Ascent Trajectory Research Articles

Successive Convexification for Ascent Trajectory Replanning of a Multistage Launch Vehicle Experiencing Nonfatal Dynamic Faults

Successive Convexification for Ascent Trajectory Replanning of a Multistage Launch Vehicle Experiencing Nonfatal Dynamic Faults

Optimization of the Conceptual Design of a Multistage Rocket Launcher

The design of a vehicle launch comprises many factors, including the optimization of the climb path and the distribution of the mass in stages. The optimization process has been addressed historically from different points of view, using proprietary software solutions to obtain an ideal mass distribution among stages. In this research, we propose software for the separate optimization of the trajectory of a launch rocket, maximizing the payload weight and the global design, while varying the power plant selection. The launch is mathematically modeled considering its propulsive, gravitational, and aerodynamical aspects. The ascent trajectory is optimized by discretizing the trajectory using structural and physical constraints, and the design accounts for the mass and power plant of each stage. The optimization algorithm is checked against various real rockets and other modeling algorithms, obtaining differences of up to 9%.

Finding multiple local solutions to optimal control problems via saddle points and its application to the ascent trajectory of a winged rocket

Finding multiple local solutions to optimal control problems via saddle points and its application to the ascent trajectory of a winged rocket

Ascent trajectory design and optimization of a two-stage throttleable liquid rocket

This paper explores in detail the design and optimization of the ascent trajectory for a two-stage liquid rocket capable of providing variable thrust. Thrust and steering profiles play an essential role in designing a launch vehicle mission. Variable thrusting or throttling requirement arises in the thrust profile of a liquid rocket due to structural, aerodynamic and mission-related constraints. The current study investigates this aspect for two-stage liquid rockets, which deliver a 500 kg payload to 500 km circular orbit and 180×36000 km Geostationary Transfer Orbit (GTO). As payload mass is kept constant in both cases, the lift-off mass varies. The study found that for both missions mentioned earlier, the required variation of thrust (throttling) is in the range of 100% to 33% for the first stage to achieve an optimal trajectory. For the second stage, a considerable requirement for throttling is absent. Optimal trajectory with throttling capability reduced the lift-off mass of the launch vehicle for a mission to circular orbit by 41.6% and to the GTO by 16.3% compared to the constant thrusting cases. Also, a two-stage liquid rocket using semi cryogenic engines with variable thrusting capability and stage structural factors as low as 0.1 can replace a three-stage rocket which delivers the same payload to the 500 km circular orbit. Kulasekharapattinam and Sriharikota were the launch sites considered for the circular orbit and GTO. The study did not consider spent stage impact constraints. Canonical Differential Evolution based solver is used for global optimization. Thrust profile, steering profile and lift-off mass are the outputs of the optimization strategy.

Physical insights into multi-point global optimum design of scramjet intakes for ascent flight

Scramjet propulsion offers promise for flexible and sustainable space transportation. Hypersonic airflow compression through the intake plays a major role in successful scramjet-powered ascent. It is thus of crucial importance to design high-performance intakes that can work efficiently and robustly throughout the ascent trajectory. The present research is conducted to gain physical insight into multi-point global optimum scramjet intake design and associated requirements by means of multi-point design optimization and analytical investigation in inviscid and viscous regimes. It has been found and verified theoretically that at any operating conditions, compression efficiency, drag, compression ratio, and mean exit temperature can theoretically be determined uniquely in an interrelated manner under calorically perfect and adiabatic flow assumptions, provided one of them is given or known except for compression efficiency. The multi-point optimization study in the inviscid regime has verified that the multi-point global optimum design can exist with respect to compression efficiency and drag. On the other hand, while the theoretical analysis also prescribes that the global optimum design can exist for a single operating condition in viscous flow, trade-off relations have been found to exist between two operating conditions in the viscous regime, where boundary layer is responsible for counteracting behaviors between compression efficiency and drag at different operating conditions. The insights gained contribute to the guiding principle for intake design of scramjet engines for access-to-space.

