Direct kinematics of the exechon tripod

This paper presents a solution to the forward kinematics problem of the Exechon parallel mechanism, a three-legged three-degree-of-freedom spatial mechanism with a complex motion pattern of the platform. In series with a universal or a spherical wrist, it has been used in a number of PKM (parallel kinematic machine) designs, and more recently as a mobile robotic fixture. The inverse kinematics solution has been known for a number of years. However, past publications on the Exechon tripod have not presented a method successfully solving the direct kinematics problem. To achieve this here, we first reduce the forward problem to a system of four non-linear equations and then use a standard numerical solver to obtain all sets of possible real roots. This solution allows the calculation of all joint displacements and from there the transformation matrix describing the pose of the end-effector. The obtained solutions divide into two groups, each for a different assembly mode of the mechanism. The method is easy to implement and can potentially be applied to other types of parallel manipulators with revolute joints at the mobile platform.