Product of Exponentials (POE) Splines on Lie-Groups: Limitations, Extensions, and Application to SO(3) and SE(3)

Existing methods for constructing splines and Bézier curves on a Lie group (Formula presented.) involve repeated products of exponentials deduced from local geodesics, w.r.t. a Riemannian metric, or rely on general polynomials. Moreover, each of these local curves is supposed to start at the identity of (Formula presented.). Both assumptions may not reflect the actual curve to be interpolated. This paper pursues a different approach to construct splines on (Formula presented.). Local curves are expressed as solutions of the Poisson equation on (Formula presented.). Therewith, the local interpolations satisfies the boundary conditions while respecting the geometry of (Formula presented.). A (Formula presented.) -order approximation of the solutions gives rise to a (Formula presented.) -order product of exponential (POE) spline. Algorithms for constructing 3rd- and 4th-order splines are derived from closed form expressions for the approximate solutions. Additionally, spline algorithms are introduced that allow prescribing a vector field the curve must follow at the interpolation points. It is shown that the established algorithms, where (Formula presented.) -order POE-splines are constructed by concatenating local curves starting at the identity, cannot exactly reconstruct a (Formula presented.) -order motion. To tackle this issue, the formulations are extended by allowing for local curves between arbitrary points, rather than curves emanating from the identity. This gives rise to a global (Formula presented.) -order spline with arbitrary initial conditions. Several examples are presented, in particular the shape reconstruction of slender rods modeled as geometrically nonlinear Cosserat rods.