Ascent Trajectory Research Articles

Refined mode classification and performance analysis method of dual-mode scramjet vehicles for flight trajectory optimization

Hypersonic air-breathing vehicles, utilizing ramjet and scramjet propulsion systems, are emerging as a promising option for future hypersonic transportation due to high specific impulse. The performance of hypersonic aircraft is significantly influenced by their flight trajectory, as combustion efficiency and aerodynamic forces vary markedly with altitude. Although numerous studies have focused on optimizing these trajectories, they have not adequately considered the potential for operational failures in the propulsion system, such as unstart and blowout. This study introduces a trajectory optimization approach for hypersonic aircraft that proactively addresses and mitigates these operational failures. This is achieved by establishing an operational classification process for a dual-mode scramjet engine and proposing a failure mode avoidance strategy using a Gaussian process classifier. Optimized ascent trajectories are achieved through the development of a two-stage robust sequential convex programming approach. The results of trajectory optimization indicate that the failure mode avoidance strategy proposed is crucial in obtaining minimal time trajectories, and that the optimal trajectories include drop motion in the initial stage flight in order to increase combustion efficiency.

Multi-disciplinary optimization of single-stage hybrid rockets for lunar ascent

This study aims to determine the optimal design and ascent trajectory for a single-stage hybrid rocket in the context of a lunar ascent mission, with the objective of reaching a 100 km circular orbit on the Moon equatorial plane. A multi-disciplinary optimization problem is formulated, integrating flight design variables and launch vehicle design variables to achieve the best mission scenario. A zero-dimensional internal ballistics model is employed to estimate key parameters such as thrust, mass flow rate, and gross mass of the hybrid rocket, while a three-degree of freedom dynamical model is utilized to simulate the ascent trajectory. The resulting integrated optimization problem is solved using an in-house heuristic algorithm that combines particle swarm and genetic algorithms. Two propellant combinations, liquid oxygen/paraffin-wax and hydrogen peroxide (90%)/high-density polyethylene, are evaluated, considering both their performance in the integrated system and the feasibility of the proposed mission. After conducting a sensitivity analysis to investigate the influence of particle count in the optimization process, results for both propellant combinations are presented. While paraffin and oxygen provide the highest payload ratio, hydrogen peroxide and polyethylene entail a more feasible and flexible design for a lunar mission.

Approximate analytical solutions for three-dimensional ascent trajectory of a solid-fuel launch vehicle with time-varying mass flow rate

In the scenario that a solid-fuel launch vehicle maneuvers in outer space at high angles of attack and sideslip for energy management, Approximate Analytical Solutions (AAS) for the three-dimensional (3D) ascent flight states are derived, which are the only solutions capable of considering time-varying Mass Flow Rate (MFR) at present. The uneven MFR makes the thrust vary nonlinearly and thus increases the difficulty of the problem greatly. The AAS are derived based on a 3D Generalized Ascent Dynamics Model (GADM) with a normalized mass as the independent variable. To simplify some highly nonlinear terms in the GADM, several approximate functions are introduced carefully, while the errors of the approximations relative to the original terms are regarded as minor perturbations. Notably, a finite series with positive and negative exponents, called Exponent-Symmetry Series (ESS), is proposed for function approximation to decrease the highest exponent in the AAS so as to reduce computer round-off errors. To calculate the ESS coefficients, a method of seeking the Optimal Interpolation Points (OIP) is proposed using the least-squares-approximation theory. Due to the artful design of the approximations, the GADM can be decomposed into two analytically solvable subsystems by a perturbation method, and thus the AAS are obtained successfully. Finally, to help implement the AAS, two indirect methods for measuring the remaining mass and predicting the burnout time in flight are put forward using information from accelerometers. Simulation results verify the superiority of the AAS under the condition of time-varying MFR.

A three-stage sequential convex programming approach for trajectory optimization

Recently, sequential convex programming (SCP) has become a potential approach in trajectory optimization because of its high efficiency. To improve stability and discretization accuracy, a three-stage SCP approach based on the hp-adaptive Radau pseudospectral discretization is proposed in this paper. In most instances, the initial subproblem may risk infeasibility due to the undesignated initial guess. Therefore, we design a constraint relaxation stage for the SCP to enhance the feasibility of the subproblem as much as possible. Once the subproblem can be directly solved, the iteration enters the second stage, during which a mesh refinement algorithm based on discretization error analysis is utilized to decrease the discretization error to the tolerance. In the final stage, the damping term is introduced into the objective of the subproblem to suppress the oscillation of the solution and accelerate the convergence. A dual-channel control reentry trajectory optimization and an ascent trajectory optimization are taken as examples, and the simulation results show that the proposed approach outperforms conventional SCP approaches in terms of accuracy and efficiency.

