A novel simplified analytical three-spheres intersection algorithm is presented for use with forward pose kinematics solutions of a four-cable-suspended robot (the method is applicable to various other cable-suspended robots with equal pole heights and three cables intersecting in one point). It is required that the vertical center heights of all three spheres are equal (otherwise one can use the existing more-complicated algorithm). We derive this new algorithm and show that the multiple solutions, algorithmic singularity, and imaginary solutions do not cause any trouble in practical implementation. The algorithmic singularity of the original three-spheres intersection algorithm regarding equal Z heights is eliminated with the new algorithm. The new algorithm requires significantly less computation compared with the original algorithm. Examples are presented to demonstrate the new three-spheres intersection algorithm for a 4-cable robot.
A line/trimmed NURBS surface intersection algorithm using matrix representations
