New kinematics description of the rigid-body fixed-point rotation in a tensor frame

The current theoretical mechanics does not give a complete kinematic description of the fixed-point rotation of a rigid body. For example, the rotation angle comprising multiple fixed-point rotations is underdetermined, only the angular velocity vector about Euler angles is defined, and the angular acceleration vector composed of two rotations is an open problem. This paper demonstrates that the rotation angle, angle velocity, and angular acceleration of the rigid-body fixed-point rotation are essentially not vectors but second-order antisymmetric tensors. The latter can be expressed by the corresponding pseudovectors. The composition of ordered rotations is established to be the ordered inner product of the corresponding orthogonal tensors in the absolute reference system. General composition rules of the angular velocity and angle acceleration tensors and their vectors are formulated. Euler angles describe the attitude of a rigid body through three orderly fixed-axis rotations that are equivalent to a fixed-point rotation. The angle velocity and angular acceleration vectors in terms of Euler angles are presented on the basis of the general composition rule, and their concise form is given in the Resal coordinate system.