Robust Space Launcher Control with Time-Varying Objectives

This paper presents a novel approach to robust linear time-varying control design for space launchers. It allows the controller to explicitly account for the launcher’s time-varying dynamics and changing control objectives along the ascent trajectory. The latter are readily incorporated via time-varying weighting functions into a mixed sensitivity design. For a traceable and transparent control design, a physically motivated weighting scheme is applied, which requires little tuning effort. The controller is obtained via a novel observer-based finite horizon linear time-varying synthesis approach. It provides a transparent structure which is easy to implement. The approach is used to design a pitch controller for a flexible expendable launch vehicle in atmospheric ascent tracking the open-loop guidance reference signal.

Rapid ascent trajectory optimization of rocket-based combined cycle hypersonic vehicle

The ascent trajectory design of a rocket-based combined cycle (RBCC) hypersonic vehicle has many typical characteristics, including the complex working mode of the RBCC power system, strong coupling between thrust and flight state, strong nonlinear model and many complex constraints etc. This paper proposes a rapid trajectory optimization method based on sequential convex optimization to optimize the complex ascent trajectory with the RBCC power system. Firstly, a mathematical model for optimizing the ascent trajectory of the RBCC hypersonic vehicle is established to describe the angle of attack control system with and without the second-order lag. Then, the optimization model is made convex and discretized based on the convex optimization theory, and a trajectory optimization strategy is improved. Finally, taking the maximum terminal mechanical energy as optimization objective, the ascent trajectory optimization is simulated in cases with and without the second-order lag in the angle of attack control system. The simulation results show that the proposed mathematical model and the ascent trajectory optimization method are rapid and reliable and that the optimization results meet the characteristics of the RBCC power system and provide reference for the application of a RBCC power system and the design of an angle of attack control system.

Approximate Analytical Solutions for Launch-Vehicle Ascent Trajectory

Approximate analytical solutions for the ascent trajectory of a solid-fuel launch vehicle are derived. To the best of the authors' knowledge, the solutions are the ones achieving the highest accuracy in ascent-trajectory prediction under the condition of high angle-of-attack (AOA) maneuvers, where the magnitude of AOA can even be greater than 60 deg at high altitudes. As the first step of deriving the solutions, a reduced-order dynamics model with a normalized mass as the independent variable is put forward for longitudinal-plane flight. To further simplify the model, the sine of AOA is chosen as the key parameter for trajectory control and designed as a polynomial of the normalized mass. Meanwhile, an improved aerodynamic model with the sine of AOA as the independent variable is developed. Due to high AOA, the simplified dynamics model is still highly nonlinear and difficult to solve. To overcome this difficulty, by performing force analysis, several approximate polynomials and first-order Taylor series expansions are created to replace some highly nonlinear but relatively small terms in the dynamical equations, while the errors of the approximate formulae relative to the original nonlinear terms are retained and treated as minor perturbations. The benefit is that the modified dynamics model can be divided into two analytically solvable subsystems using a perturbation method. By solving the subsystems, the approximate analytical solutions for downrange, altitude, velocity and flight-path angle (FPA) of the vehicle are obtained successfully. Simulation results verify that the proposed solutions are much more accurate than the existing solutions in the scenario of high AOA flight.

Polytopal approximation for thrust magnitude constraints in rocket trajectory optimization

This paper proposes a linear programming approach for onboard optimization of rocket trajectories, motivated by the need for reliable and computationally efficient trajectory optimization methods in real-time applications. To reformulate rocket trajectory optimization problems within a linear programming framework, a polytopal approximation method is developed. The polytopal approximation method approximates the thrust magnitude constraint using a set of artificial variables and linear constraints. Compared with a state-of-the-art approximation method, the polytopal approximation method demonstrates a higher accuracy-to-size ratio, enabling rocket trajectory optimization problems to be cast into linear subproblems without significantly increasing problem sizes. As linear subproblems can be readily solved by linear programming solvers, the proposed trajectory optimization approach inherits the solver’s reliability and computational efficiency, making it potentially useful for critical onboard applications. An ascent trajectory optimization problem is provided as an illustrative example to demonstrate the effectiveness of the proposed linear programming approach.

