In this paper, a complete analysis of Stewart–Gough platform kinematics by unit quaternions is proposed. Even when unit quaternions have been implemented in different applications (including a kinematic analysis of the Stewart platform mechanism), the research regarding the application of this approach is limited only to the analysis of some issues related to the kinematic properties of this parallel mechanism. For this reason, a complete analysis of the Stewart–Gough platform is shown. The derivation of the inverse and forward kinematics of the Stewart platform using unit quaternions shows that they are suitable to represent the orientation of the upper platform due to their simplicity, equivalence, and compact representation as compared to rotation matrices. Then, the leg velocities are derived to compute these values under different conditions