Kinematic geometry of spatial RSSR mechanisms

Two different novel methods to derive the input-output (IO) equation of arbitrary RSSR linkages are described. Both methods share some common elements, i.e., they use the standard Denavit- Hartenberg notation to first describe the linkage as an open kinematic chain, and Study's kinematic mapping to describe the displacement of the coordinate frame attached to the end effector of the chain with respect to the relatively non-moving base frame. The kinematic closure equation is obtained in the seven-dimensional projective kinematic mapping image space by equating the eight Study soma coordinates to the identity array. Then two methods are successfully applied to eliminate the intermediate joint angle parameters leading to the degree four biquadratic implicit algebraic IO equation: (a) the linear implicitisation algorithm, which can be applied after rearranging the closure equation such that the linkage can be viewed as two serial RS chains, and (b) numerical elimination theory using pseudowitness sets. Both approaches lead to the same IO equation. The utility of this algebraic form of the IO equation is illustrated with three detailed application examples.