Fuel-optimal Trajectories Research Articles

Fuel-optimal guidance using costate supervised learning with local refinement

This work presents a novel method to compute continuous fuel-optimal guidance updates on board a spacecraft by leveraging a combination of deep learning and a local refinement using differential algebraic techniques. In order to create the large datasets necessary to train the artificial neural network, we also propose a new method using polynomial maps to rapidly and efficiently generate arbitrarily high numbers of optimal trajectories. Constructed at the initial time, such maps readily provide entire fuel-optimal trajectories for any given state deviation from the nominal, by means of the simple and efficient evaluation of polynomials. Whilst a trained deep neural network is then capable of providing continuous guidance updates, we observe that it is not capable of learning the optimal guidance policy to an accuracy required for immediate application. Further differential algebraic techniques are therefore employed to locally refine the output of the neural network into highly accurate fuel-optimal guidance updates via a lightweight and iterative-less mapping procedure, suitable for onboard implementation. Performance assessment in both an interplanetary transfer from Earth to Psyche as well as a landing scenario on Psyche demonstrates the accuracy and robustness of the proposed guidance scheme.

Thruster-Pointing-Constrained Optimal Control for Satellite Servicing Using Indirect Optimization

Future space missions such as in-space assembly of telescopes and on-orbit servicing require rendezvous and proximity operations trajectories that avoid thruster-induced contamination of delicate components onboard the client spacecraft. We present a novel technique to incorporate a hard thruster pointing constraint into the indirect optimal control formulation and solve the problem using a single-shooting method. A thruster pointing constraint is an inequality constraint that imposes a limit on the angular range over which a spacecraft thruster is permitted to operate, thus mitigating plume contamination during rendezvous and proximity operations. Through novel incorporation of the constraint directly into the dynamic model, the problem can be easily solved with no a priori knowledge of the burn sequence or information about when the constraint is active or inactive. Our formulation is capable of handling a constant thruster pointing constraint as well as one that varies as a function of distance from the target spacecraft. Results are presented under Clohessy–Wiltshire dynamics; however, the method can be easily extended to handle nonlinear dynamic systems. As expected, we see an increase in fuel consumption with increasing constraint angle for the solution of problems with the same boundary conditions and time of flight. To validate our method, we compare the performance and results to those of a direct method, sequential convex optimization, utilizing the CVX MATLAB plug-in and the MOSEK solver. We found that our approach converged more easily and required no hand-tuning of parameters. It also converged to solutions that satisfy the pointing constraints to a stricter tolerance. We anticipate that our novel approach to generating thruster-pointing-constrained fuel-optimal trajectories will be enabling to a host of future servicing space missions.

On optimal three-impulse Earth–Moon transfers in a four-body model

Within the emerging age of lunar exploration, optimizing transfer trajectories is a fundamental measure toward achieving more economical and efficient lunar missions. This study addresses the possibility of reducing the fuel cost of two-impulse Earth–Moon transfers in a four-body model with the Earth, the Moon, and the Sun as primaries. Lawden’s primer vector theory is applied within this framework to derive a set of necessary conditions for a fuel-optimal trajectory. These conditions are used to identify which trajectories from an existing database could benefit from the insertion of an additional intermediate impulse. More than 10,000 three-impulse transfers are computed with a direct numerical optimization method. These solutions contribute to enriching the database of impulsive trajectories, useful to perform trade-off analyses. While the majority of trajectories exhibit only marginal improvements, a significant breakthrough emerges for transfers featuring an initial gravity assist at the Moon. Implementing a corrective maneuver after the lunar encounter yields substantial reductions in fuel costs.

An improved convex optimization-based guidance for fuel-optimal powered landing

Convex optimization has great potential for onboard trajectory generation which can eliminate large state deviations and improve landing precision. In practical applications, the convex optimization problem is often infeasible under disturbances. In this paper, we analyze the reason for the infeasibility of the fuel-optimal trajectory optimization problem. The analysis result shows that the thrust saturation degrade the anti-disturbance performance and then lead to infeasibility. Based on this analysis, we improved the anti-disturbance performance of convex optimization-based guidance from two aspects: (1) a maximum-thrust regulation strategy is embedded into the original fuel-optimal problem to avoid thrust saturation; (2) a trajectory tracking method with a disturbance compensator is performed to attenuate disturbance effects and to make the initial state of each optimization cycle close to the existing feasible trajectory. Numerical simulations demonstrate that the proposed method can improve the infeasibility and can realize the high-precision landing under disturbances.

