A Hamilton-Jacobi Approach for Nonlinear Model Predictive Control in Applications with Navigational Uncertainty
Amit Jain, Roshan T. Eapen, Puneet Singla
TL;DR
This work addresses robust nonlinear trajectory tracking under navigational uncertainty by integrating Hamilton-Jacobi theory with nonlinear MPC. It introduces a type-2 generating function $F_2$ to transform the optimal-control problem into a transformed space $(\delta \mathbf{x}, \delta \boldsymbol{\lambda}_0)$ and solves the $HJ$ equation via a collocation framework, leveraging Conjugate Unscented Transformations to select collocation points. The generating function is approximated with basis functions $\phi_j$ and time-varying coefficients $\mathbf{c}(t)$, with sparsity enforced through iterative weighted $\ell_1$ optimization to yield a sparse polynomial expansion. Numerical validation on a PCR3BP transfer from $L_1$ to $L_2$ demonstrates accurate tracking of perturbed trajectories, with training errors consistently below testing errors and basis activity limited to low-order terms, illustrating real-time MPC viability in chaotic, nonlinear dynamics.
Abstract
This paper introduces a novel methodology that leverages the Hamilton-Jacobi solution to enhance non-linear model predictive control (MPC) in scenarios affected by navigational uncertainty. Using Hamilton-Jacobi-Theoretic approach, a methodology to improve trajectory tracking accuracy among uncertainties and non-linearities is formulated. This paper seeks to overcome the challenge of real-time computation of optimal control solutions for Model Predictive Control applications by leveraging the Hamilton-Jacobi solution in the vicinity of a nominal trajectory. The efficacy of the proposed methodology is validated within a chaotic system of the planar circular restricted three-body problem.