Ascent Trajectory Optimization Research Articles

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.

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.

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

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.

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.

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.

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.

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.

Successive Convexification for Online Ascent Trajectory Optimization

In this paper, a method based on the successive convexification is proposed to solve the ascent trajectory optimization problem, the algorithm converges to the optimal solution quickly even if the initial guess is coarse. A three-dimensional motion is formulated with complex aerodynamics and terminal constraints. Based on the modified aerodynamic coefficients, the new auxiliary control variables are designed to deal with the complex aerodynamics and non-smooth of control variables in the discrete optimization problem. The inner nonconvex constraints between the new control are relaxed to be convex without loss. The artificial infeasibility and unboundedness caused by linearization are tackled by the virtual controls and soft constraint for trust region in the successive convexification. The good convergence of the proposed method is illustrated by the iterative solutions of the ascent trajectory optimization problem for a small guided rocket, the accuracy is verified by the comparison with the optimal solution given by the typical optimal control solvers, and the feasibility and stability are demonstrated by optimal solutions of the ascent trajectory optimization problems under different missions and dispersed conditions. These excellent performances validated by the adequate simulations indicate that the proposed algorithm can be implemented online.

Ascent trajectory design of small-lift launch vehicle using hierarchical optimization

An integrated algorithm to design the optimal ascent trajectory of a target Sun-synchronous orbit (SSO) for small-lift launch vehicles is studied in this paper. A general problem description is provided, including the specific pitch law designed for an ascent trajectory to SSO and the definitions of design variables and equality/inequality constraints. Then an integrated optimization scheme is introduced to solve this problem. In order to improve its efficiency, a preliminary determination method is to analytically generate two sensitive design variables related with the coast arc. With the help of the two design variables, an initial combination of design variables can be obtained by solving separately the optimization problems of two powered phases. Based on the predetermination method, an efficient integrated algorithm using hierarchical optimization is proposed. A simulation based on a type of small-lift Electron Rocket and Rutherford engine is performed using this hierarchical optimization algorithm and the feasibility and efficiency of this algorithm is evaluated. These systematic investigations on all the possible injected orbits give a full detail of launching capability for Electron Rocket.

Minimum-Fuel Ascent of Hypersonic Vehicle considering Control Constraint Using the Improved Pigeon-Inspired Optimization Algorithm

Trajectory optimization problem for hypersonic vehicles has long been recognized as a difficult problem. This paper brings control constraints into the trajectory optimization to make the optimal trajectory meet the requirements of control performance. The strong nonlinear characteristic of the ascent phase aerodynamics makes the trajectory optimization problem difficult to be solved by the optimal control theory. A trajectory optimization algorithm based on the improved pigeon-inspired optimization (PIO) algorithm is proposed to solve the complex trajectory optimization problem under multiple constraints. To overcome the obstacle of premature convergence and deceptiveness, the evolutionary strategy of qubit in quantum evolutionary algorithm (QEA) is introduced into the PIO to maintain population diversity and judge the optimal solution. To handle constraints, the penalty function is used to construct the fitness function. The optimal ascent trajectory is obtained by utilizing the improved PIO algorithm. Then, the trajectory inverse algorithm is used to verify the feasibility of the optimal trajectory to ensure that a feasible optimal trajectory is obtained. The comparison results show that the proposed algorithm outperforms particle swarm optimization (PSO) and standard PIO on trajectory optimization. Meanwhile, the simulation result shows that the performance of the optimal ascent trajectory with control constraints is improved and the trajectory is feasible. Therefore, the method is potentially feasible for solving the ascent trajectory optimization problem under control constraint for hypersonic vehicles.

Multidisciplinary design and optimization of an innovative nano air launch vehicle with a twin-fuselage UAV as carrier aircraft

The population of nano satellites has increased dramatically in recent years, but there are few launch vehicles providing dedicated launch opportunities for them. This paper presents the conceptual design of an innovative nano air launch vehicle, dedicated to providing exclusive launch opportunity to nano satellites weighing 10 kg or less. This nano launcher is a three-stage launch vehicle carried by a twin-fuselage unmanned aerial vehicle and released at an altitude of 10 km from a velocity of 0.5Ma. The air launch vehicle is innovative in two senses: firstly, it adopts a novel air launch method. Before separation, the launch vehicle is adjusted to a large angle of attack by the carrier aircraft. The launch vehicle only needs to keep attitude stable after separation, and sequentially when first stage ignites, the orientation of the thrust is inclined upward so that the thrust can provide enough vertical force to pull up the flight path angle rapidly, without aid from large wings; secondly, a terminal lumped control unit is applied to undertake the flight control task on its own. The terminal lumped control unit is a combination of four thrusters that share the liquid propellant with the last stage's main engine. With the terminal lumped control unit, the air launch vehicle needs no movable attitude control instruments, including thrust vectoring gimbals and movable wings. This innovative flight control strategy makes the launch vehicle very simple in structure and light in weight. Multidisciplinary modeling of the nano launcher is performed and genetic algorithm is applied for optimization. Among all the disciplines, trajectory optimization is key in checking the feasibility of the design, for which the Radau pseudospectral method is successfully implemented to obtain the optimal ascent trajectory. The optimal nano launcher has a gross take-off weight of 1365 kg. Finally, a sensitivity analysis is performed to check the robustness of the result using Monte Carlo simulation method.

Ascent trajectory optimization for air‐breathing vehicles in consideration of launch window

SummaryThis paper studies the optimal ascent trajectory for a one‐stage‐to‐orbit, air‐breathing vehicle. This kind of vehicle allows endoatmospheric lateral maneuver and thus can extend the launch window. To obtain the optimal orbit‐insertion trajectory in consideration of launch time, a novel optimization model is proposed by designing longitudinal and lateral profiles separately. The longitudinal flight aims to meet the orbital shape that contains the eccentricity and the angular momentum, which are transformed to boundary constrains in altitude, path angle, and velocity. For lateral motion, the orbit inclination and right ascension of ascending node are designed to vary simultaneously as the range. By designing an altitude profile, the terminal altitude and path angle are naturally met and removed from the constraints and accurate orbit insertion is realized by searching the total range and the initial azimuth. Then, the original problem is converted to a parameter optimization problem and solved by a hybrid optimization algorithm, in which the solution is first searched by a particle swarm optimization method and then refined by a modified golden section method. Simulation tests the proposed method with various scenarios. Compared to traditional launch vehicles, a launch window over 1.5 hours is derived without consuming the propellant in the upper stage.

Rapid ascent trajectory optimization for guided rockets via sequential convex programming

A sequential convex programming algorithm is proposed to solve the complex ascent trajectory optimization problems for guided rockets in this paper. Due to the nonlinear dynamics and constraints, especially, the nonlinear thrust terms and aerodynamic drag, ascent trajectory optimization problems for guided rockets are always difficult to be solved rapidly. In this paper, first, the complex thrust terms in the dynamic equation are approximately transformed into linear (convex) functions of the angle of attack. Secondly, the nonlinear drag coefficient is transformed into a linear (convex) function of design variables by introducing two new control variables. The relaxation technique is used to relax the constraints between the control variables to avoid non- convexity, and the accuracy of the relaxation is proved using the optimal control theory. Then, nonconvex objective functions and dynamical equations are convexified by first-order Taylor expansions. At last, a sequential convex programming iterative algorithm is proposed to solve the ascent trajectory planning problem accurately and rapidly. The ascent trajectory optimization problem for the terminal velocity maximum is simulated comparing with the general pseudospectral optimal control software method, which demonstrates the effectiveness and rapidity of the proposed method.