Continuous-thrust Trajectories Research Articles

Design of impulsive asteroid flybys and scheduling of time-minimal optimal control arcs for the construction of a Dyson ring (GTOC 11)

This paper describes the approach used by the team named the Eccentric Anomalies to obtain the sixth best solution to the problem of the eleventh Global Trajectory Optimization Competition, whose futuristic scenario of Dyson sphere building remained mathematically relevant for current space mission design and engineering in general. As usual for this recurring challenge, it involved large-scale combinatorics at a high level and a multitude of optimal control problems at a lower level. Furthermore it proposed additional layers of complexity by adding a strong scheduling component to the usual flyby sequencing, and by featuring both impulsive and continuous-thrust trajectories. The authors took advantage of modern theoretical techniques and open-source tools to put together a sequential process including analysis based on analytical trajectory models, tree searches using efficient data structures, global and local finite-dimensional optimization and multi-objective trade-offs. The optimal control part was both tackled with direct transcription as well as indirect shooting methods, and the mixed-integer scheduling reformulated as a bi-level optimization. From a programming point of view, the main framework was set in an interpreted language whilst using as much as possible dependencies written in compiled ones for speed.

GTOC 11: Results from Tsinghua University and Shanghai Institute of Satellite Engineering

In this paper, the methods and results from Tsinghua University and Shanghai Institute of Satellite Engineering for the 11th Global Trajectory Optimization Competition (GTOC11) are presented. To deal with the complicated “Dyson Sphere” building problem, a three-stage procedure is conducted. First, the pre-analysis is performed to reduce search space. It is found that two-impulse maneuvers between asteroid flybys are near-optimal, the semi-major axis of the “Dyson Ring” should be better at 1.0–1.5 AU, and the larger arrival mass asteroids tend to be selected. Second, the globally optimal trajectory design problem is further divided into two sub-problems, the mothership trajectory design and the asteroid assignment to the “Dyson Ring” power stations. For the first problem, beam search is used to obtain numerous single mothership trajectories based on a pre-constructed flyby trajectory database of 3–8 asteroids. The overall trajectories and asteroids visited are obtained by selecting 10 mothership trajectories with a genetic algorithm. For the second problem, we build a database of optimal rendezvous times for all the 83,453 asteroids at different phase angles to reach power stations of different radii and phase angles, then a greedy algorithm is proposed to obtain the asteroid arrival schedule based on all the asteroids visited by motherships. Finally, local optimization of asteroid sequence and flyby epochs is conducted. The activation time adjustment in combination with indirect continuous-thrust trajectory optimization is used based on the global optimization result. In the final submission, motherships fly by 388 asteroids, and the minimum mass of twelve power stations reaches 94% of the theoretical upper bound, which is defined using the minimum-time orbital transfers with free initial and target phases.

Feedback tracking control of optimal reference trajectories for spacecraft relative motion

In this paper, first the optimal trajectories are defined by a fixed end-time, continuous-thrust trajectory which minimizes a terminal cost at the end of an inter-planetary trajectory by appending the multi-body dynamic equations defining the motion by a set of Lagrange multipliers or co-states. The solution of these co-states and the optimal trajectories for a fixed end-time are obtained by solving a two-point boundary value problem along with a steering control law, which is used to provide a reference trajectory. Inspired by the general expression for the gravitation potential and the Q-law, a tracking control law is designed to track the reference trajectory. As the steering controller is not a feedback controller, a feedback tracking control law is proposed to continuously correct the spacecraft's actual relative motion trajectory so it tracks the reference relative motion trajectory. The nonlinear relative motion tracking control law is reduced to a linear feedback control law, when the spacecraft is sufficiently close to the reference relative motion trajectory. The feedback laws demonstrate the feasibility of developing practical tracking laws for continuous trajectory correction manoeuvre control of spacecraft relative motion, for implementing low-thrust inter-planetary trajectory following control in electrically propelled spacecraft, in practice.

How Many Impulses Redux

A central problem in orbit transfer optimization is to determine the number, time, direction, and magnitude of velocity impulses that minimize the total impulse. This problem was posed in 1967 by T. N. Edelbaum, and while notable advances have been made, a rigorous means to answer Edelbaum’s question for multiple-revolution maneuvers has remained elusive for over five decades. We revisit Edelbaum’s question by taking a bottom-up approach to generate a minimum-fuel switching surface. Sweeping through time profiles of the minimum-fuel switching function for increasing admissible thrust magnitude, and in the high-thrust limit, we find that the continuous thrust switching surface reveals the N-impulse solution. It is also shown that a fundamental minimum-thrust solution plays a pivotal role in our process to determine the optimal minimum-fuel maneuver for all thrust levels. Remarkably, we find that the answer to Edelbaum’s question is not generally unique, but is frequently a set of equal-Δv extremals. We further find, when Edelbaum’s question is refined to seek the number of finite-duration thrust arcs for a specific rocket engine, that a unique extremal is usually found. Numerical results demonstrate the ideas and their utility for several interplanetary and Earth-bound optimal transfers that consist of up to eleven impulses or, for finite thrust, short thrust arcs. Another significant contribution of the paper can be viewed as a unification in astrodynamics where the connection between impulsive and continuous-thrust trajectories are demonstrated through the notion of optimal switching surfaces.

