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Tham khảo tài liệu 'robot manipulators trends and development 2010 part 12', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 432 Robot Manipulators Trends and Development Eq. 23 . Each of the three polynomials T T and T3 can be written as follows T TQ R 26 where Qi and Ri are the quotient and the remainder of a polynomial division on Ti through the divisor T4 with respect to a given variable Xj. At every point where all Ti vanish along with T4 all remainders Ri must vanish too. Therefore the equation set R1 R2 R3 T4 0 27 is always equivalent to Eq. 23 along with the condition T4 0 . If R1 R2 and R3 are remainders of polynomial divisions with respect to variables X2 X1 and X1 respectively R1 R2 and R3 are polynomials of degree four in the two variables x2 and x3. Therefore the equation set R1 R2 R3 0 28 can be solved with a method similar to that used for spherical wrists. Variable X1 can be hidden in the coefficients and a partial homogenization with respect to X2 and X3 yields a set of three homogeneous equations in three unknowns of degree four. A resultant polynomial in X1 can then be found through classical elimination methods. a Fig. 10. a Positive critical points of manipulator T1 b All critical points of manipulator T1. b Fig. 11. The steepest ascent and descent paths are all singularity-free. Topological Methods for Singularity-Free Path-Planning 433 Unfortunately in this way the condition T4 0 has not been directly imposed Eq. 28 is not completely equivalent to Eq. 27 which introduces extraneous solutions. The author has found no way to factor out such extraneous solutions from the resultant polynomial however they can be easily detected for they do not satisfy the condition T4 0 . Once all real solutions have been found by numerically solving the resultant polynomial and all extraneous solutions have been cancelled all critical points of J are known. The classification of critical points and the determination of steepest ascent paths is then analogous to the one proposed in Section 4.1 for spherical wrists. Manipulator T1 is now considered as a numerical example. According to .