Implementations of the Universal Birkhoff Theory for Fast Trajectory Optimization

This is part II of a two-part paper. Part I presented a universal Birkhoff theory for fast and accurate trajectory optimization. The theory rested on two main hypotheses. In this paper, it is shown that if the computational grid is selected from any one of the Legendre and Chebyshev family of node points, be it Lobatto, Radau or Gauss, then, the resulting collection of trajectory optimization methods satisfy the hypotheses required for the universal Birkhoff theory to hold. All of these grid points can be generated at an $\mathcal{O}(1)$ computational speed. Furthermore, all Birkhoff-generated solutions can be tested for optimality by a joint application of Pontryagin's- and Covector-Mapping Principles, where the latter was developed in Part~I. More importantly, the optimality checks can be performed without resorting to an indirect method or even explicitly producing the full differential-algebraic boundary value problem that results from an application of Pontryagin's Principle. Numerical problems are solved to illustrate all these ideas. The examples are chosen to particularly highlight three practically useful features of Birkhoff methods: (1) bang-bang optimal controls can be produced without suffering any Gibbs phenomenon, (2) discontinuous and even Dirac delta covector trajectories can be well approximated, and (3) extremal solutions over dense grids can be computed in a stable and efficient manner

Numerical optimal control with applications in aerospace

This thesis explores various computational aspects of solving nonlinear, continuous-time dynamic optimization problems (DOPs) numerically. Firstly, a direct transcription method for solving DOPs is proposed, named the integrated residual method (IRM). Instead of forcing the dynamic constraints to be satisfied only at a selected number of points as in direct collocation, this new approach alternates between minimizing and constraining the squared norm of the dynamic constraint residuals integrated along the whole solution trajectories. The method is capable of obtaining solutions of higher accuracy for the same mesh compared to direct collocation methods, enabling a flexible trade-off between solution accuracy and optimality, and providing reliable solutions for challenging problems, including those with singular arcs and high-index differential-algebraic equations. A number of techniques have also been proposed in this work for efficient numerical solution of large scale and challenging DOPs. A general approach for direct implementation of rate constraints on the discretization mesh is proposed. Unlike conventional approaches that may lead to singular control arcs, the solution of this on-mesh implementation has better numerical properties, while achieving computational speedups. Another development is related to the handling of inactive constraints, which do not contribute to the solution of DOPs, but increase the problem size and burden the numerical computations. A strategy to systematically remove the inactive and redundant constraints under a mesh refinement framework is proposed. The last part of this work focuses on the use of DOPs in aerospace applications, with a number of topics studied. Using example scenarios of intercontinental flights, the benefits of formulating DOPs directly according to problem specifications are demonstrated, with notable savings in fuel usage. The numerical challenges with direct collocation are also identified, with the IRM obtaining solutions of higher accuracy, and at the same time suppressing the singular arc fluctuations.Open Acces

New near-optimal feedback guidance algorithms for space missions

This dissertation describes several different spacecraft guidance algorithms, with applications including asteroid intercept and rendezvous, planetary landing, and orbital transfer. A comprehensive review of spacecraft guidance algorithms for asteroid intercept and rendezvous. Zero-Effort-Miss/Zero-Effort-Velocity (ZEM/ZEV) guidance is introduced and applied to asteroid intercept and rendezvous, and to a wealth of different example problems, including missile intercept, planetary landing, and orbital transfer. It is seen that the ZEM/ZEV guidance law can be used in many different scenarios, and that it provides near-optimal performance where an analytical optimal guidance law does not exist, such as in a non-linear gravity field

Hybrid Direct-Indirect Strategy for Optimal Landing Guidance of Reusable Rockets

The present work focuses on developing a fast and accurate algorithm to find feasible trajectories for reusable rockets’ landing while minimizing their fuel consumption. Trajectory exploits aerodynamic forces, thus an Optimal Aerodynamic Powered Landing Problem is faced. A hybrid strategy is adopted, combining convex direct optimization with a novel indirect collocation scheme. A Covector Mapping Theorem is exploited to bridge the two methods. Development of the algorithm is organized in two steps: firstly, the structure of the optimal solution is derived solving the problem with a single shooting indirect method combined with a double homotopic continuation scheme; in second instance, an algorithm tailored on the optimal solution structure is presented and discussed. The suggested strategy is finally compared with the homotopic continuation scheme considering accuracy and computational times. Outcome is a net superiority of the designed algorithm over the homotopic technique; the power of a hybrid approach is therefore demonstrated over traditional solution methods

Autonomous Trajectory Planning and Guidance Control for Launch Vehicles

This open access book highlights the autonomous and intelligent flight control of future launch vehicles for improving flight autonomy to plan ascent and descent trajectories onboard, and autonomously handle unexpected events or failures during the flight. Since the beginning of the twenty-first century, space launch activities worldwide have grown vigorously. Meanwhile, commercial launches also account for the booming trend. Unfortunately, the risk of space launches still exists and is gradually increasing in line with the rapidly rising launch activities and commercial rockets. In the history of space launches, propulsion and control systems are the two main contributors to launch failures. With the development of information technologies, the increase of the functional density of hardware products, the application of redundant or fault-tolerant solutions, and the improvement of the testability of avionics, the launch losses caused by control systems exhibit a downward trend, and the failures induced by propulsion systems become the focus of attention. Under these failures, the autonomous planning and guidance control may save the missions. This book focuses on the latest progress of relevant projects and academic studies of autonomous guidance, especially on some advanced methods which can be potentially real-time implemented in the future control system of launch vehicles. In Chapter 1, the prospect and technical challenges are summarized by reviewing the development of launch vehicles. Chapters 2 to 4 mainly focus on the flight in the ascent phase, in which the autonomous guidance is mainly reflected in the online planning. Chapters 5 and 6 mainly discuss the powered descent guidance technologies. Finally, since aerodynamic uncertainties exert a significant impact on the performance of the ascent / landing guidance control systems, the estimation of aerodynamic parameters, which are helpful to improve flight autonomy, is discussed in Chapter 7. The book serves as a valuable reference for researchers and engineers working on launch vehicles. It is also a timely source of information for graduate students interested in the subject

