perm filename ABSTR.ROM[NET,GUE] blob sn#877473
filedate 1989-09-21 generic text, type T, neo UTF8
ABSTRACT for ROMANSY-90
A General Solution for the Inverse Kinematics of All Series Chains
Dr. Madhusudan Raghavan Prof. Bernard Roth
General Motors Research Laboratories and Dept. of Mech. Eng.
Warren, Michigan, USA Stanford University
Stanford, CA 94305, USA
The inverse kinematics problem is the name given to the problem of
determining those joint coordinates for a manipulator for which its
end-effector takes on a desired position and orientation. It is known
that, for a general manipulator geometry, a manipulator with 6
revolute joints (a 6R) can have at most 16 sets of joint coordinates
corresponding to a single end-effector position and orientation.
Recently we, as well as other researchers, have been able to reduce
the inverse kinematics problem for a 6R manipulator to a single
polynomial of degree 16 in one of the joint-angle variables. Our
paper describing these results was present at the ISRR meeting in
Tokyo this August, it is titled "Kinematic Analysis of the 6R
Manipulator of General Geometry."
What we proposed for ROMANSY is to show that the elimination
techniques that were developed for the 6R are in fact universally
applicable to any series chain. The argument goes as follows: first,
we restrict ourselves to 6-degree-of-freedom chains with any
combination of revolute and 3 or less prismatic joints. In this case
we go through all the possibilities in this paper and show the reader
exactly which variables need to be eliminated, how to do it, and in
which cases the resultant polynomial will have degree less than 16.
Our results include the well known fact that if 3 adjacent revolute
axes intersect we get a polynomial of degree 8.
Second, we show that anytime we have less than 6-degrees-of-freedom or
more than 3 prismatic joints, the result follows trivially from the
above by simply setting certain values to zero or some other
convenient numerical value. Third, we mention that any system with
more than 6-degrees-of-freedom allows for arbitrary choices which
reduces the problem to one of the 6-degree-of-freedom problems we have
solved. Finally we mention that any type of lower pair joint can be
represented as a combination of revolute and prismatic joints. For
these reasons we can claim our analysis is universal for any
combination of lower pair joints arranged in series.
The heart of this paper will be the material mentioned as the first
point in the foregoing. All the other arguments are very short and
obvious. For the first point, we will be presenting new results that
show that the original set of closure equations may be supplemented by
an additional set of equations which lie in the ideal of the closure
set and have exactly the same number of power products as the set of
closure equations. This new set of equations together with the
original form the basis group from which we can obtain a single
polynomial using the method of dyalitic elimination. It will be shown
that in every case our resultant polynomial is of exactly the minimum
degree and contains no extraneous solutions.