A Unified Projective Mapping Framework for Deforming Kinematic Links and Soft Robotics

Motivated by applications in soft robotics and deformable mechanisms, this paper introduces a linear deformation matrix framework that provides an elegant and unified means of approximating complex geometric transformations. The approach represents deformation in homogeneous coordinates using a single 4×4 matrix that simultaneously captures rotation, nonuniform scaling, shear, translation, and perspective effects. The deformation matrix is estimated through a linear least-squares from point correspondences, with each correspondence contributing linear constraints and an explicit per-point projective scale. The resulting homogeneous system is solved in closed form via singular value decomposition, yielding a stable, computationally efficient, and geometrically interpretable mapping. Despite its linearity, the method accurately approximates strongly nonlinear deformations while remaining translation-invariant after simple centroid normalization. It offers a lightweight, transparent alternative to dense or learning-based techniques and establishes an analytical foundation for soft kinematics — extending classical kinematic concepts to continuously deformable links. Applications include real-time deformation tracking in soft robotics, fast registration in computer vision, and large-scale structural monitoring.