Ascent Trajectory Optimization With Nonlinearity-Kept Convexification

Ascent Trajectory Optimization With Nonlinearity-Kept Convexification

Convex optimization for post-fault ascent trajectory replanning using auxiliary phases

For ascent trajectory replanning after non-fatal dynamic faults, the terminal orbital constraints and optimization indexes under different replanning strategies usually have very strong nonconvexity, which brings significant challenges to real-time guidance. This paper introduces auxiliary phases to convexify the terminal constraints and objective functions losslessly, significantly improving the convergence and injection accuracy. After lossless and successive convexification, the original problems are converted into a series of convex subproblems. The replanning algorithm is then presented, including the fault detection module, the residual carrying capacity determination module, and the task degradation module. Two typical launch missions are simulated to verify the proposed method's pmethoderformance. Numerical results show that the method can always converge efficiently and reliably in different scenarios and has great potential for real-time onboard applications.

Mixed-Compression Supersonic Intake and Engine–Airframe Integration

Hypersonic flight is gaining increasing attention by aerospace companies interested in designing and developing reusable aircraftlike vehicles (spaceplanes). Advanced concepts include a combination of space and aviation approaches able to fly few minutes in space for sightseeing or fast enough to enable point-to-point transportation. In this framework, a concept of hypersonic systems for space travel and high-speed transportation is presented, with a particular focus on the intake and engine–airframe configurations. The adopted intake is designed to ensure an efficient functioning, not only in nominal conditions but along the supersonic ascent trajectory thanks to a movable inner spike. A nonzero angle of attack resulted in a little degradation of the performances in terms of efficiency, spillage, and flow distortion. In the second part, the interaction between the engine and airframe is also investigated. An initial configuration, characterized by the propulsion system located on the fuselage side near the wing, is characterized by a strong interference responsible for an off-design functioning with a high percentage of air spillage. An alternative solution with engine–wing integration exhibits a completely different behavior with interference minimized, resulting in a design intake functioning without air spillage and a considerable drag reduction.

Design and evaluation of INS/GNSS loose and tight coupling applied to launch vehicle integrated navigation

<abstract><p>A satellite launcher requires accurate and reliable navigation to reach orbit safely. This challenging application includes vertical launch, flying through diverse layers of the atmosphere, discontinuous acceleration and stages separation, high terminal velocities and hostile temperature and vibration ranges. The Inertial Navigation System (INS) is typically coupled with a Global Navigation Satellite System (GNSS) receiver to achieve the desired solution quality along the complete ascent trajectory. Here, several INS/GNSS integrated navigation strategies are described and evaluated through numerical simulations. In particular, we evaluate different INS/GNSS integration using: loose coupling, tight coupling and an intermediate Kalman filter on the receiver to enhance the loosely coupled navigation solution. The main contributions are: a compensation model of tropospheric delays on the GNSS observables, a compensation of processing/communication delays on GNSS receiver outputs, covariance adaptations to consider vehicle's high dynamics, and an specific way to integrate the filtered solution and its covariance provided by the GNSS receiver. The tightly coupled integrated navigation proved to be the best choice to handle possible GNSS receiver positioning solution outages, to exploit more degrees of freedom for the compensation of GNSS observables and the possibility to make a cross validation of individual GNSS observables with INS-based information. This is specially important during the periods with a low number of satellites in view when the GNSS positioning solution can not be validated or even computed by the GNSS receiver alone. Finally, a test with hardware in the loop is provided to validate the numerical result for the selected tightly coupled INS/GNSS.</p></abstract>

Emission-Driven Hybrid Rocket Engine Optimization for Small Launchers

Hybrid rocket engines are a green alternative to solid rocket motors and may represent a low-cost alternative to kerosene fueled rockets, while granting performance and control features similar to that of typical storable liquid rocket engines. In this work, the design of a three-stage hybrid launcher is optimized by means of a coupled procedure: an evolutionary algorithm optimizes the engine design, whereas an indirect optimization method optimizes the corresponding ascent trajectory. The trajectory integration also provides the vertical emission profiles required for the evaluation of the environmental impact of the launch. The propellants are a paraffin-based wax and liquid oxygen. The vehicle is launched from the ground and uses an electric turbo pump feed system. The initial mass is given (5000 kg) and the insertion of the payload into a 600-km circular, and polar orbit is considered as a reference mission. Clusters of similar hybrid rocket engines, with only few differences, are employed in all stages to reduce the development and operational costs of the launcher. Optimization is carried out with the aim of maximizing the payload mass and then minimizing the overall environmental impact of the launch. The results show that satisfactory performance is achievable also considering rocket polluting emissions: the carbon footprint of the launch can be reduced by one fourth at the cost of a 5-kg payload mass reduction.

