joint axes and tool velocity) is called the Jacobian matrix of the