SPARTAN: A Novel Pseudospectral Algorithm for Entry, Descent, and Landing Analysis

Abstract

In the last decades the theoretical development of more and more refined direct methods, together with a new generation of CPUs, led to a significant improvement of numerical approaches for solving optimal-control problems. One of the most promising class of methods is based on Pseudospectral Optimal Control. These methods not only provide an efficient algorithm to solve optimal-control problems, but also define a theoretical framework for linking the discrete numerical solution to the analytical one in virtue of the covector-mapping theorem. However, several aspects in their implementation can be refined. In this framework SPARTAN, the first European tool based on flipped Radau pseudospectral methods, has been developed. The tool, and the method behind it include two novel aspects. First, the discretized problem is automatically scaled with a novel technique, called Projected-Rows Jacobian Normalization. This avoids ill-conditioned problems, which could lead to non-reliable solutions. Second, the structure of the Jacobian matrix is exploited, and the dual-number theory is used for its computation. This yields faster and more accurate solutions, since the associated Jacobian matrix computed in this way is exact. Two concrete examples show the validity of the proposed approach, and the quality of the results obtained with SPARTAN.

Apollo 11 Reloaded: Optimization-based Trajectory Reconstruction

“…Results were obtained with SPARTAN [20,24,26], a tool developed by the German Aerospace Center implementing pseudospectral methods, and are depicted in Figs. 4-8.…”

“…In this paper we want to reconstruct three critical phases of the Apollo 11 mission by employing SPARTAN [19,24,25] a tool developed by the German Aerospace Center implementing multi-phase pseudospectral methods based on flipped Legendre-Gauss-Radau polynomials. For a more thorough description we suggest the reader to refer to [26], while in this context we will focus on the modeling of the phases of Apollo 11 mission, based on a critical comparison and integration of the different sources provided by NASA over the years.…”

Pseudospectral Convex Optimization for Powered Descent and Landing

“…It can be shown that Eqs. (16) and (17) yield exact results for polynomials of order at most equal to 2n − 2 and 2n − 3 for fRPm and LPm, respectively [21]. Once again, the presence of the term…”

Review of guidance techniques for landing on small bodies

“…Application examples using more sophisticated algorithms, such as the flipped Radau pseudospectral method (RPM), can be found in (Arslantas et al, 2014) and (Sagliano et al, 2017) for lunar landing and Earth re-entry guidance, respectively.…”