On the Integrability of Screw Spaces

In the development of a method for generating the dynamic equations of a chain of rigid bodies in elliptic line space, it has been discovered that the joint freedom spaces from the tangent space of the system’s joint space, called the configuration manifold by dynamicists. The freedom space of a joint can be calculated from the joint contact geometry using reciprocal screw theory. The present paper describes the relationship of the joint freedom spaces to the tangent space of the configuration manifold, and determines the integrability conditions which must be satisfied if the freedom space represents a valid configuration manifold. The paper also shows that the integrability of the tangent space of a chain of rigid bodies is established once the integrability of each joint freedom space has been demonstrated. These conditions are used to define valid generalized coordinates for describing the chain’s configuration. Cylindrical and spherical joints are treated as examples.