The solution of this problem is required in Computer Graphics in order to animate objects, as well as in Robotics, e.g. for path planning of robot manipulators. Because of their fast and stable algorithms, the methods which interpolate rotations using quaternions seem to become very popular. The results on the spherical rational spline motions are extended to the spatial ones, by combining them with rational trajectories of the origin. An interpolation scheme based on cubic spline functions will be discussed since it possesses some features which make it a suitable tool in many industrial applications.