Dual-bell nozzle with fluidic control of transition for space launchers

The dual-bell nozzle is a promising concept for improving the performance of space launchers. It is characterised by the presence of two altitude-dependent working modes which allow to reduce non-adaptation losses. However, the transition between the two working modes usually takes place prematurely and dangerous side loads might be observed. In this work, fluidic control is investigated as a potential method to delay the transition and limit the risk of side loads. A launcher configuration similar to the Ariane 5 with a dual-bell nozzle in the core engine is considered. First, a parametric optimisation is performed to identify the dual-bell geometry that maximises the payload mass delivered into geostationary transfer: a preliminary model is adopted to describe the dual-bell mode transition and a fast and reliable in-house trajectory optimisation code is used to optimise the ascent trajectory. The flow field in the optimal geometry is then investigated by CFD simulations to verify the effectiveness of fluidic control. Finally, the CFD study results are used to model the dual-bell mode transition and trajectory optimisation is performed again. The proposed solution is characterised by a large payload gain (approximately 1.5 metric tons) with respect to the reference launcher. The simulations showed that fluidic control reduces the order of magnitude of side loads which can arise during transition, showing its potential as enabling technology for the application of dual-bell nozzles on real launchers.

Feasible Sequential Convex Programming With Inexact Restoration for Multistage Ascent Trajectory Optimization

This paper presents a feasible sequential convex programming method to solve the multistage ascent trajectory optimization problem. The proposed method is based on the inexact restoration technique, in which a more feasible intermediate iterate is first produced by solving a constrained least-squares problem, and then a more optimal iterate is generated by solving a convex programming problem constructed around the newly found feasible solution. By virtue of the inexact restoration idea, the proposed method prevents the common artificial infeasibility issue and can provide the intermediate iterate as a feasible suboptimal solution if the algorithmic procedure terminates before convergence. In addition, the Picard iteration-based convexification and Chebyshev polynomial-based discretization methods are employed in the proposed method, given their benefits in terms of robustness, efficiency, and solution accuracy. Numerical simulations for a minimum-time launch ascent problem are conducted, and the results show that the proposed method exhibits better practical performance than other sequential convex programming methods.

Convex Optimization of Launch Vehicle Ascent Trajectory with Heat-Flux and Splash-Down Constraints

This paper presents a convex programming approach to the optimization of a multistage launch vehicle ascent trajectory, from the liftoff to the payload injection into the target orbit, taking into account multiple nonconvex constraints, such as the maximum heat flux after fairing jettisoning and the splash-down of the burned-out stages. Lossless and successive convexification methods are employed to convert the problem into a sequence of convex subproblems. Virtual controls and buffer zones are included to ensure the recursive feasibility of the process, and a state-of-the-art method for updating the reference solution is implemented to filter out undesired phenomena that may hinder convergence. A pseudospectral discretization scheme is used to accurately capture the complex ascent and return dynamics with a limited computational effort. The convergence properties, computational efficiency, and robustness of the algorithm are discussed on the basis of numerical results. The ascent of a VEGA-like launch vehicle toward a polar orbit is used as a case study to discuss the interaction between the heat flux and splash-down constraints. Finally, a sensitivity analysis of the launch vehicle carrying capacity to different splash-down locations is presented.

Polybaric fractional crystallisation of arc magmas: an experimental study simulating trans-crustal magmatic systems

