For the general spatial case, the solution
is not so trivial. This is because the joint angles do not
simply add as they do in the planar case.
Denavit and Hartenberg used
screw theory in the 1950's to show that the most compact representation of a
general transformation between two robot joints required four
parameters. These are now known as the Denavit and Hartenberg
parameters (D-H parameters) and they are the de-facto standard
for describing a robot's geometry. Here is a description of
the four D-H parameters:
a - the perpendicular distance between two
joint axes measured along the mutual perpendicular. The mutual
perpendicular is designated the x-axis.
a - the relative
twist between two joint axes measured about the mutual
perpendicular
d - the distance between the two
perpendiculars measured along the joint axis
Q - joint angle
about the z axis measured between the two perpendiculars
Learning the proper procedure for assigning
the D-H parameters is a typical exercise in an upper-level
undergraduate or first graduate course in robotics.
Once the parameters have been assigned we
can solve the forward kinematics problem by moving from the
base of the robot out to the hand using the following
transformations at each joint: |