A new polynomial solution to the synthesis of the body guided by plane-sphere contacts through six assigned poses will be presented. The proposed procedure starts from a system of seven quadratic compatibility equations in seven unknowns and computes a 10th-degree eliminant without passing through a fictitious polynomial equation with degree greater than ten. In particular, the proposed solution procedure requires the computation of the determinant of a 30×30 matrix that has 20 columns with constant entries and the remaining 10 columns with entries that are linear in one out of the seven unknowns. The entries of this matrix are obtained by simply permuting the coefficients appearing in the compatibility equations. Moreover, it will be shown that, due to the special structure of this matrix, the solution procedure can be reduced further so that it involves only the computation of the determinant of a 10×10 matrix. Finally, the proposed procedure is applied to a real case.

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