Crystallisation-driven differentiation is one fundamental mechanism proposed to control the compositional evolution of magmas. In this experimental study, we simulated polybaric fractional crystallisation of mantle-derived arc magmas. Various pressure–temperature trajectories were explored to cover a range of potential magma ascent paths and to investigate the role of decompression on phase equilibria and liquid lines of descent (LLD). Fractional crystallisation was approached in a step-wise manner by repetitively synthesising new starting materials chemically corresponding to liquids formed in previous runs. Experiments were performed at temperatures ranging from 1140 to 870 °C with 30 °C steps, and pressure was varied between 0.8 and 0.2 GPa with 0.2 GPa steps. For most fractionation paths, oxygen fugacity (fO2) was buffered close to the Ni-NiO equilibrium (NNO). An additional fractionation series was conducted at fO2 corresponding to the Re-ReO2 buffer (RRO ≈ NNO+2). High-pressure experiments (0.4–0.8 GPa) were run in piston cylinder apparatus while 0.2 GPa runs were conducted in externally heated pressure vessels. Resulting liquid lines of descent follow calc-alkaline differentiation trends where the onset of pronounced silica enrichment coincides with the saturation of amphibole and/or Fe–Ti–oxide. Both pressure and fO2 exert crucial control on the stability fields of olivine, pyroxene, amphibole, plagioclase, and Fe–Ti–oxide phases and on the differentiation behaviour of arc magmas. Key observations are a shift of the olivine–clinopyroxene cotectic towards more clinopyroxene-rich liquid composition, an expansion of the plagioclase stability field and a decrease of amphibole stability with decreasing pressure. Decompression-dominated ascent trajectories result in liquid lines of descent approaching the metaluminous compositional range observed for typical arc volcanic rocks, while differentiation trends obtained for cooling-dominated trajectories evolve to peraluminous compositions, similar to isobaric liquid lines of descent at elevated pressures. Experiments buffered at RRO provide a closer match with natural calc-alkaline differentiation trends compared to fO2 conditions close to NNO. We conclude that decompression-dominated fractionation at oxidising conditions represents one possible scenario for arc magma differentiation.

Trajectory Determination of Chang’E-5 during Landing and Ascending

Chang’E-5 (CE-5) is China’s first lunar sample return mission. This paper focuses on the trajectory determination of the CE-5 lander and ascender during the landing and ascending phases, and the positioning of the CE-5 lander on the Moon. Based on the kinematic statistical orbit determination method using B-spline and polynomial functions, the descent and ascent trajectories of the lander and ascender are determined by using ground-based radiometric ranging, Doppler and interferometry data. The results show that a B-spline function is suitable for a trajectory with complex maneuvers. For a smooth trajectory, B-spline and polynomial functions can reach almost the same solutions. The positioning of the CE-5 lander on the Moon is also investigated here. Using the kinematic statistical positioning method, the landing site of the lander is 43.0590°N, 51.9208°W with an elevation of −2480.26 m, which is less than 200 m different from the LRO (Lunar Reconnaissance Orbiter) image data.

Improved sequential convex programming using modified Chebyshev–Picard iteration for ascent trajectory optimization

This paper presents an improved sequential convex programming (SCP) algorithm for ascent trajectory optimization of launch vehicles, by exploiting the state-of-the-art modified Chebyshev–Picard iteration (MCPI) technique. In the proposed algorithm, the MCPI technique is first utilized to transcribe the continuous-time optimization problem into a sequence of finite-dimensional subproblems. The lossless and successive convexification techniques are then employed to deal with the nonconvexity in optimization. The resulting convex subproblems can be reliably and efficiently solved via a primal-dual interior-point method solver. Numerical simulations for a minimum-time ascent trajectory optimization problem are conducted and the results show that the proposed algorithm has significant improvements over the standard SCP (which uses the Euler or trapezoidal rule for discretization), pseudospectral SCP, and GPOPS.

Single-stage to orbit ascent trajectory optimisation with reliable evolutionary initial guess

In this paper, the ascent trajectory optimization of a lifting body Single-Stage To Orbit (SSTO) reusable launch vehicle is investigated. The work is carried out using a Direct Multiple Shooting method to solve the Optimal Control problem. The crucial initialisation of the optimisation process is performed by using a combination of two evolutionary algorithms, namely a Multi-Objective Parzen-based Estimation of Distribution (MOPED) algorithm and a Multi-Population Adaptive Inflationary Differential Evolution Algorithm (MP-AIDEA). Multi-Objective Parzen-based Estimation of Distribution (MOPED) belongs to the class of Estimation of Distribution Algorithms (EDAs) and it is used in the first phase of the initial guess research to explore the search space, then Multi-Population Adaptive Inflationary Differential Evolution Algorithm (MP-AIDEA) is used to refine the obtained results, and better fulfill the imposed constraints. The initial guesses obtained with this evolutionary framework were tested on different multiple shooting configurations. The importance of the continuity properties of the employed mathematical models was also quantitatively addressed.

