On the Construction of Kinematic Hull for N-Poses of a Bounded Spatial Object

In this study, we extend the concept of the average squared distance (ASD)-minimizing motion sweeps from two poses to four poses for the construction of kinematic hulls of bounded spatial objects. Drawing an analogy to a tetrahedron, the simplest unit of a convex hull in three-dimensional space, we propose a four-pose motion sweep, called Tetrasweep, to serve as the basic building block for a kinematic hull. Our methodology integrates ASD minimization as both an optimization tool and a constraint mechanism, enabling the construction of a tightest fitting kinematic hull for four given poses. This approach ensures a precise representation of the motion, with the ASD metric defining the boundary of the motion sweep. The framework is extended to address the general N-pose problem, where the kinematic hull is constructed by combining multiple Tetrasweeps. By systematically applying the ASD-minimizing formulation to each tetrahedron formed by the poses, an overall kinematic hull is achieved, providing a cohesive and geometrically optimal representation of motion. Numerical examples are provided to demonstrate the accuracy and robustness of the method, showcasing its ability to handle complex pose configurations. The ASD-based methodology offers a novel way to model and analyze motion sweeps, with potential for further applications in fields requiring precise motion representation.