On Closed-Form Preintegration for a Class of Mixed-Invariant Systems in SEₙ(3)

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 ${\mathrm { SE}}_{\mathrm { n}}(3)$ . 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 ${\mathrm { SE}}_{\mathrm { n}}(3)$ Lie group. Through numerical simulations, we show that our closed-form solution is more efficient than the Runge-Kutta $4^{\mathrm { th}}$ -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.