Surrogate-based entire trajectory optimization for full space mission from launch to reentry

A surrogate-based optimization method is proposed in this paper to realize the optimization of the entire trajectory from launch to reentry. Different from traditional strategy of multiphase trajectory optimization that directly utilizes the original dynamical model, a surrogate model is substituted for the original time-consuming model to improve the iterative efficiency and decrease the computational cost. The procedures of modeling, solving, and optimizing in each substage are addressed, including air-launch ascent trajectory optimization, orbital maneuver planning, deorbit braking, and atmospheric reentry. Instead of optimizing every substage separately and setting interfaces by experience, in this study, substages are patched as an entire system in advance and then the surrogate model is utilized to replace it to perform a systematic optimization. Additionally, the Fourier series is applied to fit the air launch stage during the iteration. Numerical simulations show that the proposed method is acceptable for the pre-design of a full space mission and the results are optimal.

Fuel-Optimal Ascent Trajectory Problem for Launch Vehicle via Theory of Functional Connections

In this study, we develop a method based on the Theory of Functional Connections (TFC) to solve the fuel-optimal problem in the ascending phase of the launch vehicle. The problem is first transformed into a nonlinear two-point boundary value problem (TPBVP) using the indirect method. Then, using the function interpolation technique called the TFC, the problem’s constraints are analytically embedded into a functional, and the TPBVP is transformed into an unconstrained optimization problem that includes orthogonal polynomials with unknown coefficients. This process effectively reduces the search space of the solution because the original constrained problem transformed into an unconstrained problem, and thus, the unknown coefficients of the unconstrained expression can be solved using simple numerical methods. Finally, the proposed algorithm is validated by comparing to a general nonlinear optimal control software GPOPS-II and the traditional indirect numerical method. The results demonstrated that the proposed algorithm is robust to poor initial values, and solutions can be solved in less than 300 ms within the MATLAB implementation. Consequently, the proposed method has the potential to generate optimal trajectories on-board in real time.

Multipoint Design Optimization of Busemann-Based Intakes for Scramjet-Powered Ascent Flight

Scramjet engines are one of the most promising hypersonic airbreathing propulsion technologies for robust, efficient, and economical access to space. Multi-objective design optimization has been conducted for Busemann-based intakes in inviscid and viscous regimes at single and multiple design points, respectively, by means of surrogate-assisted evolutionary algorithms coupled with computational fluid dynamics in the present research. Intake geometries are generated by applying geometric alterations to the full Busemann intake via leading-edge truncation, axial stunting, and radial contraction, aiming to simultaneously minimize intake drag and maximize the compression efficiency at two different design conditions (i.e., Mach 7.7 at an altitude of 30 km and Mach 10 at 33.5 km on a constant dynamic pressure ascent trajectory). The single-point inviscid optimization study has found that the nondominated solutions obtained from minimizing drag and maximizing compression efficiency are approximately the same as those obtained from maximizing total pressure recovery and minimizing static pressure ratio. From the multipoint viscous optimization study, the optimal solutions have been found to retain the original advantages of the inviscid full Busemann intakes in terms of high compression efficiency with shorter intakes and higher static pressure as well as adequately high mean exit temperature for both design conditions.

Deterministic and Robust Optimization of Hybrid Rocket Engines for Small Satellite Launchers

In this paper, design and optimization approaches are applied to a hybrid-powered small-satellite launcher, namely a deterministic and a robust-based method. A unique hybrid rocket engine is considered, employed in different numbers in each stage of the three-stage launcher: six, three, and one, respectively, in the first, second, and third stage. Liquid oxygen and paraffin-based fuel are considered as propellant combination. A blowdown feed system is assumed to ensure system simplicity and reduce the overall launcher cost. An airborne launch is proposed at a given altitude and speed, whereas the velocity angle on the horizon at first stage ignition is set free. The optimization procedure of the engine design parameters employs a direct method and an evolutionary algorithm in the deterministic and robust-based approach, respectively. An indirect method is used to optimize the ascent trajectory, given the engine design, in both the approaches. In the robust-based approach, uncertainties in the fuel regression rate are taken into account. Small-launcher initial mass is given and payload mass is maximized for a given insertion orbit. A mission-specific objective function is used in the robust-based optimization process to grant the actual achievement of mission goals, despite the uncertainties in the design. Two different design strategies are compared. In the first case the acceleration at first-stage ignition is fixed, whereas in the second case the initial thrust is optimized. Results show that a 5-ton hybrid-rocket launcher can be viable for launch of satellites in the 50–100 kg range and that a small payload reduction, with respect to deterministic optimized solutions, is sufficient to ensure mission accomplishment and also grant the required design robustness.

