Fourth-order kinematic analysis: Advanced decomposition methods for particle motion in modified orthogonal frame

The analysis of higher-order derivatives of position, particularly the snap (fourth derivative), provides critical insights into complex motion dynamics in fields such as robotics, aerospace, and biomechanics. This research investigates novel decomposition techniques for the fourth-order derivative of position (snap vector) of a particle traveling along spatial curves within three-dimensional Euclidean geometry. We establish comprehensive analytical expressions that decompose the snap vector into components corresponding to modified orthonormal basis vectors. Furthermore, we present an innovative alternative formulation that partitions the snap vector into tangential and radial components within the osculating and rectifying planes. The developed theoretical framework is validated through computational examples, illustrating the practical implementation and effectiveness of the proposed decomposition methodologies.