Learning Fuel-Optimal Trajectories for Space Applications via Pontryagin Neural Networks

This paper demonstrates the utilization of Pontryagin Neural Networks (PoNNs) to acquire control strategies for achieving fuel-optimal trajectories. PoNNs, a subtype of Physics-Informed Neural Networks (PINNs), are tailored for solving optimal control problems through indirect methods. Specifically, PoNNs learn to solve the Two-Point Boundary Value Problem derived from the application of the Pontryagin Minimum Principle to the problem’s Hamiltonian. Within PoNNs, the Extreme Theory of Functional Connections (X-TFC) is leveraged to approximate states and costates using constrained expressions (CEs). These CEs comprise a free function, modeled by a shallow neural network trained via Extreme Learning Machine, and a functional component that consistently satisfies boundary conditions analytically. Addressing discontinuous control, a smoothing technique is employed, substituting the sign function with a hyperbolic tangent function and implementing a continuation procedure on the smoothing parameter. The proposed methodology is applied to scenarios involving fuel-optimal Earth−Mars interplanetary transfers and Mars landing trajectories. Remarkably, PoNNs exhibit convergence to solutions even with randomly initialized parameters, determining the number and timing of control switches without prior information. Additionally, an analytical approximation of the solution allows for optimal control computation at unencountered points during training. Comparative analysis reveals the efficacy of the proposed approach, which rivals state-of-the-art methods such as the shooting technique and the adaptive Gaussian quadrature collocation method.

An analytical solution of fuel-near-optimal trajectory for three-dimensional planetary landings

The existing solutions for three-dimensional fuel-optimal trajectory generation are inadequate in terms of real-time performance and convergence property, which are vital to successful and reliable powered planetary landings. In this study, the fast generation of three-dimensional fuel-near-optimal landing trajectories is achieved based on problem decomposition and analytical iterative techniques. Specifically, the following three contributions are emphasized. First, the original three-dimensional (3D) thrust-constrained powered planetary landing problem is decoupled into three analytically solvable one-dimensional (1D) sub-problems: one along the vertical direction, and two along the horizon direction. These three 1D problems are re-coupled together by six to-be-determined coordination coefficients. Second, an analytical iteration algorithm is developed to determine these six coordination coefficients and to ensure the satisfaction of thrust magnitude and direction constraints. Third, three necessary proofs are provided to validate the guaranteed convergence of the proposed analytical iteration algorithm. Simulation results of various planetary landing missions on Earth demonstrate that the proposed analytical algorithm can achieve a significant fast generation of 3D fuel-near-optimal landing trajectories with acceptable optimality (2.6% more fuel on average) and guaranteed convergence rate (100%) for planetary landings.

A Novel Framework for Trajectory Planning and Safe Navigation of Satellite Swarms

This paper proposes a new framework for near-optimal, collision-free navigation of a swarm of identical satellites from an initial formation to a desired final formation. The framework combines a centralized optimization problem for location assignment with collision avoidance constraints, a decentralized fuel-optimal trajectory planner, a decentralized Artificial Potential Field (APF)-based real-time controller to ensure robustness, and a Shrinking Horizon Model Predictive Control logic to prevent deadlocks and livelocks and to improve robustness. To ensure high performance and low fuel cost, a novel optimization problem is formulated to tune the APF parameters based on mission requirements. The proposed framework provides a balance between optimality and scalability while keeping computational requirements low. The effectiveness of the frame work is demonstrated using numerical simulations.

