This article illustrates the use of unsupervised probabilistic learning techniques for the analysis of planetary reentry trajectories. A three-degree-of-freedom model was employed to generate optimal trajectories that comprise the training datasets. The algorithm first extracts the intrinsic structure in the data via a diffusion map approach. We find that data resides on manifolds of much lower dimensionality compared to the high-dimensional state space that describes each trajectory. Using the diffusion coordinates on the graph of training samples, the probabilistic framework subsequently augments the original data with samples that are statistically consistent with the original set. The augmented samples are then used to construct conditional statistics that are ultimately assembled in a path planning algorithm. In this framework, the controls are determined stage by stage during the flight to adapt to changing mission objectives in real-time.
This work presents the development of a multistaged stabilized continuation for the three-dimensional unpowered hypersonic trajectory planning problem using indirect optimal control methods. The stabilized continuation method is noniterative and guaranteed to terminate within a finite number of floating point operations, thereby making it well suited for onboard autonomous implementations. Here, we present a multistage formulation of the stabilized continuation scheme that involves starting with a “loose” integration tolerance during the first stage and ramping up toward a “strict” integration tolerance through subsequent stages. An important benefit of this approach is that even when the solution to the underlying optimal control problem is numerically unstable, such as with the hypersonic vehicle footprint generation problem, the stabilized continuation algorithm is shown to be successful in finding a solution while providing some additional insights into the underlying cause of the numerical instability.
This report investigates the use of unsupervised probabilistic learning techniques for the analysis of hypersonic trajectories. The algorithm first extracts the intrinsic structure in the data via a diffusion map approach. Using the diffusion coordinates on the graph of training samples, the probabilistic framework augments the original data with samples that are statistically consistent with the original set. The augmented samples are then used to construct conditional statistics that are ultimately assembled in a path-planing algorithm. In this framework the controls are determined stage by stage during the flight to adapt to changing mission objectives in real-time. A 3DOF model was employed to generate optimal hypersonic trajectories that comprise the training datasets. The diffusion map algorithm identfied that data resides on manifolds of much lower dimensionality compared to the high-dimensional state space that describes each trajectory. In addition to the path-planing worflow we also propose an algorithm that utilizes the diffusion map coordinates along the manifold to label and possibly remove outlier samples from the training data. This algorithm can be used to both identify edge cases for further analysis as well as to remove them from the training set to create a more robust set of samples to be used for the path-planing process.
Problems in optimal control may exhibit a bang-bang or singular control structure. These qualities pose challenges with indirect solution methods when the control law is discontinuous or indefinite. Recent efforts in control regularization strategies have sought to overcome these difficulties. These methods approximate a smoothed mapping of the constrained multi-stage Hamiltonian boundary value problem, resolving the singular/bang arcs into a single-stage problem. This work investigates the use of control saturation functions for error-control regularization. A key feature of the new approach is to eliminate ambiguity of the control law derived from the necessary conditions for optimality. The method is shown to have improved stability in numerical continuation due to the removal of small error terms from the control law. A well-known classical problem with analytical solutions is studied, as well as a more applied problem involving atmospheric flight of a maneuvering reentry vehicle.
This work presents the application of stabilized continuation to the planar unpowered hypersonic trajectory generation problem using indirect optimal control methods. This scheme is non-iterative and guaranteed to terminate within a finite number of floating point operations-thereby making it well-suited for onboard autonomous operations. This algorithm involves using the stabilized continuation method in multiple stages starting with a “loose” integration tolerance and subsequently ramping up toward a “strict” integration tolerance. An important outcome of this approach is that even when the underlying optimal control problem governing the planar hypersonic trajectory becomes numerically stiff, our studies indicate that the stabilized continuation scheme terminates successfully with a converged solution.