Fast Initial Trajectory Design for Low-Thrust Restricted-Three-Body Problems

This paper presents an approach to generate initial trajectories in a three-body dynamical framework, assuming the use of a low-thrust propulsion system. Finite Fourier series were previously implemented successfully in approximating two-dimensional continuous-thrust trajectories in two-body dynamic models. In this paper, Finite Fourier series are implemented in a dual-level-solver strategy for low-thrust trajectory approximation, in the presence of thrust acceleration constraints, in the three-body dynamic model. This approximation enables the feasibility assessment of low-thrust trajectories, especially in the presence of thrust level constraint. The developed method demonstrates capability in generating trajectories using a fully automated procedure for various levels of thrust acceleration with a moderate to high number of revolutions. The suitability of using the approximated trajectories as initial guesses for high-fidelity solvers is demonstrated.

Optimal Continuous-Thrust Responsive Theater Maneuvers

This research investigates a continuous-thrust implementation of the responsive theater maneuver, which is designed to alter a spacecraft’s entry conditions as it overflies a specified geographic region, called the exclusion zone. A particle swarm optimization algorithm employing shape-based continuous-thrust trajectory approximation is used to seed a direct orthogonal collocation routine employing a nonlinear programming problem solver. This approach is used to generate optimal continuous-thrust responsive theater maneuver trajectories. The combination of particle swarm optimization, shape-based continuous-thrust trajectory approximation, and direct orthogonal collocation is shown to generate fuel-optimal trajectories for single-, double-, and triple-pass cases of the responsive theater maneuver problem. Further, these continuous-thrust trajectories are shown to satisfy the analytical necessary conditions for an optimal control and require delta velocities only slightly larger than those required for impulsive responsive theater maneuvers delivering the same effects. As continuous-thrust propulsion technology improves, the continuous-thrust responsive theater maneuvers can provide propellant mass expenditure savings over their impulsive counterparts despite requiring more delta velocity.

Spacecraft Trajectory Optimization Based on Discrete Sets of Pseudoimpulses

A brief review of new methods for continuous-thrust trajectory optimization is presented. The methods use discretization of the spacecraft trajectory on segments and sets of pseudoimpulses for each segment. Boundary conditions are presented as a linear matrix equation. A matrix inequality on the sum of the characteristic velocities for the pseudoimpulses is used to transform the problem into a large-scale linear programming form. Present-day linear programming methods use interior-point algorithms to solve such problems. In the general case, the continuous burns include a number of adjacent segments and a postprocessing of the linear programming solutions is needed to form a sequence of the burns. An optimal number of the burns is automatically determined in the postprocessing. The methods provide flexible opportunities for the trajectory computation in complex missions with various requirements and constraints. A systematic mathematical representation of these problems is considered. A summary of examples for orbit transfer, rendezvous, and moon ascent trajectories is presented. Application examples of lunar landing trajectories are examined. The examples represent a set of optimal unconstrained trajectories for different thrust-to-weight ratios and trajectories with safety descent profile, thrust level, and attitude constraints.

Second-Order Necessary Conditions and Sufficient Conditions Applied to Continuous-Thrust Trajectories

Introduction I N optimal control theory, requiring the first variation of the performance functional to vanish leads to well-known first-order necessary conditions (NC) for an optimal solution.1 These NC allow one to identify candidates for optimality, called stationary or extremal solutions, to distinguish them from solutions that have been proven to be optimal. To determine whether a stationary solution is indeed optimal, one must also test the second-order Jacobi no-conjugate-point NC, which applies if the trajectory is smooth. Also, one can formulate sufficient conditions (SC) that, if satisfied, guarantee that the solution is at least locally optimal. In this Note, a procedure developed by Jo and Prussing2−4 for testing second-order NC and SC is streamlined and applied to an example optimal continuous-thrust trajectory with multiple terminal constraints that yields a different type of result compared to previous examples in Refs. 2, 4, and 5. The procedure is based on earlier work by Wood5,6 that derives new, less restrictive SC for a weak local minimum of the Bolza optimal control problem. However, those SC require that the solution of a matrix Riccati equation be bounded. This is difficult to test numerically because a bounded but rapidly increasing solution can stop the numerical integration and give the false impression that the solution is unbounded. The procedure described and illustrated in this Note replaces the test for an unbounded matrix5,6 by a test for a (scalar) determinant being zero.