Linearity is an assumption often underlying the algorithms commonly used in spacecraft guidance, navigation, and control (GNC). However, the dynamics encountered in spaceflight can be highly nonlinear. This nonlinearity becomes more pressing as larger state distributions are considered, or on longer arcs without corrective maneuvers or adequate measurement updates. In particular, it is also a well-known challenge for spacecraft GNC in cislunar space. In such environments, linearization-based strategies are not always appropriate, and nonlinear analogs and extensions are a topic of much interest to the spaceflight community.

Many relevant works have investigated nonlinear expansion-based methods for efficient state and uncertainty propagation in spaceflight. We provide a compressed discussion here. Ref. 1 introduces the “state transition tensors" (STTs), which are a nonlinear extension of the state transition matrix with their own higher-order variational equations. Ref. 2 derives the “directional state transition tensors" (DSTTs) which are a compact representation of the most important information in the STTs, noteworthy for their reduced computation time and storage requirements. In addition to an exploration of their application to efficient trajectory propagation, Ref. 2  also explores their application to computationally efficient nonlinear updates of the covariance matrix. 

Similarly to the STTs, differential algebra (DA) can be used to compute identical nonlinear maps of the flow of the dynamics, via a form of automatic differentiation [3]. This has been applied extensively in the literature on spaceflight navigation. For instance, Ref. 4 explores the application of DA to uncertainty quantification in the case of non-Gaussian processes, via both a sampling-based and a direct moment computation-based approach. A large body of past relevant works have applied STTs and DA to spaceflight, and the full paper will provide a more comprehensive discussion.

The main contribution of this paper is an exploration of the ways that nonlinear Taylor expansions of the flow can be used for uncertainty quantification and uncertainty propagation in the case of nonlinear dynamics (which generally corrupt initially Gaussian statistics), mainly via DA. To enable a class of particularly computationally efficient methods, this paper also introduces a DA-based analog of the DSTTs. This so-called “directional differential algebra" leverages the fact that DA expansions can be performed for any choice of independent variables, producing identical maps to the DSTTs. The DA-powered methods are evaluated in a broad study of many popular UQ methods, including LinCov, unscented transformation (“UT”), conjugate unscented transformation (“CUT”), polynomial chaos expansion (“PCE”), and the Gaussian mixture model (“GMM”) [5, 6, 7]. This study contextualizes the strengths, weaknesses, and relative runtimes of the methods in a cislunar application, of great relevance on the eve of humanity’s return to the Moon. The directional DA method could be promising for extending the use of DA methods to circumstances with limited computational resources.

References
1. R. Park and D. Scheeres, JGCD 2006
2. S. Boone and J. McMahon, JGCD 2023
3. M. Berz, Academic Press 1999
4. Valli et al., JGCD 2013
5. Uhlmann 1995
6. Adurthi et al., ACC 2012
7. Luo et al., Prog. Aero. Sci. 2017