Closing Trajectories: Equation-Free Cyclic Animation via Koopman Surrogates

2026-05-22

AI Summary

This work addresses the challenge of generating physically plausible, seamless looping animations when the underlying system dynamics are unknown and initial–final states are mismatched. The authors propose a general framework that circumvents the need for explicit dynamical equations by constructing a Koopman-based surrogate model from observed trajectories. A Fourier-parameterized time-varying control force is introduced, and under strict temporal periodicity constraints, the loop synthesis problem is reformulated as a quadratically constrained optimization with linear equality constraints. This formulation enables efficient solution via a structured KKT system. The method demonstrates robust performance across diverse scenarios—including N-body systems, cloth, deformable objects, and shallow water—producing high-quality, long-duration seamless loops and significantly advancing the generality and efficiency of periodic trajectory closure.

Abstract

Cyclic animation is widely used in computer graphics and interactive content.It supports seamless playback in games, VR, and interactive simulation,where short clips must repeat smoothly over long durations. Achievingphysically plausible cyclic synthesis from an input sequence is challengingbecause the endpoint states of the observed sequence rarely match exactly,and the governing equations of the underlying system are often unavailable.We therefore propose an equation-free framework that identiffes a Koopmansurrogate from the observed trajectory and computes a cyclic trajectory byapplying a Fourier-parameterized, time-varying control force under a hardtemporal periodicity constraint. The resulting formulation reduces cyclicsynthesis to a linearly constrained quadratic program that can be solvedefffciently through a structured KKT system. Our method is applicable toa diverse range of examples, including N-body systems, cloth, deformableobjects, shallow water, etc.

Problem

• cyclic animation
• physically plausible
• endpoint mismatch
• equation-free
• temporal periodicity

Innovation

• Koopman operator
• equation-free
• cyclic animation
• Fourier-parameterized control
• quadratic programming