This paper proposes a novel methodology for analyzing rigid body motion’s hyper-state, including pose, velocity, and acceleration vector fields, through automatic differentiation applied to trident nilpotent algebra. With their automatic differentiation capabilities, we show that trident functions are highly effective for quantifying higher-order kinematic properties of rigid body motion, avoiding further differentiation concerning time. The outcomes derived from trident nilpotent algebra offer a comprehensive solution for identifying the hyper-state vector fields in rigid body and multibody motion. Underlying this framework is the principle that the space of rigid body motions constitutes a Lie group paired with its Lie algebra.