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.


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