Optimal solution method of multi-degree-of-freedom manipulator vector polynomial system
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A polynomial and robotic arm technology, applied in complex mathematical operations, geometric CAD, special data processing applications, etc., can solve problems such as high computational complexity, not using the half-angle tangent form of joint angles, and singularity in the solution process
Active Publication Date: 2021-04-27
居鹤华
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Problems solved by technology
For the solution of general polynomial equations, the solution method based on the friendly matrix is used, which has a large amount of calculation and low solution accuracy.
Currently, The basis theory is a possible way to solve the problem of multivariate polynomial equations, but its computational complexity is usually extremely high, which cannot meet the requirements of high-precision, high-real-time inverse solution of the manipulator
[0003] At present, the kinematic equation used to solve the inverse solution of the decoupled manipulator does not use the half-angle tangent form of the joint angle, and the solution process is singular
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Embodiment 1
[0351] Solve the polynomial p(x)=x 3 -10x 2 +31x-30=0,
[0352] From formula (152) and (154) get
[0353]
[0354] All solutions of polynomial equations are sequences of characteristic roots [2,3,5]. And all solutions of univariate polynomials can be obtained from the adjoint matrix.
[0355] Multilinear Polynomial Equation Solving
[0356] Solving a 2-variable 2nd-order multilinear polynomial f 2 (x 1 ,x 2 )=0 2 :
[0357]
[0358] means f 2 The nth subequation of . 2-variable 2nd-order multilinear polynomial f 2 (x 1 ,x 2 ) is abbreviated as f 2 .
[0359] Step 1: Calculate Dixon polynomials. Introducing the substitution variable y 2 to replace the original variable x 2 , denoted as |2, then the reduced polynomial matrix is:
[0360]
[0361] call the second column f 2 downscaling substitution. determinant called f 2 Dixon polynomials
[0362]
[0363] Equation (157) is a necessary condition for the solution of Equation (155).
[0364]...
Embodiment 2
[0375] Given a 3-fold linear polynomial, we have
[0376]
[0377] second order polynomial system
[0378] The n-fold linear order sequence W n and the original variable sequence X n The second-order form of is denoted as and thus obtaining a quadratic polynomial and have a one-to-one mapping relationship. The reduced-order substitution matrix of is denoted as
[0379]
[0380]
[0381]
[0382] is the original variable sequence, is the second-order form of the original variable sequence.
Embodiment 3
[0384] From formula (162), we get
[0385]
[0386]
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Abstract
The invention discloses a multi-degree-of-freedom mechanical arm vector polynomial system optimal solution method, comprising the following steps: constructing the Dixon knot of the multivariate polynomial system, obtaining the necessary conditions for the solution of the Dixon polynomial; obtaining the optimal Dixon solution of the multivariate vector polynomial system Necessary conditions of the element; the optimal elimination method of the vector polynomial system is applied to the kinematic equation of the 3R, 5R or 6R manipulator to obtain the inverse solution. This method establishes a polynomial symbol system by introducing word-order sequences in computer science to meet the stylization requirements of polynomial systems; it reduces the calculation amount of Dixon polynomials, and the calculation amount of vector polynomial optimal Dixon elimination method is only linear complexity. And when it is used for the inverse solution calculation of a multi-degree-of-freedom manipulator, there will be no singularity problems; it has a fast solution speed, and there is no combination explosion problem; it ensures the real-time and accuracy of the inverse solution of the vector polynomial system.
Description
technical field [0001] The invention relates to a method for establishing a vector polynomial system of a multi-degree-of-freedom manipulator and an optimal solution method thereof, which is applicable to obtaining an inverse solution of a multi-degree-of-freedom manipulator kinematics model, and belongs to the fields of robots and precision machinery. Background technique [0002] An important aspect of autonomous robot research is the need to solve the kinematics modeling and solving problems of variable topology robots. By modeling the kinematics of a multi-degree-of-freedom manipulator, the established model is usually a multivariate second-order polynomial equation. For the solution of general polynomial equations, the solution method based on friendly array is used, which has a large amount of calculation and low solution accuracy. Currently, The basis theory is a possible way to solve the problem of multivariate polynomial equations, but its computational complexit...
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