This paper presents a new systematic formulation method to describe all kinds of lower and higher kinematic pairs and to express the pertaining kinematic relationships. The method can be applied to any mechanical system, which may be a mechanism or a manipulator, but it may especially be convenient for a system with multijoint links and multiaxis joints, such as a parallel manipulator. The method is based on the kinematic elements of the joints. In the first stage of the method, a joint frame is attached to every kinematic element hosted by the links of the system. It is attached in such a way that its relative position with respect to the link frame of the hosting link is described by a minimal number of essential parameters that are necessary and sufficient to represent all the characteristic features of the kinematic element. To systematize the attachment of the joint frames, the kinematic elements are classified into six types according to their geometric complexity. The link frames may also be attached judiciously to further minimize the total number of parameters required by the whole system. In the second stage of the method, the necessary equations are written to express the relative position between the mating kinematic elements of each joint. In the paper, such equations are written for a set of typical lower and higher kinematic pairs including samples ranging from a revolute joint up to a spatial cam joint. The application of the method is demonstrated on two mechanisms. One of them is a two-loop spatial mechanism with five different joints and the other one is a single-loop spatial cam mechanism with ellipsoidal and cylindrical cams. For each mechanism, the loop closure equations are first written and then simplified to prepare for solution to determine the unspecified joint variables. Afterward, the semi-analytical solutions of the loop closure equations are described and discussed.