Some results on counting roots of polynomials and the Sylvester resultant.

International audience We present two results, the first on the distribution of the roots of a polynomial over the ring of integers modulo n and the second on the distribution of the roots of the Sylvester resultant of two multivariate polynomials. The second result has application to polynomial GCD computation and solving polynomial diophantine equations.

Forward Displacement Analysis of a General 6-3 Stewart Platform Using Conformal Geometric Algebra

In this paper, a new algorithm for the forward displacement analysis of a general 6-3 Stewart platform (6-3SPS) based on conformal geometric algebra (CGA) is presented. First, a 6-3SPS structure is changed into an equivalent 2RPS-2SPS structure. Then, two kinematic constraint equations are established based on the geometric characteristics, one of which is built according to the point characteristic four-ball intersection in CGA. A 16th-degree univariate polynomial equation is derived from the aforementioned two equations by the Sylvester resultant elimination. Finally, a numerical example is given to verify the algorithm.

New Analytical Approach to Linear Intersection and its Application

Linear intersection is a frequently encountered issue in surveying engineering. The paper firstly reviews the common analytical approaches, and then introduces a new analytical approach based on Sylvester resultant, which does not use differencing and substitution, thus is direct and faster, lastly it is very efficient to make the adjustment of more than two known points’ linear intersection with the initial values computed by the new approach. Numerical case is demonstrated.

The Position Analysis of Spherical Seven-Bar Mechanism

In this paper, we turn plane seven-bar mechanism into spherical seven-bar mechanism, using quaternion to construct mathematical model for spherical seven-bar mechanism. Three constraint equations are obtained according to the angles constraint. Using Sylvester resultant elimination by two steps, a 32 degree univariate polynomial equation can be obtained. A numerical example confirms that analytical solutions of spherical seven-bar mechanism are 32 and with the help of Mathematic software to solve the location parameters.

A New Method on the Position Analysis of 3-Dof 3-PC(RR)N Spherical Parallel Mechanism

By applying quaternion and Sylvester resultant elimination method, and with the help of Mathematica 8.0 software, the 3-Dof 3-PC(RR)N spherical parallel mechanism location is analyzed. In solving inverse solution, there are 512-group input angle solutions, and 64-group real solutions. In the example of this paper, there are a total of 8-group real solutions by removing the extraneous roots.