Closed-Form Solutions to the Direct Kinematics of a Family of 3- X PS Parallel Mechanisms

We derive closed-form solutions to the direct-kinematics problem of a family of three- and six-degree-of-freedom (DOF) 3-XPS parallel manipulators. In this class, the direct kinematics reduces to the classical “three-points-on-three-lines” (3P3L) problem, where the three lines have direction vectors that are mutually separated by 120° and are all horizontal. In some earlier works, the problem was reduced to solving a univariate polynomial of degree four. Here, we use both ZYZ Euler angles and quaternions to derive two alternative solutions, each involving only simple trigonometric or quadratic equations. These new formulations provide insight into the separation of the four assembly modes and the robot's singular configurations.