**Abstract.**

**To solve symbolically the inverse kinematic problem usually requires
to compute the roots of the so-called sine-cosine polynomials. A functional
decomposition
of a sine-cosine polynomial equation ***f*(*s*,*c*)=0** reduces this task to the
solution of two
equations of degree less or equal than half the degree of ***f*(*s*,*c*)**. Methods
for simplifying the
symbolic solution of kinematics problems through functional decomposition
were initiated by G. Hommel and
P. Kovacs, well documented with applications in the recent book of the
latter author.
They addressed the
problem of simplifying the characteristic equation ***f*(*s*,*c*)=0** for a
revolute joint variable ****( ***s*** and ***c*** stand for **** and ****), whose
solution provides the possible value(s) of the joint angle **** in a
robot manipulator
for each given position of the end effector.
**

**given a sine-cosine characteristic equation ***f*(*s*,*c*)** for a
revolute joint variable ****,
we want to know if there exist a univariate polynomial ***g*(*x*)** and a
sine-cosine polynomial
***h*(*s*,*c*)** with ***deg*(*NF*(*h*(*s*,*c*)) < *deg*(*NF*(*h*(*s*,*c*))** such that:
**

**It is very easy to see that if a sine-cosine polynomial ***f*(*s*,*c*)** is
decomposable, then the ***f*(*t*)** associated univariate polynomial has a bivariate
homogeneous decomposition, but not conversely.
Nevertheless there is a serious limitation for
efficient robotic applications to ***t***-polynomials of degree bigger than six,
because the BHD algorithm requires factorization procedures over
algebraic extensions
of the field ***K*(*t*)**. **

**In this talk, we present the implementation of the the method by
Gutierrez****Recio(1997) for decomposing sine-cosine equations with
numerical or parametric
coefficients. The algorithm is in polynomial time. Factoring is not
required in the
algorithm, but rather it just need finding one root in the field
***K*** of a certain polynomial ***p*(*Z*)** in ***K*[*Z*]** . The implementation on MAPLE
V, allowing
decomposition over the rationals, algebraic extension of the rationals or
parametric coefficient
fields. The perfomance of the algorithm allows decomposing, almost
instantaneously,
sine-cosine polynomials of degree **16

**Gathen, J.****Weiss, J.: Homogeneous bivariate decompositions.
Preprint, Dep. of Computer Science, University of Toronto, 1993.
**

**Gutierrez, J.****Recio, T.: Advances on the Simplification of
Sine-Cosine Equations.
To appear, J of Symbolic Computation, 40 pages, 1997.
Preprint, Dep. of Computer Science, University of Toronto, 1993.
**

IMACS ACA'98 Electronic Proceedings