This paper describes the theory of Euler parameters with the concept of the basis, and uses it to handle coordinate transformations of spatial mechanical motion. A robot is selected as an illustrative example. Applying the theory of Euler parameters the kinematic and dynamic equations of robots in the earlier works can also be derived. For the orientation error functions of robots with Euler parameters, the result of this paper is a necessary mathematical background in the design of the hand Cartesian control of robots.