Multi-phase and dual aero/propulsive rocket landing guidance using successive convex programming

This paper aims to suggest a new landing guidance algorithm for reusable launch vehicles (RLVs) to enable generation of fuel-efficient trajectories based on successive convex programming. To this end, a dual aero/propulsive landing guidance problem is first formulated to fully exploit the additional moment generated by the aerodynamic control to reduce the propulsion demand required for attitude control. As the result of the aerodynamic landing phase could greatly affect the fuel-optimal trajectory during the vertical landing phase, the formulation is further extended to the multi-phase optimal guidance problem using state-triggered constraints. The proposed guidance strategy is then obtained by solving the formulated optimal control problem based on the successive convex optimization framework using an interior point method The main contribution of this study lies in forming new RLV landing guidance problems to get an optimal trajectory and transforming the corresponding nonconvex problem into a convex optimization problem by introducing appropriate combinations for convexification techniques. Additionally, this paper introduces several practical constraints, such as the maximum slew rate of aerodynamic control fins and nozzle angles, which are not considered in previous works. In this paper, the performance of the proposed method with the potential for online computation is investigated through numerical simulations.

Performance Assessment of Convex Low-Thrust Trajectory Optimization Methods

Different discretization and trust-region methods are compared for the low-thrust fuel-optimal trajectory optimization problem using successive convex programming. In particular, the differential and integral formulations of the adaptive pseudospectral Legendre–Gauss–Radau method, an arbitrary-order Legendre–Gauss–Lobatto technique based on Hermite interpolation, and a first-order-hold discretization are considered. The number of discretization points and segments is varied. Moreover, two hard-trust-region methods and a soft-trust-region strategy are compared. It is briefly discussed whether these methods, if implemented on relevant hardware, would fulfill the general requirements for onboard guidance. A perturbed cubic interpolation and the propagation of the nonlinear dynamics are used to generate initial guesses of varying quality. Interplanetary transfers to a near-Earth asteroid, Venus, and asteroid Dionysus are chosen to assess the overall performance.

Stochastic learning and extremal-field map based autonomous guidance of low-thrust spacecraft

A supervised stochastic learning method called the Gaussian Process Regression (GPR) is used to design an autonomous guidance law for low-thrust spacecraft. The problems considered are both of the time- and fuel-optimal regimes and a methodology based on “perturbed back-propagation” approach is presented to generate optimal control along neighboring optimal trajectories which form the extremal bundle constituting the training data-set. The use of this methodology coupled with a GPR approximation of the spacecraft control via prediction of the costate n-tuple or the primer vector respectively for time- and fuel-optimal trajectories at discrete time-steps is demonstrated to be effective in designing an autonomous guidance law using the open-loop bundle of trajectories to-go. The methodology is applied to the Earth-3671 Dionysus time-optimal interplanetary transfer of a low-thrust spacecraft with off-nominal thruster performance and the resulting guidance law is evaluated under different design parameters using case-studies. The results highlight the utility and applicability of the proposed framework with scope for further improvements.

Robust Low-Thrust Trajectory Optimization Using Convex Programming and a Homotopic Approach

A robust algorithm to solve the low-thrust fuel-optimal trajectory optimization problem for interplanetary spacecraft is developed in this article. The original nonlinear optimal control problem is convexified and transformed into a parameter optimization problem using an arbitrary-order Gauss–Lobatto discretization scheme with nonlinear control interpolation. A homotopic approach that considers the energy-to-fuel smoothing path is combined with an adaptive second-order trust-region mechanism to increase performance. The overall robustness is assessed in several fuel-optimal transfers with poor initial guesses. The results show a superior performance in terms of convergence and computational time compared to standard convex programming approaches in the literature.

Rapid Generation of the Fuel-Optimal Trajectory for Landing on Irregularly Shaped Asteroids

In an asteroid soft landing mission, the irregularly distributed gravitational field and unexpected external disturbances may lead to a significant deviation of the actual landing trajectory to the predesigned trajectory. Therefore, the capability of trajectory onboard generation is essential to meet the requirements of landing accuracy and safety. To this end, this article proposes a rapid trajectory generation algorithm for the fuel-optimal asteroid landing. First, two important properties of the fuel-optimal landing trajectory are revealed in detail with the presented theorems. Then, based on these properties, the problem of landing trajectory generation is transformed into an onboard parameter searching problem, which is of significant benefit for the real-time performance and convergence. The effectiveness of the proposed algorithm is verified through the 433 Eros soft landing mission based numerical simulations. It is shown that the proposed fuel optimal asteroid landing trajectory generation algorithm is valid with deviated initial conditions and irregularly distributed gravitational field.