A novel linear uncertainty propagation method for nonlinear dynamics with interval process

Interval process is a preferable model for time-varying uncertainty propagation of dynamic systems when only the range of uncertainties can be obtained. However, for nonlinear systems, except for Monte Carlo (MC) simulation, there are still few efficient uncertainty propagation methods under the interval process model. This paper develops a non-intrusive and semi-analytical uncertainty propagation method, named the “convex model linearization method (CMLM),” by constructing a linearization formulation of a nonlinear system in a non-probabilistic sense. First, the criterion to evaluate the difference between the original system and the linearization formulation is derived, represented by the discrepancy of middle point, radius and correlations of response. By minimizing these three parameters, the coefficients of linear equations will be optimized to obtain the linearization formulation of the original system. Then, analytical equations are built to calculate uncertainty response under the interval process, without time-consuming analysis of the original system. To further improve the efficiency of the linearization process, Chebyshev polynomial is introduced to approximate the nonlinear dynamic analysis. Two numerical examples of duffing oscillators and vehicle rides are set to test the proposed CMLM. Compared to the MC method, with comparable uncertainty response precision, the CMLM just needs 1–10% times of dynamic analyses of the nonlinear system. Furthermore, a practical launch vehicle ascent trajectory problem with black-box dynamics is solved by, respectively, the CMLM and MC method. The results verify the capacity of the CMLM to deal with black-box problems and show that the CMLM performs better in terms of accuracy, efficiency and robustness.

Mars Ascent Trajectory Optimization Based on Convex Optimization

In this paper, a Mars ascent trajectory optimization algorithm based on convex optimization is designed for the Mars ascent trajectory optimization problem. The main accomplishment of this paper is the trajectory optimization of the second-stage ignition based on the convex optimization. For the first-stage ignition, open-loop guidance is used. In this stage, the ascender flies using the maximum thrust to track the attitude profile, thus completing the ascent flight control. In the presence of errors in the first-stage ignition, the experiment generates multiple sets of first-stage shutdown point data and completes simulation verification experiments based on them. The simulation verifies that the second-stage ignition trajectory optimization based on convex optimization algorithm can achieve high accuracy into orbit despite the large starting point deviation.

Surrogate Neural Network for Rapid Flight Performance Evaluation of Hybrid Rocket Engines

This paper investigates the use of neural networks as surrogate models for fast and accurate performance prediction of hybrid rocket engines (HREs). To this aim, a deep feedforward neural network is trained to estimate the thrust and mass flow rate laws of a liquid oxygen/paraffin-wax HRE (as well as other motor characteristics, including the burning time as well as dry and propellant masses), starting from a limited number of motor design parameters, which define the fuel grain, the nozzle geometry, and the initial oxidizer mass flow rate. A medium-fidelity HRE ballistic model, accounting for nozzle throat erosion and two alternative oxidizer feed systems (a pump-fed and a pressure-fed system), is used to generate the sample data the network is trained on. A comprehensive analysis is carried out to assess the neural network accuracy and its generalization capability using several network topologies and varying the size of the training set. Finally, to demonstrate the effectiveness of the proposed approach, the trained network is used as a surrogate HRE model within a multidisciplinary optimization process, which is aimed at determining the optimal HRE design and ascent trajectory for a Vega-derived three-stage launch vehicle, where a HRE replaces the last two stages of the original four-stage rocket.

