Order-free rotations and transformations

A serial multibody with a common point and n controllable revolute joints, or a serial multibody, such as a manipulator, with n controllable joints can be moved with any arbitrary order of joint actuations. The final orientation and configuration of the bodies are independent of the order of activation of the joints. We introduce two theorems and develop order-free transformation matrices to show that under specific conditions, the order of transformations of such a multibody is immaterial. The theorems will be applied to a detector camera, and an Eulerian wrist. To apply the order-free transformations to a spherical wrist, we present a classification for the spherical wrists and show that there are only three types of spherical wrists. The Eulerian spherical wrist is mathematically and physically the most applied one