Fast and accurate estimation of fuel-optimal trajectories to Near-Earth Asteroids

This paper proposes an improved method for the preliminary evaluation of minimum-propellant trajectories to Near-Earth Asteroids (NEAs). The method applies to missions from Earth to asteroids with small eccentricity and inclination. A planar and a plane-change problem can be distinguished. In the planar problem, the solution assumes that multiple burn arcs are performed in correspondence of the apsides of the target asteroid in order to change the initial spacecraft orbit (i.e., Earth’s orbit) into the target one. The number of arcs is established once the time of flight is given (1 burn at each apsis per revolution, 1 revolution per year can be assumed). The length and propellant consumption of each arc to attain the required changes of semi-major axis and eccentricity are computed by a procedure based on Edelbaum’s approximation, which is well-suited to the problem at hand, as eccentricity changes are expected to be small for feasible missions. No numerical integration is required, but only the numerical solution of a three-unknown algebraic system is needed, making the procedure extremely fast. Plane change is taken into account assuming a constant out-of-plane thrust angle during each burn. A previous simple formulation used an averaged thrust effect over one revolution and neglected the fact that plane changes are more effective at the nodes. Several improvements are here introduced, which greatly increase the method accuracy. The influences of the eccentricity change, the angle between the asteroid line of nodes and line of apsides, and the expected length of the arc are considered: In fact, when the eccentricity is small, the thrust arc can be performed at the nodes where the inclination is efficiently changed, with little penalty in the planar maneuver. An efficient plane change is also performed when the angle between the asteroid line of nodes and line of apsides is small and/or the length of the arc is large, because, in this case, the node is comprised in the apsidal burn. A simple corrective formula accounts for this effect. The new method shows remarkable accuracy. The results comparison with solutions obtained with an indirect optimization method for a set of more than 60 NEAs shows a 0.95 correlation coefficient in the propellant masses. The estimation error is below 10% for 75% of the targets, below 15% for 95% of the targets, and always below 20%.

Low-Thrust Gravity-Assist Trajectory Design Using Optimal Multimode Propulsion Models

The complexities in using indirect optimization methods get compounded for practical co-optimization problems in the presence of continuous and discrete design variables. In this paper, realistic multimode electric propulsion systems are incorporated within the formulation of gravity-assist, low-thrust trajectory design problems. Electric thrusters operate over a large set of discrete operation modes, and each mode is characterized by specific values for power, thrust, specific impulse, and mass flow rate. Unlike the traditional methods that approximate the aggregate behavior of electric thrusters’ performance (using polynomials fits), a novel construct is proposed to incorporate optimal discrete operating modes within indirect optimization. The capability of the tool and the efficacy of the proposed methodology are demonstrated by solving a multiyear, fuel-optimal trajectory problem from Earth to asteroid Psyche via a Mars gravity-assist maneuver and using a Hall-effect SPT-140 thruster with 21 operating modes. Comparisons with the traditional polynomial-based thruster modeling are presented.

Central Difference Convexification Method for Soft-Landing Trajectory Optimization in Planetary Powered Descending Phase

Aiming at the mission of real-time online guidance for fixed-point soft landing on planet, the paper designs an algorithm based on sequential convex optimization that is designed to solve the fuel-optimal trajectory. The pre-labeled central difference algorithm is used to linearize the dynamics equations and the termination condition based on the deviation of index is proposed to judge whether it converges, which can generate a fuel-optimal trajectory quickly. Besides, a fitting function is given to approximately estimate the optimal terminal time by analyzing the relationship between the terminal time of optimal trajectories and their remaining fuel, which can reduce the amount of unknown variables. The simulation results of this algorithm show the weak sensitivity to initial guess, good convergence and small terminal error compared to the traditional convexification method of linearizing the dynamics equations whose variables are substituted at first.

