Abstract
A large class of nonlinear systems in engineering and robotics evolves on geometric manifolds, such as fixed-wing aircraft, quadrotors, etc., whose kinematic motion can be described using Lie groups, in particular SEn(3) . The existing techniques involve numerical integration methods such as Runge-Kutta integration to propagate their motion forward in time. However, these methods are approximate solutions and do not respect the geometric constraints of the nonlinear differential equation, leading to higher computational costs and numerical integration errors. To this end, we propose a geometric closed-form approach to solve the initial value problem for a class of nonlinear systems that evolves on the SEn(3) Lie group. Through numerical simulations, we show that our closed-form solution is more efficient than the Runge-Kutta 4th -order (RK-4) integrator and reduces floating-point operations by 45% in a strapdown-inertial-navigation (SINS) case study. We believe that the efficiency and accuracy gains of our approach warrant the general adoption of this method for numerical integration in applications such as SINS.
Keywords
Lie groups; Closed-form solutions; Robots; Algebra; Differential equations; Vectors; Simultaneous localization and mapping; Manifolds; Nonlinear systems; Legged locomotion; Numerical algorithms; aerospace; algebraic/geometric methods
Date of this Version
6-5-2025
Recommended Citation
Lin, Li-Yu; Pant, Kartik A.; Perseghetti, Benjamin; and Goppert, James, "On Closed-Form Preintegration for a Class of Mixed-Invariant Systems in SEₙ(3)" (2025). School of Aeronautics and Astronautics Faculty Publications. Paper 61.
https://docs.lib.purdue.edu/aaepubs/61
Comments
This is the author-accepted manuscript version of L. -Y. Lin, K. A. Pant, B. Perseghetti and J. Goppert, "On Closed-Form Preintegration for a Class of Mixed-Invariant Systems in SEₙ(3)," in IEEE Control Systems Letters, vol. 9, pp. 667-672, 2025. The version of record is available at DOI: 10.1109/LCSYS.2025.3577074.
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