Optimal Design of Hybrid Rocket Small Satellite Launchers: Ground Versus Airborne Launch

A three-stage hybrid rocket is considered as a small satellite launcher. The same engine is used in different numbers in each stage: six, three, and one in the first, second, and third stages, respectively. This design choice aims at an overall reduction of the launcher cost. The propellants are liquid oxygen and a paraffin-based fuel. The performance of different feed systems and launch options are evaluated: the feasibility of a ground launch is analyzed and compared to similar three-stage launchers with airborne launch using both a gas pressurized feed system and an electric turbopump feed system. The optimization procedure exploits a direct method to evaluate the best values of engine design parameters, whereas an indirect method optimizes the ascent trajectory once the engine design is given. Constant power and blowdown operation are, respectively, assumed for the electrical feed system and the gas pressurized feed system. The initial mass of the launcher is given (5000 kg), and the payload mass is maximized for a given insertion orbit. The initial thrust is fixed in order to have an initial acceleration equal to . The nozzle expansion ratio in the first-stage engines is reduced to avoid separation at liftoff in the ground case, and the third-stage engines are used at a lower vacuum thrust level to satisfy maximum acceleration constraints. The results show that the proposed small satellite launcher concepts are able to deliver payload masses in the range of 40–100 kg into the desired orbit.

An online generation method of ascent trajectory based on feedforward neural networks

With the development of advanced guidance technology and onboard calculation ability, a drawback of poor real-time performance of traditional trajectory optimization methods has emerged. In this paper, a neural network is applied to generate the online ascent trajectory of a solid rocket. With the advantage of its nonlinear mapping capability, the neural network is used to approximate the trajectory computation model to reduce the burden of onboard computation. In particular, according to the different environment characteristics of the atmospheric and exoatmospheric portions, two neural networks (a control-network and a costate-network) are trained offline. The former generates the pitch angle command in real time during the atmospheric portion. The latter provides good initial costate guesses for the indirect method in exoatmosphere, and then the trajectory deviation after flight through the atmosphere is corrected by solving the optimal control of the remaining time online. Finally, numerical simulations are performed to show the effectiveness and robustness of the proposed scheme.

Online trajectory optimization and guidance algorithm for space interceptors with nonlinear terminal constraints via convex programming

PurposeIn this paper, an online convex optimization method for the exoatmospheric ascent trajectory of space interceptors is proposed. The purpose of this paper is to transform the original trajectory optimization problem into a sequence of convex optimization subproblems.Design/methodology/approachFor convenience in calculating accuracy and efficiency, the complex nonlinear terminal orbital elements constraints are converted into several quadratic equality constraints, which can be better computed by a two-step correction method during the iteration. First, the nonconvex thrust magnitude constraint is convexified by the lossless convexification technique. Then, discretization and successive linearization are introduced to transform the original problem into a sequence of one convex optimization subproblem, considering different flight phases. Parameters of trust-region and penalty are also applied to improve the computation performance. To correct the deviation in real time, the iterative guidance method is applied before orbit injection.FindingsNumerical experiments show that the algorithm proposed in this paper has good convergence and accuracy. The successive progress can converge in a few steps and 3–4 s of CPU time. Even under engine failure or mission change, the algorithm can yield satisfactory results.Practical implicationsThe convex optimization method presented in this paper is expected to generate a reliable optimal trajectory rapidly in different situations and has great potential for onboard applications of space interceptors.Originality/valueThe originality of this paper lies in the proposed online trajectory optimization method and guidance algorithm of the space inceptors, especially for onboard applications in emergency situations.

Analytic Derivation of Ascent Trajectories and Performance of Launch Vehicles

This papers introduces an analytic method to define multistage launcher trajectories to determine the payload mass that can be inserted in orbits of different semimajor axes and inclinations. This method can evaluate the gravity loss, which is the main term to be subtracted to the Tziolkowski evaluation of the velocity provided by the thrust of a launcher. In the method, the trajectories are dependent on two parameters only: the final flight-path angle γf at the end of the gravity-turn arc of the launcher trajectory and the duration tc of the coasting arc following the gravity-turn phase. The analytic formulas for the gravity-turn phase, being solutions of differential equations with a singularity, allow us to identify the trajectory with a required final flight-path angle γf in infinite solutions with the same initial vertical launch condition. This can also drive the selection of the parameters of the pitch manoeuvre needed to turn the launcher from the initial vertical arc. For any pair γf and tc, a launcher trajectory is determined. A numerical solver is used to identify the values γf and tc, allowing for the insertion of the payload mass into the required orbit. The analytic method is compared with a numerical code including the drag effect, which is the only effect overlooked in the analytic formulas. The analytical method is proven to predict the payload mass with an error never exceeding the 10% of the actual payload mass, found through numerical propagation.