Ascent Trajectory Optimization for Air-Breathing Hypersonic Vehicles Based on IGS-MPSP

This paper proposes an improved Generalized Quasi-Spectral Model Predictive Static Programming (GS-MPSP) algorithm for the ascent trajectory optimization for hypersonic vehicles in a complex flight environment. The proposed method guarantees the satisfaction of constraints related to the state and control vector while retaining its high computational efficiency. The spectral representation technique is used to describe the control variables, which reduces the number of decision variables and makes the control input smooth enough. Through Taylor expansion, the constraints are transformed into an inequality containing only decision variables, such that it can be added into GS-MPSP framework. By Gauss quadrature collocation method, only a few collocation points are needed to solve the sensitivity matrix, which greatly accelerates the calculation. Subsequently, the analytical expression is obtained by combining the static optimization with the penalty function method. Finally, the simulation results demonstrate that the proposed improved GS-MPSP algorithm can achieve both high computational efficiency and high terminal precision under the constraints.

Convex Approach to Three-Dimensional Launch Vehicle Ascent Trajectory Optimization

This paper deals with the optimization of the ascent trajectory of a multistage launch vehicle, from liftoff to the payload injection into the target orbit, considering inverse-square gravity acceleration and aerodynamic forces. A combination of lossless and successive convexification techniques is adopted to generate a sequence of convex problems that rapidly converges to the original problem solution. An automatic initialization strategy is proposed to make the solution process completely autonomous. In particular, a novel three-step continuation procedure is developed and proved to be more efficient than simpler strategies. This approach relies on the solution of intermediate problems, which either neglect atmospheric drag or fix the time-lengths of the launch vehicle ascent phases, that are solved in succession, gradually passing from easier instances of the optimization problem to the originally intended problem. State-of-the-art techniques to deal with such a complex problem are adopted to enhance the convergence rate, including safeguarding modifications, such as virtual controls and an adaptive trust region. To assess the validity of the proposed approach in a practical scenario, numerical results are presented for two representative practical applications, using as reference a Falcon 9 launch vehicle.

Integrated Optimization of First-Stage SRM and Ascent Trajectory of Multistage Launch Vehicles

This paper presents a methodology for the concurrent first-stage preliminary design and ascent trajectory optimization, with application to a Vega-derived light launch vehicle. The reuse as first stage of an existing upper-stage (Zefiro 40) requires a redesign of the propellant grain geometry, in order to account for the mutated operating conditions. An optimization code based on the parallel running of several differential evolution algorithms is used to find the optimal law of the internal pressure during first-stage operation, together with the optimal thrust direction and other relevant flight parameters of the entire ascent trajectory. The payload injected into a target orbit is maximized, while meeting multiple design constraints, either involving the alone solid rocket motor or dependent on the actual flight trajectory. Numerical results for sun-synchronous orbit injection are presented.

Multiconstrained Ascent Trajectory Optimization Using an Improved Particle Swarm Optimization Method

This paper researches the ascent trajectory optimization problem in view of multiple constraints that effect on the launch vehicle. First, a series of common constraints that effect on the ascent trajectory are formulated for the trajectory optimization problem. Then, in order to reduce the computational burden on the optimal solution, the restrictions on the angular momentum and the eccentricity of the target orbit are converted into constraints on the terminal altitude, velocity, and flight path angle. In this way, the requirement on accurate orbit insertion can be easily realized by solving a three-parameter optimization problem. Next, an improved particle swarm optimization algorithm is developed based on the Gaussian perturbation method to generate the optimal trajectory. Finally, the algorithm is verified by numerical simulation.