Analysis of Dimensions of Constraint Varieties for the Inverse Kinematics of General P6R Serial Manipulators

This paper investigates the inverse kinematics of general P6R serial manipulators by analyzing the dimensions of the constraint varieties that contain the workspaces of its divided chain. The approach uses the algebra of dual quaternions and algebraic-geometric techniques to model rigid motions in SE (3), which are identified with a quasi-projective variety called the Study set in the seven-dimensional real projective space. The Study parameters associated with this variety are used to compute the linear spaces representing the workspaces of the left $\mathbf{P 3 R}$ and right $\mathbf{3 R}$ chains. The intersection of these workspaces with the Study quadric is analyzed to determine the dimension of the variety that contains the inverse kinematics solutions using results in dimension theory. Our analysis reveals that the kinematic image of 7 -seven chain manipulator lies in a one-dimensional variety, indicating that the inverse kinematics problem for a general P6R manipulator has infinite solutions.