Abstract
We discuss the symbolic computation of inverse kinematics for serial 6R manipulators with arbitrary geometries (general 6R manipulators) based on Raghavan and Roth’s solution. The elements of the matrices required in the solution were symbolically calculated. In the symbolic computation, an algorithm for simplifying polynomials upon considering the symbolic constraints (constraints of the trigonometric functions and those of the rotation matrix), a method for symbolic elimination of the joint variables, and an efficient computation of the rational polynomials are presented. The elements of the matrix whose determinant produces a 16th-order single variable polynomial (characteristic polynomial) were symbolically calculated by using structural parameters (parameters that define the geometry of the manipulator) and hand configuration parameters (parameters that define the hand configuration). The symbolic determinant of the matrix consists of huge number of terms even when each element is replaced by a single symbol. Instead of expressing the coefficients in a characteristic polynomial by structural parameters and hand configuration parameters, we substituted appropriate rational numbers that strictly satisfy the constraints of the symbols for the elements of the matrix and calculated the determinant (numerical error free calculation). By numerically calculating the real roots of the rational characteristic polynomial and the joint angles for each root, we verified the formulation for the symbolic computation.
Efficient computation of the characteristic polynomial of a threshold graph
