A Systems Approach to Modeling Uncertain ODE Systems

Incorporating the impacts of system uncertainty such as navigation errors, disturbance accelerations, burn execution biases and misalignments, orbit insertion dispersions, and a host of other influential factors into the design of a trajectory profile to make it robust to these unknown perturbations is becoming more feasible, particularly with the maturation of linear covariance analysis (LinCov) for closed-loop guidance, navigation, and control (GNC) systems. LinCov is an analytic technique for predicting stochastic behavior of dynamical systems. Previous studies interfaced a genetic algorithm (GA) with a linear covariance analysis tool to provide the initial demonstration of applying this non-traditional approach to robust trajectory optimization. These robust trajectory design techniques were originally developed for rendezvous applications in low Earth orbit. It was then extended to cis-lunar outbound trajectories to a Near Rectilinear Halo Orbit (NRHO) and introduced for a simple rendezvous approach trajectory in the NRHO for mid-course correction placement. Recently, these robust trajectory optimization principles have been applied to solve cislunar transfers to low-lunar orbit, NRHO rendezvous and docking, angles-only navigation for NRHO rendezvous, lunar powered descent and landing, along with Mars aerocapture problems. They were also utilized to determine the optimized trajectory correction burn placement for the upcoming Artemis II free-return cis-lunar trajectory profile and exercised to determine the optimized trajectory correction burn placement for NRHO orbit maintenance anticipated for the Artemis III mission. LinCov has also been used with a sweeping gradient method to perform robust optimal trajectory design for rendezvous in low-earth orbits and near rectilinear halo orbits. Recent developments in techniques generalize some of the modeling concepts and derive the corresponding governing equations. In this work, we present the software architecture for the Swift Integrated GNC and Mission Analysis (SIGMA) tool. In existing tools, the linear covariance analysis governing equations for particular models were implemented manually, involving significant development effort to derive and code partial derivatives and ensure proper bookkeeping for a variety of scenario configurations. SIGMA takes a different approach, modeling scenarios by connecting individual system models representing physical or GNC subsystems. By leveraging Condor for automatic differentiation and an engineering-focused modeling language, SIGMA is able to generate the functions required to perform covariance analysis. We have found that this reduces the number of configuration-specific special cases and reduces software development for performing analysis. The rest of the paper is organized as follows. In the next section, we review the scenario models used for analyzing closed-loop GNC systems and review the linear covariance analysis techniques and the core governing equations. Then we re-derive these equations from a block-diagram algebra. Finally, we describe a rendezvous scenario that is implemented in SIGMA.