General inverse solution modeling and solving method of 7R manipulator based on axis invariant
A robotic arm and variable technology, applied in the field of robotics, can solve problems such as the inability to calculate the inverse solution of a 7R robotic arm
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Embodiment 1
[0197] Given 2 2D row vector polynomials On the one hand, from formula (4) we get
[0198]
[0199] on the other hand,
[0200]
[0201] The above results verify the correctness of formula (4).
[0202] Give the determinant computation theorem for block matrices:
[0203] If the square matrix whose size is (n+m)·(n+m) is recorded as M, the matrix whose size is n·n It is a sub-matrix composed of the first n rows and any n columns of elements of the square matrix M, and the size of the matrix is m m It is a sub-matrix composed of m rows and remaining m columns after the square matrix M; the sequence cn and cm composed of the matrix column numbers arranged in ascending order is a subset of the sequence [1:m+n], [cn,cm]∈, and there is cm∪cn=[1:m+n]; then the square matrix M determinant and block matrix and The determinant relationship of
[0204]
[0205] The principle of row stepping calculation of determinant:
[0206] For an S×S matrix, each item is abo...
Embodiment 2
[0209] Through the elementary row transformation of the matrix, we get The row echelon matrix of .
[0210] The steps are: rk represents the kth row. have to
[0211]
[0212] then have
[0213] N-order polynomial system based on "N-carry word":
[0214] If n "n-ary 1st order" polynomial power product If the independent variable appears repeatedly N times, then n "n-element N-order" polynomial systems are obtained "N-element N-order polynomial system" and "n-bit N-carry word" isomorphic.
[0215]
[0216]
[0217]
[0218]
[0219] Dixon polynomials of a system of n "n-ary n-order" polynomials:
[0220] Introduce auxiliary variable [y 2 ,y 3 ,...,y n ], with
[0221]
[0222] In the multivariate polynomial (8), with the auxiliary variable Y m The first m in turn replace the original variables (OriginalVariables) X n For the m variables in , record "|" as the replacement operator to get the extended (Extended) polynomial
[0223]
[0224...
Embodiment 3
[0287] Perform Dixon elimination on the polynomial system (39).
[0288]
[0289] The steps are as follows: the formula is a polynomial system of 4 "quaternary 1st order", which satisfies the Dixon elimination condition. From formula (19) and formula (22), get
[0290]
[0291] in:
[0292]
[0293] Five solutions are obtained from formula (34) and formula (40):
[0294]
[0295]
[0296]
[0297] in: is not a solution to this system of equations. Substitute other solutions into formula (35). when , from equation (35) we get
[0298]
[0299] Solved: τ 3 = 1, τ 4 =-2. Will τ 3 and τ 4 Substitute into equation (39) to get τ 2 =1. Similarly, the other three sets of solutions can be obtained. Obviously, the dependent variable does not satisfy the formula (26), and the Dixon matrix shown in the formula (40) is not symmetrical. This example shows that the Dixon determinant being zero is sufficient for multilinear polynomial systems.
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