Trajectory Design via Convex Optimization for Six-Degree-of-Freedom Asteroid Powered Landing

This paper presents a convex programming algorithm based on modified Rodrigues parameters for a six-degree-of-freedom asteroid powered landing trajectory design problem. The trajectory design problem is formulated as a nonconvex fuel-optimal control problem with nonlinear coupled translational and rotational dynamics, nonconvex state and control constraints. In order to tackle the fuel-optimal control problem in the convexification framework, the original nonconvex continuous-time optimization problem is converted into a convex discrete-time optimization subproblem by linearizing and discretizing the dynamics and constraints. The effect of the coupling of translational and rotational dynamics on the convexification of nonconvex control constraints is discussed. The successive convexification method of solving a sequence of constrained convex subproblem is used to generate the optimal trajectory. The validity of the proposed algorithm for generating fuel-optimal trajectories and the effect of asteroid gravity on generated trajectories are examined through simulations of landing on different irregular asteroids. The Monte Carlo simulation is performed to examine the calculation performance of the proposed algorithm comparing to the Radau pseudospectral method algorithm and the proposed algorithm with the rotational motion parameterized by quaternions.

A homotopy approach connecting time-optimal with fuel-optimal trajectories

This paper describes a novel homotopy method to compute fuel-optimal trajectories starting from a time-optimal solution. The time-optimal problem is proposed to serve as a gateway for solving the minimum thrust problem. Homotopy is used to link the original low-thrust fuel-optimal problem with the minimum thrust problem. Two new variables are introduced in the dynamic model. The first is a logarithm of mass variable and the second is an acceleration magnitude variable. The analytic expression of the logarithm of mass co-state is solved. For the time-optimal problem, initial co-state of logarithm of mass can be expressed by thrust magnitude and transfer time. Then, the number of unknown initial co-states decreased. The effectiveness and optimality of the proposed method is validated through simulations of two rendezvous missions.

Robust Bang-Off-Bang Low-Thrust Guidance Using Model Predictive Static Programming

Model Predictive Static Programming (MPSP) has been always used under the assumption of continuous or impulsive control, but never in low-thrust transfers featuring bang-off-bang control. Following the observation that the sensitivity matrix (SM) is discontinuous across the switching time, this work introduces a two-loop MPSP guidance scheme using fuel-optimal trajectories as nominal solutions. In our method, the equations of motion are augmented by the mass costate equation, while the velocity costate is the MPSP control, expressed by a weighted sum of Fourier basis functions. The SM is computed at the switching time by using calculus of variations. Both the MPSP control and the initial mass costate are updated in an inner loop using Newton’s method, and continuation is employed in an outer loop to face large perturbations. A sample interplanetary CubeSat mission to an asteroid is used as study case to illustrate the effectiveness and robustness of the method developed.

Throttled explicit guidance for pinpoint landing under bounded thrust acceleration

Onboard computation of a fuel-optimal trajectory is an indispensable technology for future lunar and planetary missions with pinpoint landings. This paper proposes a throttled explicit guidance (TEG) scheme under bounded constant thrust acceleration. TEG is capable of achieving fuel-optimal large diversions with good accuracy and can find optimal solutions. Thus far, the TEG algorithm is unique as it offers an explicit and simultaneous search method for the fuel-optimal thrust direction and thrust magnitude switching in predictor-corrector iterations. Fast numerical search is realized with a straightforward computation of seven final states (position, velocity, and the Hamiltonian) from seven unknowns (six adjoint variables for position and velocity and one final time). In addition, global convergence capability is enhanced by implementing the damped Newton's method. A number of simulations of large diversions show the excellent convergence of the TEG algorithm within at most 15 iterations from a cold start. The experimental results of the runtime measurement of the TEG algorithm support its real-time feasibility on a flight processor. These features of the TEG are suitable for onboard guidance of pinpoint landings.

Low-thrust spacecraft trajectory optimization via a DNN-based method

Initial solution guess has a significant impact on the convergence of indirect methods, especially for continuous low-thrust trajectory optimization problems. In this study, an intelligent initial solution supplying approach based on deep neural networks (DNNs) is proposed to help achieve the fast generation of optimal trajectories for low-thrust orbit transfers. Energy-optimal and fuel-optimal trajectories with three different terminal conditions are considered. Based on the training dataset obtained by an indirect method, DNNs are constructed to approximate the solutions corresponding to different flight states. Based on the trained DNNs, an intelligent trajectory optimization method named DNN-based method is developed with the help of the homotopy technique. Numerical simulations are conducted to evaluate the performance of the proposed method on success rates and time consumptions. Simulation results demonstrate that the combination of traditional techniques and the new DNN technology can achieve the fast generation of low-thrust optimal trajectories with advantages on computational efficiency and reliability.