Genetic Programming to Optimise 3D Trajectories

Dissertation submitted in partial fulfilment of the requirements for the Degree of Master of Science in Geospatial TechnologiesTrajectory optimisation is a method of finding the optimal route connecting a start and end point. The suitability of a trajectory depends on non-intersection with any obstacles as well as predefined performance metrics. In the context of UAVs, the goal is to minimise the cost of the route, in terms of energy or time, while avoiding restricted flight zones. Artificial intelligence techniques including evolutionary computation have been applied to trajectory optimisation with various degrees of success. This thesis explores the use of genetic programming (GP) to optimise trajectories in 3D space, by encoding 3D geographic trajectories as syntax trees representing a curve. A comprehensive review of the relevant literature is presented, covering the theory and techniques of GP, as well as the principles and challenges of 3D trajectory optimisation. The main contribution of this work is the development and implementation of a novel GP algorithm using function trees to encode 3D geographical trajectories. The trajectories are validated and evaluated using a realworld dataset and multiple objectives. The results demonstrate the effectiveness of the proposed algorithm, which outperforms existing methods in terms of speed, automaticity, and robustness. Finally, insights and recommendations for future research in this area are provided, highlighting the potential for GP to be applied to other complex optimisation problems in engineering and science

Natural motion around the Martian moon Phobos : The dynamical substitutes of the libration point orbits in an elliptical three-body problem with gravity harmonics

The Martian moon Phobos is becoming an appealing destination for future scientific missions. The orbital dynamics around this planetary satellite is particularly complex due to the unique combination of both small mass-ratio and length-scale of the Mars-Phobos couple: the resulting sphere of influence of the moon is very close to its surface, therefore both the classical two-body problem and circular restricted three-body problem (CR3BP) do not provide an accurate approximation to describe the spacecraft’s dynamics in the vicinity of Phobos. The aim of this paper is to extend the model of the CR3BP to consider the orbital eccentricity and the highly-inhomogeneous gravity field of Phobos, by incorporating the gravity harmonics series expansion into an elliptic R3BP, named ER3BP-GH. Following this, the dynamical substitutes of the Libration Point Orbits (LPOs) are computed in this more realistic model of the relative dynamics around Phobos, combining methodologies from dynamical systems theory and numerical continuation techniques. Results obtained show that the structure of the periodic and quasi-periodic LPOs differs substantially from the classical case without harmonics. Several potential applications of these natural orbits are presented to enable unique low-cost operations in the proximity of Phobos, such as close-range observation, communication, and passive radiation shielding for human spaceflight. Furthermore, their invariant manifolds are demonstrated to provide high-performance natural landing and take-off pathways to and from Phobos’ surface, and transfers from and to Martian orbits. These orbits could be exploited in upcoming and future space missions targeting the exploration of this Martian moon

Commonwealth of Independent States aerospace science and technology, 1992: A bibliography with indexes

This bibliography contains 1237 annotated references to reports and journal articles of Commonwealth of Independent States (CIS) intellectual origin entered into the NASA Scientific and Technical Information System during 1992. Representative subject areas include the following: aeronautics, astronautics, chemistry and materials, engineering, geosciences, life sciences, mathematical and computer sciences, physics, social sciences, and space sciences

Rigid Body Constrained Motion Optimization and Control on Lie Groups and Their Tangent Bundles

Rigid body motion requires formulations where rotational and translational motion are accounted for appropriately. Two Lie groups, the special orthogonal group SO(3) and the space of quaternions H, are commonly used to represent attitude. When considering rigid body pose, that is spacecraft position and attitude, the special Euclidean group SE(3) and the space of dual quaternions DH are frequently utilized. All these groups are Lie groups and Riemannian manifolds, and these identifications have profound implications for dynamics and controls. The trajectory optimization and optimal control problem on Riemannian manifolds presents significant opportunities for theoretical development. Riemannian optimization is an attractive approach to tackling these problems because it does not require the imposition of additional space-preserving constraints on the solver. Rather, these constraints are accounted for in the optimization algorithm. As such, implementing these solvers in trajectory optimization and optimal controls problems through direct transcription offers a reduction in the number of constraints imposed on the problem. The on-manifold optimization methodologies are applied to the Lie groups listed above. Then, the direct transcription of the optimal control and trajectory optimization problems on these Riemannian optimization is presented and applied. These trajectories are utilized in an unscented Kalman filter to show how these generated reference trajectories interface with state estimation. Finally, the fundamental equation of mechanics is utilized for generating initial guess trajectories which satisfy path and terminal time constraints. The results contained herein show, for the first time, that direct transcription of trajectory optimization and optimal control problems on Riemannian manifolds may be effectively conducted with on-manifold optimization techniques using relatively simple optimization algorithms