Linear Approximation of Deformation in Soft Robotics and Kinematic Links

Modeling large shape changes efficiently is an important challenge in deformable kinematics, soft robotics, compliant mechanisms, geometric modeling, and deformation-based path planning. This paper presents Projective Estimation with Per-Point Scales (PEPS), a family of linear estimation methodologies for obtaining a compact global approximation of the deformation between two configurations of a body from corresponding points. The deformation is represented by a single projective deformation matrix, 3 by 3 in two dimensions and 4 by 4 in three dimensions, which provides a unified representation of translation, rotation, scaling, shearing, and projective effects. Three related formulations are developed and compared. PEPS-1 explicitly introduces an independent homogeneous scale for each point correspondence. PEPS-2 eliminates these additional variables by enforcing projective collinearity and has an algebraic structure closely related to the classical Direct Linear Transform. PEPS-3 augments the collinearity formulation with equivalent constraints derived in a translated coordinate frame to investigate improvements in numerical conditioning. Isotropic coordinate normalization is applied to all three formulations to reduce sensitivity to coordinate magnitude, reference-frame placement, and point distribution. The resulting systems are estimated efficiently using singular value decomposition. The methodologies are evaluated on representative nonlinear shape transformations, including square-to-circle, cube-to-sphere, and cube-to-ellipsoid mappings. Reconstruction accuracy is assessed both at the correspondences used for estimation and at additional points, allowing fitting performance to be distinguished from generalization over the complete shape. The results show that a single projective deformation matrix can effectively capture the dominant global characteristics of nonlinear shape changes. PEPS-2 and PEPS-3 generally provide smaller linear systems, improved numerical conditioning, and lower computational cost than PEPS-1, while PEPS-1 can offer greater fitting flexibility for certain redundant or geometrically dependent correspondence sets. Comparisons with iterative nonlinear optimization demonstrate that the proposed linear formulations achieve comparable global approximations at substantially lower computational cost. These characteristics make the PEPS framework particularly suitable for fast deformation representation, deformation-aware kinematics, soft and continuum robotics, compliant mechanisms, and geometry-based path planning and control.