Groebner bases is an important concept in polynomial ideals. In this paper the method of Groebner bases is applied to solving spatial Burmester problem for the first time, and the symbolic “triangular” Groebner base...Groebner bases is an important concept in polynomial ideals. In this paper the method of Groebner bases is applied to solving spatial Burmester problem for the first time, and the symbolic “triangular” Groebner bases, i.e. the closed form solution for the problem is obtained. An example of the synthesis of rigid body guidance of a spatial 5 s s mechanism which can realize spatial Burmester points is given to demonstrate the efficiency of the method.展开更多
This article proposes a new algorithm of quaternion and dual quaternion in matrix form. It applies quaternion in special cases of rotated plane, transforming the sine and cosine of the rotation angle into matrix form,...This article proposes a new algorithm of quaternion and dual quaternion in matrix form. It applies quaternion in special cases of rotated plane, transforming the sine and cosine of the rotation angle into matrix form, then exporting flat quaternions base in two matrix form. It establishes serial 6R manipulator kinematic equations in the form of quaternion matrix. Then five variables are eliminated through linear elimination and application of lexicographic Groebner base. Thus, upper bound of the degree of the equation is determined, which is 16. In this way, a 16-degree equation with single variable is obtained without any extraneous root. This is the first time that quaternion matrix modeling has been used in 6R robot inverse kinematics analysis.展开更多
A new method for the mechanical elementary geometry theorem proving is presented by using Groebner bases of polynomial ideals. It has two main advantages over the approach proposed in literature: (i) It is complete an...A new method for the mechanical elementary geometry theorem proving is presented by using Groebner bases of polynomial ideals. It has two main advantages over the approach proposed in literature: (i) It is complete and not a refutational procedure; (ii) The subcases of the geometry statements which are not generally true can be differentiated clearly.展开更多
A method for determining symbolic and all numerical solutions in design optimization based on monotonicity analysis and solving polynomial systems is presented in this paper. Groebner Bases of the algebraic system equ...A method for determining symbolic and all numerical solutions in design optimization based on monotonicity analysis and solving polynomial systems is presented in this paper. Groebner Bases of the algebraic system equivalent to the subproblem of the design optimization is taken as the symbolic (analytical) expression of the optimum solution for the symbolic optimization, i.e. the problem with symbolic coefficients. A method based on substituting and eliminating for determining Groebner Bases is also proposed, and method for finding all numerical optimum solutions is discussed. Finally an example is given, demonstrating the strategy and efficiency of the method.展开更多
文摘Groebner bases is an important concept in polynomial ideals. In this paper the method of Groebner bases is applied to solving spatial Burmester problem for the first time, and the symbolic “triangular” Groebner bases, i.e. the closed form solution for the problem is obtained. An example of the synthesis of rigid body guidance of a spatial 5 s s mechanism which can realize spatial Burmester points is given to demonstrate the efficiency of the method.
文摘This article proposes a new algorithm of quaternion and dual quaternion in matrix form. It applies quaternion in special cases of rotated plane, transforming the sine and cosine of the rotation angle into matrix form, then exporting flat quaternions base in two matrix form. It establishes serial 6R manipulator kinematic equations in the form of quaternion matrix. Then five variables are eliminated through linear elimination and application of lexicographic Groebner base. Thus, upper bound of the degree of the equation is determined, which is 16. In this way, a 16-degree equation with single variable is obtained without any extraneous root. This is the first time that quaternion matrix modeling has been used in 6R robot inverse kinematics analysis.
文摘A new method for the mechanical elementary geometry theorem proving is presented by using Groebner bases of polynomial ideals. It has two main advantages over the approach proposed in literature: (i) It is complete and not a refutational procedure; (ii) The subcases of the geometry statements which are not generally true can be differentiated clearly.
文摘A method for determining symbolic and all numerical solutions in design optimization based on monotonicity analysis and solving polynomial systems is presented in this paper. Groebner Bases of the algebraic system equivalent to the subproblem of the design optimization is taken as the symbolic (analytical) expression of the optimum solution for the symbolic optimization, i.e. the problem with symbolic coefficients. A method based on substituting and eliminating for determining Groebner Bases is also proposed, and method for finding all numerical optimum solutions is discussed. Finally an example is given, demonstrating the strategy and efficiency of the method.
